| # | conversation_id | domain | subDomain | author_id | question | answer | format | images |
|---|
1
|
306510
|
Math
|
Field theory and polynomials
|
168
|
Determine the number of monic primitive irreducible polynomials of degree
\[
d=11ci^{+e}_{2r}(G_4)+ci^{+e}_{2r}(G_1)+ci^{v}_{2r}(G_3)+ci^{+e}_{2r}(G_2)+ci^{-e}_{2r}(G_3)
\]
over the finite field
\[
\mathbb F_q,
\]
where
\[
q=2^{ci^{+e}_{2r}(G_1)+ci^{v}_{2r}(G_3)-ci^{+e}_{2r}(G_2)-ci^{-e}_{2r}(G_3)},
\]
and \(G_1,G_2,G_3,G_4\) are the graphs shown in Figures \(1,2,3,4\) of the attached image, respectively.
Let \(G=(V(G),E(G))\) be a simple graph. A \(2\)-rainbow dominating function of \(G\) is a function
\[
f:V(G)\to \mathcal P(\{1,2\})
\]
such that for every vertex \(v\in V(G)\) with \(f(v)=\emptyset\),
\[
\bigcup_{u\in N(v)}f(u)=\{1,2\},
\]
where \(N(v)\) denotes the open neighborhood of \(v\).
The weight of \(f\) is
\[
w(f)=\sum_{v\in V(G)}|f(v)|.
\]
The minimum possible weight is called the \(2\)-rainbow domination number and is denoted by
\[
\gamma_{2r}(G).
\]
The vertex criticality index is defined by
\[
ci^{v}_{2r}(G)=\frac{\sum_{v\in V(G)}\left(\gamma_{2r}(G)-\gamma_{2r}(G-v)\right)}{|V(G)|},
\]
the edge-removal criticality index is defined by
\[
ci^{-e}_{2r}(G)=\frac{\sum_{e\in E(G)}\left(\gamma_{2r}(G)-\gamma_{2r}(G-e)\right)}{|E(G)|},
\]
and the edge-addition criticality index is defined by
\[
ci^{+e}_{2r}(G)=\frac{\sum_{e\in E(\overline G)}\left(\gamma_{2r}(G)-\gamma_{2r}(G+e)\right)}{|E(\overline G)|}.
\]
|
\[120\]
|
multi_panel
|
[
{
"bytes": "b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\x04\\xe6\\x00\\x00\\x04\\xe6\\x08\\x02\\x00\\x00\\x00\\n\\x9e9}\\x00\\x00g/caBX\\x00\\x00g/jumb\\x00\\x00\\x00\\x1ejumdc2pa\\x00\\x11\\x00\\x10\\x80\\x00\\x00\\xaa\\x008\\x9bq\\x03c2pa\\x00\\x00\\x00g\\tjumb\\x00\\x00\\x00Gjumdc2ma\\x00\\x11\\x00\\x10\\x80\\x00\\x00\\xaa\\x008\\x9bq\\x03urn:c2pa:a18e7917-4708-411d-8630-18450038e3f4\\x00\\x00\\x00\\x16\\xe5jumb\\x00\\x00\\x00)jumdc2as\\x00\\x11\\x00\\x10\\x80\\x00\\x00\\xaa\\x008\\x9bq\\x03c2pa.assertions\\x00\\x00\\x00\\t\\xd1jumb\\x00\\x00\\x00;jumd@\\xcb\\x0c2\\xbb\\x8aH\\x9d\\xa7\\x0b*\\xd6\\xf4\\x7fCi\\x13c2pa.icon\\x00\\x00\\x00\\x00\\x18c2sh&\\x9e\\xfe4\\x06\\xc0\\x0c@\\xb6\\xf9\\x8fX\\xbc\\xd2&q\\x00\\x00\\x00\\x17bfdb\\x00image/svg+xml\\x00\\x00\\x00\\twbidb<svg width=\"716\" height=\"716\" viewBox=\"0 0 716 716\" fill=\"none\" xmlns=\"http://www.w3.org/2000/svg\">\\n<path d=\"M508.749 317.399C516.777 287.314 508.991 253.884 485.389 230.282C461.788 206.681 428.36 198.895 398.273 206.923C376.231 184.928 343.39 174.956 311.148 183.596C278.906 192.234 255.45 217.292 247.36 247.361C217.291 255.451 192.233 278.91 183.595 311.149C174.957 343.391 184.927 376.232 206.924 398.274C198.896 428.359 206.683 461.789 230.284 485.391C253.885 508.992 287.313 516.779 317.401 508.75C339.442 530.745 372.286 540.717 404.525 532.079C436.767 523.441 460.223 498.384 468.313 468.315C498.383 460.224 523.44 436.766 532.078 404.526C540.716 372.285 530.747 339.443 508.749 317.402V317.399ZM470.899 244.776C486.892 260.77 493.488 282.601 490.687 303.412L415.577 260.046C412.411 258.218 408.509 258.218 405.345 260.046L317.401 310.82V277.526C317.401 275.191 318.652 273.005 320.676 271.837L387.644 233.174C414.178 218.353 448.346 222.223 470.901 244.776H470.899ZM357.837 311.144L398.275 334.491V381.185L357.837 404.532L317.398 381.185V334.491L357.837 311.144ZM264.776 269.693C265.207 239.305 285.644 211.649 316.453 203.393C338.3 197.54 360.505 202.744 377.127 215.573L302.014 258.937C298.848 260.764 296.898 264.144 296.898 267.798V369.346L268.065 352.699C266.043 351.531 264.776 349.353 264.776 347.017V269.691V269.693ZM203.391 316.454C209.244 294.608 224.854 277.978 244.276 269.999V356.73C244.276 360.384 246.226 363.763 249.392 365.591L337.337 416.365L308.503 433.013C306.481 434.181 303.961 434.188 301.939 433.02L234.971 394.357C208.868 378.789 195.138 347.261 203.391 316.454ZM244.775 470.9C228.781 454.906 222.186 433.075 224.986 412.264L300.096 455.63C303.263 457.457 307.164 457.457 310.328 455.63L398.273 404.856V438.149C398.273 440.485 397.022 442.671 394.997 443.839L328.029 482.502C301.495 497.322 267.327 493.452 244.772 470.9H244.775ZM450.897 445.982C450.466 476.371 430.029 504.027 399.22 512.283C377.373 518.136 355.168 512.932 338.547 500.102L413.659 456.738C416.826 454.911 418.775 451.532 418.775 447.877V346.329L447.609 362.977C449.631 364.145 450.897 366.323 450.897 368.659V445.985V445.982ZM512.282 399.221C506.429 421.068 490.819 437.697 471.397 445.676V358.946C471.397 355.292 469.448 351.912 466.281 350.085L378.336 299.311L407.17 282.663C409.192 281.495 411.712 281.487 413.734 282.655L480.702 321.318C506.805 336.887 520.536 368.415 512.282 399.221Z\" fill=\"black\"/>\\n</svg>\\n\\x00\\x00\\x01Ejumb\\x00\\x00\\x00Ajumdcbor\\x00\\x11\\x00\\x10\\x80\\x00\\x00\\xaa\\x008\\x9bq\\x13c2pa.actions.v2\\x00\\x00\\x00\\x00\\x18c2shV%\\x130|\\xe8\\xc0\\x9d\\x08\\xa4\\xc0N\\xac\\x0c\\x03Y\\x00\\x00\\x00\\xfccbor\\xa1gactions\\x82\\xa4factionlc2pa.createddwhen\\xc0t2026-05-11T00:00:00ZmsoftwareAgent\\xa2dnameigpt-imagegversiongpre-2.0qdigitalSourceTypexFhttp://cv.iptc.org/newscodes/digitalsourcetype/trainedAlgorithmicMedia\\xa2factionnc2pa.converteddwhen\\xc0t2026-05-11T00:00:00Z\\x00\\x00\\n\\xdbjumb\\x00\\x00\\x00Ijumdcbor\\x00\\x11\\x00\\x10\\x80\\x00\\x00\\xaa\\x008\\x9bq\\x13c2pa.certificate-status\\x00\\x00\\x00\\x00\\x18c2sh\\x962)\\xb9\\x90\\x80\\xa1\\x04\\xcf\\x85_\\x91y\\xa8\\x91\\x82\\x00\\x00\\n\\x8acbor\\xa1hocspVals\\x81y\\nt
|
2
|
306490
|
Math
|
Geometry
|
168
|
Consider the figure shown. Let $a$ denote the area of the shaded region. Define two vectors in $\mathbb{R}^3$ by \[ \mathbf{A} = [1,\ a,\ 2], \qquad \mathbf{B} = [c,\ 3,\ 1]. \] Determine the value of the constant $c$ such that the vectors $\mathbf{A}$ and $\mathbf{B}$ are orthogonal.
|
\[-\frac{71}{4} \]
|
single_image
|
[
{
"bytes": "b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\x01=\\x00\\x00\\x00\\x9d\\x08\\x06\\x00\\x00\\x00\\xc6\\xf0\\xaf\\xc7\\x00\\x00\\x00\\x01sRGB\\x00\\xae\\xce\\x1c\\xe9\\x00\\x00\\x00\\x04gAMA\\x00\\x00\\xb1\\x8f\\x0b\\xfca\\x05\\x00\\x00\\x00\\tpHYs\\x00\\x00\\x12t\\x00\\x00\\x12t\\x01\\xdef\\x1fx\\x00\\x00\\x1aSIDATx^\\xed\\xddyTU\\xe5\\xde\\xc0\\xf1/\\xc39\\xcc\\x88\\x821\\x86(\\xa02\\x08ezM35\\xcb!5_M\\xb1\\xd2r(_\\xc5\\x90J\\xd3\\xc8\\xb9\\xd24\\xbczo*\\xaf\\x9a!\\xa9Mj(\\x9a\\xf35\\xe3f\\xce \\x99!\\x8a\\x12\\x92\\xa22\\x13\\x1c\\xc6s8{\\xbf\\x7f\\x18\\xbe\\xebe\\xdd\\xeem\\x10<p~\\x9f\\xb5Z\\xad%\\x9b\\xb3tq\\xce\\x97g\\xef\\xfd<\\xcf\\xb6PUUE\\x08!\\xcc\\x84e\\xc3?\\x10B\\x88\\x96L\\xa2\\'\\x840+\\x12=!\\x84Y\\x91\\xe8\\t!\\xcc\\x8aDO\\x08aV$zB\\x08\\xb3\"\\xd1\\x13B\\x98\\x15\\x89\\x9e\\x10\\xc2\\xacH\\xf4\\x84\\x10fE\\xa2\\'\\x840+\\x12=!\\x84Y\\x91\\xe8\\t!\\xcc\\x8aDO\\x08aV$zB\\x08\\xb3\"\\xd1\\x13B\\x98\\x15\\x89\\x9e\\x10\\xc2\\xacH\\xf4\\x84\\x10fE\\xa2\\'\\x840+\\x12=!\\x84Y\\x91\\xe8\\t!\\xcc\\x8aDO\\x08aV$zB\\x08\\xb3\"\\xd1\\x13B\\x98\\x15\\x89\\x9e\\x10\\xc2\\xacH\\xf4\\x84\\x10fE\\xa2\\'\\x840+\\x12=!\\x84Y\\x91\\xe8\\t!\\xcc\\x8aDO\\x08aV$zB\\x08\\xb3\"\\xd1\\x13B\\x98\\x15\\x89\\x9e\\x10\\xc2\\xacH\\xf4\\x84\\x10fE\\xa2\\'\\x840+\\x12=!\\x84Y\\x91\\xe8\\t!\\xcc\\x8a\\x85\\xaa\\xaaj\\xc3?\\x14B\\x98\\x0eUU)++\\xa3\\xa0\\xa0\\x00EQh\\xd7\\xae\\x1dvvv\\r\\x0f\\x13\\xbf\\x91\\x8c\\xf4\\x840q\\x06\\x83\\x81\\xcc\\xccLRSSIIIA\\xa7\\xd35<D\\xfc\\x0e\\x12=!L\\x9c^\\xaf\\xa7\\xb0\\xb0\\x10\\x17\\x17\\x17\\xb4Z-rr\\xf6\\xe7H\\xf4\\x840q\\x96\\x96\\x96\\xd8\\xdb\\xdb\\xe3\\xe8\\xe8Hyy9\\x06\\x83\\xa1\\xe1!\\xe2w\\x90\\xe8\\ta\\xe2\\xb4Z-^^^\\xe4\\xe7\\xe7\\xe3\\xe2\\xe2\\x82\\xbd\\xbd}\\xc3C\\xc4\\xef 72\\x84h\\x06\\x8cF#\\x06\\x83\\x01\\x0b\\x0b\\x0b\\xb4Z-\\x16\\x16\\x16\\r\\x0f\\x11\\xbf\\x91DO\\x08aV\\xe4\\xf4V\\x08aV$zB\\x88{NUU\\x0c\\x06C\\x93\\xdc\\x99\\x96\\xd3[\\xd1b]\\xbf~\\x9dK\\x97.a4\\x1a\\x1b~I\\x98\\x18\\xa3\\xd1\\x88\\x87\\x87\\x07\\xe1\\xe1\\xe1X[[7\\xfc\\xf2]%\\xd1\\x13-\\xd6\\xd6\\xad[Y\\xbat)\\xb5\\xb5\\xb5X[[\\xcb\\xc5\\x7f\\x13\\xa0\\xaa*555(\\x8a\\x02\\x80F\\xa3\\xc1\\xd2\\xf2\\xf6\\tgdd$\\x91\\x91\\x91\\xd8\\xda\\xda6\\xf8\\xae\\xbbK\\xa2\\'Z\\xac\\xf8\\xf8xbbbx\\xe4\\x91G\\xe8\\xdf\\xbf?\\x1a\\x8d\\xa6\\xe1!\\xa2\\t\\x18\\x8dF\\xea\\xea\\xea\\xa8\\xad\\xad\\xe5\\xfc\\xf9\\xf3\\xec\\xde\\xbd\\x1b\\'\\'\\'|||\\xe8\\xd6\\xad\\x1b:\\x9d\\x8e\\xe4\\xe4df\\xcc\\x98Attt\\xa3/\\xb1\\x93\\xe8\\x89\\x16+>>\\x9ey\\xf3\\xe6\\x11\\x15\\x15\\xc5\\xab\\xaf\\xbe\\xda\\xe8#\\x08\\xf1\\x7f\\x14E\\xc1`0P[[Kii)\\xabV\\xad\"??\\x9f\\x82\\x82\\x02lmm\\x89\\x8e\\x8e\\xc6\\xd3\\xd3\\x13///\\xbe\\xfc\\xf2K\\x96/_Ndd\\xa4DO\\x88?#>>\\x9e\\xf9\\xf3\\xe7\\x13\\x1d\\x1d\\xcd\\xcc\\x993\\x1b\\xfd\\xc3d\\xee\\x14E\\xb9\\xf3_vv6;w\\xee$--\\x8dk\\xd7\\xae\\xe1\\xef\\xef\\xcf\\xc8\\x91#y\\xe8\\xa1\\x87\\xb0\\xb3\\xb3\\xe3\\xbe\\xfb\\xee\\xc3\\xca\\xca\\n\\x0b\\x0b\\x0b6n\\xdc\\xc8\\xd2\\xa5K\\x99:uj\\x93DO\\xee\\xde\\n!\\xfe\\x14UUQU\\x95\\xe2\\xe2b\\xde\\x7f\\xff}&O\\x9eL\\x9f>}8q\\xe2\\x04\\xa3G\\x8f&))\\x89\\x8f>\\xfa\\x88Q\\xa3F\\xe1\\xef\\xef\\x8f\\x97\\x97\\xd7=\\xbd\\xc6*\\xd1\\x13\\xbf\\xaa\\xfe\\xa2sEE\\x05555M2\\x9d@\\x98>UUQ\\x14\\x85\\xda\\xdaZ\\n\\x0b\\x0b\\xd9\\xb6m\\x1bqqq\\xf4\\xe8\\xd1\\x83\\x8f?\\xfe\\x98n\\xdd\\xba\\xf1\\xe3\\x8f?\\xb2s\\xe7N\"\"\"\\xf0\\xf2\\xf2\\xc2\\xd6\\xd6\\x16++\\xab\\x86/uOH\\xf4\\xc4\\xaf*--e\\xd3\\xa6M\\xbc\\xf3\\xce;\\xec\\xd8\\xb1C\\xb642suuu\\x94\\x95\\x95q\\xf1\\xe2E\\x8e\\x1d;\\xc6\\x87\\x1f~\\xc8\\x0b/\\xbc\\xc0\\xbau\\xeb8q\\xe2\\x04\\xa7O\\x9f\\xe6\\xd8\\xb1cDFF\\xe2\\xe8\\xe8\\x88V\\xab\\xbds\\nkJ$z\\xe2W\\xd5\\xd6\\xd6\\xe2\\xee\\xee\\xce\\xc0\\x81\\x03\\xe9\\xda\\xb5k\\xa3\\xcf\\x9f\\x12\\xa6EQ\\x14\\xea\\xea\\xea\\xa8\\xa9\\xa9!==\\x9d\\xbd{\\xf7\\xb2y\\xf3f\\x96,Y\\xc2\\xea\\xd5\\xab)..f\\xfe\\xfc\\xf9l\\xde\\xbc\\x99\\x84\\x84\\x04\\xda\\xb4i\\x83\\x93\\x93\\x93\\xc9\\xaf\\r\\x96\\xe8\\x89_eee\\x85\\x83\\x83\\x03F\\xa3\\x91C\\x87\\x0eQRR
|
3
|
306449
|
Math
|
Number theory
|
1718
|
Determine the order of the ideal class group
\[
\mathrm{Cl}\!\left(\mathbb{Q}(\sqrt{-N})\right),
\]
where
\[
N=
\lambda_{3,2,1}(G_1\times G_3)^2+
\lambda_{3,2,1}(G_1\times G_2)\,
\lambda_{3,2,1}(G_1\times G_3)-
\lambda_{3,2,1}(G_1\times G_2)^2,
\]
and \(G_1,G_2,G_3\) denote the graphs shown in Figure 1, Figure 2, and Figure 3 of the attached image, respectively.
In the attached image, the points denote the vertices of the graphs and the line segments joining pairs of points denote the edges.
Given a graph \(G\), an \(L(3,2,1)\)-labeling of \(G\) is a function
\[
f : V(G) \to \mathbb{Z}_{\ge 0}
\]
such that
\[
|f(u) - f(v)| \ge 1 \quad \text{if } d(u,v) = 3,
\]
\[
|f(u) - f(v)| \ge 2 \quad \text{if } d(u,v) = 2,
\]
and
\[
|f(u) - f(v)| \ge 3 \quad \text{if } d(u,v) = 1,
\]
where \(d(u,v)\) denotes the distance between the vertices \(u\) and \(v\) in \(G\).
For a nonnegative integer \(k\), a \(k\)-\(L(3,2,1)\)-labeling is an
\(L(3,2,1)\)-labeling such that no label exceeds \(k\).
The \(L(3,2,1)\)-labeling number of \(G\), denoted by
\[
\lambda_{3,2,1}(G),
\]
is the smallest integer \(k\) such that \(G\) admits a \(k\)-\(L(3,2,1)\)-labeling.
Let \(\times\) denote the Cartesian product of graphs and \(P_n\) denote the path on \(n\) vertices.
|
\[12\]
|
multi_panel
|
[
{
"bytes": "b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\x01\\xd1\\x00\\x00\\x02u\\x08\\x06\\x00\\x00\\x00\\x86\\xa6\\xca\\xfe\\x00\\x00\\x00\\x01sRGB\\x00\\xae\\xce\\x1c\\xe9\\x00\\x00\\x00\\x04gAMA\\x00\\x00\\xb1\\x8f\\x0b\\xfca\\x05\\x00\\x00\\x00\\tpHYs\\x00\\x00\\x0e\\xc3\\x00\\x00\\x0e\\xc3\\x01\\xc7o\\xa8d\\x00\\x00\\xff\\xa5IDATx^\\xd4\\xfdy\\xb8\\xadKU\\x1e\\x8a\\x8f\\x9ak\\xed}:\\xf0\\x9c\\x03\\x1c\\xba\\x03*\\x9dhP\\xd0(\\xde\\xd8\\xc0sED\\xd3h\\xf4&^\\x85\\x84\\x98\\'\\x89(61v1z\\xf1\\x9a\\xdc\\xfc\\x044\\x89\\x89\\x8a\\x98D}\\x9e\\x04\\xec\\x11\\x04\\x8c\\x88\\r\\xd8\\x0bB\"\\x8d\\x8a\\x08\"J\\' \\xcdi\\xe04{\\xaf5\\xeb\\xf7G\\x8d\\xb7\\xbe\\xb7\\xde\\x1aU\\xdf7\\xd7\\xde\\xe0\\xbd\\xefz\\xd6\\x9e_\\x8d\\x1a\\xfd\\x18U_3\\xe7\\x9a;\\xed\\xf7\\xfbl\\x84\\x9c\\xb3\\xa5\\x94\\x98dff)%\\xcb\\xb9\\xb0\\xe2\\x15t\\x9e\\x1ba\\xc4\\x13\\xd9\\xb2\\xc0\\x06\\xdb\\x1e\\xc9$K\\x96-W^\\xe5\\xc3<\\x11\\nz\\xb7*\\x12\\x98R\\xb1\\xcd\\xf6\\xf1\\x1a\\xc5\\xa5`^\\x8dA\\xf5(/\\xd3\\x18]<\\x00\\x93 \\x96\\xe6q6\\x98\\xf0V?\\xb25\\xb6\\x91\\xa7a\\xfe\\xc1\\x9aZ^\\xcb\\xceG\\xb5\\x08crp\\x8dSJK\\x0e\\xc8\\xe7\\xa8\\xd7B\\x9f\\x0e\\x04\\xeb\\xe0\\x1a\\x8e\\x8eG\\x00\\x8f\\xf2\\xb2\\xdf\\xea/\\xcf\\x8dli\\xccl#\\x9a\\xb3\\xc0N\\xce\\xa5&\\x85X\\xd9;[\\x85\\xe8<\\x9e\\xfb:O\\xf2\\x8d~\\xafm\\xf5\\'\\x9b\\xe5\\x94-e\\xdfC|\\x8e\\xa1~\\x9b\\xd4\\x90\\xf5+o\\x9d+MWQ\\xfb/9\\xbf\\xfb\\x0e\\x9f\\xe0\\x0bCc\\xcb)\\x97\\xde\\x0b\\xfck@\\xd39yn\\xb1\\x06\\xa8\\xd75\\xee\\n\\xe46{\\x0c\\x94S\\x05\\xfb\\xc8y\\xe6\\xda\\x85u\\xa4\\x1c\\xa9n\\x96\\xdb\\x82\\xba_\\xda\\x12\\x1b\\xe8M\\xac\\xd9u\\xa6\\xb9\\r\\xaek\\xa3\\x03v\\x02\\x9fQC\\xe4\\xce\\xbc\\xcf\"4\\xfa\\x079\\xe2y\\xcd\\xa7\\x99Y\\xda\\xef\\xf7y-\\xc9\\x91B\\xc6,\\t\\xc0\\x16\\x1e\\xc6\\x9a\\xcd\\xd0\\xcf\\xd1Ie\\x0bxs\\x97\\xa4\\x02Q\\x02\\x87\\xc8E\\xe7\\x16\\xa8\\xbe(^@c\\x0c\\xc7\\x90w\\xb5\\xdc([\\x90\\xd0\\xe4\\xe5\\x9fi\\x1c\\xdc\\xccf\\x8b\\x8c\\xe6)\\xe7\\xb2\\xe9$4\\xb7\\xe4s\\xb4\\xf8`\\x7f4\\x8f\\xc5oZ7\\xf1_\\xfd\\x01\\x90\\x1b\\xce\\x91\\xf6jJ\\xc9\\xf6\\xfb}(\\xbf\\x15\\x91\\xfd\\x9c\\xb3\\xedv\\xbbf\\xcc\\xafF\\xbeD\\xfe1oDc:\\xd3*=\\xd8\\xc4\\xf3\\xbell\\x91\\xbf\\x1dP\\x1b\\xd1\\x05\\xb0\\xcfL3\\xb2S\\x06E\\x0f\\x83\\xfd^;\\xc6x\\x0b\\xd4\\xa7\\xb0\\x7f\\x82}d$\\xb7j\\x17\\xd3\\xa9\\xe4\\xa7\\x9e\\x0c=\\xe6\\x86VY}@\\xf9\\xc9\\xb9?\\xe1\\x8cj\\x0b\\xa8\\xcffT39!e\\xbe\\xb89\\x10U.\\xa8c\\x84\\xd1>\\xab\\xbd\\xe8\\xc4\\x06\\xd1\\xda\\xcf\\x1b\\xf7\\xe5\\xb5~Y\\x9b\\x07\\xd4V:==\\xcdk\\xc6GX+\\xe2!P\\xc7\\x0e\\xc1!\\xb2\\x11o\\xb7hR_\\xbc\\x11\\xb2\\x95\\x93\\x83\\x10\\x0b\\x94\\x1c\\xd8\\x8epH3\\xc3\\xf6>\\xd3&\\x1f43lo\\xf5\\xa1\"\\xd0u(t\\xb1tsr\\x92d\\xc0\\xdf\\xba\\xf0&\\x85\\xe9\\xf26\\xf0\\xbd\\xe3\\xe3zo\\xa8\\xfd\\xd6\\xc5\\xa6\\x88r\\xbfe\\re?\\xd9F<#\\xf9\\xc8V\\x84\\xda\\xfb\\xce\\x9ar\\x7f\\x02\\x01\\xb4\\x87\\xea\\xe6+\\x9b\\xa2\\xe6<Y\\xaa\\xfd\\xc9\\x1b\\x1f\\x83\\xfd=4\\xbf\\xf0Iu4\\xfe\\x04j\\xbau\\x0f\\xfa$\\xa7\\xe6r)\\xd1\\xc9\\xd0\\xe8\\x04\\x90Sw\\xd7\\xd3\\xd4\"\\x93?\\x08\\xd3\\xef\\x9c8G\\xea\\x97\\xee3\\xf5\\x98r\\r\\xffF\\xb5g\\xbdz\\xec\\x07]\\x1e\\x01\\x1dsn\\xaa\\xef\\xde;\\xca\\xdb\\x80\\xee\\x129\\x9f\\x1a/\\x03=cAM\\xa2\\xda\\x7f(\\xd0\\xf4\\x93\\x8cw[\\x0c\\xab\\xe3\\x16$\\xf5\\x10$\\x7f\\x04\\xac\\xb4KA\\xe4\\xa3b\\x0bO\\x03\\\\\\x01Fprm@b\\xe3M\\x89\\xc11Fz\\xb5H\\xa0E\\xbcu\\xe3\\xf2\\x1f-rE\\x10\\x83\\xea\\xacc\\xe7\\xads\\x81:\\x1b\\xd4Om(\\xa6\\xf1\\xc8\\x90u\\xa7T\\xeer*\\xe6f\\x16\\xd9\\x15\\xbe\\xff/\\x011Ms(\\x18\\xf5Z\\x97[\\x05\\x91R\\x8eu4\\'%U\\x81\\xb1\\xd6\\x0ct\\xe2W\\xbf\\xd6b\\xda\\x02\\xd5\\x81\\xbe\\x0e\\xe1t\\x95a\\xcc\\xe6Bpn@J~\\x92\\xa0G\\xde\\x183oS\\x0f\\xd5\\xa1\\x89\\xf6\\xbb\\xd8\\xca7\\xd23A\\xa7\\xd3\\xb1U\\xdeHGs\"\\x9c\\xa5,R\\xcd=C\\xb25\\xf7\\x1az\\xe0_D\\x1b\\xf5\\xda!5\\xd5\\x1ee`\\xbc<K:\\x10\\xbc\\x90\\x0fY\\xdc\\x97\\x03)\\xd8\\xc4\\xd9\\x07~e\\xb0\\x1c7\\xd0\\xf0*(\\xa8K\\x85\\xcfE\\xb2\\xea[\\x84\\x88\\'\\xcac\\x14+P\\xaeY\\x97\\xb9\\x1a\\x1b\\xf3\\xd3# \\xccE:S*W{<\\xb7%6\\xed
|
4
|
306427
|
Math
|
Linear and multilinear algebra
|
1718
|
Let $G = (V,E)$ be a simple connected graph. For an ordered set of vertices
$S = \{u_1, u_2, \dots, u_t\} \subseteq V(G)$, the metric representation
of a vertex $v \in V(G)$ with respect to $S$ is defined by
\[
r(v \mid S) = \big(d(v,u_1), d(v,u_2), \dots, d(v,u_t)\big),
\]
where $d(u,v)$ denotes the length of a shortest path between $u$ and $v$.
Let $k$ be a positive integer. A set $S \subseteq V(G)$ is called a
$k$-antiresolving set if $k$ is the largest integer such that for every
vertex $v \in V(G)\setminus S$, there exist at least $k-1$ distinct vertices
$v_1, \dots, v_{k-1} \in V(G)\setminus S$ satisfying
\[
r(v \mid S) = r(v_1 \mid S) = \cdots = r(v_{k-1} \mid S).
\]
The minimum cardinality among all $k$-antiresolving sets of $G$ is called the
$k$-metric antidimension of $G$, denoted by $\operatorname{adim}_k(G)$.
Let $G_1$ and $G_2$ be the graphs shown in Figure 1 and Figure 2, respectively.
Define
\[
\alpha = \sum_{k=1}^{3} \operatorname{adim}_k(G_1), \qquad
\beta = \sum_{k=1}^{3} \operatorname{adim}_k(G_2).
\]
Let $n = \alpha + \beta$.
Consider the vector space $V = \mathbb{R}^n$ and define a linear operator $T : V \to V$ whose matrix representation with respect to the standard basis is the block matrix
\[
T =
\begin{pmatrix}
\alpha I_{\alpha} & J_{\alpha \times \beta} \\
0 & \beta I_{\beta}
\end{pmatrix},
\]
where $I_m$ denotes the $m \times m$ identity matrix and $J_{\alpha \times \beta}$ denotes the $\alpha \times \beta$ matrix with all entries equal to $1$.
Determine the dimension of the eigenspace corresponding to the eigenvalue $\beta$.
|
\[5\]
|
multi_image
|
[
{
"bytes": "b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\x05\\x10\\x00\\x00\\x04\\xbd\\x08\\x02\\x00\\x00\\x00\\xb0\\x0b\\xe2W\\x00\\x00g/caBX\\x00\\x00g/jumb\\x00\\x00\\x00\\x1ejumdc2pa\\x00\\x11\\x00\\x10\\x80\\x00\\x00\\xaa\\x008\\x9bq\\x03c2pa\\x00\\x00\\x00g\\tjumb\\x00\\x00\\x00Gjumdc2ma\\x00\\x11\\x00\\x10\\x80\\x00\\x00\\xaa\\x008\\x9bq\\x03urn:c2pa:a36c7d11-261c-4e8c-8581-8b67f170790b\\x00\\x00\\x00\\x16\\xe5jumb\\x00\\x00\\x00)jumdc2as\\x00\\x11\\x00\\x10\\x80\\x00\\x00\\xaa\\x008\\x9bq\\x03c2pa.assertions\\x00\\x00\\x00\\t\\xd1jumb\\x00\\x00\\x00;jumd@\\xcb\\x0c2\\xbb\\x8aH\\x9d\\xa7\\x0b*\\xd6\\xf4\\x7fCi\\x13c2pa.icon\\x00\\x00\\x00\\x00\\x18c2sh\\x7f\\xf1\\xfd\\xbboX\\xf2\\xbfB\\xb3\\xec\\xaax\\x89\\\\G\\x00\\x00\\x00\\x17bfdb\\x00image/svg+xml\\x00\\x00\\x00\\twbidb<svg width=\"716\" height=\"716\" viewBox=\"0 0 716 716\" fill=\"none\" xmlns=\"http://www.w3.org/2000/svg\">\\n<path d=\"M508.749 317.399C516.777 287.314 508.991 253.884 485.389 230.282C461.788 206.681 428.36 198.895 398.273 206.923C376.231 184.928 343.39 174.956 311.148 183.596C278.906 192.234 255.45 217.292 247.36 247.361C217.291 255.451 192.233 278.91 183.595 311.149C174.957 343.391 184.927 376.232 206.924 398.274C198.896 428.359 206.683 461.789 230.284 485.391C253.885 508.992 287.313 516.779 317.401 508.75C339.442 530.745 372.286 540.717 404.525 532.079C436.767 523.441 460.223 498.384 468.313 468.315C498.383 460.224 523.44 436.766 532.078 404.526C540.716 372.285 530.747 339.443 508.749 317.402V317.399ZM470.899 244.776C486.892 260.77 493.488 282.601 490.687 303.412L415.577 260.046C412.411 258.218 408.509 258.218 405.345 260.046L317.401 310.82V277.526C317.401 275.191 318.652 273.005 320.676 271.837L387.644 233.174C414.178 218.353 448.346 222.223 470.901 244.776H470.899ZM357.837 311.144L398.275 334.491V381.185L357.837 404.532L317.398 381.185V334.491L357.837 311.144ZM264.776 269.693C265.207 239.305 285.644 211.649 316.453 203.393C338.3 197.54 360.505 202.744 377.127 215.573L302.014 258.937C298.848 260.764 296.898 264.144 296.898 267.798V369.346L268.065 352.699C266.043 351.531 264.776 349.353 264.776 347.017V269.691V269.693ZM203.391 316.454C209.244 294.608 224.854 277.978 244.276 269.999V356.73C244.276 360.384 246.226 363.763 249.392 365.591L337.337 416.365L308.503 433.013C306.481 434.181 303.961 434.188 301.939 433.02L234.971 394.357C208.868 378.789 195.138 347.261 203.391 316.454ZM244.775 470.9C228.781 454.906 222.186 433.075 224.986 412.264L300.096 455.63C303.263 457.457 307.164 457.457 310.328 455.63L398.273 404.856V438.149C398.273 440.485 397.022 442.671 394.997 443.839L328.029 482.502C301.495 497.322 267.327 493.452 244.772 470.9H244.775ZM450.897 445.982C450.466 476.371 430.029 504.027 399.22 512.283C377.373 518.136 355.168 512.932 338.547 500.102L413.659 456.738C416.826 454.911 418.775 451.532 418.775 447.877V346.329L447.609 362.977C449.631 364.145 450.897 366.323 450.897 368.659V445.985V445.982ZM512.282 399.221C506.429 421.068 490.819 437.697 471.397 445.676V358.946C471.397 355.292 469.448 351.912 466.281 350.085L378.336 299.311L407.17 282.663C409.192 281.495 411.712 281.487 413.734 282.655L480.702 321.318C506.805 336.887 520.536 368.415 512.282 399.221Z\" fill=\"black\"/>\\n</svg>\\n\\x00\\x00\\x01Ejumb\\x00\\x00\\x00Ajumdcbor\\x00\\x11\\x00\\x10\\x80\\x00\\x00\\xaa\\x008\\x9bq\\x13c2pa.actions.v2\\x00\\x00\\x00\\x00\\x18c2sh3\\xbb`9\\xa4\\xd2\"m\\xda\\x84`\\xdd\\xe1u\\x85\\x01\\x00\\x00\\x00\\xfccbor\\xa1gactions\\x82\\xa4factionlc2pa.createddwhen\\xc0t2026-05-11T00:00:00ZmsoftwareAgent\\xa2dnameigpt-imagegversiongpre-2.0qdigitalSourceTypexFhttp://cv.iptc.org/newscodes/digitalsourcetype/trainedAlgorithmicMedia\\xa2factionnc2pa.converteddwhen\\xc0t2026-05-11T00:00:00Z\\x00\\x00\\n\\xdbjumb\\x00\\x00\\x00Ijumdcbor\\x00\\x11\\x00\\x10\\x80\\x00\\x00\\xaa\\x008\\x9bq\\x13c2pa.certificate-status\\x00\\x00\\x00\\x00\\x18c2sh9\\xe3\\xc8n\\xf9;\\xd1\\xcb\\x88\\xd8\\xaf\\xdc\\x1dk\\xc9\\xfd\\x00\\x00\\n\\x8acbor\\xa1hocspVals\\x8
|
5
|
306425
|
Math
|
Differential geometry
|
1718
|
Determine the value of the surface integral
\[
\iint_{S}\left(x^2+y^2+z^2\right)\,dA,
\]
where
\[
S:\quad x^2+y^2+z^2=N,
\]
and
\[
N=C_4(G)\cdot C_5(G)+1,
\]
with \(G\) denoting the graph shown in the attached image.
Consider a simple and undirected graph \(G\) with vertex set \(V=V(G)\). For an integer \(k\ge1\), a \(k\)-dominating set of \(G\) is a set \(S\subseteq V(G)\) such that each vertex in \(V(G)\setminus S\) is adjacent to at least \(k\) vertices in \(S\).
A \(k\)-coalition refers to a pair of disjoint vertex sets that jointly constitute a \(k\)-dominating set of the graph, meaning that every vertex not in the set has at least \(k\) neighbors in the set.
A \(k\)-coalition partition of a graph is a partition of the vertex set in which each set is either a \(k\)-dominating set with exactly \(k\) vertices or forms a \(k\)-coalition with another set in the partition.
The maximum number of sets in a \(k\)-coalition partition is called the \(k\)-coalition number of the graph and is denoted by
\[
C_k(G).
\]
|
\[7396\pi\]
|
single_image
|
[
{
"bytes": "b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\x02.\\x00\\x00\\x02+\\x08\\x06\\x00\\x00\\x00\\xc8\\x87\\x12l\\x00\\x00\\x00\\x01sRGB\\x00\\xae\\xce\\x1c\\xe9\\x00\\x00\\x00\\x04gAMA\\x00\\x00\\xb1\\x8f\\x0b\\xfca\\x05\\x00\\x00\\x00\\tpHYs\\x00\\x00\\x0e\\xc3\\x00\\x00\\x0e\\xc3\\x01\\xc7o\\xa8d\\x00\\x00\\xff\\xa5IDATx^\\xec\\xfdw\\x98\\x1c\\xc5\\xd56\\x8c\\xdf53\\x1b\\x95\\x03\\x12\\x02\\x04\"\\x1b\\x10\"\\x0b09\\xdb\\xe4`\\x1b\\xe3@N\\xf6c0\\xc6\\x11?\\xb6\\t\\xb6\\x01c\\xc08\\x91L\\x0e\\x02L\\x0e\\x02\\x9136\\x19\\x93\\x11 \\xb2\"J\\xbb\\xd2\\xa6\\x99\\xe9\\xfa\\xfdQ\\xe7T\\x9d:]=\\xbb\\xe2y\\xbe\\xf7\\xfa\\xde\\xdf\\xa7{\\xaf\\xde\\xe9\\xae:ur\\x85\\x0e\\xd3c\\xfa\\xfa\\xfa,\\x96c9\\x96c9\\x96c9\\x96c9\\xfe/@I\\x17,\\xc7r,\\xc7r,\\xc7r,\\xc7r\\xfc\\xbf\\x15\\xcb\\x17.\\xcb\\xb1\\x1c\\xcb\\xb1\\x1c\\xcb\\xb1\\x1c\\xcb\\xf1\\x7f\\r\\xcc\\xff\\x8d\\xb7\\x8a\\xac\\xfd\\xbfN\\xe5\\xe5X\\x8e\\xe5X\\x8e\\xe5X\\x8e\\xff\\xc7`\\x8c\\xd1E\\xff\\x7f\\x8b\\xff\\xab\\xae\\xb8Xk\\x97/Z\\x96c9\\x96c9\\x96c9\\x14\\xfe\\xbf4?\\x9a\\x05\\x0b\\x16\\xfc\\x7f\\xc3\\xd2\\xe5X\\x8e\\xe5X\\x8e\\xe5X\\x8e\\xe5\\xf8\\x7f\\x1d\\x06\\x0f\\x1e\\xac\\x8b\\x1a\\xc2\\xd4\\xb3\\xba\\x85\\xcd_f\\xe2\\x95\\x9b1\\x06\\xc6\\x00z!\\'Wv\\xc6\\x18Xk\\x93\\xb4\\\\\\'\\xf91B\\x1b\\x03X\\x00&\\xf0\\xd5\\xf4L[\\xb4\\xcfr\\xb8Mj\\xbf\\x88^C\\xf2g\\xbd\\n\\xeb\\x95\\x8d\\x92\\xb7\\xa6K\\xc1\\x8bI\\xab\\x92\\xb4\\xd5)\\x94o mJ\\xd9V\\xdc\\xd6\\xc2R\\x0eH\\xbf\\xc1\\xf3qm\\xd86.\\x8f\\xf5\\xc9\\x88W\\x89\\xca\\x9dm\\t5<\\x82\\xbf$\\x1d\\xe7R>vL\\xd7\\x9f\\x1fR>+\\xe2\\xc54\\xae\\x8c\\xe9 |\"\\xe5r\\xbc\\xf2r\\xa5n\\xcc\\x0bJ\\x17\\xa2$\\xde\\x92.\\xcf\\x8f\\xeb\\x1c\\x9d\\x8bA\\x9e\\x97\\xa6+.\\x93\\xf2\\x8a|\"\\xe9\\x98V\\x96s\\x9b\\xa0\\x0f\"\\xbd\\xe38\\xa6\\xc1|\\xb4-)]$\\x82\\xefeY\\xac\\xab\\xac\\x0fU\\xb1\\x0f\\x99&\\xf6\\x7f>\\x07\\xb2\\x8c\\xf39\\x94I\\xdd\\x19\\xd2\\x8e\\xc6<Y\\'\\xd6#e/\\x1b\\xa0\\xcbc[\\x19\\xd2\\x7fE9\\xa4!u\\x0b1\\xcd\\xc7M\\xeb\\x1b>\\x03\\xe2\\x18\\x86\\xfe\\xc2t\\xda7\\xecG\\xa6\\x05\\xe9#u\\x19\\xa8\\x1dRww\\x9c\\xf7Q\\xde6\\xb6!\\x96!u\\x08\\xed\\xe22.\\x87\\x90-m\\x8ad\\x1b\\xc0\\xf8\\xbc\\x93`\\xdb\\xd3\\xb6j_8\\x7f\\xc64^&\\x0c,28I\\x8e\"\\xae\\x13m`a\\nsn\\xd9\\xa1}\\xc3~\\xd1(\\x96\\xe5\\xe8\\r\\x8d\\xad\\x0b\\x17.Z\\xf6\\x85K\\x96e\\xb9\\xb0jp2F\\x81\\xa4\\xe0hH\\x1b\\x8a\\xf4\\x0e\\x81s0\\xa6\\xff\\x85\\xcb\\xb2` \\x01Z\\x16\\xe7\\xfb:\\xe1 )C\\xcb\\x93\\xc7\\xb2\\xbd\\xa6a\\xbf\\xea\\xc1\\xa1?\\xdd%4O.\\x93\\xf6h\\x1d\\xa4l\\xdd^\\x1ek>L#\\x07\\x9e\\xfe\\xa0\\xfd\\x0c\\x00\\xa5\\x12\\xeb\\xe2\\x8e\\xa5\\x1f\\xff\\xb7\\xa0m\\xe9\\xef8\\xf8%\\xe8\\xeb\\xf410&\\xb6W\\xf3\\xd1\\xbc4\\r\\x84<I+\\xed\\xd5e\\xecg\\xee\\x9e\\xb2\\xbd\\x86\\x96\\x1fl\\xe1c_\\x03\\xd9\\xc3\\x8d\\x18\\xd4\\xf3v\\xa7\\xa1\\xeb$}\\xdef\\xbf\\x97\\xab\\xd3\\xd060\\x82\\xcf\\x1c?\\xf6K\\x8ag\\xd0M\\xc6\\x8b\\xe9\\x03\\x06b#\\xf3\\x0e\\xbe\\x0c6Zk\\xfd\\xf8\\xa7y\\xe9c\\r\\xe6\\xabi\\\\;x\\xbb\\xd2\\xf5\\xc5\\xb9\\xd3\\xa8\\x8e\\xa1\\xed\\x90\\xe5\\xb2\\x8cy\\x05\\x9a\\xc0\\xaf\\x81iIp\\xdc\\x18\\xbc\\x9f\\x08\\xb5GJ\\xb7\"H\\xbb\\x8b\\xe1\\xf2%*\\x11\\xb6J\\x1e\\xec\\xcbTY\\x11\\x8cq\\xf3\\x17/\\x16x\\xe1\\x90\\x8aG\\x114\\xed\\xb2\\xf8\\xc0\\xa7\\xb7\\'3\\xae\\xc8\\xd2\\x1e\\x95\\xcb\\x05\\x8d\\xb6I\\xfb1\\xe5\\x9f\\x94n\\x12\\x9ag\\x8c|\\x0c\\x18\\x8b\\x16-\\xfb\\xc2\\xa5\\xc4\\xcc\\x8a\\xd9\\xe6\\x93\\xcc\\xd0\\xaa\\x8e\\xf7U\\xad\\xd8\\xa8\\x84\\x8cgZ\\xfdi\\xadu\\x0e\\x15rd\\x1bcLR9\\xa7E\\xe0\\xcb\\xd0\\x8e\\xb5j\\xe2\\xe42\\x089\\xbc\\x9f\\xae\\xa72\\xa1\\xa3N4\\x861.9\\x8a\\xfd#\\xcbx\\x80`\\x1d\\xf3\\xba\\xa7\\xa0i\\xb4\\x0cm\\x8f\\xac\\xe7})+\\x94\\xe5\\x13\\x97\\xeb\\xd9\\x0f\\xae\\xbc\\xd8\\xdfz_\\xcb\\x90\\xf2eYZ\\xc7|\\xdcRH\\xd1\\x17\\xb5+*G\\xa1\\xbe\\xf0\\xf6\\xeaz\\xa9\\xb3F\\xf0Y\\x9c_6\\xd7\\xb9\\xb5_\\x82\\xfeZ\\xd5\"y\\x9ag\\xd8\\xe7\\t\\x9fu\\x90\\xb2\\xc21\\x84\\\\\\xd6QC\\xdb+i\\xa4_b8\\xf9\\xb2\\xce\\xf7\\xd9T\\x87n\\x00\\x16\\xcd~I\\xc5\\x81\\xf5`\\x9b\\xa9\\x05\\xb3\\xf0H\\xf9\\xb1\\xc8\\xb6\\xa8\\x
|
6
|
306423
|
Math
|
Group theory
|
1718
|
Let $G=(V(G),E(G))$ be a finite, simple, connected graph. A function
$f:V(G)\to \{0,1,2,3\}$ is called a modern Roman dominating function if every vertex labeled $0$ is adjacent to two vertices, one labeled $2$ and one labeled $3$, and every vertex labeled $1$ is adjacent to at least one vertex labeled $2$ or $3$. The minimum possible value of $\sum_{v\in V(G)} f(v)$ is called the modern Roman domination number and is denoted by $\gamma_{mR}(G)$.
Let $\odot$ denote the corona product of graphs. Let the graph in Figure 1 be denoted by $H_1$, the graph in Figure 2 by $H_2$, and the graph in Figure 3 by $K$.
Let $|V(H_1)|=n_1$ and $|V(H_2)|=n_2$. Define
\[
\mathcal{R} =
\mathbb{Z}_{\gamma_{mR}(H_1\odot K)} \times
\mathbb{Z}_{\gamma_{mR}(H_2\odot K)} \times
\mathbb{Z}_{\gamma_{mR}(C_{n_1+2}\odot K)} \times
\mathbb{Z}_{\gamma_{mR}(C_{n_2+2}\odot K)}.
\]
Determine the exact value of $|\mathcal{R}|$.
|
\[4423680\]
|
multi_image
|
[
{
"bytes": "b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\x04\\xe6\\x00\\x00\\x04\\xe6\\x08\\x02\\x00\\x00\\x00\\n\\x9e9}\\x00\\x00\\\\JcaBX\\x00\\x00\\\\Jjumb\\x00\\x00\\x00\\x1ejumdc2pa\\x00\\x11\\x00\\x10\\x80\\x00\\x00\\xaa\\x008\\x9bq\\x03c2pa\\x00\\x00\\x00\\\\$jumb\\x00\\x00\\x00Gjumdc2ma\\x00\\x11\\x00\\x10\\x80\\x00\\x00\\xaa\\x008\\x9bq\\x03urn:c2pa:c6bf8530-b689-4c77-8d34-f868f09d94c1\\x00\\x00\\x00\\x12\\xedjumb\\x00\\x00\\x00)jumdc2as\\x00\\x11\\x00\\x10\\x80\\x00\\x00\\xaa\\x008\\x9bq\\x03c2pa.assertions\\x00\\x00\\x00\\t\\xd1jumb\\x00\\x00\\x00;jumd@\\xcb\\x0c2\\xbb\\x8aH\\x9d\\xa7\\x0b*\\xd6\\xf4\\x7fCi\\x13c2pa.icon\\x00\\x00\\x00\\x00\\x18c2sh-fiI\\xd9Et\\x12\\x90=\\\\\\xc6,\\xb6\\x19\\xc5\\x00\\x00\\x00\\x17bfdb\\x00image/svg+xml\\x00\\x00\\x00\\twbidb<svg width=\"716\" height=\"716\" viewBox=\"0 0 716 716\" fill=\"none\" xmlns=\"http://www.w3.org/2000/svg\">\\n<path d=\"M508.749 317.399C516.777 287.314 508.991 253.884 485.389 230.282C461.788 206.681 428.36 198.895 398.273 206.923C376.231 184.928 343.39 174.956 311.148 183.596C278.906 192.234 255.45 217.292 247.36 247.361C217.291 255.451 192.233 278.91 183.595 311.149C174.957 343.391 184.927 376.232 206.924 398.274C198.896 428.359 206.683 461.789 230.284 485.391C253.885 508.992 287.313 516.779 317.401 508.75C339.442 530.745 372.286 540.717 404.525 532.079C436.767 523.441 460.223 498.384 468.313 468.315C498.383 460.224 523.44 436.766 532.078 404.526C540.716 372.285 530.747 339.443 508.749 317.402V317.399ZM470.899 244.776C486.892 260.77 493.488 282.601 490.687 303.412L415.577 260.046C412.411 258.218 408.509 258.218 405.345 260.046L317.401 310.82V277.526C317.401 275.191 318.652 273.005 320.676 271.837L387.644 233.174C414.178 218.353 448.346 222.223 470.901 244.776H470.899ZM357.837 311.144L398.275 334.491V381.185L357.837 404.532L317.398 381.185V334.491L357.837 311.144ZM264.776 269.693C265.207 239.305 285.644 211.649 316.453 203.393C338.3 197.54 360.505 202.744 377.127 215.573L302.014 258.937C298.848 260.764 296.898 264.144 296.898 267.798V369.346L268.065 352.699C266.043 351.531 264.776 349.353 264.776 347.017V269.691V269.693ZM203.391 316.454C209.244 294.608 224.854 277.978 244.276 269.999V356.73C244.276 360.384 246.226 363.763 249.392 365.591L337.337 416.365L308.503 433.013C306.481 434.181 303.961 434.188 301.939 433.02L234.971 394.357C208.868 378.789 195.138 347.261 203.391 316.454ZM244.775 470.9C228.781 454.906 222.186 433.075 224.986 412.264L300.096 455.63C303.263 457.457 307.164 457.457 310.328 455.63L398.273 404.856V438.149C398.273 440.485 397.022 442.671 394.997 443.839L328.029 482.502C301.495 497.322 267.327 493.452 244.772 470.9H244.775ZM450.897 445.982C450.466 476.371 430.029 504.027 399.22 512.283C377.373 518.136 355.168 512.932 338.547 500.102L413.659 456.738C416.826 454.911 418.775 451.532 418.775 447.877V346.329L447.609 362.977C449.631 364.145 450.897 366.323 450.897 368.659V445.985V445.982ZM512.282 399.221C506.429 421.068 490.819 437.697 471.397 445.676V358.946C471.397 355.292 469.448 351.912 466.281 350.085L378.336 299.311L407.17 282.663C409.192 281.495 411.712 281.487 413.734 282.655L480.702 321.318C506.805 336.887 520.536 368.415 512.282 399.221Z\" fill=\"black\"/>\\n</svg>\\n\\x00\\x00\\x01Ejumb\\x00\\x00\\x00Ajumdcbor\\x00\\x11\\x00\\x10\\x80\\x00\\x00\\xaa\\x008\\x9bq\\x13c2pa.actions.v2\\x00\\x00\\x00\\x00\\x18c2sh\\x98059\\x1e~\\xf4\\x9e\\xfe?\\xf1g\\x8a\\xb9S-\\x00\\x00\\x00\\xfccbor\\xa1gactions\\x82\\xa4factionlc2pa.createddwhen\\xc0t2026-04-28T00:00:00ZmsoftwareAgent\\xa2dnameigpt-imagegversiongpre-2.0qdigitalSourceTypexFhttp://cv.iptc.org/newscodes/digitalsourcetype/trainedAlgorithmicMedia\\xa2factionnc2pa.converteddwhen\\xc0t2026-04-28T00:00:00Z\\x00\\x00\\x06\\xe3jumb\\x00\\x00\\x00Ijumdcbor\\x00\\x11\\x00\\x10\\x80\\x00\\x00\\xaa\\x008\\x9bq\\x13c2pa.certificate-status\\x00\\x00\\x00\\x00\\x18c2sh\\x07\\xc3!\\x8dU\\x03^\\x18g{\\xa0\\xbc\\xc3~\\xa3;\\x00\\x00\\x06\\x92cbor\\xa1hocspVals\\x81y\\x06|MIIE2AoBAKCCBNEw
|
7
|
306419
|
Math
|
Combinatorics
|
1428
|
Let \(\lambda\) be the Young diagram shown on the left in the figure, and let \(\mu\) be the Young diagram shown on the right in the figure. For a Young diagram \(\nu\), let \(\nu^\vee_j\) denote the length of its \(j\)-th column, with \(\nu^\vee_j=0\) for all sufficiently large \(j\). Define
\[
[z]:=z^{-1/2}-z^{1/2}.
\]
Define
\[
N^{(0\mid 2)}_{\lambda,\mu}(u\mid q,\kappa)
:=
\prod_{\substack{1\le i\le j\\0\le \ell<\lambda^\vee_j-\lambda^\vee_{j+1}\\\lambda^\vee_{j+1}-\mu^\vee_i+\ell\equiv 0\pmod 2}}
\left[u\,q^{\,j-i}\kappa^{\,\lambda^\vee_{j+1}-\mu^\vee_i+\ell}\right]
\;
\prod_{\substack{1\le i\le j\\0\le \ell<\mu^\vee_j-\mu^\vee_{j+1}\\\lambda^\vee_i-\mu^\vee_j+\ell\equiv 0\pmod 2}}
\left[u\,q^{\,i-j-1}\kappa^{\,\lambda^\vee_i-\mu^\vee_j+\ell}\right].
\]
Determine the exact fully factorized expression for \(N^{(0\mid 2)}_{\lambda,\mu}(u\mid q,\kappa)\).
|
\[
[u]^2\,[u q^{-1}]\,[u q]\,[u q\,\kappa^{-2}]\,[u q^{2}\kappa^{-2}]\,[u q^{-2}\kappa^{2}]
\]
|
multi_image
|
[
{
"bytes": "b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\x03\\xb1\\x00\\x00\\x02/\\x08\\x06\\x00\\x00\\x00\\xa28\\\\\\xde\\x00\\x00\\x00\\x01sRGB\\x00\\xae\\xce\\x1c\\xe9\\x00\\x00\\x00\\x04gAMA\\x00\\x00\\xb1\\x8f\\x0b\\xfca\\x05\\x00\\x00\\x00\\tpHYs\\x00\\x00\\x0e\\xc3\\x00\\x00\\x0e\\xc3\\x01\\xc7o\\xa8d\\x00\\x006mIDATx^\\xed\\xddm\\xac\\xadg]\\xe7\\xf1E\\xa1<\\x14t\\xda\\xc28@\\xad\\xc28\\xd0N\\x0f\\xa7L\\xd5)\\xc5\\x84Q!X\\x1ej\\x0b%\\x82\\x14*\\xf2P\\x88J\\xdaR\\xa00\\xd0\\x16[(\\x05;LS\\x04\\xa1\\x10%\\xc3\\xe8\\x10b\\xc4D\\xd4\\x04\\x94\\x18%1&\\x1a}c\\xd4W\\x92a^\\xb4\\x01464\\x87\\n\\xb5\\xf7d\\xed\\xf6\\xde\\xe7\\xe6\\x7fLfMr\\xad\\xebZ?\\xd6\\xe7\\x93\\xfc\\x02\\xe9\\xde\\xaf\\xf6\\x9d\\xfb\\xbf\\xfb\\xed9g\\x9f\\xd5\\x04\\x00\\xc0\\xff\\xd3j\\xb5:a\\x00\\xf4\\xe7\\xfa\\x02\\x00l\\xa0\\x06\\xac\\x88\\x05\\x18\\xc3\\xf5\\x05\\x00\\xd8@\\rX\\x11\\x0b0\\x86\\xeb\\x0b\\x00\\xb0\\x81\\x1a\\xb0\"\\x16`\\x0c\\xd7\\x17\\x00`\\x035`E,\\xc0\\x18\\xae/\\x00\\xc0\\x06j\\xc0\\x8aX\\x801\\\\_\\x00\\x80\\r\\xd4\\x80\\x15\\xb1\\x00c\\xb8\\xbe\\x00\\x00\\x1b\\xa8\\x01+b\\x01\\xc6p}\\x01\\x006P\\x03V\\xc4\\x02\\x8c\\xe1\\xfa\\x02\\x00l\\xa0\\x06\\xac\\x88\\x05\\x18\\xc3\\xf5\\x05\\x00\\xd8@\\rX\\x11\\x0b0\\x86\\xeb\\x0b\\x00\\xb0\\x81\\x1a\\xb0\"\\x16`\\x0c\\xd7\\x17\\x00`\\x035`E,\\xc0\\x18\\xae/\\x00\\xc0\\x06j\\xc0\\x8aX\\x801\\\\_\\x00\\x80\\r\\xd4\\x80\\x15\\xb1\\x00c\\xb8\\xbe\\x00\\x00\\x1b\\xa8\\x01+b\\x01\\xc6p}\\x01\\x006P\\x03V\\xc4\\x02\\x8c\\xe1\\xfa\\x02\\x00l\\xa0\\x06\\xac\\x88\\x05\\x18\\xc3\\xf5\\x05\\x00\\xd8@\\rX\\x11\\x0b0\\x86\\xeb\\x0b\\x00\\xb0\\x81\\x1a\\xb0\"\\x16`\\x0c\\xd7\\x17\\x00`\\x035`E,\\xc0\\x18\\xae/\\x00\\xc0\\x06j\\xc0\\x8aX\\x801\\\\_\\x00\\x80\\r\\xd4\\x80\\x15\\xb1\\x00c\\xb8\\xbe\\x00\\x00\\x1b\\xa8\\x01+b\\x01\\xc6p}\\x01\\x006P\\x03V\\xc4\\x02\\x8c\\xe1\\xfa\\x02\\x00l\\xa0\\x06\\xac\\x88\\x05\\x18\\xc3\\xf5\\x05\\x00\\xd8@\\rX\\x11\\x0b0\\x86\\xeb\\x0b\\x00\\xb0\\x81\\x1a\\xb0\"\\x16`\\x0c\\xd7\\x17\\x00`\\x035`E,\\xc0\\x18\\xae/\\x00\\xc0\\x06j\\xc0\\x8aX\\x801\\\\_\\x00\\x80\\r\\xd4\\x80\\x15\\xb1\\x00c\\xb8\\xbe\\x00\\x00\\x1b\\xa8\\x01+b\\x01\\xc6p}\\x01\\x006P\\x03V\\xc4\\x02\\x8c\\xe1\\xfa\\x02\\x00l\\xa0\\x06\\xac\\x88\\x05\\x18\\xc3\\xf5\\x05\\x00\\xd8@\\rX\\x11\\x0b0\\x86\\xeb\\x0b\\x00\\xb0\\x81\\x1a\\xb0\"\\x16`\\x0c\\xd7\\x17\\x00`\\x035`E,\\xc0\\x18\\xae/\\x00\\xc0\\x06j\\xc0\\x8aX\\x801\\\\_\\x00\\x80\\r\\xd4\\x80\\x15\\xb1\\x00c\\xb8\\xbe\\x00\\x00\\x1b\\xa8\\x01+b\\x01\\xc6p}\\x01\\x006P\\x03V\\xc4\\x02\\x8c\\xe1\\xfa\\x02\\x00l\\xa0\\x06\\xac\\x88\\x05\\x18\\xc3\\xf5\\x05\\x00\\xd8@\\rX\\x11\\x0b0\\x86\\xeb\\x0b\\x00\\xb0\\x81\\x1a\\xb0\"\\x16`\\x0c\\xd7\\x17\\x00`\\x035`E,\\xc0\\x18\\xae/\\x00\\xc0\\x06j\\xc0\\x8aX\\x801\\\\_\\x00\\x80\\r\\xd4\\x80\\x15\\xb1\\x00c\\xb8\\xbe\\x00\\x00\\x1b\\xa8\\x01+b\\x01\\xc6p}\\x01\\x006P\\x03V\\xc4\\x02\\x8c\\xe1\\xfa\\x02\\x00l\\xa0\\x06\\xac\\x88\\x05\\x18\\xc3\\xf5\\x05\\x00\\xd8@\\rX\\x11\\x0b0\\x86\\xeb\\x0b\\x00\\xb0\\x81\\x1a\\xb0\"\\x16`\\x0c\\xd7\\x17\\x00`\\x035`E,\\xc0\\x18\\xae/\\x00\\xc0\\x06j\\xc0\\x8aX\\x801\\\\_\\x00\\x80\\r\\xd4\\x80\\x15\\xb1\\x00c\\xb8\\xbe\\x00\\x00\\x1b\\xa8\\x01+b\\x01\\xc6p}\\x01\\x006P\\x03V\\xc4\\x02\\x8c\\xe1\\xfa\\x02\\x00l\\xa0\\x06\\xac\\x88\\x05\\x18\\xc3\\xf5\\x05\\x00\\xd8@\\rX\\x11\\x0b0\\x86\\xeb\\xcb0\\xf7\\xdf\\x7f\\x7f\\xfdG\\x04\\xf1\\xfcr\\xad\\x9f\\x9d\\xe7\\x07\\xff\\xffj\\xc0\\x8a\\xd8\\xf6\\xbe\\xfc\\xe5/O\\x7f\\xf4G\\x7f4\\xfd\\xc9\\x9f\\xfc\\x89\\x85\\xed\\x8b_\\xfc\\xe2\\xf4{\\xbf\\xf7{\\xd3\\'?\\xf9\\xc9\\x83\\xfd\\xee\\xef\\xfe\\xee\\xc1?\\xab\\x9fg\\xbb\\xbf\\xf5;\\xf8\\xf7\\x7f\\xff\\xf7\\xf5\\xf5\\xdc)\\x91\\xd7w}\\xe0n\\xba\\xe9\\xa6\\xe9\\xedo\\x7f\\xfb\\xf4\\x8ew\\xbc\\xc3\\x82\\xf6\\xcew\\xbesz\\xf3\\x9b\\xdf<\\xbd\\xe2\\x15\\xaf\\x98^\\xfa\\xd2\\x97N\\xafz\\xd5\\xab\\x0e\\xfeY\\xfd<\\xdb\\xcd\\xad\\x9f\\xd5\\xd5W_=\\xbd\\xfc\\xe5/?x~\\xaf{\\xdd\\xeb\\xa6\\xeb\\xae\\xbb\\xee\\x84\\xcf\\xb3\\xdd\\xdc\\xfa\\xf9\\xbd\\xf1\\x8do\\x9c^\\xf6\\xb2\
|
8
|
306415
|
Math
|
Number theory
|
1718
|
Determine the number of primitive Dirichlet characters modulo
\[
N=\chi'_{NK}(G)^{\,3}+2,
\]
where \(G\) denotes the graph shown in the attached image.
In the attached image, the colored points denote the vertices of the graph and the line segments joining pairs of points denote the edges of the graph.
Let \(G=(V(G),E(G))\) be a simple graph, and let
\[
c_E:E(G)\to\mathbb{N}
\]
be a proper edge coloring of \(G\), that is, no two adjacent edges receive the same color.
For a positive integer \(n\), define a mapping
\[
c_V':V(G)\to\mathbb{Z}_n
\]
by
\[
c_V'(v)\equiv \sum_{e\in E_v} c_E(e)\pmod n,
\]
where \(E_v\) denotes the set of edges incident with the vertex \(v\in V(G)\).
If the induced vertex labeling \(c_V'\) is a proper vertex coloring of \(G\), that is,
\[
c_V'(u)\neq c_V'(v)
\]
for every edge \(uv\in E(G)\), then the edge coloring \(c_E\) is called an NK-labeling of \(G\).
The smallest positive integer \(n\) for which \(G\) admits an NK-labeling is called the NK-chromatic index of \(G\), denoted by
\[
\chi'_{NK}(G).
\]
|
\[125\]
|
single_image
|
[
{
"bytes": "b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\x02,\\x00\\x00\\x02&\\x08\\x06\\x00\\x00\\x00p\\xec\\xd1\\x8f\\x00\\x00\\x00\\x01sRGB\\x00\\xae\\xce\\x1c\\xe9\\x00\\x00\\x00\\x04gAMA\\x00\\x00\\xb1\\x8f\\x0b\\xfca\\x05\\x00\\x00\\x00\\tpHYs\\x00\\x00\\x0e\\xc3\\x00\\x00\\x0e\\xc3\\x01\\xc7o\\xa8d\\x00\\x00\\xff\\xa5IDATx^\\xec\\xfdy\\xbcmWU&\\x0c?s\\xed\\xd3\\xdc\\xdc\\xdc67=\\xa4!\\rI $@\\x12\\x04$\\xf4\\x84 A-\\x9b\\xa2\\x14\\xab\\xa0P\\xb1\\x17\\xf5+Q\\xc4\\xaa\\xc2\\xa6T@T@\\x91`\\xbd\\xa5\"\\x8ah\\x89\\x1d =\\xa4#-\\tm:H\\xdf\\xdc\\x9b\\xdc\\xdc\\xe6\\x9c\\xb3\\xf7\\x9a\\xef\\x1f\\xa3\\x99\\xcf\\x1ck\\xae}N\\xf4\\xfd\\xbe\\xef\\xf7\\xd6\\x8f\\xe7fg\\xaf5\\xe7\\x98\\xa3y\\xc6\\x98s\\xcd\\xbdvs\\xd2\\xea\\xeaj\\xc6\\xffA\\xc8\\xf9\\xff\\xa8p\\xbe\\x89o\\xe2\\x9b\\xf8&\\xbe\\x89o\\xe2\\xdf\\x84\\x94Rl\\xfa\\x7f%\\xba\\xd8\\xf0\\xffV\\xe4\\x9c\\xbf\\xb9Y\\xf9&\\xbe\\x89o\\xe2\\x9b\\xf8&\\xbe\\x89\\x80\\xffS\\xae\\x8f\\xa9u\\x87\\xe5\\xa1\\x87\\x1e\\x8aM\\xdf\\xc47\\xf1M|\\x13\\xdf\\xc47\\xf1M|\\x13\\xff?\\xc5\\x96-[\\xfcxt\\xc3\\xb2c\\xc7\\x0ed\\x00\\xc8\\x99n\\'\\xc9\\xb3\\xed\\xd4R\\x92\\xc7F7n)%\\x88R C\\xf4\\xda\\xceo\\xec\\x96\\x15\\xb7\\xf3\\x0e\\x91\\xc7\\xacwlv\\xa0\\xfa\\xe2qBB6\\xc7Fd\\x0c\\xf3|\\x1d\\xc3<}\\xf3\\x10\\xe3\\x88\\xe3Z\\xbexLA>\\x9eG]\\xac\\x87\\xed\\xdas\\xf4\\xa5\\x05\\x961\\xb9\\x96\\xddV\\xbb!\\xda\\x8erq||f=\\xac\\xc3\\xc6J\\x03\\x80<\\xd4\\xd5\\x1ak\\xe3L\\xce\\xceY.\\xb6G\\x98\\\\\\xcb\\x1f;\\x9e\\'\\xc7\\xfd\\x98S\\x0f-\\xb4\\xe4\\xd8o\\xeb\\x1b\\xf3\\x85\\x11\\xfd\\x8a2\\xd1_\\xb6\\x11e\\x19\\x1cW\\x0b<v=}c1\\xb1o\\xdc\\x9e\\x1b|\\xb7\\xceYO\\xdf\\xf7\\xb5\\xde\\\\\\xafg\\x06\\x1e\\xd7:7\\xc4q\\xf1|\\x1eb\\xbc\\xdc\\xbeQ=\\xb6\\xfe%$ \\r\\xf5X|\\xdc6/Nn3\\xd9\\x96\\x8c\\x81e\\xed\\x1cd\\xa7\\xd2%\\xce\\x8c\"A}K\\xb5\\xef\\x91\\x871~b\\x9b\\x9f[S\\xd2\\x07\\x80\\x94I\\xb6\\x1d\\xda\\x86\\xf8ka\\x8c\\x8b(c\\xdcT>\\xd3\\xf5\\xcc9\\xd4\\xfa4~\\x12\\x80\\x9c\\xe0qE\\x1d\\x1b\\x85\\xf9\\t\\xcacn\\xcc\\xa9\\x08\\xf1AcD\\x82\\x88e\\xdfK\\x88\\xaf\\x19\\xbb\\xf7\\xec\\xa96,\\xa3o\\t\\x99\\xba\\xda\\xa09!\\x0fA)4C3\\xf0\\x86\\xdf\\xad\\xa0*\\xe2\\xad=\\xa83b\\x0c~lE\\x1a\\xdb\\x03\\x9c(\\xb5\\xc5\\xc5\\xe4\\xc8\\xf2\\xc8\\xfd\\xb0H\\x1c\\x8da0\\x99\\xb6\\xe96\\x1a\\xb2\\xe6#r\\xb1\\x19\\xe3\\x89\\xe7hp\\x1f\\xcf\\rI\\'\\xac=\\xd0\\x925ZIfT\\x165\\x1fcr\\xdc\\x1e\\xfd\\x8f\\xb2\\x00\\xd0\\xf7}\\xb3&<w\\xe1\\xd9\\x10k+\\xf6s\\xdfz`\\xb9\\x1cn\\xad\\xa60A\\xa3\\x1d\\xeb\\xb7c>G\\x90\\x8fr\\xdc\\xce\\x0f\\x8ei\\xcc\\xeez\\x88\\xf9\\x1cpA\\xea\\xb8\\xaf\\x92\\xcbT\\xa7s\\x10c\\xb66\\xe6#\\xf6\\x8fa^\\xbc1\\x96AL\\x04\\x1e\\xcf|\\x1b\\xbf\\xad\\x98\\xc7\\x9e[h\\xf9\\x17\\x11\\xc7\\xb7r\\x0c\\x8c\\xacO\\x8a\\x18\\x07?G\\x1f\\xc6\\xcey\\xb3\\x02`\\xb8\\x1e\\xd19\\xe7\\xcb\\xc6W\\\\\\xa9p\\xf4!\\xc6\\xca\\xf0\\xbe\\\\\\xd7]\\x1c\\x9bs\\x96\\xf5X_l\\xb4\\x90\\xb3lV\\xe4\\xa4\\xb4[N\\xed\\xe1\\xb2\\x01cqU\\x1c\\xe4$\\x9b\\x15\\xbf\\xda\\x97\\xbe&\\x1a\\xfc!\\xd8\\x8a\\x88}}\\xdfW\\xfd\\xe0\\xfc\\x85\\x1a\\xf0\\xcd\\n\\x12\\xa0\\xc3\\x12\\xf4fA\\x96\\xb3\\x9c\\x93\\xeeX\\x144\\x97\\xa3O\\xb1\\xae\\x10d\\x98/>N\\xbaA\\x8a\\xf2\\xd6\\x8f$\\xdc$J|\\xa2\\xbdE\\x8b\\x9a\\xd1\\r\\x8b(P\\x8d\\x83v:\\x1bq\\x9c\\x9f\\xa5\\xb3\\xad2\\x16\\x11?[\\xd2\\xe2\\x84\\xb5`#)\\xa5~\\xea\\xe2\\x8c>\\xdan\\x93\\xdb\\x1c\\xf6\\n\\xc3\\xfcT\\xb1*\\x16\\x83\\x13^\\xee\\x1c\\xc1\\xf4\\xe9F\\x83\\xed{l\\x91S\\x1d\\xeb~X8\\xea\\xab\\xf3\\x10\\n\\xb3\\x05\\x93\\x896\\xab\\x18G\\xc6\\xb3,\\xc7`m\\x8e\\x8a.\\xe29\\xd56\\x10\\xc6E\\x1fb\\x1f\\xfb\\xe4\\\\\\x8d\\\\<\\xd0\\xd0\\x07\\xd493\\x0c\\xe4\\x1a\\xc3*\\x0c\\xc4K\\x1e0\\xe2C\\xabo\\x0cc\\xb1E?Y\\x8e\\xfb=?\\r\\xd9\\x08\\xe3\\xb5\\xd5\\xdf\\xf29\\xa5\\xb2\\x90\\x8c\\xc9\\xc5\\xfa\\x8d\\xfac^\\xa3\\xaf\\x86\\xd8\\xd7\\xf2q\\x1e\\xa2^\\xd3\\xc7\\xfe2w\\xf6\\x88|\\xb2\\\\|\\x8e\\xfd16\\x11\\x90G\\x8c%\\xf2\\xe5:\\xf5i=\\x7f\\x93\\xe6\\xc2\\xd6\\x17_\\x07\\xdd\\xec\\x90\\xaf\\x8a\\xc3\\xd0\\x1dcsY\\xbb\\x0ba5N\\xb4\\xb2\\x7fv\\x8e\\xe0\\xab!\\x8d\\xac\\x01\\x18\\x
|
9
|
306407
|
Math
|
Differential geometry
|
1428
|
Use the right-hand panel of the given figure.
Let $({\rm CP}^{2},\omega)$ be complex projective $2$-space with the Fubini--Study form normalized by
$$
\int_{{\rm CP}^{1}}\omega=1.
$$
Define
$$
\mu([z_{1}:z_{2}:z_{3}])
=
\frac{1}{2\sum_{j=1}^{3}|z_{j}|^{2}}
\bigl(|z_{1}|^{2},|z_{2}|^{2}\bigr),
$$
let
$$
\Delta=\mu({\rm CP}^{2}),
\qquad
\ell=\Delta\cap\{(x,x)\mid x\in {\bf R}\}.
$$
The hypersurface $\mu^{-1}(\ell)$ is preserved by the circle action
$$
e^{i\theta}\cdot[z_{1}:z_{2}:z_{3}]
=
[e^{i\theta}z_{1}:e^{-i\theta}z_{2}:z_{3}],
$$
and the corresponding symplectic reduction is a singular $2$-sphere $S$ with singular points
$$
p,q\in S
$$
lying over the two endpoints of $\ell$.
Let $\gamma_{B}\subset S\setminus\{p,q\}$ be the solid, non-dotted, simple closed curve shown in the right-hand panel of the figure. Let $P$ be the connected component of $S\setminus\gamma_{B}$ containing $p$, and assume that
$$
{\rm area}(P)=\frac13\,{\rm area}(S).
$$
Let $Y\subset{\rm CP}^{2}$ be the full circle-orbit preimage of $\gamma_{B}$.
For each $t\in {\bf R}/{\bf Z}$, define
$$
L_{t}
=
\left\{
\bigl[
(a_{1}+ia_{2})e^{i\pi t/3}:
(a_{1}-ia_{2})e^{i\pi t/3}:
a_{3}e^{-2\pi i t/3}
\bigr]
\ \Big|\
[a_{1}:a_{2}:a_{3}]\in{\rm RP}^{2}
\right\}
\subset{\rm CP}^{2}.
$$
Let $\mathrm{Ham}({\rm CP}^{2},\omega)$ denote the Hamiltonian diffeomorphism group. For each
$$
\phi\in\mathrm{Ham}({\rm CP}^{2},\omega),
$$
set
$$
K_{\phi}:=\phi(Y),
\qquad
\Sigma_{\phi}
=
\{\,t\in {\bf R}/{\bf Z}\mid K_{\phi}\cap L_{t}\neq\emptyset\,\}.
$$
Let $\lambda$ be Lebesgue measure on ${\bf R}/{\bf Z}$ normalized by
$$
\lambda({\bf R}/{\bf Z})=1.
$$
Determine the ordered pair $(m,N)$, where
$$
m
=
\inf_{\phi\in\mathrm{Ham}({\rm CP}^{2},\omega)}
\lambda(\Sigma_{\phi}),
$$
and $N$ is the largest positive integer such that the following statement holds:
for every
$$
\phi\in\mathrm{Ham}({\rm CP}^{2},\omega)
\qquad\hbox{and}\qquad
t\in {\bf R}/{\bf Z},
$$
if
$$
K_{\phi}\cap L_{t}\quad\hbox{and}\quad K_{\phi}\cap L_{t+1/3}
$$
are both transverse, and
$$
K_{\phi}\cap(L_{t}\cap L_{t+1/3})=\emptyset,
$$
then
$$
\#\bigl(K_{\phi}\cap(L_{t}\cup L_{t+1/3})\bigr)\ge N.
$$
Give the exact ordered pair $(m,N)$.
|
\[
\left(\frac{2}{3},\,4\right)
\]
|
multi_image
|
[
{
"bytes": "b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\x03\\xba\\x00\\x00\\x01\\xd7\\x08\\x06\\x00\\x00\\x00\\x92\\xf1&\\xc0\\x00\\x00\\x00\\x01sRGB\\x00\\xae\\xce\\x1c\\xe9\\x00\\x00\\x00\\x04gAMA\\x00\\x00\\xb1\\x8f\\x0b\\xfca\\x05\\x00\\x00\\x00\\tpHYs\\x00\\x00\\x16%\\x00\\x00\\x16%\\x01IR$\\xf0\\x00\\x00\\xbdsIDATx^\\xec\\xddwTT\\xd7\\x1a\\x05\\xf0M\\x13\\x15\\xec]l`\\x01\\xb1 \\x02vl\\xd8\\x1b(\\nv\\x8d\\xbd\\xc4\\xc4Dc\\xef\\x89\\x1aM,\\x89\\xbdwA\\xec\\xd8\\xa3\\xa0\\x88\\rAE\\x01\\x95\"(J\\xb3\"E)3s\\xdf\\x1f\\x1a_<b\\x82\\n8sg\\xff\\xd6bi\\xce\\xfe\\x86\\xe7Sf\\xce\\xf9n;:\\x92$I \"\"\"\"\"\"\\x92\\t]q\\x80\\x88\\x88\\x88\\x88\\x88\\x88H\\x93\\xb1\\xd1%\"\"\"\"\"\"Ya\\xa3KDDDDDD\\xb2\\xc2F\\x97\\x88\\x88\\x88\\x88\\x88\\x88d\\x85\\x8d.\\x11\\x11\\x11\\x11\\x11\\x11\\xc9\\n\\x1b]\"\"\"\"\"\"\\x92\\x156\\xbaDDDDDD$+lt\\x89\\x88\\x88\\x88\\x88\\x88HV\\xd8\\xe8\\x12\\x11\\x11\\x11\\x11\\x11\\x91\\xac\\xb0\\xd1%\"\"\"\"\"\"Ya\\xa3KDDDDDD\\xb2\\xc2F\\x97\\x88\\x88\\x88\\x88\\x88\\x88d\\x85\\x8d.\\x11\\x11\\x11\\x11\\x11\\x11\\xc9\\n\\x1b]\"\"\"\"\"\"\\x92\\x156\\xbaDDDDDD$+lt\\x89\\x88\\x88\\x88\\x88\\x88HV\\xd8\\xe8\\x12\\x11\\x11\\x11\\x11\\x11\\x91\\xac\\xb0\\xd1%\"\"\"\"\"\"Ya\\xa3KDDDDDD\\xb2\\xc2F\\x97\\x88\\x88\\x88\\x88\\x88\\x88d\\x85\\x8d.\\x11\\x11\\x11\\x11\\x11\\x11\\xc9\\n\\x1b]\"\"\"\"\"\"\\x92\\x156\\xbaDDDDDD$+lt\\x89\\x88\\x88\\x88\\x88\\x88HV\\xd8\\xe8\\x12\\x11\\x11\\x11\\x11\\x11\\x91\\xac\\xb0\\xd1%\"\"\"\"\"\"Ya\\xa3KDDDDDD\\xb2\\xc2F\\x97\\x88\\x88\\x88\\x88\\x88\\x88d\\x85\\x8d.\\x11\\x11\\x11\\x11\\x11\\x11\\xc9\\n\\x1b]\"\"\"\"\"\"\\x92\\x156\\xbaDDDDDD$+lt\\x89\\x88\\x88\\x88\\x88\\x88HV\\xd8\\xe8\\x12\\x11\\x11\\x11\\x11\\x11\\x91\\xac\\xb0\\xd1%\"\"\"\"\"\"Ya\\xa3KDDDDDD\\xb2\\xc2F\\x97\\x88\\x88\\x88\\x88\\x88\\x88d\\x85\\x8d.\\x11\\x11\\x11\\x11\\x11\\x11\\xc9\\n\\x1b]\"\"\"\"\"\"\\x92\\x156\\xbaDDDDDD$+lt\\x89\\x88\\x88\\x88\\x88\\x88HV\\xd8\\xe8\\x12\\x11\\x11\\x11\\x11\\x11\\x91\\xac\\xe8H\\x92$\\x89\\x83D\\x94C\\xd2\\xa3q\\xe1\\xc0\\x11\\\\}l\\x84Z\\xed\\x9d\\xd1\\xd6\\xa2\\xd0\\xbb\\xa3K\\xaa\\xe4\\x08\\xf8\\x1c\\xf3B`B>\\x98\\xda4A\\xf3\\x06\\xe6(\\x9eOx=\\x11\\x11\\x11i\\x94\\xf4\\xe8\\x0b8p\\xe4*\\x1e\\x1b\\xd5B{\\xe7\\xb6\\xb0(\\xf4n\\xe6Gr\\x84\\x0f\\x8ey\\x05\"!\\x9f)l\\x9a4G\\x03\\xf3\\xe2\\xe0\\xd4O\\x94;xF\\x97(\\xb7d\\x84`\\x95\\x8b+\\x96\\x86\\xe5\\x83q\\xdcv\\x0c\\xaeo\\x8d\\x81\\xbb\\xa3\\xa0D*B\\xb6\\x8fE\\x87\\xaeSp\\xec\\xb11*\\x95|\\x89\\x93?\\xd9\\xa3F\\xa3Qp\\xbf\\x97!~\\x17\"\"\"\\xd2\\x10\\x19!\\xab\\xe0\\xe2\\xba\\x14a\\xf9\\x8c\\x11\\xb7}0\\xea[\\x0f\\xc4\\xee(%\\x90\\x1a\\x82\\xedc;\\xa0\\xeb\\x94cxl\\\\\\t%_\\x9e\\xc4O\\xf65\\xd0h\\x94;8\\xf5\\x13\\xe5\\x0e\\x9e\\xd1%\\xca\\x15*\\xc4os\\x81\\xf3\\x8d\\xb18\\xb5\\xbc\\x15\\x8cSw\\xa2G\\x99\\x818\\\\\\xc4\\x05s\\x86\\xbc\\xc0\\xa9\\xe8>X\\xbbz0j\\x1b\\xbd\\xad\\x8e\\xd9\\x84n\\xb5G\\xe0l\\xd5\\x198wa.\\xec\\xf2\\x8b\\xdf\\x8f\\x88\\x88\\x88\\xd4\\x9a*\\x1e\\xdb\\\\\\x9cqc\\xec),oe\\x8c\\xd4\\x9d=Pf\\xe0a\\x14q\\x99\\x83!/N!\\xba\\xcfZ\\xac\\x1e\\\\\\x1bo\\xa6~\\x15b6uC\\xed\\x11gQu\\xc69\\\\\\x98k\\x07N\\xfdD9\\x8bgt\\x89r\\x832\\n;\\xb7?E\\xe7\\x11\\xf60\\x06\\x80\\xf44\\xa4)%\\xa8b\\xf7cCX\\x7f\\xec\\xd8\\xf4\\xff&\\x17\\x00t\\xcb\\xb6E\\x9bzzxuc\\r~?\\xfc\\xfc\\x1f\\xdf\\x88\\x88\\x88\\x884\\x812j\\'\\xb6?\\xed\\x8c\\x11\\xf6\\xc6\\x00\\x80\\xf4\\xb44(%\\x15b\\xf7o@X\\xff\\x1d\\xd8\\xf4\\xae\\xc9\\x05\\x00]\\x94m\\xdb\\x06\\xf5\\xf4^\\xe1\\xc6\\x9a\\xdfq\\xf8\\xb9\\xea\\x1f\\xdf\\x89\\x88r\\x02\\x1b]\\xa2\\\\\\xa0|t\\x0c\\x7f\\xa5\\xda\\xa3c\\r}\\x00@\\xfa\\xcd@\\xdc\\xcd\\x04t\\x8a\\xb4\\xc5\\x8c\\xa5\\xfd`\\xfaf\\xf8\\xff\\xf4\\x8a\\xa3D\\x11}@\\xf5\\x0c\\x17O\\xfb\\t!\\x11\\x11\\x11\\xa97%\\x1e\\x1d\\xfb\\x0b\\xa9\\xf6\\x1d\\xf1f\\xeaO\\xc7\\xcd\\xc0\\xbb\\xc8\\x84\\x0e\\x8a\\xb4\\x9d\\x81\\xa5\\xfdL\\xf1\\xe1\\xd4_\\x02o\\xa6\\xfe\\x8b8\\xed\\xc7\\xeb\\x97\\x89r\\x1a\\x1b]\\xa2\\\\\\xa0\\x88x\\x84B\\x9d\\xbb\\xa2\\x96>\\x00(\\x10v\\xfe\\n\\x1e)\\x81\\xfc6\\x9d\\xd0\\xb1\\\\\\x16o;\\xe
|
10
|
306296
|
Biology
|
Neurobiology
|
899
|
In this multi-compartment model of leech T-cells, different spike initiation zone (SIZ) distributions are tested on reconstructed morphologies while holding ion channel parameters fixed.
Using the Figure $1$, identify which SIZ placement produces the second lowest height across the reconstructed morphologies?
|
Anterior
|
single_image
|
[
{
"bytes": "b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\x02\\xd4\\x00\\x00\\x02\\xd0\\x08\\x06\\x00\\x00\\x00\\xb0\\xbc\\xe7\\xe6\\x00\\x00\\x00\\x01sRGB\\x00\\xae\\xce\\x1c\\xe9\\x00\\x00\\x00\\x04gAMA\\x00\\x00\\xb1\\x8f\\x0b\\xfca\\x05\\x00\\x00\\x00\\tpHYs\\x00\\x00\\x0e\\xc3\\x00\\x00\\x0e\\xc3\\x01\\xc7o\\xa8d\\x00\\x00\\xff\\xa5IDATx^\\xec\\x9d\\x05x\\x14\\xe7\\xd7\\xc5gV\\x92\\xa0-\\x10O \\x01*HK\\r*X\\x8bT\\xfe\\xa5\\xfa\\xb5h\\xa1.\\xb8\\'\\xb8\\x16(Vw\\x8a;\\x04\\x08\\t\\xee\\x12\\\\\\x13\\xdc\\xdd\\xdd5\\xd9\\xf3=\\xe7\\x9d\\x99\\xdd\\xc9d\\xa3$\\x84\\x849<\\x87\\xd9\\x1d\\xdd\\xdd\\x8c\\xfc\\xe6\\xce}\\xef+\\xc1\\x94)S\\xa6L\\x992e\\xca\\x94)S\\x19\\x96d\\x1ca\\xca\\x94)S\\xa6L\\x992e\\xca\\x94\\xa9\\xb4+\\xdd@\\x1d\\x1f\\x1f\\x8f\\xfe\\xfd\\xfb\\xa3}\\xfb\\xf6\\x08\\x0f\\x0fG\\x87\\x0e\\x1dp\\xf4\\xe8Q\\xe3l\\xa6L\\x992e\\xca\\x94)S\\xa6L=\\x14J7Po\\xd8\\xb0\\x01\\xf9\\xf3\\xe7\\x87$IN\\xff\\xf4\\xd3O\\xc6\\xd9L\\x992e\\xca\\x94)S\\xa6L\\x99z(\\x94n\\xa0\\x0e\\x0b\\x0bK\\x04\\xd3\\xf4\\x0b/\\xbc\\x80\\xbbw\\xef\\x1ag5e\\xca\\x94)S\\xa6L\\x992e*\\xd7+]@}\\xe5\\xca\\x15\\x01\\xcf\\x84h\\x1f\\x1f\\x1f<\\xfe\\xf8\\xe3\\xe2\\xb5\\x87\\x87\\x07V\\xacXa\\x9c\\xdd\\x94)S\\xa6L\\x992e\\xca\\x94\\xa9\\\\\\xaft\\x01uDD\\x04\\x1e}\\xf4Q\\x01\\xd1\\x1f~\\xf8!~\\xf9\\xe5\\x17g\\x94\\xfa\\xdbo\\xbf5\\xcen\\xca\\x94)S\\xa6L\\x992e\\xcaT\\xaeW\\xba\\x80\\xfa\\xbb\\xef\\xbe\\x13\\xf0l\\xb7\\xdb\\xf1\\xeb\\xaf\\xbf\"66\\x16\\xa1\\xa1\\xa1b\\\\\\xa9R\\xa5p\\xea\\xd4)\\xe3\"\\xa6L\\x992e\\xca\\x94)S\\xa6L\\xe5j\\xa5\\x19\\xa8w\\xee\\xdc\\x89r\\xe5\\xca\\tx\\xf6\\xf6\\xf6\\xc6\\xde\\xbd{\\xc5\\xf8\\x0f>\\xf8@\\x8c\\xcb\\x9b7/F\\x8c\\x18!\\xc6%$$\\xc0\\xe1p\\x18\\xd6`\\xca\\x94)S\\xa6L\\x992e\\xcaT\\xeeS\\x9a\\x81z\\xd4\\xa8Q\\xce\\xf4\\x8e\\xd7_\\x7f\\x1d\\xd7\\xae]\\xc3\\xed\\xdb\\xb7\\xd1\\xaf_?\\x11\\xb1\\xe6\\xf8\\xbau\\xeb\\n\\x98\\xa6L\\xa06e\\xca\\x94)S\\xa6L\\x992\\xf50(M@Mx\\xfe\\xe4\\x93O\\x9c@]\\xb6lY\\xd4\\xaf_\\x1f\\xf5\\xea\\xd5C\\xf5\\xea\\xd5\\xe1\\xe9\\xe9)\\xc6\\xb3\\x91\\xe2\\xc6\\r\\x1b\\xc52&P\\x9b2e\\xca\\x94)S\\xa6L\\x99z\\x18\\x94&\\xa0\\xde\\xbau+\\n\\x16,\\xe8\\x04\\xea\\x94\\xfc\\xfb\\xef\\xbf\\x1b\\x177e\\xca\\x94)S\\xa6L\\x992e*\\xd7*M@\\xcd\\x8e[4`~\\xe6\\x99g\\xd0\\xb8qc4k\\xda\\x0c\\x8d\\xbfk\\x8c\\xe6\\xcd\\x9b\\xe3\\xfd\\xf7\\xdf\\x87,\\xcbb\\xfa\\x87\\x1f|\\x88\\x8b\\x17.\\x1aWa\\xca\\x94)S\\xa6L\\x992e\\xcaT\\xaeT\\xaa@}\\xf5\\xeaU<\\xfd\\xf4\\xd3N\\xa0\\x1e7n\\x9c\\x18\\xcf\\x8e\\\\h\\xe6L\\xb3\\xeb\\xf1G\\x1eyDLg\\xfaGLL\\x8cq5\\xa6L\\x992e\\xca\\x94)S\\xa6L\\xe5J\\xa5\\n\\xd4\\xdb\\xb7o\\xc7\\xb3\\xcf>\\x8b\\xa7\\x9ezJ4F\\xdc\\xbf\\x7f\\xbfq\\x16\\xc4\\xc7\\xc7\\xa3u\\xeb\\xd6(S\\xa6\\x8c(\\x9f7t\\xe8Pg\\xe3\\xc4\\x07U\\xf7;\\xc7\\x9b\\xdb\\xbb\\xdf\\xdb4e\\xca\\x94)S\\xa6L\\x992\\x95\\xf5J\\x15\\xa8/^\\xbc\\x88\\x03\\x07\\x0e\\xe0\\xe0\\xc1\\x83\\xa2\\xce\\xf4\\xdd;w\\x13\\xc1\\xa1\\xf6\\xfa\\xf2\\xe5\\xcb\\xa2\\x94\\xde\\xa1C\\x87p\\xe6\\xcc\\x99\\x07\\xbet\\x1e?\\xdf\\xf1\\xe3\\xc7q\\xe4\\xc8\\x91,\\xef6\\x9d\\xbf\\x03\\xb7w\\xfa\\xf4i\\xf1[f\\xf5\\xf6L\\x992e\\xca\\x94)S\\xa6L\\xdd?\\xa5\\n\\xd4\\xeet/\\xa0L\\xb0\\xbc~\\xfd:n\\xde\\xbc\\x99\\xadQ\\xdb[\\xb7n\\xa1i\\xd3\\xa6\\xf8\\xfc\\xf3\\xcfq\\xe1\\xc2\\x05\\xe3\\xe4L\\x17\\xbfg\\xdf\\xbe}Q\\xa3F\\r\\x9c;w\\xce8\\xd9\\x94)S\\xa6L\\x992e\\xcaT\\x0eU\\xaa@\\x9dV\\xe0M+\\x1c3*\\xcc\\xce`Z\\xb6l\\xe96R\\x9b\\x96ud\\x86\\x08\\xf5o\\xbe\\xf9&\\xaaV\\xad*\"\\xc7Y-~/\\x02\\xbc\\xbf\\xbf?N\\x9e<i\\x9cl\\xca\\x94)S\\xa6L\\x992e*\\x87*U\\xa0\\xce\\xa8\\x12\\xe2\\x13D\\x14\\xd8\\x08\\xc8\\x84\\xc9\\x97_~\\x19\\x8d\\x1a5r\\x0b\\xd4\\x9a8\\x8d\\x1d\\xc7d\\x95\\x18!\\x7f\\xfb\\xed\\xb7\\xf1\\xdak\\xaf\\x89\\x14\\x15\\xa3\\x18IO\\xe9\\xf3%\\xab\\x14\\xee\\x07\\x9a5k\\x86\\xa2E\\x8b\\x9a]\\xb4\\x9b2e\\xca\\x94)S\\xa6L\\xe5\"e:P\\x9f={VT\\x02a#\\xc5/\\xbe\\xf8\\x02m\\xda\\xb4\\xc1\\x9f\\x7f\\xfe) \\x92\\x15CX\\xa7\\xbax\\xf1\\xe2x\\xe5\\x95W0d\\xc8\\x10\\xfc\\xfd\\xf7\\xdf\\x983g\\x8e\\x80gB\\xec\\xc2\\x85\\x0b1p\\xc0@Q\\x92\\xef\\xdbo\
|