[{"data":1,"prerenderedAt":55251},["ShallowReactive",2],{"post-\u002Fblog\u002F2026\u002F2026-04-06-nonlinearity-with-activations":3},{"post":4,"nextPost":11698,"prevPost":27131},{"id":5,"title":6,"body":7,"description":11668,"draft":11684,"enableComment":11685,"extension":11686,"image":11668,"important":11684,"location":11687,"meta":11688,"navigation":11685,"ogImage":11687,"onday":11689,"path":11690,"seo":11691,"stem":11692,"summary":11693,"tags":11694,"__hash__":11697},"blog\u002Fblog\u002F2026\u002F2026-04-06-nonlinearity-with-activations.md","Nonlinearity with Activations",{"type":8,"value":9,"toc":11667},"minimark",[10,15,27,30,145,150,156,315,1142,1146,1152,1311,2082,2085,2088,2155,2401,2404,2452,2957,2968,2972,3118,3216,3278,3281,3556,3777,3780,4509,4516,4519,4854,5120,5183,5187,5190,5193,5376,5379,5540,5543,5777,5782,5785,6006,6009,6158,6169,6172,6270,6440,6627,6663,6666,6906,6980,7014,7054,7057,7060,7066,7069,7072,7077,7081,7083,7306,7396,7710,7713,7716,8413,8471,8474,8681,8684,8687,9210,9213,9432,9435,10081,10084,10087,10287,10290,10349,10352,10593,10596,10599,11350,11353,11427,11430,11434,11437,11441,11444,11447,11640,11643,11646,11650,11657,11661,11664],[11,12,14],"h2",{"id":13},"linear-combinations","Linear Combinations",[16,17,18,19,23,24,26],"p",{},"The concept of ",[20,21,22],"strong",{},"linearity"," runs through mathematics, physics, engineering, and many other fields. Simply put, ",[20,25,22],{}," describes two fundamental operational properties: additivity and homogeneity.",[16,28,29],{},"A relationship, function, or system is said to be linear if it satisfies the following two conditions simultaneously.",[16,31,32,33,81,82,112,113,144],{},"Consider a transformation or function ",[34,35,39,61],"span",{"className":36,"translate":38},[37],"katex","no",[34,40,43],{"className":41},[42],"katex-mathml",[44,45,47],"math",{"xmlns":46},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[48,49,50,57],"semantics",{},[51,52,53],"mrow",{},[54,55,56],"mi",{},"f",[58,59,56],"annotation",{"encoding":60},"application\u002Fx-tex",[34,62,66],{"className":63,"ariaHidden":65},[64],"katex-html","true",[34,67,70,75],{"className":68},[69],"base",[34,71],{"className":72,"style":74},[73],"strut","height:0.8889em;vertical-align:-0.1944em;",[34,76,56],{"className":77,"style":80},[78,79],"mord","mathnormal","margin-right:0.1076em;"," acting on inputs ",[34,83,85,99],{"className":84,"translate":38},[37],[34,86,88],{"className":87},[42],[44,89,90],{"xmlns":46},[48,91,92,97],{},[51,93,94],{},[54,95,96],{},"x",[58,98,96],{"encoding":60},[34,100,102],{"className":101,"ariaHidden":65},[64],[34,103,105,109],{"className":104},[69],[34,106],{"className":107,"style":108},[73],"height:0.4306em;",[34,110,96],{"className":111},[78,79]," and ",[34,114,116,130],{"className":115,"translate":38},[37],[34,117,119],{"className":118},[42],[44,120,121],{"xmlns":46},[48,122,123,128],{},[51,124,125],{},[54,126,127],{},"y",[58,129,127],{"encoding":60},[34,131,133],{"className":132,"ariaHidden":65},[64],[34,134,136,140],{"className":135},[69],[34,137],{"className":138,"style":139},[73],"height:0.625em;vertical-align:-0.1944em;",[34,141,127],{"className":142,"style":143},[78,79],"margin-right:0.0359em;",".",[146,147,149],"h3",{"id":148},"additivity","Additivity",[16,151,152,155],{},[20,153,154],{},"The sum of parts equals the whole."," That is, processing two inputs separately and then adding the results is the same as adding the inputs first and then processing the sum.",[34,157,160],{"className":158,"translate":38},[159],"katex-display",[34,161,163,214],{"className":162,"translate":38},[37],[34,164,166],{"className":165},[42],[44,167,169],{"xmlns":46,"display":168},"block",[48,170,171,211],{},[51,172,173,175,180,182,185,187,190,193,195,197,199,201,203,205,207,209],{},[54,174,56],{},[176,177,179],"mo",{"stretchy":178},"false","(",[54,181,96],{},[176,183,184],{},"+",[54,186,127],{},[176,188,189],{"stretchy":178},")",[176,191,192],{},"=",[54,194,56],{},[176,196,179],{"stretchy":178},[54,198,96],{},[176,200,189],{"stretchy":178},[176,202,184],{},[54,204,56],{},[176,206,179],{"stretchy":178},[54,208,127],{},[176,210,189],{"stretchy":178},[58,212,213],{"encoding":60},"f(x + y) = f(x) + f(y)",[34,215,217,246,270,297],{"className":216,"ariaHidden":65},[64],[34,218,220,224,227,231,234,239,243],{"className":219},[69],[34,221],{"className":222,"style":223},[73],"height:1em;vertical-align:-0.25em;",[34,225,56],{"className":226,"style":80},[78,79],[34,228,179],{"className":229},[230],"mopen",[34,232,96],{"className":233},[78,79],[34,235],{"className":236,"style":238},[237],"mspace","margin-right:0.2222em;",[34,240,184],{"className":241},[242],"mbin",[34,244],{"className":245,"style":238},[237],[34,247,249,252,255,259,263,267],{"className":248},[69],[34,250],{"className":251,"style":223},[73],[34,253,127],{"className":254,"style":143},[78,79],[34,256,189],{"className":257},[258],"mclose",[34,260],{"className":261,"style":262},[237],"margin-right:0.2778em;",[34,264,192],{"className":265},[266],"mrel",[34,268],{"className":269,"style":262},[237],[34,271,273,276,279,282,285,288,291,294],{"className":272},[69],[34,274],{"className":275,"style":223},[73],[34,277,56],{"className":278,"style":80},[78,79],[34,280,179],{"className":281},[230],[34,283,96],{"className":284},[78,79],[34,286,189],{"className":287},[258],[34,289],{"className":290,"style":238},[237],[34,292,184],{"className":293},[242],[34,295],{"className":296,"style":238},[237],[34,298,300,303,306,309,312],{"className":299},[69],[34,301],{"className":302,"style":223},[73],[34,304,56],{"className":305,"style":80},[78,79],[34,307,179],{"className":308},[230],[34,310,127],{"className":311,"style":143},[78,79],[34,313,189],{"className":314},[258],[316,317,318,399,726,842],"blockquote",{},[16,319,320,323,324,398],{},[20,321,322],{},"Example",": Multiplication ",[34,325,327,355],{"className":326,"translate":38},[37],[34,328,330],{"className":329},[42],[44,331,332],{"xmlns":46},[48,333,334,352],{},[51,335,336,338,340,342,344,346,350],{},[54,337,56],{},[176,339,179],{"stretchy":178},[54,341,96],{},[176,343,189],{"stretchy":178},[176,345,192],{},[347,348,349],"mn",{},"5",[54,351,96],{},[58,353,354],{"encoding":60},"f(x) = 5x",[34,356,358,385],{"className":357,"ariaHidden":65},[64],[34,359,361,364,367,370,373,376,379,382],{"className":360},[69],[34,362],{"className":363,"style":223},[73],[34,365,56],{"className":366,"style":80},[78,79],[34,368,179],{"className":369},[230],[34,371,96],{"className":372},[78,79],[34,374,189],{"className":375},[258],[34,377],{"className":378,"style":262},[237],[34,380,192],{"className":381},[266],[34,383],{"className":384,"style":262},[237],[34,386,388,392,395],{"className":387},[69],[34,389],{"className":390,"style":391},[73],"height:0.6444em;",[34,393,349],{"className":394},[78],[34,396,96],{"className":397},[78,79]," satisfies additivity.",[400,401,402],"ul",{},[403,404,405,406,575,576,725],"li",{},"Left: ",[34,407,409,457],{"className":408,"translate":38},[37],[34,410,412],{"className":411},[42],[44,413,414],{"xmlns":46},[48,415,416,454],{},[51,417,418,420,422,425,427,430,432,434,436,439,441,443,445,447,449,451],{},[54,419,56],{},[176,421,179],{"stretchy":178},[347,423,424],{},"2",[176,426,184],{},[347,428,429],{},"3",[176,431,189],{"stretchy":178},[176,433,192],{},[347,435,349],{},[176,437,438],{},"×",[176,440,179],{"stretchy":178},[347,442,424],{},[176,444,184],{},[347,446,429],{},[176,448,189],{"stretchy":178},[176,450,192],{},[347,452,453],{},"25",[58,455,456],{"encoding":60},"f(2+3) = 5 \\times (2+3) = 25",[34,458,460,484,505,524,545,566],{"className":459,"ariaHidden":65},[64],[34,461,463,466,469,472,475,478,481],{"className":462},[69],[34,464],{"className":465,"style":223},[73],[34,467,56],{"className":468,"style":80},[78,79],[34,470,179],{"className":471},[230],[34,473,424],{"className":474},[78],[34,476],{"className":477,"style":238},[237],[34,479,184],{"className":480},[242],[34,482],{"className":483,"style":238},[237],[34,485,487,490,493,496,499,502],{"className":486},[69],[34,488],{"className":489,"style":223},[73],[34,491,429],{"className":492},[78],[34,494,189],{"className":495},[258],[34,497],{"className":498,"style":262},[237],[34,500,192],{"className":501},[266],[34,503],{"className":504,"style":262},[237],[34,506,508,512,515,518,521],{"className":507},[69],[34,509],{"className":510,"style":511},[73],"height:0.7278em;vertical-align:-0.0833em;",[34,513,349],{"className":514},[78],[34,516],{"className":517,"style":238},[237],[34,519,438],{"className":520},[242],[34,522],{"className":523,"style":238},[237],[34,525,527,530,533,536,539,542],{"className":526},[69],[34,528],{"className":529,"style":223},[73],[34,531,179],{"className":532},[230],[34,534,424],{"className":535},[78],[34,537],{"className":538,"style":238},[237],[34,540,184],{"className":541},[242],[34,543],{"className":544,"style":238},[237],[34,546,548,551,554,557,560,563],{"className":547},[69],[34,549],{"className":550,"style":223},[73],[34,552,429],{"className":553},[78],[34,555,189],{"className":556},[258],[34,558],{"className":559,"style":262},[237],[34,561,192],{"className":562},[266],[34,564],{"className":565,"style":262},[237],[34,567,569,572],{"className":568},[69],[34,570],{"className":571,"style":391},[73],[34,573,453],{"className":574},[78],", Right: ",[34,577,579,623],{"className":578,"translate":38},[37],[34,580,582],{"className":581},[42],[44,583,584],{"xmlns":46},[48,585,586,620],{},[51,587,588,590,592,594,596,598,600,602,604,606,608,611,613,616,618],{},[54,589,56],{},[176,591,179],{"stretchy":178},[347,593,424],{},[176,595,189],{"stretchy":178},[176,597,184],{},[54,599,56],{},[176,601,179],{"stretchy":178},[347,603,429],{},[176,605,189],{"stretchy":178},[176,607,192],{},[347,609,610],{},"10",[176,612,184],{},[347,614,615],{},"15",[176,617,192],{},[347,619,453],{},[58,621,622],{"encoding":60},"f(2) + f(3) = 10 + 15 = 25",[34,624,626,653,680,698,716],{"className":625,"ariaHidden":65},[64],[34,627,629,632,635,638,641,644,647,650],{"className":628},[69],[34,630],{"className":631,"style":223},[73],[34,633,56],{"className":634,"style":80},[78,79],[34,636,179],{"className":637},[230],[34,639,424],{"className":640},[78],[34,642,189],{"className":643},[258],[34,645],{"className":646,"style":238},[237],[34,648,184],{"className":649},[242],[34,651],{"className":652,"style":238},[237],[34,654,656,659,662,665,668,671,674,677],{"className":655},[69],[34,657],{"className":658,"style":223},[73],[34,660,56],{"className":661,"style":80},[78,79],[34,663,179],{"className":664},[230],[34,666,429],{"className":667},[78],[34,669,189],{"className":670},[258],[34,672],{"className":673,"style":262},[237],[34,675,192],{"className":676},[266],[34,678],{"className":679,"style":262},[237],[34,681,683,686,689,692,695],{"className":682},[69],[34,684],{"className":685,"style":511},[73],[34,687,610],{"className":688},[78],[34,690],{"className":691,"style":238},[237],[34,693,184],{"className":694},[242],[34,696],{"className":697,"style":238},[237],[34,699,701,704,707,710,713],{"className":700},[69],[34,702],{"className":703,"style":391},[73],[34,705,615],{"className":706},[78],[34,708],{"className":709,"style":262},[237],[34,711,192],{"className":712},[266],[34,714],{"className":715,"style":262},[237],[34,717,719,722],{"className":718},[69],[34,720],{"className":721,"style":391},[73],[34,723,453],{"className":724},[78],", equal.",[16,727,728,731,732,841],{},[20,729,730],{},"Counterexample",": The square function ",[34,733,735,764],{"className":734,"translate":38},[37],[34,736,738],{"className":737},[42],[44,739,740],{"xmlns":46},[48,741,742,761],{},[51,743,744,746,748,750,752,754],{},[54,745,56],{},[176,747,179],{"stretchy":178},[54,749,96],{},[176,751,189],{"stretchy":178},[176,753,192],{},[755,756,757,759],"msup",{},[54,758,96],{},[347,760,424],{},[58,762,763],{"encoding":60},"f(x) = x^2",[34,765,767,794],{"className":766,"ariaHidden":65},[64],[34,768,770,773,776,779,782,785,788,791],{"className":769},[69],[34,771],{"className":772,"style":223},[73],[34,774,56],{"className":775,"style":80},[78,79],[34,777,179],{"className":778},[230],[34,780,96],{"className":781},[78,79],[34,783,189],{"className":784},[258],[34,786],{"className":787,"style":262},[237],[34,789,192],{"className":790},[266],[34,792],{"className":793,"style":262},[237],[34,795,797,801],{"className":796},[69],[34,798],{"className":799,"style":800},[73],"height:0.8141em;",[34,802,804,807],{"className":803},[78],[34,805,96],{"className":806},[78,79],[34,808,811],{"className":809},[810],"msupsub",[34,812,815],{"className":813},[814],"vlist-t",[34,816,819],{"className":817},[818],"vlist-r",[34,820,823],{"className":821,"style":800},[822],"vlist",[34,824,826,831],{"style":825},"top:-3.063em;margin-right:0.05em;",[34,827],{"className":828,"style":830},[829],"pstrut","height:2.7em;",[34,832,838],{"className":833},[834,835,836,837],"sizing","reset-size6","size3","mtight",[34,839,424],{"className":840},[78,837]," does not satisfy additivity.",[400,843,844],{},[403,845,405,846,575,960,1141],{},[34,847,849,879],{"className":848,"translate":38},[37],[34,850,852],{"className":851},[42],[44,853,854],{"xmlns":46},[48,855,856,876],{},[51,857,858,860,862,864,866,872,874],{},[176,859,179],{"stretchy":178},[347,861,424],{},[176,863,184],{},[347,865,429],{},[755,867,868,870],{},[176,869,189],{"stretchy":178},[347,871,424],{},[176,873,192],{},[347,875,453],{},[58,877,878],{"encoding":60},"(2+3)^2 = 25",[34,880,882,903,951],{"className":881,"ariaHidden":65},[64],[34,883,885,888,891,894,897,900],{"className":884},[69],[34,886],{"className":887,"style":223},[73],[34,889,179],{"className":890},[230],[34,892,424],{"className":893},[78],[34,895],{"className":896,"style":238},[237],[34,898,184],{"className":899},[242],[34,901],{"className":902,"style":238},[237],[34,904,906,910,913,942,945,948],{"className":905},[69],[34,907],{"className":908,"style":909},[73],"height:1.0641em;vertical-align:-0.25em;",[34,911,429],{"className":912},[78],[34,914,916,919],{"className":915},[258],[34,917,189],{"className":918},[258],[34,920,922],{"className":921},[810],[34,923,925],{"className":924},[814],[34,926,928],{"className":927},[818],[34,929,931],{"className":930,"style":800},[822],[34,932,933,936],{"style":825},[34,934],{"className":935,"style":830},[829],[34,937,939],{"className":938},[834,835,836,837],[34,940,424],{"className":941},[78,837],[34,943],{"className":944,"style":262},[237],[34,946,192],{"className":947},[266],[34,949],{"className":950,"style":262},[237],[34,952,954,957],{"className":953},[69],[34,955],{"className":956,"style":391},[73],[34,958,453],{"className":959},[78],[34,961,963,1004],{"className":962,"translate":38},[37],[34,964,966],{"className":965},[42],[44,967,968],{"xmlns":46},[48,969,970,1001],{},[51,971,972,978,980,986,988,991,993,996,998],{},[755,973,974,976],{},[347,975,424],{},[347,977,424],{},[176,979,184],{},[755,981,982,984],{},[347,983,429],{},[347,985,424],{},[176,987,192],{},[347,989,990],{},"4",[176,992,184],{},[347,994,995],{},"9",[176,997,192],{},[347,999,1000],{},"13",[58,1002,1003],{"encoding":60},"2^2 + 3^2 = 4+9=13",[34,1005,1007,1052,1096,1114,1132],{"className":1006,"ariaHidden":65},[64],[34,1008,1010,1014,1043,1046,1049],{"className":1009},[69],[34,1011],{"className":1012,"style":1013},[73],"height:0.8974em;vertical-align:-0.0833em;",[34,1015,1017,1020],{"className":1016},[78],[34,1018,424],{"className":1019},[78],[34,1021,1023],{"className":1022},[810],[34,1024,1026],{"className":1025},[814],[34,1027,1029],{"className":1028},[818],[34,1030,1032],{"className":1031,"style":800},[822],[34,1033,1034,1037],{"style":825},[34,1035],{"className":1036,"style":830},[829],[34,1038,1040],{"className":1039},[834,835,836,837],[34,1041,424],{"className":1042},[78,837],[34,1044],{"className":1045,"style":238},[237],[34,1047,184],{"className":1048},[242],[34,1050],{"className":1051,"style":238},[237],[34,1053,1055,1058,1087,1090,1093],{"className":1054},[69],[34,1056],{"className":1057,"style":800},[73],[34,1059,1061,1064],{"className":1060},[78],[34,1062,429],{"className":1063},[78],[34,1065,1067],{"className":1066},[810],[34,1068,1070],{"className":1069},[814],[34,1071,1073],{"className":1072},[818],[34,1074,1076],{"className":1075,"style":800},[822],[34,1077,1078,1081],{"style":825},[34,1079],{"className":1080,"style":830},[829],[34,1082,1084],{"className":1083},[834,835,836,837],[34,1085,424],{"className":1086},[78,837],[34,1088],{"className":1089,"style":262},[237],[34,1091,192],{"className":1092},[266],[34,1094],{"className":1095,"style":262},[237],[34,1097,1099,1102,1105,1108,1111],{"className":1098},[69],[34,1100],{"className":1101,"style":511},[73],[34,1103,990],{"className":1104},[78],[34,1106],{"className":1107,"style":238},[237],[34,1109,184],{"className":1110},[242],[34,1112],{"className":1113,"style":238},[237],[34,1115,1117,1120,1123,1126,1129],{"className":1116},[69],[34,1118],{"className":1119,"style":391},[73],[34,1121,995],{"className":1122},[78],[34,1124],{"className":1125,"style":262},[237],[34,1127,192],{"className":1128},[266],[34,1130],{"className":1131,"style":262},[237],[34,1133,1135,1138],{"className":1134},[69],[34,1136],{"className":1137,"style":391},[73],[34,1139,1000],{"className":1140},[78],", not equal.",[146,1143,1145],{"id":1144},"homogeneity","Homogeneity",[16,1147,1148,1151],{},[20,1149,1150],{},"Scaling the input scales the output by the same factor."," That is, multiplying an input by a constant (scalar) yields the original output multiplied by the same constant.",[34,1153,1155],{"className":1154,"translate":38},[159],[34,1156,1158,1209],{"className":1157,"translate":38},[37],[34,1159,1161],{"className":1160},[42],[44,1162,1163],{"xmlns":46,"display":168},[48,1164,1165,1206],{},[51,1166,1167,1169,1171,1174,1177,1179,1181,1183,1185,1187,1189,1191,1193,1195,1198,1200,1204],{},[54,1168,56],{},[176,1170,179],{"stretchy":178},[54,1172,1173],{},"c",[176,1175,1176],{},"⋅",[54,1178,96],{},[176,1180,189],{"stretchy":178},[176,1182,192],{},[54,1184,1173],{},[176,1186,1176],{},[54,1188,56],{},[176,1190,179],{"stretchy":178},[54,1192,96],{},[176,1194,189],{"stretchy":178},[237,1196],{"width":1197},"1em",[176,1199,179],{"stretchy":178},[1201,1202,1203],"mtext",{},"c is any constant",[176,1205,189],{"stretchy":178},[58,1207,1208],{"encoding":60},"f(c \\cdot x) = c \\cdot f(x) \\quad (\\text{c is any constant})",[34,1210,1212,1236,1257,1276],{"className":1211,"ariaHidden":65},[64],[34,1213,1215,1218,1221,1224,1227,1230,1233],{"className":1214},[69],[34,1216],{"className":1217,"style":223},[73],[34,1219,56],{"className":1220,"style":80},[78,79],[34,1222,179],{"className":1223},[230],[34,1225,1173],{"className":1226},[78,79],[34,1228],{"className":1229,"style":238},[237],[34,1231,1176],{"className":1232},[242],[34,1234],{"className":1235,"style":238},[237],[34,1237,1239,1242,1245,1248,1251,1254],{"className":1238},[69],[34,1240],{"className":1241,"style":223},[73],[34,1243,96],{"className":1244},[78,79],[34,1246,189],{"className":1247},[258],[34,1249],{"className":1250,"style":262},[237],[34,1252,192],{"className":1253},[266],[34,1255],{"className":1256,"style":262},[237],[34,1258,1260,1264,1267,1270,1273],{"className":1259},[69],[34,1261],{"className":1262,"style":1263},[73],"height:0.4445em;",[34,1265,1173],{"className":1266},[78,79],[34,1268],{"className":1269,"style":238},[237],[34,1271,1176],{"className":1272},[242],[34,1274],{"className":1275,"style":238},[237],[34,1277,1279,1282,1285,1288,1291,1294,1298,1301,1308],{"className":1278},[69],[34,1280],{"className":1281,"style":223},[73],[34,1283,56],{"className":1284,"style":80},[78,79],[34,1286,179],{"className":1287},[230],[34,1289,96],{"className":1290},[78,79],[34,1292,189],{"className":1293},[258],[34,1295],{"className":1296,"style":1297},[237],"margin-right:1em;",[34,1299,179],{"className":1300},[230],[34,1302,1305],{"className":1303},[78,1304],"text",[34,1306,1203],{"className":1307},[78],[34,1309,189],{"className":1310},[258],[316,1312,1313,1389,1659,1758],{},[16,1314,1315,1317,1318,1388],{},[20,1316,322],{},": ",[34,1319,1321,1346],{"className":1320,"translate":38},[37],[34,1322,1324],{"className":1323},[42],[44,1325,1326],{"xmlns":46},[48,1327,1328,1344],{},[51,1329,1330,1332,1334,1336,1338,1340,1342],{},[54,1331,56],{},[176,1333,179],{"stretchy":178},[54,1335,96],{},[176,1337,189],{"stretchy":178},[176,1339,192],{},[347,1341,349],{},[54,1343,96],{},[58,1345,354],{"encoding":60},[34,1347,1349,1376],{"className":1348,"ariaHidden":65},[64],[34,1350,1352,1355,1358,1361,1364,1367,1370,1373],{"className":1351},[69],[34,1353],{"className":1354,"style":223},[73],[34,1356,56],{"className":1357,"style":80},[78,79],[34,1359,179],{"className":1360},[230],[34,1362,96],{"className":1363},[78,79],[34,1365,189],{"className":1366},[258],[34,1368],{"className":1369,"style":262},[237],[34,1371,192],{"className":1372},[266],[34,1374],{"className":1375,"style":262},[237],[34,1377,1379,1382,1385],{"className":1378},[69],[34,1380],{"className":1381,"style":391},[73],[34,1383,349],{"className":1384},[78],[34,1386,96],{"className":1387},[78,79]," satisfies homogeneity.",[400,1390,1391],{},[403,1392,405,1393,575,1527,725],{},[34,1394,1396,1434],{"className":1395,"translate":38},[37],[34,1397,1399],{"className":1398},[42],[44,1400,1401],{"xmlns":46},[48,1402,1403,1431],{},[51,1404,1405,1407,1409,1411,1413,1415,1417,1419,1421,1423,1426,1428],{},[54,1406,56],{},[176,1408,179],{"stretchy":178},[347,1410,990],{},[176,1412,438],{},[347,1414,429],{},[176,1416,189],{"stretchy":178},[176,1418,192],{},[347,1420,349],{},[176,1422,438],{},[347,1424,1425],{},"12",[176,1427,192],{},[347,1429,1430],{},"60",[58,1432,1433],{"encoding":60},"f(4 \\times 3) = 5 \\times 12 = 60",[34,1435,1437,1461,1482,1500,1518],{"className":1436,"ariaHidden":65},[64],[34,1438,1440,1443,1446,1449,1452,1455,1458],{"className":1439},[69],[34,1441],{"className":1442,"style":223},[73],[34,1444,56],{"className":1445,"style":80},[78,79],[34,1447,179],{"className":1448},[230],[34,1450,990],{"className":1451},[78],[34,1453],{"className":1454,"style":238},[237],[34,1456,438],{"className":1457},[242],[34,1459],{"className":1460,"style":238},[237],[34,1462,1464,1467,1470,1473,1476,1479],{"className":1463},[69],[34,1465],{"className":1466,"style":223},[73],[34,1468,429],{"className":1469},[78],[34,1471,189],{"className":1472},[258],[34,1474],{"className":1475,"style":262},[237],[34,1477,192],{"className":1478},[266],[34,1480],{"className":1481,"style":262},[237],[34,1483,1485,1488,1491,1494,1497],{"className":1484},[69],[34,1486],{"className":1487,"style":511},[73],[34,1489,349],{"className":1490},[78],[34,1492],{"className":1493,"style":238},[237],[34,1495,438],{"className":1496},[242],[34,1498],{"className":1499,"style":238},[237],[34,1501,1503,1506,1509,1512,1515],{"className":1502},[69],[34,1504],{"className":1505,"style":391},[73],[34,1507,1425],{"className":1508},[78],[34,1510],{"className":1511,"style":262},[237],[34,1513,192],{"className":1514},[266],[34,1516],{"className":1517,"style":262},[237],[34,1519,1521,1524],{"className":1520},[69],[34,1522],{"className":1523,"style":391},[73],[34,1525,1430],{"className":1526},[78],[34,1528,1530,1566],{"className":1529,"translate":38},[37],[34,1531,1533],{"className":1532},[42],[44,1534,1535],{"xmlns":46},[48,1536,1537,1563],{},[51,1538,1539,1541,1543,1545,1547,1549,1551,1553,1555,1557,1559,1561],{},[347,1540,990],{},[176,1542,438],{},[54,1544,56],{},[176,1546,179],{"stretchy":178},[347,1548,429],{},[176,1550,189],{"stretchy":178},[176,1552,192],{},[347,1554,990],{},[176,1556,438],{},[347,1558,615],{},[176,1560,192],{},[347,1562,1430],{},[58,1564,1565],{"encoding":60},"4 \\times f(3) = 4 \\times 15 = 60",[34,1567,1569,1587,1614,1632,1650],{"className":1568,"ariaHidden":65},[64],[34,1570,1572,1575,1578,1581,1584],{"className":1571},[69],[34,1573],{"className":1574,"style":511},[73],[34,1576,990],{"className":1577},[78],[34,1579],{"className":1580,"style":238},[237],[34,1582,438],{"className":1583},[242],[34,1585],{"className":1586,"style":238},[237],[34,1588,1590,1593,1596,1599,1602,1605,1608,1611],{"className":1589},[69],[34,1591],{"className":1592,"style":223},[73],[34,1594,56],{"className":1595,"style":80},[78,79],[34,1597,179],{"className":1598},[230],[34,1600,429],{"className":1601},[78],[34,1603,189],{"className":1604},[258],[34,1606],{"className":1607,"style":262},[237],[34,1609,192],{"className":1610},[266],[34,1612],{"className":1613,"style":262},[237],[34,1615,1617,1620,1623,1626,1629],{"className":1616},[69],[34,1618],{"className":1619,"style":511},[73],[34,1621,990],{"className":1622},[78],[34,1624],{"className":1625,"style":238},[237],[34,1627,438],{"className":1628},[242],[34,1630],{"className":1631,"style":238},[237],[34,1633,1635,1638,1641,1644,1647],{"className":1634},[69],[34,1636],{"className":1637,"style":391},[73],[34,1639,615],{"className":1640},[78],[34,1642],{"className":1643,"style":262},[237],[34,1645,192],{"className":1646},[266],[34,1648],{"className":1649,"style":262},[237],[34,1651,1653,1656],{"className":1652},[69],[34,1654],{"className":1655,"style":391},[73],[34,1657,1430],{"className":1658},[78],[16,1660,1661,1663,1664,1757],{},[20,1662,730],{},": Adding a constant ",[34,1665,1667,1697],{"className":1666,"translate":38},[37],[34,1668,1670],{"className":1669},[42],[44,1671,1672],{"xmlns":46},[48,1673,1674,1694],{},[51,1675,1676,1678,1680,1682,1684,1686,1688,1690,1692],{},[54,1677,56],{},[176,1679,179],{"stretchy":178},[54,1681,96],{},[176,1683,189],{"stretchy":178},[176,1685,192],{},[347,1687,349],{},[54,1689,96],{},[176,1691,184],{},[347,1693,424],{},[58,1695,1696],{"encoding":60},"f(x) = 5x + 2",[34,1698,1700,1727,1748],{"className":1699,"ariaHidden":65},[64],[34,1701,1703,1706,1709,1712,1715,1718,1721,1724],{"className":1702},[69],[34,1704],{"className":1705,"style":223},[73],[34,1707,56],{"className":1708,"style":80},[78,79],[34,1710,179],{"className":1711},[230],[34,1713,96],{"className":1714},[78,79],[34,1716,189],{"className":1717},[258],[34,1719],{"className":1720,"style":262},[237],[34,1722,192],{"className":1723},[266],[34,1725],{"className":1726,"style":262},[237],[34,1728,1730,1733,1736,1739,1742,1745],{"className":1729},[69],[34,1731],{"className":1732,"style":511},[73],[34,1734,349],{"className":1735},[78],[34,1737,96],{"className":1738},[78,79],[34,1740],{"className":1741,"style":238},[237],[34,1743,184],{"className":1744},[242],[34,1746],{"className":1747,"style":238},[237],[34,1749,1751,1754],{"className":1750},[69],[34,1752],{"className":1753,"style":391},[73],[34,1755,424],{"className":1756},[78]," does not satisfy homogeneity.",[400,1759,1760],{},[403,1761,405,1762,575,1917,1141],{},[34,1763,1765,1806],{"className":1764,"translate":38},[37],[34,1766,1768],{"className":1767},[42],[44,1769,1770],{"xmlns":46},[48,1771,1772,1803],{},[51,1773,1774,1776,1778,1780,1782,1784,1786,1788,1790,1792,1794,1796,1798,1800],{},[54,1775,56],{},[176,1777,179],{"stretchy":178},[347,1779,990],{},[176,1781,438],{},[347,1783,429],{},[176,1785,189],{"stretchy":178},[176,1787,192],{},[347,1789,349],{},[176,1791,438],{},[347,1793,1425],{},[176,1795,184],{},[347,1797,424],{},[176,1799,192],{},[347,1801,1802],{},"62",[58,1804,1805],{"encoding":60},"f(4 \\times 3) = 5 \\times 12 + 2 = 62",[34,1807,1809,1833,1854,1872,1890,1908],{"className":1808,"ariaHidden":65},[64],[34,1810,1812,1815,1818,1821,1824,1827,1830],{"className":1811},[69],[34,1813],{"className":1814,"style":223},[73],[34,1816,56],{"className":1817,"style":80},[78,79],[34,1819,179],{"className":1820},[230],[34,1822,990],{"className":1823},[78],[34,1825],{"className":1826,"style":238},[237],[34,1828,438],{"className":1829},[242],[34,1831],{"className":1832,"style":238},[237],[34,1834,1836,1839,1842,1845,1848,1851],{"className":1835},[69],[34,1837],{"className":1838,"style":223},[73],[34,1840,429],{"className":1841},[78],[34,1843,189],{"className":1844},[258],[34,1846],{"className":1847,"style":262},[237],[34,1849,192],{"className":1850},[266],[34,1852],{"className":1853,"style":262},[237],[34,1855,1857,1860,1863,1866,1869],{"className":1856},[69],[34,1858],{"className":1859,"style":511},[73],[34,1861,349],{"className":1862},[78],[34,1864],{"className":1865,"style":238},[237],[34,1867,438],{"className":1868},[242],[34,1870],{"className":1871,"style":238},[237],[34,1873,1875,1878,1881,1884,1887],{"className":1874},[69],[34,1876],{"className":1877,"style":511},[73],[34,1879,1425],{"className":1880},[78],[34,1882],{"className":1883,"style":238},[237],[34,1885,184],{"className":1886},[242],[34,1888],{"className":1889,"style":238},[237],[34,1891,1893,1896,1899,1902,1905],{"className":1892},[69],[34,1894],{"className":1895,"style":391},[73],[34,1897,424],{"className":1898},[78],[34,1900],{"className":1901,"style":262},[237],[34,1903,192],{"className":1904},[266],[34,1906],{"className":1907,"style":262},[237],[34,1909,1911,1914],{"className":1910},[69],[34,1912],{"className":1913,"style":391},[73],[34,1915,1802],{"className":1916},[78],[34,1918,1920,1965],{"className":1919,"translate":38},[37],[34,1921,1923],{"className":1922},[42],[44,1924,1925],{"xmlns":46},[48,1926,1927,1962],{},[51,1928,1929,1931,1933,1935,1937,1939,1941,1943,1945,1947,1949,1951,1953,1955,1957,1959],{},[347,1930,990],{},[176,1932,438],{},[54,1934,56],{},[176,1936,179],{"stretchy":178},[347,1938,429],{},[176,1940,189],{"stretchy":178},[176,1942,192],{},[347,1944,990],{},[176,1946,438],{},[176,1948,179],{"stretchy":178},[347,1950,615],{},[176,1952,184],{},[347,1954,424],{},[176,1956,189],{"stretchy":178},[176,1958,192],{},[347,1960,1961],{},"68",[58,1963,1964],{"encoding":60},"4 \\times f(3) = 4 \\times (15+2) = 68",[34,1966,1968,1986,2013,2031,2052,2073],{"className":1967,"ariaHidden":65},[64],[34,1969,1971,1974,1977,1980,1983],{"className":1970},[69],[34,1972],{"className":1973,"style":511},[73],[34,1975,990],{"className":1976},[78],[34,1978],{"className":1979,"style":238},[237],[34,1981,438],{"className":1982},[242],[34,1984],{"className":1985,"style":238},[237],[34,1987,1989,1992,1995,1998,2001,2004,2007,2010],{"className":1988},[69],[34,1990],{"className":1991,"style":223},[73],[34,1993,56],{"className":1994,"style":80},[78,79],[34,1996,179],{"className":1997},[230],[34,1999,429],{"className":2000},[78],[34,2002,189],{"className":2003},[258],[34,2005],{"className":2006,"style":262},[237],[34,2008,192],{"className":2009},[266],[34,2011],{"className":2012,"style":262},[237],[34,2014,2016,2019,2022,2025,2028],{"className":2015},[69],[34,2017],{"className":2018,"style":511},[73],[34,2020,990],{"className":2021},[78],[34,2023],{"className":2024,"style":238},[237],[34,2026,438],{"className":2027},[242],[34,2029],{"className":2030,"style":238},[237],[34,2032,2034,2037,2040,2043,2046,2049],{"className":2033},[69],[34,2035],{"className":2036,"style":223},[73],[34,2038,179],{"className":2039},[230],[34,2041,615],{"className":2042},[78],[34,2044],{"className":2045,"style":238},[237],[34,2047,184],{"className":2048},[242],[34,2050],{"className":2051,"style":238},[237],[34,2053,2055,2058,2061,2064,2067,2070],{"className":2054},[69],[34,2056],{"className":2057,"style":223},[73],[34,2059,424],{"className":2060},[78],[34,2062,189],{"className":2063},[258],[34,2065],{"className":2066,"style":262},[237],[34,2068,192],{"className":2069},[266],[34,2071],{"className":2072,"style":262},[237],[34,2074,2076,2079],{"className":2075},[69],[34,2077],{"className":2078,"style":391},[73],[34,2080,1961],{"className":2081},[78],[16,2083,2084],{},"In the definition above, satisfying both additivity and homogeneity is sufficient and necessary for linearity.",[16,2086,2087],{},"Many people mistakenly think that \"linear\" simply means \"the graph is a straight line.\" This understanding is inaccurate.",[16,2089,2090,2093,2094,2097,2098,2154],{},[20,2091,2092],{},"Truly linear",": A straight line that ",[20,2095,2096],{},"must pass through the origin",". For example, ",[34,2099,2101,2121],{"className":2100,"translate":38},[37],[34,2102,2104],{"className":2103},[42],[44,2105,2106],{"xmlns":46},[48,2107,2108,2118],{},[51,2109,2110,2112,2114,2116],{},[54,2111,127],{},[176,2113,192],{},[347,2115,424],{},[54,2117,96],{},[58,2119,2120],{"encoding":60},"y = 2x",[34,2122,2124,2142],{"className":2123,"ariaHidden":65},[64],[34,2125,2127,2130,2133,2136,2139],{"className":2126},[69],[34,2128],{"className":2129,"style":139},[73],[34,2131,127],{"className":2132,"style":143},[78,79],[34,2134],{"className":2135,"style":262},[237],[34,2137,192],{"className":2138},[266],[34,2140],{"className":2141,"style":262},[237],[34,2143,2145,2148,2151],{"className":2144},[69],[34,2146],{"className":2147,"style":391},[73],[34,2149,424],{"className":2150},[78],[34,2152,96],{"className":2153},[78,79],". It satisfies both additivity and homogeneity. If the input is 0, the output must also be 0.",[16,2156,2157,2158,2188,2189,2268,2269,2400],{},"Strictly speaking, if a bias term ",[34,2159,2161,2175],{"className":2160,"translate":38},[37],[34,2162,2164],{"className":2163},[42],[44,2165,2166],{"xmlns":46},[48,2167,2168,2173],{},[51,2169,2170],{},[54,2171,2172],{},"b",[58,2174,2172],{"encoding":60},[34,2176,2178],{"className":2177,"ariaHidden":65},[64],[34,2179,2181,2185],{"className":2180},[69],[34,2182],{"className":2183,"style":2184},[73],"height:0.6944em;",[34,2186,2172],{"className":2187},[78,79]," is added, it becomes affine. It is a straight line, but not necessarily passing through the origin. For example, ",[34,2190,2192,2217],{"className":2191,"translate":38},[37],[34,2193,2195],{"className":2194},[42],[44,2196,2197],{"xmlns":46},[48,2198,2199,2214],{},[51,2200,2201,2203,2205,2207,2209,2211],{},[54,2202,127],{},[176,2204,192],{},[347,2206,424],{},[54,2208,96],{},[176,2210,184],{},[347,2212,2213],{},"1",[58,2215,2216],{"encoding":60},"y = 2x + 1",[34,2218,2220,2238,2259],{"className":2219,"ariaHidden":65},[64],[34,2221,2223,2226,2229,2232,2235],{"className":2222},[69],[34,2224],{"className":2225,"style":139},[73],[34,2227,127],{"className":2228,"style":143},[78,79],[34,2230],{"className":2231,"style":262},[237],[34,2233,192],{"className":2234},[266],[34,2236],{"className":2237,"style":262},[237],[34,2239,2241,2244,2247,2250,2253,2256],{"className":2240},[69],[34,2242],{"className":2243,"style":511},[73],[34,2245,424],{"className":2246},[78],[34,2248,96],{"className":2249},[78,79],[34,2251],{"className":2252,"style":238},[237],[34,2254,184],{"className":2255},[242],[34,2257],{"className":2258,"style":238},[237],[34,2260,2262,2265],{"className":2261},[69],[34,2263],{"className":2264,"style":391},[73],[34,2266,2213],{"className":2267},[78],". It does not satisfy homogeneity (since ",[34,2270,2272,2303],{"className":2271,"translate":38},[37],[34,2273,2275],{"className":2274},[42],[44,2276,2277],{"xmlns":46},[48,2278,2279,2300],{},[51,2280,2281,2283,2285,2288,2290,2292,2294,2298],{},[54,2282,56],{},[176,2284,179],{"stretchy":178},[347,2286,2287],{},"0",[176,2289,189],{"stretchy":178},[176,2291,192],{},[347,2293,2213],{},[176,2295,2297],{"mathvariant":2296},"normal","≠",[347,2299,2287],{},[58,2301,2302],{"encoding":60},"f(0)=1 \\neq 0",[34,2304,2306,2333,2391],{"className":2305,"ariaHidden":65},[64],[34,2307,2309,2312,2315,2318,2321,2324,2327,2330],{"className":2308},[69],[34,2310],{"className":2311,"style":223},[73],[34,2313,56],{"className":2314,"style":80},[78,79],[34,2316,179],{"className":2317},[230],[34,2319,2287],{"className":2320},[78],[34,2322,189],{"className":2323},[258],[34,2325],{"className":2326,"style":262},[237],[34,2328,192],{"className":2329},[266],[34,2331],{"className":2332,"style":262},[237],[34,2334,2336,2339,2342,2345,2388],{"className":2335},[69],[34,2337],{"className":2338,"style":74},[73],[34,2340,2213],{"className":2341},[78],[34,2343],{"className":2344,"style":262},[237],[34,2346,2348,2381,2385],{"className":2347},[266],[34,2349,2351],{"className":2350},[266],[34,2352,2355],{"className":2353},[78,2354],"vbox",[34,2356,2359],{"className":2357},[2358],"thinbox",[34,2360,2363,2366,2377],{"className":2361},[2362],"rlap",[34,2364],{"className":2365,"style":74},[73],[34,2367,2370],{"className":2368},[2369],"inner",[34,2371,2373],{"className":2372},[78],[34,2374,2376],{"className":2375},[266],"",[34,2378],{"className":2379},[2380],"fix",[34,2382],{"className":2383},[237,2384],"nobreak",[34,2386,192],{"className":2387},[266],[34,2389],{"className":2390,"style":262},[237],[34,2392,2394,2397],{"className":2393},[69],[34,2395],{"className":2396,"style":391},[73],[34,2398,2287],{"className":2399},[78],"). Although it is often called a \"linear function,\" in the strict definition it is not linear, but \"linear + translation,\" also known as affine.",[16,2402,2403],{},"Now we can understand why it's called a \"linear\" combination. This combination allows only two types of \"linear\" operations:",[2405,2406,2407,2444],"ol",{},[403,2408,2409,2412,2413,2415,2416,144],{},[20,2410,2411],{},"Scalar multiplication"," (corresponding to ",[20,2414,1144],{},"): Stretching or compressing a vector by a factor ",[34,2417,2419,2432],{"className":2418,"translate":38},[37],[34,2420,2422],{"className":2421},[42],[44,2423,2424],{"xmlns":46},[48,2425,2426,2430],{},[51,2427,2428],{},[54,2429,1173],{},[58,2431,1173],{"encoding":60},[34,2433,2435],{"className":2434,"ariaHidden":65},[64],[34,2436,2438,2441],{"className":2437},[69],[34,2439],{"className":2440,"style":108},[73],[34,2442,1173],{"className":2443},[78,79],[403,2445,2446,2412,2449,2451],{},[20,2447,2448],{},"Addition",[20,2450,148],{},"): Adding the processed vectors together.",[16,2453,2454,2455,2707,2708,2956],{},"Nonlinear operations like squaring, taking absolute values, or logarithms are not allowed. Therefore, expressions like ",[34,2456,2458,2501],{"className":2457,"translate":38},[37],[34,2459,2461],{"className":2460},[42],[44,2462,2463],{"xmlns":46},[48,2464,2465,2498],{},[51,2466,2467,2474,2484,2486,2492],{},[2468,2469,2470,2472],"msub",{},[54,2471,1173],{},[347,2473,2213],{},[2475,2476,2477,2480,2482],"msubsup",{},[54,2478,2479],{},"v",[347,2481,2213],{},[347,2483,424],{},[176,2485,184],{},[2468,2487,2488,2490],{},[54,2489,1173],{},[347,2491,424],{},[2468,2493,2494,2496],{},[54,2495,2479],{},[347,2497,424],{},[58,2499,2500],{"encoding":60},"c_1 v_1^2 + c_2 v_2",[34,2502,2504,2619],{"className":2503,"ariaHidden":65},[64],[34,2505,2507,2511,2557,2610,2613,2616],{"className":2506},[69],[34,2508],{"className":2509,"style":2510},[73],"height:1.0622em;vertical-align:-0.2481em;",[34,2512,2514,2517],{"className":2513},[78],[34,2515,1173],{"className":2516},[78,79],[34,2518,2520],{"className":2519},[810],[34,2521,2524,2548],{"className":2522},[814,2523],"vlist-t2",[34,2525,2527,2543],{"className":2526},[818],[34,2528,2531],{"className":2529,"style":2530},[822],"height:0.3011em;",[34,2532,2534,2537],{"style":2533},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[34,2535],{"className":2536,"style":830},[829],[34,2538,2540],{"className":2539},[834,835,836,837],[34,2541,2213],{"className":2542},[78,837],[34,2544,2547],{"className":2545},[2546],"vlist-s","​",[34,2549,2551],{"className":2550},[818],[34,2552,2555],{"className":2553,"style":2554},[822],"height:0.15em;",[34,2556],{},[34,2558,2560,2563],{"className":2559},[78],[34,2561,2479],{"className":2562,"style":143},[78,79],[34,2564,2566],{"className":2565},[810],[34,2567,2569,2601],{"className":2568},[814,2523],[34,2570,2572,2598],{"className":2571},[818],[34,2573,2575,2587],{"className":2574,"style":800},[822],[34,2576,2578,2581],{"style":2577},"top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;",[34,2579],{"className":2580,"style":830},[829],[34,2582,2584],{"className":2583},[834,835,836,837],[34,2585,2213],{"className":2586},[78,837],[34,2588,2589,2592],{"style":825},[34,2590],{"className":2591,"style":830},[829],[34,2593,2595],{"className":2594},[834,835,836,837],[34,2596,424],{"className":2597},[78,837],[34,2599,2547],{"className":2600},[2546],[34,2602,2604],{"className":2603},[818],[34,2605,2608],{"className":2606,"style":2607},[822],"height:0.2481em;",[34,2609],{},[34,2611],{"className":2612,"style":238},[237],[34,2614,184],{"className":2615},[242],[34,2617],{"className":2618,"style":238},[237],[34,2620,2622,2626,2666],{"className":2621},[69],[34,2623],{"className":2624,"style":2625},[73],"height:0.5806em;vertical-align:-0.15em;",[34,2627,2629,2632],{"className":2628},[78],[34,2630,1173],{"className":2631},[78,79],[34,2633,2635],{"className":2634},[810],[34,2636,2638,2658],{"className":2637},[814,2523],[34,2639,2641,2655],{"className":2640},[818],[34,2642,2644],{"className":2643,"style":2530},[822],[34,2645,2646,2649],{"style":2533},[34,2647],{"className":2648,"style":830},[829],[34,2650,2652],{"className":2651},[834,835,836,837],[34,2653,424],{"className":2654},[78,837],[34,2656,2547],{"className":2657},[2546],[34,2659,2661],{"className":2660},[818],[34,2662,2664],{"className":2663,"style":2554},[822],[34,2665],{},[34,2667,2669,2672],{"className":2668},[78],[34,2670,2479],{"className":2671,"style":143},[78,79],[34,2673,2675],{"className":2674},[810],[34,2676,2678,2699],{"className":2677},[814,2523],[34,2679,2681,2696],{"className":2680},[818],[34,2682,2684],{"className":2683,"style":2530},[822],[34,2685,2687,2690],{"style":2686},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[34,2688],{"className":2689,"style":830},[829],[34,2691,2693],{"className":2692},[834,835,836,837],[34,2694,424],{"className":2695},[78,837],[34,2697,2547],{"className":2698},[2546],[34,2700,2702],{"className":2701},[818],[34,2703,2705],{"className":2704,"style":2554},[822],[34,2706],{}," or ",[34,2709,2711,2753],{"className":2710,"translate":38},[37],[34,2712,2714],{"className":2713},[42],[44,2715,2716],{"xmlns":46},[48,2717,2718,2750],{},[51,2719,2720,2726,2732,2734,2740,2746,2748],{},[2468,2721,2722,2724],{},[54,2723,1173],{},[347,2725,2213],{},[2468,2727,2728,2730],{},[54,2729,2479],{},[347,2731,2213],{},[176,2733,184],{},[2468,2735,2736,2738],{},[54,2737,1173],{},[347,2739,424],{},[2468,2741,2742,2744],{},[54,2743,2479],{},[347,2745,424],{},[176,2747,184],{},[347,2749,429],{},[58,2751,2752],{"encoding":60},"c_1 v_1 + c_2 v_2 + 3",[34,2754,2756,2852,2947],{"className":2755,"ariaHidden":65},[64],[34,2757,2759,2763,2803,2843,2846,2849],{"className":2758},[69],[34,2760],{"className":2761,"style":2762},[73],"height:0.7333em;vertical-align:-0.15em;",[34,2764,2766,2769],{"className":2765},[78],[34,2767,1173],{"className":2768},[78,79],[34,2770,2772],{"className":2771},[810],[34,2773,2775,2795],{"className":2774},[814,2523],[34,2776,2778,2792],{"className":2777},[818],[34,2779,2781],{"className":2780,"style":2530},[822],[34,2782,2783,2786],{"style":2533},[34,2784],{"className":2785,"style":830},[829],[34,2787,2789],{"className":2788},[834,835,836,837],[34,2790,2213],{"className":2791},[78,837],[34,2793,2547],{"className":2794},[2546],[34,2796,2798],{"className":2797},[818],[34,2799,2801],{"className":2800,"style":2554},[822],[34,2802],{},[34,2804,2806,2809],{"className":2805},[78],[34,2807,2479],{"className":2808,"style":143},[78,79],[34,2810,2812],{"className":2811},[810],[34,2813,2815,2835],{"className":2814},[814,2523],[34,2816,2818,2832],{"className":2817},[818],[34,2819,2821],{"className":2820,"style":2530},[822],[34,2822,2823,2826],{"style":2686},[34,2824],{"className":2825,"style":830},[829],[34,2827,2829],{"className":2828},[834,835,836,837],[34,2830,2213],{"className":2831},[78,837],[34,2833,2547],{"className":2834},[2546],[34,2836,2838],{"className":2837},[818],[34,2839,2841],{"className":2840,"style":2554},[822],[34,2842],{},[34,2844],{"className":2845,"style":238},[237],[34,2847,184],{"className":2848},[242],[34,2850],{"className":2851,"style":238},[237],[34,2853,2855,2858,2898,2938,2941,2944],{"className":2854},[69],[34,2856],{"className":2857,"style":2762},[73],[34,2859,2861,2864],{"className":2860},[78],[34,2862,1173],{"className":2863},[78,79],[34,2865,2867],{"className":2866},[810],[34,2868,2870,2890],{"className":2869},[814,2523],[34,2871,2873,2887],{"className":2872},[818],[34,2874,2876],{"className":2875,"style":2530},[822],[34,2877,2878,2881],{"style":2533},[34,2879],{"className":2880,"style":830},[829],[34,2882,2884],{"className":2883},[834,835,836,837],[34,2885,424],{"className":2886},[78,837],[34,2888,2547],{"className":2889},[2546],[34,2891,2893],{"className":2892},[818],[34,2894,2896],{"className":2895,"style":2554},[822],[34,2897],{},[34,2899,2901,2904],{"className":2900},[78],[34,2902,2479],{"className":2903,"style":143},[78,79],[34,2905,2907],{"className":2906},[810],[34,2908,2910,2930],{"className":2909},[814,2523],[34,2911,2913,2927],{"className":2912},[818],[34,2914,2916],{"className":2915,"style":2530},[822],[34,2917,2918,2921],{"style":2686},[34,2919],{"className":2920,"style":830},[829],[34,2922,2924],{"className":2923},[834,835,836,837],[34,2925,424],{"className":2926},[78,837],[34,2928,2547],{"className":2929},[2546],[34,2931,2933],{"className":2932},[818],[34,2934,2936],{"className":2935,"style":2554},[822],[34,2937],{},[34,2939],{"className":2940,"style":238},[237],[34,2942,184],{"className":2943},[242],[34,2945],{"className":2946,"style":238},[237],[34,2948,2950,2953],{"className":2949},[69],[34,2951],{"className":2952,"style":391},[73],[34,2954,429],{"className":2955},[78]," are not called linear combinations.",[16,2958,2959,2960,2963,2964,2967],{},"It can be said that \"linear\" is the simplest and most core relationship in mathematics and science because it allows us to ",[20,2961,2962],{},"decompose"," complex problems into a ",[20,2965,2966],{},"superposition"," of simple parts.",[11,2969,2971],{"id":2970},"affine","Affine",[16,2973,2974,2975,3117],{},"An affine function ",[34,2976,2978,3014],{"className":2977,"translate":38},[37],[34,2979,2981],{"className":2980},[42],[44,2982,2983],{"xmlns":46},[48,2984,2985,3011],{},[51,2986,2987,2989,2992,3001,3004],{},[54,2988,56],{},[176,2990,2991],{},":",[755,2993,2994,2998],{},[54,2995,2997],{"mathvariant":2996},"double-struck","R",[54,2999,3000],{},"n",[176,3002,3003],{},"→",[755,3005,3006,3008],{},[54,3007,2997],{"mathvariant":2996},[54,3009,3010],{},"m",[58,3012,3013],{"encoding":60},"f: \\mathbb{R}^n \\to \\mathbb{R}^m",[34,3015,3017,3035,3082],{"className":3016,"ariaHidden":65},[64],[34,3018,3020,3023,3026,3029,3032],{"className":3019},[69],[34,3021],{"className":3022,"style":74},[73],[34,3024,56],{"className":3025,"style":80},[78,79],[34,3027],{"className":3028,"style":262},[237],[34,3030,2991],{"className":3031},[266],[34,3033],{"className":3034,"style":262},[237],[34,3036,3038,3042,3073,3076,3079],{"className":3037},[69],[34,3039],{"className":3040,"style":3041},[73],"height:0.6889em;",[34,3043,3045,3049],{"className":3044},[78],[34,3046,2997],{"className":3047},[78,3048],"mathbb",[34,3050,3052],{"className":3051},[810],[34,3053,3055],{"className":3054},[814],[34,3056,3058],{"className":3057},[818],[34,3059,3062],{"className":3060,"style":3061},[822],"height:0.6644em;",[34,3063,3064,3067],{"style":825},[34,3065],{"className":3066,"style":830},[829],[34,3068,3070],{"className":3069},[834,835,836,837],[34,3071,3000],{"className":3072},[78,79,837],[34,3074],{"className":3075,"style":262},[237],[34,3077,3003],{"className":3078},[266],[34,3080],{"className":3081,"style":262},[237],[34,3083,3085,3088],{"className":3084},[69],[34,3086],{"className":3087,"style":3041},[73],[34,3089,3091,3094],{"className":3090},[78],[34,3092,2997],{"className":3093},[78,3048],[34,3095,3097],{"className":3096},[810],[34,3098,3100],{"className":3099},[814],[34,3101,3103],{"className":3102},[818],[34,3104,3106],{"className":3105,"style":3061},[822],[34,3107,3108,3111],{"style":825},[34,3109],{"className":3110,"style":830},[829],[34,3112,3114],{"className":3113},[834,835,836,837],[34,3115,3010],{"className":3116},[78,79,837]," is defined as:",[34,3119,3121],{"className":3120,"translate":38},[159],[34,3122,3124,3155],{"className":3123,"translate":38},[37],[34,3125,3127],{"className":3126},[42],[44,3128,3129],{"xmlns":46,"display":168},[48,3130,3131,3152],{},[51,3132,3133,3135,3137,3139,3141,3143,3146,3148,3150],{},[54,3134,56],{},[176,3136,179],{"stretchy":178},[54,3138,96],{},[176,3140,189],{"stretchy":178},[176,3142,192],{},[54,3144,3145],{},"A",[54,3147,96],{},[176,3149,184],{},[54,3151,2172],{},[58,3153,3154],{"encoding":60},"f(x) = Ax + b",[34,3156,3158,3185,3207],{"className":3157,"ariaHidden":65},[64],[34,3159,3161,3164,3167,3170,3173,3176,3179,3182],{"className":3160},[69],[34,3162],{"className":3163,"style":223},[73],[34,3165,56],{"className":3166,"style":80},[78,79],[34,3168,179],{"className":3169},[230],[34,3171,96],{"className":3172},[78,79],[34,3174,189],{"className":3175},[258],[34,3177],{"className":3178,"style":262},[237],[34,3180,192],{"className":3181},[266],[34,3183],{"className":3184,"style":262},[237],[34,3186,3188,3192,3195,3198,3201,3204],{"className":3187},[69],[34,3189],{"className":3190,"style":3191},[73],"height:0.7667em;vertical-align:-0.0833em;",[34,3193,3145],{"className":3194},[78,79],[34,3196,96],{"className":3197},[78,79],[34,3199],{"className":3200,"style":238},[237],[34,3202,184],{"className":3203},[242],[34,3205],{"className":3206,"style":238},[237],[34,3208,3210,3213],{"className":3209},[69],[34,3211],{"className":3212,"style":2184},[73],[34,3214,2172],{"className":3215},[78,79],[16,3217,3218,3219,3248,3249,3277],{},"where ",[34,3220,3222,3235],{"className":3221,"translate":38},[37],[34,3223,3225],{"className":3224},[42],[44,3226,3227],{"xmlns":46},[48,3228,3229,3233],{},[51,3230,3231],{},[54,3232,3145],{},[58,3234,3145],{"encoding":60},[34,3236,3238],{"className":3237,"ariaHidden":65},[64],[34,3239,3241,3245],{"className":3240},[69],[34,3242],{"className":3243,"style":3244},[73],"height:0.6833em;",[34,3246,3145],{"className":3247},[78,79]," is a matrix and ",[34,3250,3252,3265],{"className":3251,"translate":38},[37],[34,3253,3255],{"className":3254},[42],[44,3256,3257],{"xmlns":46},[48,3258,3259,3263],{},[51,3260,3261],{},[54,3262,2172],{},[58,3264,2172],{"encoding":60},[34,3266,3268],{"className":3267,"ariaHidden":65},[64],[34,3269,3271,3274],{"className":3270},[69],[34,3272],{"className":3273,"style":2184},[73],[34,3275,2172],{"className":3276},[78,79]," is a constant vector. It is a linear function plus a bias.",[16,3279,3280],{},"If you take a weighted linear combination (coefficients can be scalars) of affine functions:",[34,3282,3284],{"className":3283,"translate":38},[159],[34,3285,3287,3347],{"className":3286,"translate":38},[37],[34,3288,3290],{"className":3289},[42],[44,3291,3292],{"xmlns":46,"display":168},[48,3293,3294,3344],{},[51,3295,3296,3299,3301,3303,3305,3307,3325,3332,3338,3340,3342],{},[54,3297,3298],{},"g",[176,3300,179],{"stretchy":178},[54,3302,96],{},[176,3304,189],{"stretchy":178},[176,3306,192],{},[3308,3309,3310,3313,3322],"munderover",{},[176,3311,3312],{},"∑",[51,3314,3315,3318,3320],{},[54,3316,3317],{},"i",[176,3319,192],{},[347,3321,2213],{},[54,3323,3324],{},"k",[2468,3326,3327,3330],{},[54,3328,3329],{},"α",[54,3331,3317],{},[2468,3333,3334,3336],{},[54,3335,56],{},[54,3337,3317],{},[176,3339,179],{"stretchy":178},[54,3341,96],{},[176,3343,189],{"stretchy":178},[58,3345,3346],{"encoding":60},"g(x) = \\sum_{i=1}^k \\alpha_i f_i(x)",[34,3348,3350,3377],{"className":3349,"ariaHidden":65},[64],[34,3351,3353,3356,3359,3362,3365,3368,3371,3374],{"className":3352},[69],[34,3354],{"className":3355,"style":223},[73],[34,3357,3298],{"className":3358,"style":143},[78,79],[34,3360,179],{"className":3361},[230],[34,3363,96],{"className":3364},[78,79],[34,3366,189],{"className":3367},[258],[34,3369],{"className":3370,"style":262},[237],[34,3372,192],{"className":3373},[266],[34,3375],{"className":3376,"style":262},[237],[34,3378,3380,3384,3459,3463,3506,3547,3550,3553],{"className":3379},[69],[34,3381],{"className":3382,"style":3383},[73],"height:3.1138em;vertical-align:-1.2777em;",[34,3385,3389],{"className":3386},[3387,3388],"mop","op-limits",[34,3390,3392,3450],{"className":3391},[814,2523],[34,3393,3395,3447],{"className":3394},[818],[34,3396,3399,3421,3434],{"className":3397,"style":3398},[822],"height:1.8361em;",[34,3400,3402,3406],{"style":3401},"top:-1.8723em;margin-left:0em;",[34,3403],{"className":3404,"style":3405},[829],"height:3.05em;",[34,3407,3409],{"className":3408},[834,835,836,837],[34,3410,3412,3415,3418],{"className":3411},[78,837],[34,3413,3317],{"className":3414},[78,79,837],[34,3416,192],{"className":3417},[266,837],[34,3419,2213],{"className":3420},[78,837],[34,3422,3424,3427],{"style":3423},"top:-3.05em;",[34,3425],{"className":3426,"style":3405},[829],[34,3428,3429],{},[34,3430,3312],{"className":3431},[3387,3432,3433],"op-symbol","large-op",[34,3435,3437,3440],{"style":3436},"top:-4.3em;margin-left:0em;",[34,3438],{"className":3439,"style":3405},[829],[34,3441,3443],{"className":3442},[834,835,836,837],[34,3444,3324],{"className":3445,"style":3446},[78,79,837],"margin-right:0.0315em;",[34,3448,2547],{"className":3449},[2546],[34,3451,3453],{"className":3452},[818],[34,3454,3457],{"className":3455,"style":3456},[822],"height:1.2777em;",[34,3458],{},[34,3460],{"className":3461,"style":3462},[237],"margin-right:0.1667em;",[34,3464,3466,3470],{"className":3465},[78],[34,3467,3329],{"className":3468,"style":3469},[78,79],"margin-right:0.0037em;",[34,3471,3473],{"className":3472},[810],[34,3474,3476,3498],{"className":3475},[814,2523],[34,3477,3479,3495],{"className":3478},[818],[34,3480,3483],{"className":3481,"style":3482},[822],"height:0.3117em;",[34,3484,3486,3489],{"style":3485},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[34,3487],{"className":3488,"style":830},[829],[34,3490,3492],{"className":3491},[834,835,836,837],[34,3493,3317],{"className":3494},[78,79,837],[34,3496,2547],{"className":3497},[2546],[34,3499,3501],{"className":3500},[818],[34,3502,3504],{"className":3503,"style":2554},[822],[34,3505],{},[34,3507,3509,3512],{"className":3508},[78],[34,3510,56],{"className":3511,"style":80},[78,79],[34,3513,3515],{"className":3514},[810],[34,3516,3518,3539],{"className":3517},[814,2523],[34,3519,3521,3536],{"className":3520},[818],[34,3522,3524],{"className":3523,"style":3482},[822],[34,3525,3527,3530],{"style":3526},"top:-2.55em;margin-left:-0.1076em;margin-right:0.05em;",[34,3528],{"className":3529,"style":830},[829],[34,3531,3533],{"className":3532},[834,835,836,837],[34,3534,3317],{"className":3535},[78,79,837],[34,3537,2547],{"className":3538},[2546],[34,3540,3542],{"className":3541},[818],[34,3543,3545],{"className":3544,"style":2554},[822],[34,3546],{},[34,3548,179],{"className":3549},[230],[34,3551,96],{"className":3552},[78,79],[34,3554,189],{"className":3555},[258],[16,3557,3558,3559,144],{},"where each ",[34,3560,3562,3604],{"className":3561,"translate":38},[37],[34,3563,3565],{"className":3564},[42],[44,3566,3567],{"xmlns":46},[48,3568,3569,3601],{},[51,3570,3571,3577,3579,3581,3583,3585,3591,3593,3595],{},[2468,3572,3573,3575],{},[54,3574,56],{},[54,3576,3317],{},[176,3578,179],{"stretchy":178},[54,3580,96],{},[176,3582,189],{"stretchy":178},[176,3584,192],{},[2468,3586,3587,3589],{},[54,3588,3145],{},[54,3590,3317],{},[54,3592,96],{},[176,3594,184],{},[2468,3596,3597,3599],{},[54,3598,2172],{},[54,3600,3317],{},[58,3602,3603],{"encoding":60},"f_i(x) = A_i x + b_i",[34,3605,3607,3671,3730],{"className":3606,"ariaHidden":65},[64],[34,3608,3610,3613,3653,3656,3659,3662,3665,3668],{"className":3609},[69],[34,3611],{"className":3612,"style":223},[73],[34,3614,3616,3619],{"className":3615},[78],[34,3617,56],{"className":3618,"style":80},[78,79],[34,3620,3622],{"className":3621},[810],[34,3623,3625,3645],{"className":3624},[814,2523],[34,3626,3628,3642],{"className":3627},[818],[34,3629,3631],{"className":3630,"style":3482},[822],[34,3632,3633,3636],{"style":3526},[34,3634],{"className":3635,"style":830},[829],[34,3637,3639],{"className":3638},[834,835,836,837],[34,3640,3317],{"className":3641},[78,79,837],[34,3643,2547],{"className":3644},[2546],[34,3646,3648],{"className":3647},[818],[34,3649,3651],{"className":3650,"style":2554},[822],[34,3652],{},[34,3654,179],{"className":3655},[230],[34,3657,96],{"className":3658},[78,79],[34,3660,189],{"className":3661},[258],[34,3663],{"className":3664,"style":262},[237],[34,3666,192],{"className":3667},[266],[34,3669],{"className":3670,"style":262},[237],[34,3672,3674,3678,3718,3721,3724,3727],{"className":3673},[69],[34,3675],{"className":3676,"style":3677},[73],"height:0.8333em;vertical-align:-0.15em;",[34,3679,3681,3684],{"className":3680},[78],[34,3682,3145],{"className":3683},[78,79],[34,3685,3687],{"className":3686},[810],[34,3688,3690,3710],{"className":3689},[814,2523],[34,3691,3693,3707],{"className":3692},[818],[34,3694,3696],{"className":3695,"style":3482},[822],[34,3697,3698,3701],{"style":2533},[34,3699],{"className":3700,"style":830},[829],[34,3702,3704],{"className":3703},[834,835,836,837],[34,3705,3317],{"className":3706},[78,79,837],[34,3708,2547],{"className":3709},[2546],[34,3711,3713],{"className":3712},[818],[34,3714,3716],{"className":3715,"style":2554},[822],[34,3717],{},[34,3719,96],{"className":3720},[78,79],[34,3722],{"className":3723,"style":238},[237],[34,3725,184],{"className":3726},[242],[34,3728],{"className":3729,"style":238},[237],[34,3731,3733,3737],{"className":3732},[69],[34,3734],{"className":3735,"style":3736},[73],"height:0.8444em;vertical-align:-0.15em;",[34,3738,3740,3743],{"className":3739},[78],[34,3741,2172],{"className":3742},[78,79],[34,3744,3746],{"className":3745},[810],[34,3747,3749,3769],{"className":3748},[814,2523],[34,3750,3752,3766],{"className":3751},[818],[34,3753,3755],{"className":3754,"style":3482},[822],[34,3756,3757,3760],{"style":2533},[34,3758],{"className":3759,"style":830},[829],[34,3761,3763],{"className":3762},[834,835,836,837],[34,3764,3317],{"className":3765},[78,79,837],[34,3767,2547],{"className":3768},[2546],[34,3770,3772],{"className":3771},[818],[34,3773,3775],{"className":3774,"style":2554},[822],[34,3776],{},[16,3778,3779],{},"Expanding:",[34,3781,3783],{"className":3782,"translate":38},[159],[34,3784,3786,3912],{"className":3785,"translate":38},[37],[34,3787,3789],{"className":3788},[42],[44,3790,3791],{"xmlns":46,"display":168},[48,3792,3793,3909],{},[51,3794,3795,3797,3799,3801,3803,3805,3819,3825,3827,3833,3835,3837,3843,3845,3847,3879,3881,3883,3897,3903],{},[54,3796,3298],{},[176,3798,179],{"stretchy":178},[54,3800,96],{},[176,3802,189],{"stretchy":178},[176,3804,192],{},[3308,3806,3807,3809,3817],{},[176,3808,3312],{},[51,3810,3811,3813,3815],{},[54,3812,3317],{},[176,3814,192],{},[347,3816,2213],{},[54,3818,3324],{},[2468,3820,3821,3823],{},[54,3822,3329],{},[54,3824,3317],{},[176,3826,179],{"stretchy":178},[2468,3828,3829,3831],{},[54,3830,3145],{},[54,3832,3317],{},[54,3834,96],{},[176,3836,184],{},[2468,3838,3839,3841],{},[54,3840,2172],{},[54,3842,3317],{},[176,3844,189],{"stretchy":178},[176,3846,192],{},[51,3848,3849,3851,3865,3871,3877],{},[176,3850,179],{"fence":65},[3308,3852,3853,3855,3863],{},[176,3854,3312],{},[51,3856,3857,3859,3861],{},[54,3858,3317],{},[176,3860,192],{},[347,3862,2213],{},[54,3864,3324],{},[2468,3866,3867,3869],{},[54,3868,3329],{},[54,3870,3317],{},[2468,3872,3873,3875],{},[54,3874,3145],{},[54,3876,3317],{},[176,3878,189],{"fence":65},[54,3880,96],{},[176,3882,184],{},[3308,3884,3885,3887,3895],{},[176,3886,3312],{},[51,3888,3889,3891,3893],{},[54,3890,3317],{},[176,3892,192],{},[347,3894,2213],{},[54,3896,3324],{},[2468,3898,3899,3901],{},[54,3900,3329],{},[54,3902,3317],{},[2468,3904,3905,3907],{},[54,3906,2172],{},[54,3908,3317],{},[58,3910,3911],{"encoding":60},"g(x) = \\sum_{i=1}^k \\alpha_i (A_i x + b_i) = \\left(\\sum_{i=1}^k \\alpha_i A_i \\right) x + \\sum_{i=1}^k \\alpha_i b_i",[34,3913,3915,3942,4110,4168,4356],{"className":3914,"ariaHidden":65},[64],[34,3916,3918,3921,3924,3927,3930,3933,3936,3939],{"className":3917},[69],[34,3919],{"className":3920,"style":223},[73],[34,3922,3298],{"className":3923,"style":143},[78,79],[34,3925,179],{"className":3926},[230],[34,3928,96],{"className":3929},[78,79],[34,3931,189],{"className":3932},[258],[34,3934],{"className":3935,"style":262},[237],[34,3937,192],{"className":3938},[266],[34,3940],{"className":3941,"style":262},[237],[34,3943,3945,3948,4012,4015,4055,4058,4098,4101,4104,4107],{"className":3944},[69],[34,3946],{"className":3947,"style":3383},[73],[34,3949,3951],{"className":3950},[3387,3388],[34,3952,3954,4004],{"className":3953},[814,2523],[34,3955,3957,4001],{"className":3956},[818],[34,3958,3960,3980,3990],{"className":3959,"style":3398},[822],[34,3961,3962,3965],{"style":3401},[34,3963],{"className":3964,"style":3405},[829],[34,3966,3968],{"className":3967},[834,835,836,837],[34,3969,3971,3974,3977],{"className":3970},[78,837],[34,3972,3317],{"className":3973},[78,79,837],[34,3975,192],{"className":3976},[266,837],[34,3978,2213],{"className":3979},[78,837],[34,3981,3982,3985],{"style":3423},[34,3983],{"className":3984,"style":3405},[829],[34,3986,3987],{},[34,3988,3312],{"className":3989},[3387,3432,3433],[34,3991,3992,3995],{"style":3436},[34,3993],{"className":3994,"style":3405},[829],[34,3996,3998],{"className":3997},[834,835,836,837],[34,3999,3324],{"className":4000,"style":3446},[78,79,837],[34,4002,2547],{"className":4003},[2546],[34,4005,4007],{"className":4006},[818],[34,4008,4010],{"className":4009,"style":3456},[822],[34,4011],{},[34,4013],{"className":4014,"style":3462},[237],[34,4016,4018,4021],{"className":4017},[78],[34,4019,3329],{"className":4020,"style":3469},[78,79],[34,4022,4024],{"className":4023},[810],[34,4025,4027,4047],{"className":4026},[814,2523],[34,4028,4030,4044],{"className":4029},[818],[34,4031,4033],{"className":4032,"style":3482},[822],[34,4034,4035,4038],{"style":3485},[34,4036],{"className":4037,"style":830},[829],[34,4039,4041],{"className":4040},[834,835,836,837],[34,4042,3317],{"className":4043},[78,79,837],[34,4045,2547],{"className":4046},[2546],[34,4048,4050],{"className":4049},[818],[34,4051,4053],{"className":4052,"style":2554},[822],[34,4054],{},[34,4056,179],{"className":4057},[230],[34,4059,4061,4064],{"className":4060},[78],[34,4062,3145],{"className":4063},[78,79],[34,4065,4067],{"className":4066},[810],[34,4068,4070,4090],{"className":4069},[814,2523],[34,4071,4073,4087],{"className":4072},[818],[34,4074,4076],{"className":4075,"style":3482},[822],[34,4077,4078,4081],{"style":2533},[34,4079],{"className":4080,"style":830},[829],[34,4082,4084],{"className":4083},[834,835,836,837],[34,4085,3317],{"className":4086},[78,79,837],[34,4088,2547],{"className":4089},[2546],[34,4091,4093],{"className":4092},[818],[34,4094,4096],{"className":4095,"style":2554},[822],[34,4097],{},[34,4099,96],{"className":4100},[78,79],[34,4102],{"className":4103,"style":238},[237],[34,4105,184],{"className":4106},[242],[34,4108],{"className":4109,"style":238},[237],[34,4111,4113,4116,4156,4159,4162,4165],{"className":4112},[69],[34,4114],{"className":4115,"style":223},[73],[34,4117,4119,4122],{"className":4118},[78],[34,4120,2172],{"className":4121},[78,79],[34,4123,4125],{"className":4124},[810],[34,4126,4128,4148],{"className":4127},[814,2523],[34,4129,4131,4145],{"className":4130},[818],[34,4132,4134],{"className":4133,"style":3482},[822],[34,4135,4136,4139],{"style":2533},[34,4137],{"className":4138,"style":830},[829],[34,4140,4142],{"className":4141},[834,835,836,837],[34,4143,3317],{"className":4144},[78,79,837],[34,4146,2547],{"className":4147},[2546],[34,4149,4151],{"className":4150},[818],[34,4152,4154],{"className":4153,"style":2554},[822],[34,4155],{},[34,4157,189],{"className":4158},[258],[34,4160],{"className":4161,"style":262},[237],[34,4163,192],{"className":4164},[266],[34,4166],{"className":4167,"style":262},[237],[34,4169,4171,4174,4341,4344,4347,4350,4353],{"className":4170},[69],[34,4172],{"className":4173,"style":3383},[73],[34,4175,4178,4188,4252,4255,4295,4335],{"className":4176},[4177],"minner",[34,4179,4183],{"className":4180,"style":4182},[230,4181],"delimcenter","top:0em;",[34,4184,179],{"className":4185},[4186,4187],"delimsizing","size4",[34,4189,4191],{"className":4190},[3387,3388],[34,4192,4194,4244],{"className":4193},[814,2523],[34,4195,4197,4241],{"className":4196},[818],[34,4198,4200,4220,4230],{"className":4199,"style":3398},[822],[34,4201,4202,4205],{"style":3401},[34,4203],{"className":4204,"style":3405},[829],[34,4206,4208],{"className":4207},[834,835,836,837],[34,4209,4211,4214,4217],{"className":4210},[78,837],[34,4212,3317],{"className":4213},[78,79,837],[34,4215,192],{"className":4216},[266,837],[34,4218,2213],{"className":4219},[78,837],[34,4221,4222,4225],{"style":3423},[34,4223],{"className":4224,"style":3405},[829],[34,4226,4227],{},[34,4228,3312],{"className":4229},[3387,3432,3433],[34,4231,4232,4235],{"style":3436},[34,4233],{"className":4234,"style":3405},[829],[34,4236,4238],{"className":4237},[834,835,836,837],[34,4239,3324],{"className":4240,"style":3446},[78,79,837],[34,4242,2547],{"className":4243},[2546],[34,4245,4247],{"className":4246},[818],[34,4248,4250],{"className":4249,"style":3456},[822],[34,4251],{},[34,4253],{"className":4254,"style":3462},[237],[34,4256,4258,4261],{"className":4257},[78],[34,4259,3329],{"className":4260,"style":3469},[78,79],[34,4262,4264],{"className":4263},[810],[34,4265,4267,4287],{"className":4266},[814,2523],[34,4268,4270,4284],{"className":4269},[818],[34,4271,4273],{"className":4272,"style":3482},[822],[34,4274,4275,4278],{"style":3485},[34,4276],{"className":4277,"style":830},[829],[34,4279,4281],{"className":4280},[834,835,836,837],[34,4282,3317],{"className":4283},[78,79,837],[34,4285,2547],{"className":4286},[2546],[34,4288,4290],{"className":4289},[818],[34,4291,4293],{"className":4292,"style":2554},[822],[34,4294],{},[34,4296,4298,4301],{"className":4297},[78],[34,4299,3145],{"className":4300},[78,79],[34,4302,4304],{"className":4303},[810],[34,4305,4307,4327],{"className":4306},[814,2523],[34,4308,4310,4324],{"className":4309},[818],[34,4311,4313],{"className":4312,"style":3482},[822],[34,4314,4315,4318],{"style":2533},[34,4316],{"className":4317,"style":830},[829],[34,4319,4321],{"className":4320},[834,835,836,837],[34,4322,3317],{"className":4323},[78,79,837],[34,4325,2547],{"className":4326},[2546],[34,4328,4330],{"className":4329},[818],[34,4331,4333],{"className":4332,"style":2554},[822],[34,4334],{},[34,4336,4338],{"className":4337,"style":4182},[258,4181],[34,4339,189],{"className":4340},[4186,4187],[34,4342],{"className":4343,"style":3462},[237],[34,4345,96],{"className":4346},[78,79],[34,4348],{"className":4349,"style":238},[237],[34,4351,184],{"className":4352},[242],[34,4354],{"className":4355,"style":238},[237],[34,4357,4359,4362,4426,4429,4469],{"className":4358},[69],[34,4360],{"className":4361,"style":3383},[73],[34,4363,4365],{"className":4364},[3387,3388],[34,4366,4368,4418],{"className":4367},[814,2523],[34,4369,4371,4415],{"className":4370},[818],[34,4372,4374,4394,4404],{"className":4373,"style":3398},[822],[34,4375,4376,4379],{"style":3401},[34,4377],{"className":4378,"style":3405},[829],[34,4380,4382],{"className":4381},[834,835,836,837],[34,4383,4385,4388,4391],{"className":4384},[78,837],[34,4386,3317],{"className":4387},[78,79,837],[34,4389,192],{"className":4390},[266,837],[34,4392,2213],{"className":4393},[78,837],[34,4395,4396,4399],{"style":3423},[34,4397],{"className":4398,"style":3405},[829],[34,4400,4401],{},[34,4402,3312],{"className":4403},[3387,3432,3433],[34,4405,4406,4409],{"style":3436},[34,4407],{"className":4408,"style":3405},[829],[34,4410,4412],{"className":4411},[834,835,836,837],[34,4413,3324],{"className":4414,"style":3446},[78,79,837],[34,4416,2547],{"className":4417},[2546],[34,4419,4421],{"className":4420},[818],[34,4422,4424],{"className":4423,"style":3456},[822],[34,4425],{},[34,4427],{"className":4428,"style":3462},[237],[34,4430,4432,4435],{"className":4431},[78],[34,4433,3329],{"className":4434,"style":3469},[78,79],[34,4436,4438],{"className":4437},[810],[34,4439,4441,4461],{"className":4440},[814,2523],[34,4442,4444,4458],{"className":4443},[818],[34,4445,4447],{"className":4446,"style":3482},[822],[34,4448,4449,4452],{"style":3485},[34,4450],{"className":4451,"style":830},[829],[34,4453,4455],{"className":4454},[834,835,836,837],[34,4456,3317],{"className":4457},[78,79,837],[34,4459,2547],{"className":4460},[2546],[34,4462,4464],{"className":4463},[818],[34,4465,4467],{"className":4466,"style":2554},[822],[34,4468],{},[34,4470,4472,4475],{"className":4471},[78],[34,4473,2172],{"className":4474},[78,79],[34,4476,4478],{"className":4477},[810],[34,4479,4481,4501],{"className":4480},[814,2523],[34,4482,4484,4498],{"className":4483},[818],[34,4485,4487],{"className":4486,"style":3482},[822],[34,4488,4489,4492],{"style":2533},[34,4490],{"className":4491,"style":830},[829],[34,4493,4495],{"className":4494},[834,835,836,837],[34,4496,3317],{"className":4497},[78,79,837],[34,4499,2547],{"className":4500},[2546],[34,4502,4504],{"className":4503},[818],[34,4505,4507],{"className":4506,"style":2554},[822],[34,4508],{},[16,4510,4511,4512,4515],{},"This is still an ",[20,4513,4514],{},"affine function",". Thus, weighted combinations do not destroy affinity.",[16,4517,4518],{},"If you combine affine functions using nonlinear polynomial operations, such as squaring or higher powers:",[34,4520,4522],{"className":4521,"translate":38},[159],[34,4523,4525,4600],{"className":4524,"translate":38},[37],[34,4526,4528],{"className":4527},[42],[44,4529,4530],{"xmlns":46,"display":168},[48,4531,4532,4597],{},[51,4533,4534,4537,4539,4541,4543,4545,4547,4553,4555,4557,4559,4565,4567,4569,4575,4577,4579,4581,4587,4589,4591,4593,4595],{},[54,4535,4536],{},"h",[176,4538,179],{"stretchy":178},[54,4540,96],{},[176,4542,189],{"stretchy":178},[176,4544,192],{},[176,4546,179],{"stretchy":178},[2468,4548,4549,4551],{},[54,4550,56],{},[347,4552,2213],{},[176,4554,179],{"stretchy":178},[54,4556,96],{},[176,4558,189],{"stretchy":178},[755,4560,4561,4563],{},[176,4562,189],{"stretchy":178},[347,4564,424],{},[176,4566,184],{},[347,4568,429],{},[2468,4570,4571,4573],{},[54,4572,56],{},[347,4574,424],{},[176,4576,179],{"stretchy":178},[54,4578,96],{},[176,4580,189],{"stretchy":178},[2468,4582,4583,4585],{},[54,4584,56],{},[347,4586,429],{},[176,4588,179],{"stretchy":178},[54,4590,96],{},[176,4592,189],{"stretchy":178},[176,4594,184],{},[347,4596,349],{},[58,4598,4599],{"encoding":60},"h(x) = (f_1(x))^2 + 3 f_2(x) f_3(x) + 5",[34,4601,4603,4630,4729,4845],{"className":4602,"ariaHidden":65},[64],[34,4604,4606,4609,4612,4615,4618,4621,4624,4627],{"className":4605},[69],[34,4607],{"className":4608,"style":223},[73],[34,4610,4536],{"className":4611},[78,79],[34,4613,179],{"className":4614},[230],[34,4616,96],{"className":4617},[78,79],[34,4619,189],{"className":4620},[258],[34,4622],{"className":4623,"style":262},[237],[34,4625,192],{"className":4626},[266],[34,4628],{"className":4629,"style":262},[237],[34,4631,4633,4637,4640,4680,4683,4686,4689,4720,4723,4726],{"className":4632},[69],[34,4634],{"className":4635,"style":4636},[73],"height:1.1141em;vertical-align:-0.25em;",[34,4638,179],{"className":4639},[230],[34,4641,4643,4646],{"className":4642},[78],[34,4644,56],{"className":4645,"style":80},[78,79],[34,4647,4649],{"className":4648},[810],[34,4650,4652,4672],{"className":4651},[814,2523],[34,4653,4655,4669],{"className":4654},[818],[34,4656,4658],{"className":4657,"style":2530},[822],[34,4659,4660,4663],{"style":3526},[34,4661],{"className":4662,"style":830},[829],[34,4664,4666],{"className":4665},[834,835,836,837],[34,4667,2213],{"className":4668},[78,837],[34,4670,2547],{"className":4671},[2546],[34,4673,4675],{"className":4674},[818],[34,4676,4678],{"className":4677,"style":2554},[822],[34,4679],{},[34,4681,179],{"className":4682},[230],[34,4684,96],{"className":4685},[78,79],[34,4687,189],{"className":4688},[258],[34,4690,4692,4695],{"className":4691},[258],[34,4693,189],{"className":4694},[258],[34,4696,4698],{"className":4697},[810],[34,4699,4701],{"className":4700},[814],[34,4702,4704],{"className":4703},[818],[34,4705,4708],{"className":4706,"style":4707},[822],"height:0.8641em;",[34,4709,4711,4714],{"style":4710},"top:-3.113em;margin-right:0.05em;",[34,4712],{"className":4713,"style":830},[829],[34,4715,4717],{"className":4716},[834,835,836,837],[34,4718,424],{"className":4719},[78,837],[34,4721],{"className":4722,"style":238},[237],[34,4724,184],{"className":4725},[242],[34,4727],{"className":4728,"style":238},[237],[34,4730,4732,4735,4738,4778,4781,4784,4787,4827,4830,4833,4836,4839,4842],{"className":4731},[69],[34,4733],{"className":4734,"style":223},[73],[34,4736,429],{"className":4737},[78],[34,4739,4741,4744],{"className":4740},[78],[34,4742,56],{"className":4743,"style":80},[78,79],[34,4745,4747],{"className":4746},[810],[34,4748,4750,4770],{"className":4749},[814,2523],[34,4751,4753,4767],{"className":4752},[818],[34,4754,4756],{"className":4755,"style":2530},[822],[34,4757,4758,4761],{"style":3526},[34,4759],{"className":4760,"style":830},[829],[34,4762,4764],{"className":4763},[834,835,836,837],[34,4765,424],{"className":4766},[78,837],[34,4768,2547],{"className":4769},[2546],[34,4771,4773],{"className":4772},[818],[34,4774,4776],{"className":4775,"style":2554},[822],[34,4777],{},[34,4779,179],{"className":4780},[230],[34,4782,96],{"className":4783},[78,79],[34,4785,189],{"className":4786},[258],[34,4788,4790,4793],{"className":4789},[78],[34,4791,56],{"className":4792,"style":80},[78,79],[34,4794,4796],{"className":4795},[810],[34,4797,4799,4819],{"className":4798},[814,2523],[34,4800,4802,4816],{"className":4801},[818],[34,4803,4805],{"className":4804,"style":2530},[822],[34,4806,4807,4810],{"style":3526},[34,4808],{"className":4809,"style":830},[829],[34,4811,4813],{"className":4812},[834,835,836,837],[34,4814,429],{"className":4815},[78,837],[34,4817,2547],{"className":4818},[2546],[34,4820,4822],{"className":4821},[818],[34,4823,4825],{"className":4824,"style":2554},[822],[34,4826],{},[34,4828,179],{"className":4829},[230],[34,4831,96],{"className":4832},[78,79],[34,4834,189],{"className":4835},[258],[34,4837],{"className":4838,"style":238},[237],[34,4840,184],{"className":4841},[242],[34,4843],{"className":4844,"style":238},[237],[34,4846,4848,4851],{"className":4847},[69],[34,4849],{"className":4850,"style":391},[73],[34,4852,349],{"className":4853},[78],[16,4855,4856,4857,4972,4973,5119],{},"Note: ",[34,4858,4860,4888],{"className":4859,"translate":38},[37],[34,4861,4863],{"className":4862},[42],[44,4864,4865],{"xmlns":46},[48,4866,4867,4885],{},[51,4868,4869,4875,4877,4879],{},[2468,4870,4871,4873],{},[54,4872,56],{},[347,4874,2213],{},[176,4876,179],{"stretchy":178},[54,4878,96],{},[755,4880,4881,4883],{},[176,4882,189],{"stretchy":178},[347,4884,424],{},[58,4886,4887],{"encoding":60},"f_1(x)^2",[34,4889,4891],{"className":4890,"ariaHidden":65},[64],[34,4892,4894,4897,4937,4940,4943],{"className":4893},[69],[34,4895],{"className":4896,"style":909},[73],[34,4898,4900,4903],{"className":4899},[78],[34,4901,56],{"className":4902,"style":80},[78,79],[34,4904,4906],{"className":4905},[810],[34,4907,4909,4929],{"className":4908},[814,2523],[34,4910,4912,4926],{"className":4911},[818],[34,4913,4915],{"className":4914,"style":2530},[822],[34,4916,4917,4920],{"style":3526},[34,4918],{"className":4919,"style":830},[829],[34,4921,4923],{"className":4922},[834,835,836,837],[34,4924,2213],{"className":4925},[78,837],[34,4927,2547],{"className":4928},[2546],[34,4930,4932],{"className":4931},[818],[34,4933,4935],{"className":4934,"style":2554},[822],[34,4936],{},[34,4938,179],{"className":4939},[230],[34,4941,96],{"className":4942},[78,79],[34,4944,4946,4949],{"className":4945},[258],[34,4947,189],{"className":4948},[258],[34,4950,4952],{"className":4951},[810],[34,4953,4955],{"className":4954},[814],[34,4956,4958],{"className":4957},[818],[34,4959,4961],{"className":4960,"style":800},[822],[34,4962,4963,4966],{"style":825},[34,4964],{"className":4965,"style":830},[829],[34,4967,4969],{"className":4968},[834,835,836,837],[34,4970,424],{"className":4971},[78,837]," is element-wise squaring, ",[34,4974,4976,5012],{"className":4975,"translate":38},[37],[34,4977,4979],{"className":4978},[42],[44,4980,4981],{"xmlns":46},[48,4982,4983,5009],{},[51,4984,4985,4991,4993,4995,4997,5003,5005,5007],{},[2468,4986,4987,4989],{},[54,4988,56],{},[347,4990,424],{},[176,4992,179],{"stretchy":178},[54,4994,96],{},[176,4996,189],{"stretchy":178},[2468,4998,4999,5001],{},[54,5000,56],{},[347,5002,429],{},[176,5004,179],{"stretchy":178},[54,5006,96],{},[176,5008,189],{"stretchy":178},[58,5010,5011],{"encoding":60},"f_2(x) f_3(x)",[34,5013,5015],{"className":5014,"ariaHidden":65},[64],[34,5016,5018,5021,5061,5064,5067,5070,5110,5113,5116],{"className":5017},[69],[34,5019],{"className":5020,"style":223},[73],[34,5022,5024,5027],{"className":5023},[78],[34,5025,56],{"className":5026,"style":80},[78,79],[34,5028,5030],{"className":5029},[810],[34,5031,5033,5053],{"className":5032},[814,2523],[34,5034,5036,5050],{"className":5035},[818],[34,5037,5039],{"className":5038,"style":2530},[822],[34,5040,5041,5044],{"style":3526},[34,5042],{"className":5043,"style":830},[829],[34,5045,5047],{"className":5046},[834,835,836,837],[34,5048,424],{"className":5049},[78,837],[34,5051,2547],{"className":5052},[2546],[34,5054,5056],{"className":5055},[818],[34,5057,5059],{"className":5058,"style":2554},[822],[34,5060],{},[34,5062,179],{"className":5063},[230],[34,5065,96],{"className":5066},[78,79],[34,5068,189],{"className":5069},[258],[34,5071,5073,5076],{"className":5072},[78],[34,5074,56],{"className":5075,"style":80},[78,79],[34,5077,5079],{"className":5078},[810],[34,5080,5082,5102],{"className":5081},[814,2523],[34,5083,5085,5099],{"className":5084},[818],[34,5086,5088],{"className":5087,"style":2530},[822],[34,5089,5090,5093],{"style":3526},[34,5091],{"className":5092,"style":830},[829],[34,5094,5096],{"className":5095},[834,835,836,837],[34,5097,429],{"className":5098},[78,837],[34,5100,2547],{"className":5101},[2546],[34,5103,5105],{"className":5104},[818],[34,5106,5108],{"className":5107,"style":2554},[822],[34,5109],{},[34,5111,179],{"className":5112},[230],[34,5114,96],{"className":5115},[78,79],[34,5117,189],{"className":5118},[258]," is element-wise multiplication or dot product.",[16,5121,5122,5123,5179,5180,144],{},"These operations introduce quadratic or higher-order terms and no longer maintain the ",[34,5124,5126,5146],{"className":5125,"translate":38},[37],[34,5127,5129],{"className":5128},[42],[44,5130,5131],{"xmlns":46},[48,5132,5133,5143],{},[51,5134,5135,5137,5139,5141],{},[54,5136,3145],{},[54,5138,96],{},[176,5140,184],{},[54,5142,2172],{},[58,5144,5145],{"encoding":60},"Ax+b",[34,5147,5149,5170],{"className":5148,"ariaHidden":65},[64],[34,5150,5152,5155,5158,5161,5164,5167],{"className":5151},[69],[34,5153],{"className":5154,"style":3191},[73],[34,5156,3145],{"className":5157},[78,79],[34,5159,96],{"className":5160},[78,79],[34,5162],{"className":5163,"style":238},[237],[34,5165,184],{"className":5166},[242],[34,5168],{"className":5169,"style":238},[237],[34,5171,5173,5176],{"className":5172},[69],[34,5174],{"className":5175,"style":2184},[73],[34,5177,2172],{"className":5178},[78,79]," form. Therefore, nonlinear polynomial combinations are ",[20,5181,5182],{},"generally not affine functions",[11,5184,5186],{"id":5185},"affine-composition","Affine Composition",[16,5188,5189],{},"This question is very important and key to understanding the nature of neural networks. Let's carefully analyze whether the composition of affine functions remains affine.",[16,5191,5192],{},"Suppose you have two affine functions:",[34,5194,5196],{"className":5195,"translate":38},[159],[34,5197,5199,5254],{"className":5198,"translate":38},[37],[34,5200,5202],{"className":5201},[42],[44,5203,5204],{"xmlns":46,"display":168},[48,5205,5206,5251],{},[51,5207,5208,5210,5212,5214,5216,5218,5220,5222,5224,5226,5229,5231,5233,5235,5237,5239,5241,5244,5246,5248],{},[54,5209,56],{},[176,5211,179],{"stretchy":178},[54,5213,96],{},[176,5215,189],{"stretchy":178},[176,5217,192],{},[54,5219,3145],{},[54,5221,96],{},[176,5223,184],{},[54,5225,2172],{},[176,5227,5228],{"separator":65},",",[237,5230],{"width":1197},[54,5232,3298],{},[176,5234,179],{"stretchy":178},[54,5236,96],{},[176,5238,189],{"stretchy":178},[176,5240,192],{},[54,5242,5243],{},"C",[54,5245,96],{},[176,5247,184],{},[54,5249,5250],{},"d",[58,5252,5253],{"encoding":60},"f(x) = A x + b, \\quad g(x) = C x + d",[34,5255,5257,5284,5305,5345,5367],{"className":5256,"ariaHidden":65},[64],[34,5258,5260,5263,5266,5269,5272,5275,5278,5281],{"className":5259},[69],[34,5261],{"className":5262,"style":223},[73],[34,5264,56],{"className":5265,"style":80},[78,79],[34,5267,179],{"className":5268},[230],[34,5270,96],{"className":5271},[78,79],[34,5273,189],{"className":5274},[258],[34,5276],{"className":5277,"style":262},[237],[34,5279,192],{"className":5280},[266],[34,5282],{"className":5283,"style":262},[237],[34,5285,5287,5290,5293,5296,5299,5302],{"className":5286},[69],[34,5288],{"className":5289,"style":3191},[73],[34,5291,3145],{"className":5292},[78,79],[34,5294,96],{"className":5295},[78,79],[34,5297],{"className":5298,"style":238},[237],[34,5300,184],{"className":5301},[242],[34,5303],{"className":5304,"style":238},[237],[34,5306,5308,5311,5314,5318,5321,5324,5327,5330,5333,5336,5339,5342],{"className":5307},[69],[34,5309],{"className":5310,"style":223},[73],[34,5312,2172],{"className":5313},[78,79],[34,5315,5228],{"className":5316},[5317],"mpunct",[34,5319],{"className":5320,"style":1297},[237],[34,5322],{"className":5323,"style":3462},[237],[34,5325,3298],{"className":5326,"style":143},[78,79],[34,5328,179],{"className":5329},[230],[34,5331,96],{"className":5332},[78,79],[34,5334,189],{"className":5335},[258],[34,5337],{"className":5338,"style":262},[237],[34,5340,192],{"className":5341},[266],[34,5343],{"className":5344,"style":262},[237],[34,5346,5348,5351,5355,5358,5361,5364],{"className":5347},[69],[34,5349],{"className":5350,"style":3191},[73],[34,5352,5243],{"className":5353,"style":5354},[78,79],"margin-right:0.0715em;",[34,5356,96],{"className":5357},[78,79],[34,5359],{"className":5360,"style":238},[237],[34,5362,184],{"className":5363},[242],[34,5365],{"className":5366,"style":238},[237],[34,5368,5370,5373],{"className":5369},[69],[34,5371],{"className":5372,"style":2184},[73],[34,5374,5250],{"className":5375},[78,79],[16,5377,5378],{},"Composition is:",[34,5380,5382],{"className":5381,"translate":38},[159],[34,5383,5385,5437],{"className":5384,"translate":38},[37],[34,5386,5388],{"className":5387},[42],[44,5389,5390],{"xmlns":46,"display":168},[48,5391,5392,5434],{},[51,5393,5394,5396,5398,5400,5402,5404,5406,5408,5410,5412,5414,5416,5418,5420,5422,5424,5426,5428,5430,5432],{},[54,5395,4536],{},[176,5397,179],{"stretchy":178},[54,5399,96],{},[176,5401,189],{"stretchy":178},[176,5403,192],{},[54,5405,56],{},[176,5407,179],{"stretchy":178},[54,5409,3298],{},[176,5411,179],{"stretchy":178},[54,5413,96],{},[176,5415,189],{"stretchy":178},[176,5417,189],{"stretchy":178},[176,5419,192],{},[54,5421,56],{},[176,5423,179],{"stretchy":178},[54,5425,5243],{},[54,5427,96],{},[176,5429,184],{},[54,5431,5250],{},[176,5433,189],{"stretchy":178},[58,5435,5436],{"encoding":60},"h(x) = f(g(x)) = f(Cx + d)",[34,5438,5440,5467,5501,5528],{"className":5439,"ariaHidden":65},[64],[34,5441,5443,5446,5449,5452,5455,5458,5461,5464],{"className":5442},[69],[34,5444],{"className":5445,"style":223},[73],[34,5447,4536],{"className":5448},[78,79],[34,5450,179],{"className":5451},[230],[34,5453,96],{"className":5454},[78,79],[34,5456,189],{"className":5457},[258],[34,5459],{"className":5460,"style":262},[237],[34,5462,192],{"className":5463},[266],[34,5465],{"className":5466,"style":262},[237],[34,5468,5470,5473,5476,5479,5482,5485,5488,5492,5495,5498],{"className":5469},[69],[34,5471],{"className":5472,"style":223},[73],[34,5474,56],{"className":5475,"style":80},[78,79],[34,5477,179],{"className":5478},[230],[34,5480,3298],{"className":5481,"style":143},[78,79],[34,5483,179],{"className":5484},[230],[34,5486,96],{"className":5487},[78,79],[34,5489,5491],{"className":5490},[258],"))",[34,5493],{"className":5494,"style":262},[237],[34,5496,192],{"className":5497},[266],[34,5499],{"className":5500,"style":262},[237],[34,5502,5504,5507,5510,5513,5516,5519,5522,5525],{"className":5503},[69],[34,5505],{"className":5506,"style":223},[73],[34,5508,56],{"className":5509,"style":80},[78,79],[34,5511,179],{"className":5512},[230],[34,5514,5243],{"className":5515,"style":5354},[78,79],[34,5517,96],{"className":5518},[78,79],[34,5520],{"className":5521,"style":238},[237],[34,5523,184],{"className":5524},[242],[34,5526],{"className":5527,"style":238},[237],[34,5529,5531,5534,5537],{"className":5530},[69],[34,5532],{"className":5533,"style":223},[73],[34,5535,5250],{"className":5536},[78,79],[34,5538,189],{"className":5539},[258],[16,5541,5542],{},"Expanding to see the form:",[34,5544,5546],{"className":5545,"translate":38},[159],[34,5547,5549,5615],{"className":5548,"translate":38},[37],[34,5550,5552],{"className":5551},[42],[44,5553,5554],{"xmlns":46,"display":168},[48,5555,5556,5612],{},[51,5557,5558,5560,5562,5564,5566,5568,5570,5572,5574,5576,5578,5580,5582,5584,5586,5588,5590,5592,5594,5596,5598,5600,5602,5604,5606,5608,5610],{},[54,5559,4536],{},[176,5561,179],{"stretchy":178},[54,5563,96],{},[176,5565,189],{"stretchy":178},[176,5567,192],{},[54,5569,3145],{},[176,5571,179],{"stretchy":178},[54,5573,5243],{},[54,5575,96],{},[176,5577,184],{},[54,5579,5250],{},[176,5581,189],{"stretchy":178},[176,5583,184],{},[54,5585,2172],{},[176,5587,192],{},[176,5589,179],{"stretchy":178},[54,5591,3145],{},[54,5593,5243],{},[176,5595,189],{"stretchy":178},[54,5597,96],{},[176,5599,184],{},[176,5601,179],{"stretchy":178},[54,5603,3145],{},[54,5605,5250],{},[176,5607,184],{},[54,5609,2172],{},[176,5611,189],{"stretchy":178},[58,5613,5614],{"encoding":60},"h(x) = A(Cx + d) + b = (AC)x + (Ad + b)",[34,5616,5618,5645,5672,5693,5711,5741,5765],{"className":5617,"ariaHidden":65},[64],[34,5619,5621,5624,5627,5630,5633,5636,5639,5642],{"className":5620},[69],[34,5622],{"className":5623,"style":223},[73],[34,5625,4536],{"className":5626},[78,79],[34,5628,179],{"className":5629},[230],[34,5631,96],{"className":5632},[78,79],[34,5634,189],{"className":5635},[258],[34,5637],{"className":5638,"style":262},[237],[34,5640,192],{"className":5641},[266],[34,5643],{"className":5644,"style":262},[237],[34,5646,5648,5651,5654,5657,5660,5663,5666,5669],{"className":5647},[69],[34,5649],{"className":5650,"style":223},[73],[34,5652,3145],{"className":5653},[78,79],[34,5655,179],{"className":5656},[230],[34,5658,5243],{"className":5659,"style":5354},[78,79],[34,5661,96],{"className":5662},[78,79],[34,5664],{"className":5665,"style":238},[237],[34,5667,184],{"className":5668},[242],[34,5670],{"className":5671,"style":238},[237],[34,5673,5675,5678,5681,5684,5687,5690],{"className":5674},[69],[34,5676],{"className":5677,"style":223},[73],[34,5679,5250],{"className":5680},[78,79],[34,5682,189],{"className":5683},[258],[34,5685],{"className":5686,"style":238},[237],[34,5688,184],{"className":5689},[242],[34,5691],{"className":5692,"style":238},[237],[34,5694,5696,5699,5702,5705,5708],{"className":5695},[69],[34,5697],{"className":5698,"style":2184},[73],[34,5700,2172],{"className":5701},[78,79],[34,5703],{"className":5704,"style":262},[237],[34,5706,192],{"className":5707},[266],[34,5709],{"className":5710,"style":262},[237],[34,5712,5714,5717,5720,5723,5726,5729,5732,5735,5738],{"className":5713},[69],[34,5715],{"className":5716,"style":223},[73],[34,5718,179],{"className":5719},[230],[34,5721,3145],{"className":5722},[78,79],[34,5724,5243],{"className":5725,"style":5354},[78,79],[34,5727,189],{"className":5728},[258],[34,5730,96],{"className":5731},[78,79],[34,5733],{"className":5734,"style":238},[237],[34,5736,184],{"className":5737},[242],[34,5739],{"className":5740,"style":238},[237],[34,5742,5744,5747,5750,5753,5756,5759,5762],{"className":5743},[69],[34,5745],{"className":5746,"style":223},[73],[34,5748,179],{"className":5749},[230],[34,5751,3145],{"className":5752},[78,79],[34,5754,5250],{"className":5755},[78,79],[34,5757],{"className":5758,"style":238},[237],[34,5760,184],{"className":5761},[242],[34,5763],{"className":5764,"style":238},[237],[34,5766,5768,5771,5774],{"className":5767},[69],[34,5769],{"className":5770,"style":223},[73],[34,5772,2172],{"className":5773},[78,79],[34,5775,189],{"className":5776},[258],[16,5778,5779,5780,144],{},"The result is still an ",[20,5781,4514],{},[16,5783,5784],{},"Thus, the composition of affine functions remains affine. This can be generalized: composing any number of affine functions:",[34,5786,5788],{"className":5787,"translate":38},[159],[34,5789,5791,5843],{"className":5790,"translate":38},[37],[34,5792,5794],{"className":5793},[42],[44,5795,5796],{"xmlns":46,"display":168},[48,5797,5798,5840],{},[51,5799,5800,5806,5808,5811,5817,5819,5825,5827,5829,5831,5833,5835,5838],{},[2468,5801,5802,5804],{},[54,5803,56],{},[54,5805,3000],{},[176,5807,179],{"stretchy":178},[176,5809,5810],{},"…",[2468,5812,5813,5815],{},[54,5814,56],{},[347,5816,424],{},[176,5818,179],{"stretchy":178},[2468,5820,5821,5823],{},[54,5822,56],{},[347,5824,2213],{},[176,5826,179],{"stretchy":178},[54,5828,96],{},[176,5830,189],{"stretchy":178},[176,5832,189],{"stretchy":178},[176,5834,5810],{},[1201,5836,5837],{}," ",[176,5839,189],{"stretchy":178},[58,5841,5842],{"encoding":60},"f_n(\\dots f_2(f_1(x))\\dots)",[34,5844,5846],{"className":5845,"ariaHidden":65},[64],[34,5847,5849,5852,5893,5896,5899,5902,5942,5945,5985,5988,5991,5994,5997,6000,6003],{"className":5848},[69],[34,5850],{"className":5851,"style":223},[73],[34,5853,5855,5858],{"className":5854},[78],[34,5856,56],{"className":5857,"style":80},[78,79],[34,5859,5861],{"className":5860},[810],[34,5862,5864,5885],{"className":5863},[814,2523],[34,5865,5867,5882],{"className":5866},[818],[34,5868,5871],{"className":5869,"style":5870},[822],"height:0.1514em;",[34,5872,5873,5876],{"style":3526},[34,5874],{"className":5875,"style":830},[829],[34,5877,5879],{"className":5878},[834,835,836,837],[34,5880,3000],{"className":5881},[78,79,837],[34,5883,2547],{"className":5884},[2546],[34,5886,5888],{"className":5887},[818],[34,5889,5891],{"className":5890,"style":2554},[822],[34,5892],{},[34,5894,179],{"className":5895},[230],[34,5897,5810],{"className":5898},[4177],[34,5900],{"className":5901,"style":3462},[237],[34,5903,5905,5908],{"className":5904},[78],[34,5906,56],{"className":5907,"style":80},[78,79],[34,5909,5911],{"className":5910},[810],[34,5912,5914,5934],{"className":5913},[814,2523],[34,5915,5917,5931],{"className":5916},[818],[34,5918,5920],{"className":5919,"style":2530},[822],[34,5921,5922,5925],{"style":3526},[34,5923],{"className":5924,"style":830},[829],[34,5926,5928],{"className":5927},[834,835,836,837],[34,5929,424],{"className":5930},[78,837],[34,5932,2547],{"className":5933},[2546],[34,5935,5937],{"className":5936},[818],[34,5938,5940],{"className":5939,"style":2554},[822],[34,5941],{},[34,5943,179],{"className":5944},[230],[34,5946,5948,5951],{"className":5947},[78],[34,5949,56],{"className":5950,"style":80},[78,79],[34,5952,5954],{"className":5953},[810],[34,5955,5957,5977],{"className":5956},[814,2523],[34,5958,5960,5974],{"className":5959},[818],[34,5961,5963],{"className":5962,"style":2530},[822],[34,5964,5965,5968],{"style":3526},[34,5966],{"className":5967,"style":830},[829],[34,5969,5971],{"className":5970},[834,835,836,837],[34,5972,2213],{"className":5973},[78,837],[34,5975,2547],{"className":5976},[2546],[34,5978,5980],{"className":5979},[818],[34,5981,5983],{"className":5982,"style":2554},[822],[34,5984],{},[34,5986,179],{"className":5987},[230],[34,5989,96],{"className":5990},[78,79],[34,5992,5491],{"className":5993},[258],[34,5995],{"className":5996,"style":3462},[237],[34,5998,5810],{"className":5999},[4177],[34,6001],{"className":6002,"style":3462},[237],[34,6004,189],{"className":6005},[258],[16,6007,6008],{},"can always be written as:",[34,6010,6012],{"className":6011,"translate":38},[159],[34,6013,6015,6044],{"className":6014,"translate":38},[37],[34,6016,6018],{"className":6017},[42],[44,6019,6020],{"xmlns":46,"display":168},[48,6021,6022,6041],{},[51,6023,6024,6031,6033,6035],{},[2468,6025,6026,6028],{},[54,6027,3145],{},[1201,6029,6030],{},"total",[54,6032,96],{},[176,6034,184],{},[2468,6036,6037,6039],{},[54,6038,2172],{},[1201,6040,6030],{},[58,6042,6043],{"encoding":60},"A_\\text{total} x + b_\\text{total}",[34,6045,6047,6109],{"className":6046,"ariaHidden":65},[64],[34,6048,6050,6053,6097,6100,6103,6106],{"className":6049},[69],[34,6051],{"className":6052,"style":3677},[73],[34,6054,6056,6059],{"className":6055},[78],[34,6057,3145],{"className":6058},[78,79],[34,6060,6062],{"className":6061},[810],[34,6063,6065,6089],{"className":6064},[814,2523],[34,6066,6068,6086],{"className":6067},[818],[34,6069,6072],{"className":6070,"style":6071},[822],"height:0.3361em;",[34,6073,6074,6077],{"style":2533},[34,6075],{"className":6076,"style":830},[829],[34,6078,6080],{"className":6079},[834,835,836,837],[34,6081,6083],{"className":6082},[78,1304,837],[34,6084,6030],{"className":6085},[78,837],[34,6087,2547],{"className":6088},[2546],[34,6090,6092],{"className":6091},[818],[34,6093,6095],{"className":6094,"style":2554},[822],[34,6096],{},[34,6098,96],{"className":6099},[78,79],[34,6101],{"className":6102,"style":238},[237],[34,6104,184],{"className":6105},[242],[34,6107],{"className":6108,"style":238},[237],[34,6110,6112,6115],{"className":6111},[69],[34,6113],{"className":6114,"style":3736},[73],[34,6116,6118,6121],{"className":6117},[78],[34,6119,2172],{"className":6120},[78,79],[34,6122,6124],{"className":6123},[810],[34,6125,6127,6150],{"className":6126},[814,2523],[34,6128,6130,6147],{"className":6129},[818],[34,6131,6133],{"className":6132,"style":6071},[822],[34,6134,6135,6138],{"style":2533},[34,6136],{"className":6137,"style":830},[829],[34,6139,6141],{"className":6140},[834,835,836,837],[34,6142,6144],{"className":6143},[78,1304,837],[34,6145,6030],{"className":6146},[78,837],[34,6148,2547],{"className":6149},[2546],[34,6151,6153],{"className":6152},[818],[34,6154,6156],{"className":6155,"style":2554},[822],[34,6157],{},[16,6159,6160,6161,6164,6165,6168],{},"This is also why: ",[20,6162,6163],{},"if each layer of a neural network is affine without activation functions",", no matter how many layers are stacked, the network's expressive power is ",[20,6166,6167],{},"the same as a single layer",". We can think of an affine layer as an operation of stretching, rotating, or translating, or a combination of these. Composing multiple affine transformations is equivalent to continuous stretching + rotation + translation → ultimately, a single overall stretching + rotation + translation. Therefore, composing affine transformations does not increase nonlinear degrees of freedom and cannot represent curves or complex functions.",[16,6170,6171],{},"Each layer of a neural network is an affine mapping:",[34,6173,6175],{"className":6174,"translate":38},[159],[34,6176,6178,6209],{"className":6177,"translate":38},[37],[34,6179,6181],{"className":6180},[42],[44,6182,6183],{"xmlns":46,"display":168},[48,6184,6185,6206],{},[51,6186,6187,6189,6191,6193,6195,6197,6200,6202,6204],{},[54,6188,56],{},[176,6190,179],{"stretchy":178},[54,6192,96],{},[176,6194,189],{"stretchy":178},[176,6196,192],{},[54,6198,6199],{},"W",[54,6201,96],{},[176,6203,184],{},[54,6205,2172],{},[58,6207,6208],{"encoding":60},"f(x) = Wx + b",[34,6210,6212,6239,6261],{"className":6211,"ariaHidden":65},[64],[34,6213,6215,6218,6221,6224,6227,6230,6233,6236],{"className":6214},[69],[34,6216],{"className":6217,"style":223},[73],[34,6219,56],{"className":6220,"style":80},[78,79],[34,6222,179],{"className":6223},[230],[34,6225,96],{"className":6226},[78,79],[34,6228,189],{"className":6229},[258],[34,6231],{"className":6232,"style":262},[237],[34,6234,192],{"className":6235},[266],[34,6237],{"className":6238,"style":262},[237],[34,6240,6242,6245,6249,6252,6255,6258],{"className":6241},[69],[34,6243],{"className":6244,"style":3191},[73],[34,6246,6199],{"className":6247,"style":6248},[78,79],"margin-right:0.1389em;",[34,6250,96],{"className":6251},[78,79],[34,6253],{"className":6254,"style":238},[237],[34,6256,184],{"className":6257},[242],[34,6259],{"className":6260,"style":238},[237],[34,6262,6264,6267],{"className":6263},[69],[34,6265],{"className":6266,"style":2184},[73],[34,6268,2172],{"className":6269},[78,79],[400,6271,6272,6437],{},[403,6273,6274,6275,6303,6304,6407,6408,6436],{},"If ",[34,6276,6278,6291],{"className":6277,"translate":38},[37],[34,6279,6281],{"className":6280},[42],[44,6282,6283],{"xmlns":46},[48,6284,6285,6289],{},[51,6286,6287],{},[54,6288,6199],{},[58,6290,6199],{"encoding":60},[34,6292,6294],{"className":6293,"ariaHidden":65},[64],[34,6295,6297,6300],{"className":6296},[69],[34,6298],{"className":6299,"style":3244},[73],[34,6301,6199],{"className":6302,"style":6248},[78,79]," is invertible (i.e., ",[34,6305,6307,6335],{"className":6306,"translate":38},[37],[34,6308,6310],{"className":6309},[42],[44,6311,6312],{"xmlns":46},[48,6313,6314,6332],{},[51,6315,6316,6319,6322,6324,6326,6328,6330],{},[54,6317,6318],{},"det",[176,6320,6321],{},"⁡",[176,6323,179],{"stretchy":178},[54,6325,6199],{},[176,6327,189],{"stretchy":178},[176,6329,2297],{"mathvariant":2296},[347,6331,2287],{},[58,6333,6334],{"encoding":60},"\\det(W) \\neq 0",[34,6336,6338,6398],{"className":6337,"ariaHidden":65},[64],[34,6339,6341,6344,6347,6350,6353,6356,6359,6395],{"className":6340},[69],[34,6342],{"className":6343,"style":223},[73],[34,6345,6318],{"className":6346},[3387],[34,6348,179],{"className":6349},[230],[34,6351,6199],{"className":6352,"style":6248},[78,79],[34,6354,189],{"className":6355},[258],[34,6357],{"className":6358,"style":262},[237],[34,6360,6362,6389,6392],{"className":6361},[266],[34,6363,6365],{"className":6364},[266],[34,6366,6368],{"className":6367},[78,2354],[34,6369,6371],{"className":6370},[2358],[34,6372,6374,6377,6386],{"className":6373},[2362],[34,6375],{"className":6376,"style":74},[73],[34,6378,6380],{"className":6379},[2369],[34,6381,6383],{"className":6382},[78],[34,6384,2376],{"className":6385},[266],[34,6387],{"className":6388},[2380],[34,6390],{"className":6391},[237,2384],[34,6393,192],{"className":6394},[266],[34,6396],{"className":6397,"style":262},[237],[34,6399,6401,6404],{"className":6400},[69],[34,6402],{"className":6403,"style":391},[73],[34,6405,2287],{"className":6406},[78],"), then ",[34,6409,6411,6424],{"className":6410,"translate":38},[37],[34,6412,6414],{"className":6413},[42],[44,6415,6416],{"xmlns":46},[48,6417,6418,6422],{},[51,6419,6420],{},[54,6421,56],{},[58,6423,56],{"encoding":60},[34,6425,6427],{"className":6426,"ariaHidden":65},[64],[34,6428,6430,6433],{"className":6429},[69],[34,6431],{"className":6432,"style":74},[73],[34,6434,56],{"className":6435,"style":80},[78,79]," is bijective and continuous.",[403,6438,6439],{},"The inverse mapping is:",[34,6441,6443],{"className":6442,"translate":38},[159],[34,6444,6446,6497],{"className":6445,"translate":38},[37],[34,6447,6449],{"className":6448},[42],[44,6450,6451],{"xmlns":46,"display":168},[48,6452,6453,6494],{},[51,6454,6455,6466,6468,6470,6472,6474,6484,6486,6488,6490,6492],{},[755,6456,6457,6459],{},[54,6458,56],{},[51,6460,6461,6464],{},[176,6462,6463],{},"−",[347,6465,2213],{},[176,6467,179],{"stretchy":178},[54,6469,127],{},[176,6471,189],{"stretchy":178},[176,6473,192],{},[755,6475,6476,6478],{},[54,6477,3145],{},[51,6479,6480,6482],{},[176,6481,6463],{},[347,6483,2213],{},[176,6485,179],{"stretchy":178},[54,6487,127],{},[176,6489,6463],{},[54,6491,2172],{},[176,6493,189],{"stretchy":178},[58,6495,6496],{"encoding":60},"f^{-1}(y) = A^{-1}(y - b)",[34,6498,6500,6559,6615],{"className":6499,"ariaHidden":65},[64],[34,6501,6503,6506,6541,6544,6547,6550,6553,6556],{"className":6502},[69],[34,6504],{"className":6505,"style":4636},[73],[34,6507,6509,6512],{"className":6508},[78],[34,6510,56],{"className":6511,"style":80},[78,79],[34,6513,6515],{"className":6514},[810],[34,6516,6518],{"className":6517},[814],[34,6519,6521],{"className":6520},[818],[34,6522,6524],{"className":6523,"style":4707},[822],[34,6525,6526,6529],{"style":4710},[34,6527],{"className":6528,"style":830},[829],[34,6530,6532],{"className":6531},[834,835,836,837],[34,6533,6535,6538],{"className":6534},[78,837],[34,6536,6463],{"className":6537},[78,837],[34,6539,2213],{"className":6540},[78,837],[34,6542,179],{"className":6543},[230],[34,6545,127],{"className":6546,"style":143},[78,79],[34,6548,189],{"className":6549},[258],[34,6551],{"className":6552,"style":262},[237],[34,6554,192],{"className":6555},[266],[34,6557],{"className":6558,"style":262},[237],[34,6560,6562,6565,6600,6603,6606,6609,6612],{"className":6561},[69],[34,6563],{"className":6564,"style":4636},[73],[34,6566,6568,6571],{"className":6567},[78],[34,6569,3145],{"className":6570},[78,79],[34,6572,6574],{"className":6573},[810],[34,6575,6577],{"className":6576},[814],[34,6578,6580],{"className":6579},[818],[34,6581,6583],{"className":6582,"style":4707},[822],[34,6584,6585,6588],{"style":4710},[34,6586],{"className":6587,"style":830},[829],[34,6589,6591],{"className":6590},[834,835,836,837],[34,6592,6594,6597],{"className":6593},[78,837],[34,6595,6463],{"className":6596},[78,837],[34,6598,2213],{"className":6599},[78,837],[34,6601,179],{"className":6602},[230],[34,6604,127],{"className":6605,"style":143},[78,79],[34,6607],{"className":6608,"style":238},[237],[34,6610,6463],{"className":6611},[242],[34,6613],{"className":6614,"style":238},[237],[34,6616,6618,6621,6624],{"className":6617},[69],[34,6619],{"className":6620,"style":223},[73],[34,6622,2172],{"className":6623},[78,79],[34,6625,189],{"className":6626},[258],[16,6628,6629,6630,6633,6634,6662],{},"which is also affine and continuous → a ",[20,6631,6632],{},"homeomorphism",". However, there is a core condition: ",[34,6635,6637,6650],{"className":6636,"translate":38},[37],[34,6638,6640],{"className":6639},[42],[44,6641,6642],{"xmlns":46},[48,6643,6644,6648],{},[51,6645,6646],{},[54,6647,6199],{},[58,6649,6199],{"encoding":60},[34,6651,6653],{"className":6652,"ariaHidden":65},[64],[34,6654,6656,6659],{"className":6655},[69],[34,6657],{"className":6658,"style":3244},[73],[34,6660,6199],{"className":6661,"style":6248},[78,79]," must be invertible.",[16,6664,6665],{},"Composing multiple affine transformations is equivalent to composing multiple homeomorphisms. Suppose you have multiple consecutive invertible affine maps:",[34,6667,6669],{"className":6668,"translate":38},[159],[34,6670,6672,6718],{"className":6671,"translate":38},[37],[34,6673,6675],{"className":6674},[42],[44,6676,6677],{"xmlns":46,"display":168},[48,6678,6679,6715],{},[51,6680,6681,6687,6690,6702,6704,6707,6709],{},[2468,6682,6683,6685],{},[54,6684,56],{},[54,6686,3000],{},[176,6688,6689],{},"∘",[2468,6691,6692,6694],{},[54,6693,56],{},[51,6695,6696,6698,6700],{},[54,6697,3000],{},[176,6699,6463],{},[347,6701,2213],{},[176,6703,6689],{},[176,6705,6706],{},"⋯",[176,6708,6689],{},[2468,6710,6711,6713],{},[54,6712,56],{},[347,6714,2213],{},[58,6716,6717],{"encoding":60},"f_n \\circ f_{n-1} \\circ \\cdots \\circ f_1",[34,6719,6721,6776,6842,6860],{"className":6720,"ariaHidden":65},[64],[34,6722,6724,6727,6767,6770,6773],{"className":6723},[69],[34,6725],{"className":6726,"style":74},[73],[34,6728,6730,6733],{"className":6729},[78],[34,6731,56],{"className":6732,"style":80},[78,79],[34,6734,6736],{"className":6735},[810],[34,6737,6739,6759],{"className":6738},[814,2523],[34,6740,6742,6756],{"className":6741},[818],[34,6743,6745],{"className":6744,"style":5870},[822],[34,6746,6747,6750],{"style":3526},[34,6748],{"className":6749,"style":830},[829],[34,6751,6753],{"className":6752},[834,835,836,837],[34,6754,3000],{"className":6755},[78,79,837],[34,6757,2547],{"className":6758},[2546],[34,6760,6762],{"className":6761},[818],[34,6763,6765],{"className":6764,"style":2554},[822],[34,6766],{},[34,6768],{"className":6769,"style":238},[237],[34,6771,6689],{"className":6772},[242],[34,6774],{"className":6775,"style":238},[237],[34,6777,6779,6783,6833,6836,6839],{"className":6778},[69],[34,6780],{"className":6781,"style":6782},[73],"height:0.9028em;vertical-align:-0.2083em;",[34,6784,6786,6789],{"className":6785},[78],[34,6787,56],{"className":6788,"style":80},[78,79],[34,6790,6792],{"className":6791},[810],[34,6793,6795,6824],{"className":6794},[814,2523],[34,6796,6798,6821],{"className":6797},[818],[34,6799,6801],{"className":6800,"style":2530},[822],[34,6802,6803,6806],{"style":3526},[34,6804],{"className":6805,"style":830},[829],[34,6807,6809],{"className":6808},[834,835,836,837],[34,6810,6812,6815,6818],{"className":6811},[78,837],[34,6813,3000],{"className":6814},[78,79,837],[34,6816,6463],{"className":6817},[242,837],[34,6819,2213],{"className":6820},[78,837],[34,6822,2547],{"className":6823},[2546],[34,6825,6827],{"className":6826},[818],[34,6828,6831],{"className":6829,"style":6830},[822],"height:0.2083em;",[34,6832],{},[34,6834],{"className":6835,"style":238},[237],[34,6837,6689],{"className":6838},[242],[34,6840],{"className":6841,"style":238},[237],[34,6843,6845,6848,6851,6854,6857],{"className":6844},[69],[34,6846],{"className":6847,"style":1263},[73],[34,6849,6706],{"className":6850},[4177],[34,6852],{"className":6853,"style":238},[237],[34,6855,6689],{"className":6856},[242],[34,6858],{"className":6859,"style":238},[237],[34,6861,6863,6866],{"className":6862},[69],[34,6864],{"className":6865,"style":74},[73],[34,6867,6869,6872],{"className":6868},[78],[34,6870,56],{"className":6871,"style":80},[78,79],[34,6873,6875],{"className":6874},[810],[34,6876,6878,6898],{"className":6877},[814,2523],[34,6879,6881,6895],{"className":6880},[818],[34,6882,6884],{"className":6883,"style":2530},[822],[34,6885,6886,6889],{"style":3526},[34,6887],{"className":6888,"style":830},[829],[34,6890,6892],{"className":6891},[834,835,836,837],[34,6893,2213],{"className":6894},[78,837],[34,6896,2547],{"className":6897},[2546],[34,6899,6901],{"className":6900},[818],[34,6902,6904],{"className":6903,"style":2554},[822],[34,6905],{},[16,6907,6908,6909,6979],{},"Each ",[34,6910,6912,6930],{"className":6911,"translate":38},[37],[34,6913,6915],{"className":6914},[42],[44,6916,6917],{"xmlns":46},[48,6918,6919,6927],{},[51,6920,6921],{},[2468,6922,6923,6925],{},[54,6924,56],{},[54,6926,3317],{},[58,6928,6929],{"encoding":60},"f_i",[34,6931,6933],{"className":6932,"ariaHidden":65},[64],[34,6934,6936,6939],{"className":6935},[69],[34,6937],{"className":6938,"style":74},[73],[34,6940,6942,6945],{"className":6941},[78],[34,6943,56],{"className":6944,"style":80},[78,79],[34,6946,6948],{"className":6947},[810],[34,6949,6951,6971],{"className":6950},[814,2523],[34,6952,6954,6968],{"className":6953},[818],[34,6955,6957],{"className":6956,"style":3482},[822],[34,6958,6959,6962],{"style":3526},[34,6960],{"className":6961,"style":830},[829],[34,6963,6965],{"className":6964},[834,835,836,837],[34,6966,3317],{"className":6967},[78,79,837],[34,6969,2547],{"className":6970},[2546],[34,6972,6974],{"className":6973},[818],[34,6975,6977],{"className":6976,"style":2554},[822],[34,6978],{}," is a homeomorphism, and the composition of homeomorphisms remains a homeomorphism. Hence, the final mapping is also a homeomorphism.",[16,6981,6982,6983,7013],{},"Thus, we have the following conclusion: Let ",[34,6984,6986,7000],{"className":6985,"translate":38},[37],[34,6987,6989],{"className":6988},[42],[44,6990,6991],{"xmlns":46},[48,6992,6993,6998],{},[51,6994,6995],{},[54,6996,6997],{},"M",[58,6999,6997],{"encoding":60},[34,7001,7003],{"className":7002,"ariaHidden":65},[64],[34,7004,7006,7009],{"className":7005},[69],[34,7007],{"className":7008,"style":3244},[73],[34,7010,6997],{"className":7011,"style":7012},[78,79],"margin-right:0.109em;"," be the original manifold:",[400,7015,7016,7019],{},[403,7017,7018],{},"An affine mapping that is continuous and bijective → the manifold structure is preserved (homeomorphic).",[403,7020,7021,7022,7025,7026,144],{},"Composing multiple such affine maps continuously → the resulting manifold is ",[20,7023,7024],{},"homeomorphic"," to ",[34,7027,7029,7042],{"className":7028,"translate":38},[37],[34,7030,7032],{"className":7031},[42],[44,7033,7034],{"xmlns":46},[48,7035,7036,7040],{},[51,7037,7038],{},[54,7039,6997],{},[58,7041,6997],{"encoding":60},[34,7043,7045],{"className":7044,"ariaHidden":65},[64],[34,7046,7048,7051],{"className":7047},[69],[34,7049],{"className":7050,"style":3244},[73],[34,7052,6997],{"className":7053,"style":7012},[78,79],[16,7055,7056],{},"In other words: \"Continuous linear + bias transformations do not change the topology of the manifold; they only stretch\u002Frotate\u002Ftranslate\u002Fshear it.\"",[16,7058,7059],{},"This is why using only linear layers cannot \"bend\" the data manifold: the data retains its \"original shape\" (topologically invariant) under nested linear layers. If a decision boundary needs to cut a complex manifold, nonlinearity (activation functions) must be introduced.",[16,7061,7062,7063,144],{},"This is also the goal in deep learning. If the original manifold's structure is inseparable for a given task, but separability is required for a learning objective, then the original manifold and the target manifold ",[20,7064,7065],{},"must not be homeomorphic",[16,7067,7068],{},"In the transformation from the original manifold to the target manifold, losing homeomorphism requires breaking away from linear, affine transformations. Because affine transformations (or more generally, homeomorphic transformations) alone are usually insufficient for disentangling classes. Deep networks often need to reconstruct the data manifold through non-homeomorphic nonlinear operations to make it ultimately linearly separable.",[16,7070,7071],{},"We need to consider how to make the composition of affine transformations nonlinear. This insight is the core of the geometric interpretation of neural networks:",[316,7073,7074],{},[16,7075,7076],{},"Linear layers can only perform topology-preserving affine transformations; nonlinear layers allow topological changes, enabling neural networks to represent complex functions.",[11,7078,7080],{"id":7079},"activation-for-affine-layers","Activation for Affine Layers",[16,7082,6171],{},[34,7084,7086],{"className":7085,"translate":38},[159],[34,7087,7089,7132],{"className":7088,"translate":38},[37],[34,7090,7092],{"className":7091},[42],[44,7093,7094],{"xmlns":46,"display":168},[48,7095,7096,7129],{},[51,7097,7098,7105,7107,7109,7111,7113,7119,7121,7123],{},[2468,7099,7100,7103],{},[54,7101,7102],{},"l",[54,7104,3317],{},[176,7106,179],{"stretchy":178},[54,7108,96],{},[176,7110,189],{"stretchy":178},[176,7112,192],{},[2468,7114,7115,7117],{},[54,7116,6199],{},[54,7118,3317],{},[54,7120,96],{},[176,7122,184],{},[2468,7124,7125,7127],{},[54,7126,2172],{},[54,7128,3317],{},[58,7130,7131],{"encoding":60},"l_i(x) = W_i x + b_i",[34,7133,7135,7201,7260],{"className":7134,"ariaHidden":65},[64],[34,7136,7138,7141,7183,7186,7189,7192,7195,7198],{"className":7137},[69],[34,7139],{"className":7140,"style":223},[73],[34,7142,7144,7148],{"className":7143},[78],[34,7145,7102],{"className":7146,"style":7147},[78,79],"margin-right:0.0197em;",[34,7149,7151],{"className":7150},[810],[34,7152,7154,7175],{"className":7153},[814,2523],[34,7155,7157,7172],{"className":7156},[818],[34,7158,7160],{"className":7159,"style":3482},[822],[34,7161,7163,7166],{"style":7162},"top:-2.55em;margin-left:-0.0197em;margin-right:0.05em;",[34,7164],{"className":7165,"style":830},[829],[34,7167,7169],{"className":7168},[834,835,836,837],[34,7170,3317],{"className":7171},[78,79,837],[34,7173,2547],{"className":7174},[2546],[34,7176,7178],{"className":7177},[818],[34,7179,7181],{"className":7180,"style":2554},[822],[34,7182],{},[34,7184,179],{"className":7185},[230],[34,7187,96],{"className":7188},[78,79],[34,7190,189],{"className":7191},[258],[34,7193],{"className":7194,"style":262},[237],[34,7196,192],{"className":7197},[266],[34,7199],{"className":7200,"style":262},[237],[34,7202,7204,7207,7248,7251,7254,7257],{"className":7203},[69],[34,7205],{"className":7206,"style":3677},[73],[34,7208,7210,7213],{"className":7209},[78],[34,7211,6199],{"className":7212,"style":6248},[78,79],[34,7214,7216],{"className":7215},[810],[34,7217,7219,7240],{"className":7218},[814,2523],[34,7220,7222,7237],{"className":7221},[818],[34,7223,7225],{"className":7224,"style":3482},[822],[34,7226,7228,7231],{"style":7227},"top:-2.55em;margin-left:-0.1389em;margin-right:0.05em;",[34,7229],{"className":7230,"style":830},[829],[34,7232,7234],{"className":7233},[834,835,836,837],[34,7235,3317],{"className":7236},[78,79,837],[34,7238,2547],{"className":7239},[2546],[34,7241,7243],{"className":7242},[818],[34,7244,7246],{"className":7245,"style":2554},[822],[34,7247],{},[34,7249,96],{"className":7250},[78,79],[34,7252],{"className":7253,"style":238},[237],[34,7255,184],{"className":7256},[242],[34,7258],{"className":7259,"style":238},[237],[34,7261,7263,7266],{"className":7262},[69],[34,7264],{"className":7265,"style":3736},[73],[34,7267,7269,7272],{"className":7268},[78],[34,7270,2172],{"className":7271},[78,79],[34,7273,7275],{"className":7274},[810],[34,7276,7278,7298],{"className":7277},[814,2523],[34,7279,7281,7295],{"className":7280},[818],[34,7282,7284],{"className":7283,"style":3482},[822],[34,7285,7286,7289],{"style":2533},[34,7287],{"className":7288,"style":830},[829],[34,7290,7292],{"className":7291},[834,835,836,837],[34,7293,3317],{"className":7294},[78,79,837],[34,7296,2547],{"className":7297},[2546],[34,7299,7301],{"className":7300},[818],[34,7302,7304],{"className":7303,"style":2554},[822],[34,7305],{},[16,7307,3218,7308,7337,7338,7366,7367,7395],{},[34,7309,7311,7324],{"className":7310,"translate":38},[37],[34,7312,7314],{"className":7313},[42],[44,7315,7316],{"xmlns":46},[48,7317,7318,7322],{},[51,7319,7320],{},[54,7321,3317],{},[58,7323,3317],{"encoding":60},[34,7325,7327],{"className":7326,"ariaHidden":65},[64],[34,7328,7330,7334],{"className":7329},[69],[34,7331],{"className":7332,"style":7333},[73],"height:0.6595em;",[34,7335,3317],{"className":7336},[78,79]," denotes the ",[34,7339,7341,7354],{"className":7340,"translate":38},[37],[34,7342,7344],{"className":7343},[42],[44,7345,7346],{"xmlns":46},[48,7347,7348,7352],{},[51,7349,7350],{},[54,7351,3317],{},[58,7353,3317],{"encoding":60},[34,7355,7357],{"className":7356,"ariaHidden":65},[64],[34,7358,7360,7363],{"className":7359},[69],[34,7361],{"className":7362,"style":7333},[73],[34,7364,3317],{"className":7365},[78,79],"-th layer. The nested composition of affine mappings across ",[34,7368,7370,7383],{"className":7369,"translate":38},[37],[34,7371,7373],{"className":7372},[42],[44,7374,7375],{"xmlns":46},[48,7376,7377,7381],{},[51,7378,7379],{},[54,7380,3000],{},[58,7382,3000],{"encoding":60},[34,7384,7386],{"className":7385,"ariaHidden":65},[64],[34,7387,7389,7392],{"className":7388},[69],[34,7390],{"className":7391,"style":108},[73],[34,7393,3000],{"className":7394},[78,79]," layers is:",[34,7397,7399],{"className":7398,"translate":38},[159],[34,7400,7402,7460],{"className":7401,"translate":38},[37],[34,7403,7405],{"className":7404},[42],[44,7406,7407],{"xmlns":46,"display":168},[48,7408,7409,7457],{},[51,7410,7411,7417,7419,7431,7433,7445,7447,7449,7451],{},[2468,7412,7413,7415],{},[54,7414,7102],{},[54,7416,3000],{},[176,7418,6689],{},[2468,7420,7421,7423],{},[54,7422,7102],{},[51,7424,7425,7427,7429],{},[54,7426,3000],{},[176,7428,6463],{},[347,7430,2213],{},[176,7432,6689],{},[2468,7434,7435,7437],{},[54,7436,7102],{},[51,7438,7439,7441,7443],{},[54,7440,3000],{},[176,7442,6463],{},[347,7444,424],{},[176,7446,6689],{},[176,7448,6706],{},[176,7450,6689],{},[2468,7452,7453,7455],{},[54,7454,7102],{},[347,7456,2287],{},[58,7458,7459],{"encoding":60},"l_n \\circ l_{n-1} \\circ l_{n-2} \\circ \\dots \\circ l_0",[34,7461,7463,7518,7582,7646,7664],{"className":7462,"ariaHidden":65},[64],[34,7464,7466,7469,7509,7512,7515],{"className":7465},[69],[34,7467],{"className":7468,"style":3736},[73],[34,7470,7472,7475],{"className":7471},[78],[34,7473,7102],{"className":7474,"style":7147},[78,79],[34,7476,7478],{"className":7477},[810],[34,7479,7481,7501],{"className":7480},[814,2523],[34,7482,7484,7498],{"className":7483},[818],[34,7485,7487],{"className":7486,"style":5870},[822],[34,7488,7489,7492],{"style":7162},[34,7490],{"className":7491,"style":830},[829],[34,7493,7495],{"className":7494},[834,835,836,837],[34,7496,3000],{"className":7497},[78,79,837],[34,7499,2547],{"className":7500},[2546],[34,7502,7504],{"className":7503},[818],[34,7505,7507],{"className":7506,"style":2554},[822],[34,7508],{},[34,7510],{"className":7511,"style":238},[237],[34,7513,6689],{"className":7514},[242],[34,7516],{"className":7517,"style":238},[237],[34,7519,7521,7524,7573,7576,7579],{"className":7520},[69],[34,7522],{"className":7523,"style":6782},[73],[34,7525,7527,7530],{"className":7526},[78],[34,7528,7102],{"className":7529,"style":7147},[78,79],[34,7531,7533],{"className":7532},[810],[34,7534,7536,7565],{"className":7535},[814,2523],[34,7537,7539,7562],{"className":7538},[818],[34,7540,7542],{"className":7541,"style":2530},[822],[34,7543,7544,7547],{"style":7162},[34,7545],{"className":7546,"style":830},[829],[34,7548,7550],{"className":7549},[834,835,836,837],[34,7551,7553,7556,7559],{"className":7552},[78,837],[34,7554,3000],{"className":7555},[78,79,837],[34,7557,6463],{"className":7558},[242,837],[34,7560,2213],{"className":7561},[78,837],[34,7563,2547],{"className":7564},[2546],[34,7566,7568],{"className":7567},[818],[34,7569,7571],{"className":7570,"style":6830},[822],[34,7572],{},[34,7574],{"className":7575,"style":238},[237],[34,7577,6689],{"className":7578},[242],[34,7580],{"className":7581,"style":238},[237],[34,7583,7585,7588,7637,7640,7643],{"className":7584},[69],[34,7586],{"className":7587,"style":6782},[73],[34,7589,7591,7594],{"className":7590},[78],[34,7592,7102],{"className":7593,"style":7147},[78,79],[34,7595,7597],{"className":7596},[810],[34,7598,7600,7629],{"className":7599},[814,2523],[34,7601,7603,7626],{"className":7602},[818],[34,7604,7606],{"className":7605,"style":2530},[822],[34,7607,7608,7611],{"style":7162},[34,7609],{"className":7610,"style":830},[829],[34,7612,7614],{"className":7613},[834,835,836,837],[34,7615,7617,7620,7623],{"className":7616},[78,837],[34,7618,3000],{"className":7619},[78,79,837],[34,7621,6463],{"className":7622},[242,837],[34,7624,424],{"className":7625},[78,837],[34,7627,2547],{"className":7628},[2546],[34,7630,7632],{"className":7631},[818],[34,7633,7635],{"className":7634,"style":6830},[822],[34,7636],{},[34,7638],{"className":7639,"style":238},[237],[34,7641,6689],{"className":7642},[242],[34,7644],{"className":7645,"style":238},[237],[34,7647,7649,7652,7655,7658,7661],{"className":7648},[69],[34,7650],{"className":7651,"style":1263},[73],[34,7653,6706],{"className":7654},[4177],[34,7656],{"className":7657,"style":238},[237],[34,7659,6689],{"className":7660},[242],[34,7662],{"className":7663,"style":238},[237],[34,7665,7667,7670],{"className":7666},[69],[34,7668],{"className":7669,"style":3736},[73],[34,7671,7673,7676],{"className":7672},[78],[34,7674,7102],{"className":7675,"style":7147},[78,79],[34,7677,7679],{"className":7678},[810],[34,7680,7682,7702],{"className":7681},[814,2523],[34,7683,7685,7699],{"className":7684},[818],[34,7686,7688],{"className":7687,"style":2530},[822],[34,7689,7690,7693],{"style":7162},[34,7691],{"className":7692,"style":830},[829],[34,7694,7696],{"className":7695},[834,835,836,837],[34,7697,2287],{"className":7698},[78,837],[34,7700,2547],{"className":7701},[2546],[34,7703,7705],{"className":7704},[818],[34,7706,7708],{"className":7707,"style":2554},[822],[34,7709],{},[16,7711,7712],{},"If no activation function is introduced, the result above is still an affine function.",[16,7714,7715],{},"Proof is simple:",[34,7717,7719],{"className":7718,"translate":38},[159],[34,7720,7722,7838],{"className":7721,"translate":38},[37],[34,7723,7725],{"className":7724},[42],[44,7726,7727],{"xmlns":46,"display":168},[48,7728,7729,7835],{},[51,7730,7731,7737,7739,7745,7747,7749,7751,7753,7755,7761,7763,7769,7771,7773,7779,7781,7783,7789,7791,7793,7799,7805,7807,7809,7811,7813,7819,7825,7827,7833],{},[2468,7732,7733,7735],{},[54,7734,7102],{},[347,7736,424],{},[176,7738,179],{"stretchy":178},[2468,7740,7741,7743],{},[54,7742,7102],{},[347,7744,2213],{},[176,7746,179],{"stretchy":178},[54,7748,96],{},[176,7750,189],{"stretchy":178},[176,7752,189],{"stretchy":178},[176,7754,192],{},[2468,7756,7757,7759],{},[54,7758,6199],{},[347,7760,424],{},[176,7762,179],{"stretchy":178},[2468,7764,7765,7767],{},[54,7766,6199],{},[347,7768,2213],{},[54,7770,96],{},[176,7772,184],{},[2468,7774,7775,7777],{},[54,7776,2172],{},[347,7778,2213],{},[176,7780,189],{"stretchy":178},[176,7782,184],{},[2468,7784,7785,7787],{},[54,7786,2172],{},[347,7788,424],{},[176,7790,192],{},[176,7792,179],{"stretchy":178},[2468,7794,7795,7797],{},[54,7796,6199],{},[347,7798,424],{},[2468,7800,7801,7803],{},[54,7802,6199],{},[347,7804,2213],{},[176,7806,189],{"stretchy":178},[54,7808,96],{},[176,7810,184],{},[176,7812,179],{"stretchy":178},[2468,7814,7815,7817],{},[54,7816,6199],{},[347,7818,424],{},[2468,7820,7821,7823],{},[54,7822,2172],{},[347,7824,2213],{},[176,7826,184],{},[2468,7828,7829,7831],{},[54,7830,2172],{},[347,7832,424],{},[176,7834,189],{"stretchy":178},[58,7836,7837],{"encoding":60},"l_2(l_1(x)) = W_2(W_1x+b_1)+b_2 = (W_2W_1)x+(W_2b_1+b_2)",[34,7839,7841,7948,8049,8107,8162,8266,8364],{"className":7840,"ariaHidden":65},[64],[34,7842,7844,7847,7887,7890,7930,7933,7936,7939,7942,7945],{"className":7843},[69],[34,7845],{"className":7846,"style":223},[73],[34,7848,7850,7853],{"className":7849},[78],[34,7851,7102],{"className":7852,"style":7147},[78,79],[34,7854,7856],{"className":7855},[810],[34,7857,7859,7879],{"className":7858},[814,2523],[34,7860,7862,7876],{"className":7861},[818],[34,7863,7865],{"className":7864,"style":2530},[822],[34,7866,7867,7870],{"style":7162},[34,7868],{"className":7869,"style":830},[829],[34,7871,7873],{"className":7872},[834,835,836,837],[34,7874,424],{"className":7875},[78,837],[34,7877,2547],{"className":7878},[2546],[34,7880,7882],{"className":7881},[818],[34,7883,7885],{"className":7884,"style":2554},[822],[34,7886],{},[34,7888,179],{"className":7889},[230],[34,7891,7893,7896],{"className":7892},[78],[34,7894,7102],{"className":7895,"style":7147},[78,79],[34,7897,7899],{"className":7898},[810],[34,7900,7902,7922],{"className":7901},[814,2523],[34,7903,7905,7919],{"className":7904},[818],[34,7906,7908],{"className":7907,"style":2530},[822],[34,7909,7910,7913],{"style":7162},[34,7911],{"className":7912,"style":830},[829],[34,7914,7916],{"className":7915},[834,835,836,837],[34,7917,2213],{"className":7918},[78,837],[34,7920,2547],{"className":7921},[2546],[34,7923,7925],{"className":7924},[818],[34,7926,7928],{"className":7927,"style":2554},[822],[34,7929],{},[34,7931,179],{"className":7932},[230],[34,7934,96],{"className":7935},[78,79],[34,7937,5491],{"className":7938},[258],[34,7940],{"className":7941,"style":262},[237],[34,7943,192],{"className":7944},[266],[34,7946],{"className":7947,"style":262},[237],[34,7949,7951,7954,7994,7997,8037,8040,8043,8046],{"className":7950},[69],[34,7952],{"className":7953,"style":223},[73],[34,7955,7957,7960],{"className":7956},[78],[34,7958,6199],{"className":7959,"style":6248},[78,79],[34,7961,7963],{"className":7962},[810],[34,7964,7966,7986],{"className":7965},[814,2523],[34,7967,7969,7983],{"className":7968},[818],[34,7970,7972],{"className":7971,"style":2530},[822],[34,7973,7974,7977],{"style":7227},[34,7975],{"className":7976,"style":830},[829],[34,7978,7980],{"className":7979},[834,835,836,837],[34,7981,424],{"className":7982},[78,837],[34,7984,2547],{"className":7985},[2546],[34,7987,7989],{"className":7988},[818],[34,7990,7992],{"className":7991,"style":2554},[822],[34,7993],{},[34,7995,179],{"className":7996},[230],[34,7998,8000,8003],{"className":7999},[78],[34,8001,6199],{"className":8002,"style":6248},[78,79],[34,8004,8006],{"className":8005},[810],[34,8007,8009,8029],{"className":8008},[814,2523],[34,8010,8012,8026],{"className":8011},[818],[34,8013,8015],{"className":8014,"style":2530},[822],[34,8016,8017,8020],{"style":7227},[34,8018],{"className":8019,"style":830},[829],[34,8021,8023],{"className":8022},[834,835,836,837],[34,8024,2213],{"className":8025},[78,837],[34,8027,2547],{"className":8028},[2546],[34,8030,8032],{"className":8031},[818],[34,8033,8035],{"className":8034,"style":2554},[822],[34,8036],{},[34,8038,96],{"className":8039},[78,79],[34,8041],{"className":8042,"style":238},[237],[34,8044,184],{"className":8045},[242],[34,8047],{"className":8048,"style":238},[237],[34,8050,8052,8055,8095,8098,8101,8104],{"className":8051},[69],[34,8053],{"className":8054,"style":223},[73],[34,8056,8058,8061],{"className":8057},[78],[34,8059,2172],{"className":8060},[78,79],[34,8062,8064],{"className":8063},[810],[34,8065,8067,8087],{"className":8066},[814,2523],[34,8068,8070,8084],{"className":8069},[818],[34,8071,8073],{"className":8072,"style":2530},[822],[34,8074,8075,8078],{"style":2533},[34,8076],{"className":8077,"style":830},[829],[34,8079,8081],{"className":8080},[834,835,836,837],[34,8082,2213],{"className":8083},[78,837],[34,8085,2547],{"className":8086},[2546],[34,8088,8090],{"className":8089},[818],[34,8091,8093],{"className":8092,"style":2554},[822],[34,8094],{},[34,8096,189],{"className":8097},[258],[34,8099],{"className":8100,"style":238},[237],[34,8102,184],{"className":8103},[242],[34,8105],{"className":8106,"style":238},[237],[34,8108,8110,8113,8153,8156,8159],{"className":8109},[69],[34,8111],{"className":8112,"style":3736},[73],[34,8114,8116,8119],{"className":8115},[78],[34,8117,2172],{"className":8118},[78,79],[34,8120,8122],{"className":8121},[810],[34,8123,8125,8145],{"className":8124},[814,2523],[34,8126,8128,8142],{"className":8127},[818],[34,8129,8131],{"className":8130,"style":2530},[822],[34,8132,8133,8136],{"style":2533},[34,8134],{"className":8135,"style":830},[829],[34,8137,8139],{"className":8138},[834,835,836,837],[34,8140,424],{"className":8141},[78,837],[34,8143,2547],{"className":8144},[2546],[34,8146,8148],{"className":8147},[818],[34,8149,8151],{"className":8150,"style":2554},[822],[34,8152],{},[34,8154],{"className":8155,"style":262},[237],[34,8157,192],{"className":8158},[266],[34,8160],{"className":8161,"style":262},[237],[34,8163,8165,8168,8171,8211,8251,8254,8257,8260,8263],{"className":8164},[69],[34,8166],{"className":8167,"style":223},[73],[34,8169,179],{"className":8170},[230],[34,8172,8174,8177],{"className":8173},[78],[34,8175,6199],{"className":8176,"style":6248},[78,79],[34,8178,8180],{"className":8179},[810],[34,8181,8183,8203],{"className":8182},[814,2523],[34,8184,8186,8200],{"className":8185},[818],[34,8187,8189],{"className":8188,"style":2530},[822],[34,8190,8191,8194],{"style":7227},[34,8192],{"className":8193,"style":830},[829],[34,8195,8197],{"className":8196},[834,835,836,837],[34,8198,424],{"className":8199},[78,837],[34,8201,2547],{"className":8202},[2546],[34,8204,8206],{"className":8205},[818],[34,8207,8209],{"className":8208,"style":2554},[822],[34,8210],{},[34,8212,8214,8217],{"className":8213},[78],[34,8215,6199],{"className":8216,"style":6248},[78,79],[34,8218,8220],{"className":8219},[810],[34,8221,8223,8243],{"className":8222},[814,2523],[34,8224,8226,8240],{"className":8225},[818],[34,8227,8229],{"className":8228,"style":2530},[822],[34,8230,8231,8234],{"style":7227},[34,8232],{"className":8233,"style":830},[829],[34,8235,8237],{"className":8236},[834,835,836,837],[34,8238,2213],{"className":8239},[78,837],[34,8241,2547],{"className":8242},[2546],[34,8244,8246],{"className":8245},[818],[34,8247,8249],{"className":8248,"style":2554},[822],[34,8250],{},[34,8252,189],{"className":8253},[258],[34,8255,96],{"className":8256},[78,79],[34,8258],{"className":8259,"style":238},[237],[34,8261,184],{"className":8262},[242],[34,8264],{"className":8265,"style":238},[237],[34,8267,8269,8272,8275,8315,8355,8358,8361],{"className":8268},[69],[34,8270],{"className":8271,"style":223},[73],[34,8273,179],{"className":8274},[230],[34,8276,8278,8281],{"className":8277},[78],[34,8279,6199],{"className":8280,"style":6248},[78,79],[34,8282,8284],{"className":8283},[810],[34,8285,8287,8307],{"className":8286},[814,2523],[34,8288,8290,8304],{"className":8289},[818],[34,8291,8293],{"className":8292,"style":2530},[822],[34,8294,8295,8298],{"style":7227},[34,8296],{"className":8297,"style":830},[829],[34,8299,8301],{"className":8300},[834,835,836,837],[34,8302,424],{"className":8303},[78,837],[34,8305,2547],{"className":8306},[2546],[34,8308,8310],{"className":8309},[818],[34,8311,8313],{"className":8312,"style":2554},[822],[34,8314],{},[34,8316,8318,8321],{"className":8317},[78],[34,8319,2172],{"className":8320},[78,79],[34,8322,8324],{"className":8323},[810],[34,8325,8327,8347],{"className":8326},[814,2523],[34,8328,8330,8344],{"className":8329},[818],[34,8331,8333],{"className":8332,"style":2530},[822],[34,8334,8335,8338],{"style":2533},[34,8336],{"className":8337,"style":830},[829],[34,8339,8341],{"className":8340},[834,835,836,837],[34,8342,2213],{"className":8343},[78,837],[34,8345,2547],{"className":8346},[2546],[34,8348,8350],{"className":8349},[818],[34,8351,8353],{"className":8352,"style":2554},[822],[34,8354],{},[34,8356],{"className":8357,"style":238},[237],[34,8359,184],{"className":8360},[242],[34,8362],{"className":8363,"style":238},[237],[34,8365,8367,8370,8410],{"className":8366},[69],[34,8368],{"className":8369,"style":223},[73],[34,8371,8373,8376],{"className":8372},[78],[34,8374,2172],{"className":8375},[78,79],[34,8377,8379],{"className":8378},[810],[34,8380,8382,8402],{"className":8381},[814,2523],[34,8383,8385,8399],{"className":8384},[818],[34,8386,8388],{"className":8387,"style":2530},[822],[34,8389,8390,8393],{"style":2533},[34,8391],{"className":8392,"style":830},[829],[34,8394,8396],{"className":8395},[834,835,836,837],[34,8397,424],{"className":8398},[78,837],[34,8400,2547],{"className":8401},[2546],[34,8403,8405],{"className":8404},[818],[34,8406,8408],{"className":8407,"style":2554},[822],[34,8409],{},[34,8411,189],{"className":8412},[258],[16,8414,8415,8416,144],{},"which remains in the form ",[34,8417,8419,8438],{"className":8418,"translate":38},[37],[34,8420,8422],{"className":8421},[42],[44,8423,8424],{"xmlns":46},[48,8425,8426,8436],{},[51,8427,8428,8430,8432,8434],{},[54,8429,3145],{},[54,8431,96],{},[176,8433,184],{},[54,8435,2172],{},[58,8437,5145],{"encoding":60},[34,8439,8441,8462],{"className":8440,"ariaHidden":65},[64],[34,8442,8444,8447,8450,8453,8456,8459],{"className":8443},[69],[34,8445],{"className":8446,"style":3191},[73],[34,8448,3145],{"className":8449},[78,79],[34,8451,96],{"className":8452},[78,79],[34,8454],{"className":8455,"style":238},[237],[34,8457,184],{"className":8458},[242],[34,8460],{"className":8461,"style":238},[237],[34,8463,8465,8468],{"className":8464},[69],[34,8466],{"className":8467,"style":2184},[73],[34,8469,2172],{"className":8470},[78,79],[16,8472,8473],{},"By induction:",[34,8475,8477],{"className":8476,"translate":38},[159],[34,8478,8480,8520],{"className":8479,"translate":38},[37],[34,8481,8483],{"className":8482},[42],[44,8484,8485],{"xmlns":46,"display":168},[48,8486,8487,8517],{},[51,8488,8489,8495,8497,8499,8501,8507,8509,8511,8513,8515],{},[2468,8490,8491,8493],{},[54,8492,7102],{},[54,8494,3000],{},[176,8496,6689],{},[176,8498,6706],{},[176,8500,6689],{},[2468,8502,8503,8505],{},[54,8504,7102],{},[347,8506,2287],{},[176,8508,192],{},[54,8510,6199],{},[54,8512,96],{},[176,8514,184],{},[54,8516,2172],{},[58,8518,8519],{"encoding":60},"l_n \\circ \\cdots \\circ l_0 = Wx + b",[34,8521,8523,8578,8596,8651,8672],{"className":8522,"ariaHidden":65},[64],[34,8524,8526,8529,8569,8572,8575],{"className":8525},[69],[34,8527],{"className":8528,"style":3736},[73],[34,8530,8532,8535],{"className":8531},[78],[34,8533,7102],{"className":8534,"style":7147},[78,79],[34,8536,8538],{"className":8537},[810],[34,8539,8541,8561],{"className":8540},[814,2523],[34,8542,8544,8558],{"className":8543},[818],[34,8545,8547],{"className":8546,"style":5870},[822],[34,8548,8549,8552],{"style":7162},[34,8550],{"className":8551,"style":830},[829],[34,8553,8555],{"className":8554},[834,835,836,837],[34,8556,3000],{"className":8557},[78,79,837],[34,8559,2547],{"className":8560},[2546],[34,8562,8564],{"className":8563},[818],[34,8565,8567],{"className":8566,"style":2554},[822],[34,8568],{},[34,8570],{"className":8571,"style":238},[237],[34,8573,6689],{"className":8574},[242],[34,8576],{"className":8577,"style":238},[237],[34,8579,8581,8584,8587,8590,8593],{"className":8580},[69],[34,8582],{"className":8583,"style":1263},[73],[34,8585,6706],{"className":8586},[4177],[34,8588],{"className":8589,"style":238},[237],[34,8591,6689],{"className":8592},[242],[34,8594],{"className":8595,"style":238},[237],[34,8597,8599,8602,8642,8645,8648],{"className":8598},[69],[34,8600],{"className":8601,"style":3736},[73],[34,8603,8605,8608],{"className":8604},[78],[34,8606,7102],{"className":8607,"style":7147},[78,79],[34,8609,8611],{"className":8610},[810],[34,8612,8614,8634],{"className":8613},[814,2523],[34,8615,8617,8631],{"className":8616},[818],[34,8618,8620],{"className":8619,"style":2530},[822],[34,8621,8622,8625],{"style":7162},[34,8623],{"className":8624,"style":830},[829],[34,8626,8628],{"className":8627},[834,835,836,837],[34,8629,2287],{"className":8630},[78,837],[34,8632,2547],{"className":8633},[2546],[34,8635,8637],{"className":8636},[818],[34,8638,8640],{"className":8639,"style":2554},[822],[34,8641],{},[34,8643],{"className":8644,"style":262},[237],[34,8646,192],{"className":8647},[266],[34,8649],{"className":8650,"style":262},[237],[34,8652,8654,8657,8660,8663,8666,8669],{"className":8653},[69],[34,8655],{"className":8656,"style":3191},[73],[34,8658,6199],{"className":8659,"style":6248},[78,79],[34,8661,96],{"className":8662},[78,79],[34,8664],{"className":8665,"style":238},[237],[34,8667,184],{"className":8668},[242],[34,8670],{"className":8671,"style":238},[237],[34,8673,8675,8678],{"className":8674},[69],[34,8676],{"className":8677,"style":2184},[73],[34,8679,2172],{"className":8680},[78,79],[16,8682,8683],{},"Therefore, network depth loses its meaning entirely.",[16,8685,8686],{},"To introduce nonlinearity, we can add activation functions to each 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= \\sigma_n \\Bigl( W_n \\sigma_{n-1} \\bigl( W_{n-1} \\cdots \\sigma_0(W_0x+b_0) \\cdots +b_{n-1} \\bigr) + b_n \\Bigr)",[34,9562,9564,9591,9894,9958,10029],{"className":9563,"ariaHidden":65},[64],[34,9565,9567,9570,9573,9576,9579,9582,9585,9588],{"className":9566},[69],[34,9568],{"className":9569,"style":223},[73],[34,9571,56],{"className":9572,"style":80},[78,79],[34,9574,179],{"className":9575},[230],[34,9577,96],{"className":9578},[78,79],[34,9580,189],{"className":9581},[258],[34,9583],{"className":9584,"style":262},[237],[34,9586,192],{"className":9587},[266],[34,9589],{"className":9590,"style":262},[237],[34,9592,9594,9598,9638,9645,9685,9734,9741,9790,9793,9796,9799,9839,9842,9882,9885,9888,9891],{"className":9593},[69],[34,9595],{"className":9596,"style":9597},[73],"height:1.8em;vertical-align:-0.65em;",[34,9599,9601,9604],{"className":9600},[78],[34,9602,8718],{"className":9603,"style":143},[78,79],[34,9605,9607],{"className":9606},[810],[34,9608,9610,9630],{"className":9609},[814,2523],[34,9611,9613,9627],{"className":9612},[818],[34,9614,9616],{"className":9615,"style":5870},[822],[34,9617,9618,9621],{"style":2686},[34,9619],{"className":9620,"style":830},[829],[34,9622,9624],{"className":9623},[834,835,836,837],[34,9625,3000],{"className":9626},[78,79,837],[34,9628,2547],{"className":9629},[2546],[34,9631,9633],{"className":9632},[818],[34,9634,9636],{"className":9635,"style":2554},[822],[34,9637],{},[34,9639,9641],{"className":9640},[230],[34,9642,179],{"className":9643},[4186,9644],"size2",[34,9646,9648,9651],{"className":9647},[78],[34,9649,6199],{"className":9650,"style":6248},[78,79],[34,9652,9654],{"className":9653},[810],[34,9655,9657,9677],{"className":9656},[814,2523],[34,9658,9660,9674],{"className":9659},[818],[34,9661,9663],{"className":9662,"style":5870},[822],[34,9664,9665,9668],{"style":7227},[34,9666],{"className":9667,"style":830},[829],[34,9669,9671],{"className":9670},[834,835,836,837],[34,9672,3000],{"className":9673},[78,79,837],[34,9675,2547],{"className":9676},[2546],[34,9678,9680],{"className":9679},[818],[34,9681,9683],{"className":9682,"style":2554},[822],[34,9684],{},[34,9686,9688,9691],{"className":9687},[78],[34,9689,8718],{"className":9690,"style":143},[78,79],[34,9692,9694],{"className":9693},[810],[34,9695,9697,9726],{"className":9696},[814,2523],[34,9698,9700,9723],{"className":9699},[818],[34,9701,9703],{"className":9702,"style":2530},[822],[34,9704,9705,9708],{"style":2686},[34,9706],{"className":9707,"style":830},[829],[34,9709,9711],{"className":9710},[834,835,836,837],[34,9712,9714,9717,9720],{"className":9713},[78,837],[34,9715,3000],{"className":9716},[78,79,837],[34,9718,6463],{"className":9719},[242,837],[34,9721,2213],{"className":9722},[78,837],[34,9724,2547],{"className":9725},[2546],[34,9727,9729],{"className":9728},[818],[34,9730,9732],{"className":9731,"style":6830},[822],[34,9733],{},[34,9735,9737],{"className":9736},[230],[34,9738,179],{"className":9739},[4186,9740],"size1",[34,9742,9744,9747],{"className":9743},[78],[34,9745,6199],{"className":9746,"style":6248},[78,79],[34,9748,9750],{"className":9749},[810],[34,9751,9753,9782],{"className":9752},[814,2523],[34,9754,9756,9779],{"className":9755},[818],[34,9757,9759],{"className":9758,"style":2530},[822],[34,9760,9761,9764],{"style":7227},[34,9762],{"className":9763,"style":830},[829],[34,9765,9767],{"className":9766},[834,835,836,837],[34,9768,9770,9773,9776],{"className":9769},[78,837],[34,9771,3000],{"className":9772},[78,79,837],[34,9774,6463],{"className":9775},[242,837],[34,9777,2213],{"className":9778},[78,837],[34,9780,2547],{"className":9781},[2546],[34,9783,9785],{"className":9784},[818],[34,9786,9788],{"className":9787,"style":6830},[822],[34,9789],{},[34,9791],{"className":9792,"style":3462},[237],[34,9794,6706],{"className":9795},[4177],[34,9797],{"className":9798,"style":3462},[237],[34,9800,9802,9805],{"className":9801},[78],[34,9803,8718],{"className":9804,"style":143},[78,79],[34,9806,9808],{"className":9807},[810],[34,9809,9811,9831],{"className":9810},[814,2523],[34,9812,9814,9828],{"className":9813},[818],[34,9815,9817],{"className":9816,"style":2530},[822],[34,9818,9819,9822],{"style":2686},[34,9820],{"className":9821,"style":830},[829],[34,9823,9825],{"className":9824},[834,835,836,837],[34,9826,2287],{"className":9827},[78,837],[34,9829,2547],{"className":9830},[2546],[34,9832,9834],{"className":9833},[818],[34,9835,9837],{"className":9836,"style":2554},[822],[34,9838],{},[34,9840,179],{"className":9841},[230],[34,9843,9845,9848],{"className":9844},[78],[34,9846,6199],{"className":9847,"style":6248},[78,79],[34,9849,9851],{"className":9850},[810],[34,9852,9854,9874],{"className":9853},[814,2523],[34,9855,9857,9871],{"className":9856},[818],[34,9858,9860],{"className":9859,"style":2530},[822],[34,9861,9862,9865],{"style":7227},[34,9863],{"className":9864,"style":830},[829],[34,9866,9868],{"className":9867},[834,835,836,837],[34,9869,2287],{"className":9870},[78,837],[34,9872,2547],{"className":9873},[2546],[34,9875,9877],{"className":9876},[818],[34,9878,9880],{"className":9879,"sty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is the standard form of a neural network.",[16,10085,10086],{},"With activation functions added:",[34,10088,10090],{"className":10089,"translate":38},[159],[34,10091,10093,10135],{"className":10092,"translate":38},[37],[34,10094,10096],{"className":10095},[42],[44,10097,10098],{"xmlns":46,"display":168},[48,10099,10100,10132],{},[51,10101,10102,10104,10106,10108,10114,10116,10118,10120,10122,10124,10130],{},[176,10103,179],{"stretchy":178},[54,10105,8718],{},[176,10107,6689],{},[2468,10109,10110,10112],{},[54,10111,7102],{},[347,10113,424],{},[176,10115,189],{"stretchy":178},[176,10117,6689],{},[176,10119,179],{"stretchy":178},[54,10121,8718],{},[176,10123,6689],{},[2468,10125,10126,10128],{},[54,10127,7102],{},[347,10129,2213],{},[176,10131,189],{"stretchy":178},[58,10133,10134],{"encoding":60},"(\\sigma \\circ l_2) \\circ (\\sigma \\circ l_1)",[34,10136,10138,10159,10217,10238],{"className":10137,"ariaHidden":65},[64],[34,10139,10141,10144,10147,10150,10153,10156],{"className":10140},[69],[34,10142],{"className":10143,"style":223},[73],[34,10145,179],{"className":10146},[230],[34,10148,8718],{"className":10149,"style":143},[78,79],[34,10151],{"className":10152,"style":238},[237],[34,10154,6689],{"className":10155},[242],[34,10157],{"className":10158,"style":238},[237],[34,10160,10162,10165,10205,10208,10211,10214],{"className":10161},[69],[34,10163],{"className":10164,"style":223},[73],[34,10166,10168,10171],{"className":10167},[78],[34,10169,7102],{"className":10170,"style":7147},[78,79],[34,10172,10174],{"className":10173},[810],[34,10175,10177,10197],{"className":10176},[814,2523],[34,10178,10180,10194],{"className":10179},[818],[34,10181,10183],{"className":10182,"style":2530},[822],[34,10184,10185,10188],{"style":7162},[34,10186],{"className":10187,"style":830},[829],[34,10189,10191],{"className":10190},[834,835,836,837],[34,10192,424],{"className":10193},[78,837],[34,10195,2547],{"className":10196},[2546],[34,10198,10200],{"className":10199},[818],[34,10201,10203],{"className":10202,"style":2554},[822],[34,10204],{},[34,10206,189],{"className":10207},[258],[34,10209],{"className":10210,"style":238},[237],[34,10212,6689],{"className":10213},[242],[34,10215],{"className":10216,"style":238},[237],[34,10218,10220,10223,10226,10229,10232,10235],{"className":10219},[69],[34,10221],{"className":10222,"style":223},[73],[34,10224,179],{"className":10225},[230],[34,10227,8718],{"className":10228,"style":143},[78,79],[34,10230],{"className":10231,"style":238},[237],[34,10233,6689],{"className":10234},[242],[34,10236],{"className":10237,"style":238},[237],[34,10239,10241,10244,10284],{"className":10240},[69],[34,10242],{"className":10243,"style":223},[73],[34,10245,10247,10250],{"className":10246},[78],[34,10248,7102],{"className":10249,"style":7147},[78,79],[34,10251,10253],{"className":10252},[810],[34,10254,10256,10276],{"className":10255},[814,2523],[34,10257,10259,10273],{"className":10258},[818],[34,10260,10262],{"className":10261,"style":2530},[822],[34,10263,10264,10267],{"style":7162},[34,10265],{"className":10266,"style":830},[829],[34,10268,10270],{"className":10269},[834,835,836,837],[34,10271,2213],{"className":10272},[78,837],[34,10274,2547],{"className":10275},[2546],[34,10277,10279],{"className":10278},[818],[34,10280,10282],{"className":10281,"style":2554},[822],[34,10283],{},[34,10285,189],{"className":10286},[258],[16,10288,10289],{},"cannot be simplified to",[34,10291,10293],{"className":10292,"translate":38},[159],[34,10294,10296,10316],{"className":10295,"translate":38},[37],[34,10297,10299],{"className":10298},[42],[44,10300,10301],{"xmlns":46,"display":168},[48,10302,10303,10313],{},[51,10304,10305,10307,10309,10311],{},[54,10306,6199],{},[54,10308,96],{},[176,10310,184],{},[54,10312,2172],{},[58,10314,10315],{"encoding":60},"Wx+b",[34,10317,10319,10340],{"className":10318,"ariaHidden":65},[64],[34,10320,10322,10325,10328,10331,10334,10337],{"className":10321},[69],[34,10323],{"className":10324,"style":3191},[73],[34,10326,6199],{"className":10327,"style":6248},[78,79],[34,10329,96],{"className":10330},[78,79],[34,10332],{"className":10333,"style":238},[237],[34,10335,184],{"className":10336},[242],[34,10338],{"className":10339,"style":238},[237],[34,10341,10343,10346],{"className":10342},[69],[34,10344],{"className":10345,"style":2184},[73],[34,10347,2172],{"className":10348},[78,79],[16,10350,10351],{},"because generally:",[34,10353,10355],{"className":10354,"translate":38},[159],[34,10356,10358,10410],{"className":10357,"translate":38},[37],[34,10359,10361],{"className":10360},[42],[44,10362,10363],{"xmlns":46,"display":168},[48,10364,10365,10407],{},[51,10366,10367,10369,10371,10373,10375,10377,10379,10381,10383,10391,10393,10395,10397,10399,10401],{},[54,10368,8718],{},[176,10370,179],{"stretchy":178},[54,10372,3145],{},[54,10374,96],{},[176,10376,184],{},[54,10378,2172],{},[176,10380,189],{"stretchy":178},[176,10382,2297],{"mathvariant":2296},[755,10384,10385,10387],{},[54,10386,3145],{},[176,10388,10390],{"mathvariant":2296,"lspace":10389,"rspace":10389},"0em","′",[54,10392,8718],{},[176,10394,179],{"stretchy":178},[54,10396,96],{},[176,10398,189],{"stretchy":178},[176,10400,184],{},[755,10402,10403,10405],{},[54,10404,2172],{},[176,10406,10390],{"mathvariant":2296,"lspace":10389,"rspace":10389},[58,10408,10409],{"encoding":60},"\\sigma(Ax+b) \\neq A'\\sigma(x)+b'",[34,10411,10413,10440,10494,10555],{"className":10412,"ariaHidden":65},[64],[34,10414,10416,10419,10422,10425,10428,10431,10434,10437],{"className":10415},[69],[34,10417],{"className":10418,"style":223},[73],[34,10420,8718],{"className":10421,"style":143},[78,79],[34,10423,179],{"className":10424},[230],[34,10426,3145],{"className":10427},[78,79],[34,10429,96],{"className":10430},[78,79],[34,10432],{"className":10433,"style":238},[237],[34,10435,184],{"className":10436},[242],[34,10438],{"className":10439,"style":238},[237],[34,10441,10443,10446,10449,10452,10455,10491],{"className":10442},[69],[34,10444],{"className":10445,"style":223},[73],[34,10447,2172],{"className":10448},[78,79],[34,10450,189],{"className":10451},[258],[34,10453],{"className":10454,"style":262},[237],[34,10456,10458,10485,10488],{"className":10457},[266],[34,10459,10461],{"className":10460},[266],[34,10462,10464],{"className":10463},[78,2354],[34,10465,10467],{"className":10466},[2358],[34,10468,10470,10473,10482],{"className":10469},[2362],[34,10471],{"className":10472,"style":74},[73],[34,10474,10476],{"className":10475},[2369],[34,10477,10479],{"className":10478},[78],[34,10480,2376],{"className":10481},[266],[34,10483],{"className":10484},[2380],[34,10486],{"className":10487},[237,2384],[34,10489,192],{"className":10490},[266],[34,10492],{"className":10493,"style":262},[237],[34,10495,10497,10501,10534,10537,10540,10543,10546,10549,10552],{"className":10496},[69],[34,10498],{"className":10499,"style":10500},[73],"height:1.0519em;vertical-align:-0.25em;",[34,10502,10504,10507],{"className":10503},[78],[34,10505,3145],{"className":10506},[78,79],[34,10508,10510],{"className":10509},[810],[34,10511,10513],{"className":10512},[814],[34,10514,10516],{"className":10515},[818],[34,10517,10520],{"className":10518,"style":10519},[822],"height:0.8019em;",[34,10521,10522,10525],{"style":4710},[34,10523],{"className":10524,"style":830},[829],[34,10526,10528],{"className":10527},[834,835,836,837],[34,10529,10531],{"className":10530},[78,837],[34,10532,10390],{"className":10533},[78,837],[34,10535,8718],{"className":10536,"style":143},[78,79],[34,10538,179],{"className":10539},[230],[34,10541,96],{"className":10542},[78,79],[34,10544,189],{"className":10545},[258],[34,10547],{"className":10548,"style":238},[237],[34,10550,184],{"className":10551},[242],[34,10553],{"className":10554,"style":238},[237],[34,10556,10558,10561],{"className":10557},[69],[34,10559],{"className":10560,"style":10519},[73],[34,10562,10564,10567],{"className":10563},[78],[34,10565,2172],{"className":10566},[78,79],[34,10568,10570],{"className":10569},[810],[34,10571,10573],{"className":10572},[814],[34,10574,10576],{"className":10575},[818],[34,10577,10579],{"className":10578,"style":10519},[822],[34,10580,10581,10584],{"style":4710},[34,10582],{"className":10583,"style":830},[829],[34,10585,10587],{"className":10586},[834,835,836,837],[34,10588,10590],{"className":10589},[78,837],[34,10591,10390],{"className":10592},[78,837],[16,10594,10595],{},"The activation function breaks the closure property of affine mappings under composition. In other words, it breaks the two key characteristics of additivity and homogeneity. This is essentially the source of the nonlinear expressive power of neural networks.",[16,10597,10598],{},"From the manifold perspective discussed earlier, this can be further written as:",[34,10600,10602],{"className":10601,"translate":38},[159],[34,10603,10605,10704],{"className":10604,"translate":38},[37],[34,10606,10608],{"className":10607},[42],[44,10609,10610],{"xmlns":46,"display":168},[48,10611,10612,10701],{},[51,10613,10614,10616,10633,10639,10651,10659,10671,10677,10689,10697,10699],{},[54,10615,6997],{},[10617,10618,10619,10622],"mover",{},[176,10620,3003],{"stretchy":65,"minsize":10621},"3.0em",[10623,10624,10627],"mpadded",{"width":10625,"lspace":10626},"+0.6em","0.3em",[2468,10628,10629,10631],{},[54,10630,7102],{},[347,10632,2287],{},[2468,10634,10635,10637],{},[54,10636,6997],{},[347,10638,2213],{},[10617,10640,10641,10643],{},[176,10642,3003],{"stretchy":65,"minsize":10621},[10623,10644,10645],{"width":10625,"lspace":10626},[2468,10646,10647,10649],{},[54,10648,8718],{},[347,10650,2287],{},[2475,10652,10653,10655,10657],{},[54,10654,6997],{},[347,10656,2213],{},[176,10658,10390],{"mathvariant":2296,"lspace":10389,"rspace":10389},[10617,10660,10661,10663],{},[176,10662,3003],{"stretchy":65,"minsize":10621},[10623,10664,10665],{"width":10625,"lspace":10626},[2468,10666,10667,10669],{},[54,10668,7102],{},[347,10670,2213],{},[2468,10672,10673,10675],{},[54,10674,6997],{},[347,10676,424],{},[10617,10678,10679,10681],{},[176,10680,3003],{"stretchy":65,"minsize":10621},[10623,10682,10683],{"width":10625,"lspace":10626},[2468,10684,10685,10687],{},[54,10686,8718],{},[347,10688,2213],{},[2475,10690,10691,10693,10695],{},[54,10692,6997],{},[347,10694,424],{},[176,10696,10390],{"mathvariant":2296,"lspace":10389,"rspace":10389},[176,10698,3003],{},[176,10700,6706],{},[58,10702,10703],{"encoding":60},"M \\xrightarrow{l_0} M_1 \\xrightarrow{\\sigma_0} M_1' \\xrightarrow{l_1} M_2 \\xrightarrow{\\sigma_1} M_2' \\rightarrow \\cdots",[34,10705,10707,10832,10975,11131,11270,11340],{"className":10706,"ariaHidden":65},[64],[34,10708,10710,10714,10717,10720,10829],{"className":10709},[69],[34,10711],{"className":10712,"style":10713},[73],"height:1.1191em;vertical-align:-0.011em;",[34,10715,6997],{"className":10716,"style":7012},[78,79],[34,10718],{"className":10719,"style":262},[237],[34,10721,10724],{"className":10722},[266,10723],"x-arrow",[34,10725,10727,10820],{"className":10726},[814,2523],[34,10728,10730,10817],{"className":10729},[818],[34,10731,10734,10792],{"className":10732,"style":10733},[822],"height:1.1081em;",[34,10735,10737,10740],{"style":10736},"top:-3.322em;",[34,10738],{"className":10739,"style":830},[829],[34,10741,10744],{"className":10742},[834,835,836,837,10743],"x-arrow-pad",[34,10745,10747],{"className":10746},[78,837],[34,10748,10750,10753],{"className":10749},[78,837],[34,10751,7102],{"className":10752,"style":7147},[78,79,837],[34,10754,10756],{"className":10755},[810],[34,10757,10759,10783],{"className":10758},[814,2523],[34,10760,10762,10780],{"className":10761},[818],[34,10763,10766],{"className":10764,"style":10765},[822],"height:0.3173em;",[34,10767,10769,10773],{"style":10768},"top:-2.357em;margin-left:-0.0197em;margin-right:0.0714em;",[34,10770],{"className":10771,"style":10772},[829],"height:2.5em;",[34,10774,10777],{"className":10775},[834,10776,9740,837],"reset-size3",[34,10778,2287],{"className":10779},[78,837],[34,10781,2547],{"className":10782},[2546],[34,10784,10786],{"className":10785},[818],[34,10787,10790],{"className":10788,"style":10789},[822],"height:0.143em;",[34,10791],{},[34,10793,10797,10800],{"className":10794,"style":10796},[10795],"svg-align","top:-2.689em;",[34,10798],{"className":10799,"style":830},[829],[34,10801,10805],{"className":10802,"style":10804},[10803],"hide-tail","height:0.522em;min-width:1.469em;",[10806,10807,10813],"svg",{"xmlns":10808,"width":10809,"height":10810,"viewBox":10811,"preserveAspectRatio":10812},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","0.522em","0 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affine layers are responsible for stretching, rotating, translating, shearing; activation layers are responsible for folding, cropping, compressing, and restructuring the manifold.",[16,11354,11355,11356,11426],{},"If all ",[34,11357,11359,11377],{"className":11358,"translate":38},[37],[34,11360,11362],{"className":11361},[42],[44,11363,11364],{"xmlns":46},[48,11365,11366,11374],{},[51,11367,11368],{},[2468,11369,11370,11372],{},[54,11371,6199],{},[54,11373,3317],{},[58,11375,11376],{"encoding":60},"W_i",[34,11378,11380],{"className":11379,"ariaHidden":65},[64],[34,11381,11383,11386],{"className":11382},[69],[34,11384],{"className":11385,"style":3677},[73],[34,11387,11389,11392],{"className":11388},[78],[34,11390,6199],{"className":11391,"style":6248},[78,79],[34,11393,11395],{"className":11394},[810],[34,11396,11398,11418],{"className":11397},[814,2523],[34,11399,11401,11415],{"className":11400},[818],[34,11402,11404],{"className":11403,"style":3482},[822],[34,11405,11406,11409],{"style":7227},[34,11407],{"className":11408,"style":830},[829],[34,11410,11412],{"className":11411},[834,835,836,837],[34,11413,3317],{"className":11414},[78,79,837],[34,11416,2547],{"className":11417},[2546],[34,11419,11421],{"className":11420},[818],[34,11422,11424],{"className":11423,"style":2554},[822],[34,11425],{}," are invertible, then the affine layers themselves are only homeomorphic transformations and do not change the topology of the manifold. What gradually makes the manifold separable is often precisely the nonlinear deformation introduced by activation functions.",[16,11428,11429],{},"Therefore, from a geometric perspective, a neural network can be understood as an alternating combination of affine transformations and nonlinear manifold operations acting on the data manifold. This perspective aligns with many interpretations of modern manifold learning and geometric deep learning.",[11,11431,11433],{"id":11432},"activations","Activations",[16,11435,11436],{},"There is no \"single best\" choice for an activation function. However, from the perspective of trainability and expressive power of neural networks, a good activation function generally should satisfy several core characteristics:",[146,11438,11440],{"id":11439},"nonlinearity","Nonlinearity",[16,11442,11443],{},"As discussed earlier, nonlinearity is needed to break the linear closure of stacked affine layers, allowing multi-layer networks to approximate arbitrary functions.",[16,11445,11446],{},"Mathematical requirement: It must not satisfy linear combination closure:",[34,11448,11450],{"className":11449,"translate":38},[159],[34,11451,11453,11506],{"className":11452,"translate":38},[37],[34,11454,11456],{"className":11455},[42],[44,11457,11458],{"xmlns":46,"display":168},[48,11459,11460,11503],{},[51,11461,11462,11464,11466,11469,11471,11473,11475,11477,11479,11481,11483,11485,11487,11489,11491,11493,11495,11497,11499,11501],{},[54,11463,8718],{},[176,11465,179],{"stretchy":178},[54,11467,11468],{},"a",[54,11470,96],{},[176,11472,184],{},[54,11474,2172],{},[54,11476,127],{},[176,11478,189],{"stretchy":178},[176,11480,2297],{"mathvariant":2296},[54,11482,11468],{},[54,11484,8718],{},[176,11486,179],{"stretchy":178},[54,11488,96],{},[176,11490,189],{"stretchy":178},[176,11492,184],{},[54,11494,2172],{},[54,11496,8718],{},[176,11498,179],{"stretchy":178},[54,11500,127],{},[176,11502,189],{"stretchy":178},[58,11504,11505],{"encoding":60},"\\sigma(a x + b y) \\neq a \\sigma(x) + b \\sigma(y)",[34,11507,11509,11536,11593,11621],{"className":11508,"ariaHidden":65},[64],[34,11510,11512,11515,11518,11521,11524,11527,11530,11533],{"className":11511},[69],[34,11513],{"className":11514,"style":223},[73],[34,11516,8718],{"className":11517,"style":143},[78,79],[34,11519,179],{"className":11520},[230],[34,11522,11468],{"className":11523},[78,79],[34,11525,96],{"className":11526},[78,79],[34,11528],{"className":11529,"style":238},[237],[34,11531,184],{"className":11532},[242],[34,11534],{"className":11535,"style":238},[237],[34,11537,11539,11542,11545,11548,11551,11554,11590],{"className":11538},[69],[34,11540],{"className":11541,"style":223},[73],[34,11543,2172],{"className":11544},[78,79],[34,11546,127],{"className":11547,"style":143},[78,79],[34,11549,189],{"className":11550},[258],[34,11552],{"className":11553,"style":262},[237],[34,11555,11557,11584,11587],{"className":11556},[266],[34,11558,11560],{"className":11559},[266],[34,11561,11563],{"className":11562},[78,2354],[34,11564,11566],{"className":11565},[2358],[34,11567,11569,11572,11581],{"className":11568},[2362],[34,11570],{"className":11571,"style":74},[73],[34,11573,11575],{"className":11574},[2369],[34,11576,11578],{"className":11577},[78],[34,11579,2376],{"className":11580},[266],[34,11582],{"className":11583},[2380],[34,11585],{"className":11586},[237,2384],[34,11588,192],{"className":11589},[266],[34,11591],{"className":11592,"style":262},[237],[34,11594,11596,11599,11603,11606,11609,11612,11615,11618],{"className":11595},[69],[34,11597],{"className":11598,"style":223},[73],[34,11600,11602],{"className":11601,"style":143},[78,79],"aσ",[34,11604,179],{"className":11605},[230],[34,11607,96],{"className":11608},[78,79],[34,11610,189],{"className":11611},[258],[34,11613],{"className":11614,"style":238},[237],[34,11616,184],{"className":11617},[242],[34,11619],{"className":11620,"style":238},[237],[34,11622,11624,11627,11631,11634,11637],{"className":11623},[69],[34,11625],{"className":11626,"style":223},[73],[34,11628,11630],{"className":11629,"style":143},[78,79],"bσ",[34,11632,179],{"className":11633},[230],[34,11635,127],{"className":11636,"style":143},[78,79],[34,11638,189],{"className":11639},[258],[16,11641,11642],{},"Examples: ReLU, Tanh, Sigmoid, GELU.",[16,11644,11645],{},"Linear functions are not suitable because the composition of multiple linear layers remains linear, which would degenerate back to affine composition and limit expressive power.",[146,11647,11649],{"id":11648},"differentiability","Differentiability",[16,11651,11652,11653,11656],{},"Because the core of neural networks relies on backpropagation to determine weights, the activation function must be differentiable over most of its domain to facilitate gradient calculation during backpropagation. For non-differentiable points (e.g., ReLU at 0), ",[20,11654,11655],{},"subgradients"," are typically used. Moreover, the derivative properties affect gradient flow: gradients that are too small can cause vanishing gradients (Sigmoid), while gradients that are too large can cause exploding gradients.",[146,11658,11660],{"id":11659},"continuity","Continuity",[16,11662,11663],{},"Ensures that the network output changes continuously with the input; otherwise, training becomes unstable. Having few and controllable discontinuities is usually acceptable (e.g., ReLU has only one point of discontinuity, which has little practical impact).",[16,11665,11666],{},"Among the properties above, meeting the conditions is merely an \"entry ticket,\" not a \"guarantee of success.\"",{"title":11668,"searchDepth":11669,"depth":11669,"links":11670},"",2,[11671,11676,11677,11678,11679],{"id":13,"depth":11669,"text":14,"children":11672},[11673,11675],{"id":148,"depth":11674,"text":149},3,{"id":1144,"depth":11674,"text":1145},{"id":2970,"depth":11669,"text":2971},{"id":5185,"depth":11669,"text":5186},{"id":7079,"depth":11669,"text":7080},{"id":11432,"depth":11669,"text":11433,"children":11680},[11681,11682,11683],{"id":11439,"depth":11674,"text":11440},{"id":11648,"depth":11674,"text":11649},{"id":11659,"depth":11674,"text":11660},false,true,"md",null,{},"2026-04-06","\u002Fblog\u002F2026\u002F2026-04-06-nonlinearity-with-activations",{"title":6,"description":11668},"blog\u002F2026\u002F2026-04-06-nonlinearity-with-activations","Linearity requires additivity and scaling, but neural networks need nonlinearity to break free from simple affine transformations. While affine layers only stretch and rotate data without changing its topology, activation functions enable complex reshaping—allowing networks to untangle data manifolds and learn intricate patterns effectively.",[11695,11696],"mathematics","machine-learning","gMnqBt12ndLnmzHMqRT-BOrdIWtEdi3Cj-ufnCu4l-A",{"id":11699,"title":11700,"body":11701,"description":27122,"draft":11684,"enableComment":11685,"extension":11686,"image":11668,"important":11684,"location":11687,"meta":27123,"navigation":11685,"ogImage":11687,"onday":27124,"path":27125,"seo":27126,"stem":27127,"summary":27128,"tags":27129,"__hash__":27130},"blog\u002Fblog\u002F2026\u002F2026-04-09-introduction-of-v-mf-distribution-on-hypersphere.md","Introduction of vMF Distribution on Hypersphere",{"type":8,"value":11702,"toc":27099},[11703,11802,11806,11871,12094,12097,12266,12269,12511,12513,13034,13111,13115,13383,13386,13488,13492,13496,13499,13502,13695,13698,13757,13760,13771,13774,13843,13847,13980,13982,14312,14401,14405,14867,14871,14903,14906,14993,14996,15091,15094,15098,15101,15104,15392,15395,15399,15402,15733,15765,15769,15772,16003,16076,16080,16115,16340,16441,16444,16542,16573,16650,16653,17147,17236,17487,17588,17642,17645,17766,17803,17979,17982,18665,18668,18671,18732,18912,18915,19154,19157,19320,19352,19625,19699,19702,19814,19817,20030,20062,20417,20419,20741,20744,21190,21193,21359,21361,21636,21714,21717,21951,21954,22138,22170,22306,22309,22341,22344,22467,22587,22591,22623,22655,22939,22942,23239,23242,23475,23507,23649,23651,23973,23976,23980,23986,24040,24369,24417,24420,24424,24430,24433,24798,24801,25578,25581,25584,25588,25591,25594,26069,26072,26105,26136,26356,26359,26561,26564,26597,26628,26803,26806,26897,26900],[16,11704,11705,11706,11736,11737,11796,11797,11801],{},"The von Mises–Fisher distribution (vMF) distribution is often regarded as the hyperspherical analogue of the Normal distribution, as it models data concentrated around a mean direction on the unit hypersphere. In Euclidean space, a normal distribution is typically parameterized by a mean ",[34,11707,11709,11724],{"className":11708,"translate":38},[37],[34,11710,11712],{"className":11711},[42],[44,11713,11714],{"xmlns":46},[48,11715,11716,11721],{},[51,11717,11718],{},[54,11719,11720],{},"μ",[58,11722,11723],{"encoding":60},"\\mu",[34,11725,11727],{"className":11726,"ariaHidden":65},[64],[34,11728,11730,11733],{"className":11729},[69],[34,11731],{"className":11732,"style":139},[73],[34,11734,11720],{"className":11735},[78,79]," and variance ",[34,11738,11740,11758],{"className":11739,"translate":38},[37],[34,11741,11743],{"className":11742},[42],[44,11744,11745],{"xmlns":46},[48,11746,11747,11755],{},[51,11748,11749],{},[755,11750,11751,11753],{},[54,11752,8718],{},[347,11754,424],{},[58,11756,11757],{"encoding":60},"\\sigma^2",[34,11759,11761],{"className":11760,"ariaHidden":65},[64],[34,11762,11764,11767],{"className":11763},[69],[34,11765],{"className":11766,"style":800},[73],[34,11768,11770,11773],{"className":11769},[78],[34,11771,8718],{"className":11772,"style":143},[78,79],[34,11774,11776],{"className":11775},[810],[34,11777,11779],{"className":11778},[814],[34,11780,11782],{"className":11781},[818],[34,11783,11785],{"className":11784,"style":800},[822],[34,11786,11787,11790],{"style":825},[34,11788],{"className":11789,"style":830},[829],[34,11791,11793],{"className":11792},[834,835,836,837],[34,11794,424],{"className":11795},[78,837],", and in one dimension its density forms the familiar bell-shaped curve. Just as the normal distribution is the distribution with the maximum entropy given the mean and variance constraints, the vMF distribution is the distribution with the maximum entropy on a sphere given the average direction constraint. According to the ",[11798,11799,11800],"em",{},"Central Limit Theorem",": the limiting distribution of a random walk or accumulated direction vector on a sphere will also tend towards the vMF distribution.",[11,11803,11805],{"id":11804},"basic-concepts","Basic Concepts",[16,11807,11808,11809,11870],{},"Domain: ",[34,11810,11812,11834],{"className":11811,"translate":38},[37],[34,11813,11815],{"className":11814},[42],[44,11816,11817],{"xmlns":46},[48,11818,11819,11831],{},[51,11820,11821,11823,11825,11827,11829],{},[176,11822,179],{"stretchy":178},[54,11824,5250],{},[176,11826,6463],{},[347,11828,2213],{},[176,11830,189],{"stretchy":178},[58,11832,11833],{"encoding":60},"(d-1)",[34,11835,11837,11858],{"className":11836,"ariaHidden":65},[64],[34,11838,11840,11843,11846,11849,11852,11855],{"className":11839},[69],[34,11841],{"className":11842,"style":223},[73],[34,11844,179],{"className":11845},[230],[34,11847,5250],{"className":11848},[78,79],[34,11850],{"className":11851,"style":238},[237],[34,11853,6463],{"className":11854},[242],[34,11856],{"className":11857,"style":238},[237],[34,11859,11861,11864,11867],{"className":11860},[69],[34,11862],{"className":11863,"style":223},[73],[34,11865,2213],{"className":11866},[78],[34,11868,189],{"className":11869},[258],"-dimensional unit sphere",[34,11872,11874],{"className":11873,"translate":38},[159],[34,11875,11877,11935],{"className":11876,"translate":38},[37],[34,11878,11880],{"className":11879},[42],[44,11881,11882],{"xmlns":46,"display":168},[48,11883,11884,11932],{},[51,11885,11886,11899,11901,11904,11907,11910,11916,11918,11921,11923,11925,11927,11929],{},[755,11887,11888,11891],{},[54,11889,11890],{"mathvariant":2996},"S",[51,11892,11893,11895,11897],{},[54,11894,5250],{},[176,11896,6463],{},[347,11898,2213],{},[176,11900,192],{},[176,11902,11903],{"stretchy":178},"{",[54,11905,96],{"mathvariant":11906},"bold",[176,11908,11909],{},"∈",[755,11911,11912,11914],{},[54,11913,2997],{"mathvariant":2996},[54,11915,5250],{},[176,11917,2991],{},[54,11919,11920],{"mathvariant":2296},"∥",[54,11922,96],{"mathvariant":11906},[54,11924,11920],{"mathvariant":2296},[176,11926,192],{},[347,11928,2213],{},[176,11930,11931],{"stretchy":178},"}",[58,11933,11934],{"encoding":60},"\\mathbb{S}^{d-1} = \\{ \\mathbf{x} \\in \\mathbb{R}^d : \\|\\mathbf{x}\\|=1 \\}",[34,11936,11938,11992,12014,12058,12082],{"className":11937,"ariaHidden":65},[64],[34,11939,11941,11945,11983,11986,11989],{"className":11940},[69],[34,11942],{"className":11943,"style":11944},[73],"height:0.8991em;",[34,11946,11948,11951],{"className":11947},[78],[34,11949,11890],{"className":11950},[78,3048],[34,11952,11954],{"className":11953},[810],[34,11955,11957],{"className":11956},[814],[34,11958,11960],{"className":11959},[818],[34,11961,11963],{"className":11962,"style":11944},[822],[34,11964,11965,11968],{"style":4710},[34,11966],{"className":11967,"style":830},[829],[34,11969,11971],{"className":11970},[834,835,836,837],[34,11972,11974,11977,11980],{"className":11973},[78,837],[34,11975,5250],{"className":11976},[78,79,837],[34,11978,6463],{"className":11979},[242,837],[34,11981,2213],{"className":11982},[78,837],[34,11984],{"className":11985,"style":262},[237],[34,11987,192],{"className":11988},[266],[34,11990],{"className":11991,"style":262},[237],[34,11993,11995,11998,12001,12005,12008,12011],{"className":11994},[69],[34,11996],{"className":11997,"style":223},[73],[34,11999,11903],{"className":12000},[230],[34,12002,96],{"className":12003},[78,12004],"mathbf",[34,12006],{"className":12007,"style":262},[237],[34,12009,11909],{"className":12010},[266],[34,12012],{"className":12013,"style":262},[237],[34,12015,12017,12020,12049,12052,12055],{"className":12016},[69],[34,12018],{"className":12019,"style":11944},[73],[34,12021,12023,12026],{"className":12022},[78],[34,12024,2997],{"className":12025},[78,3048],[34,12027,12029],{"className":12028},[810],[34,12030,12032],{"className":12031},[814],[34,12033,12035],{"className":12034},[818],[34,12036,12038],{"className":12037,"style":11944},[822],[34,12039,12040,12043],{"style":4710},[34,12041],{"className":12042,"style":830},[829],[34,12044,12046],{"className":12045},[834,835,836,837],[34,12047,5250],{"className":12048},[78,79,837],[34,12050],{"className":12051,"style":262},[237],[34,12053,2991],{"className":12054},[266],[34,12056],{"className":12057,"style":262},[237],[34,12059,12061,12064,12067,12070,12073,12076,12079],{"className":12060},[69],[34,12062],{"className":12063,"style":223},[73],[34,12065,11920],{"className":12066},[78],[34,12068,96],{"className":12069},[78,12004],[34,12071,11920],{"className":12072},[78],[34,12074],{"className":12075,"style":262},[237],[34,12077,192],{"className":12078},[266],[34,12080],{"className":12081,"style":262},[237],[34,12083,12085,12088,12091],{"className":12084},[69],[34,12086],{"className":12087,"style":223},[73],[34,12089,2213],{"className":12090},[78],[34,12092,11931],{"className":12093},[258],[16,12095,12096],{},"Parameters:",[400,12098,12099,12209],{},[403,12100,12101,12102],{},"Mean direction ",[34,12103,12105,12134],{"className":12104,"translate":38},[37],[34,12106,12108],{"className":12107},[42],[44,12109,12110],{"xmlns":46},[48,12111,12112,12131],{},[51,12113,12114,12117,12119],{},[54,12115,11720],{"mathvariant":12116},"bold-italic",[176,12118,11909],{},[755,12120,12121,12123],{},[54,12122,11890],{},[51,12124,12125,12127,12129],{},[54,12126,5250],{},[176,12128,6463],{},[347,12130,2213],{},[58,12132,12133],{"encoding":60},"\\boldsymbol{\\mu} \\in S^{d-1}",[34,12135,12137,12163],{"className":12136,"ariaHidden":65},[64],[34,12138,12140,12144,12154,12157,12160],{"className":12139},[69],[34,12141],{"className":12142,"style":12143},[73],"height:0.7335em;vertical-align:-0.1944em;",[34,12145,12147],{"className":12146},[78],[34,12148,12150],{"className":12149},[78],[34,12151,11720],{"className":12152},[78,12153],"boldsymbol",[34,12155],{"className":12156,"style":262},[237],[34,12158,11909],{"className":12159},[266],[34,12161],{"className":12162,"style":262},[237],[34,12164,12166,12170],{"className":12165},[69],[34,12167],{"className":12168,"style":12169},[73],"height:0.8491em;",[34,12171,12173,12177],{"className":12172},[78],[34,12174,11890],{"className":12175,"style":12176},[78,79],"margin-right:0.0576em;",[34,12178,12180],{"className":12179},[810],[34,12181,12183],{"className":12182},[814],[34,12184,12186],{"className":12185},[818],[34,12187,12189],{"className":12188,"style":12169},[822],[34,12190,12191,12194],{"style":825},[34,12192],{"className":12193,"style":830},[829],[34,12195,12197],{"className":12196},[834,835,836,837],[34,12198,12200,12203,12206],{"className":12199},[78,837],[34,12201,5250],{"className":12202},[78,79,837],[34,12204,6463],{"className":12205},[242,837],[34,12207,2213],{"className":12208},[78,837],[403,12210,12211,12212],{},"Concentration parameter ",[34,12213,12215,12235],{"className":12214,"translate":38},[37],[34,12216,12218],{"className":12217},[42],[44,12219,12220],{"xmlns":46},[48,12221,12222,12232],{},[51,12223,12224,12227,12230],{},[54,12225,12226],{},"κ",[176,12228,12229],{},"≥",[347,12231,2287],{},[58,12233,12234],{"encoding":60},"\\kappa \\ge 0",[34,12236,12238,12257],{"className":12237,"ariaHidden":65},[64],[34,12239,12241,12245,12248,12251,12254],{"className":12240},[69],[34,12242],{"className":12243,"style":12244},[73],"height:0.7719em;vertical-align:-0.136em;",[34,12246,12226],{"className":12247},[78,79],[34,12249],{"className":12250,"style":262},[237],[34,12252,12229],{"className":12253},[266],[34,12255],{"className":12256,"style":262},[237],[34,12258,12260,12263],{"className":12259},[69],[34,12261],{"className":12262,"style":391},[73],[34,12264,2287],{"className":12265},[78],[16,12267,12268],{},"Probability density function (with respect to the uniform measure on the sphere):",[34,12270,12272],{"className":12271,"translate":38},[159],[34,12273,12275,12338],{"className":12274,"translate":38},[37],[34,12276,12278],{"className":12277},[42],[44,12279,12280],{"xmlns":46,"display":168},[48,12281,12282,12335],{},[51,12283,12284,12286,12288,12290,12293,12295,12297,12299,12301,12303,12309,12311,12313,12315,12318,12320,12322,12324,12331,12333],{},[54,12285,56],{},[176,12287,179],{"stretchy":178},[54,12289,96],{"mathvariant":11906},[176,12291,12292],{},"∣",[54,12294,11720],{"mathvariant":12116},[176,12296,5228],{"separator":65},[54,12298,12226],{},[176,12300,189],{"stretchy":178},[176,12302,192],{},[2468,12304,12305,12307],{},[54,12306,5243],{},[54,12308,5250],{},[176,12310,179],{"stretchy":178},[54,12312,12226],{},[176,12314,189],{"stretchy":178},[54,12316,12317],{},"exp",[176,12319,6321],{},[176,12321,179],{"stretchy":178},[54,12323,12226],{},[755,12325,12326,12328],{},[54,12327,11720],{"mathvariant":12116},[54,12329,12330],{"mathvariant":2296},"⊤",[54,12332,96],{"mathvariant":11906},[176,12334,189],{"stretchy":178},[58,12336,12337],{"encoding":60},"f(\\mathbf{x} \\mid \\boldsymbol{\\mu}, \\kappa) = C_d(\\kappa) \\exp(\\kappa \\boldsymbol{\\mu}^\\top \\mathbf{x})",[34,12339,12341,12365,12401],{"className":12340,"ariaHidden":65},[64],[34,12342,12344,12347,12350,12353,12356,12359,12362],{"className":12343},[69],[34,12345],{"className":12346,"style":223},[73],[34,12348,56],{"className":12349,"style":80},[78,79],[34,12351,179],{"className":12352},[230],[34,12354,96],{"className":12355},[78,12004],[34,12357],{"className":12358,"style":262},[237],[34,12360,12292],{"className":12361},[266],[34,12363],{"className":12364,"style":262},[237],[34,12366,12368,12371,12380,12383,12386,12389,12392,12395,12398],{"className":12367},[69],[34,12369],{"className":12370,"style":223},[73],[34,12372,12374],{"className":12373},[78],[34,12375,12377],{"className":12376},[78],[34,12378,11720],{"className":12379},[78,12153],[34,12381,5228],{"className":12382},[5317],[34,12384],{"className":12385,"style":3462},[237],[34,12387,12226],{"className":12388},[78,79],[34,12390,189],{"className":12391},[258],[34,12393],{"className":12394,"style":262},[237],[34,12396,192],{"className":12397},[266],[34,12399],{"className":12400,"style":262},[237],[34,12402,12404,12408,12449,12452,12455,12458,12461,12464,12467,12470,12505,12508],{"className":12403},[69],[34,12405],{"className":12406,"style":12407},[73],"height:1.1491em;vertical-align:-0.25em;",[34,12409,12411,12414],{"className":12410},[78],[34,12412,5243],{"className":12413,"style":5354},[78,79],[34,12415,12417],{"className":12416},[810],[34,12418,12420,12441],{"className":12419},[814,2523],[34,12421,12423,12438],{"className":12422},[818],[34,12424,12426],{"className":12425,"style":6071},[822],[34,12427,12429,12432],{"style":12428},"top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;",[34,12430],{"className":12431,"style":830},[829],[34,12433,12435],{"className":12434},[834,835,836,837],[34,12436,5250],{"className":12437},[78,79,837],[34,12439,2547],{"className":12440},[2546],[34,12442,12444],{"className":12443},[818],[34,12445,12447],{"className":12446,"style":2554},[822],[34,12448],{},[34,12450,179],{"className":12451},[230],[34,12453,12226],{"className":12454},[78,79],[34,12456,189],{"className":12457},[258],[34,12459],{"className":12460,"style":3462},[237],[34,12462,12317],{"className":12463},[3387],[34,12465,179],{"className":12466},[230],[34,12468,12226],{"className":12469},[78,79],[34,12471,12473,12482],{"className":12472},[78],[34,12474,12476],{"className":12475},[78],[34,12477,12479],{"className":12478},[78],[34,12480,11720],{"className":12481},[78,12153],[34,12483,12485],{"className":12484},[810],[34,12486,12488],{"className":12487},[814],[34,12489,12491],{"className":12490},[818],[34,12492,12494],{"className":12493,"style":11944},[822],[34,12495,12496,12499],{"style":4710},[34,12497],{"className":12498,"style":830},[829],[34,12500,12502],{"className":12501},[834,835,836,837],[34,12503,12330],{"className":12504},[78,837],[34,12506,96],{"className":12507},[78,12004],[34,12509,189],{"className":12510},[258],[16,12512,9212],{},[34,12514,12516],{"className":12515,"translate":38},[159],[34,12517,12519,12609],{"className":12518,"translate":38},[37],[34,12520,12522],{"className":12521},[42],[44,12523,12524],{"xmlns":46,"display":168},[48,12525,12526,12606],{},[51,12527,12528,12534,12536,12538,12540,12542],{},[2468,12529,12530,12532],{},[54,12531,5243],{},[54,12533,5250],{},[176,12535,179],{"stretchy":178},[54,12537,12226],{},[176,12539,189],{"stretchy":178},[176,12541,192],{},[12543,12544,12545,12561],"mfrac",{},[755,12546,12547,12549],{},[54,12548,12226],{},[51,12550,12551,12557,12559],{},[12543,12552,12553,12555],{},[54,12554,5250],{},[347,12556,424],{},[176,12558,6463],{},[347,12560,2213],{},[51,12562,12563,12565,12567,12570,12583,12600,12602,12604],{},[176,12564,179],{"stretchy":178},[347,12566,424],{},[54,12568,12569],{},"π",[755,12571,12572,12574],{},[176,12573,189],{"stretchy":178},[51,12575,12576,12578,12581],{},[54,12577,5250],{},[54,12579,12580],{"mathvariant":2296},"\u002F",[347,12582,424],{},[2468,12584,12585,12588],{},[54,12586,12587],{},"I",[51,12589,12590,12596,12598],{},[12543,12591,12592,12594],{},[54,12593,5250],{},[347,12595,424],{},[176,12597,6463],{},[347,12599,2213],{},[176,12601,179],{"stretchy":178},[54,12603,12226],{},[176,12605,189],{"stretchy":178},[58,12607,12608],{"encoding":60},"C_d(\\kappa) = \\frac{\\kappa^{\\frac{d}{2} - 1}}{(2\\pi)^{d\u002F2} I_{\\frac{d}{2} - 1}(\\kappa)}",[34,12610,12612,12676],{"className":12611,"ariaHidden":65},[64],[34,12613,12615,12618,12658,12661,12664,12667,12670,12673],{"className":12614},[69],[34,12616],{"className":12617,"style":223},[73],[34,12619,12621,12624],{"className":12620},[78],[34,12622,5243],{"className":12623,"style":5354},[78,79],[34,12625,12627],{"className":12626},[810],[34,12628,12630,12650],{"className":12629},[814,2523],[34,12631,12633,12647],{"className":12632},[818],[34,12634,12636],{"className":12635,"style":6071},[822],[34,12637,12638,12641],{"style":12428},[34,12639],{"className":12640,"style":830},[829],[34,12642,12644],{"className":12643},[834,835,836,837],[34,12645,5250],{"className":12646},[78,79,837],[34,12648,2547],{"className":12649},[2546],[34,12651,12653],{"className":12652},[818],[34,12654,12656],{"className":12655,"style":2554},[822],[34,12657],{},[34,12659,179],{"className":12660},[230],[34,12662,12226],{"className":12663},[78,79],[34,12665,189],{"className":12666},[258],[34,12668],{"className":12669,"style":262},[237],[34,12671,192],{"className":12672},[266],[34,12674],{"className":12675,"style":262},[237],[34,12677,12679,12683],{"className":12678},[69],[34,12680],{"className":12681,"style":12682},[73],"height:2.872em;vertical-align:-1.216em;",[34,12684,12686,12690,13031],{"className":12685},[78],[34,12687],{"className":12688},[230,12689],"nulldelimiter",[34,12691,12693],{"className":12692},[12543],[34,12694,12696,13022],{"className":12695},[814,2523],[34,12697,12699,13019],{"className":12698},[818],[34,12700,12703,12895,12905],{"className":12701,"style":12702},[822],"height:1.656em;",[34,12704,12706,12710],{"style":12705},"top:-2.296em;",[34,12707],{"className":12708,"style":12709},[829],"height:3em;",[34,12711,12713,12716,12719,12722,12760,12886,12889,12892],{"className":12712},[78],[34,12714,179],{"className":12715},[230],[34,12717,424],{"className":12718},[78],[34,12720,12569],{"className":12721,"style":143},[78,79],[34,12723,12725,12728],{"className":12724},[258],[34,12726,189],{"className":12727},[258],[34,12729,12731],{"className":12730},[810],[34,12732,12734],{"className":12733},[814],[34,12735,12737],{"className":12736},[818],[34,12738,12741],{"className":12739,"style":12740},[822],"height:0.814em;",[34,12742,12744,12747],{"style":12743},"top:-2.989em;margin-right:0.05em;",[34,12745],{"className":12746,"style":830},[829],[34,12748,12750],{"className":12749},[834,835,836,837],[34,12751,12753,12756],{"className":12752},[78,837],[34,12754,5250],{"className":12755},[78,79,837],[34,12757,12759],{"className":12758},[78,837],"\u002F2",[34,12761,12763,12767],{"className":12762},[78],[34,12764,12587],{"className":12765,"style":12766},[78,79],"margin-right:0.0785em;",[34,12768,12770],{"className":12769},[810],[34,12771,12773,12877],{"className":12772},[814,2523],[34,12774,12776,12874],{"className":12775},[818],[34,12777,12780],{"className":12778,"style":12779},[822],"height:0.3448em;",[34,12781,12783,12786],{"style":12782},"top:-2.7288em;margin-left:-0.0785em;margin-right:0.05em;",[34,12784],{"className":12785,"style":12709},[829],[34,12787,12789],{"className":12788},[834,835,836,837],[34,12790,12792,12868,12871],{"className":12791},[78,837],[34,12793,12795,12799,12865],{"className":12794},[78,837],[34,12796],{"className":12797},[230,12689,834,10776,12798],"size6",[34,12800,12802],{"className":12801},[12543],[34,12803,12805,12856],{"className":12804},[814,2523],[34,12806,12808,12853],{"className":12807},[818],[34,12809,12812,12827,12838],{"className":12810,"style":12811},[822],"height:0.88em;",[34,12813,12815,12818],{"style":12814},"top:-2.656em;",[34,12816],{"className":12817,"style":12709},[829],[34,12819,12821],{"className":12820},[834,10776,9740,837],[34,12822,12824],{"className":12823},[78,837],[34,12825,424],{"className":12826},[78,837],[34,12828,12830,12833],{"style":12829},"top:-3.2255em;",[34,12831],{"className":12832,"style":12709},[829],[34,12834],{"className":12835,"style":12837},[12836,837],"frac-line","border-bottom-width:0.049em;",[34,12839,12841,12844],{"style":12840},"top:-3.384em;",[34,12842],{"className":12843,"style":12709},[829],[34,12845,12847],{"className":12846},[834,10776,9740,837],[34,12848,12850],{"className":12849},[78,837],[34,12851,5250],{"className":12852},[78,79,837],[34,12854,2547],{"className":12855},[2546],[34,12857,12859],{"className":12858},[818],[34,12860,12863],{"className":12861,"style":12862},[822],"height:0.344em;",[34,12864],{},[34,12866],{"className":12867},[258,12689,834,10776,12798],[34,12869,6463],{"className":12870},[242,837],[34,12872,2213],{"className":12873},[78,837],[34,12875,2547],{"className":12876},[2546],[34,12878,12880],{"className":12879},[818],[34,12881,12884],{"className":12882,"style":12883},[822],"height:0.512em;",[34,12885],{},[34,12887,179],{"className":12888},[230],[34,12890,12226],{"className":12891},[78,79],[34,12893,189],{"className":12894},[258],[34,12896,12898,12901],{"style":12897},"top:-3.23em;",[34,12899],{"className":12900,"style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",[34,13038,13040,13059],{"className":13039,"translate":38},[37],[34,13041,13043],{"className":13042},[42],[44,13044,13045],{"xmlns":46},[48,13046,13047,13056],{},[51,13048,13049],{},[2468,13050,13051,13053],{},[54,13052,12587],{},[54,13054,13055],{},"ν",[58,13057,13058],{"encoding":60},"I_\\nu",[34,13060,13062],{"className":13061,"ariaHidden":65},[64],[34,13063,13065,13068],{"className":13064},[69],[34,13066],{"className":13067,"style":3677},[73],[34,13069,13071,13074],{"className":13070},[78],[34,13072,12587],{"className":13073,"style":12766},[78,79],[34,13075,13077],{"className":13076},[810],[34,13078,13080,13102],{"className":13079},[814,2523],[34,13081,13083,13099],{"className":13082},[818],[34,13084,13086],{"className":13085,"style":5870},[822],[34,13087,13089,13092],{"style":13088},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[34,13090],{"className":13091,"style":830},[829],[34,13093,13095],{"className":13094},[834,835,836,837],[34,13096,13055],{"className":13097,"style":13098},[78,79,837],"margin-right:0.0637em;",[34,13100,2547],{"className":13101},[2546],[34,13103,13105],{"className":13104},[818],[34,13106,13108],{"className":13107,"style":2554},[822],[34,13109],{}," is the modified Bessel function of the first kind.",[11,13112,13114],{"id":13113},"intuitive-understanding","Intuitive Understanding",[400,13116,13117,13171,13294],{},[403,13118,13119,13170],{},[34,13120,13122,13140],{"className":13121,"translate":38},[37],[34,13123,13125],{"className":13124},[42],[44,13126,13127],{"xmlns":46},[48,13128,13129,13137],{},[51,13130,13131,13133,13135],{},[54,13132,12226],{},[176,13134,192],{},[347,13136,2287],{},[58,13138,13139],{"encoding":60},"\\kappa = 0",[34,13141,13143,13161],{"className":13142,"ariaHidden":65},[64],[34,13144,13146,13149,13152,13155,13158],{"className":13145},[69],[34,13147],{"className":13148,"style":108},[73],[34,13150,12226],{"className":13151},[78,79],[34,13153],{"className":13154,"style":262},[237],[34,13156,192],{"className":13157},[266],[34,13159],{"className":13160,"style":262},[237],[34,13162,13164,13167],{"className":13163},[69],[34,13165],{"className":13166,"style":391},[73],[34,13168,2287],{"className":13169},[78],": uniform distribution on the sphere (no directional preference)",[403,13172,13173,13226,13227,13263,13264,13293],{},[34,13174,13176,13195],{"className":13175,"translate":38},[37],[34,13177,13179],{"className":13178},[42],[44,13180,13181],{"xmlns":46},[48,13182,13183,13192],{},[51,13184,13185,13187,13190],{},[54,13186,12226],{},[176,13188,13189],{},">",[347,13191,2287],{},[58,13193,13194],{"encoding":60},"\\kappa > 0",[34,13196,13198,13217],{"className":13197,"ariaHidden":65},[64],[34,13199,13201,13205,13208,13211,13214],{"className":13200},[69],[34,13202],{"className":13203,"style":13204},[73],"height:0.5782em;vertical-align:-0.0391em;",[34,13206,12226],{"className":13207},[78,79],[34,13209],{"className":13210,"style":262},[237],[34,13212,13189],{"className":13213},[266],[34,13215],{"className":13216,"style":262},[237],[34,13218,13220,13223],{"className":13219},[69],[34,13221],{"className":13222,"style":391},[73],[34,13224,2287],{"className":13225},[78],": distribution concentrated around ",[34,13228,13230,13244],{"className":13229,"translate":38},[37],[34,13231,13233],{"className":13232},[42],[44,13234,13235],{"xmlns":46},[48,13236,13237,13241],{},[51,13238,13239],{},[54,13240,11720],{"mathvariant":12116},[58,13242,13243],{"encoding":60},"\\boldsymbol{\\mu}",[34,13245,13247],{"className":13246,"ariaHidden":65},[64],[34,13248,13250,13254],{"className":13249},[69],[34,13251],{"className":13252,"style":13253},[73],"height:0.6389em;vertical-align:-0.1944em;",[34,13255,13257],{"className":13256},[78],[34,13258,13260],{"className":13259},[78],[34,13261,11720],{"className":13262},[78,12153],", with larger ",[34,13265,13267,13281],{"className":13266,"translate":38},[37],[34,13268,13270],{"className":13269},[42],[44,13271,13272],{"xmlns":46},[48,13273,13274,13278],{},[51,13275,13276],{},[54,13277,12226],{},[58,13279,13280],{"encoding":60},"\\kappa",[34,13282,13284],{"className":13283,"ariaHidden":65},[64],[34,13285,13287,13290],{"className":13286},[69],[34,13288],{"className":13289,"style":108},[73],[34,13291,12226],{"className":13292},[78,79]," implying higher concentration",[403,13295,13296,13348,13349],{},[34,13297,13299,13318],{"className":13298,"translate":38},[37],[34,13300,13302],{"className":13301},[42],[44,13303,13304],{"xmlns":46},[48,13305,13306,13315],{},[51,13307,13308,13310,13312],{},[54,13309,12226],{},[176,13311,3003],{},[54,13313,13314],{"mathvariant":2296},"∞",[58,13316,13317],{"encoding":60},"\\kappa \\to \\infty",[34,13319,13321,13339],{"className":13320,"ariaHidden":65},[64],[34,13322,13324,13327,13330,13333,13336],{"className":13323},[69],[34,13325],{"className":13326,"style":108},[73],[34,13328,12226],{"className":13329},[78,79],[34,13331],{"className":13332,"style":262},[237],[34,13334,3003],{"className":13335},[266],[34,13337],{"className":13338,"style":262},[237],[34,13340,13342,13345],{"className":13341},[69],[34,13343],{"className":13344,"style":108},[73],[34,13346,13314],{"className":13347},[78],": tends to a point mass at ",[34,13350,13352,13365],{"className":13351,"translate":38},[37],[34,13353,13355],{"className":13354},[42],[44,13356,13357],{"xmlns":46},[48,13358,13359,13363],{},[51,13360,13361],{},[54,13362,11720],{"mathvariant":12116},[58,13364,13243],{"encoding":60},[34,13366,13368],{"className":13367,"ariaHidden":65},[64],[34,13369,13371,13374],{"className":13370},[69],[34,13372],{"className":13373,"style":13253},[73],[34,13375,13377],{"className":13376},[78],[34,13378,13380],{"className":13379},[78],[34,13381,11720],{"className":13382},[78,12153],[16,13384,13385],{},"Thus, the vMF distribution provides an analogy:",[400,13387,13388,13451],{},[403,13389,13390,13391,13450],{},"In ",[34,13392,13394,13412],{"className":13393,"translate":38},[37],[34,13395,13397],{"className":13396},[42],[44,13398,13399],{"xmlns":46},[48,13400,13401,13409],{},[51,13402,13403],{},[755,13404,13405,13407],{},[54,13406,2997],{"mathvariant":2996},[54,13408,16],{},[58,13410,13411],{"encoding":60},"\\mathbb{R}^p",[34,13413,13415],{"className":13414,"ariaHidden":65},[64],[34,13416,13418,13421],{"className":13417},[69],[34,13419],{"className":13420,"style":3041},[73],[34,13422,13424,13427],{"className":13423},[78],[34,13425,2997],{"className":13426},[78,3048],[34,13428,13430],{"className":13429},[810],[34,13431,13433],{"className":13432},[814],[34,13434,13436],{"className":13435},[818],[34,13437,13439],{"className":13438,"style":3061},[822],[34,13440,13441,13444],{"style":825},[34,13442],{"className":13443,"style":830},[829],[34,13445,13447],{"className":13446},[834,835,836,837],[34,13448,16],{"className":13449},[78,79,837],", the normal distribution controls concentration via the precision matrix",[403,13452,13453,13454],{},"On the sphere, the vMF distribution controls concentration around ",[34,13455,13457,13470],{"className":13456,"translate":38},[37],[34,13458,13460],{"className":13459},[42],[44,13461,13462],{"xmlns":46},[48,13463,13464,13468],{},[51,13465,13466],{},[54,13467,11720],{"mathvariant":12116},[58,13469,13243],{"encoding":60},[34,13471,13473],{"className":13472,"ariaHidden":65},[64],[34,13474,13476,13479],{"className":13475},[69],[34,13477],{"className":13478,"style":13253},[73],[34,13480,13482],{"className":13481},[78],[34,13483,13485],{"className":13484},[78],[34,13486,11720],{"className":13487},[78,12153],[11,13489,13491],{"id":13490},"properties","Properties",[146,13493,13495],{"id":13494},"exponential-family-property","Exponential Family Property",[16,13497,13498],{},"The vMF distribution belongs to the exponential family.",[16,13500,13501],{},"Its form is:",[34,13503,13505],{"className":13504,"translate":38},[159],[34,13506,13508,13570],{"className":13507,"translate":38},[37],[34,13509,13511],{"className":13510},[42],[44,13512,13513],{"xmlns":46,"display":168},[48,13514,13515,13567],{},[51,13516,13517,13519,13521,13523,13525,13528,13530,13532,13534,13536,13538,13540,13542,13544,13546,13553,13555,13557,13559,13561,13563,13565],{},[54,13518,56],{},[176,13520,179],{"stretchy":178},[54,13522,96],{},[54,13524,12292],{"mathvariant":2296},[54,13526,13527],{},"θ",[176,13529,189],{"stretchy":178},[176,13531,192],{},[54,13533,4536],{},[176,13535,179],{"stretchy":178},[54,13537,96],{},[176,13539,189],{"stretchy":178},[54,13541,12317],{},[176,13543,6321],{},[176,13545,179],{"stretchy":178},[755,13547,13548,13550],{},[54,13549,13527],{},[54,13551,13552],{},"T",[54,13554,96],{},[176,13556,6463],{},[54,13558,3145],{},[176,13560,179],{"stretchy":178},[54,13562,13527],{},[176,13564,189],{"stretchy":178},[176,13566,189],{"stretchy":178},[58,13568,13569],{"encoding":60},"f(x|\\theta) = h(x)\\exp(\\theta^T x - A(\\theta))",[34,13571,13573,13607,13677],{"className":13572,"ariaHidden":65},[64],[34,13574,13576,13579,13582,13585,13588,13591,13595,13598,13601,13604],{"className":13575},[69],[34,13577],{"className":13578,"style":223},[73],[34,13580,56],{"className":13581,"style":80},[78,79],[34,13583,179],{"className":13584},[230],[34,13586,96],{"className":13587},[78,79],[34,13589,12292],{"className":13590},[78],[34,13592,13527],{"className":13593,"style":13594},[78,79],"margin-right:0.0278em;",[34,13596,189],{"className":13597},[258],[34,13599],{"className":13600,"style":262},[237],[34,13602,192],{"className":13603},[266],[34,13605],{"className":13606,"style":262},[237],[34,13608,13610,13614,13617,13620,13623,13626,13629,13632,13635,13665,13668,13671,13674],{"className":13609},[69],[34,13611],{"className":13612,"style":13613},[73],"height:1.1413em;vertical-align:-0.25em;",[34,13615,4536],{"className":13616},[78,79],[34,13618,179],{"className":13619},[230],[34,13621,96],{"className":13622},[78,79],[34,13624,189],{"className":13625},[258],[34,13627],{"className":13628,"style":3462},[237],[34,13630,12317],{"className":13631},[3387],[34,13633,179],{"className":13634},[230],[34,13636,13638,13641],{"className":13637},[78],[34,13639,13527],{"className":13640,"style":13594},[78,79],[34,13642,13644],{"className":13643},[810],[34,13645,13647],{"className":13646},[814],[34,13648,13650],{"className":13649},[818],[34,13651,13654],{"className":13652,"style":13653},[822],"height:0.8913em;",[34,13655,13656,13659],{"style":4710},[34,13657],{"className":13658,"style":830},[829],[34,13660,13662],{"className":13661},[834,835,836,837],[34,13663,13552],{"className":13664,"style":6248},[78,79,837],[34,13666,96],{"className":13667},[78,79],[34,13669],{"className":13670,"style":238},[237],[34,13672,6463],{"className":13673},[242],[34,13675],{"className":13676,"style":238},[237],[34,13678,13680,13683,13686,13689,13692],{"className":13679},[69],[34,13681],{"className":13682,"style":223},[73],[34,13684,3145],{"className":13685},[78,79],[34,13687,179],{"className":13688},[230],[34,13690,13527],{"className":13691,"style":13594},[78,79],[34,13693,5491],{"className":13694},[258],[16,13696,13697],{},"with",[34,13699,13701],{"className":13700,"translate":38},[159],[34,13702,13704,13724],{"className":13703,"translate":38},[37],[34,13705,13707],{"className":13706},[42],[44,13708,13709],{"xmlns":46,"display":168},[48,13710,13711,13721],{},[51,13712,13713,13715,13717,13719],{},[54,13714,13527],{},[176,13716,192],{},[54,13718,12226],{},[54,13720,11720],{},[58,13722,13723],{"encoding":60},"\\theta = \\kappa \\mu",[34,13725,13727,13745],{"className":13726,"ariaHidden":65},[64],[34,13728,13730,13733,13736,13739,13742],{"className":13729},[69],[34,13731],{"className":13732,"style":2184},[73],[34,13734,13527],{"className":13735,"style":13594},[78,79],[34,13737],{"className":13738,"style":262},[237],[34,13740,192],{"className":13741},[266],[34,13743],{"className":13744,"style":262},[237],[34,13746,13748,13751,13754],{"className":13747},[69],[34,13749],{"className":13750,"style":139},[73],[34,13752,12226],{"className":13753},[78,79],[34,13755,11720],{"className":13756},[78,79],[16,13758,13759],{},"Hence, vMF enjoys standard exponential family properties:",[400,13761,13762,13765,13768],{},[403,13763,13764],{},"Existence of a sufficient statistic",[403,13766,13767],{},"Simple maximum likelihood estimation",[403,13769,13770],{},"Convenient for use in EM algorithms",[16,13772,13773],{},"Sufficient statistic:",[34,13775,13777],{"className":13776,"translate":38},[159],[34,13778,13780,13804],{"className":13779,"translate":38},[37],[34,13781,13783],{"className":13782},[42],[44,13784,13785],{"xmlns":46,"display":168},[48,13786,13787,13801],{},[51,13788,13789,13791,13793,13795,13797,13799],{},[54,13790,13552],{},[176,13792,179],{"stretchy":178},[54,13794,96],{},[176,13796,189],{"stretchy":178},[176,13798,192],{},[54,13800,96],{},[58,13802,13803],{"encoding":60},"T(x) = x",[34,13805,13807,13834],{"className":13806,"ariaHidden":65},[64],[34,13808,13810,13813,13816,13819,13822,13825,13828,13831],{"className":13809},[69],[34,13811],{"className":13812,"style":223},[73],[34,13814,13552],{"className":13815,"style":6248},[78,79],[34,13817,179],{"className":13818},[230],[34,13820,96],{"className":13821},[78,79],[34,13823,189],{"className":13824},[258],[34,13826],{"className":13827,"style":262},[237],[34,13829,192],{"className":13830},[266],[34,13832],{"className":13833,"style":262},[237],[34,13835,13837,13840],{"className":13836},[69],[34,13838],{"className":13839,"style":108},[73],[34,13841,96],{"className":13842},[78,79],[146,13844,13846],{"id":13845},"first-moment-mean","First Moment (Mean)",[34,13848,13850],{"className":13849,"translate":38},[159],[34,13851,13853,13892],{"className":13852,"translate":38},[37],[34,13854,13856],{"className":13855},[42],[44,13857,13858],{"xmlns":46,"display":168},[48,13859,13860,13889],{},[51,13861,13862,13865,13868,13870,13873,13875,13881,13883,13885,13887],{},[54,13863,13864],{},"E",[176,13866,13867],{"stretchy":178},"[",[54,13869,96],{},[176,13871,13872],{"stretchy":178},"]",[176,13874,192],{},[2468,13876,13877,13879],{},[54,13878,3145],{},[54,13880,5250],{},[176,13882,179],{"stretchy":178},[54,13884,12226],{},[176,13886,189],{"stretchy":178},[54,13888,11720],{},[58,13890,13891],{"encoding":60},"E[x] = A_d(\\kappa)\\mu",[34,13893,13895,13922],{"className":13894,"ariaHidden":65},[64],[34,13896,13898,13901,13904,13907,13910,13913,13916,13919],{"className":13897},[69],[34,13899],{"className":13900,"style":223},[73],[34,13902,13864],{"className":13903,"style":12176},[78,79],[34,13905,13867],{"className":13906},[230],[34,13908,96],{"className":13909},[78,79],[34,13911,13872],{"className":13912},[258],[34,13914],{"className":13915,"style":262},[237],[34,13917,192],{"className":13918},[266],[34,13920],{"className":13921,"style":262},[237],[34,13923,13925,13928,13968,13971,13974,13977],{"className":13924},[69],[34,13926],{"className":13927,"style":223},[73],[34,13929,13931,13934],{"className":13930},[78],[34,13932,3145],{"className":13933},[78,79],[34,13935,13937],{"className":13936},[810],[34,13938,13940,13960],{"className":13939},[814,2523],[34,13941,13943,13957],{"className":13942},[818],[34,13944,13946],{"className":13945,"style":6071},[822],[34,13947,13948,13951],{"style":2533},[34,13949],{"className":13950,"style":830},[829],[34,13952,13954],{"className":13953},[834,835,836,837],[34,13955,5250],{"className":13956},[78,79,837],[34,13958,2547],{"className":13959},[2546],[34,13961,13963],{"className":13962},[818],[34,13964,13966],{"className":13965,"style":2554},[822],[34,13967],{},[34,13969,179],{"className":13970},[230],[34,13972,12226],{"className":13973},[78,79],[34,13975,189],{"className":13976},[258],[34,13978,11720],{"className":13979},[78,79],[16,13981,9212],{},[34,13983,13985],{"className":13984,"translate":38},[159],[34,13986,13988,14060],{"className":13987,"translate":38},[37],[34,13989,13991],{"className":13990},[42],[44,13992,13993],{"xmlns":46,"display":168},[48,13994,13995,14057],{},[51,13996,13997,14003,14005,14007,14009,14011],{},[2468,13998,13999,14001],{},[54,14000,3145],{},[54,14002,5250],{},[176,14004,179],{"stretchy":178},[54,14006,12226],{},[176,14008,189],{"stretchy":178},[176,14010,192],{},[12543,14012,14013,14033],{},[51,14014,14015,14027,14029,14031],{},[2468,14016,14017,14019],{},[54,14018,12587],{},[51,14020,14021,14023,14025],{},[54,14022,5250],{},[54,14024,12580],{"mathvariant":2296},[347,14026,424],{},[176,14028,179],{"stretchy":178},[54,14030,12226],{},[176,14032,189],{"stretchy":178},[51,14034,14035,14051,14053,14055],{},[2468,14036,14037,14039],{},[54,14038,12587],{},[51,14040,14041,14043,14045,14047,14049],{},[54,14042,5250],{},[54,14044,12580],{"mathvariant":2296},[347,14046,424],{},[176,14048,6463],{},[347,14050,2213],{},[176,14052,179],{"stretchy":178},[54,14054,12226],{},[176,14056,189],{"stretchy":178},[58,14058,14059],{"encoding":60},"A_d(\\kappa) = \\frac{I_{d\u002F2}(\\kappa)}{I_{d\u002F2-1}(\\kappa)}",[34,14061,14063,14127],{"className":14062,"ariaHidden":65},[64],[34,14064,14066,14069,14109,14112,14115,14118,14121,14124],{"className":14065},[69],[34,14067],{"className":14068,"style":223},[73],[34,14070,14072,14075],{"className":14071},[78],[34,14073,3145],{"className":14074},[78,79],[34,14076,14078],{"className":14077},[810],[34,14079,14081,14101],{"className":14080},[814,2523],[34,14082,14084,14098],{"className":14083},[818],[34,14085,14087],{"className":14086,"style":6071},[822],[34,14088,14089,14092],{"style":2533},[34,14090],{"className":14091,"style":830},[829],[34,14093,14095],{"className":14094},[834,835,836,837],[34,14096,5250],{"className":14097},[78,79,837],[34,14099,2547],{"className":14100},[2546],[34,14102,14104],{"className":14103},[818],[34,14105,14107],{"className":14106,"style":2554},[822],[34,14108],{},[34,14110,179],{"className":14111},[230],[34,14113,12226],{"className":14114},[78,79],[34,14116,189],{"className":14117},[258],[34,14119],{"className":14120,"style":262},[237],[34,14122,192],{"className":14123},[266],[34,14125],{"className":14126,"style":262},[237],[34,14128,14130,14134],{"className":14129},[69],[34,14131],{"className":14132,"style":14133},[73],"height:2.5364em;vertical-align:-1.0412em;",[34,14135,14137,14140,14309],{"className":14136},[78],[34,14138],{"className":14139},[230,12689],[34,14141,14143],{"className":14142},[12543],[34,14144,14146,14300],{"className":14145},[814,2523],[34,14147,14149,14297],{"className":14148},[818],[34,14150,14153,14225,14233],{"className":14151,"style":14152},[822],"height:1.4952em;",[34,14154,14156,14159],{"style":14155},"top:-2.314em;",[34,14157],{"className":14158,"style":12709},[829],[34,14160,14162,14216,14219,14222],{"className":14161},[78],[34,14163,14165,14168],{"className":14164},[78],[34,14166,12587],{"className":14167,"style":12766},[78,79],[34,14169,14171],{"className":14170},[810],[34,14172,14174,14207],{"className":14173},[814,2523],[34,14175,14177,14204],{"className":14176},[818],[34,14178,14180],{"className":14179,"style":12779},[822],[34,14181,14183,14186],{"style":14182},"top:-2.5198em;margin-left:-0.0785em;margin-right:0.05em;",[34,14184],{"className":14185,"style":830},[829],[34,14187,14189],{"className":14188},[834,835,836,837],[34,14190,14192,14195,14198,14201],{"className":14191},[78,837],[34,14193,5250],{"className":14194},[78,79,837],[34,14196,12759],{"className":14197},[78,837],[34,14199,6463],{"className":14200},[242,837],[34,14202,2213],{"className":14203},[78,837],[34,14205,2547],{"className":14206},[2546],[34,14208,14210],{"className":14209},[818],[34,14211,14214],{"className":14212,"style":14213},[822],"height:0.3552em;",[34,14215],{},[34,14217,179],{"className":14218},[230],[34,14220,12226],{"className":14221},[78,79],[34,14223,189],{"className":14224},[258],[34,14226,14227,14230],{"style":12897},[34,14228],{"className":14229,"style":12709},[829],[34,14231],{"className":14232,"style":12904},[12836],[34,14234,14236,14239],{"style":14235},"top:-3.7452em;",[34,14237],{"className":14238,"style":12709},[829],[34,14240,14242,14288,14291,14294],{"className":14241},[78],[34,14243,14245,14248],{"className":14244},[78],[34,14246,12587],{"className":14247,"style":12766},[78,79],[34,14249,14251],{"className":14250},[810],[34,14252,14254,14280],{"className":14253},[814,2523],[34,14255,14257,14277],{"className":14256},[818],[34,14258,14260],{"className":14259,"style":12779},[822],[34,14261,14262,14265],{"style":14182},[34,14263],{"className":14264,"style":830},[829],[34,14266,14268],{"className":14267},[834,835,836,837],[34,14269,14271,14274],{"className":14270},[78,837],[34,14272,5250],{"className":14273},[78,79,837],[34,14275,12759],{"className":14276},[78,837],[34,14278,2547],{"className":14279},[2546],[34,14281,14283],{"className":14282},[818],[34,14284,14286],{"className":14285,"style":14213},[822],[34,14287],{},[34,14289,179],{"className":14290},[230],[34,14292,12226],{"className":14293},[78,79],[34,14295,189],{"className":14296},[258],[34,14298,2547],{"className":14299},[2546],[34,14301,14303],{"className":14302},[818],[34,14304,14307],{"className":14305,"style":14306},[822],"height:1.0412em;",[34,14308],{},[34,14310],{"className":14311},[258,12689],[16,14313,14314,14315,14400],{},"The function ",[34,14316,14318,14342],{"className":14317,"translate":38},[37],[34,14319,14321],{"className":14320},[42],[44,14322,14323],{"xmlns":46},[48,14324,14325,14339],{},[51,14326,14327,14333,14335,14337],{},[2468,14328,14329,14331],{},[54,14330,3145],{},[54,14332,5250],{},[176,14334,179],{"stretchy":178},[54,14336,12226],{},[176,14338,189],{"stretchy":178},[58,14340,14341],{"encoding":60},"A_d(\\kappa)",[34,14343,14345],{"className":14344,"ariaHidden":65},[64],[34,14346,14348,14351,14391,14394,14397],{"className":14347},[69],[34,14349],{"className":14350,"style":223},[73],[34,14352,14354,14357],{"className":14353},[78],[34,14355,3145],{"className":14356},[78,79],[34,14358,14360],{"className":14359},[810],[34,14361,14363,14383],{"className":14362},[814,2523],[34,14364,14366,14380],{"className":14365},[818],[34,14367,14369],{"className":14368,"style":6071},[822],[34,14370,14371,14374],{"style":2533},[34,14372],{"className":14373,"style":830},[829],[34,14375,14377],{"className":14376},[834,835,836,837],[34,14378,5250],{"className":14379},[78,79,837],[34,14381,2547],{"className":14382},[2546],[34,14384,14386],{"className":14385},[818],[34,14387,14389],{"className":14388,"style":2554},[822],[34,14390],{},[34,14392,179],{"className":14393},[230],[34,14395,12226],{"className":14396},[78,79],[34,14398,189],{"className":14399},[258]," is called the mean resultant length function.",[146,14402,14404],{"id":14403},"second-moment","Second Moment",[34,14406,14408],{"className":14407,"translate":38},[159],[34,14409,14411,14499],{"className":14410,"translate":38},[37],[34,14412,14414],{"className":14413},[42],[44,14415,14416],{"xmlns":46,"display":168},[48,14417,14418,14496],{},[51,14419,14420,14422,14424,14426,14432,14434,14436,14454,14456,14458,14488,14490],{},[54,14421,13864],{},[176,14423,13867],{"stretchy":178},[54,14425,96],{},[755,14427,14428,14430],{},[54,14429,96],{},[54,14431,13552],{},[176,14433,13872],{"stretchy":178},[176,14435,192],{},[12543,14437,14438,14452],{},[51,14439,14440,14446,14448,14450],{},[2468,14441,14442,14444],{},[54,14443,3145],{},[54,14445,5250],{},[176,14447,179],{"stretchy":178},[54,14449,12226],{},[176,14451,189],{"stretchy":178},[54,14453,12226],{},[54,14455,12587],{},[176,14457,184],{},[51,14459,14460,14462,14464,14466,14486],{},[176,14461,179],{"fence":65},[347,14463,2213],{},[176,14465,6463],{},[12543,14467,14468,14484],{},[51,14469,14470,14472,14478,14480,14482],{},[54,14471,5250],{},[2468,14473,14474,14476],{},[54,14475,3145],{},[54,14477,5250],{},[176,14479,179],{"stretchy":178},[54,14481,12226],{},[176,14483,189],{"stretchy":178},[54,14485,12226],{},[176,14487,189],{"fence":65},[54,14489,11720],{},[755,14491,14492,14494],{},[54,14493,11720],{},[54,14495,13552],{},[58,14497,14498],{"encoding":60},"E[xx^T] = \\frac{A_d(\\kappa)}{\\kappa} I + \\left(1 - \\frac{dA_d(\\kappa)}{\\kappa}\\right)\\mu\\mu^T",[34,14500,14502,14558,14687],{"className":14501,"ariaHidden":65},[64],[34,14503,14505,14508,14511,14514,14517,14546,14549,14552,14555],{"className":14504},[69],[34,14506],{"className":14507,"style":13613},[73],[34,14509,13864],{"className":14510,"style":12176},[78,79],[34,14512,13867],{"className":14513},[230],[34,14515,96],{"className":14516},[78,79],[34,14518,14520,14523],{"className":14519},[78],[34,14521,96],{"className":14522},[78,79],[34,14524,14526],{"className":14525},[810],[34,14527,14529],{"className":14528},[814],[34,14530,14532],{"className":14531},[818],[34,14533,14535],{"className":14534,"style":13653},[822],[34,14536,14537,14540],{"style":4710},[34,14538],{"className":14539,"style":830},[829],[34,14541,14543],{"className":14542},[834,835,836,837],[34,14544,13552],{"className":14545,"style":6248},[78,79,837],[34,14547,13872],{"className":14548},[258],[34,14550],{"className":14551,"style":262},[237],[34,14553,192],{"className":14554},[266],[34,14556],{"className":14557,"style":262},[237],[34,14559,14561,14565,14675,14678,14681,14684],{"className":14560},[69],[34,14562],{"className":14563,"style":14564},[73],"height:2.113em;vertical-align:-0.686em;",[34,14566,14568,14571,14672],{"className":14567},[78],[34,14569],{"className":14570},[230,12689],[34,14572,14574],{"className":14573},[12543],[34,14575,14577,14663],{"className":14576},[814,2523],[34,14578,14580,14660],{"className":14579},[818],[34,14581,14584,14595,14603],{"className":14582,"style":14583},[822],"height:1.427em;",[34,14585,14586,14589],{"style":14155},[34,14587],{"className":14588,"style":12709},[829],[34,14590,14592],{"className":14591},[78],[34,14593,12226],{"className":14594},[78,79],[34,14596,14597,14600],{"style":12897},[34,14598],{"className":14599,"style":12709},[829],[34,14601],{"className":14602,"style":12904},[12836],[34,14604,14605,14608],{"style":12907},[34,14606],{"className":14607,"style":12709},[829],[34,14609,14611,14651,14654,14657],{"className":14610},[78],[34,14612,14614,14617],{"className":14613},[78],[34,14615,3145],{"className":14616},[78,79],[34,14618,14620],{"className":14619},[810],[34,14621,14623,14643],{"className":14622},[814,2523],[34,14624,14626,14640],{"className":14625},[818],[34,14627,14629],{"className":14628,"style":6071},[822],[34,14630,14631,14634],{"style":2533},[34,14632],{"className":14633,"style":830},[829],[34,14635,14637],{"className":14636},[834,835,836,837],[34,14638,5250],{"className":14639},[78,79,837],[34,14641,2547],{"className":14642},[2546],[34,14644,14646],{"className":14645},[818],[34,14647,14649],{"className":14648,"style":2554},[822],[34,14650],{},[34,14652,179],{"className":14653},[230],[34,14655,12226],{"className":14656},[78,79],[34,14658,189],{"className":14659},[258],[34,14661,2547],{"className":14662},[2546],[34,14664,14666],{"className":14665},[818],[34,14667,14670],{"className":14668,"style":14669},[822],"height:0.686em;",[34,14671],{},[34,14673],{"className":14674},[258,12689],[34,14676,12587],{"className":14677,"style":12766},[78,79],[34,14679],{"className":14680,"style":238},[237],[34,14682,184],{"className":14683},[242],[34,14685],{"className":14686,"style":238},[237],[34,14688,14690,14694,14832,14835,14838],{"className":14689},[69],[34,14691],{"className":14692,"style":14693},[73],"height:2.4em;vertical-align:-0.95em;",[34,14695,14697,14703,14706,14709,14712,14715,14826],{"className":14696},[4177],[34,14698,14700],{"className":14699,"style":4182},[230,4181],[34,14701,179],{"className":14702},[4186,836],[34,14704,2213],{"className":14705},[78],[34,14707],{"className":14708,"style":238},[237],[34,14710,6463],{"className":14711},[242],[34,14713],{"className":14714,"style":238},[237],[34,14716,14718,14721,14823],{"className":14717},[78],[34,14719],{"className":14720},[230,12689],[34,14722,14724],{"className":14723},[12543],[34,14725,14727,14815],{"className":14726},[814,2523],[34,14728,14730,14812],{"className":14729},[818],[34,14731,14733,14744,14752],{"className":14732,"style":14583},[822],[34,14734,14735,14738],{"style":14155},[34,14736],{"className":14737,"style":12709},[829],[34,14739,14741],{"className":14740},[78],[34,14742,12226],{"className":14743},[78,79],[34,14745,14746,14749],{"style":12897},[34,14747],{"className":14748,"style":12709},[829],[34,14750],{"className":14751,"style":12904},[12836],[34,14753,14754,14757],{"style":12907},[34,14755],{"className":14756,"style":12709},[829],[34,14758,14760,14763,14803,14806,14809],{"className":14759},[78],[34,14761,5250],{"className":14762},[78,79],[34,14764,14766,14769],{"className":14765},[78],[34,14767,3145],{"className":14768},[78,79],[34,14770,14772],{"className":14771},[810],[34,14773,14775,14795],{"className":14774},[814,2523],[34,14776,14778,14792],{"className":14777},[818],[34,14779,14781],{"className":14780,"style":6071},[822],[34,14782,14783,14786],{"style":2533},[34,14784],{"className":14785,"style":830},[829],[34,14787,14789],{"className":14788},[834,835,836,837],[34,14790,5250],{"className":14791},[78,79,837],[34,14793,2547],{"className":14794},[2546],[34,14796,14798],{"className":14797},[818],[34,14799,14801],{"className":14800,"style":2554},[822],[34,14802],{},[34,14804,179],{"className":14805},[230],[34,14807,12226],{"className":14808},[78,79],[34,14810,189],{"className":14811},[258],[34,14813,2547],{"className":14814},[2546],[34,14816,14818],{"className":14817},[818],[34,14819,14821],{"className":14820,"style":14669},[822],[34,14822],{},[34,14824],{"className":14825},[258,12689],[34,14827,14829],{"className":14828,"style":4182},[258,4181],[34,14830,189],{"className":14831},[4186,836],[34,14833],{"className":14834,"style":3462},[237],[34,14836,11720],{"className":14837},[78,79],[34,14839,14841,14844],{"className":14840},[78],[34,14842,11720],{"className":14843},[78,79],[34,14845,14847],{"className":14846},[810],[34,14848,14850],{"className":14849},[814],[34,14851,14853],{"className":14852},[818],[34,14854,14856],{"className":14855,"style":13653},[822],[34,14857,14858,14861],{"style":4710},[34,14859],{"className":14860,"style":830},[829],[34,14862,14864],{"className":14863},[834,835,836,837],[34,14865,13552],{"className":14866,"style":6248},[78,79,837],[146,14868,14870],{"id":14869},"rotation-invariance","Rotation Invariance",[16,14872,14873,14874,5228],{},"For any orthogonal matrix ",[34,14875,14877,14890],{"className":14876,"translate":38},[37],[34,14878,14880],{"className":14879},[42],[44,14881,14882],{"xmlns":46},[48,14883,14884,14888],{},[51,14885,14886],{},[54,14887,2997],{},[58,14889,2997],{"encoding":60},[34,14891,14893],{"className":14892,"ariaHidden":65},[64],[34,14894,14896,14899],{"className":14895},[69],[34,14897],{"className":14898,"style":3244},[73],[34,14900,2997],{"className":14901,"style":14902},[78,79],"margin-right:0.0077em;",[16,14904,14905],{},"if",[34,14907,14909],{"className":14908,"translate":38},[159],[34,14910,14912,14942],{"className":14911,"translate":38},[37],[34,14913,14915],{"className":14914},[42],[44,14916,14917],{"xmlns":46,"display":168},[48,14918,14919,14939],{},[51,14920,14921,14923,14926,14929,14931,14933,14935,14937],{},[54,14922,96],{},[176,14924,14925],{},"∼",[1201,14927,14928],{},"vMF",[176,14930,179],{"stretchy":178},[54,14932,11720],{},[176,14934,5228],{"separator":65},[54,14936,12226],{},[176,14938,189],{"stretchy":178},[58,14940,14941],{"encoding":60},"x \\sim \\text{vMF}(\\mu,\\kappa)",[34,14943,14945,14963],{"className":14944,"ariaHidden":65},[64],[34,14946,14948,14951,14954,14957,14960],{"className":14947},[69],[34,14949],{"className":14950,"style":108},[73],[34,14952,96],{"className":14953},[78,79],[34,14955],{"className":14956,"style":262},[237],[34,14958,14925],{"className":14959},[266],[34,14961],{"className":14962,"style":262},[237],[34,14964,14966,14969,14975,14978,14981,14984,14987,14990],{"className":14965},[69],[34,14967],{"className":14968,"style":223},[73],[34,14970,14972],{"className":14971},[78,1304],[34,14973,14928],{"className":14974},[78],[34,14976,179],{"className":14977},[230],[34,14979,11720],{"className":14980},[78,79],[34,14982,5228],{"className":14983},[5317],[34,14985],{"className":14986,"style":3462},[237],[34,14988,12226],{"className":14989},[78,79],[34,14991,189],{"className":14992},[258],[16,14994,14995],{},"then",[34,14997,14999],{"className":14998,"translate":38},[159],[34,15000,15002,15034],{"className":15001,"translate":38},[37],[34,15003,15005],{"className":15004},[42],[44,15006,15007],{"xmlns":46,"display":168},[48,15008,15009,15031],{},[51,15010,15011,15013,15015,15017,15019,15021,15023,15025,15027,15029],{},[54,15012,2997],{},[54,15014,96],{},[176,15016,14925],{},[1201,15018,14928],{},[176,15020,179],{"stretchy":178},[54,15022,2997],{},[54,15024,11720],{},[176,15026,5228],{"separator":65},[54,15028,12226],{},[176,15030,189],{"stretchy":178},[58,15032,15033],{"encoding":60},"Rx \\sim \\text{vMF}(R\\mu,\\kappa)",[34,15035,15037,15058],{"className":15036,"ariaHidden":65},[64],[34,15038,15040,15043,15046,15049,15052,15055],{"className":15039},[69],[34,15041],{"className":15042,"style":3244},[73],[34,15044,2997],{"className":15045,"style":14902},[78,79],[34,15047,96],{"className":15048},[78,79],[34,15050],{"className":15051,"style":262},[237],[34,15053,14925],{"className":15054},[266],[34,15056],{"className":15057,"style":262},[237],[34,15059,15061,15064,15070,15073,15076,15079,15082,15085,15088],{"className":15060},[69],[34,15062],{"className":15063,"style":223},[73],[34,15065,15067],{"className":15066},[78,1304],[34,15068,14928],{"className":15069},[78],[34,15071,179],{"className":15072},[230],[34,15074,2997],{"className":15075,"style":14902},[78,79],[34,15077,11720],{"className":15078},[78,79],[34,15080,5228],{"className":15081},[5317],[34,15083],{"className":15084,"style":3462},[237],[34,15086,12226],{"className":15087},[78,79],[34,15089,189],{"className":15090},[258],[16,15092,15093],{},"This shows that the vMF distribution is invariant under the rotation group.",[146,15095,15097],{"id":15096},"information-geometry-properties","Information Geometry Properties",[16,15099,15100],{},"The family of vMF distributions forms an information-geometric manifold.",[16,15102,15103],{},"Fisher information:",[34,15105,15107],{"className":15106,"translate":38},[159],[34,15108,15110,15170],{"className":15109,"translate":38},[37],[34,15111,15113],{"className":15112},[42],[44,15114,15115],{"xmlns":46,"display":168},[48,15116,15117,15167],{},[51,15118,15119,15121,15123,15125,15127,15129,15131,15150,15153,15155,15161,15163,15165],{},[54,15120,12587],{},[176,15122,179],{"stretchy":178},[54,15124,12226],{},[176,15126,189],{"stretchy":178},[176,15128,192],{},[176,15130,6463],{},[12543,15132,15133,15140],{},[755,15134,15135,15138],{},[54,15136,15137],{"mathvariant":2296},"∂",[347,15139,424],{},[51,15141,15142,15144],{},[54,15143,15137],{"mathvariant":2296},[755,15145,15146,15148],{},[54,15147,12226],{},[347,15149,424],{},[54,15151,15152],{},"log",[176,15154,6321],{},[2468,15156,15157,15159],{},[54,15158,5243],{},[54,15160,5250],{},[176,15162,179],{"stretchy":178},[54,15164,12226],{},[176,15166,189],{"stretchy":178},[58,15168,15169],{"encoding":60},"I(\\kappa) = -\\frac{\\partial^2}{\\partial \\kappa^2} \\log C_d(\\kappa)",[34,15171,15173,15200],{"className":15172,"ariaHidden":65},[64],[34,15174,15176,15179,15182,15185,15188,15191,15194,15197],{"className":15175},[69],[34,15177],{"className":15178,"style":223},[73],[34,15180,12587],{"className":15181,"style":12766},[78,79],[34,15183,179],{"className":15184},[230],[34,15186,12226],{"className":15187},[78,79],[34,15189,189],{"className":15190},[258],[34,15192],{"className":15193,"style":262},[237],[34,15195,192],{"className":15196},[266],[34,15198],{"className":15199,"style":262},[237],[34,15201,15203,15207,15210,15330,15333,15340,15343,15383,15386,15389],{"className":15202},[69],[34,15204],{"className":15205,"style":15206},[73],"height:2.1771em;vertical-align:-0.686em;",[34,15208,6463],{"className":15209},[78],[34,15211,15213,15216,15327],{"className":15212},[78],[34,15214],{"className":15215},[230,12689],[34,15217,15219],{"className":15218},[12543],[34,15220,15222,15319],{"className":15221},[814,2523],[34,15223,15225,15316],{"className":15224},[818],[34,15226,15229,15271,15279],{"className":15227,"style":15228},[822],"height:1.4911em;",[34,15230,15231,15234],{"style":14155},[34,15232],{"className":15233,"style":12709},[829],[34,15235,15237,15241],{"className":15236},[78],[34,15238,15137],{"className":15239,"style":15240},[78],"margin-right:0.0556em;",[34,15242,15244,15247],{"className":15243},[78],[34,15245,12226],{"className":15246},[78,79],[34,15248,15250],{"className":15249},[810],[34,15251,15253],{"className":15252},[814],[34,15254,15256],{"className":15255},[818],[34,15257,15260],{"className":15258,"style":15259},[822],"height:0.7401em;",[34,15261,15262,15265],{"style":12743},[34,15263],{"className":15264,"style":830},[829],[34,15266,15268],{"className":15267},[834,835,836,837],[34,15269,424],{"className":15270},[78,837],[34,15272,15273,15276],{"style":12897},[34,15274],{"className":15275,"style":12709},[829],[34,15277],{"className":15278,"style":12904},[12836],[34,15280,15281,15284],{"style":12907},[34,15282],{"className":15283,"style":12709},[829],[34,15285,15287],{"className":15286},[78],[34,15288,15290,15293],{"className":15289},[78],[34,15291,15137],{"className":15292,"style":15240},[78],[34,15294,15296],{"className":15295},[810],[34,15297,15299],{"className":15298},[814],[34,15300,15302],{"className":15301},[818],[34,15303,15305],{"className":15304,"style":800},[822],[34,15306,15307,15310],{"style":825},[34,15308],{"className":15309,"style":830},[829],[34,15311,15313],{"className":15312},[834,835,836,837],[34,15314,424],{"className":15315},[78,837],[34,15317,2547],{"className":15318},[2546],[34,15320,15322],{"className":15321},[818],[34,15323,15325],{"className":15324,"style":14669},[822],[34,15326],{},[34,15328],{"className":15329},[258,12689],[34,15331],{"className":15332,"style":3462},[237],[34,15334,15336,15337],{"className":15335},[3387],"lo",[34,15338,3298],{"style":15339},"margin-right:0.0139em;",[34,15341],{"className":15342,"style":3462},[237],[34,15344,15346,15349],{"className":15345},[78],[34,15347,5243],{"className":15348,"style":5354},[78,79],[34,15350,15352],{"className":15351},[810],[34,15353,15355,15375],{"className":15354},[814,2523],[34,15356,15358,15372],{"className":15357},[818],[34,15359,15361],{"className":15360,"style":6071},[822],[34,15362,15363,15366],{"style":12428},[34,15364],{"className":15365,"style":830},[829],[34,15367,15369],{"className":15368},[834,835,836,837],[34,15370,5250],{"className":15371},[78,79,837],[34,15373,2547],{"className":15374},[2546],[34,15376,15378],{"className":15377},[818],[34,15379,15381],{"className":15380,"style":2554},[822],[34,15382],{},[34,15384,179],{"className":15385},[230],[34,15387,12226],{"className":15388},[78,79],[34,15390,189],{"className":15391},[258],[16,15393,15394],{},"This allows the use of natural gradient and information-geometric optimization.",[146,15396,15398],{"id":15397},"entropy","Entropy",[16,15400,15401],{},"The entropy of the vMF distribution is:",[34,15403,15405],{"className":15404,"translate":38},[159],[34,15406,15408,15479],{"className":15407,"translate":38},[37],[34,15409,15411],{"className":15410},[42],[44,15412,15413],{"xmlns":46,"display":168},[48,15414,15415,15476],{},[51,15416,15417,15420,15422,15424,15426,15432,15434,15436,15438,15440,15442,15444,15450,15452,15454,15456,15458,15464,15466,15468,15470,15472,15474],{},[54,15418,15419],{},"H",[176,15421,192],{},[176,15423,6463],{},[54,15425,12226],{},[2468,15427,15428,15430],{},[54,15429,3145],{},[54,15431,5250],{},[176,15433,179],{"stretchy":178},[54,15435,12226],{},[176,15437,189],{"stretchy":178},[176,15439,184],{},[54,15441,15152],{},[176,15443,6321],{},[2468,15445,15446,15448],{},[54,15447,5243],{},[54,15449,5250],{},[176,15451,179],{"stretchy":178},[54,15453,12226],{},[176,15455,189],{"stretchy":178},[176,15457,184],{},[12543,15459,15460,15462],{},[54,15461,5250],{},[347,15463,424],{},[54,15465,15152],{},[176,15467,6321],{},[176,15469,179],{"stretchy":178},[347,15471,424],{},[54,15473,12569],{},[176,15475,189],{"stretchy":178},[58,15477,15478],{"encoding":60},"H = -\\kappa A_d(\\kappa) + \\log C_d(\\kappa) + \\frac{d}{2}\\log(2\\pi)",[34,15480,15482,15501,15571,15643],{"className":15481,"ariaHidden":65},[64],[34,15483,15485,15488,15492,15495,15498],{"className":15484},[69],[34,15486],{"className":15487,"style":3244},[73],[34,15489,15419],{"className":15490,"style":15491},[78,79],"margin-right:0.0813em;",[34,15493],{"className":15494,"style":262},[237],[34,15496,192],{"className":15497},[266],[34,15499],{"className":15500,"style":262},[237],[34,15502,15504,15507,15510,15513,15553,15556,15559,15562,15565,15568],{"className":15503},[69],[34,15505],{"className":15506,"style":223},[73],[34,15508,6463],{"className":15509},[78],[34,15511,12226],{"className":15512},[78,79],[34,15514,15516,15519],{"className":15515},[78],[34,15517,3145],{"className":15518},[78,79],[34,15520,15522],{"className":15521},[810],[34,15523,15525,15545],{"className":15524},[814,2523],[34,15526,15528,15542],{"className":15527},[818],[34,15529,15531],{"className":15530,"style":6071},[822],[34,15532,15533,15536],{"style":2533},[34,15534],{"className":15535,"style":830},[829],[34,15537,15539],{"className":15538},[834,835,836,837],[34,15540,5250],{"className":15541},[78,79,837],[34,15543,2547],{"className":15544},[2546],[34,15546,15548],{"className":15547},[818],[34,15549,15551],{"className":15550,"style":2554},[822],[34,15552],{},[34,15554,179],{"className":15555},[230],[34,15557,12226],{"className":15558},[78,79],[34,15560,189],{"className":15561},[258],[34,15563],{"className":15564,"style":238},[237],[34,15566,184],{"className":15567},[242],[34,15569],{"className":15570,"style":238},[237],[34,15572,15574,15577,15582,15585,15625,15628,15631,15634,15637,15640],{"className":15573},[69],[34,15575],{"className":15576,"style":223},[73],[34,15578,15336,15580],{"className":15579},[3387],[34,15581,3298],{"style":15339},[34,15583],{"className":15584,"style":3462},[237],[34,15586,15588,15591],{"className":15587},[78],[34,15589,5243],{"className":15590,"style":5354},[78,79],[34,15592,15594],{"className":15593},[810],[34,15595,15597,15617],{"className":15596},[814,2523],[34,15598,15600,15614],{"className":15599},[818],[34,15601,15603],{"className":15602,"style":6071},[822],[34,15604,15605,15608],{"style":12428},[34,15606],{"className":15607,"style":830},[829],[34,15609,15611],{"className":15610},[834,835,836,837],[34,15612,5250],{"className":15613},[78,79,837],[34,15615,2547],{"className":15616},[2546],[34,15618,15620],{"className":15619},[818],[34,15621,15623],{"className":15622,"style":2554},[822],[34,15624],{},[34,15626,179],{"className":15627},[230],[34,15629,12226],{"className":15630},[78,79],[34,15632,189],{"className":15633},[258],[34,15635],{"className":15636,"style":238},[237],[34,15638,184],{"className":15639},[242],[34,15641],{"className":15642,"style":238},[237],[34,15644,15646,15650,15713,15716,15721,15724,15727,15730],{"className":15645},[69],[34,15647],{"className":15648,"style":15649},[73],"height:2.0574em;vertical-align:-0.686em;",[34,15651,15653,15656,15710],{"className":15652},[78],[34,15654],{"className":15655},[230,12689],[34,15657,15659],{"className":15658},[12543],[34,15660,15662,15702],{"className":15661},[814,2523],[34,15663,15665,15699],{"className":15664},[818],[34,15666,15669,15680,15688],{"className":15667,"style":15668},[822],"height:1.3714em;",[34,15670,15671,15674],{"style":14155},[34,15672],{"className":15673,"style":12709},[829],[34,15675,15677],{"className":15676},[78],[34,15678,424],{"className":15679},[78],[34,15681,15682,15685],{"style":12897},[34,15683],{"className":15684,"style":12709},[829],[34,15686],{"className":15687,"style":12904},[12836],[34,15689,15690,15693],{"style":12907},[34,15691],{"className":15692,"style":12709},[829],[34,15694,15696],{"className":15695},[78],[34,15697,5250],{"className":15698},[78,79],[34,15700,2547],{"className":15701},[2546],[34,15703,15705],{"className":15704},[818],[34,15706,15708],{"className":15707,"style":14669},[822],[34,15709],{},[34,15711],{"className":15712},[258,12689],[34,15714],{"className":15715,"style":3462},[237],[34,15717,15336,15719],{"className":15718},[3387],[34,15720,3298],{"style":15339},[34,15722,179],{"className":15723},[230],[34,15725,424],{"className":15726},[78],[34,15728,12569],{"className":15729,"style":143},[78,79],[34,15731,189],{"className":15732},[258],[16,15734,15735,15736,15764],{},"Property: larger ",[34,15737,15739,15752],{"className":15738,"translate":38},[37],[34,15740,15742],{"className":15741},[42],[44,15743,15744],{"xmlns":46},[48,15745,15746,15750],{},[51,15747,15748],{},[54,15749,12226],{},[58,15751,13280],{"encoding":60},[34,15753,15755],{"className":15754,"ariaHidden":65},[64],[34,15756,15758,15761],{"className":15757},[69],[34,15759],{"className":15760,"style":108},[73],[34,15762,12226],{"className":15763},[78,79]," gives smaller entropy.",[146,15766,15768],{"id":15767},"spherical-harmonic-expansion","Spherical Harmonic Expansion",[16,15770,15771],{},"The vMF distribution can be expanded as:",[34,15773,15775],{"className":15774,"translate":38},[159],[34,15776,15778,15826],{"className":15777,"translate":38},[37],[34,15779,15781],{"className":15780},[42],[44,15782,15783],{"xmlns":46,"display":168},[48,15784,15785,15823],{},[51,15786,15787,15789,15791,15793,15795,15797,15804,15810,15817,15819,15821],{},[54,15788,56],{},[176,15790,179],{"stretchy":178},[54,15792,96],{},[176,15794,189],{"stretchy":178},[176,15796,192],{},[15798,15799,15800,15802],"munder",{},[176,15801,3312],{},[54,15803,7102],{},[2468,15805,15806,15808],{},[54,15807,11468],{},[54,15809,7102],{},[2468,15811,15812,15815],{},[54,15813,15814],{},"Y",[54,15816,7102],{},[176,15818,179],{"stretchy":178},[54,15820,96],{},[176,15822,189],{"stretchy":178},[58,15824,15825],{"encoding":60},"f(x) = \\sum_l a_l Y_l(x)",[34,15827,15829,15856],{"className":15828,"ariaHidden":65},[64],[34,15830,15832,15835,15838,15841,15844,15847,15850,15853],{"className":15831},[69],[34,15833],{"className":15834,"style":223},[73],[34,15836,56],{"className":15837,"style":80},[78,79],[34,15839,179],{"className":15840},[230],[34,15842,96],{"className":15843},[78,79],[34,15845,189],{"className":15846},[258],[34,15848],{"className":15849,"style":262},[237],[34,15851,192],{"className":15852},[266],[34,15854],{"className":15855,"style":262},[237],[34,15857,15859,15863,15910,15913,15953,15994,15997,16000],{"className":15858},[69],[34,15860],{"className":15861,"style":15862},[73],"height:2.3521em;vertical-align:-1.3021em;",[34,15864,15866],{"className":15865},[3387,3388],[34,15867,15869,15901],{"className":15868},[814,2523],[34,15870,15872,15898],{"className":15871},[818],[34,15873,15876,15888],{"className":15874,"style":15875},[822],"height:1.05em;",[34,15877,15879,15882],{"style":15878},"top:-1.8479em;margin-left:0em;",[34,15880],{"className":15881,"style":3405},[829],[34,15883,15885],{"className":15884},[834,835,836,837],[34,15886,7102],{"className":15887,"style":7147},[78,79,837],[34,15889,15890,15893],{"style":3423},[34,15891],{"className":15892,"style":3405},[829],[34,15894,15895],{},[34,15896,3312],{"className":15897},[3387,3432,3433],[34,15899,2547],{"className":15900},[2546],[34,15902,15904],{"className":15903},[818],[34,15905,15908],{"className":15906,"style":15907},[822],"height:1.3021em;",[34,15909],{},[34,15911],{"className":15912,"style":3462},[237],[34,15914,15916,15919],{"className":15915},[78],[34,15917,11468],{"className":15918},[78,79],[34,15920,15922],{"className":15921},[810],[34,15923,15925,15945],{"className":15924},[814,2523],[34,15926,15928,15942],{"className":15927},[818],[34,15929,15931],{"className":15930,"style":6071},[822],[34,15932,15933,15936],{"style":2533},[34,15934],{"className":15935,"style":830},[829],[34,15937,15939],{"className":15938},[834,835,836,837],[34,15940,7102],{"className":15941,"style":7147},[78,79,837],[34,15943,2547],{"className":15944},[2546],[34,15946,15948],{"className":15947},[818],[34,15949,15951],{"className":15950,"style":2554},[822],[34,15952],{},[34,15954,15956,15959],{"className":15955},[78],[34,15957,15814],{"className":15958,"style":238},[78,79],[34,15960,15962],{"className":15961},[810],[34,15963,15965,15986],{"className":15964},[814,2523],[34,15966,15968,15983],{"className":15967},[818],[34,15969,15971],{"className":15970,"style":6071},[822],[34,15972,15974,15977],{"style":15973},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[34,15975],{"className":15976,"style":830},[829],[34,15978,15980],{"className":15979},[834,835,836,837],[34,15981,7102],{"className":15982,"style":7147},[78,79,837],[34,15984,2547],{"className":15985},[2546],[34,15987,15989],{"className":15988},[818],[34,15990,15992],{"className":15991,"style":2554},[822],[34,15993],{},[34,15995,179],{"className":15996},[230],[34,15998,96],{"className":15999},[78,79],[34,16001,189],{"className":16002},[258],[16,16004,3218,16005,16075],{},[34,16006,16008,16026],{"className":16007,"translate":38},[37],[34,16009,16011],{"className":16010},[42],[44,16012,16013],{"xmlns":46},[48,16014,16015,16023],{},[51,16016,16017],{},[2468,16018,16019,16021],{},[54,16020,15814],{},[54,16022,7102],{},[58,16024,16025],{"encoding":60},"Y_l",[34,16027,16029],{"className":16028,"ariaHidden":65},[64],[34,16030,16032,16035],{"className":16031},[69],[34,16033],{"className":16034,"style":3677},[73],[34,16036,16038,16041],{"className":16037},[78],[34,16039,15814],{"className":16040,"style":238},[78,79],[34,16042,16044],{"className":16043},[810],[34,16045,16047,16067],{"className":16046},[814,2523],[34,16048,16050,16064],{"className":16049},[818],[34,16051,16053],{"className":16052,"style":6071},[822],[34,16054,16055,16058],{"style":15973},[34,16056],{"className":16057,"style":830},[829],[34,16059,16061],{"className":16060},[834,835,836,837],[34,16062,7102],{"className":16063,"style":7147},[78,79,837],[34,16065,2547],{"className":16066},[2546],[34,16068,16070],{"className":16069},[818],[34,16071,16073],{"className":16072,"style":2554},[822],[34,16074],{}," are spherical harmonics.",[146,16077,16079],{"id":16078},"spherical-cap","Spherical Cap",[16,16081,16082,16083,144,16111,16114],{},"In the von Mises–Fisher distribution, probability mass is mainly concentrated around the mean direction ",[34,16084,16086,16099],{"className":16085,"translate":38},[37],[34,16087,16089],{"className":16088},[42],[44,16090,16091],{"xmlns":46},[48,16092,16093,16097],{},[51,16094,16095],{},[54,16096,11720],{},[58,16098,11723],{"encoding":60},[34,16100,16102],{"className":16101,"ariaHidden":65},[64],[34,16103,16105,16108],{"className":16104},[69],[34,16106],{"className":16107,"style":139},[73],[34,16109,11720],{"className":16110},[78,79],[16112,16113],"br",{},"\nHence, a spherical cap is commonly used to describe confidence regions.",[16,16116,16117,16118,16199,16200,16309,16310,16339],{},"Let ",[34,16119,16121,16148],{"className":16120,"translate":38},[37],[34,16122,16124],{"className":16123},[42],[44,16125,16126],{"xmlns":46},[48,16127,16128,16146],{},[51,16129,16130,16132,16134,16136,16138,16140,16142,16144],{},[54,16131,96],{},[176,16133,14925],{},[1201,16135,14928],{},[176,16137,179],{"stretchy":178},[54,16139,11720],{},[176,16141,5228],{"separator":65},[54,16143,12226],{},[176,16145,189],{"stretchy":178},[58,16147,14941],{"encoding":60},[34,16149,16151,16169],{"className":16150,"ariaHidden":65},[64],[34,16152,16154,16157,16160,16163,16166],{"className":16153},[69],[34,16155],{"className":16156,"style":108},[73],[34,16158,96],{"className":16159},[78,79],[34,16161],{"className":16162,"style":262},[237],[34,16164,14925],{"className":16165},[266],[34,16167],{"className":16168,"style":262},[237],[34,16170,16172,16175,16181,16184,16187,16190,16193,16196],{"className":16171},[69],[34,16173],{"className":16174,"style":223},[73],[34,16176,16178],{"className":16177},[78,1304],[34,16179,14928],{"className":16180},[78],[34,16182,179],{"className":16183},[230],[34,16185,11720],{"className":16186},[78,79],[34,16188,5228],{"className":16189},[5317],[34,16191],{"className":16192,"style":3462},[237],[34,16194,12226],{"className":16195},[78,79],[34,16197,189],{"className":16198},[258],", ",[34,16201,16203,16235],{"className":16202,"translate":38},[37],[34,16204,16206],{"className":16205},[42],[44,16207,16208],{"xmlns":46},[48,16209,16210,16232],{},[51,16211,16212,16214,16216,16218,16220],{},[54,16213,96],{},[176,16215,5228],{"separator":65},[54,16217,11720],{},[176,16219,11909],{},[755,16221,16222,16224],{},[54,16223,11890],{},[51,16225,16226,16228,16230],{},[54,16227,5250],{},[176,16229,6463],{},[347,16231,2213],{},[58,16233,16234],{"encoding":60},"x,\\mu \\in S^{d-1}",[34,16236,16238,16265],{"className":16237,"ariaHidden":65},[64],[34,16239,16241,16244,16247,16250,16253,16256,16259,16262],{"className":16240},[69],[34,16242],{"className":16243,"style":12143},[73],[34,16245,96],{"className":16246},[78,79],[34,16248,5228],{"className":16249},[5317],[34,16251],{"className":16252,"style":3462},[237],[34,16254,11720],{"className":16255},[78,79],[34,16257],{"className":16258,"style":262},[237],[34,16260,11909],{"className":16261},[266],[34,16263],{"className":16264,"style":262},[237],[34,16266,16268,16271],{"className":16267},[69],[34,16269],{"className":16270,"style":12169},[73],[34,16272,16274,16277],{"className":16273},[78],[34,16275,11890],{"className":16276,"style":12176},[78,79],[34,16278,16280],{"className":16279},[810],[34,16281,16283],{"className":16282},[814],[34,16284,16286],{"className":16285},[818],[34,16287,16289],{"className":16288,"style":12169},[822],[34,16290,16291,16294],{"style":825},[34,16292],{"className":16293,"style":830},[829],[34,16295,16297],{"className":16296},[834,835,836,837],[34,16298,16300,16303,16306],{"className":16299},[78,837],[34,16301,5250],{"className":16302},[78,79,837],[34,16304,6463],{"className":16305},[242,837],[34,16307,2213],{"className":16308},[78,837],", and let ",[34,16311,16313,16327],{"className":16312,"translate":38},[37],[34,16314,16316],{"className":16315},[42],[44,16317,16318],{"xmlns":46},[48,16319,16320,16324],{},[51,16321,16322],{},[54,16323,13527],{},[58,16325,16326],{"encoding":60},"\\theta",[34,16328,16330],{"className":16329,"ariaHidden":65},[64],[34,16331,16333,16336],{"className":16332},[69],[34,16334],{"className":16335,"style":2184},[73],[34,16337,13527],{"className":16338,"style":13594},[78,79]," be the angle between the vectors:",[34,16341,16343],{"className":16342,"translate":38},[159],[34,16344,16346,16375],{"className":16345,"translate":38},[37],[34,16347,16349],{"className":16348},[42],[44,16350,16351],{"xmlns":46,"display":168},[48,16352,16353,16372],{},[51,16354,16355,16358,16360,16362,16364,16370],{},[54,16356,16357],{},"cos",[176,16359,6321],{},[54,16361,13527],{},[176,16363,192],{},[755,16365,16366,16368],{},[54,16367,11720],{},[54,16369,13552],{},[54,16371,96],{},[58,16373,16374],{"encoding":60},"\\cos\\theta = \\mu^T x",[34,16376,16378,16402],{"className":16377,"ariaHidden":65},[64],[34,16379,16381,16384,16387,16390,16393,16396,16399],{"className":16380},[69],[34,16382],{"className":16383,"style":2184},[73],[34,16385,16357],{"className":16386},[3387],[34,16388],{"className":16389,"style":3462},[237],[34,16391,13527],{"className":16392,"style":13594},[78,79],[34,16394],{"className":16395,"style":262},[237],[34,16397,192],{"className":16398},[266],[34,16400],{"className":16401,"style":262},[237],[34,16403,16405,16409,16438],{"className":16404},[69],[34,16406],{"className":16407,"style":16408},[73],"height:1.0858em;vertical-align:-0.1944em;",[34,16410,16412,16415],{"className":16411},[78],[34,16413,11720],{"className":16414},[78,79],[34,16416,16418],{"className":16417},[810],[34,16419,16421],{"className":16420},[814],[34,16422,16424],{"className":16423},[818],[34,16425,16427],{"className":16426,"style":13653},[822],[34,16428,16429,16432],{"style":4710},[34,16430],{"className":16431,"style":830},[829],[34,16433,16435],{"className":16434},[834,835,836,837],[34,16436,13552],{"className":16437,"style":6248},[78,79,837],[34,16439,96],{"className":16440},[78,79],[16,16442,16443],{},"A confidence region can be written as",[34,16445,16447],{"className":16446,"translate":38},[159],[34,16448,16450,16473],{"className":16449,"translate":38},[37],[34,16451,16453],{"className":16452},[42],[44,16454,16455],{"xmlns":46,"display":168},[48,16456,16457,16470],{},[51,16458,16459,16461,16464],{},[54,16460,13527],{},[176,16462,16463],{},"≤",[2468,16465,16466,16468],{},[54,16467,13527],{},[54,16469,1173],{},[58,16471,16472],{"encoding":60},"\\theta \\le \\theta_c",[34,16474,16476,16495],{"className":16475,"ariaHidden":65},[64],[34,16477,16479,16483,16486,16489,16492],{"className":16478},[69],[34,16480],{"className":16481,"style":16482},[73],"height:0.8304em;vertical-align:-0.136em;",[34,16484,13527],{"className":16485,"style":13594},[78,79],[34,16487],{"className":16488,"style":262},[237],[34,16490,16463],{"className":16491},[266],[34,16493],{"className":16494,"style":262},[237],[34,16496,16498,16501],{"className":16497},[69],[34,16499],{"className":16500,"style":3736},[73],[34,16502,16504,16507],{"className":16503},[78],[34,16505,13527],{"className":16506,"style":13594},[78,79],[34,16508,16510],{"className":16509},[810],[34,16511,16513,16534],{"className":16512},[814,2523],[34,16514,16516,16531],{"className":16515},[818],[34,16517,16519],{"className":16518,"style":5870},[822],[34,16520,16522,16525],{"style":16521},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[34,16523],{"className":16524,"style":830},[829],[34,16526,16528],{"className":16527},[834,835,836,837],[34,16529,1173],{"className":16530},[78,79,837],[34,16532,2547],{"className":16533},[2546],[34,16535,16537],{"className":16536},[818],[34,16538,16540],{"className":16539,"style":2554},[822],[34,16541],{},[16,16543,16544,16545,144],{},"i.e., a spherical cap with axis ",[34,16546,16548,16561],{"className":16547,"translate":38},[37],[34,16549,16551],{"className":16550},[42],[44,16552,16553],{"xmlns":46},[48,16554,16555,16559],{},[51,16556,16557],{},[54,16558,11720],{},[58,16560,11723],{"encoding":60},[34,16562,16564],{"className":16563,"ariaHidden":65},[64],[34,16565,16567,16570],{"className":16566},[69],[34,16568],{"className":16569,"style":139},[73],[34,16571,11720],{"className":16572},[78,79],[16,16574,16575,16576,5228],{},"On the unit sphere ",[34,16577,16579,16603],{"className":16578,"translate":38},[37],[34,16580,16582],{"className":16581},[42],[44,16583,16584],{"xmlns":46},[48,16585,16586,16600],{},[51,16587,16588],{},[755,16589,16590,16592],{},[54,16591,11890],{"mathvariant":2996},[51,16593,16594,16596,16598],{},[54,16595,5250],{},[176,16597,6463],{},[347,16599,2213],{},[58,16601,16602],{"encoding":60},"\\mathbb{S}^{d-1}",[34,16604,16606],{"className":16605,"ariaHidden":65},[64],[34,16607,16609,16612],{"className":16608},[69],[34,16610],{"className":16611,"style":12169},[73],[34,16613,16615,16618],{"className":16614},[78],[34,16616,11890],{"className":16617},[78,3048],[34,16619,16621],{"className":16620},[810],[34,16622,16624],{"className":16623},[814],[34,16625,16627],{"className":16626},[818],[34,16628,16630],{"className":16629,"style":12169},[822],[34,16631,16632,16635],{"style":825},[34,16633],{"className":16634,"style":830},[829],[34,16636,16638],{"className":16637},[834,835,836,837],[34,16639,16641,16644,16647],{"className":16640},[78,837],[34,16642,5250],{"className":16643},[78,79,837],[34,16645,6463],{"className":16646},[242,837],[34,16648,2213],{"className":16649},[78,837],[16,16651,16652],{},"the area of a spherical cap is:",[34,16654,16656],{"className":16655,"translate":38},[159],[34,16657,16659,16746],{"className":16658,"translate":38},[37],[34,16660,16662],{"className":16661},[42],[44,16663,16664],{"xmlns":46,"display":168},[48,16665,16666,16743],{},[51,16667,16668,16670,16672,16674,16676,16678,16684,16696,16698,16717],{},[54,16669,3145],{},[176,16671,179],{"stretchy":178},[54,16673,13527],{},[176,16675,189],{"stretchy":178},[176,16677,192],{},[12543,16679,16680,16682],{},[347,16681,2213],{},[347,16683,424],{},[2468,16685,16686,16688],{},[54,16687,3145],{},[51,16689,16690,16692,16694],{},[54,16691,5250],{},[176,16693,6463],{},[347,16695,2213],{},[1201,16697,5837],{},[2468,16699,16700,16702],{},[54,16701,12587],{},[51,16703,16704,16715],{},[755,16705,16706,16713],{},[51,16707,16708,16711],{},[54,16709,16710],{},"sin",[176,16712,6321],{},[347,16714,424],{},[54,16716,13527],{},[51,16718,16719,16721,16733,16735,16741],{},[176,16720,179],{"fence":65},[12543,16722,16723,16731],{},[51,16724,16725,16727,16729],{},[54,16726,5250],{},[176,16728,6463],{},[347,16730,2213],{},[347,16732,424],{},[176,16734,5228],{"separator":65},[12543,16736,16737,16739],{},[347,16738,2213],{},[347,16740,424],{},[176,16742,189],{"fence":65},[58,16744,16745],{"encoding":60},"A(\\theta) = \\frac{1}{2} A_{d-1} \\, I_{\\sin^2\\theta}\\left(\\frac{d-1}{2},\\frac12\\right)",[34,16747,16749,16776],{"className":16748,"ariaHidden":65},[64],[34,16750,16752,16755,16758,16761,16764,16767,16770,16773],{"className":16751},[69],[34,16753],{"className":16754,"style":223},[73],[34,16756,3145],{"className":16757},[78,79],[34,16759,179],{"className":16760},[230],[34,16762,13527],{"className":16763,"style":13594},[78,79],[34,16765,189],{"className":16766},[258],[34,16768],{"className":16769,"style":262},[237],[34,16771,192],{"className":16772},[266],[34,16774],{"className":16775,"style":262},[237],[34,16777,16779,16782,16845,16894,16897,16987,16990],{"className":16778},[69],[34,16780],{"className":16781,"style":14693},[73],[34,16783,16785,16788,16842],{"className":16784},[78],[34,16786],{"className":16787},[230,12689],[34,16789,16791],{"className":16790},[12543],[34,16792,16794,16834],{"className":16793},[814,2523],[34,16795,16797,16831],{"className":16796},[818],[34,16798,16801,16812,16820],{"className":16799,"style":16800},[822],"height:1.3214em;",[34,16802,16803,16806],{"style":14155},[34,16804],{"className":16805,"style":12709},[829],[34,16807,16809],{"className":16808},[78],[34,16810,424],{"className":16811},[78],[34,16813,16814,16817],{"style":12897},[34,16815],{"className":16816,"style":12709},[829],[34,16818],{"className":16819,"style":12904},[12836],[34,16821,16822,16825],{"style":12907},[34,16823],{"className":16824,"style":12709},[829],[34,16826,16828],{"className":16827},[78],[34,16829,2213],{"className":16830},[78],[34,16832,2547],{"className":16833},[2546],[34,16835,16837],{"className":16836},[818],[34,16838,16840],{"className":16839,"style":14669},[822],[34,16841],{},[34,16843],{"className":16844},[258,12689],[34,16846,16848,16851],{"className":16847},[78],[34,16849,3145],{"className":16850},[78,79],[34,16852,16854],{"className":16853},[810],[34,16855,16857,16886],{"className":16856},[814,2523],[34,16858,16860,16883],{"className":16859},[818],[34,16861,16863],{"className":16862,"style":6071},[822],[34,16864,16865,16868],{"style":2533},[34,16866],{"className":16867,"style":830},[829],[34,16869,16871],{"className":16870},[834,835,836,837],[34,16872,16874,16877,16880],{"className":16873},[78,837],[34,16875,5250],{"className":16876},[78,79,837],[34,16878,6463],{"className":16879},[242,837],[34,16881,2213],{"className":16882},[78,837],[34,16884,2547],{"className":16885},[2546],[34,16887,16889],{"className":16888},[818],[34,16890,16892],{"className":16891,"style":6830},[822],[34,16893],{},[34,16895],{"className":16896,"style":3462},[237],[34,16898,16900,16903],{"className":16899},[78],[34,16901,12587],{"className":16902,"style":12766},[78,79],[34,16904,16906],{"className":16905},[810],[34,16907,16909,16978],{"className":16908},[814,2523],[34,16910,16912,16975],{"className":16911},[818],[34,16913,16915],{"className":16914,"style":12779},[822],[34,16916,16918,16921],{"style":16917},"top:-2.5021em;margin-left:-0.0785em;margin-right:0.05em;",[34,16919],{"className":16920,"style":830},[829],[34,16922,16924],{"className":16923},[834,835,836,837],[34,16925,16927,16968,16972],{"className":16926},[78,837],[34,16928,16930,16943],{"className":16929},[3387,837],[34,16931,16933,16937,16940],{"className":16932},[3387,837],[34,16934,16936],{"className":16935},[837],"s",[34,16938,3317],{"className":16939},[837],[34,16941,3000],{"className":16942},[837],[34,16944,16946],{"className":16945},[810],[34,16947,16949],{"className":16948},[814],[34,16950,16952],{"className":16951},[818],[34,16953,16956],{"className":16954,"style":16955},[822],"height:0.7753em;",[34,16957,16959,16962],{"style":16958},"top:-2.815em;margin-right:0.0714em;",[34,16960],{"className":16961,"style":10772},[829],[34,16963,16965],{"className":16964},[834,10776,9740,837],[34,16966,424],{"className":16967},[78,837],[34,16969],{"className":16970,"style":16971},[237,837],"margin-right:0.1952em;",[34,16973,13527],{"className":16974,"style":13594},[78,79,837],[34,16976,2547],{"className":16977},[2546],[34,16979,16981],{"className":16980},[818],[34,16982,16985],{"className":16983,"style":16984},[822],"height:0.1979em;",[34,16986],{},[34,16988],{"className":16989,"style":3462},[237],[34,16991,16993,16999,17073,17076,17079,17141],{"className":16992},[4177],[34,16994,16996],{"className":16995,"style":4182},[230,4181],[34,16997,179],{"className":16998},[4186,836],[34,17000,17002,17005,17070],{"className":17001},[78],[34,17003],{"className":17004},[230,12689],[34,17006,17008],{"className":17007},[12543],[34,17009,17011,17062],{"className":17010},[814,2523],[34,17012,17014,17059],{"className":17013},[818],[34,17015,17017,17028,17036],{"className":17016,"style":15668},[822],[34,17018,17019,17022],{"style":14155},[34,17020],{"className":17021,"style":12709},[829],[34,17023,17025],{"className":17024},[78],[34,17026,424],{"className":17027},[78],[34,17029,17030,17033],{"style":12897},[34,17031],{"className":17032,"style":12709},[829],[34,17034],{"className":17035,"style":12904},[12836],[34,17037,17038,17041],{"style":12907},[34,17039],{"className":17040,"style":12709},[829],[34,17042,17044,17047,17050,17053,17056],{"className":17043},[78],[34,17045,5250],{"className":17046},[78,79],[34,17048],{"className":17049,"style":238},[237],[34,17051,6463],{"className":17052},[242],[34,17054],{"className":17055,"style":238},[237],[34,17057,2213],{"className":17058},[78],[34,17060,2547],{"className":17061},[2546],[34,17063,17065],{"className":17064},[818],[34,17066,17068],{"className":17067,"style":14669},[822],[34,17069],{},[34,17071],{"className":17072},[258,12689],[34,17074,5228],{"className":17075},[5317],[34,17077],{"className":17078,"style":3462},[237],[34,17080,17082,17085,17138],{"className":17081},[78],[34,17083],{"className":17084},[230,12689],[34,17086,17088],{"className":17087},[12543],[34,17089,17091,17130],{"className":17090},[814,2523],[34,17092,17094,17127],{"className":17093},[818],[34,17095,17097,17108,17116],{"className":17096,"style":16800},[822],[34,17098,17099,17102],{"style":14155},[34,17100],{"className":17101,"style":12709},[829],[34,17103,17105],{"className":17104},[78],[34,17106,424],{"className":17107},[78],[34,17109,17110,17113],{"style":12897},[34,17111],{"className":17112,"style":12709},[829],[34,17114],{"className":17115,"style":12904},[12836],[34,17117,17118,17121],{"style":12907},[34,17119],{"className":17120,"style":12709},[829],[34,17122,17124],{"className":17123},[78],[34,17125,2213],{"className":17126},[78],[34,17128,2547],{"className":17129},[2546],[34,17131,17133],{"className":17132},[818],[34,17134,17136],{"className":17135,"style":14669},[822],[34,17137],{},[34,17139],{"className":17140},[258,12689],[34,17142,17144],{"className":17143,"style":4182},[258,4181],[34,17145,189],{"className":17146},[4186,836],[16,17148,3218,17149,17235],{},[34,17150,17152,17176],{"className":17151,"translate":38},[37],[34,17153,17155],{"className":17154},[42],[44,17156,17157],{"xmlns":46},[48,17158,17159,17173],{},[51,17160,17161],{},[2468,17162,17163,17165],{},[54,17164,3145],{},[51,17166,17167,17169,17171],{},[54,17168,5250],{},[176,17170,6463],{},[347,17172,2213],{},[58,17174,17175],{"encoding":60},"A_{d-1}",[34,17177,17179],{"className":17178,"ariaHidden":65},[64],[34,17180,17182,17186],{"className":17181},[69],[34,17183],{"className":17184,"style":17185},[73],"height:0.8917em;vertical-align:-0.2083em;",[34,17187,17189,17192],{"className":17188},[78],[34,17190,3145],{"className":17191},[78,79],[34,17193,17195],{"className":17194},[810],[34,17196,17198,17227],{"className":17197},[814,2523],[34,17199,17201,17224],{"className":17200},[818],[34,17202,17204],{"className":17203,"style":6071},[822],[34,17205,17206,17209],{"style":2533},[34,17207],{"className":17208,"style":830},[829],[34,17210,17212],{"className":17211},[834,835,836,837],[34,17213,17215,17218,17221],{"className":17214},[78,837],[34,17216,5250],{"className":17217},[78,79,837],[34,17219,6463],{"className":17220},[242,837],[34,17222,2213],{"className":17223},[78,837],[34,17225,2547],{"className":17226},[2546],[34,17228,17230],{"className":17229},[818],[34,17231,17233],{"className":17232,"style":6830},[822],[34,17234],{}," is the total surface area of the sphere, given by:",[34,17237,17239],{"className":17238,"translate":38},[159],[34,17240,17242,17301],{"className":17241,"translate":38},[37],[34,17243,17245],{"className":17244},[42],[44,17246,17247],{"xmlns":46,"display":168},[48,17248,17249,17298],{},[51,17250,17251,17263,17265],{},[2468,17252,17253,17255],{},[54,17254,3145],{},[51,17256,17257,17259,17261],{},[54,17258,5250],{},[176,17260,6463],{},[347,17262,2213],{},[176,17264,192],{},[12543,17266,17267,17283],{},[51,17268,17269,17271],{},[347,17270,424],{},[755,17272,17273,17275],{},[54,17274,12569],{},[51,17276,17277,17279,17281],{},[54,17278,5250],{},[54,17280,12580],{"mathvariant":2296},[347,17282,424],{},[51,17284,17285,17288,17290,17292,17294,17296],{},[54,17286,17287],{"mathvariant":2296},"Γ",[176,17289,179],{"stretchy":178},[54,17291,5250],{},[54,17293,12580],{"mathvariant":2296},[347,17295,424],{},[176,17297,189],{"stretchy":178},[58,17299,17300],{"encoding":60},"A_{d-1} = \\frac{2\\pi^{d\u002F2}}{\\Gamma(d\u002F2)}",[34,17302,17304,17368],{"className":17303,"ariaHidden":65},[64],[34,17305,17307,17310,17359,17362,17365],{"className":17306},[69],[34,17308],{"className":17309,"style":17185},[73],[34,17311,17313,17316],{"className":17312},[78],[34,17314,3145],{"className":17315},[78,79],[34,17317,17319],{"className":17318},[810],[34,17320,17322,17351],{"className":17321},[814,2523],[34,17323,17325,17348],{"className":17324},[818],[34,17326,17328],{"className":17327,"style":6071},[822],[34,17329,17330,17333],{"style":2533},[34,17331],{"className":17332,"style":830},[829],[34,17334,17336],{"className":17335},[834,835,836,837],[34,17337,17339,17342,17345],{"className":17338},[78,837],[34,17340,5250],{"className":17341},[78,79,837],[34,17343,6463],{"className":17344},[242,837],[34,17346,2213],{"className":17347},[78,837],[34,17349,2547],{"className":17350},[2546],[34,17352,17354],{"className":17353},[818],[34,17355,17357],{"className":17356,"style":6830},[822],[34,17358],{},[34,17360],{"className":17361,"style":262},[237],[34,17363,192],{"className":17364},[266],[34,17366],{"className":17367,"style":262},[237],[34,17369,17371,17375],{"className":17370},[69],[34,17372],{"className":17373,"style":17374},[73],"height:2.501em;vertical-align:-0.936em;",[34,17376,17378,17381,17484],{"className":17377},[78],[34,17379],{"className":17380},[230,12689],[34,17382,17384],{"className":17383},[12543],[34,17385,17387,17475],{"className":17386},[814,2523],[34,17388,17390,17472],{"className":17389},[818],[34,17391,17394,17417,17425],{"className":17392,"style":17393},[822],"height:1.565em;",[34,17395,17396,17399],{"style":14155},[34,17397],{"className":17398,"style":12709},[829],[34,17400,17402,17405,17408,17411,17414],{"className":17401},[78],[34,17403,17287],{"className":17404},[78],[34,17406,179],{"className":17407},[230],[34,17409,5250],{"className":17410},[78,79],[34,17412,12759],{"className":17413},[78],[34,17415,189],{"className":17416},[258],[34,17418,17419,17422],{"style":12897},[34,17420],{"className":17421,"style":12709},[829],[34,17423],{"className":17424,"style":12904},[12836],[34,17426,17427,17430],{"style":12907},[34,17428],{"className":17429,"style":12709},[829],[34,17431,17433,17436],{"className":17432},[78],[34,17434,424],{"className":17435},[78],[34,17437,17439,17442],{"className":17438},[78],[34,17440,12569],{"className":17441,"style":143},[78,79],[34,17443,17445],{"className":17444},[810],[34,17446,17448],{"className":17447},[814],[34,17449,17451],{"className":17450},[818],[34,17452,17455],{"className":17453,"style":17454},[822],"height:0.888em;",[34,17456,17457,17460],{"style":825},[34,17458],{"className":17459,"style":830},[829],[34,17461,17463],{"className":17462},[834,835,836,837],[34,17464,17466,17469],{"className":17465},[78,837],[34,17467,5250],{"className":17468},[78,79,837],[34,17470,12759],{"className":17471},[78,837],[34,17473,2547],{"className":17474},[2546],[34,17476,17478],{"className":17477},[818],[34,17479,17482],{"className":17480,"style":17481},[822],"height:0.936em;",[34,17483],{},[34,17485],{"className":17486},[258,12689],[16,17488,13036,17489,17587],{},[34,17490,17492,17520],{"className":17491,"translate":38},[37],[34,17493,17495],{"className":17494},[42],[44,17496,17497],{"xmlns":46},[48,17498,17499,17517],{},[51,17500,17501,17507,17509,17511,17513,17515],{},[2468,17502,17503,17505],{},[54,17504,12587],{},[54,17506,96],{},[176,17508,179],{"stretchy":178},[54,17510,11468],{},[176,17512,5228],{"separator":65},[54,17514,2172],{},[176,17516,189],{"stretchy":178},[58,17518,17519],{"encoding":60},"I_x(a,b)",[34,17521,17523],{"className":17522,"ariaHidden":65},[64],[34,17524,17526,17529,17569,17572,17575,17578,17581,17584],{"className":17525},[69],[34,17527],{"className":17528,"style":223},[73],[34,17530,17532,17535],{"className":17531},[78],[34,17533,12587],{"className":17534,"style":12766},[78,79],[34,17536,17538],{"className":17537},[810],[34,17539,17541,17561],{"className":17540},[814,2523],[34,17542,17544,17558],{"className":17543},[818],[34,17545,17547],{"className":17546,"style":5870},[822],[34,17548,17549,17552],{"style":13088},[34,17550],{"className":17551,"style":830},[829],[34,17553,17555],{"className":17554},[834,835,836,837],[34,17556,96],{"className":17557},[78,79,837],[34,17559,2547],{"className":17560},[2546],[34,17562,17564],{"className":17563},[818],[34,17565,17567],{"className":17566,"style":2554},[822],[34,17568],{},[34,17570,179],{"className":17571},[230],[34,17573,11468],{"className":17574},[78,79],[34,17576,5228],{"className":17577},[5317],[34,17579],{"className":17580,"style":3462},[237],[34,17582,2172],{"className":17583},[78,79],[34,17585,189],{"className":17586},[258]," is the regularized incomplete beta function.",[16,17589,17590,17591,2991],{},"For ",[34,17592,17594,17612],{"className":17593,"translate":38},[37],[34,17595,17597],{"className":17596},[42],[44,17598,17599],{"xmlns":46},[48,17600,17601,17609],{},[51,17602,17603,17605,17607],{},[54,17604,5250],{},[176,17606,192],{},[347,17608,429],{},[58,17610,17611],{"encoding":60},"d=3",[34,17613,17615,17633],{"className":17614,"ariaHidden":65},[64],[34,17616,17618,17621,17624,17627,17630],{"className":17617},[69],[34,17619],{"className":17620,"style":2184},[73],[34,17622,5250],{"className":17623},[78,79],[34,17625],{"className":17626,"style":262},[237],[34,17628,192],{"className":17629},[266],[34,17631],{"className":17632,"style":262},[237],[34,17634,17636,17639],{"className":17635},[69],[34,17637],{"className":17638,"style":391},[73],[34,17640,429],{"className":17641},[78],[16,17643,17644],{},"Cap area:",[34,17646,17648],{"className":17647,"translate":38},[159],[34,17649,17651,17691],{"className":17650,"translate":38},[37],[34,17652,17654],{"className":17653},[42],[44,17655,17656],{"xmlns":46,"display":168},[48,17657,17658,17688],{},[51,17659,17660,17662,17664,17666,17668,17670,17672,17674,17676,17678,17680,17682,17684,17686],{},[54,17661,3145],{},[176,17663,179],{"stretchy":178},[54,17665,13527],{},[176,17667,189],{"stretchy":178},[176,17669,192],{},[347,17671,424],{},[54,17673,12569],{},[176,17675,179],{"stretchy":178},[347,17677,2213],{},[176,17679,6463],{},[54,17681,16357],{},[176,17683,6321],{},[54,17685,13527],{},[176,17687,189],{"stretchy":178},[58,17689,17690],{"encoding":60},"A(\\theta) = 2\\pi (1-\\cos\\theta)",[34,17692,17694,17721,17748],{"className":17693,"ariaHidden":65},[64],[34,17695,17697,17700,17703,17706,17709,17712,17715,17718],{"className":17696},[69],[34,17698],{"className":17699,"style":223},[73],[34,17701,3145],{"className":17702},[78,79],[34,17704,179],{"className":17705},[230],[34,17707,13527],{"className":17708,"style":13594},[78,79],[34,17710,189],{"className":17711},[258],[34,17713],{"className":17714,"style":262},[237],[34,17716,192],{"className":17717},[266],[34,17719],{"className":17720,"style":262},[237],[34,17722,17724,17727,17730,17733,17736,17739,17742,17745],{"className":17723},[69],[34,17725],{"className":17726,"style":223},[73],[34,17728,424],{"className":17729},[78],[34,17731,12569],{"className":17732,"style":143},[78,79],[34,17734,179],{"className":17735},[230],[34,17737,2213],{"className":17738},[78],[34,17740],{"className":17741,"style":238},[237],[34,17743,6463],{"className":17744},[242],[34,17746],{"className":17747,"style":238},[237],[34,17749,17751,17754,17757,17760,17763],{"className":17750},[69],[34,17752],{"className":17753,"style":223},[73],[34,17755,16357],{"className":17756},[3387],[34,17758],{"className":17759,"style":3462},[237],[34,17761,13527],{"className":17762,"style":13594},[78,79],[34,17764,189],{"className":17765},[258],[16,17767,17768,17769],{},"Total sphere area: ",[34,17770,17772,17788],{"className":17771,"translate":38},[37],[34,17773,17775],{"className":17774},[42],[44,17776,17777],{"xmlns":46},[48,17778,17779,17785],{},[51,17780,17781,17783],{},[347,17782,990],{},[54,17784,12569],{},[58,17786,17787],{"encoding":60},"4\\pi",[34,17789,17791],{"className":17790,"ariaHidden":65},[64],[34,17792,17794,17797,17800],{"className":17793},[69],[34,17795],{"className":17796,"style":391},[73],[34,17798,990],{"className":17799},[78],[34,17801,12569],{"className":17802,"style":143},[78,79],[16,17804,17805,17806,144],{},"Area proportion: ",[34,17807,17809,17848],{"className":17808,"translate":38},[37],[34,17810,17812],{"className":17811},[42],[44,17813,17814],{"xmlns":46},[48,17815,17816,17845],{},[51,17817,17818,17821,17823,17825,17827,17829],{},[54,17819,17820],{},"P",[176,17822,179],{"stretchy":178},[54,17824,13527],{},[176,17826,189],{"stretchy":178},[176,17828,192],{},[12543,17830,17831,17843],{},[51,17832,17833,17835,17837,17839,17841],{},[347,17834,2213],{},[176,17836,6463],{},[54,17838,16357],{},[176,17840,6321],{},[54,17842,13527],{},[347,17844,424],{},[58,17846,17847],{"encoding":60},"P(\\theta) = \\frac{1-\\cos\\theta}{2}",[34,17849,17851,17878],{"className":17850,"ariaHidden":65},[64],[34,17852,17854,17857,17860,17863,17866,17869,17872,17875],{"className":17853},[69],[34,17855],{"className":17856,"style":223},[73],[34,17858,17820],{"className":17859,"style":6248},[78,79],[34,17861,179],{"className":17862},[230],[34,17864,13527],{"className":17865,"style":13594},[78,79],[34,17867,189],{"className":17868},[258],[34,17870],{"className":17871,"style":262},[237],[34,17873,192],{"className":17874},[266],[34,17876],{"className":17877,"style":262},[237],[34,17879,17881,17885],{"className":17880},[69],[34,17882],{"className":17883,"style":17884},[73],"height:1.2251em;vertical-align:-0.345em;",[34,17886,17888,17891,17976],{"className":17887},[78],[34,17889],{"className":17890},[230,12689],[34,17892,17894],{"className":17893},[12543],[34,17895,17897,17967],{"className":17896},[814,2523],[34,17898,17900,17964],{"className":17899},[818],[34,17901,17904,17919,17927],{"className":17902,"style":17903},[822],"height:0.8801em;",[34,17905,17907,17910],{"style":17906},"top:-2.655em;",[34,17908],{"className":17909,"style":12709},[829],[34,17911,17913],{"className":17912},[834,835,836,837],[34,17914,17916],{"className":17915},[78,837],[34,17917,424],{"className":17918},[78,837],[34,17920,17921,17924],{"style":12897},[34,17922],{"className":17923,"style":12709},[829],[34,17925],{"className":17926,"style":12904},[12836],[34,17928,17930,17933],{"style":17929},"top:-3.394em;",[34,17931],{"className":17932,"style":12709},[829],[34,17934,17936],{"className":17935},[834,835,836,837],[34,17937,17939,17942,17945,17958,17961],{"className":17938},[78,837],[34,17940,2213],{"className":17941},[78,837],[34,17943,6463],{"className":17944},[242,837],[34,17946,17948,17951,17955],{"className":17947},[3387,837],[34,17949,1173],{"className":17950},[837],[34,17952,17954],{"className":17953},[837],"o",[34,17956,16936],{"className":17957},[837],[34,17959],{"className":17960,"style":16971},[237,837],[34,17962,13527],{"className":17963,"style":13594},[78,79,837],[34,17965,2547],{"className":17966},[2546],[34,17968,17970],{"className":17969},[818],[34,17971,17974],{"className":17972,"style":17973},[822],"height:0.345em;",[34,17975],{},[34,17977],{"className":17978},[258,12689],[16,17980,17981],{},"In high dimensions, the probability within a spherical cap for a vMF distribution is:",[34,17983,17985],{"className":17984,"translate":38},[159],[34,17986,17988,18126],{"className":17987,"translate":38},[37],[34,17989,17991],{"className":17990},[42],[44,17992,17993],{"xmlns":46,"display":168},[48,17994,17995,18123],{},[51,17996,17997,17999,18001,18003,18005,18011,18013,18015],{},[54,17998,17820],{},[176,18000,179],{"stretchy":178},[54,18002,13527],{},[176,18004,16463],{},[2468,18006,18007,18009],{},[54,18008,13527],{},[54,18010,1173],{},[176,18012,189],{"stretchy":178},[176,18014,192],{},[12543,18016,18017,18073],{},[51,18018,18019,18032,18047,18049,18051,18053,18055,18067,18069,18071],{},[2475,18020,18021,18024,18026],{},[176,18022,18023],{},"∫",[347,18025,2287],{},[2468,18027,18028,18030],{},[54,18029,13527],{},[54,18031,1173],{},[755,18033,18034,18037],{},[54,18035,18036],{},"e",[51,18038,18039,18041,18043,18045],{},[54,18040,12226],{},[54,18042,16357],{},[176,18044,6321],{},[54,18046,13527],{},[176,18048,179],{"stretchy":178},[54,18050,16710],{},[176,18052,6321],{},[54,18054,13527],{},[755,18056,18057,18059],{},[176,18058,189],{"stretchy":178},[51,18060,18061,18063,18065],{},[54,18062,5250],{},[176,18064,6463],{},[347,18066,424],{},[1201,18068,5837],{},[54,18070,5250],{},[54,18072,13527],{},[51,18074,18075,18083,18097,18099,18101,18103,18105,18117,18119,18121],{},[2475,18076,18077,18079,18081],{},[176,18078,18023],{},[347,18080,2287],{},[54,18082,12569],{},[755,18084,18085,18087],{},[54,18086,18036],{},[51,18088,18089,18091,18093,18095],{},[54,18090,12226],{},[54,18092,16357],{},[176,18094,6321],{},[54,18096,13527],{},[176,18098,179],{"stretchy":178},[54,18100,16710],{},[176,18102,6321],{},[54,18104,13527],{},[755,18106,18107,18109],{},[176,18108,189],{"stretchy":178},[51,18110,18111,18113,18115],{},[54,18112,5250],{},[176,18114,6463],{},[347,18116,424],{},[1201,18118,5837],{},[54,18120,5250],{},[54,18122,13527],{},[58,18124,18125],{"encoding":60},"P(\\theta \\le \\theta_c) = \\frac{ \\int_0^{\\theta_c} e^{\\kappa\\cos\\theta} (\\sin\\theta)^{d-2} \\,d\\theta }{ \\int_0^{\\pi} e^{\\kappa\\cos\\theta} (\\sin\\theta)^{d-2} \\,d\\theta 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is no simple closed-form solution.",[16,18669,18670],{},"Approximations are commonly used.",[16,18672,18673,18674,18702,18703,18731],{},"When ",[34,18675,18677,18690],{"className":18676,"translate":38},[37],[34,18678,18680],{"className":18679},[42],[44,18681,18682],{"xmlns":46},[48,18683,18684,18688],{},[51,18685,18686],{},[54,18687,12226],{},[58,18689,13280],{"encoding":60},[34,18691,18693],{"className":18692,"ariaHidden":65},[64],[34,18694,18696,18699],{"className":18695},[69],[34,18697],{"className":18698,"style":108},[73],[34,18700,12226],{"className":18701},[78,79]," is large, the distribution is concentrated around 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Using",[34,18733,18735],{"className":18734,"translate":38},[159],[34,18736,18738,18773],{"className":18737,"translate":38},[37],[34,18739,18741],{"className":18740},[42],[44,18742,18743],{"xmlns":46,"display":168},[48,18744,18745,18770],{},[51,18746,18747,18749,18751,18753,18756,18758,18760],{},[54,18748,16357],{},[176,18750,6321],{},[54,18752,13527],{},[176,18754,18755],{},"≈",[347,18757,2213],{},[176,18759,6463],{},[12543,18761,18762,18768],{},[755,18763,18764,18766],{},[54,18765,13527],{},[347,18767,424],{},[347,18769,424],{},[58,18771,18772],{"encoding":60},"\\cos\\theta \\approx 1 - \\frac{\\theta^2}{2}",[34,18774,18776,18800,18818],{"className":18775,"ariaHidden":65},[64],[34,18777,18779,18782,18785,18788,18791,18794,18797],{"className":18778},[69],[34,18780],{"className":18781,"style":2184},[73],[34,18783,16357],{"className":18784},[3387],[34,18786],{"className":18787,"style":3462},[237],[34,18789,13527],{"className":18790,"style":13594},[78,79],[34,18792],{"className":18793,"style":262},[237],[34,18795,18755],{"className":18796},[266],[34,18798],{"className":18799,"style":262},[237],[34,18801,18803,18806,18809,18812,18815],{"className":18802},[69],[34,18804],{"className":18805,"style":511},[73],[34,18807,2213],{"className":18808},[78],[34,18810],{"className":18811,"style":238},[237],[34,18813,6463],{"className":18814},[242],[34,18816],{"className":18817,"style":238},[237],[34,18819,18821,18824],{"className":18820},[69],[34,18822],{"className":18823,"style":15206},[73],[34,18825,18827,18830,18909],{"className":18826},[78],[34,18828],{"className":18829},[230,12689],[34,18831,18833],{"className":18832},[12543],[34,18834,18836,18901],{"className":18835},[814,2523],[34,18837,18839,18898],{"className":18838},[818],[34,18840,18842,18853,18861],{"className":18841,"style":15228},[822],[34,18843,18844,18847],{"style":14155},[34,18845],{"className":18846,"style":12709},[829],[34,18848,18850],{"className":18849},[78],[34,18851,424],{"className":18852},[78],[34,18854,18855,18858],{"style":12897},[34,18856],{"className":18857,"style":12709},[829],[34,18859],{"className":18860,"style":12904},[12836],[34,18862,18863,18866],{"style":12907},[34,18864],{"className":18865,"style":12709},[829],[34,18867,18869],{"className":18868},[78],[34,18870,18872,18875],{"className":18871},[78],[34,18873,13527],{"className":18874,"style":13594},[78,79],[34,18876,18878],{"className":18877},[810],[34,18879,18881],{"className":18880},[814],[34,18882,18884],{"className":18883},[818],[34,18885,18887],{"className":18886,"style":800},[822],[34,18888,18889,18892],{"style":825},[34,18890],{"className":18891,"style":830},[829],[34,18893,18895],{"className":18894},[834,835,836,837],[34,18896,424],{"className":18897},[78,837],[34,18899,2547],{"className":18900},[2546],[34,18902,18904],{"className":18903},[818],[34,18905,18907],{"className":18906,"style":14669},[822],[34,18908],{},[34,18910],{"className":18911},[258,12689],[16,18913,18914],{},"we obtain",[34,18916,18918],{"className":18917,"translate":38},[159],[34,18919,18921,18975],{"className":18920,"translate":38},[37],[34,18922,18924],{"className":18923},[42],[44,18925,18926],{"xmlns":46,"display":168},[48,18927,18928,18972],{},[51,18929,18930,18944,18946,18952],{},[755,18931,18932,18934],{},[54,18933,18036],{},[51,18935,18936,18938,18940,18942],{},[54,18937,12226],{},[54,18939,16357],{},[176,18941,6321],{},[54,18943,13527],{},[176,18945,18755],{},[755,18947,18948,18950],{},[54,18949,18036],{},[54,18951,12226],{},[755,18953,18954,18956],{},[54,18955,18036],{},[51,18957,18958,18960,18962,18968,18970],{},[176,18959,6463],{},[54,18961,12226],{},[755,18963,18964,18966],{},[54,18965,13527],{},[347,18967,424],{},[54,18969,12580],{"mathvariant":2296},[347,18971,424],{},[58,18973,18974],{"encoding":60},"e^{\\kappa\\cos\\theta} \\approx e^{\\kappa} e^{-\\kappa\\theta^2\u002F2}",[34,18976,18978,19046],{"className":18977,"ariaHidden":65},[64],[34,18979,18981,18984,19037,19040,19043],{"className":18980},[69],[34,18982],{"className":18983,"style":11944},[73],[34,18985,18987,18990],{"className":18986},[78],[34,18988,18036],{"className":18989},[78,79],[34,18991,18993],{"className":18992},[810],[34,18994,18996],{"className":18995},[814],[34,18997,18999],{"className":18998},[818],[34,19000,19002],{"className":19001,"style":11944},[822],[34,19003,19004,19007],{"style":4710},[34,19005],{"className":19006,"style":830},[829],[34,19008,19010],{"className":19009},[834,835,836,837],[34,19011,19013,19016,19019,19031,19034],{"className":19012},[78,837],[34,19014,12226],{"className":19015},[78,79,837],[34,19017],{"className":19018,"style":16971},[237,837],[34,19020,19022,19025,19028],{"className":19021},[3387,837],[34,19023,1173],{"className":19024},[837],[34,19026,17954],{"className":19027},[837],[34,19029,16936],{"className":19030},[837],[34,19032],{"className":19033,"style":16971},[237,837],[34,19035,13527],{"className":19036,"style":13594},[78,79,837],[34,19038],{"className":19039,"style":262},[237],[34,19041,18755],{"className":19042},[266],[34,19044],{"className":19045,"style":262},[237],[34,19047,19049,19053,19086],{"className":19048},[69],[34,19050],{"className":19051,"style":19052},[73],"height:1.0369em;",[34,19054,19056,19059],{"className":19055},[78],[34,19057,18036],{"className":19058},[78,79],[34,19060,19062],{"className":19061},[810],[34,19063,19065],{"className":19064},[814],[34,19066,19068],{"className":19067},[818],[34,19069,19072],{"className":19070,"style":19071},[822],"height:0.7144em;",[34,19073,19074,19077],{"style":4710},[34,19075],{"className":19076,"style":830},[829],[34,19078,19080],{"className":19079},[834,835,836,837],[34,19081,19083],{"className":19082},[78,837],[34,19084,12226],{"className":19085},[78,79,837],[34,19087,19089,19092],{"className":19088},[78],[34,19090,18036],{"className":19091},[78,79],[34,19093,19095],{"className":19094},[810],[34,19096,19098],{"className":19097},[814],[34,19099,19101],{"className":19100},[818],[34,19102,19104],{"className":19103,"style":19052},[822],[34,19105,19106,19109],{"style":4710},[34,19107],{"className":19108,"style":830},[829],[34,19110,19112],{"className":19111},[834,835,836,837],[34,19113,19115,19118,19121,19151],{"className":19114},[78,837],[34,19116,6463],{"className":19117},[78,837],[34,19119,12226],{"className":19120},[78,79,837],[34,19122,19124,19127],{"className":19123},[78,837],[34,19125,13527],{"className":19126,"style":13594},[78,79,837],[34,19128,19130],{"className":19129},[810],[34,19131,19133],{"className":19132},[814],[34,19134,19136],{"className":19135},[818],[34,19137,19139],{"className":19138,"style":13653},[822],[34,19140,19142,19145],{"style":19141},"top:-2.931em;margin-right:0.0714em;",[34,19143],{"className":19144,"style":10772},[829],[34,19146,19148],{"className":19147},[834,10776,9740,837],[34,19149,424],{"className":19150},[78,837],[34,19152,12759],{"className":19153},[78,837],[16,19155,19156],{},"Thus, locally, the vMF distribution approximates a Gaussian distribution:",[34,19158,19160],{"className":19159,"translate":38},[159],[34,19161,19163,19199],{"className":19162,"translate":38},[37],[34,19164,19166],{"className":19165},[42],[44,19167,19168],{"xmlns":46,"display":168},[48,19169,19170,19196],{},[51,19171,19172,19174,19176,19180],{},[54,19173,13527],{},[176,19175,14925],{},[54,19177,19179],{"mathvariant":19178},"script","N",[51,19181,19182,19184,19186,19188,19194],{},[176,19183,179],{"fence":65},[347,19185,2287],{},[176,19187,5228],{"separator":65},[12543,19189,19190,19192],{},[347,19191,2213],{},[54,19193,12226],{},[176,19195,189],{"fence":65},[58,19197,19198],{"encoding":60},"\\theta \\sim \\mathcal{N}\\left(0,\\frac1\\kappa\\right)",[34,19200,19202,19220],{"className":19201,"ariaHidden":65},[64],[34,19203,19205,19208,19211,19214,19217],{"className":19204},[69],[34,19206],{"className":19207,"style":2184},[73],[34,19209,13527],{"className":19210,"style":13594},[78,79],[34,19212],{"className":19213,"style":262},[237],[34,19215,14925],{"className":19216},[266],[34,19218],{"className":19219,"style":262},[237],[34,19221,19223,19226,19231,19234],{"className":19222},[69],[34,19224],{"className":19225,"style":14693},[73],[34,19227,19179],{"className":19228,"style":19230},[78,19229],"mathcal","margin-right:0.1474em;",[34,19232],{"className":19233,"style":3462},[237],[34,19235,19237,19243,19246,19249,19252,19314],{"className":19236},[4177],[34,19238,19240],{"className":19239,"style":4182},[230,4181],[34,19241,179],{"className":19242},[4186,836],[34,19244,2287],{"className":19245},[78],[34,19247,5228],{"className":19248},[5317],[34,19250],{"className":19251,"style":3462},[237],[34,19253,19255,19258,19311],{"className":19254},[78],[34,19256],{"className":19257},[230,12689],[34,19259,19261],{"className":19260},[12543],[34,19262,19264,19303],{"className":19263},[814,2523],[34,19265,19267,19300],{"className":19266},[818],[34,19268,19270,19281,19289],{"className":19269,"style":16800},[822],[34,19271,19272,19275],{"style":14155},[34,19273],{"className":19274,"style":12709},[829],[34,19276,19278],{"className":19277},[78],[34,19279,12226],{"className":19280},[78,79],[34,19282,19283,19286],{"style":12897},[34,19284],{"className":19285,"style":12709},[829],[34,19287],{"className":19288,"style":12904},[12836],[34,19290,19291,19294],{"style":12907},[34,19292],{"className":19293,"style":12709},[829],[34,19295,19297],{"className":19296},[78],[34,19298,2213],{"className":19299},[78],[34,19301,2547],{"className":19302},[2546],[34,19304,19306],{"className":19305},[818],[34,19307,19309],{"className":19308,"style":14669},[822],[34,19310],{},[34,19312],{"className":19313},[258,12689],[34,19315,19317],{"className":19316,"style":4182},[258,4181],[34,19318,189],{"className":19319},[4186,836],[16,19321,19322,19323,19351],{},"If the confidence probability is ",[34,19324,19326,19339],{"className":19325,"translate":38},[37],[34,19327,19329],{"className":19328},[42],[44,19330,19331],{"xmlns":46},[48,19332,19333,19337],{},[51,19334,19335],{},[54,19336,16],{},[58,19338,16],{"encoding":60},[34,19340,19342],{"className":19341,"ariaHidden":65},[64],[34,19343,19345,19348],{"className":19344},[69],[34,19346],{"className":19347,"style":139},[73],[34,19349,16],{"className":19350},[78,79],", then",[34,19353,19355],{"className":19354,"translate":38},[159],[34,19356,19358,19396],{"className":19357,"translate":38},[37],[34,19359,19361],{"className":19360},[42],[44,19362,19363],{"xmlns":46,"display":168},[48,19364,19365,19393],{},[51,19366,19367,19373,19375,19384,19386],{},[2468,19368,19369,19371],{},[54,19370,13527],{},[54,19372,1173],{},[176,19374,18755],{},[19376,19377,19378],"msqrt",{},[12543,19379,19380,19382],{},[347,19381,424],{},[54,19383,12226],{},[1201,19385,5837],{},[2468,19387,19388,19391],{},[54,19389,19390],{},"z",[54,19392,16],{},[58,19394,19395],{"encoding":60},"\\theta_c \\approx \\sqrt{\\frac{2}{\\kappa}} \\, z_p",[34,19397,19399,19454],{"className":19398,"ariaHidden":65},[64],[34,19400,19402,19405,19445,19448,19451],{"className":19401},[69],[34,19403],{"className":19404,"style":3736},[73],[34,19406,19408,19411],{"className":19407},[78],[34,19409,13527],{"className":19410,"style":13594},[78,79],[34,19412,19414],{"className":19413},[810],[34,19415,19417,19437],{"className":19416},[814,2523],[34,19418,19420,19434],{"className":19419},[818],[34,19421,19423],{"className":19422,"style":5870},[822],[34,19424,19425,19428],{"style":16521},[34,19426],{"className":19427,"style":830},[829],[34,19429,19431],{"className":19430},[834,835,836,837],[34,19432,1173],{"className":19433},[78,79,837],[34,19435,2547],{"className":19436},[2546],[34,19438,19440],{"className":19439},[818],[34,19441,19443],{"className":19442,"style":2554},[822],[34,19444],{},[34,19446],{"className":19447,"style":262},[237],[34,19449,18755],{"className":19450},[266],[34,19452],{"className":19453,"style":262},[237],[34,19455,19457,19461,19579,19582],{"className":19456},[69],[34,19458],{"className":19459,"style":19460},[73],"height:2.44em;vertical-align:-0.7884em;",[34,19462,19465],{"className":19463},[78,19464],"sqrt",[34,19466,19468,19570],{"className":19467},[814,2523],[34,19469,19471,19567],{"className":19470},[818],[34,19472,19475,19549],{"className":19473,"style":19474},[822],"height:1.6516em;",[34,19476,19479,19483],{"className":19477,"style":19478},[10795],"top:-4.4em;",[34,19480],{"className":19481,"style":19482},[829],"height:4.4em;",[34,19484,19487],{"className":19485,"style":19486},[78],"padding-left:1em;",[34,19488,19490,19493,19546],{"className":19489},[78],[34,19491],{"className":19492},[230,12689],[34,19494,19496],{"className":19495},[12543],[34,19497,19499,19538],{"className":19498},[814,2523],[34,19500,19502,19535],{"className":19501},[818],[34,19503,19505,19516,19524],{"className":19504,"style":16800},[822],[34,19506,19507,19510],{"style":14155},[34,19508],{"className":19509,"style":12709},[829],[34,19511,19513],{"className":19512},[78],[34,19514,12226],{"className":19515},[78,79],[34,19517,19518,19521],{"style":12897},[34,19519],{"className":19520,"style":12709},[829],[34,19522],{"className":19523,"style":12904},[12836],[34,19525,19526,19529],{"style":12907},[34,19527],{"className":19528,"style":12709},[829],[34,19530,19532],{"className":19531},[78],[34,19533,424],{"className":19534},[78],[34,19536,2547],{"className":19537},[2546],[34,19539,19541],{"className":19540},[818],[34,19542,19544],{"className":19543,"style":14669},[822],[34,19545],{},[34,19547],{"className":19548},[258,12689],[34,19550,19552,19555],{"style":19551},"top:-3.6116em;",[34,19553],{"className":19554,"style":19482},[829],[34,19556,19559],{"className":19557,"style":19558},[10803],"min-width:1.02em;height:2.48em;",[10806,19560,19564],{"xmlns":10808,"width":10809,"height":19561,"viewBox":19562,"preserveAspectRatio":19563},"2.48em","0 0 400000 2592","xMinYMin slice",[10814,19565],{"d":19566},"M424,2478\nc-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514\nc0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20\ns-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121\ns209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081\nl0 -0c4,-6.7,10,-10,18,-10 H400000\nv40H1014.6\ns-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185\nc-2,6,-10,9,-24,9\nc-8,0,-12,-0.7,-12,-2z M1001 80\nh400000v40h-400000z",[34,19568,2547],{"className":19569},[2546],[34,19571,19573],{"className":19572},[818],[34,19574,19577],{"className":19575,"style":19576},[822],"height:0.7884em;",[34,19578],{},[34,19580],{"className":19581,"style":3462},[237],[34,19583,19585,19589],{"className":19584},[78],[34,19586,19390],{"className":19587,"style":19588},[78,79],"margin-right:0.044em;",[34,19590,19592],{"className":19591},[810],[34,19593,19595,19616],{"className":19594},[814,2523],[34,19596,19598,19613],{"className":19597},[818],[34,19599,19601],{"className":19600,"style":5870},[822],[34,19602,19604,19607],{"style":19603},"top:-2.55em;margin-left:-0.044em;margin-right:0.05em;",[34,19605],{"className":19606,"style":830},[829],[34,19608,19610],{"className":19609},[834,835,836,837],[34,19611,16],{"className":19612},[78,79,837],[34,19614,2547],{"className":19615},[2546],[34,19617,19619],{"className":19618},[818],[34,19620,19623],{"className":19621,"style":19622},[822],"height:0.2861em;",[34,19624],{},[16,19626,3218,19627,19698],{},[34,19628,19630,19648],{"className":19629,"translate":38},[37],[34,19631,19633],{"className":19632},[42],[44,19634,19635],{"xmlns":46},[48,19636,19637,19645],{},[51,19638,19639],{},[2468,19640,19641,19643],{},[54,19642,19390],{},[54,19644,16],{},[58,19646,19647],{"encoding":60},"z_p",[34,19649,19651],{"className":19650,"ariaHidden":65},[64],[34,19652,19654,19658],{"className":19653},[69],[34,19655],{"className":19656,"style":19657},[73],"height:0.7167em;vertical-align:-0.2861em;",[34,19659,19661,19664],{"className":19660},[78],[34,19662,19390],{"className":19663,"style":19588},[78,79],[34,19665,19667],{"className":19666},[810],[34,19668,19670,19690],{"className":19669},[814,2523],[34,19671,19673,19687],{"className":19672},[818],[34,19674,19676],{"className":19675,"style":5870},[822],[34,19677,19678,19681],{"style":19603},[34,19679],{"className":19680,"style":830},[829],[34,19682,19684],{"className":19683},[834,835,836,837],[34,19685,16],{"className":19686},[78,79,837],[34,19688,2547],{"className":19689},[2546],[34,19691,19693],{"className":19692},[818],[34,19694,19696],{"className":19695,"style":19622},[822],[34,19697],{}," is the quantile of the normal distribution.",[16,19700,19701],{},"Examples:",[19703,19704,19705,19786],"table",{},[19706,19707,19708],"thead",{},[19709,19710,19711,19715],"tr",{},[19712,19713,19714],"th",{},"Confidence",[19712,19716,19717],{},[34,19718,19720,19737],{"className":19719,"translate":38},[37],[34,19721,19723],{"className":19722},[42],[44,19724,19725],{"xmlns":46},[48,19726,19727,19735],{},[51,19728,19729],{},[2468,19730,19731,19733],{},[54,19732,19390],{},[54,19734,16],{},[58,19736,19647],{"encoding":60},[34,19738,19740],{"className":19739,"ariaHidden":65},[64],[34,19741,19743,19746],{"className":19742},[69],[34,19744],{"className":19745,"style":19657},[73],[34,19747,19749,19752],{"className":19748},[78],[34,19750,19390],{"className":19751,"style":19588},[78,79],[34,19753,19755],{"className":19754},[810],[34,19756,19758,19778],{"className":19757},[814,2523],[34,19759,19761,19775],{"className":19760},[818],[34,19762,19764],{"className":19763,"style":5870},[822],[34,19765,19766,19769],{"style":19603},[34,19767],{"className":19768,"style":830},[829],[34,19770,19772],{"className":19771},[834,835,836,837],[34,19773,16],{"className":19774},[78,79,837],[34,19776,2547],{"className":19777},[2546],[34,19779,19781],{"className":19780},[818],[34,19782,19784],{"className":19783,"style":19622},[822],[34,19785],{},[19787,19788,19789,19798,19806],"tbody",{},[19709,19790,19791,19795],{},[19792,19793,19794],"td",{},"90%",[19792,19796,19797],{},"1.64",[19709,19799,19800,19803],{},[19792,19801,19802],{},"95%",[19792,19804,19805],{},"1.96",[19709,19807,19808,19811],{},[19792,19809,19810],{},"99%",[19792,19812,19813],{},"2.58",[16,19815,19816],{},"Thus, the 95% confidence cone angle is:",[34,19818,19820],{"className":19819,"translate":38},[159],[34,19821,19823,19854],{"className":19822,"translate":38},[37],[34,19824,19826],{"className":19825},[42],[44,19827,19828],{"xmlns":46,"display":168},[48,19829,19830,19851],{},[51,19831,19832,19839,19841,19843],{},[2468,19833,19834,19836],{},[54,19835,13527],{},[347,19837,19838],{},"95",[176,19840,18755],{},[347,19842,19805],{},[19376,19844,19845],{},[12543,19846,19847,19849],{},[347,19848,424],{},[54,19850,12226],{},[58,19852,19853],{"encoding":60},"\\theta_{95} \\approx 1.96\\sqrt{\\frac{2}{\\kappa}}",[34,19855,19857,19915],{"className":19856,"ariaHidden":65},[64],[34,19858,19860,19863,19906,19909,19912],{"className":19859},[69],[34,19861],{"className":19862,"style":3736},[73],[34,19864,19866,19869],{"className":19865},[78],[34,19867,13527],{"className":19868,"style":13594},[78,79],[34,19870,19872],{"className":19871},[810],[34,19873,19875,19898],{"className":19874},[814,2523],[34,19876,19878,19895],{"className":19877},[818],[34,19879,19881],{"className":19880,"style":2530},[822],[34,19882,19883,19886],{"style":16521},[34,19884],{"className":19885,"style":830},[829],[34,19887,19889],{"className":19888},[834,835,836,837],[34,19890,19892],{"className":19891},[78,837],[34,19893,19838],{"className":19894},[78,837],[34,19896,2547],{"className":19897},[2546],[34,19899,19901],{"className":19900},[818],[34,19902,19904],{"className":19903,"style":2554},[822],[34,19905],{},[34,19907],{"className":19908,"style":262},[237],[34,19910,18755],{"className":19911},[266],[34,19913],{"className":19914,"style":262},[237],[34,19916,19918,19921,19924],{"className":19917},[69],[34,19919],{"className":19920,"style":19460},[73],[34,19922,19805],{"className":19923},[78],[34,19925,19927],{"className":19926},[78,19464],[34,19928,19930,20022],{"className":19929},[814,2523],[34,19931,19933,20019],{"className":19932},[818],[34,19934,19936,20007],{"className":19935,"style":19474},[822],[34,19937,19939,19942],{"className":19938,"style":19478},[10795],[34,19940],{"className":19941,"style":19482},[829],[34,19943,19945],{"className":19944,"style":19486},[78],[34,19946,19948,19951,20004],{"className":19947},[78],[34,19949],{"className":19950},[230,12689],[34,19952,19954],{"className":19953},[12543],[34,19955,19957,19996],{"className":19956},[814,2523],[34,19958,19960,19993],{"className":19959},[818],[34,19961,19963,19974,19982],{"className":19962,"style":16800},[822],[34,19964,19965,19968],{"style":14155},[34,19966],{"className":19967,"style":12709},[829],[34,19969,19971],{"className":19970},[78],[34,19972,12226],{"className":19973},[78,79],[34,19975,19976,19979],{"style":12897},[34,19977],{"className":19978,"style":12709},[829],[34,19980],{"className":19981,"style":12904},[12836],[34,19983,19984,19987],{"style":12907},[34,19985],{"className":19986,"style":12709},[829],[34,19988,19990],{"className":19989},[78],[34,19991,424],{"className":19992},[78],[34,19994,2547],{"className":19995},[2546],[34,19997,19999],{"className":19998},[818],[34,20000,20002],{"className":20001,"style":14669},[822],[34,20003],{},[34,20005],{"className":20006},[258,12689],[34,20008,20009,20012],{"style":19551},[34,20010],{"className":20011,"style":19482},[829],[34,20013,20015],{"className":20014,"style":19558},[10803],[10806,20016,20017],{"xmlns":10808,"width":10809,"height":19561,"viewBox":19562,"preserveAspectRatio":19563},[10814,20018],{"d":19566},[34,20020,2547],{"className":20021},[2546],[34,20023,20025],{"className":20024},[818],[34,20026,20028],{"className":20027,"style":19576},[822],[34,20029],{},[16,20031,20032,20033,20061],{},"In high dimensions ",[34,20034,20036,20049],{"className":20035,"translate":38},[37],[34,20037,20039],{"className":20038},[42],[44,20040,20041],{"xmlns":46},[48,20042,20043,20047],{},[51,20044,20045],{},[54,20046,5250],{},[58,20048,5250],{"encoding":60},[34,20050,20052],{"className":20051,"ariaHidden":65},[64],[34,20053,20055,20058],{"className":20054},[69],[34,20056],{"className":20057,"style":2184},[73],[34,20059,5250],{"className":20060},[78,79],", we have",[34,20063,20065],{"className":20064,"translate":38},[159],[34,20066,20068,20138],{"className":20067,"translate":38},[37],[34,20069,20071],{"className":20070},[42],[44,20072,20073],{"xmlns":46,"display":168},[48,20074,20075,20135],{},[51,20076,20077,20083,20085,20087,20089],{},[755,20078,20079,20081],{},[54,20080,11720],{},[54,20082,13552],{},[54,20084,96],{},[176,20086,18755],{},[54,20088,19179],{"mathvariant":19178},[51,20090,20091,20093,20099,20101,20103,20105,20107,20133],{},[176,20092,179],{"fence":65},[2468,20094,20095,20097],{},[54,20096,3145],{},[54,20098,5250],{},[176,20100,179],{"stretchy":178},[54,20102,12226],{},[176,20104,189],{"stretchy":178},[176,20106,5228],{"separator":65},[12543,20108,20109,20131],{},[51,20110,20111,20113,20115,20121,20123,20125],{},[347,20112,2213],{},[176,20114,6463],{},[2468,20116,20117,20119],{},[54,20118,3145],{},[54,20120,5250],{},[176,20122,179],{"stretchy":178},[54,20124,12226],{},[755,20126,20127,20129],{},[176,20128,189],{"stretchy":178},[347,20130,424],{},[54,20132,5250],{},[176,20134,189],{"fence":65},[58,20136,20137],{"encoding":60},"\\mu^T x \\approx \\mathcal{N}\\left(A_d(\\kappa), \\frac{1-A_d(\\kappa)^2}{d}\\right)",[34,20139,20141,20188],{"className":20140,"ariaHidden":65},[64],[34,20142,20144,20147,20176,20179,20182,20185],{"className":20143},[69],[34,20145],{"className":20146,"style":16408},[73],[34,20148,20150,20153],{"className":20149},[78],[34,20151,11720],{"className":20152},[78,79],[34,20154,20156],{"className":20155},[810],[34,20157,20159],{"className":20158},[814],[34,20160,20162],{"className":20161},[818],[34,20163,20165],{"className":20164,"style":13653},[822],[34,20166,20167,20170],{"style":4710},[34,20168],{"className":20169,"style":830},[829],[34,20171,20173],{"className":20172},[834,835,836,837],[34,20174,13552],{"className":20175,"style":6248},[78,79,837],[34,20177,96],{"className":20178},[78,79],[34,20180],{"className":20181,"style":262},[237],[34,20183,18755],{"className":20184},[266],[34,20186],{"className":20187,"style":262},[237],[34,20189,20191,20195,20198,20201],{"className":20190},[69],[34,20192],{"className":20193,"style":20194},[73],"height:2.4411em;vertical-align:-0.95em;",[34,20196,19179],{"className":20197,"style":19230},[78,19229],[34,20199],{"className":20200,"style":3462},[237],[34,20202,20204,20210,20250,20253,20256,20259,20262,20265,20411],{"className":20203},[4177],[34,20205,20207],{"className":20206,"style":4182},[230,4181],[34,20208,179],{"className":20209},[4186,836],[34,20211,20213,20216],{"className":20212},[78],[34,20214,3145],{"className":20215},[78,79],[34,20217,20219],{"className":20218},[810],[34,20220,20222,20242],{"className":20221},[814,2523],[34,20223,20225,20239],{"className":20224},[818],[34,20226,20228],{"className":20227,"style":6071},[822],[34,20229,20230,20233],{"style":2533},[34,20231],{"className":20232,"style":830},[829],[34,20234,20236],{"className":20235},[834,835,836,837],[34,20237,5250],{"className":20238},[78,79,837],[34,20240,2547],{"className":20241},[2546],[34,20243,20245],{"className":20244},[818],[34,20246,20248],{"className":20247,"style":2554},[822],[34,20249],{},[34,20251,179],{"className":20252},[230],[34,20254,12226],{"className":20255},[78,79],[34,20257,189],{"className":20258},[258],[34,20260,5228],{"className":20261},[5317],[34,20263],{"className":20264,"style":3462},[237],[34,20266,20268,20271,20408],{"className":20267},[78],[34,20269],{"className":20270},[230,12689],[34,20272,20274],{"className":20273},[12543],[34,20275,20277,20400],{"className":20276},[814,2523],[34,20278,20280,20397],{"className":20279},[818],[34,20281,20283,20294,20302],{"className":20282,"style":15228},[822],[34,20284,20285,20288],{"style":14155},[34,20286],{"className":20287,"style":12709},[829],[34,20289,20291],{"className":20290},[78],[34,20292,5250],{"className":20293},[78,79],[34,20295,20296,20299],{"style":12897},[34,20297],{"className":20298,"style":12709},[829],[34,20300],{"className":20301,"style":12904},[12836],[34,20303,20304,20307],{"style":12907},[34,20305],{"className":20306,"style":12709},[829],[34,20308,20310,20313,20316,20319,20322,20362,20365,20368],{"className":20309},[78],[34,20311,2213],{"className":20312},[78],[34,20314],{"className":20315,"style":238},[237],[34,20317,6463],{"className":20318},[242],[34,20320],{"className":20321,"style":238},[237],[34,20323,20325,20328],{"className":20324},[78],[34,20326,3145],{"className":20327},[78,79],[34,20329,20331],{"className":20330},[810],[34,20332,20334,20354],{"className":20333},[814,2523],[34,20335,20337,20351],{"className":20336},[818],[34,20338,20340],{"className":20339,"style":6071},[822],[34,20341,20342,20345],{"style":2533},[34,20343],{"className":20344,"style":830},[829],[34,20346,20348],{"className":20347},[834,835,836,837],[34,20349,5250],{"className":20350},[78,79,837],[34,20352,2547],{"className":20353},[2546],[34,20355,20357],{"className":20356},[818],[34,20358,20360],{"className":20359,"style":2554},[822],[34,20361],{},[34,20363,179],{"className":20364},[230],[34,20366,12226],{"className":20367},[78,79],[34,20369,20371,20374],{"className":20370},[258],[34,20372,189],{"className":20373},[258],[34,20375,20377],{"className":20376},[810],[34,20378,20380],{"className":20379},[814],[34,20381,20383],{"className":20382},[818],[34,20384,20386],{"className":20385,"style":800},[822],[34,20387,20388,20391],{"style":825},[34,20389],{"className":20390,"style":830},[829],[34,20392,20394],{"className":20393},[834,835,836,837],[34,20395,424],{"className":20396},[78,837],[34,20398,2547],{"className":20399},[2546],[34,20401,20403],{"className":20402},[818],[34,20404,20406],{"className":20405,"style":14669},[822],[34,20407],{},[34,20409],{"className":20410},[258,12689],[34,20412,20414],{"className":20413,"style":4182},[258,4181],[34,20415,189],{"className":20416},[4186,836],[16,20418,13697],{},[34,20420,20422],{"className":20421,"translate":38},[159],[34,20423,20425,20496],{"className":20424,"translate":38},[37],[34,20426,20428],{"className":20427},[42],[44,20429,20430],{"xmlns":46,"display":168},[48,20431,20432,20494],{},[51,20433,20434,20440,20442,20444,20446,20448],{},[2468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the confidence angle satisfies:",[34,20745,20747],{"className":20746,"translate":38},[159],[34,20748,20750,20822],{"className":20749,"translate":38},[37],[34,20751,20753],{"className":20752},[42],[44,20754,20755],{"xmlns":46,"display":168},[48,20756,20757,20819],{},[51,20758,20759,20761,20763,20769,20771,20777,20779,20781,20783,20785,20791],{},[54,20760,16357],{},[176,20762,6321],{},[2468,20764,20765,20767],{},[54,20766,13527],{},[54,20768,1173],{},[176,20770,192],{},[2468,20772,20773,20775],{},[54,20774,3145],{},[54,20776,5250],{},[176,20778,179],{"stretchy":178},[54,20780,12226],{},[176,20782,189],{"stretchy":178},[176,20784,6463],{},[2468,20786,20787,20789],{},[54,20788,19390],{},[54,20790,16],{},[19376,20792,20793],{},[12543,20794,20795,20817],{},[51,20796,20797,20799,20801,20807,20809,20811],{},[347,20798,2213],{},[176,20800,6463],{},[2468,20802,20803,20805],{},[54,20804,3145],{},[54,20806,5250],{},[176,20808,179],{"stretchy":178},[54,20810,12226],{},[755,20812,20813,20815],{},[176,20814,189],{"stretchy":178},[347,20816,424],{},[54,20818,5250],{},[58,20820,20821],{"encoding":60},"\\cos\\theta_c = A_d(\\kappa) - z_p \\sqrt{\\frac{1-A_d(\\kappa)^2}{d}}",[34,20823,20825,20886,20950],{"className":20824,"ariaHidden":65},[64],[34,20826,20828,20831,20834,20837,20877,20880,20883],{"className":20827},[69],[34,20829],{"className":20830,"style":3736},[73],[34,20832,16357],{"className":20833},[3387],[34,20835],{"className":20836,"style":3462},[237],[34,20838,20840,20843],{"className":20839},[78],[34,20841,13527],{"className":20842,"style":13594},[78,79],[34,20844,20846],{"className":20845},[810],[34,20847,20849,20869],{"className":20848},[814,2523],[34,20850,20852,20866],{"className":20851},[818],[34,20853,20855],{"className":20854,"style":5870},[822],[34,20856,20857,20860],{"style":16521},[34,20858],{"className":20859,"style":830},[829],[34,20861,20863],{"className":20862},[834,835,836,837],[34,20864,1173],{"className":20865},[78,79,837],[34,20867,2547],{"className":20868},[2546],[34,20870,20872],{"className":20871},[818],[34,20873,20875],{"className":20874,"style":2554},[822],[34,20876],{},[34,20878],{"className":20879,"style":262},[237],[34,20881,192],{"className":20882},[266],[34,20884],{"className":20885,"style":262},[237],[34,20887,20889,20892,20932,20935,20938,20941,20944,20947],{"className":20888},[69],[34,20890],{"className":20891,"style":223},[73],[34,20893,20895,20898],{"className":20894},[78],[34,20896,3145],{"className":20897},[78,79],[34,20899,20901],{"className":20900},[810],[34,20902,20904,20924],{"className":20903},[814,2523],[34,20905,20907,20921],{"className":20906},[818],[34,20908,20910],{"className":20909,"style":6071},[822],[34,20911,20912,20915],{"style":2533},[34,20913],{"className":20914,"style":830},[829],[34,20916,20918],{"className":20917},[834,835,836,837],[34,20919,5250],{"className":20920},[78,79,837],[34,20922,2547],{"className":20923},[2546],[34,20925,20927],{"className":20926},[818],[34,20928,20930],{"className":20929,"style":2554},[822],[34,20931],{},[34,20933,179],{"className":20934},[230],[34,20936,12226],{"className":20937},[78,79],[34,20939,189],{"className":20940},[258],[34,20942],{"className":20943,"style":238},[237],[34,20945,6463],{"className":20946},[242],[34,20948],{"className":20949,"style":238},[237],[34,20951,20953,20957,20997],{"className":20952},[69],[34,20954],{"className":20955,"style":20956},[73],"height:2.44em;vertical-align:-0.7356em;",[34,20958,20960,20963],{"className":20959},[78],[34,20961,19390],{"className":20962,"style":19588},[78,79],[34,20964,20966],{"className":20965},[810],[34,20967,20969,20989],{"className":20968},[814,2523],[34,20970,20972,20986],{"className":20971},[818],[34,20973,20975],{"className":20974,"style":5870},[822],[34,20976,20977,20980],{"style":19603},[34,20978],{"className":20979,"style":830},[829],[34,20981,20983],{"className":20982},[834,835,836,837],[34,20984,16],{"className":20985},[78,79,837],[34,20987,2547],{"className":20988},[2546],[34,20990,20992],{"className":20991},[818],[34,20993,20995],{"className":20994,"style":19622},[822],[34,20996],{},[34,20998,21000],{"className":20999},[78,19464],[34,21001,21003,21181],{"className":21002},[814,2523],[34,21004,21006,21178],{"className":21005},[818],[34,21007,21010,21165],{"className":21008,"style":21009},[822],"height:1.7044em;",[34,21011,21013,21016],{"className":21012,"style":19478},[10795],[34,21014],{"className":21015,"style":19482},[829],[34,21017,21019],{"className":21018,"style":19486},[78],[34,21020,21022,21025,21162],{"className":21021},[78],[34,21023],{"className":21024},[230,12689],[34,21026,21028],{"className":21027},[12543],[34,21029,21031,21154],{"className":21030},[814,2523],[34,21032,21034,21151],{"className":21033},[818],[34,21035,21037,21048,21056],{"className":21036,"style":14583},[822],[34,21038,21039,21042],{"style":14155},[34,21040],{"className":21041,"style":12709},[829],[34,21043,21045],{"className":21044},[78],[34,21046,5250],{"className":21047},[78,79],[34,21049,21050,21053],{"style":12897},[34,21051],{"className":21052,"style":12709},[829],[34,21054],{"className":21055,"style":12904},[12836],[34,21057,21058,21061],{"style":12907},[34,21059],{"className":21060,"style":12709},[829],[34,21062,21064,21067,21070,21073,21076,21116,21119,21122],{"className":21063},[78],[34,21065,2213],{"className":21066},[78],[34,21068],{"className":21069,"style":238},[237],[34,21071,6463],{"className":21072},[242],[34,21074],{"className":21075,"style":238},[237],[34,21077,21079,21082],{"className":21078},[78],[34,21080,3145],{"className":21081},[78,79],[34,21083,21085],{"className":21084},[810],[34,21086,21088,21108],{"className":21087},[814,2523],[34,21089,21091,21105],{"className":21090},[818],[34,21092,21094],{"className":21093,"style":6071},[822],[34,21095,21096,21099],{"style":2533},[34,21097],{"className":21098,"style":830},[829],[34,21100,21102],{"className":21101},[834,835,836,837],[34,21103,5250],{"className":21104},[78,79,837],[34,21106,2547],{"className":21107},[2546],[34,21109,21111],{"className":21110},[818],[34,21112,21114],{"className":21113,"style":2554},[822],[34,21115],{},[34,21117,179],{"className":21118},[230],[34,21120,12226],{"className":21121},[78,79],[34,21123,21125,21128],{"className":21124},[258],[34,21126,189],{"className":21127},[258],[34,21129,21131],{"className":21130},[810],[34,21132,21134],{"className":21133},[814],[34,21135,21137],{"className":21136},[818],[34,21138,21140],{"className":21139,"style":15259},[822],[34,21141,21142,21145],{"style":12743},[34,21143],{"className":21144,"style":830},[829],[34,21146,21148],{"className":21147},[834,835,836,837],[34,21149,424],{"className":21150},[78,837],[34,21152,2547],{"className":21153},[2546],[34,21155,21157],{"className":21156},[818],[34,21158,21160],{"className":21159,"style":14669},[822],[34,21161],{},[34,21163],{"className":21164},[258,12689],[34,21166,21168,21171],{"style":21167},"top:-3.6644em;",[34,21169],{"className":21170,"style":19482},[829],[34,21172,21174],{"className":21173,"style":19558},[10803],[10806,21175,21176],{"xmlns":10808,"width":10809,"height":19561,"viewBox":19562,"preserveAspectRatio":19563},[10814,21177],{"d":19566},[34,21179,2547],{"className":21180},[2546],[34,21182,21184],{"className":21183},[818],[34,21185,21188],{"className":21186,"style":21187},[822],"height:0.7356em;",[34,21189],{},[16,21191,21192],{},"For small angles:",[34,21194,21196],{"className":21195,"translate":38},[159],[34,21197,21199,21241],{"className":21198,"translate":38},[37],[34,21200,21202],{"className":21201},[42],[44,21203,21204],{"xmlns":46,"display":168},[48,21205,21206,21238],{},[51,21207,21208,21210,21212,21214,21216,21218,21224,21226],{},[54,21209,3145],{},[176,21211,179],{"stretchy":178},[54,21213,13527],{},[176,21215,189],{"stretchy":178},[176,21217,18755],{},[2468,21219,21220,21222],{},[54,21221,5243],{},[54,21223,5250],{},[1201,21225,5837],{},[755,21227,21228,21230],{},[54,21229,13527],{},[51,21231,21232,21234,21236],{},[54,21233,5250],{},[176,21235,6463],{},[347,21237,2213],{},[58,21239,21240],{"encoding":60},"A(\\theta) \\approx C_d \\, \\theta^{d-1}",[34,21242,21244,21271],{"className":21243,"ariaHidden":65},[64],[34,21245,21247,21250,21253,21256,21259,21262,21265,21268],{"className":21246},[69],[34,21248],{"className":21249,"style":223},[73],[34,21251,3145],{"className":21252},[78,79],[34,21254,179],{"className":21255},[230],[34,21257,13527],{"className":21258,"style":13594},[78,79],[34,21260,189],{"className":21261},[258],[34,21263],{"className":21264,"style":262},[237],[34,21266,18755],{"className":21267},[266],[34,21269],{"className":21270,"style":262},[237],[34,21272,21274,21278,21318,21321],{"className":21273},[69],[34,21275],{"className":21276,"style":21277},[73],"height:1.0491em;vertical-align:-0.15em;",[34,21279,21281,21284],{"className":21280},[78],[34,21282,5243],{"className":21283,"style":5354},[78,79],[34,21285,21287],{"className":21286},[810],[34,21288,21290,21310],{"className":21289},[814,2523],[34,21291,21293,21307],{"className":21292},[818],[34,21294,21296],{"className":21295,"style":6071},[822],[34,21297,21298,21301],{"style":12428},[34,21299],{"className":21300,"style":830},[829],[34,21302,21304],{"className":21303},[834,835,836,837],[34,21305,5250],{"className":21306},[78,79,837],[34,21308,2547],{"className":21309},[2546],[34,21311,21313],{"className":21312},[818],[34,21314,21316],{"className":21315,"style":2554},[822],[34,21317],{},[34,21319],{"className":21320,"style":3462},[237],[34,21322,21324,21327],{"className":21323},[78],[34,21325,13527],{"className":21326,"style":13594},[78,79],[34,21328,21330],{"className":21329},[810],[34,21331,21333],{"className":21332},[814],[34,21334,21336],{"className":21335},[818],[34,21337,21339],{"className":21338,"style":11944},[822],[34,21340,21341,21344],{"style":4710},[34,21342],{"className":21343,"style":830},[829],[34,21345,21347],{"className":21346},[834,835,836,837],[34,21348,21350,21353,21356],{"className":21349},[78,837],[34,21351,5250],{"className":21352},[78,79,837],[34,21354,6463],{"className":21355},[242,837],[34,21357,2213],{"className":21358},[78,837],[16,21360,9212],{},[34,21362,21364],{"className":21363,"translate":38},[159],[34,21365,21367,21435],{"className":21366,"translate":38},[37],[34,21368,21370],{"className":21369},[42],[44,21371,21372],{"xmlns":46,"display":168},[48,21373,21374,21432],{},[51,21375,21376,21382,21384],{},[2468,21377,21378,21380],{},[54,21379,5243],{},[54,21381,5250],{},[176,21383,192],{},[12543,21385,21386,21410],{},[51,21387,21388,21390],{},[347,21389,424],{},[755,21391,21392,21394],{},[54,21393,12569],{},[51,21395,21396,21398,21400,21402,21404,21406,21408],{},[176,21397,179],{"stretchy":178},[54,21399,5250],{},[176,21401,6463],{},[347,21403,2213],{},[176,21405,189],{"stretchy":178},[54,21407,12580],{"mathvariant":2296},[347,21409,424],{},[51,21411,21412,21414,21416,21418,21420,21422,21424,21426,21428,21430],{},[54,21413,17287],{"mathvariant":2296},[176,21415,179],{"stretchy":178},[176,21417,179],{"stretchy":178},[54,21419,5250],{},[176,21421,6463],{},[347,21423,2213],{},[176,21425,189],{"stretchy":178},[54,21427,12580],{"mathvariant":2296},[347,21429,424],{},[176,21431,189],{"stretchy":178},[58,21433,21434],{"encoding":60},"C_d = \\frac{2\\pi^{(d-1)\u002F2}}{\\Gamma((d-1)\u002F2)}",[34,21436,21438,21493],{"className":21437,"ariaHidden":65},[64],[34,21439,21441,21444,21484,21487,21490],{"className":21440},[69],[34,21442],{"className":21443,"style":3677},[73],[34,21445,21447,21450],{"className":21446},[78],[34,21448,5243],{"className":21449,"style":5354},[78,79],[34,21451,21453],{"className":21452},[810],[34,21454,21456,21476],{"className":21455},[814,2523],[34,21457,21459,21473],{"className":21458},[818],[34,21460,21462],{"className":21461,"style":6071},[822],[34,21463,21464,21467],{"style":12428},[34,21465],{"className":21466,"style":830},[829],[34,21468,21470],{"className":21469},[834,835,836,837],[34,21471,5250],{"className":21472},[78,79,837],[34,21474,2547],{"className":21475},[2546],[34,21477,21479],{"className":21478},[818],[34,21480,21482],{"className":21481,"style":2554},[822],[34,21483],{},[34,21485],{"className":21486,"style":262},[237],[34,21488,192],{"className":21489},[266],[34,21491],{"className":21492,"style":262},[237],[34,21494,21496,21499],{"className":21495},[69],[34,21497],{"className":21498,"style":17374},[73],[34,21500,21502,21505,21633],{"className":21501},[78],[34,21503],{"className":21504},[230,12689],[34,21506,21508],{"className":21507},[12543],[34,21509,21511,21625],{"className":21510},[814,2523],[34,21512,21514,21622],{"className":21513},[818],[34,21515,21517,21556,21564],{"className":21516,"style":17393},[822],[34,21518,21519,21522],{"style":14155},[34,21520],{"className":21521,"style":12709},[829],[34,21523,21525,21528,21532,21535,21538,21541,21544,21547,21550,21553],{"className":21524},[78],[34,21526,17287],{"className":21527},[78],[34,21529,21531],{"className":21530},[230],"((",[34,21533,5250],{"className":21534},[78,79],[34,21536],{"className":21537,"style":238},[237],[34,21539,6463],{"className":21540},[242],[34,21542],{"className":21543,"style":238},[237],[34,21545,2213],{"className":21546},[78],[34,21548,189],{"className":21549},[258],[34,21551,12759],{"className":21552},[78],[34,21554,189],{"className":21555},[258],[34,21557,21558,21561],{"style":12897},[34,21559],{"className":21560,"style":12709},[829],[34,21562],{"className":21563,"style":12904},[12836],[34,21565,21566,21569],{"style":12907},[34,21567],{"className":21568,"style":12709},[829],[34,21570,21572,21575],{"className":21571},[78],[34,21573,424],{"className":21574},[78],[34,21576,21578,21581],{"className":21577},[78],[34,21579,12569],{"className":21580,"style":143},[78,79],[34,21582,21584],{"className":21583},[810],[34,21585,21587],{"className":21586},[814],[34,21588,21590],{"className":21589},[818],[34,21591,21593],{"className":21592,"style":17454},[822],[34,21594,21595,21598],{"style":825},[34,21596],{"className":21597,"style":830},[829],[34,21599,21601],{"className":21600},[834,835,836,837],[34,21602,21604,21607,21610,21613,21616,21619],{"className":21603},[78,837],[34,21605,179],{"className":21606},[230,837],[34,21608,5250],{"className":21609},[78,79,837],[34,21611,6463],{"className":21612},[242,837],[34,21614,2213],{"className":21615},[78,837],[34,21617,189],{"className":21618},[258,837],[34,21620,12759],{"className":21621},[78,837],[34,21623,2547],{"className":21624},[2546],[34,21626,21628],{"className":21627},[818],[34,21629,21631],{"className":21630,"style":17481},[822],[34,21632],{},[34,21634],{"className":21635},[258,12689],[16,21637,21638,21639,21713],{},"Hence, the spherical cap area grows as ",[34,21640,21642,21666],{"className":21641,"translate":38},[37],[34,21643,21645],{"className":21644},[42],[44,21646,21647],{"xmlns":46},[48,21648,21649,21663],{},[51,21650,21651],{},[755,21652,21653,21655],{},[54,21654,13527],{},[51,21656,21657,21659,21661],{},[54,21658,5250],{},[176,21660,6463],{},[347,21662,2213],{},[58,21664,21665],{"encoding":60},"\\theta^{d-1}",[34,21667,21669],{"className":21668,"ariaHidden":65},[64],[34,21670,21672,21675],{"className":21671},[69],[34,21673],{"className":21674,"style":12169},[73],[34,21676,21678,21681],{"className":21677},[78],[34,21679,13527],{"className":21680,"style":13594},[78,79],[34,21682,21684],{"className":21683},[810],[34,21685,21687],{"className":21686},[814],[34,21688,21690],{"className":21689},[818],[34,21691,21693],{"className":21692,"style":12169},[822],[34,21694,21695,21698],{"style":825},[34,21696],{"className":21697,"style":830},[829],[34,21699,21701],{"className":21700},[834,835,836,837],[34,21702,21704,21707,21710],{"className":21703},[78,837],[34,21705,5250],{"className":21706},[78,79,837],[34,21708,6463],{"className":21709},[242,837],[34,21711,2213],{"className":21712},[78,837],", which is a key reason why high-dimensional sphere volume concentrates near the equator.",[16,21715,21716],{},"In many embedding papers, the following is used:",[34,21718,21720],{"className":21719,"translate":38},[159],[34,21721,21723,21758],{"className":21722,"translate":38},[37],[34,21724,21726],{"className":21725},[42],[44,21727,21728],{"xmlns":46,"display":168},[48,21729,21730,21755],{},[51,21731,21732,21739,21741],{},[2468,21733,21734,21736],{},[54,21735,13527],{},[1201,21737,21738],{},"typical",[176,21740,18755],{},[19376,21742,21743],{},[12543,21744,21745,21753],{},[51,21746,21747,21749,21751],{},[54,21748,5250],{},[176,21750,6463],{},[347,21752,2213],{},[54,21754,12226],{},[58,21756,21757],{"encoding":60},"\\theta_{\\text{typical}} \\approx \\sqrt{\\frac{d-1}{\\kappa}}",[34,21759,21761,21823],{"className":21760,"ariaHidden":65},[64],[34,21762,21764,21768,21814,21817,21820],{"className":21763},[69],[34,21765],{"className":21766,"style":21767},[73],"height:0.9805em;vertical-align:-0.2861em;",[34,21769,21771,21774],{"className":21770},[78],[34,21772,13527],{"className":21773,"style":13594},[78,79],[34,21775,21777],{"className":21776},[810],[34,21778,21780,21806],{"className":21779},[814,2523],[34,21781,21783,21803],{"className":21782},[818],[34,21784,21786],{"className":21785,"style":6071},[822],[34,21787,21788,21791],{"style":16521},[34,21789],{"className":21790,"style":830},[829],[34,21792,21794],{"className":21793},[834,835,836,837],[34,21795,21797],{"className":21796},[78,837],[34,21798,21800],{"className":21799},[78,1304,837],[34,21801,21738],{"className":21802},[78,837],[34,21804,2547],{"className":21805},[2546],[34,21807,21809],{"className":21808},[818],[34,21810,21812],{"className":21811,"style":19622},[822],[34,21813],{},[34,21815],{"className":21816,"style":262},[237],[34,21818,18755],{"className":21819},[266],[34,21821],{"className":21822,"style":262},[237],[34,21824,21826,21830],{"className":21825},[69],[34,21827],{"className":21828,"style":21829},[73],"height:2.44em;vertical-align:-0.7634em;",[34,21831,21833],{"className":21832},[78,19464],[34,21834,21836,21942],{"className":21835},[814,2523],[34,21837,21839,21939],{"className":21838},[818],[34,21840,21843,21926],{"className":21841,"style":21842},[822],"height:1.6766em;",[34,21844,21846,21849],{"className":21845,"style":19478},[10795],[34,21847],{"className":21848,"style":19482},[829],[34,21850,21852],{"className":21851,"style":19486},[78],[34,21853,21855,21858,21923],{"className":21854},[78],[34,21856],{"className":21857},[230,12689],[34,21859,21861],{"className":21860},[12543],[34,21862,21864,21915],{"className":21863},[814,2523],[34,21865,21867,21912],{"className":21866},[818],[34,21868,21870,21881,21889],{"className":21869,"style":15668},[822],[34,21871,21872,21875],{"style":14155},[34,21873],{"className":21874,"style":12709},[829],[34,21876,21878],{"className":21877},[78],[34,21879,12226],{"className":21880},[78,79],[34,21882,21883,21886],{"style":12897},[34,21884],{"className":21885,"style":12709},[829],[34,21887],{"className":21888,"style":12904},[12836],[34,21890,21891,21894],{"style":12907},[34,21892],{"className":21893,"style":12709},[829],[34,21895,21897,21900,21903,21906,21909],{"className":21896},[78],[34,21898,5250],{"className":21899},[78,79],[34,21901],{"className":21902,"style":238},[237],[34,21904,6463],{"className":21905},[242],[34,21907],{"className":21908,"style":238},[237],[34,21910,2213],{"className":21911},[78],[34,21913,2547],{"className":21914},[2546],[34,21916,21918],{"className":21917},[818],[34,21919,21921],{"className":21920,"style":14669},[822],[34,21922],{},[34,21924],{"className":21925},[258,12689],[34,21927,21929,21932],{"style":21928},"top:-3.6366em;",[34,21930],{"className":21931,"style":19482},[829],[34,21933,21935],{"className":21934,"style":19558},[10803],[10806,21936,21937],{"xmlns":10808,"width":10809,"height":19561,"viewBox":19562,"preserveAspectRatio":19563},[10814,21938],{"d":19566},[34,21940,2547],{"className":21941},[2546],[34,21943,21945],{"className":21944},[818],[34,21946,21949],{"className":21947,"style":21948},[822],"height:0.7634em;",[34,21950],{},[16,21952,21953],{},"Interpretation: most probability mass of the vMF distribution lies in",[34,21955,21957],{"className":21956,"translate":38},[159],[34,21958,21960,21991],{"className":21959,"translate":38},[37],[34,21961,21963],{"className":21962},[42],[44,21964,21965],{"xmlns":46,"display":168},[48,21966,21967,21988],{},[51,21968,21969,21971,21974],{},[54,21970,13527],{},[176,21972,21973],{},"≲",[19376,21975,21976],{},[12543,21977,21978,21986],{},[51,21979,21980,21982,21984],{},[54,21981,5250],{},[176,21983,6463],{},[347,21985,2213],{},[54,21987,12226],{},[58,21989,21990],{"encoding":60},"\\theta \\lesssim \\sqrt{\\frac{d-1}{\\kappa}}",[34,21992,21994,22014],{"className":21993,"ariaHidden":65},[64],[34,21995,21997,22001,22004,22007,22011],{"className":21996},[69],[34,21998],{"className":21999,"style":22000},[73],"height:0.9592em;vertical-align:-0.2296em;",[34,22002,13527],{"className":22003,"style":13594},[78,79],[34,22005],{"className":22006,"style":262},[237],[34,22008,21973],{"className":22009},[266,22010],"amsrm",[34,22012],{"className":22013,"style":262},[237],[34,22015,22017,22020],{"className":22016},[69],[34,22018],{"className":22019,"style":21829},[73],[34,22021,22023],{"className":22022},[78,19464],[34,22024,22026,22130],{"className":22025},[814,2523],[34,22027,22029,22127],{"className":22028},[818],[34,22030,22032,22115],{"className":22031,"style":21842},[822],[34,22033,22035,22038],{"className":22034,"style":19478},[10795],[34,22036],{"className":22037,"style":19482},[829],[34,22039,22041],{"className":22040,"style":19486},[78],[34,22042,22044,22047,22112],{"className":22043},[78],[34,22045],{"className":22046},[230,12689],[34,22048,22050],{"className":22049},[12543],[34,22051,22053,22104],{"className":22052},[814,2523],[34,22054,22056,22101],{"className":22055},[818],[34,22057,22059,22070,22078],{"className":22058,"style":15668},[822],[34,22060,22061,22064],{"style":14155},[34,22062],{"className":22063,"style":12709},[829],[34,22065,22067],{"className":22066},[78],[34,22068,12226],{"className":22069},[78,79],[34,22071,22072,22075],{"style":12897},[34,22073],{"className":22074,"style":12709},[829],[34,22076],{"className":22077,"style":12904},[12836],[34,22079,22080,22083],{"style":12907},[34,22081],{"className":22082,"style":12709},[829],[34,22084,22086,22089,22092,22095,22098],{"className":22085},[78],[34,22087,5250],{"className":22088},[78,79],[34,22090],{"className":22091,"style":238},[237],[34,22093,6463],{"className":22094},[242],[34,22096],{"className":22097,"style":238},[237],[34,22099,2213],{"className":22100},[78],[34,22102,2547],{"className":22103},[2546],[34,22105,22107],{"className":22106},[818],[34,22108,22110],{"className":22109,"style":14669},[822],[34,22111],{},[34,22113],{"className":22114},[258,12689],[34,22116,22117,22120],{"style":21928},[34,22118],{"className":22119,"style":19482},[829],[34,22121,22123],{"className":22122,"style":19558},[10803],[10806,22124,22125],{"xmlns":10808,"width":10809,"height":19561,"viewBox":19562,"preserveAspectRatio":19563},[10814,22126],{"d":19566},[34,22128,2547],{"className":22129},[2546],[34,22131,22133],{"className":22132},[818],[34,22134,22136],{"className":22135,"style":21948},[822],[34,22137],{},[16,22139,22140,22141,22169],{},"Intuition: ",[34,22142,22144,22157],{"className":22143,"translate":38},[37],[34,22145,22147],{"className":22146},[42],[44,22148,22149],{"xmlns":46},[48,22150,22151,22155],{},[51,22152,22153],{},[54,22154,12226],{},[58,22156,13280],{"encoding":60},[34,22158,22160],{"className":22159,"ariaHidden":65},[64],[34,22161,22163,22166],{"className":22162},[69],[34,22164],{"className":22165,"style":108},[73],[34,22167,12226],{"className":22168},[78,79]," determines the directional noise magnitude:",[34,22171,22173],{"className":22172,"translate":38},[159],[34,22174,22176,22209],{"className":22175,"translate":38},[37],[34,22177,22179],{"className":22178},[42],[44,22180,22181],{"xmlns":46,"display":168},[48,22182,22183,22206],{},[51,22184,22185,22187,22189,22192,22194,22204],{},[54,22186,13527],{},[176,22188,14925],{},[54,22190,22191],{},"O",[176,22193,179],{"stretchy":178},[19376,22195,22196],{},[51,22197,22198,22200,22202],{},[347,22199,2213],{},[54,22201,12580],{"mathvariant":2296},[54,22203,12226],{},[176,22205,189],{"stretchy":178},[58,22207,22208],{"encoding":60},"\\theta \\sim O(\\sqrt{1\u002F\\kappa})",[34,22210,22212,22230],{"className":22211,"ariaHidden":65},[64],[34,22213,22215,22218,22221,22224,22227],{"className":22214},[69],[34,22216],{"className":22217,"style":2184},[73],[34,22219,13527],{"className":22220,"style":13594},[78,79],[34,22222],{"className":22223,"style":262},[237],[34,22225,14925],{"className":22226},[266],[34,22228],{"className":22229,"style":262},[237],[34,22231,22233,22237,22240,22243,22303],{"className":22232},[69],[34,22234],{"className":22235,"style":22236},[73],"height:1.24em;vertical-align:-0.2561em;",[34,22238,22191],{"className":22239,"style":13594},[78,79],[34,22241,179],{"className":22242},[230],[34,22244,22246],{"className":22245},[78,19464],[34,22247,22249,22294],{"className":22248},[814,2523],[34,22250,22252,22291],{"className":22251},[818],[34,22253,22256,22274],{"className":22254,"style":22255},[822],"height:0.9839em;",[34,22257,22260,22264],{"className":22258,"style":22259},[10795],"top:-3.2em;",[34,22261],{"className":22262,"style":22263},[829],"height:3.2em;",[34,22265,22267,22271],{"className":22266,"style":19486},[78],[34,22268,22270],{"className":22269},[78],"1\u002F",[34,22272,12226],{"className":22273},[78,79],[34,22275,22277,22280],{"style":22276},"top:-2.9439em;",[34,22278],{"className":22279,"style":22263},[829],[34,22281,22284],{"className":22282,"style":22283},[10803],"min-width:1.02em;height:1.28em;",[10806,22285,22288],{"xmlns":10808,"width":10809,"height":22286,"viewBox":22287,"preserveAspectRatio":19563},"1.28em","0 0 400000 1296",[10814,22289],{"d":22290},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[34,22292,2547],{"className":22293},[2546],[34,22295,22297],{"className":22296},[818],[34,22298,22301],{"className":22299,"style":22300},[822],"height:0.2561em;",[34,22302],{},[34,22304,189],{"className":22305},[258],[16,22307,22308],{},"These formulas are frequently used in sphere embedding, CLIP embedding, contrastive learning, spherical clustering, and directional statistics.",[16,22310,22311,22312,22340],{},"For example, ",[34,22313,22315,22328],{"className":22314,"translate":38},[37],[34,22316,22318],{"className":22317},[42],[44,22319,22320],{"xmlns":46},[48,22321,22322,22326],{},[51,22323,22324],{},[54,22325,12226],{},[58,22327,13280],{"encoding":60},[34,22329,22331],{"className":22330,"ariaHidden":65},[64],[34,22332,22334,22337],{"className":22333},[69],[34,22335],{"className":22336,"style":108},[73],[34,22338,12226],{"className":22339},[78,79]," can be used to estimate: angular noise in embeddings, angular radius of clusters, confidence cone of prototypes.",[16,22342,22343],{},"The probability mass of the vMF distribution is concentrated at",[34,22345,22347],{"className":22346,"translate":38},[159],[34,22348,22350,22381],{"className":22349,"translate":38},[37],[34,22351,22353],{"className":22352},[42],[44,22354,22355],{"xmlns":46,"display":168},[48,22356,22357,22379],{},[51,22358,22359,22361,22363,22365,22367,22377],{},[54,22360,13527],{},[176,22362,14925],{},[54,22364,22191],{},[176,22366,179],{"stretchy":178},[19376,22368,22369],{},[51,22370,22371,22373,22375],{},[347,22372,2213],{},[54,22374,12580],{"mathvariant":2296},[54,22376,12226],{},[176,22378,189],{"stretchy":178},[58,22380,22208],{"encoding":60},[34,22382,22384,22402],{"className":22383,"ariaHidden":65},[64],[34,22385,22387,22390,22393,22396,22399],{"className":22386},[69],[34,22388],{"className":22389,"style":2184},[73],[34,22391,13527],{"className":22392,"style":13594},[78,79],[34,22394],{"className":22395,"style":262},[237],[34,22397,14925],{"className":22398},[266],[34,22400],{"className":22401,"style":262},[237],[34,22403,22405,22408,22411,22414,22464],{"className":22404},[69],[34,22406],{"className":22407,"style":22236},[73],[34,22409,22191],{"className":22410,"style":13594},[78,79],[34,22412,179],{"className":22413},[230],[34,22415,22417],{"className":22416},[78,19464],[34,22418,22420,22456],{"className":22419},[814,2523],[34,22421,22423,22453],{"className":22422},[818],[34,22424,22426,22441],{"className":22425,"style":22255},[822],[34,22427,22429,22432],{"className":22428,"style":22259},[10795],[34,22430],{"className":22431,"style":22263},[829],[34,22433,22435,22438],{"className":22434,"style":19486},[78],[34,22436,22270],{"className":22437},[78],[34,22439,12226],{"className":22440},[78,79],[34,22442,22443,22446],{"style":22276},[34,22444],{"className":22445,"style":22263},[829],[34,22447,22449],{"className":22448,"style":22283},[10803],[10806,22450,22451],{"xmlns":10808,"width":10809,"height":22286,"viewBox":22287,"preserveAspectRatio":19563},[10814,22452],{"d":22290},[34,22454,2547],{"className":22455},[2546],[34,22457,22459],{"className":22458},[818],[34,22460,22462],{"className":22461,"style":22300},[822],[34,22463],{},[34,22465,189],{"className":22466},[258],[16,22468,22469,22470,22498,22499,144],{},"The corresponding confidence region is a spherical cap with axis ",[34,22471,22473,22486],{"className":22472,"translate":38},[37],[34,22474,22476],{"className":22475},[42],[44,22477,22478],{"xmlns":46},[48,22479,22480,22484],{},[51,22481,22482],{},[54,22483,11720],{},[58,22485,11723],{"encoding":60},[34,22487,22489],{"className":22488,"ariaHidden":65},[64],[34,22490,22492,22495],{"className":22491},[69],[34,22493],{"className":22494,"style":139},[73],[34,22496,11720],{"className":22497},[78,79]," and angular radius roughly ",[34,22500,22502,22524],{"className":22501,"translate":38},[37],[34,22503,22505],{"className":22504},[42],[44,22506,22507],{"xmlns":46},[48,22508,22509,22521],{},[51,22510,22511],{},[19376,22512,22513],{},[51,22514,22515,22517,22519],{},[347,22516,2213],{},[54,22518,12580],{"mathvariant":2296},[54,22520,12226],{},[58,22522,22523],{"encoding":60},"\\sqrt{1\u002F\\kappa}",[34,22525,22527],{"className":22526,"ariaHidden":65},[64],[34,22528,22530,22534],{"className":22529},[69],[34,22531],{"className":22532,"style":22533},[73],"height:1.24em;vertical-align:-0.305em;",[34,22535,22537],{"className":22536},[78,19464],[34,22538,22540,22578],{"className":22539},[814,2523],[34,22541,22543,22575],{"className":22542},[818],[34,22544,22547,22562],{"className":22545,"style":22546},[822],"height:0.935em;",[34,22548,22550,22553],{"className":22549,"style":22259},[10795],[34,22551],{"className":22552,"style":22263},[829],[34,22554,22556,22559],{"className":22555,"style":19486},[78],[34,22557,22270],{"className":22558},[78],[34,22560,12226],{"className":22561},[78,79],[34,22563,22565,22568],{"style":22564},"top:-2.895em;",[34,22566],{"className":22567,"style":22263},[829],[34,22569,22571],{"className":22570,"style":22283},[10803],[10806,22572,22573],{"xmlns":10808,"width":10809,"height":22286,"viewBox":22287,"preserveAspectRatio":19563},[10814,22574],{"d":22290},[34,22576,2547],{"className":22577},[2546],[34,22579,22581],{"className":22580},[818],[34,22582,22585],{"className":22583,"style":22584},[822],"height:0.305em;",[34,22586],{},[11,22588,22590],{"id":22589},"parameter-estimation","Parameter Estimation",[16,22592,22593,22594,22622],{},"Estimating ",[34,22595,22597,22610],{"className":22596,"translate":38},[37],[34,22598,22600],{"className":22599},[42],[44,22601,22602],{"xmlns":46},[48,22603,22604,22608],{},[51,22605,22606],{},[54,22607,12226],{},[58,22609,13280],{"encoding":60},[34,22611,22613],{"className":22612,"ariaHidden":65},[64],[34,22614,22616,22619],{"className":22615},[69],[34,22617],{"className":22618,"style":108},[73],[34,22620,12226],{"className":22621},[78,79]," is the most challenging part, as it requires solving an equation involving modified Bessel functions.",[16,22624,22625,22626,22654],{},"Below are several practical approximation algorithms\u002Fformulas for estimating the vMF ",[34,22627,22629,22642],{"className":22628,"translate":38},[37],[34,22630,22632],{"className":22631},[42],[44,22633,22634],{"xmlns":46},[48,22635,22636,22640],{},[51,22637,22638],{},[54,22639,12226],{},[58,22641,13280],{"encoding":60},[34,22643,22645],{"className":22644,"ariaHidden":65},[64],[34,22646,22648,22651],{"className":22647},[69],[34,22649],{"className":22650,"style":108},[73],[34,22652,12226],{"className":22653},[78,79]," parameter, listed in approximate order of common usage.",[16,22656,22657,22658,22800,22801,144],{},"Given samples ",[34,22659,22661,22695],{"className":22660,"translate":38},[37],[34,22662,22664],{"className":22663},[42],[44,22665,22666],{"xmlns":46},[48,22667,22668,22692],{},[51,22669,22670,22676,22678,22680,22682,22684,22686],{},[2468,22671,22672,22674],{},[54,22673,96],{},[347,22675,2213],{},[176,22677,5228],{"separator":65},[54,22679,144],{"mathvariant":2296},[54,22681,144],{"mathvariant":2296},[54,22683,144],{"mathvariant":2296},[176,22685,5228],{"separator":65},[2468,22687,22688,22690],{},[54,22689,96],{},[54,22691,3000],{},[58,22693,22694],{"encoding":60},"x_1,...,x_n",[34,22696,22698],{"className":22697,"ariaHidden":65},[64],[34,22699,22701,22704,22744,22747,22750,22754,22757,22760],{"className":22700},[69],[34,22702],{"className":22703,"style":139},[73],[34,22705,22707,22710],{"className":22706},[78],[34,22708,96],{"className":22709},[78,79],[34,22711,22713],{"className":22712},[810],[34,22714,22716,22736],{"className":22715},[814,2523],[34,22717,22719,22733],{"className":22718},[818],[34,22720,22722],{"className":22721,"style":2530},[822],[34,22723,22724,22727],{"style":2533},[34,22725],{"className":22726,"style":830},[829],[34,22728,22730],{"className":22729},[834,835,836,837],[34,22731,2213],{"className":22732},[78,837],[34,22734,2547],{"className":22735},[2546],[34,22737,22739],{"className":22738},[818],[34,22740,22742],{"className":22741,"style":2554},[822],[34,22743],{},[34,22745,5228],{"className":22746},[5317],[34,22748],{"className":22749,"style":3462},[237],[34,22751,22753],{"className":22752},[78],"...",[34,22755,5228],{"className":22756},[5317],[34,22758],{"className":22759,"style":3462},[237],[34,22761,22763,22766],{"className":22762},[78],[34,22764,96],{"className":22765},[78,79],[34,22767,22769],{"className":22768},[810],[34,22770,22772,22792],{"className":22771},[814,2523],[34,22773,22775,22789],{"className":22774},[818],[34,22776,22778],{"className":22777,"style":5870},[822],[34,22779,22780,22783],{"style":2533},[34,22781],{"className":22782,"style":830},[829],[34,22784,22786],{"className":22785},[834,835,836,837],[34,22787,3000],{"className":22788},[78,79,837],[34,22790,2547],{"className":22791},[2546],[34,22793,22795],{"className":22794},[818],[34,22796,22798],{"className":22797,"style":2554},[822],[34,22799],{},", each ",[34,22802,22804,22836],{"className":22803,"translate":38},[37],[34,22805,22807],{"className":22806},[42],[44,22808,22809],{"xmlns":46},[48,22810,22811,22833],{},[51,22812,22813,22819,22821],{},[2468,22814,22815,22817],{},[54,22816,96],{},[54,22818,3317],{},[176,22820,11909],{},[755,22822,22823,22825],{},[54,22824,11890],{"mathvariant":2996},[51,22826,22827,22829,22831],{},[54,22828,5250],{},[176,22830,6463],{},[347,22832,2213],{},[58,22834,22835],{"encoding":60},"x_i \\in \\mathbb{S}^{d-1}",[34,22837,22839,22895],{"className":22838,"ariaHidden":65},[64],[34,22840,22842,22846,22886,22889,22892],{"className":22841},[69],[34,22843],{"className":22844,"style":22845},[73],"height:0.6891em;vertical-align:-0.15em;",[34,22847,22849,22852],{"className":22848},[78],[34,22850,96],{"className":22851},[78,79],[34,22853,22855],{"className":22854},[810],[34,22856,22858,22878],{"className":22857},[814,2523],[34,22859,22861,22875],{"className":22860},[818],[34,22862,22864],{"className":22863,"style":3482},[822],[34,22865,22866,22869],{"style":2533},[34,22867],{"className":22868,"style":830},[829],[34,22870,22872],{"className":22871},[834,835,836,837],[34,22873,3317],{"className":22874},[78,79,837],[34,22876,2547],{"className":22877},[2546],[34,22879,22881],{"className":22880},[818],[34,22882,22884],{"className":22883,"style":2554},[822],[34,22885],{},[34,22887],{"className":22888,"style":262},[237],[34,22890,11909],{"className":22891},[266],[34,22893],{"className":22894,"style":262},[237],[34,22896,22898,22901],{"className":22897},[69],[34,22899],{"className":22900,"style":12169},[73],[34,22902,22904,22907],{"className":22903},[78],[34,22905,11890],{"className":22906},[78,3048],[34,22908,22910],{"className":22909},[810],[34,22911,22913],{"className":22912},[814],[34,22914,22916],{"className":22915},[818],[34,22917,22919],{"className":22918,"style":12169},[822],[34,22920,22921,22924],{"style":825},[34,22922],{"className":22923,"style":830},[829],[34,22925,22927],{"className":22926},[834,835,836,837],[34,22928,22930,22933,22936],{"className":22929},[78,837],[34,22931,5250],{"className":22932},[78,79,837],[34,22934,6463],{"className":22935},[242,837],[34,22937,2213],{"className":22938},[78,837],[16,22940,22941],{},"First compute:",[34,22943,22945],{"className":22944,"translate":38},[159],[34,22946,22948,22999],{"className":22947,"translate":38},[37],[34,22949,22951],{"className":22950},[42],[44,22952,22953],{"xmlns":46,"display":168},[48,22954,22955,22996],{},[51,22956,22957,22964,22966],{},[10617,22958,22959,22961],{"accent":65},[54,22960,2997],{},[176,22962,22963],{},"ˉ",[176,22965,192],{},[12543,22967,22968,22994],{},[51,22969,22970,22972,22986,22992],{},[54,22971,12292],{"mathvariant":2296},[3308,22973,22974,22976,22984],{},[176,22975,3312],{},[51,22977,22978,22980,22982],{},[54,22979,3317],{},[176,22981,192],{},[347,22983,2213],{},[54,22985,3000],{},[2468,22987,22988,22990],{},[54,22989,96],{},[54,22991,3317],{},[54,22993,12292],{"mathvariant":2296},[54,22995,3000],{},[58,22997,22998],{"encoding":60},"\\bar{R} = \\frac{|\\sum_{i=1}^n x_i|}{n}",[34,23000,23002,23054],{"className":23001,"ariaHidden":65},[64],[34,23003,23005,23009,23045,23048,23051],{"className":23004},[69],[34,23006],{"className":23007,"style":23008},[73],"height:0.8201em;",[34,23010,23013],{"className":23011},[78,23012],"accent",[34,23014,23016],{"className":23015},[814],[34,23017,23019],{"className":23018},[818],[34,23020,23022,23031],{"className":23021,"style":23008},[822],[34,23023,23025,23028],{"style":23024},"top:-3em;",[34,23026],{"className":23027,"style":12709},[829],[34,23029,2997],{"className":23030,"style":14902},[78,79],[34,23032,23034,23037],{"style":23033},"top:-3.2523em;",[34,23035],{"className":23036,"style":12709},[829],[34,23038,23042],{"className":23039,"style":23041},[23040],"accent-body","left:-0.1667em;",[34,23043,22963],{"className":23044},[78],[34,23046],{"className":23047,"style":262},[237],[34,23049,192],{"className":23050},[266],[34,23052],{"className":23053,"style":262},[237],[34,23055,23057,23061],{"className":23056},[69],[34,23058],{"className":23059,"style":23060},[73],"height:2.18em;vertical-align:-0.686em;",[34,23062,23064,23067,23236],{"className":23063},[78],[34,23065],{"className":23066},[230,12689],[34,23068,23070],{"className":23069},[12543],[34,23071,23073,23228],{"className":23072},[814,2523],[34,23074,23076,23225],{"className":23075},[818],[34,23077,23080,23091,23099],{"className":23078,"style":23079},[822],"height:1.494em;",[34,23081,23082,23085],{"style":14155},[34,23083],{"className":23084,"style":12709},[829],[34,23086,23088],{"className":23087},[78],[34,23089,3000],{"className":23090},[78,79],[34,23092,23093,23096],{"style":12897},[34,23094],{"className":23095,"style":12709},[829],[34,23097],{"className":23098,"style":12904},[12836],[34,23100,23102,23105],{"style":23101},"top:-3.6897em;",[34,23103],{"className":23104,"style":12709},[829],[34,23106,23108,23111,23114,23179,23182,23222],{"className":23107},[78],[34,23109,12292],{"className":23110},[78],[34,23112],{"className":23113,"style":3462},[237],[34,23115,23117,23121],{"className":23116},[3387],[34,23118,3312],{"className":23119,"style":23120},[3387,3432,18253],"position:relative;top:0em;",[34,23122,23124],{"className":23123},[810],[34,23125,23127,23170],{"className":23126},[814,2523],[34,23128,23130,23167],{"className":23129},[818],[34,23131,23134,23155],{"className":23132,"style":23133},[822],"height:0.8043em;",[34,23135,23137,23140],{"style":23136},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[34,23138],{"className":23139,"style":830},[829],[34,23141,23143],{"className":23142},[834,835,836,837],[34,23144,23146,23149,23152],{"className":23145},[78,837],[34,23147,3317],{"className":23148},[78,79,837],[34,23150,192],{"className":23151},[266,837],[34,23153,2213],{"className":23154},[78,837],[34,23156,23158,23161],{"style":23157},"top:-3.2029em;margin-right:0.05em;",[34,23159],{"className":23160,"style":830},[829],[34,23162,23164],{"className":23163},[834,835,836,837],[34,23165,3000],{"className":23166},[78,79,837],[34,23168,2547],{"className":23169},[2546],[34,23171,23173],{"className":23172},[818],[34,23174,23177],{"className":23175,"style":23176},[822],"height:0.2997em;",[34,23178],{},[34,23180],{"className":23181,"style":3462},[237],[34,23183,23185,23188],{"className":23184},[78],[34,23186,96],{"className":23187},[78,79],[34,23189,23191],{"className":23190},[810],[34,23192,23194,23214],{"className":23193},[814,2523],[34,23195,23197,23211],{"className":23196},[818],[34,23198,23200],{"className":23199,"style":3482},[822],[34,23201,23202,23205],{"style":2533},[34,23203],{"className":23204,"style":830},[829],[34,23206,23208],{"className":23207},[834,835,836,837],[34,23209,3317],{"className":23210},[78,79,837],[34,23212,2547],{"className":23213},[2546],[34,23215,23217],{"className":23216},[818],[34,23218,23220],{"className":23219,"style":2554},[822],[34,23221],{},[34,23223,12292],{"className":23224},[78],[34,23226,2547],{"className":23227},[2546],[34,23229,23231],{"className":23230},[818],[34,23232,23234],{"className":23233,"style":14669},[822],[34,23235],{},[34,23237],{"className":23238},[258,12689],[16,23240,23241],{},"Mean direction:",[34,23243,23245],{"className":23244,"translate":38},[159],[34,23246,23248,23290],{"className":23247,"translate":38},[37],[34,23249,23251],{"className":23250},[42],[44,23252,23253],{"xmlns":46,"display":168},[48,23254,23255,23287],{},[51,23256,23257,23259,23261],{},[54,23258,11720],{},[176,23260,192],{},[12543,23262,23263,23273],{},[51,23264,23265,23267],{},[176,23266,3312],{},[2468,23268,23269,23271],{},[54,23270,96],{},[54,23272,3317],{},[51,23274,23275,23277,23279,23285],{},[54,23276,12292],{"mathvariant":2296},[176,23278,3312],{},[2468,23280,23281,23283],{},[54,23282,96],{},[54,23284,3317],{},[54,23286,12292],{"mathvariant":2296},[58,23288,23289],{"encoding":60},"\\mu = \\frac{\\sum x_i}{|\\sum x_i|}",[34,23291,23293,23311],{"className":23292,"ariaHidden":65},[64],[34,23294,23296,23299,23302,23305,23308],{"className":23295},[69],[34,23297],{"className":23298,"style":139},[73],[34,23300,11720],{"className":23301},[78,79],[34,23303],{"className":23304,"style":262},[237],[34,23306,192],{"className":23307},[266],[34,23309],{"className":23310,"style":262},[237],[34,23312,23314,23318],{"className":23313},[69],[34,23315],{"className":23316,"style":23317},[73],"height:2.363em;vertical-align:-0.936em;",[34,23319,23321,23324,23472],{"className":23320},[78],[34,23322],{"className":23323},[230,12689],[34,23325,23327],{"className":23326},[12543],[34,23328,23330,23464],{"className":23329},[814,2523],[34,23331,23333,23461],{"className":23332},[818],[34,23334,23336,23399,23407],{"className":23335,"style":14583},[822],[34,23337,23338,23341],{"style":14155},[34,23339],{"className":23340,"style":12709},[829],[34,23342,23344,23347,23350,23353,23356,23396],{"className":23343},[78],[34,23345,12292],{"className":23346},[78],[34,23348],{"className":23349,"style":3462},[237],[34,23351,3312],{"className":23352,"style":23120},[3387,3432,18253],[34,23354],{"className":23355,"style":3462},[237],[34,23357,23359,23362],{"className":23358},[78],[34,23360,96],{"className":23361},[78,79],[34,23363,23365],{"className":23364},[810],[34,23366,23368,23388],{"className":23367},[814,2523],[34,23369,23371,23385],{"className":23370},[818],[34,23372,23374],{"className":23373,"style":3482},[822],[34,23375,23376,23379],{"style":2533},[34,23377],{"className":23378,"style":830},[829],[34,23380,23382],{"className":23381},[834,835,836,837],[34,23383,3317],{"className":23384},[78,79,837],[34,23386,2547],{"className":23387},[2546],[34,23389,23391],{"className":23390},[818],[34,23392,23394],{"className":23393,"style":2554},[822],[34,23395],{},[34,23397,12292],{"className":23398},[78],[34,23400,23401,23404],{"style":12897},[34,23402],{"className":23403,"style":12709},[829],[34,23405],{"className":23406,"style":12904},[12836],[34,23408,23409,23412],{"style":12907},[34,23410],{"className":23411,"style":12709},[829],[34,23413,23415,23418,23421],{"className":23414},[78],[34,23416,3312],{"className":23417,"style":23120},[3387,3432,18253],[34,23419],{"className":23420,"style":3462},[237],[34,23422,23424,23427],{"className":23423},[78],[34,23425,96],{"className":23426},[78,79],[34,23428,23430],{"className":23429},[810],[34,23431,23433,23453],{"className":23432},[814,2523],[34,23434,23436,23450],{"className":23435},[818],[34,23437,23439],{"className":23438,"style":3482},[822],[34,23440,23441,23444],{"style":2533},[34,23442],{"className":23443,"style":830},[829],[34,23445,23447],{"className":23446},[834,835,836,837],[34,23448,3317],{"className":23449},[78,79,837],[34,23451,2547],{"className":23452},[2546],[34,23454,23456],{"className":23455},[818],[34,23457,23459],{"className":23458,"style":2554},[822],[34,23460],{},[34,23462,2547],{"className":23463},[2546],[34,23465,23467],{"className":23466},[818],[34,23468,23470],{"className":23469,"style":17481},[822],[34,23471],{},[34,23473],{"className":23474},[258,12689],[16,23476,23477,23478,23506],{},"The exact equation for ",[34,23479,23481,23494],{"className":23480,"translate":38},[37],[34,23482,23484],{"className":23483},[42],[44,23485,23486],{"xmlns":46},[48,23487,23488,23492],{},[51,23489,23490],{},[54,23491,12226],{},[58,23493,13280],{"encoding":60},[34,23495,23497],{"className":23496,"ariaHidden":65},[64],[34,23498,23500,23503],{"className":23499},[69],[34,23501],{"className":23502,"style":108},[73],[34,23504,12226],{"className":23505},[78,79]," is:",[34,23508,23510],{"className":23509,"translate":38},[159],[34,23511,23513,23545],{"className":23512,"translate":38},[37],[34,23514,23516],{"className":23515},[42],[44,23517,23518],{"xmlns":46,"display":168},[48,23519,23520,23542],{},[51,23521,23522,23528,23530,23532,23534,23536],{},[2468,23523,23524,23526],{},[54,23525,3145],{},[54,23527,5250],{},[176,23529,179],{"stretchy":178},[54,23531,12226],{},[176,23533,189],{"stretchy":178},[176,23535,192],{},[10617,23537,23538,23540],{"accent":65},[54,23539,2997],{},[176,23541,22963],{},[58,23543,23544],{"encoding":60},"A_d(\\kappa) = \\bar{R}",[34,23546,23548,23612],{"className":23547,"ariaHidden":65},[64],[34,23549,23551,23554,23594,23597,23600,23603,23606,23609],{"className":23550},[69],[34,23552],{"className":23553,"style":223},[73],[34,23555,23557,23560],{"className":23556},[78],[34,23558,3145],{"className":23559},[78,79],[34,23561,23563],{"className":23562},[810],[34,23564,23566,23586],{"className":23565},[814,2523],[34,23567,23569,23583],{"className":23568},[818],[34,23570,23572],{"className":23571,"style":6071},[822],[34,23573,23574,23577],{"style":2533},[34,23575],{"className":23576,"style":830},[829],[34,23578,23580],{"className":23579},[834,835,836,837],[34,23581,5250],{"className":23582},[78,79,837],[34,23584,2547],{"className":23585},[2546],[34,23587,23589],{"className":23588},[818],[34,23590,23592],{"className":23591,"style":2554},[822],[34,23593],{},[34,23595,179],{"className":23596},[230],[34,23598,12226],{"className":23599},[78,79],[34,23601,189],{"className":23602},[258],[34,23604],{"className":23605,"style":262},[237],[34,23607,192],{"className":23608},[266],[34,23610],{"className":23611,"style":262},[237],[34,23613,23615,23618],{"className":23614},[69],[34,23616],{"className":23617,"style":23008},[73],[34,23619,23621],{"className":23620},[78,23012],[34,23622,23624],{"className":23623},[814],[34,23625,23627],{"className":23626},[818],[34,23628,23630,23638],{"className":23629,"style":23008},[822],[34,23631,23632,23635],{"style":23024},[34,23633],{"className":23634,"style":12709},[829],[34,23636,2997],{"className":23637,"style":14902},[78,79],[34,23639,23640,23643],{"style":23033},[34,23641],{"className":23642,"style":12709},[829],[34,23644,23646],{"className":23645,"style":23041},[23040],[34,23647,22963],{"className":23648},[78],[16,23650,9212],{},[34,23652,23654],{"className":23653,"translate":38},[159],[34,23655,23657,23728],{"className":23656,"translate":38},[37],[34,23658,23660],{"className":23659},[42],[44,23661,23662],{"xmlns":46,"display":168},[48,23663,23664,23726],{},[51,23665,23666,23672,23674,23676,23678,23680],{},[2468,23667,23668,23670],{},[54,23669,3145],{},[54,23671,5250],{},[176,23673,179],{"stretchy":178},[54,23675,12226],{},[176,23677,189],{"stretchy":178},[176,23679,192],{},[12543,23681,23682,23702],{},[51,23683,23684,23696,23698,23700],{},[2468,23685,23686,23688],{},[54,23687,12587],{},[51,23689,23690,23692,23694],{},[54,23691,5250],{},[54,23693,12580],{"mathvariant":2296},[347,23695,424],{},[176,23697,179],{"stretchy":178},[54,23699,12226],{},[176,23701,189],{"stretchy":178},[51,23703,23704,23720,23722,23724],{},[2468,23705,23706,23708],{},[54,23707,12587],{},[51,23709,23710,23712,23714,23716,23718],{},[54,23711,5250],{},[54,23713,12580],{"mathvariant":2296},[347,23715,424],{},[176,23717,6463],{},[347,23719,2213],{},[176,23721,179],{"stretchy":178},[54,23723,12226],{},[176,23725,189],{"stretchy":178},[58,23727,14059],{"encoding":60},[34,23729,23731,23795],{"className":23730,"ariaHidden":65},[64],[34,23732,23734,23737,23777,23780,23783,23786,23789,23792],{"className":23733},[69],[34,23735],{"className":23736,"style":223},[73],[34,23738,23740,23743],{"className":23739},[78],[34,23741,3145],{"className":23742},[78,79],[34,23744,23746],{"className":23745},[810],[34,23747,23749,23769],{"className":23748},[814,2523],[34,23750,23752,23766],{"className":23751},[818],[34,23753,23755],{"className":23754,"style":6071},[822],[34,23756,23757,23760],{"style":2533},[34,23758],{"className":23759,"style":830},[829],[34,23761,23763],{"className":23762},[834,835,836,837],[34,23764,5250],{"className":23765},[78,79,837],[34,23767,2547],{"className":23768},[2546],[34,23770,23772],{"className":23771},[818],[34,23773,23775],{"className":23774,"style":2554},[822],[34,23776],{},[34,23778,179],{"className":23779},[230],[34,23781,12226],{"className":23782},[78,79],[34,23784,189],{"className":23785},[258],[34,23787],{"className":23788,"style":262},[237],[34,23790,192],{"className":23791},[266],[34,23793],{"className":23794,"style":262},[237],[34,23796,23798,23801],{"className":23797},[69],[34,23799],{"className":23800,"style":14133},[73],[34,23802,23804,23807,23970],{"className":23803},[78],[34,23805],{"className":23806},[230,12689],[34,23808,23810],{"className":23809},[12543],[34,23811,23813,23962],{"className":23812},[814,2523],[34,23814,23816,23959],{"className":23815},[818],[34,23817,23819,23888,23896],{"className":23818,"style":14152},[822],[34,23820,23821,23824],{"style":14155},[34,23822],{"className":23823,"style":12709},[829],[34,23825,23827,23879,23882,23885],{"className":23826},[78],[34,23828,23830,23833],{"className":23829},[78],[34,23831,12587],{"className":23832,"style":12766},[78,79],[34,23834,23836],{"className":23835},[810],[34,23837,23839,23871],{"className":23838},[814,2523],[34,23840,23842,23868],{"className":23841},[818],[34,23843,23845],{"className":23844,"style":12779},[822],[34,23846,23847,23850],{"style":14182},[34,23848],{"className":23849,"style":830},[829],[34,23851,23853],{"className":23852},[834,835,836,837],[34,23854,23856,23859,23862,23865],{"className":23855},[78,837],[34,23857,5250],{"className":23858},[78,79,837],[34,23860,12759],{"className":23861},[78,837],[34,23863,6463],{"className":23864},[242,837],[34,23866,2213],{"className":23867},[78,837],[34,23869,2547],{"className":23870},[2546],[34,23872,23874],{"className":23873},[818],[34,23875,23877],{"className":23876,"style":14213},[822],[34,23878],{},[34,23880,179],{"className":23881},[230],[34,23883,12226],{"className":23884},[78,79],[34,23886,189],{"className":23887},[258],[34,23889,23890,23893],{"style":12897},[34,23891],{"className":23892,"style":12709},[829],[34,23894],{"className":23895,"style":12904},[12836],[34,23897,23898,23901],{"style":14235},[34,23899],{"className":23900,"style":12709},[829],[34,23902,23904,23950,23953,23956],{"className":23903},[78],[34,23905,23907,23910],{"className":23906},[78],[34,23908,12587],{"className":23909,"style":12766},[78,79],[34,23911,23913],{"className":23912},[810],[34,23914,23916,23942],{"className":23915},[814,2523],[34,23917,23919,23939],{"className":23918},[818],[34,23920,23922],{"className":23921,"style":12779},[822],[34,23923,23924,23927],{"style":14182},[34,23925],{"className":23926,"style":830},[829],[34,23928,23930],{"className":23929},[834,835,836,837],[34,23931,23933,23936],{"className":23932},[78,837],[34,23934,5250],{"className":23935},[78,79,837],[34,23937,12759],{"className":23938},[78,837],[34,23940,2547],{"className":23941},[2546],[34,23943,23945],{"className":23944},[818],[34,23946,23948],{"className":23947,"style":14213},[822],[34,23949],{},[34,23951,179],{"className":23952},[230],[34,23954,12226],{"className":23955},[78,79],[34,23957,189],{"className":23958},[258],[34,23960,2547],{"className":23961},[2546],[34,23963,23965],{"className":23964},[818],[34,23966,23968],{"className":23967,"style":14306},[822],[34,23969],{},[34,23971],{"className":23972},[258,12689],[16,23974,23975],{},"This involves the modified Bessel function of the first kind, so approximations or numerical solutions are necessary.",[146,23977,23979],{"id":23978},"banerjee-approximation-most-commonly-used","Banerjee Approximation (most commonly used)",[16,23981,23982,23983],{},"Source: Arindam Banerjee, 2005: ",[11798,23984,23985],{},"Clustering on the Unit Hypersphere using von Mises-Fisher",[16,23987,23988,23989],{},"Applicable for high dimensions ",[34,23990,23992,24010],{"className":23991,"translate":38},[37],[34,23993,23995],{"className":23994},[42],[44,23996,23997],{"xmlns":46},[48,23998,23999,24007],{},[51,24000,24001,24003,24005],{},[54,24002,5250],{},[176,24004,12229],{},[347,24006,429],{},[58,24008,24009],{"encoding":60},"d \\ge 3",[34,24011,24013,24031],{"className":24012,"ariaHidden":65},[64],[34,24014,24016,24019,24022,24025,24028],{"className":24015},[69],[34,24017],{"className":24018,"style":16482},[73],[34,24020,5250],{"className":24021},[78,79],[34,24023],{"className":24024,"style":262},[237],[34,24026,12229],{"className":24027},[266],[34,24029],{"className":24030,"style":262},[237],[34,24032,24034,24037],{"className":24033},[69],[34,24035],{"className":24036,"style":391},[73],[34,24038,429],{"className":24039},[78],[34,24041,24043],{"className":24042,"translate":38},[159],[34,24044,24046,24106],{"className":24045,"translate":38},[37],[34,24047,24049],{"className":24048},[42],[44,24050,24051],{"xmlns":46,"display":168},[48,24052,24053,24103],{},[51,24054,24055,24057,24059],{},[54,24056,12226],{},[176,24058,18755],{},[12543,24060,24061,24087],{},[51,24062,24063,24069,24071,24073,24075,24085],{},[10617,24064,24065,24067],{"accent":65},[54,24066,2997],{},[176,24068,22963],{},[176,24070,179],{"stretchy":178},[54,24072,5250],{},[176,24074,6463],{},[755,24076,24077,24083],{},[10617,24078,24079,24081],{"accent":65},[54,24080,2997],{},[176,24082,22963],{},[347,24084,424],{},[176,24086,189],{"stretchy":178},[51,24088,24089,24091,24093],{},[347,24090,2213],{},[176,24092,6463],{},[755,24094,24095,24101],{},[10617,24096,24097,24099],{"accent":65},[54,24098,2997],{},[176,24100,22963],{},[347,24102,424],{},[58,24104,24105],{"encoding":60},"\\kappa \\approx \\frac{\\bar{R}(d-\\bar{R}^2)}{1-\\bar{R}^2}",[34,24107,24109,24128],{"className":24108,"ariaHidden":65},[64],[34,24110,24112,24116,24119,24122,24125],{"className":24111},[69],[34,24113],{"className":24114,"style":24115},[73],"height:0.4831em;",[34,24117,12226],{"className":24118},[78,79],[34,24120],{"className":24121,"style":262},[237],[34,24123,18755],{"className":24124},[266],[34,24126],{"className":24127,"style":262},[237],[34,24129,24131,24135],{"className":24130},[69],[34,24132],{"className":24133,"style":24134},[73],"height:2.2905em;vertical-align:-0.7934em;",[34,24136,24138,24141,24366],{"className":24137},[78],[34,24139],{"className":24140},[230,12689],[34,24142,24144],{"className":24143},[12543],[34,24145,24147,24357],{"className":24146},[814,2523],[34,24148,24150,24354],{"className":24149},[818],[34,24151,24154,24232,24240],{"className":24152,"style":24153},[822],"height:1.4971em;",[34,24155,24157,24160],{"style":24156},"top:-2.2899em;",[34,24158],{"className":24159,"style":12709},[829],[34,24161,24163,24166,24169,24172,24175],{"className":24162},[78],[34,24164,2213],{"className":24165},[78],[34,24167],{"className":24168,"style":238},[237],[34,24170,6463],{"className":24171},[242],[34,24173],{"className":24174,"style":238},[237],[34,24176,24178,24209],{"className":24177},[78],[34,24179,24181],{"className":24180},[78,23012],[34,24182,24184],{"className":24183},[814],[34,24185,24187],{"className":24186},[818],[34,24188,24190,24198],{"className":24189,"style":23008},[822],[34,24191,24192,24195],{"style":23024},[34,24193],{"className":24194,"style":12709},[829],[34,24196,2997],{"className":24197,"style":14902},[78,79],[34,24199,24200,24203],{"style":23033},[34,24201],{"className":24202,"style":12709},[829],[34,24204,24206],{"className":24205,"style":23041},[23040],[34,24207,22963],{"className":24208},[78],[34,24210,24212],{"className":24211},[810],[34,24213,24215],{"className":24214},[814],[34,24216,24218],{"className":24217},[818],[34,24219,24221],{"className":24220,"style":15259},[822],[34,24222,24223,24226],{"style":12743},[34,24224],{"className":24225,"style":830},[829],[34,24227,24229],{"className":24228},[834,835,836,837],[34,24230,424],{"className":24231},[78,837],[34,24233,24234,24237],{"style":12897},[34,24235],{"className":24236,"style":12709},[829],[34,24238],{"className":24239,"style":12904},[12836],[34,24241,24242,24245],{"style":12907},[34,24243],{"className":24244,"style":12709},[829],[34,24246,24248,24279,24282,24285,24288,24291,24294,24351],{"className":24247},[78],[34,24249,24251],{"className":24250},[78,23012],[34,24252,24254],{"className":24253},[814],[34,24255,24257],{"className":24256},[818],[34,24258,24260,24268],{"className":24259,"style":23008},[822],[34,24261,24262,24265],{"style":23024},[34,24263],{"className":24264,"style":12709},[829],[34,24266,2997],{"className":24267,"style":14902},[78,79],[34,24269,24270,24273],{"style":23033},[34,24271],{"className":24272,"style":12709},[829],[34,24274,24276],{"className":24275,"style":23041},[23040],[34,24277,22963],{"className":24278},[78],[34,24280,179],{"className":24281},[230],[34,24283,5250],{"className":24284},[78,79],[34,24286],{"className":24287,"style":238},[237],[34,24289,6463],{"className":24290},[242],[34,24292],{"className":24293,"style":238},[237],[34,24295,24297,24328],{"className":24296},[78],[34,24298,24300],{"className":24299},[78,23012],[34,24301,24303],{"className":24302},[814],[34,24304,24306],{"className":24305},[818],[34,24307,24309,24317],{"className":24308,"style":23008},[822],[34,24310,24311,24314],{"style":23024},[34,24312],{"className":24313,"style":12709},[829],[34,24315,2997],{"className":24316,"style":14902},[78,79],[34,24318,24319,24322],{"style":23033},[34,24320],{"className":24321,"style":12709},[829],[34,24323,24325],{"className":24324,"style":23041},[23040],[34,24326,22963],{"className":24327},[78],[34,24329,24331],{"className":24330},[810],[34,24332,24334],{"className":24333},[814],[34,24335,24337],{"className":24336},[818],[34,24338,24340],{"className":24339,"style":800},[822],[34,24341,24342,24345],{"style":825},[34,24343],{"className":24344,"style":830},[829],[34,24346,24348],{"className":24347},[834,835,836,837],[34,24349,424],{"className":24350},[78,837],[34,24352,189],{"className":24353},[258],[34,24355,2547],{"className":24356},[2546],[34,24358,24360],{"className":24359},[818],[34,24361,24364],{"className":24362,"style":24363},[822],"height:0.7934em;",[34,24365],{},[34,24367],{"className":24368},[258,12689],[16,24370,24371,24372,24416],{},"Advantages: extremely simple, ",[34,24373,24375,24395],{"className":24374,"translate":38},[37],[34,24376,24378],{"className":24377},[42],[44,24379,24380],{"xmlns":46},[48,24381,24382,24392],{},[51,24383,24384,24386,24388,24390],{},[54,24385,22191],{},[176,24387,179],{"stretchy":178},[347,24389,2213],{},[176,24391,189],{"stretchy":178},[58,24393,24394],{"encoding":60},"O(1)",[34,24396,24398],{"className":24397,"ariaHidden":65},[64],[34,24399,24401,24404,24407,24410,24413],{"className":24400},[69],[34,24402],{"className":24403,"style":223},[73],[34,24405,22191],{"className":24406,"style":13594},[78,79],[34,24408,179],{"className":24409},[230],[34,24411,2213],{"className":24412},[78],[34,24414,189],{"className":24415},[258]," computation, works well in high dimensions. Disadvantages: larger bias in low dimensions.",[16,24418,24419],{},"This is the most common initialization formula for spherical k-means \u002F vMF mixtures.",[146,24421,24423],{"id":24422},"sra-approximation-newton-refinement-recommended","Sra Approximation + Newton Refinement (recommended)",[16,24425,24426,24427],{},"Source: Suvrit Sra, 2012: ",[11798,24428,24429],{},"A Short Note on Parameter Approximation for vMF",[16,24431,24432],{},"Initial value:",[34,24434,24436],{"className":24435,"translate":38},[159],[34,24437,24439,24503],{"className":24438,"translate":38},[37],[34,24440,24442],{"className":24441},[42],[44,24443,24444],{"xmlns":46,"display":168},[48,24445,24446,24500],{},[51,24447,24448,24454,24456],{},[2468,24449,24450,24452],{},[54,24451,12226],{},[347,24453,2287],{},[176,24455,192],{},[12543,24457,24458,24484],{},[51,24459,24460,24466,24468,24470,24472,24482],{},[10617,24461,24462,24464],{"accent":65},[54,24463,2997],{},[176,24465,22963],{},[176,24467,179],{"stretchy":178},[54,24469,5250],{},[176,24471,6463],{},[755,24473,24474,24480],{},[10617,24475,24476,24478],{"accent":65},[54,24477,2997],{},[176,24479,22963],{},[347,24481,424],{},[176,24483,189],{"stretchy":178},[51,24485,24486,24488,24490],{},[347,24487,2213],{},[176,24489,6463],{},[755,24491,24492,24498],{},[10617,24493,24494,24496],{"accent":65},[54,24495,2997],{},[176,24497,22963],{},[347,24499,424],{},[58,24501,24502],{"encoding":60},"\\kappa_0 = \\frac{\\bar{R}(d-\\bar{R}^2)}{1-\\bar{R}^2}",[34,24504,24506,24561],{"className":24505,"ariaHidden":65},[64],[34,24507,24509,24512,24552,24555,24558],{"className":24508},[69],[34,24510],{"className":24511,"style":2625},[73],[34,24513,24515,24518],{"className":24514},[78],[34,24516,12226],{"className":24517},[78,79],[34,24519,24521],{"className":24520},[810],[34,24522,24524,24544],{"className":24523},[814,2523],[34,24525,24527,24541],{"className":24526},[818],[34,24528,24530],{"className":24529,"style":2530},[822],[34,24531,24532,24535],{"style":2533},[34,24533],{"className":24534,"style":830},[829],[34,24536,24538],{"className":24537},[834,835,836,837],[34,24539,2287],{"className":24540},[78,837],[34,24542,2547],{"className":24543},[2546],[34,24545,24547],{"className":24546},[818],[34,24548,24550],{"className":24549,"style":2554},[822],[34,24551],{},[34,24553],{"className":24554,"style":262},[237],[34,24556,192],{"className":24557},[266],[34,24559],{"className":24560,"style":262},[237],[34,24562,24564,24567],{"className":24563},[69],[34,24565],{"className":24566,"style":24134},[73],[34,24568,24570,24573,24795],{"className":24569},[78],[34,24571],{"className":24572},[230,12689],[34,24574,24576],{"className":24575},[12543],[34,24577,24579,24787],{"className":24578},[814,2523],[34,24580,24582,24784],{"className":24581},[818],[34,24583,24585,24662,24670],{"className":24584,"style":24153},[822],[34,24586,24587,24590],{"style":24156},[34,24588],{"className":24589,"style":12709},[829],[34,24591,24593,24596,24599,24602,24605],{"className":24592},[78],[34,24594,2213],{"className":24595},[78],[34,24597],{"className":24598,"style":238},[237],[34,24600,6463],{"className":24601},[242],[34,24603],{"className":24604,"style":238},[237],[34,24606,24608,24639],{"className":24607},[78],[34,24609,24611],{"className":24610},[78,23012],[34,24612,24614],{"className":24613},[814],[34,24615,24617],{"className":24616},[818],[34,24618,24620,24628],{"className":24619,"style":23008},[822],[34,24621,24622,24625],{"style":23024},[34,24623],{"className":24624,"style":12709},[829],[34,24626,2997],{"className":24627,"style":14902},[78,79],[34,24629,24630,24633],{"style":23033},[34,24631],{"className":24632,"style":12709},[829],[34,24634,24636],{"className":24635,"style":23041},[23040],[34,24637,22963],{"className":24638},[78],[34,24640,24642],{"className":24641},[810],[34,24643,24645],{"className":24644},[814],[34,24646,24648],{"className":24647},[818],[34,24649,24651],{"className":24650,"style":15259},[822],[34,24652,24653,24656],{"style":12743},[34,24654],{"className":24655,"style":830},[829],[34,24657,24659],{"className":24658},[834,835,836,837],[34,24660,424],{"className":24661},[78,837],[34,24663,24664,24667],{"style":12897},[34,24665],{"className":24666,"style":12709},[829],[34,24668],{"className":24669,"style":12904},[12836],[34,24671,24672,24675],{"style":12907},[34,24673],{"className":24674,"style":12709},[829],[34,24676,24678,24709,24712,24715,24718,24721,24724,24781],{"className":24677},[78],[34,24679,24681],{"className":24680},[78,23012],[34,24682,24684],{"className":24683},[814],[34,24685,24687],{"className":24686},[818],[34,24688,24690,24698],{"className":24689,"style":23008},[822],[34,24691,24692,24695],{"style":23024},[34,24693],{"className":24694,"style":12709},[829],[34,24696,2997],{"className":24697,"style":14902},[78,79],[34,24699,24700,24703],{"style":23033},[34,24701],{"className":24702,"style":12709},[829],[34,24704,24706],{"className":24705,"style":23041},[23040],[34,24707,22963],{"className":24708},[78],[34,24710,179],{"className":24711},[230],[34,24713,5250],{"className":24714},[78,79],[34,24716],{"className":24717,"style":238},[237],[34,24719,6463],{"className":24720},[242],[34,24722],{"className":24723,"style":238},[237],[34,24725,24727,24758],{"className":24726},[78],[34,24728,24730],{"className":24729},[78,23012],[34,24731,24733],{"className":24732},[814],[34,24734,24736],{"className":24735},[818],[34,24737,24739,24747],{"className":24738,"style":23008},[822],[34,24740,24741,24744],{"style":23024},[34,24742],{"className":24743,"style":12709},[829],[34,24745,2997],{"className":24746,"style":14902},[78,79],[34,24748,24749,24752],{"style":23033},[34,24750],{"className":24751,"style":12709},[829],[34,24753,24755],{"className":24754,"style":23041},[23040],[34,24756,22963],{"className":24757},[78],[34,24759,24761],{"className":24760},[810],[34,24762,24764],{"className":24763},[814],[34,24765,24767],{"className":24766},[818],[34,24768,24770],{"className":24769,"style":800},[822],[34,24771,24772,24775],{"style":825},[34,24773],{"className":24774,"style":830},[829],[34,24776,24778],{"className":24777},[834,835,836,837],[34,24779,424],{"className":24780},[78,837],[34,24782,189],{"className":24783},[258],[34,24785,2547],{"className":24786},[2546],[34,24788,24790],{"className":24789},[818],[34,24791,24793],{"className":24792,"style":24363},[822],[34,24794],{},[34,24796],{"className":24797},[258,12689],[16,24799,24800],{},"Then apply Newton iteration:",[34,24802,24804],{"className":24803,"translate":38},[159],[34,24805,24807,24930],{"className":24806,"translate":38},[37],[34,24808,24810],{"className":24809},[42],[44,24811,24812],{"xmlns":46,"display":168},[48,24813,24814,24927],{},[51,24815,24816,24829,24831,24837,24839],{},[2468,24817,24818,24820],{},[54,24819,12226],{},[51,24821,24822,24825,24827],{},[54,24823,24824],{},"t",[176,24826,184],{},[347,24828,2213],{},[176,24830,192],{},[2468,24832,24833,24835],{},[54,24834,12226],{},[54,24836,24824],{},[176,24838,6463],{},[12543,24840,24841,24867],{},[51,24842,24843,24849,24851,24857,24859,24861],{},[2468,24844,24845,24847],{},[54,24846,3145],{},[54,24848,5250],{},[176,24850,179],{"stretchy":178},[2468,24852,24853,24855],{},[54,24854,12226],{},[54,24856,24824],{},[176,24858,189],{"stretchy":178},[176,24860,6463],{},[10617,24862,24863,24865],{"accent":65},[54,24864,2997],{},[176,24866,22963],{},[51,24868,24869,24871,24873,24879,24881,24887,24893,24895,24911,24917,24919,24925],{},[347,24870,2213],{},[176,24872,6463],{},[2468,24874,24875,24877],{},[54,24876,3145],{},[54,24878,5250],{},[176,24880,179],{"stretchy":178},[2468,24882,24883,24885],{},[54,24884,12226],{},[54,24886,24824],{},[755,24888,24889,24891],{},[176,24890,189],{"stretchy":178},[347,24892,424],{},[176,24894,6463],{},[12543,24896,24897,24905],{},[51,24898,24899,24901,24903],{},[54,24900,5250],{},[176,24902,6463],{},[347,24904,2213],{},[2468,24906,24907,24909],{},[54,24908,12226],{},[54,24910,24824],{},[2468,24912,24913,24915],{},[54,24914,3145],{},[54,24916,5250],{},[176,24918,179],{"stretchy":178},[2468,24920,24921,24923],{},[54,24922,12226],{},[54,24924,24824],{},[176,24926,189],{"stretchy":178},[58,24928,24929],{"encoding":60},"\\kappa_{t+1} = \\kappa_t - \\frac{A_d(\\kappa_t)-\\bar{R}}{1-A_d(\\kappa_t)^2-\\frac{d-1}{\\kappa_t}A_d(\\kappa_t)}",[34,24931,24933,24998,25054],{"className":24932,"ariaHidden":65},[64],[34,24934,24936,24940,24989,24992,24995],{"className":24935},[69],[34,24937],{"className":24938,"style":24939},[73],"height:0.6389em;vertical-align:-0.2083em;",[34,24941,24943,24946],{"className":24942},[78],[34,24944,12226],{"className":24945},[78,79],[34,24947,24949],{"className":24948},[810],[34,24950,24952,24981],{"className":24951},[814,2523],[34,24953,24955,24978],{"className":24954},[818],[34,24956,24958],{"className":24957,"style":2530},[822],[34,24959,24960,24963],{"style":2533},[34,24961],{"className":24962,"style":830},[829],[34,24964,24966],{"className":24965},[834,835,836,837],[34,24967,24969,24972,24975],{"className":24968},[78,837],[34,24970,24824],{"className":24971},[78,79,837],[34,24973,184],{"className":24974},[242,837],[34,24976,2213],{"className":24977},[78,837],[34,24979,2547],{"className":24980},[2546],[34,24982,24984],{"className":24983},[818],[34,24985,24987],{"className":24986,"style":6830},[822],[34,24988],{},[34,24990],{"className":24991,"style":262},[237],[34,24993,192],{"className":24994},[266],[34,24996],{"className":24997,"style":262},[237],[34,24999,25001,25004,25045,25048,25051],{"className":25000},[69],[34,25002],{"className":25003,"style":2762},[73],[34,25005,25007,25010],{"className":25006},[78],[34,25008,12226],{"className":25009},[78,79],[34,25011,25013],{"className":25012},[810],[34,25014,25016,25037],{"className":25015},[814,2523],[34,25017,25019,25034],{"className":25018},[818],[34,25020,25023],{"className":25021,"style":25022},[822],"height:0.2806em;",[34,25024,25025,25028],{"style":2533},[34,25026],{"className":25027,"style":830},[829],[34,25029,25031],{"className":25030},[834,835,836,837],[34,25032,24824],{"className":25033},[78,79,837],[34,25035,2547],{"className":25036},[2546],[34,25038,25040],{"className":25039},[818],[34,25041,25043],{"className":25042,"style":2554},[822],[34,25044],{},[34,25046],{"className":25047,"style":238},[237],[34,25049,6463],{"className":25050},[242],[34,25052],{"className":25053,"style":238},[237],[34,25055,25057,25061],{"className":25056},[69],[34,25058],{"className":25059,"style":25060},[73],"height:2.7123em;vertical-align:-1.2152em;",[34,25062,25064,25067,25575],{"className":25063},[78],[34,25065],{"className":25066},[230,12689],[34,25068,25070],{"className":25069},[12543],[34,25071,25073,25566],{"className":25072},[814,2523],[34,25074,25076,25563],{"className":25075},[818],[34,25077,25079,25421,25429],{"className":25078,"style":24153},[822],[34,25080,25082,25085],{"style":25081},"top:-2.2299em;",[34,25083],{"className":25084,"style":12709},[829],[34,25086,25088,25091,25094,25097,25100,25140,25143,25183,25212,25215,25218,25221,25335,25375,25378,25418],{"className":25087},[78],[34,25089,2213],{"className":25090},[78],[34,25092],{"className":25093,"style":238},[237],[34,25095,6463],{"className":25096},[242],[34,25098],{"className":25099,"style":238},[237],[34,25101,25103,25106],{"className":25102},[78],[34,25104,3145],{"className":25105},[78,79],[34,25107,25109],{"className":25108},[810],[34,25110,25112,25132],{"className":25111},[814,2523],[34,25113,25115,25129],{"className":25114},[818],[34,25116,25118],{"className":25117,"style":6071},[822],[34,25119,25120,25123],{"style":2533},[34,25121],{"className":25122,"style":830},[829],[34,25124,25126],{"className":25125},[834,835,836,837],[34,25127,5250],{"className":25128},[78,79,837],[34,25130,2547],{"className":25131},[2546],[34,25133,25135],{"className":25134},[818],[34,25136,25138],{"className":25137,"style":2554},[822],[34,25139],{},[34,25141,179],{"className":25142},[230],[34,25144,25146,25149],{"className":25145},[78],[34,25147,12226],{"className":25148},[78,79],[34,25150,25152],{"className":25151},[810],[34,25153,25155,25175],{"className":25154},[814,2523],[34,25156,25158,25172],{"className":25157},[818],[34,25159,25161],{"className":25160,"style":25022},[822],[34,25162,25163,25166],{"style":2533},[34,25164],{"className":25165,"style":830},[829],[34,25167,25169],{"className":25168},[834,835,836,837],[34,25170,24824],{"className":25171},[78,79,837],[34,25173,2547],{"className":25174},[2546],[34,25176,25178],{"className":25177},[818],[34,25179,25181],{"className":25180,"style":2554},[822],[34,25182],{},[34,25184,25186,25189],{"className":25185},[258],[34,25187,189],{"className":25188},[258],[34,25190,25192],{"className":25191},[810],[34,25193,25195],{"className":25194},[814],[34,25196,25198],{"className":25197},[818],[34,25199,25201],{"className":25200,"style":15259},[822],[34,25202,25203,25206],{"style":12743},[34,25204],{"className":25205,"style":830},[829],[34,25207,25209],{"className":25208},[834,835,836,837],[34,25210,424],{"className":25211},[78,837],[34,25213],{"className":25214,"style":238},[237],[34,25216,6463],{"className":25217},[242],[34,25219],{"className":25220,"style":238},[237],[34,25222,25224,25227,25332],{"className":25223},[78],[34,25225],{"className":25226},[230,12689],[34,25228,25230],{"className":25229},[12543],[34,25231,25233,25323],{"className":25232},[814,2523],[34,25234,25236,25320],{"className":25235},[818],[34,25237,25239,25292,25300],{"className":25238,"style":17903},[822],[34,25240,25241,25244],{"style":17906},[34,25242],{"className":25243,"style":12709},[829],[34,25245,25247],{"className":25246},[834,835,836,837],[34,25248,25250],{"className":25249},[78,837],[34,25251,25253,25256],{"className":25252},[78,837],[34,25254,12226],{"className":25255},[78,79,837],[34,25257,25259],{"className":25258},[810],[34,25260,25262,25284],{"className":25261},[814,2523],[34,25263,25265,25281],{"className":25264},[818],[34,25266,25269],{"className":25267,"style":25268},[822],"height:0.2963em;",[34,25270,25272,25275],{"style":25271},"top:-2.357em;margin-left:0em;margin-right:0.0714em;",[34,25273],{"className":25274,"style":10772},[829],[34,25276,25278],{"className":25277},[834,10776,9740,837],[34,25279,24824],{"className":25280},[78,79,837],[34,25282,2547],{"className":25283},[2546],[34,25285,25287],{"className":25286},[818],[34,25288,25290],{"className":25289,"style":10789},[822],[34,25291],{},[34,25293,25294,25297],{"style":12897},[34,25295],{"className":25296,"style":12709},[829],[34,25298],{"className":25299,"style":12904},[12836],[34,25301,25302,25305],{"style":17929},[34,25303],{"className":25304,"style":12709},[829],[34,25306,25308],{"className":25307},[834,835,836,837],[34,25309,25311,25314,25317],{"className":25310},[78,837],[34,25312,5250],{"className":25313},[78,79,837],[34,25315,6463],{"className":25316},[242,837],[34,25318,2213],{"className":25319},[78,837],[34,25321,2547],{"className":25322},[2546],[34,25324,25326],{"className":25325},[818],[34,25327,25330],{"className":25328,"style":25329},[822],"height:0.4451em;",[34,25331],{},[34,25333],{"className":25334},[258,12689],[34,25336,25338,25341],{"className":25337},[78],[34,25339,3145],{"className":25340},[78,79],[34,25342,25344],{"className":25343},[810],[34,25345,25347,25367],{"className":25346},[814,2523],[34,25348,25350,25364],{"className":25349},[818],[34,25351,25353],{"className":25352,"style":6071},[822],[34,25354,25355,25358],{"style":2533},[34,25356],{"className":25357,"style":830},[829],[34,25359,25361],{"className":25360},[834,835,836,837],[34,25362,5250],{"className":25363},[78,79,837],[34,25365,2547],{"className":25366},[2546],[34,25368,25370],{"className":25369},[818],[34,25371,25373],{"className":25372,"style":2554},[822],[34,25374],{},[34,25376,179],{"className":25377},[230],[34,25379,25381,25384],{"className":25380},[78],[34,25382,12226],{"className":25383},[78,79],[34,25385,25387],{"className":25386},[810],[34,25388,25390,25410],{"className":25389},[814,2523],[34,25391,25393,25407],{"className":25392},[818],[34,25394,25396],{"className":25395,"style":25022},[822],[34,25397,25398,25401],{"style":2533},[34,25399],{"className":25400,"style":830},[829],[34,25402,25404],{"className":25403},[834,835,836,837],[34,25405,24824],{"className":25406},[78,79,837],[34,25408,2547],{"className":25409},[2546],[34,25411,25413],{"className":25412},[818],[34,25414,25416],{"className":25415,"style":2554},[822],[34,25417],{},[34,25419,189],{"className":25420},[258],[34,25422,25423,25426],{"style":12897},[34,25424],{"className":25425,"style":12709},[829],[34,25427],{"className":25428,"style":12904},[12836],[34,25430,25431,25434],{"style":12907},[34,25432],{"className":25433,"style":12709},[829],[34,25435,25437,25477,25480,25520,25523,25526,25529,25532],{"className":25436},[78],[34,25438,25440,25443],{"className":25439},[78],[34,25441,3145],{"className":25442},[78,79],[34,25444,25446],{"className":25445},[810],[34,25447,25449,25469],{"className":25448},[814,2523],[34,25450,25452,25466],{"className":25451},[818],[34,25453,25455],{"className":25454,"style":6071},[822],[34,25456,25457,25460],{"style":2533},[34,25458],{"className":25459,"style":830},[829],[34,25461,25463],{"className":25462},[834,835,836,837],[34,25464,5250],{"className":25465},[78,79,837],[34,25467,2547],{"className":25468},[2546],[34,25470,25472],{"className":25471},[818],[34,25473,25475],{"className":25474,"style":2554},[822],[34,25476],{},[34,25478,179],{"className":25479},[230],[34,25481,25483,25486],{"className":25482},[78],[34,25484,12226],{"className":25485},[78,79],[34,25487,25489],{"className":25488},[810],[34,25490,25492,25512],{"className":25491},[814,2523],[34,25493,25495,25509],{"className":25494},[818],[34,25496,25498],{"className":25497,"style":25022},[822],[34,25499,25500,25503],{"style":2533},[34,25501],{"className":25502,"style":830},[829],[34,25504,25506],{"className":25505},[834,835,836,837],[34,25507,24824],{"className":25508},[78,79,837],[34,25510,2547],{"className":25511},[2546],[34,25513,25515],{"className":25514},[818],[34,25516,25518],{"className":25517,"style":2554},[822],[34,25519],{},[34,25521,189],{"className":25522},[258],[34,25524],{"className":25525,"style":238},[237],[34,25527,6463],{"className":25528},[242],[34,25530],{"className":25531,"style":238},[237],[34,25533,25535],{"className":25534},[78,23012],[34,25536,25538],{"className":25537},[814],[34,25539,25541],{"className":25540},[818],[34,25542,25544,25552],{"className":25543,"style":23008},[822],[34,25545,25546,25549],{"style":23024},[34,25547],{"className":25548,"style":12709},[829],[34,25550,2997],{"className":25551,"style":14902},[78,79],[34,25553,25554,25557],{"style":23033},[34,25555],{"className":25556,"style":12709},[829],[34,25558,25560],{"className":25559,"style":23041},[23040],[34,25561,22963],{"className":25562},[78],[34,25564,2547],{"className":25565},[2546],[34,25567,25569],{"className":25568},[818],[34,25570,25573],{"className":25571,"style":25572},[822],"height:1.2152em;",[34,25574],{},[34,25576],{"className":25577},[258,12689],[16,25579,25580],{},"Advantages: highly accurate, only 2–3 iterations needed.",[16,25582,25583],{},"Many libraries use this method.",[146,25585,25587],{"id":25586},"minka-approximation","Minka Approximation",[16,25589,25590],{},"Source: Thomas Minka",[16,25592,25593],{},"Formula:",[34,25595,25597],{"className":25596,"translate":38},[159],[34,25598,25600,25684],{"className":25599,"translate":38},[37],[34,25601,25603],{"className":25602},[42],[44,25604,25605],{"xmlns":46,"display":168},[48,25606,25607,25681],{},[51,25608,25609,25611,25613,25657,25659],{},[54,25610,12226],{},[176,25612,18755],{},[12543,25614,25615,25641],{},[51,25616,25617,25623,25625,25627,25629,25639],{},[10617,25618,25619,25621],{"accent":65},[54,25620,2997],{},[176,25622,22963],{},[176,25624,179],{"stretchy":178},[54,25626,5250],{},[176,25628,6463],{},[755,25630,25631,25637],{},[10617,25632,25633,25635],{"accent":65},[54,25634,2997],{},[176,25636,22963],{},[347,25638,424],{},[176,25640,189],{"stretchy":178},[51,25642,25643,25645,25647],{},[347,25644,2213],{},[176,25646,6463],{},[755,25648,25649,25655],{},[10617,25650,25651,25653],{"accent":65},[54,25652,2997],{},[176,25654,22963],{},[347,25656,424],{},[176,25658,184],{},[12543,25660,25661,25667],{},[10617,25662,25663,25665],{"accent":65},[54,25664,2997],{},[176,25666,22963],{},[51,25668,25669,25671,25673,25675,25677,25679],{},[347,25670,424],{},[176,25672,179],{"stretchy":178},[54,25674,5250],{},[176,25676,6463],{},[347,25678,2213],{},[176,25680,189],{"stretchy":178},[58,25682,25683],{"encoding":60},"\\kappa \\approx \\frac{\\bar{R}(d-\\bar{R}^2)}{1-\\bar{R}^2} + \\frac{\\bar{R}}{2(d-1)}",[34,25685,25687,25705,25951],{"className":25686,"ariaHidden":65},[64],[34,25688,25690,25693,25696,25699,25702],{"className":25689},[69],[34,25691],{"className":25692,"style":24115},[73],[34,25694,12226],{"className":25695},[78,79],[34,25697],{"className":25698,"style":262},[237],[34,25700,18755],{"className":25701},[266],[34,25703],{"className":25704,"style":262},[237],[34,25706,25708,25711,25942,25945,25948],{"className":25707},[69],[34,25709],{"className":25710,"style":24134},[73],[34,25712,25714,25717,25939],{"className":25713},[78],[34,25715],{"className":25716},[230,12689],[34,25718,25720],{"className":25719},[12543],[34,25721,25723,25931],{"className":25722},[814,2523],[34,25724,25726,25928],{"className":25725},[818],[34,25727,25729,25806,25814],{"className":25728,"style":24153},[822],[34,25730,25731,25734],{"style":24156},[34,25732],{"className":25733,"style":12709},[829],[34,25735,25737,25740,25743,25746,25749],{"className":25736},[78],[34,25738,2213],{"className":25739},[78],[34,25741],{"className":25742,"style":238},[237],[34,25744,6463],{"className":25745},[242],[34,25747],{"className":25748,"style":238},[237],[34,25750,25752,25783],{"className":25751},[78],[34,25753,25755],{"className":25754},[78,23012],[34,25756,25758],{"className":25757},[814],[34,25759,25761],{"className":25760},[818],[34,25762,25764,25772],{"className":25763,"style":23008},[822],[34,25765,25766,25769],{"style":23024},[34,25767],{"className":25768,"style":12709},[829],[34,25770,2997],{"className":25771,"style":14902},[78,79],[34,25773,25774,25777],{"style":23033},[34,25775],{"className":25776,"style":12709},[829],[34,25778,25780],{"className":25779,"style":23041},[23040],[34,25781,22963],{"className":25782},[78],[34,25784,25786],{"className":25785},[810],[34,25787,25789],{"className":25788},[814],[34,25790,25792],{"className":25791},[818],[34,25793,25795],{"className":25794,"style":15259},[822],[34,25796,25797,25800],{"style":12743},[34,25798],{"className":25799,"style":830},[829],[34,25801,25803],{"className":25802},[834,835,836,837],[34,25804,424],{"className":25805},[78,837],[34,25807,25808,25811],{"style":12897},[34,25809],{"className":25810,"style":12709},[829],[34,25812],{"className":25813,"style":12904},[12836],[34,25815,25816,25819],{"style":12907},[34,25817],{"className":25818,"style":12709},[829],[34,25820,25822,25853,25856,25859,25862,25865,25868,25925],{"className":25821},[78],[34,25823,25825],{"className":25824},[78,23012],[34,25826,25828],{"className":25827},[814],[34,25829,25831],{"className":25830},[818],[34,25832,25834,25842],{"className":25833,"style":23008},[822],[34,25835,25836,25839],{"style":23024},[34,25837],{"className":25838,"style":12709},[829],[34,25840,2997],{"className":25841,"style":14902},[78,79],[34,25843,25844,25847],{"style":23033},[34,25845],{"className":25846,"style":12709},[829],[34,25848,25850],{"className":25849,"style":23041},[23040],[34,25851,22963],{"className":25852},[78],[34,25854,179],{"className":25855},[230],[34,25857,5250],{"className":25858},[78,79],[34,25860],{"className":25861,"style":238},[237],[34,25863,6463],{"className":25864},[242],[34,25866],{"className":25867,"style":238},[237],[34,25869,25871,25902],{"className":25870},[78],[34,25872,25874],{"className":25873},[78,23012],[34,25875,25877],{"className":25876},[814],[34,25878,25880],{"className":25879},[818],[34,25881,25883,25891],{"className":25882,"style":23008},[822],[34,25884,25885,25888],{"style":23024},[34,25886],{"className":25887,"style":12709},[829],[34,25889,2997],{"className":25890,"style":14902},[78,79],[34,25892,25893,25896],{"style":23033},[34,25894],{"className":25895,"style":12709},[829],[34,25897,25899],{"className":25898,"style":23041},[23040],[34,25900,22963],{"className":25901},[78],[34,25903,25905],{"className":25904},[810],[34,25906,25908],{"className":25907},[814],[34,25909,25911],{"className":25910},[818],[34,25912,25914],{"className":25913,"style":800},[822],[34,25915,25916,25919],{"style":825},[34,25917],{"className":25918,"style":830},[829],[34,25920,25922],{"className":25921},[834,835,836,837],[34,25923,424],{"className":25924},[78,837],[34,25926,189],{"className":25927},[258],[34,25929,2547],{"className":25930},[2546],[34,25932,25934],{"className":25933},[818],[34,25935,25937],{"className":25936,"style":24363},[822],[34,25938],{},[34,25940],{"className":25941},[258,12689],[34,25943],{"className":25944,"style":238},[237],[34,25946,184],{"className":25947},[242],[34,25949],{"className":25950,"style":238},[237],[34,25952,25954,25958],{"className":25953},[69],[34,25955],{"className":25956,"style":25957},[73],"height:2.4331em;vertical-align:-0.936em;",[34,25959,25961,25964,26066],{"className":25960},[78],[34,25962],{"className":25963},[230,12689],[34,25965,25967],{"className":25966},[12543],[34,25968,25970,26058],{"className":25969},[814,2523],[34,25971,25973,26055],{"className":25972},[818],[34,25974,25976,26008,26016],{"className":25975,"style":24153},[822],[34,25977,25978,25981],{"style":14155},[34,25979],{"className":25980,"style":12709},[829],[34,25982,25984,25987,25990,25993,25996,25999,26002,26005],{"className":25983},[78],[34,25985,424],{"className":25986},[78],[34,25988,179],{"className":25989},[230],[34,25991,5250],{"className":25992},[78,79],[34,25994],{"className":25995,"style":238},[237],[34,25997,6463],{"className":25998},[242],[34,26000],{"className":26001,"style":238},[237],[34,26003,2213],{"className":26004},[78],[34,26006,189],{"className":26007},[258],[34,26009,26010,26013],{"style":12897},[34,26011],{"className":26012,"style":12709},[829],[34,26014],{"className":26015,"style":12904},[12836],[34,26017,26018,26021],{"style":12907},[34,26019],{"className":26020,"style":12709},[829],[34,26022,26024],{"className":26023},[78],[34,26025,26027],{"className":26026},[78,23012],[34,26028,26030],{"className":26029},[814],[34,26031,26033],{"className":26032},[818],[34,26034,26036,26044],{"className":26035,"style":23008},[822],[34,26037,26038,26041],{"style":23024},[34,26039],{"className":26040,"style":12709},[829],[34,26042,2997],{"className":26043,"style":14902},[78,79],[34,26045,26046,26049],{"style":23033},[34,26047],{"className":26048,"style":12709},[829],[34,26050,26052],{"className":26051,"style":23041},[23040],[34,26053,22963],{"className":26054},[78],[34,26056,2547],{"className":26057},[2546],[34,26059,26061],{"className":26060},[818],[34,26062,26064],{"className":26063,"style":17481},[822],[34,26065],{},[34,26067],{"className":26068},[258,12689],[16,26070,26071],{},"Characteristics: slightly more accurate than Banerjee, still closed-form.",[146,26073,26075,26076,26104],{"id":26074},"large-κkappaκ-asymptotic-approximation","Large-",[34,26077,26079,26092],{"className":26078,"translate":38},[37],[34,26080,26082],{"className":26081},[42],[44,26083,26084],{"xmlns":46},[48,26085,26086,26090],{},[51,26087,26088],{},[54,26089,12226],{},[58,26091,13280],{"encoding":60},[34,26093,26095],{"className":26094,"ariaHidden":65},[64],[34,26096,26098,26101],{"className":26097},[69],[34,26099],{"className":26100,"style":108},[73],[34,26102,12226],{"className":26103},[78,79]," Asymptotic Approximation",[16,26106,18673,26107,26135],{},[34,26108,26110,26123],{"className":26109,"translate":38},[37],[34,26111,26113],{"className":26112},[42],[44,26114,26115],{"xmlns":46},[48,26116,26117,26121],{},[51,26118,26119],{},[54,26120,12226],{},[58,26122,13280],{"encoding":60},[34,26124,26126],{"className":26125,"ariaHidden":65},[64],[34,26127,26129,26132],{"className":26128},[69],[34,26130],{"className":26131,"style":108},[73],[34,26133,12226],{"className":26134},[78,79]," is very large:",[34,26137,26139],{"className":26138,"translate":38},[159],[34,26140,26142,26188],{"className":26141,"translate":38},[37],[34,26143,26145],{"className":26144},[42],[44,26146,26147],{"xmlns":46,"display":168},[48,26148,26149,26185],{},[51,26150,26151,26157,26159,26161,26163,26165,26167,26169],{},[2468,26152,26153,26155],{},[54,26154,3145],{},[54,26156,5250],{},[176,26158,179],{"stretchy":178},[54,26160,12226],{},[176,26162,189],{"stretchy":178},[176,26164,18755],{},[347,26166,2213],{},[176,26168,6463],{},[12543,26170,26171,26179],{},[51,26172,26173,26175,26177],{},[54,26174,5250],{},[176,26176,6463],{},[347,26178,2213],{},[51,26180,26181,26183],{},[347,26182,424],{},[54,26184,12226],{},[58,26186,26187],{"encoding":60},"A_d(\\kappa) \\approx 1 - \\frac{d-1}{2\\kappa}",[34,26189,26191,26255,26273],{"className":26190,"ariaHidden":65},[64],[34,26192,26194,26197,26237,26240,26243,26246,26249,26252],{"className":26193},[69],[34,26195],{"className":26196,"style":223},[73],[34,26198,26200,26203],{"className":26199},[78],[34,26201,3145],{"className":26202},[78,79],[34,26204,26206],{"className":26205},[810],[34,26207,26209,26229],{"className":26208},[814,2523],[34,26210,26212,26226],{"className":26211},[818],[34,26213,26215],{"className":26214,"style":6071},[822],[34,26216,26217,26220],{"style":2533},[34,26218],{"className":26219,"style":830},[829],[34,26221,26223],{"className":26222},[834,835,836,837],[34,26224,5250],{"className":26225},[78,79,837],[34,26227,2547],{"className":26228},[2546],[34,26230,26232],{"className":26231},[818],[34,26233,26235],{"className":26234,"style":2554},[822],[34,26236],{},[34,26238,179],{"className":26239},[230],[34,26241,12226],{"className":26242},[78,79],[34,26244,189],{"className":26245},[258],[34,26247],{"className":26248,"style":262},[237],[34,26250,18755],{"className":26251},[266],[34,26253],{"className":26254,"style":262},[237],[34,26256,26258,26261,26264,26267,26270],{"className":26257},[69],[34,26259],{"className":26260,"style":511},[73],[34,26262,2213],{"className":26263},[78],[34,26265],{"className":26266,"style":238},[237],[34,26268,6463],{"className":26269},[242],[34,26271],{"className":26272,"style":238},[237],[34,26274,26276,26279],{"className":26275},[69],[34,26277],{"className":26278,"style":15649},[73],[34,26280,26282,26285,26353],{"className":26281},[78],[34,26283],{"className":26284},[230,12689],[34,26286,26288],{"className":26287},[12543],[34,26289,26291,26345],{"className":26290},[814,2523],[34,26292,26294,26342],{"className":26293},[818],[34,26295,26297,26311,26319],{"className":26296,"style":15668},[822],[34,26298,26299,26302],{"style":14155},[34,26300],{"className":26301,"style":12709},[829],[34,26303,26305,26308],{"className":26304},[78],[34,26306,424],{"className":26307},[78],[34,26309,12226],{"className":26310},[78,79],[34,26312,26313,26316],{"style":12897},[34,26314],{"className":26315,"style":12709},[829],[34,26317],{"className":26318,"style":12904},[12836],[34,26320,26321,26324],{"style":12907},[34,26322],{"className":26323,"style":12709},[829],[34,26325,26327,26330,26333,26336,26339],{"className":26326},[78],[34,26328,5250],{"className":26329},[78,79],[34,26331],{"className":26332,"style":238},[237],[34,26334,6463],{"className":26335},[242],[34,26337],{"className":26338,"style":238},[237],[34,26340,2213],{"className":26341},[78],[34,26343,2547],{"className":26344},[2546],[34,26346,26348],{"className":26347},[818],[34,26349,26351],{"className":26350,"style":14669},[822],[34,26352],{},[34,26354],{"className":26355},[258,12689],[16,26357,26358],{},"Inverting gives:",[34,26360,26362],{"className":26361,"translate":38},[159],[34,26363,26365,26409],{"className":26364,"translate":38},[37],[34,26366,26368],{"className":26367},[42],[44,26369,26370],{"xmlns":46,"display":168},[48,26371,26372,26406],{},[51,26373,26374,26376,26378],{},[54,26375,12226],{},[176,26377,18755],{},[12543,26379,26380,26388],{},[51,26381,26382,26384,26386],{},[54,26383,5250],{},[176,26385,6463],{},[347,26387,2213],{},[51,26389,26390,26392,26394,26396,26398,26404],{},[347,26391,424],{},[176,26393,179],{"stretchy":178},[347,26395,2213],{},[176,26397,6463],{},[10617,26399,26400,26402],{"accent":65},[54,26401,2997],{},[176,26403,22963],{},[176,26405,189],{"stretchy":178},[58,26407,26408],{"encoding":60},"\\kappa \\approx \\frac{d-1}{2(1-\\bar{R})}",[34,26410,26412,26430],{"className":26411,"ariaHidden":65},[64],[34,26413,26415,26418,26421,26424,26427],{"className":26414},[69],[34,26416],{"className":26417,"style":24115},[73],[34,26419,12226],{"className":26420},[78,79],[34,26422],{"className":26423,"style":262},[237],[34,26425,18755],{"className":26426},[266],[34,26428],{"className":26429,"style":262},[237],[34,26431,26433,26437],{"className":26432},[69],[34,26434],{"className":26435,"style":26436},[73],"height:2.3316em;vertical-align:-0.9601em;",[34,26438,26440,26443,26558],{"className":26439},[78],[34,26441],{"className":26442},[230,12689],[34,26444,26446],{"className":26445},[12543],[34,26447,26449,26549],{"className":26448},[814,2523],[34,26450,26452,26546],{"className":26451},[818],[34,26453,26455,26515,26523],{"className":26454,"style":15668},[822],[34,26456,26457,26460],{"style":24156},[34,26458],{"className":26459,"style":12709},[829],[34,26461,26463,26466,26469,26472,26475,26478,26481,26512],{"className":26462},[78],[34,26464,424],{"className":26465},[78],[34,26467,179],{"className":26468},[230],[34,26470,2213],{"className":26471},[78],[34,26473],{"className":26474,"style":238},[237],[34,26476,6463],{"className":26477},[242],[34,26479],{"className":26480,"style":238},[237],[34,26482,26484],{"className":26483},[78,23012],[34,26485,26487],{"className":26486},[814],[34,26488,26490],{"className":26489},[818],[34,26491,26493,26501],{"className":26492,"style":23008},[822],[34,26494,26495,26498],{"style":23024},[34,26496],{"className":26497,"style":12709},[829],[34,26499,2997],{"className":26500,"style":14902},[78,79],[34,26502,26503,26506],{"style":23033},[34,26504],{"className":26505,"style":12709},[829],[34,26507,26509],{"className":26508,"style":23041},[23040],[34,26510,22963],{"className":26511},[78],[34,26513,189],{"className":26514},[258],[34,26516,26517,26520],{"style":12897},[34,26518],{"className":26519,"style":12709},[829],[34,26521],{"className":26522,"style":12904},[12836],[34,26524,26525,26528],{"style":12907},[34,26526],{"className":26527,"style":12709},[829],[34,26529,26531,26534,26537,26540,26543],{"className":26530},[78],[34,26532,5250],{"className":26533},[78,79],[34,26535],{"className":26536,"style":238},[237],[34,26538,6463],{"className":26539},[242],[34,26541],{"className":26542,"style":238},[237],[34,26544,2213],{"className":26545},[78],[34,26547,2547],{"className":26548},[2546],[34,26550,26552],{"className":26551},[818],[34,26553,26556],{"className":26554,"style":26555},[822],"height:0.9601em;",[34,26557],{},[34,26559],{"className":26560},[258,12689],[16,26562,26563],{},"Applicable when embeddings are highly concentrated, e.g., in contrastive learning.",[146,26565,26567,26568,26596],{"id":26566},"small-κkappaκ-approximation","Small-",[34,26569,26571,26584],{"className":26570,"translate":38},[37],[34,26572,26574],{"className":26573},[42],[44,26575,26576],{"xmlns":46},[48,26577,26578,26582],{},[51,26579,26580],{},[54,26581,12226],{},[58,26583,13280],{"encoding":60},[34,26585,26587],{"className":26586,"ariaHidden":65},[64],[34,26588,26590,26593],{"className":26589},[69],[34,26591],{"className":26592,"style":108},[73],[34,26594,12226],{"className":26595},[78,79]," Approximation",[16,26598,18673,26599,26627],{},[34,26600,26602,26615],{"className":26601,"translate":38},[37],[34,26603,26605],{"className":26604},[42],[44,26606,26607],{"xmlns":46},[48,26608,26609,26613],{},[51,26610,26611],{},[54,26612,12226],{},[58,26614,13280],{"encoding":60},[34,26616,26618],{"className":26617,"ariaHidden":65},[64],[34,26619,26621,26624],{"className":26620},[69],[34,26622],{"className":26623,"style":108},[73],[34,26625,12226],{"className":26626},[78,79]," is very small:",[34,26629,26631],{"className":26630,"translate":38},[159],[34,26632,26634,26666],{"className":26633,"translate":38},[37],[34,26635,26637],{"className":26636},[42],[44,26638,26639],{"xmlns":46,"display":168},[48,26640,26641,26663],{},[51,26642,26643,26649,26651,26653,26655,26657],{},[2468,26644,26645,26647],{},[54,26646,3145],{},[54,26648,5250],{},[176,26650,179],{"stretchy":178},[54,26652,12226],{},[176,26654,189],{"stretchy":178},[176,26656,18755],{},[12543,26658,26659,26661],{},[54,26660,12226],{},[54,26662,5250],{},[58,26664,26665],{"encoding":60},"A_d(\\kappa) \\approx \\frac{\\kappa}{d}",[34,26667,26669,26733],{"className":26668,"ariaHidden":65},[64],[34,26670,26672,26675,26715,26718,26721,26724,26727,26730],{"className":26671},[69],[34,26673],{"className":26674,"style":223},[73],[34,26676,26678,26681],{"className":26677},[78],[34,26679,3145],{"className":26680},[78,79],[34,26682,26684],{"className":26683},[810],[34,26685,26687,26707],{"className":26686},[814,2523],[34,26688,26690,26704],{"className":26689},[818],[34,26691,26693],{"className":26692,"style":6071},[822],[34,26694,26695,26698],{"style":2533},[34,26696],{"className":26697,"style":830},[829],[34,26699,26701],{"className":26700},[834,835,836,837],[34,26702,5250],{"className":26703},[78,79,837],[34,26705,2547],{"className":26706},[2546],[34,26708,26710],{"className":26709},[818],[34,26711,26713],{"className":26712,"style":2554},[822],[34,26714],{},[34,26716,179],{"className":26717},[230],[34,26719,12226],{"className":26720},[78,79],[34,26722,189],{"className":26723},[258],[34,26725],{"className":26726,"style":262},[237],[34,26728,18755],{"className":26729},[266],[34,26731],{"className":26732,"style":262},[237],[34,26734,26736,26740],{"className":26735},[69],[34,26737],{"className":26738,"style":26739},[73],"height:1.7936em;vertical-align:-0.686em;",[34,26741,26743,26746,26800],{"className":26742},[78],[34,26744],{"className":26745},[230,12689],[34,26747,26749],{"className":26748},[12543],[34,26750,26752,26792],{"className":26751},[814,2523],[34,26753,26755,26789],{"className":26754},[818],[34,26756,26759,26770,26778],{"className":26757,"style":26758},[822],"height:1.1076em;",[34,26760,26761,26764],{"style":14155},[34,26762],{"className":26763,"style":12709},[829],[34,26765,26767],{"className":26766},[78],[34,26768,5250],{"className":26769},[78,79],[34,26771,26772,26775],{"style":12897},[34,26773],{"className":26774,"style":12709},[829],[34,26776],{"className":26777,"style":12904},[12836],[34,26779,26780,26783],{"style":12907},[34,26781],{"className":26782,"style":12709},[829],[34,26784,26786],{"className":26785},[78],[34,26787,12226],{"className":26788},[78,79],[34,26790,2547],{"className":26791},[2546],[34,26793,26795],{"className":26794},[818],[34,26796,26798],{"className":26797,"style":14669},[822],[34,26799],{},[34,26801],{"className":26802},[258,12689],[16,26804,26805],{},"Thus,",[34,26807,26809],{"className":26808,"translate":38},[159],[34,26810,26812,26836],{"className":26811,"translate":38},[37],[34,26813,26815],{"className":26814},[42],[44,26816,26817],{"xmlns":46,"display":168},[48,26818,26819,26833],{},[51,26820,26821,26823,26825,26827],{},[54,26822,12226],{},[176,26824,18755],{},[54,26826,5250],{},[10617,26828,26829,26831],{"accent":65},[54,26830,2997],{},[176,26832,22963],{},[58,26834,26835],{"encoding":60},"\\kappa \\approx d\\bar{R}",[34,26837,26839,26857],{"className":26838,"ariaHidden":65},[64],[34,26840,26842,26845,26848,26851,26854],{"className":26841},[69],[34,26843],{"className":26844,"style":24115},[73],[34,26846,12226],{"className":26847},[78,79],[34,26849],{"className":26850,"style":262},[237],[34,26852,18755],{"className":26853},[266],[34,26855],{"className":26856,"style":262},[237],[34,26858,26860,26863,26866],{"className":26859},[69],[34,26861],{"className":26862,"style":23008},[73],[34,26864,5250],{"className":26865},[78,79],[34,26867,26869],{"className":26868},[78,23012],[34,26870,26872],{"className":26871},[814],[34,26873,26875],{"className":26874},[818],[34,26876,26878,26886],{"className":26877,"style":23008},[822],[34,26879,26880,26883],{"style":23024},[34,26881],{"className":26882,"style":12709},[829],[34,26884,2997],{"className":26885,"style":14902},[78,79],[34,26887,26888,26891],{"style":23033},[34,26889],{"className":26890,"style":12709},[829],[34,26892,26894],{"className":26893,"style":23041},[23040],[34,26895,22963],{"className":26896},[78],[16,26898,26899],{},"Applicable when data are nearly uniform.",[19703,26901,26902,26918],{},[19706,26903,26904],{},[19709,26905,26906,26909,26912,26915],{},[19712,26907,26908],{},"Estimation",[19712,26910,26911],{},"Complexity",[19712,26913,26914],{},"Accuracy",[19712,26916,26917],{},"Usage",[19787,26919,26920,26934,26947,26961,27030],{},[19709,26921,26922,26925,26928,26931],{},[19792,26923,26924],{},"Banerjee",[19792,26926,26927],{},"Very low",[19792,26929,26930],{},"Medium",[19792,26932,26933],{},"Initialization",[19709,26935,26936,26939,26941,26944],{},[19792,26937,26938],{},"Sra + Newton",[19792,26940,26930],{},[19792,26942,26943],{},"High",[19792,26945,26946],{},"Recommended",[19709,26948,26949,26952,26955,26958],{},[19792,26950,26951],{},"Minka",[19792,26953,26954],{},"Low",[19792,26956,26957],{},"Medium-High",[19792,26959,26960],{},"Fast estimation",[19709,26962,26963,26994,26996,27027],{},[19792,26964,26965,26966],{},"Large ",[34,26967,26969,26982],{"className":26968,"translate":38},[37],[34,26970,26972],{"className":26971},[42],[44,26973,26974],{"xmlns":46},[48,26975,26976,26980],{},[51,26977,26978],{},[54,26979,12226],{},[58,26981,13280],{"encoding":60},[34,26983,26985],{"className":26984,"ariaHidden":65},[64],[34,26986,26988,26991],{"className":26987},[69],[34,26989],{"className":26990,"style":108},[73],[34,26992,12226],{"className":26993},[78,79],[19792,26995,26927],{},[19792,26997,26998,26999,189],{},"High (large ",[34,27000,27002,27015],{"className":27001,"translate":38},[37],[34,27003,27005],{"className":27004},[42],[44,27006,27007],{"xmlns":46},[48,27008,27009,27013],{},[51,27010,27011],{},[54,27012,12226],{},[58,27014,13280],{"encoding":60},[34,27016,27018],{"className":27017,"ariaHidden":65},[64],[34,27019,27021,27024],{"className":27020},[69],[34,27022],{"className":27023,"style":108},[73],[34,27025,12226],{"className":27026},[78,79],[19792,27028,27029],{},"High concentration",[19709,27031,27032,27063,27065,27096],{},[19792,27033,27034,27035],{},"Small ",[34,27036,27038,27051],{"className":27037,"translate":38},[37],[34,27039,27041],{"className":27040},[42],[44,27042,27043],{"xmlns":46},[48,27044,27045,27049],{},[51,27046,27047],{},[54,27048,12226],{},[58,27050,13280],{"encoding":60},[34,27052,27054],{"className":27053,"ariaHidden":65},[64],[34,27055,27057,27060],{"className":27056},[69],[34,27058],{"className":27059,"style":108},[73],[34,27061,12226],{"className":27062},[78,79],[19792,27064,26927],{},[19792,27066,27067,27068,189],{},"High (small ",[34,27069,27071,27084],{"className":27070,"translate":38},[37],[34,27072,27074],{"className":27073},[42],[44,27075,27076],{"xmlns":46},[48,27077,27078,27082],{},[51,27079,27080],{},[54,27081,12226],{},[58,27083,13280],{"encoding":60},[34,27085,27087],{"className":27086,"ariaHidden":65},[64],[34,27088,27090,27093],{"className":27089},[69],[34,27091],{"className":27092,"style":108},[73],[34,27094,12226],{"className":27095},[78,79],[19792,27097,27098],{},"Near uniformity",{"title":11668,"searchDepth":11669,"depth":11669,"links":27100},[27101,27102,27103,27113],{"id":11804,"depth":11669,"text":11805},{"id":13113,"depth":11669,"text":13114},{"id":13490,"depth":11669,"text":13491,"children":27104},[27105,27106,27107,27108,27109,27110,27111,27112],{"id":13494,"depth":11674,"text":13495},{"id":13845,"depth":11674,"text":13846},{"id":14403,"depth":11674,"text":14404},{"id":14869,"depth":11674,"text":14870},{"id":15096,"depth":11674,"text":15097},{"id":15397,"depth":11674,"text":15398},{"id":15767,"depth":11674,"text":15768},{"id":16078,"depth":11674,"text":16079},{"id":22589,"depth":11669,"text":22590,"children":27114},[27115,27116,27117,27118,27120],{"id":23978,"depth":11674,"text":23979},{"id":24422,"depth":11674,"text":24423},{"id":25586,"depth":11674,"text":25587},{"id":26074,"depth":11674,"text":27119},"Large-κ\\kappaκ Asymptotic Approximation",{"id":26566,"depth":11674,"text":27121},"Small-κ\\kappaκ Approximation","The von Mises–Fisher distribution (vMF) distribution is often regarded as the hyperspherical analogue of the Normal distribution, as it models data concentrated around a mean direction on the unit hypersphere. In Euclidean space, a normal distribution is typically parameterized by a mean μ\\muμ and variance σ2\\sigma^2σ2, and in one dimension its density forms the familiar bell-shaped curve. Just as the normal distribution is the distribution with the maximum entropy given the mean and variance constraints, the vMF distribution is the distribution with the maximum entropy on a sphere given the average direction constraint. According to the Central Limit Theorem: the limiting distribution of a random walk or accumulated direction vector on a sphere will also tend towards the vMF distribution.",{},"2026-04-09","\u002Fblog\u002F2026\u002F2026-04-09-introduction-of-v-mf-distribution-on-hypersphere",{"title":11700,"description":27122},"blog\u002F2026\u002F2026-04-09-introduction-of-v-mf-distribution-on-hypersphere","The von Mises–Fisher distribution (vMF) distribution is often regarded as the hyperspherical analogue of the Normal distribution, as it models data concentrated around a mean direction on the unit hypersphere.",[11696,11695],"kX0Iv-NpB93ngxIWKtcvLc73rGMzdLU7bXj_HI0xlkM",{"id":27132,"title":27133,"body":27134,"description":11668,"draft":11684,"enableComment":11685,"extension":11686,"image":11687,"important":11684,"location":11687,"meta":55243,"navigation":11685,"ogImage":11687,"onday":55244,"path":55245,"seo":55246,"stem":55247,"summary":55248,"tags":55249,"__hash__":55250},"blog\u002Fblog\u002F2025\u002F2025-09-18-the-stirling-formula.md","The Stirling Formula",{"type":8,"value":27135,"toc":55239},[27136,27140,27143,27378,27381,27867,27996,28851,29572,29575,30154,30157,31219,31222,31312,32841,32953,34412,34415,34560,34562,35027,35416,35817,36525,36528,37068,37308,38602,38761,38764,39303,39460,39989,40091,41535,41540,41923,41928,41932,42088,42382,42385,42805,42808,43485,43488,43821,43824,44920,44923,44925,45172,45175,45437,45440,45445,45987,46500,46632,46635,46954,47031,47728,48304,48307,48820,48850,48853,49098,49101,49408,49411,49741,49923,50308,50410,50888,50890,51481,51647,51650,51995,51998,52001,52383,52581,52925,53355,53358,53790,53793,54018,54021,54572,54575,54686,54718,54995,54998],[11,27137,27139],{"id":27138},"walliss-formula","Wallis's Formula",[16,27141,27142],{},"Let's start with a definite integral:",[34,27144,27146],{"className":27145,"translate":38},[159],[34,27147,27149,27191],{"className":27148,"translate":38},[37],[34,27150,27152],{"className":27151},[42],[44,27153,27154],{"xmlns":46,"display":168},[48,27155,27156,27188],{},[51,27157,27158,27170,27180,27182,27184,27186],{},[2475,27159,27160,27162,27164],{},[176,27161,18023],{},[347,27163,2287],{},[12543,27165,27166,27168],{},[54,27167,12569],{},[347,27169,424],{},[755,27171,27172,27178],{},[51,27173,27174,27176],{},[54,27175,16710],{},[176,27177,6321],{},[54,27179,3000],{},[54,27181,96],{},[1201,27183,5837],{},[54,27185,5250],{},[54,27187,96],{},[58,27189,27190],{"encoding":60},"\\int_{0}^{\\frac{\\pi}{2}} \\sin^n x \\, dx",[34,27192,27194],{"className":27193,"ariaHidden":65},[64],[34,27195,27197,27201,27329,27332,27363,27366,27369,27372,27375],{"className":27196},[69],[34,27198],{"className":27199,"style":27200},[73],"height:2.5089em;vertical-align:-0.9119em;",[34,27202,27204,27208],{"className":27203},[3387],[34,27205,18023],{"className":27206,"style":27207},[3387,3432,3433],"margin-right:0.4445em;position:relative;top:-0.0011em;",[34,27209,27211],{"className":27210},[810],[34,27212,27214,27320],{"className":27213},[814,2523],[34,27215,27217,27317],{"className":27216},[818],[34,27218,27221,27236],{"className":27219,"style":27220},[822],"height:1.597em;",[34,27222,27224,27227],{"style":27223},"top:-2.088em;margin-left:-0.4445em;margin-right:0.05em;",[34,27225],{"className":27226,"style":12709},[829],[34,27228,27230],{"className":27229},[834,835,836,837],[34,27231,27233],{"className":27232},[78,837],[34,27234,2287],{"className":27235},[78,837],[34,27237,27239,27242],{"style":27238},"top:-4.1129em;margin-right:0.05em;",[34,27240],{"className":27241,"style":12709},[829],[34,27243,27245],{"className":27244},[834,835,836,837],[34,27246,27248],{"className":27247},[78,837],[34,27249,27251,27254,27314],{"className":27250},[78,837],[34,27252],{"className":27253},[230,12689,834,10776,12798],[34,27255,27257],{"className":27256},[12543],[34,27258,27260,27306],{"className":27259},[814,2523],[34,27261,27263,27303],{"className":27262},[818],[34,27264,27267,27281,27289],{"className":27265,"style":27266},[822],"height:0.6915em;",[34,27268,27269,27272],{"style":12814},[34,27270],{"className":27271,"style":12709},[829],[34,27273,27275],{"className":27274},[834,10776,9740,837],[34,27276,27278],{"className":27277},[78,837],[34,27279,424],{"className":27280},[78,837],[34,27282,27283,27286],{"style":12829},[34,27284],{"className":27285,"style":12709},[829],[34,27287],{"className":27288,"style":12837},[12836,837],[34,27290,27291,27294],{"style":12840},[34,27292],{"className":27293,"style":12709},[829],[34,27295,27297],{"className":27296},[834,10776,9740,837],[34,27298,27300],{"className":27299},[78,837],[34,27301,12569],{"className":27302,"style":143},[78,79,837],[34,27304,2547],{"className":27305},[2546],[34,27307,27309],{"className":27308},[818],[34,27310,27312],{"className":27311,"style":12862},[822],[34,27313],{},[34,27315],{"className":27316},[258,12689,834,10776,12798],[34,27318,2547],{"className":27319},[2546],[34,27321,27323],{"className":27322},[818],[34,27324,27327],{"className":27325,"style":27326},[822],"height:0.9119em;",[34,27328],{},[34,27330],{"className":27331,"style":3462},[237],[34,27333,27335,27338],{"className":27334},[3387],[34,27336,16710],{"className":27337},[3387],[34,27339,27341],{"className":27340},[810],[34,27342,27344],{"className":27343},[814],[34,27345,27347],{"className":27346},[818],[34,27348,27351],{"className":27349,"style":27350},[822],"height:0.7222em;",[34,27352,27354,27357],{"style":27353},"top:-3.1208em;margin-right:0.05em;",[34,27355],{"className":27356,"style":830},[829],[34,27358,27360],{"className":27359},[834,835,836,837],[34,27361,3000],{"className":27362},[78,79,837],[34,27364],{"className":27365,"style":3462},[237],[34,27367,96],{"className":27368},[78,79],[34,27370],{"className":27371,"style":3462},[237],[34,27373,5250],{"className":27374},[78,79],[34,27376,96],{"className":27377},[78,79],[16,27379,27380],{},"We can derive the following result:",[34,27382,27384],{"className":27383,"translate":38},[159],[34,27385,27387,27477],{"className":27386,"translate":38},[37],[34,27388,27390],{"className":27389},[42],[44,27391,27392],{"xmlns":46,"display":168},[48,27393,27394,27474],{},[51,27395,27396,27408,27418,27420,27422,27424,27426,27428,27430,27442,27458,27460,27462,27464,27466,27468,27470,27472],{},[2475,27397,27398,27400,27402],{},[176,27399,18023],{},[347,27401,2287],{},[12543,27403,27404,27406],{},[54,27405,12569],{},[347,27407,424],{},[755,27409,27410,27416],{},[51,27411,27412,27414],{},[54,27413,16710],{},[176,27415,6321],{},[54,27417,3000],{},[54,27419,96],{},[1201,27421,5837],{},[54,27423,5250],{},[54,27425,96],{},[176,27427,192],{},[176,27429,6463],{},[2475,27431,27432,27434,27436],{},[176,27433,18023],{},[347,27435,2287],{},[12543,27437,27438,27440],{},[54,27439,12569],{},[347,27441,424],{},[755,27443,27444,27450],{},[51,27445,27446,27448],{},[54,27447,16710],{},[176,27449,6321],{},[51,27451,27452,27454,27456],{},[54,27453,3000],{},[176,27455,6463],{},[347,27457,2213],{},[54,27459,96],{},[1201,27461,5837],{},[54,27463,5250],{},[176,27465,179],{"stretchy":178},[54,27467,16357],{},[176,27469,6321],{},[54,27471,96],{},[176,27473,189],{"stretchy":178},[58,27475,27476],{"encoding":60},"\\int_{0}^{\\frac{\\pi}{2}} \\sin^n x \\, dx = -\\int_{0}^{\\frac{\\pi}{2}} \\sin^{n-1} x \\, d(\\cos x)",[34,27478,27480,27664],{"className":27479,"ariaHidden":65},[64],[34,27481,27483,27486,27608,27611,27640,27643,27646,27649,27652,27655,27658,27661],{"className":27482},[69],[34,27484],{"className":27485,"style":27200},[73],[34,27487,27489,27492],{"className":27488},[3387],[34,27490,18023],{"className":27491,"style":27207},[3387,3432,3433],[34,27493,27495],{"className":27494},[810],[34,27496,27498,27600],{"className":27497},[814,2523],[34,27499,27501,27597],{"className":27500},[818],[34,27502,27504,27518],{"className":27503,"style":27220},[822],[34,27505,27506,27509],{"style":27223},[34,27507],{"className":27508,"style":12709},[829],[34,27510,27512],{"className":27511},[834,835,836,837],[34,27513,27515],{"className":27514},[78,837],[34,27516,2287],{"className":27517},[78,837],[34,27519,27520,27523],{"style":27238},[34,27521],{"className":27522,"style":12709},[829],[34,27524,27526],{"className":27525},[834,835,836,837],[34,27527,27529],{"className":27528},[78,837],[34,27530,27532,27535,27594],{"className":27531},[78,837],[34,27533],{"className":27534},[230,12689,834,10776,12798],[34,27536,27538],{"className":27537},[12543],[34,27539,27541,27586],{"className":27540},[814,2523],[34,27542,27544,27583],{"className":27543},[818],[34,27545,27547,27561,27569],{"className":27546,"style":27266},[822],[34,27548,27549,27552],{"style":12814},[34,27550],{"className":27551,"style":12709},[829],[34,27553,27555],{"className":27554},[834,10776,9740,837],[34,27556,27558],{"className":27557},[78,837],[34,27559,424],{"className":27560},[78,837],[34,27562,27563,27566],{"style":12829},[34,27564],{"className":27565,"style":12709},[829],[34,27567],{"className":27568,"style":12837},[12836,837],[34,27570,27571,27574],{"style":12840},[34,27572],{"className":27573,"style":12709},[829],[34,27575,27577],{"className":27576},[834,10776,9740,837],[34,27578,27580],{"className":27579},[78,837],[34,27581,12569],{"className":27582,"style":143},[78,79,837],[34,27584,2547],{"className":27585},[2546],[34,27587,27589],{"className":27588},[818],[34,27590,27592],{"className":27591,"style":12862},[822],[34,27593],{},[34,27595],{"className":27596},[258,12689,834,10776,12798],[34,27598,2547],{"className":27599},[2546],[34,27601,27603],{"className":27602},[818],[34,27604,27606],{"className":27605,"style":27326},[822],[34,27607],{},[34,27609],{"className":27610,"style":3462},[237],[34,27612,27614,27617],{"className":27613},[3387],[34,27615,16710],{"className":27616},[3387],[34,27618,27620],{"className":27619},[810],[34,27621,27623],{"className":27622},[814],[34,27624,27626],{"className":27625},[818],[34,27627,27629],{"className":27628,"style":27350},[822],[34,27630,27631,27634],{"style":27353},[34,27632],{"className":27633,"style":830},[829],[34,27635,27637],{"className":27636},[834,835,836,837],[34,27638,3000],{"className":27639},[78,79,837],[34,27641],{"className":27642,"style":3462},[237],[34,27644,96],{"className":27645},[78,79],[34,27647],{"className":27648,"style":3462},[237],[34,27650,5250],{"className":27651},[78,79],[34,27653,96],{"className":27654},[78,79],[34,27656],{"className":27657,"style":262},[237],[34,27659,192],{"className":27660},[266],[34,27662],{"className":27663,"style":262},[237],[34,27665,27667,27670,27673,27676,27798,27801,27840,27843,27846,27849,27852,27855,27858,27861,27864],{"className":27666},[69],[34,27668],{"className":27669,"style":27200},[73],[34,27671,6463],{"className":27672},[78],[34,27674],{"className":27675,"style":3462},[237],[34,27677,27679,27682],{"className":27678},[3387],[34,27680,18023],{"className":27681,"style":27207},[3387,3432,3433],[34,27683,27685],{"className":27684},[810],[34,27686,27688,27790],{"className":27687},[814,2523],[34,27689,27691,27787],{"className":27690},[818],[34,27692,27694,27708],{"className":27693,"style":27220},[822],[34,27695,27696,27699],{"style":27223},[34,27697],{"className":27698,"style":12709},[829],[34,27700,27702],{"className":27701},[834,835,836,837],[34,27703,27705],{"className":27704},[78,837],[34,27706,2287],{"className":27707},[78,837],[34,27709,27710,27713],{"style":27238},[34,27711],{"className":27712,"style":12709},[829],[34,27714,27716],{"className":27715},[834,835,836,837],[34,27717,27719],{"className":27718},[78,837],[34,27720,27722,27725,27784],{"className":27721},[78,837],[34,27723],{"className":27724},[230,12689,834,10776,12798],[34,27726,27728],{"className":27727},[12543],[34,27729,27731,27776],{"className":27730},[814,2523],[34,27732,27734,27773],{"className":27733},[818],[34,27735,27737,27751,27759],{"className":27736,"style":27266},[822],[34,27738,27739,27742],{"style":12814},[34,27740],{"className":27741,"style":12709},[829],[34,27743,27745],{"className":27744},[834,10776,9740,837],[34,27746,27748],{"className":27747},[78,837],[34,27749,424],{"className":27750},[78,837],[34,27752,27753,27756],{"style":12829},[34,27754],{"className":27755,"style":12709},[829],[34,27757],{"className":27758,"style":12837},[12836,837],[34,27760,27761,27764],{"style":12840},[34,27762],{"className":27763,"style":12709},[829],[34,27765,27767],{"className":27766},[834,10776,9740,837],[34,27768,27770],{"className":27769},[78,837],[34,27771,12569],{"className":27772,"style":143},[78,79,837],[34,27774,2547],{"className":27775},[2546],[34,27777,27779],{"className":27778},[818],[34,27780,27782],{"className":27781,"style":12862},[822],[34,27783],{},[34,27785],{"className":27786},[258,12689,834,10776,12798],[34,27788,2547],{"className":27789},[2546],[34,27791,27793],{"className":27792},[818],[34,27794,27796],{"className":27795,"style":27326},[822],[34,27797],{},[34,27799],{"className":27800,"style":3462},[237],[34,27802,27804,27807],{"className":27803},[3387],[34,27805,16710],{"className":27806},[3387],[34,27808,27810],{"className":27809},[810],[34,27811,27813],{"className":27812},[814],[34,27814,27816],{"className":27815},[818],[34,27817,27820],{"className":27818,"style":27819},[822],"height:0.8719em;",[34,27821,27822,27825],{"style":27353},[34,27823],{"className":27824,"style":830},[829],[34,27826,27828],{"className":27827},[834,835,836,837],[34,27829,27831,27834,27837],{"className":27830},[78,837],[34,27832,3000],{"className":27833},[78,79,837],[34,27835,6463],{"className":27836},[242,837],[34,27838,2213],{"className":27839},[78,837],[34,27841],{"className":27842,"style":3462},[237],[34,27844,96],{"className":27845},[78,79],[34,27847],{"className":27848,"style":3462},[237],[34,27850,5250],{"className":27851},[78,79],[34,27853,179],{"className":27854},[230],[34,27856,16357],{"className":27857},[3387],[34,27859],{"className":27860,"style":3462},[237],[34,27862,96],{"className":27863},[78,79],[34,27865,189],{"className":27866},[258],[16,27868,27869,27870,27995],{},"Using integration by parts, where ",[34,27871,27873,27914],{"className":27872,"translate":38},[37],[34,27874,27876],{"className":27875},[42],[44,27877,27878],{"xmlns":46},[48,27879,27880,27911],{},[51,27881,27882,27884,27887,27889,27891,27893,27895,27897,27899,27901,27903,27905,27907,27909],{},[176,27883,18023],{},[54,27885,27886],{},"u",[1201,27888,5837],{},[54,27890,5250],{},[54,27892,2479],{},[176,27894,192],{},[54,27896,27886],{},[54,27898,2479],{},[176,27900,6463],{},[176,27902,18023],{},[54,27904,2479],{},[1201,27906,5837],{},[54,27908,5250],{},[54,27910,27886],{},[58,27912,27913],{"encoding":60},"\\int u \\, dv = uv - \\int v \\, du",[34,27915,27917,27951,27971],{"className":27916,"ariaHidden":65},[64],[34,27918,27920,27924,27927,27930,27933,27936,27939,27942,27945,27948],{"className":27919},[69],[34,27921],{"className":27922,"style":27923},[73],"height:1.1111em;vertical-align:-0.3061em;",[34,27925,18023],{"className":27926,"style":18254},[3387,3432,18253],[34,27928],{"className":27929,"style":3462},[237],[34,27931,27886],{"className":27932},[78,79],[34,27934],{"className":27935,"style":3462},[237],[34,27937,5250],{"className":27938},[78,79],[34,27940,2479],{"className":27941,"style":143},[78,79],[34,27943],{"className":27944,"style":262},[237],[34,27946,192],{"className":27947},[266],[34,27949],{"className":27950,"style":262},[237],[34,27952,27954,27958,27962,27965,27968],{"className":27953},[69],[34,27955],{"className":27956,"style":27957},[73],"height:0.6667em;vertical-align:-0.0833em;",[34,27959,27961],{"className":27960,"style":143},[78,79],"uv",[34,27963],{"className":27964,"style":238},[237],[34,27966,6463],{"className":27967},[242],[34,27969],{"className":27970,"style":238},[237],[34,27972,27974,27977,27980,27983,27986,27989,27992],{"className":27973},[69],[34,27975],{"className":27976,"style":27923},[73],[34,27978,18023],{"className":27979,"style":18254},[3387,3432,18253],[34,27981],{"className":27982,"style":3462},[237],[34,27984,2479],{"className":27985,"style":143},[78,79],[34,27987],{"className":27988,"style":3462},[237],[34,27990,5250],{"className":27991},[78,79],[34,27993,27886],{"className":27994},[78,79],", we get:",[34,27997,27999],{"className":27998,"translate":38},[159],[34,28000,28002,28157],{"className":28001,"translate":38},[37],[34,28003,28005],{"className":28004},[42],[44,28006,28007],{"xmlns":46,"display":168},[48,28008,28009,28154],{},[51,28010,28011,28013,28025,28041,28043,28045,28047,28049,28051,28053,28055,28057,28060,28062,28064,28094,28106,28108,28120,28122,28124,28126,28128,28130],{},[176,28012,6463],{},[2475,28014,28015,28017,28019],{},[176,28016,18023],{},[347,28018,2287],{},[12543,28020,28021,28023],{},[54,28022,12569],{},[347,28024,424],{},[755,28026,28027,28033],{},[51,28028,28029,28031],{},[54,28030,16710],{},[176,28032,6321],{},[51,28034,28035,28037,28039],{},[54,28036,3000],{},[176,28038,6463],{},[347,28040,2213],{},[54,28042,96],{},[1201,28044,5837],{},[54,28046,5250],{},[176,28048,179],{"stretchy":178},[54,28050,16357],{},[176,28052,6321],{},[54,28054,96],{},[176,28056,189],{"stretchy":178},[237,28058],{"linebreak":28059},"newline",[176,28061,192],{},[176,28063,6463],{},[51,28065,28066,28068,28084,28086,28088,28090,28092],{},[176,28067,179],{"fence":65},[755,28069,28070,28076],{},[51,28071,28072,28074],{},[54,28073,16710],{},[176,28075,6321],{},[51,28077,28078,28080,28082],{},[54,28079,3000],{},[176,28081,6463],{},[347,28083,2213],{},[54,28085,96],{},[54,28087,16357],{},[176,28089,6321],{},[54,28091,96],{},[176,28093,189],{"fence":65},[2475,28095,28096,28098,28100],{},[176,28097,12292],{"fence":178,"stretchy":65,"minsize":9468,"maxsize":9468},[347,28099,2287],{},[12543,28101,28102,28104],{},[54,28103,12569],{},[347,28105,424],{},[176,28107,184],{},[2475,28109,28110,28112,28114],{},[176,28111,18023],{},[347,28113,2287],{},[12543,28115,28116,28118],{},[54,28117,12569],{},[347,28119,424],{},[54,28121,16357],{},[176,28123,6321],{},[54,28125,96],{},[1201,28127,5837],{},[54,28129,5250],{},[51,28131,28132,28134,28150,28152],{},[176,28133,179],{"fence":65},[755,28135,28136,28142],{},[51,28137,28138,28140],{},[54,28139,16710],{},[176,28141,6321],{},[51,28143,28144,28146,28148],{},[54,28145,3000],{},[176,28147,6463],{},[347,28149,2213],{},[54,28151,96],{},[176,28153,189],{"fence":65},[58,28155,28156],{"encoding":60},"-\\int_{0}^{\\frac{\\pi}{2}} \\sin^{n-1} x \\, d(\\cos x) \\\\ = -\\left( \\sin^{n-1} x \\cos x \\right) \\Big|_{0}^{\\frac{\\pi}{2}} + \\int_{0}^{\\frac{\\pi}{2}} \\cos x \\, d\\left(\\sin^{n-1} x 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1226,31227,31255,31256,2991],{},"For even ",[34,31228,31230,31243],{"className":31229,"translate":38},[37],[34,31231,31233],{"className":31232},[42],[44,31234,31235],{"xmlns":46},[48,31236,31237,31241],{},[51,31238,31239],{},[54,31240,3000],{},[58,31242,3000],{"encoding":60},[34,31244,31246],{"className":31245,"ariaHidden":65},[64],[34,31247,31249,31252],{"className":31248},[69],[34,31250],{"className":31251,"style":108},[73],[34,31253,3000],{"className":31254},[78,79],", i.e., ",[34,31257,31259,31279],{"className":31258,"translate":38},[37],[34,31260,31262],{"className":31261},[42],[44,31263,31264],{"xmlns":46},[48,31265,31266,31276],{},[51,31267,31268,31270,31272,31274],{},[54,31269,3000],{},[176,31271,192],{},[347,31273,424],{},[54,31275,3324],{},[58,31277,31278],{"encoding":60},"n = 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we need the double factorial:",[34,34416,34418],{"className":34417,"translate":38},[159],[34,34419,34421,34466],{"className":34420,"translate":38},[37],[34,34422,34424],{"className":34423},[42],[44,34425,34426],{"xmlns":46,"display":168},[48,34427,34428,34463],{},[51,34429,34430,34432,34435,34437,34439,34441,34443,34445,34447,34449,34451,34453,34455,34457,34459,34461],{},[54,34431,3010],{},[176,34433,34434],{"stretchy":178},"!",[176,34436,34434],{"stretchy":178},[176,34438,192],{},[54,34440,3010],{},[176,34442,179],{"stretchy":178},[54,34444,3010],{},[176,34446,6463],{},[347,34448,424],{},[176,34450,189],{"stretchy":178},[176,34452,179],{"stretchy":178},[54,34454,3010],{},[176,34456,6463],{},[347,34458,990],{},[176,34460,189],{"stretchy":178},[176,34462,6706],{},[58,34464,34465],{"encoding":60},"m!! = m (m-2) (m-4) 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\\rightarrow \\infty} \\left( \\frac{(2k-2)!!}{(2k-1)!!} - \\frac{(2k)!!}{(2k+1)!!} \\right) \\\\ = \\lim_{k \\rightarrow \\infty} \\left( \\frac{(2k-2)!!}{(2k-1)!!} - \\frac{2k}{2k+1} \\cdot \\frac{(2k-2)!!}{(2k-1)!!} \\right) \\\\ = \\lim_{k \\rightarrow \\infty} \\frac{1}{2k+1} \\cdot \\frac{(2k-2)!!}{(2k-1)!!} = 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(2k)!!}{(2k-1)!!^2}",[34,39597,39599,39761,39838],{"className":39598,"ariaHidden":65},[64],[34,39600,39602,39606,39752,39755,39758],{"className":39601},[69],[34,39603],{"className":39604,"style":39605},[73],"height:2.4271em;vertical-align:-0.936em;",[34,39607,39609,39612,39749],{"className":39608},[78],[34,39610],{"className":39611},[230,12689],[34,39613,39615],{"className":39614},[12543],[34,39616,39618,39741],{"className":39617},[814,2523],[34,39619,39621,39738],{"className":39620},[818],[34,39622,39624,39680,39688],{"className":39623,"style":15228},[822],[34,39625,39626,39629],{"style":14155},[34,39627],{"className":39628,"style":12709},[829],[34,39630,39632,39635,39638,39641,39644,39647,39650,39653,39656,39659,39662,39665,39668,39671,39674,39677],{"className":39631},[78],[34,39633,179],{"className":39634},[230],[34,39636,424],{"className":39637},[78],[34,39639,3324],{"className":39640,"style":3446},[78,79],[34,39642],{"className":39643,"style":238},[237],[34,39645,184],{"className":39646},[242],[34,39648],{"className":39649,"style":238},[237],[34,39651,2213],{"className":39652},[78],[34,39654,34895],{"className":39655},[258],[34,39657,179],{"className":39658},[230],[34,39660,424],{"className":39661},[78],[34,39663,3324],{"className":39664,"style":3446},[78,79],[34,39666],{"className":39667,"style":238},[237],[34,39669,6463],{"className":39670},[242],[34,39672],{"className":39673,"style":238},[237],[34,39675,2213],{"className":39676},[78],[34,39678,34895],{"className":39679},[258],[34,39681,39682,39685],{"style":12897},[34,39683],{"className":39684,"style":12709},[829],[34,39686],{"className":39687,"style":12904},[12836],[34,39689,39690,39693],{"style":12907},[34,39691],{"className":39692,"style":12709},[829],[34,39694,39696,39699,39702,39705,39709],{"className":39695},[78],[34,39697,179],{"className":39698},[230],[34,39700,424],{"className":39701},[78],[34,39703,3324],{"className":39704,"style":3446},[78,79],[34,39706,39708],{"className":39707},[258],")!",[34,39710,39712,39715],{"className":39711},[258],[34,39713,34434],{"className":39714},[258],[34,39716,39718],{"className":39717},[810],[34,39719,39721],{"className":39720},[814],[34,39722,39724],{"className":39723},[818],[34,39725,39727],{"className":39726,"style":800},[822],[34,39728,39729,39732],{"style":825},[34,39730],{"className":39731,"style":830},[829],[34,39733,39735],{"className":39734},[834,835,836,837],[34,39736,424],{"className":39737},[78,837],[34,39739,2547],{"className":39740},[2546],[34,39742,39744],{"className":39743},[818],[34,39745,39747],{"className":39746,"style":17481},[822],[34,39748],{},[34,39750],{"className":39751},[258,12689],[34,39753],{"className":39754,"style":262},[237],[34,39756,35453],{"className":39757},[266],[34,39759],{"className":39760,"style":262},[237],[34,39762,39764,39767,39829,39832,39835],{"className":39763},[69],[34,39765],{"className":39766,"style":26739},[73],[34,39768,39770,39773,39826],{"className":39769},[78],[34,39771],{"className":39772},[230,12689],[34,39774,39776],{"className":39775},[12543],[34,39777,39779,39818],{"className":39778},[814,2523],[34,39780,39782,39815],{"className":39781},[818],[34,39783,39785,39796,39804],{"className":39784,"style":26758},[822],[34,39786,39787,39790],{"style":14155},[34,39788],{"className":39789,"style":12709},[829],[34,39791,39793],{"className":39792},[78],[34,39794,424],{"className":39795},[78],[34,39797,39798,39801],{"style":12897},[34,39799],{"className":39800,"style":12709},[829],[34,39802],{"className":39803,"style":12904},[12836],[34,39805,39806,39809],{"style":12907},[34,39807],{"className":39808,"style":12709},[829],[34,39810,39812],{"className":39811},[78],[34,39813,12569],{"className":39814,"style":143},[78,79],[34,39816,2547],{"className":39817},[2546],[34,39819,39821],{"className":39820},[818],[34,39822,39824],{"className":39823,"style":14669},[822],[34,39825],{},[34,39827],{"className":39828},[258,12689],[34,39830],{"className":39831,"style":262},[237],[34,39833,35453],{"className":39834},[266],[34,39836],{"className":39837,"style":262},[237],[34,39839,39841,39844],{"className":39840},[69],[34,39842],{"className":39843,"style":23317},[73],[34,39845,39847,39850,39986],{"className":39846},[78],[34,39848],{"className":39849},[230,12689],[34,39851,39853],{"className":39852},[12543],[34,39854,39856,39978],{"className":39855},[814,2523],[34,39857,39859,39975],{"className":39858},[818],[34,39860,39862,39923,39931],{"className":39861,"style":14583},[822],[34,39863,39864,39867],{"style":14155},[34,39865],{"className":39866,"style":12709},[829],[34,39868,39870,39873,39876,39879,39882,39885,39888,39891,39894],{"className":39869},[78],[34,39871,179],{"className":39872},[230],[34,39874,424],{"className":39875},[78],[34,39877,3324],{"className":39878,"style":3446},[78,79],[34,39880],{"className":39881,"style":238},[237],[34,39883,6463],{"className":39884},[242],[34,39886],{"className":39887,"style":238},[237],[34,39889,2213],{"className":39890},[78],[34,39892,39708],{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can further examine the difference between the bounds for ",[34,39993,39995,40013],{"className":39994,"translate":38},[37],[34,39996,39998],{"className":39997},[42],[44,39999,40000],{"xmlns":46},[48,40001,40002,40010],{},[51,40003,40004],{},[12543,40005,40006,40008],{},[54,40007,12569],{},[347,40009,424],{},[58,40011,40012],{"encoding":60},"\\frac{\\pi}{2}",[34,40014,40016],{"className":40015,"ariaHidden":65},[64],[34,40017,40019,40023],{"className":40018},[69],[34,40020],{"className":40021,"style":40022},[73],"height:1.0404em;vertical-align:-0.345em;",[34,40024,40026,40029,40088],{"className":40025},[78],[34,40027],{"className":40028},[230,12689],[34,40030,40032],{"className":40031},[12543],[34,40033,40035,40080],{"className":40034},[814,2523],[34,40036,40038,40077],{"className":40037},[818],[34,40039,40041,40055,40063],{"className":40040,"style":35762},[822],[34,40042,40043,40046],{"style":17906},[34,40044],{"className":40045,"style":12709},[829],[34,40047,40049],{"className":40048},[834,835,836,837],[34,40050,40052],{"className":40051},[78,837],[34,40053,424],{"className":40054},[78,837],[34,40056,40057,40060],{"style":12897},[34,40058],{"className":40059,"style":12709},[829],[34,40061],{"className":40062,"style":12904},[12836],[34,40064,40065,40068],{"style":17929},[34,40066],{"className":40067,"style":12709},[829],[34,40069,40071],{"className":40070},[834,835,836,837],[34,40072,40074],{"className":40073},[78,837],[34,40075,12569],{"className":40076,"style":143},[78,79,837],[34,40078,2547],{"className":40079},[2546],[34,40081,40083],{"className":40082},[818],[34,40084,40086],{"className":40085,"style":17973},[822],[34,40087],{},[34,40089],{"className":40090},[258,12689],[34,40092,40094],{"className":40093,"translate":38},[159],[34,40095,40097,40427],{"className":40096,"translate":38},[37],[34,40098,40100],{"className":40099},[42],[44,40101,40102],{"xmlns":46,"display":168},[48,40103,40104,40424],{},[51,40105,40106,40122,40238,40240,40242,40258,40290,40292,40346,40348,40350,40366,40376,40378,40420,40422],{},[15798,40107,40108,40114],{},[51,40109,40110,40112],{},[54,40111,37329],{},[176,40113,6321],{},[51,40115,40116,40118,40120],{},[54,40117,3324],{},[176,40119,3003],{},[54,40121,13314],{"mathvariant":2296},[51,40123,40124,40126,40180,40182,40236],{},[176,40125,179],{"fence":65},[12543,40127,40128,40158],{},[51,40129,40130,40132,40134,40136,40138,40140,40142,40144,40146,40148,40150,40152,40154,40156],{},[176,40131,179],{"stretchy":178},[347,40133,424],{},[54,40135,3324],{},[176,40137,6463],{},[347,40139,424],{},[176,40141,189],{"stretchy":178},[176,40143,34434],{"stretchy":178},[176,40145,34434],{"stretchy":178},[176,40147,179],{"stretchy":178},[347,40149,424],{},[54,40151,3324],{},[176,40153,189],{"stretchy":178},[176,40155,34434],{"stretchy":178},[176,40157,34434],{"stretchy":178},[51,40159,40160,40162,40164,40166,40168,40170,40172,40174],{},[176,40161,179],{"stretchy":178},[347,40163,424],{},[54,40165,3324],{},[176,40167,6463],{},[347,40169,2213],{},[176,40171,189],{"stretchy":178},[176,40173,34434],{"stretchy":178},[755,40175,40176,40178],{},[176,40177,34434],{"stretchy":178},[347,40179,424],{},[176,40181,6463],{},[12543,40183,40184,40202],{},[51,40185,40186,40188,40190,40192,40194,40196],{},[176,40187,179],{"stretchy":178},[347,40189,424],{},[54,40191,3324],{},[176,40193,189],{"stretchy":178},[176,40195,34434],{"stretchy":178},[755,40197,40198,40200],{},[176,40199,34434],{"stretchy":178},[347,40201,424],{},[51,40203,40204,40206,40208,40210,40212,40214,40216,40218,40220,40222,40224,40226,40228,40230,40232,40234],{},[176,40205,179],{"stretchy":178},[347,40207,424],{},[54,40209,3324],{},[176,40211,184],{},[347,40213,2213],{},[176,40215,189],{"stretchy":178},[176,40217,34434],{"stretchy":178},[176,40219,34434],{"stretchy":178},[176,40221,179],{"stretchy":178},[347,40223,424],{},[54,40225,3324],{},[176,40227,6463],{},[347,40229,2213],{},[176,40231,189],{"stretchy":178},[176,40233,34434],{"stretchy":178},[176,40235,34434],{"stretchy":178},[176,40237,189],{"fence":65},[237,40239],{"linebreak":28059},[176,40241,192],{},[15798,40243,40244,40250],{},[51,40245,40246,40248],{},[54,40247,37329],{},[176,40249,6321],{},[51,40251,40252,40254,40256],{},[54,40253,3324],{},[176,40255,3003],{},[54,40257,13314],{"mathvariant":2296},[51,40259,40260,40262,40272,40274,40288],{},[176,40261,179],{"fence":65},[12543,40263,40264,40266],{},[347,40265,2213],{},[51,40267,40268,40270],{},[347,40269,424],{},[54,40271,3324],{},[176,40273,6463],{},[12543,40275,40276,40278],{},[347,40277,2213],{},[51,40279,40280,40282,40284,40286],{},[347,40281,424],{},[54,40283,3324],{},[176,40285,184],{},[347,40287,2213],{},[176,40289,189],{"fence":65},[176,40291,1176],{},[12543,40293,40294,40312],{},[51,40295,40296,40298,40300,40302,40304,40306],{},[176,40297,179],{"stretchy":178},[347,40299,424],{},[54,40301,3324],{},[176,40303,189],{"stretchy":178},[176,40305,34434],{"stretchy":178},[755,40307,40308,40310],{},[176,40309,34434],{"stretchy":178},[347,40311,424],{},[51,40313,40314,40316,40318,40320,40322,4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\\rightarrow \\infty} \\left( \\frac{(2k-2)!! (2k)!!}{(2k-1)!!^2} - \\frac{(2k)!!^2}{(2k+1)!! (2k-1)!!} \\right) \\\\= \\lim_{k \\rightarrow \\infty} \\left( \\frac{1}{2k} - \\frac{1}{2k+1} \\right) \\cdot \\frac{(2k)!!^2}{(2k+1)!! (2k-1)!!} \\\\= \\lim_{k \\rightarrow \\infty} \\frac{1}{2k} \\cdot \\frac{(2k)!!^2}{(2k+1)!!^2} = 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we obtain ",[20,41539,27139],{},[34,41541,41543],{"className":41542,"translate":38},[159],[34,41544,41546,41636],{"className":41545,"translate":38},[37],[34,41547,41549],{"className":41548},[42],[44,41550,41551],{"xmlns":46,"display":168},[48,41552,41553,41633],{},[51,41554,41555,41571,41625,41627],{},[15798,41556,41557,41563],{},[51,41558,41559,41561],{},[54,41560,37329],{},[176,41562,6321],{},[51,41564,41565,41567,41569],{},[54,41566,3324],{},[176,41568,3003],{},[54,41570,13314],{"mathvariant":2296},[12543,41572,41573,41591],{},[51,41574,41575,41577,41579,41581,41583,41585],{},[176,41576,179],{"stretchy":178},[347,41578,424],{},[54,41580,3324],{},[176,41582,189],{"stretchy":178},[176,41584,34434],{"stretchy":178},[755,41586,41587,41589],{},[176,41588,34434],{"stretchy":178},[347,41590,424],{},[51,41592,41593,41595,41597,41599,41601,41603,41605,41607,41609,41611,41613,41615,41617,41619,41621,41623],{},[176,41594,179],{"stretchy":178},[347,41596,424],{},[54,41598,3324],{},[176,41600,184],{},[347,41602,2213],{},[176,41604,189],{"stretchy":178},[176,41606,34434],{"stretchy":178},[176,41608,34434],{"stretchy":178},[176,41610,179],{"stretchy":178},[347,41612,424],{},[54,41614,3324],{},[176,41616,6463],{},[347,41618,2213],{},[176,41620,189],{"stretchy":178},[176,41622,34434],{"stretchy":178},[176,41624,34434],{"stretchy":178},[176,41626,192],{},[12543,41628,41629,41631],{},[54,41630,12569],{},[347,41632,424],{},[58,41634,41635],{"encoding":60},"\\lim_{k \\rightarrow \\infty} \\frac{(2k)!!^2}{(2k+1)!! (2k-1)!!} = \\frac{\\pi}{2}",[34,41637,41639,41855],{"className":41638,"ariaHidden":65},[64],[34,41640,41642,41645,41698,41701,41846,41849,41852],{"className":41641},[69],[34,41643],{"className":41644,"style":39605},[73],[34,41646,41648],{"className":41647},[3387,3388],[34,41649,41651,41690],{"className":41650},[814,2523],[34,41652,41654,41687],{"className":41653},[818],[34,41655,41657,41677],{"className":41656,"style":2184},[822],[34,41658,41659,41662],{"style":37648},[34,41660],{"className":41661,"style":12709},[829],[34,41663,41665],{"className":41664},[834,835,836,837],[34,41666,41668,41671,41674],{"className":41667},[78,837],[34,41669,3324],{"className":41670,"style":3446},[78,79,837],[34,41672,3003],{"className":41673},[266,837],[34,41675,13314],{"className":41676},[78,837],[34,41678,41679,41682],{"style":23024},[34,41680],{"className":41681,"style":12709},[829],[34,41683,41684],{},[34,41685,37329],{"className":41686},[3387],[34,41688,2547],{"className":41689},[2546],[34,41691,41693],{"className":41692},[818],[34,41694,41696],{"className":41695,"style":37686},[822],[34,41697],{},[34,41699],{"className":41700,"style":3462},[237],[34,41702,41704,41707,41843],{"className":41703},[78],[34,41705],{"className":41706},[230,12689],[34,41708,41710],{"className":41709},[12543],[34,41711,41713,41835],{"className":41712},[814,2523],[34,41714,41716,41832],{"className":41715},[818],[34,41717,41719,41775,41783],{"className":41718,"style":15228},[822],[34,41720,41721,41724],{"style":14155},[34,41722],{"className":41723,"style":12709},[829],[34,41725,41727,41730,41733,41736,41739,41742,41745,41748,41751,41754,41757,41760,41763,41766,41769,41772],{"className":41726},[78],[34,41728,179],{"className":41729},[230],[34,41731,424],{"className":41732},[78],[34,41734,3324],{"className":41735,"style":3446},[78,79],[34,41737],{"className":41738,"style":238},[237],[34,41740,184],{"className":41741},[242],[34,41743],{"className":41744,"style":238},[237],[34,41746,2213],{"className":41747},[78],[34,41749,34895],{"className":41750},[258],[34,41752,179],{"className":41753},[230],[34,41755,424],{"className":41756},[78],[34,41758,3324],{"className":41759,"style":3446},[78,79],[34,41761],{"className":41762,"style":238},[237],[34,41764,6463],{"className":41765},[242],[34,41767],{"className":41768,"style":238},[237],[34,41770,2213],{"className":41771},[78],[34,41773,34895],{"className":41774},[258],[34,41776,41777,41780],{"style":12897},[34,41778],{"className":41779,"style":12709},[829],[34,41781],{"className":41782,"style":12904},[12836],[34,41784,41785,41788],{"style":12907},[34,41786],{"className":41787,"style":12709},[829],[34,41789,41791,41794,41797,41800,41803],{"className":41790},[78],[34,41792,179],{"className":41793},[230],[34,41795,424],{"className":41796},[78],[34,41798,3324],{"className":41799,"style":3446},[78,79],[34,41801,39708],{"className":41802},[258],[34,41804,41806,41809],{"className":41805},[258],[34,41807,34434],{"className":41808},[258],[34,41810,41812],{"className":41811},[810],[34,41813,41815],{"className":41814},[814],[34,41816,41818],{"className":41817},[818],[34,41819,41821],{"className":41820,"style":800},[822],[34,41822,41823,41826],{"style":825},[34,41824],{"className":41825,"style":830},[829],[34,41827,41829],{"className":41828},[834,835,836,837],[34,41830,424],{"className":41831},[78,837],[34,41833,2547],{"className":41834},[2546],[34,41836,41838],{"className":41837},[818],[34,41839,41841],{"className":41840,"style":17481},[822],[34,41842],{},[34,41844],{"className":41845},[258,12689],[34,41847],{"className":41848,"style":262},[237],[34,41850,192],{"className":41851},[266],[34,41853],{"className":41854,"style":262},[237],[34,41856,41858,41861],{"className":41857},[69],[34,41859],{"className":41860,"style":26739},[73],[34,41862,41864,41867,41920],{"className":41863},[78],[34,41865],{"className":41866},[230,12689],[34,41868,41870],{"className":41869},[12543],[34,41871,41873,41912],{"className":41872},[814,2523],[34,41874,41876,41909],{"className":41875},[818],[34,41877,41879,41890,41898],{"className":41878,"style":26758},[822],[34,41880,41881,41884],{"style":14155},[34,41882],{"className":41883,"style":12709},[829],[34,41885,41887],{"className":41886},[78],[34,41888,424],{"className":41889},[78],[34,41891,41892,41895],{"style":12897},[34,41893],{"className":41894,"style":12709},[829],[34,41896],{"className":41897,"style":12904},[12836],[34,41899,41900,41903],{"style":12907},[34,41901],{"className":41902,"style":12709},[829],[34,41904,41906],{"className":41905},[78],[34,41907,12569],{"className":41908,"style":143},[78,79],[34,41910,2547],{"className":41911},[2546],[34,41913,41915],{"className":41914},[818],[34,41916,41918],{"className":41917,"style":14669},[822],[34,41919],{},[34,41921],{"className":41922},[258,12689],[16,41924,41925],{},[20,41926,41927],{},"This formula introduces Wallis's result in preparation for deriving Stirling's formula.",[11,41929,41931],{"id":41930},"stirlings-formula","Stirling's Formula",[16,41933,41934,41935,16199,41969,42028,42029,144],{},"Stirling's formula reveals a relationship between ",[34,41936,41938,41954],{"className":41937,"translate":38},[37],[34,41939,41941],{"className":41940},[42],[44,41942,41943],{"xmlns":46},[48,41944,41945,41951],{},[51,41946,41947,41949],{},[54,41948,3000],{},[176,41950,34434],{"stretchy":178},[58,41952,41953],{"encoding":60},"n!",[34,41955,41957],{"className":41956,"ariaHidden":65},[64],[34,41958,41960,41963,41966],{"className":41959},[69],[34,41961],{"className":41962,"style":2184},[73],[34,41964,3000],{"className":41965},[78,79],[34,41967,34434],{"className":41968},[258],[34,41970,41972,41990],{"className":41971,"translate":38},[37],[34,41973,41975],{"className":41974},[42],[44,41976,41977],{"xmlns":46},[48,41978,41979,41987],{},[51,41980,41981],{},[755,41982,41983,41985],{},[54,41984,3000],{},[54,41986,3000],{},[58,41988,41989],{"encoding":60},"n^n",[34,41991,41993],{"className":41992,"ariaHidden":65},[64],[34,41994,41996,41999],{"className":41995},[69],[34,41997],{"className":41998,"style":3061},[73],[34,42000,42002,42005],{"className":42001},[78],[34,42003,3000],{"className":42004},[78,79],[34,42006,42008],{"className":42007},[810],[34,42009,42011],{"className":42010},[814],[34,42012,42014],{"className":42013},[818],[34,42015,42017],{"className":42016,"style":3061},[822],[34,42018,42019,42022],{"style":825},[34,42020],{"className":42021,"style":830},[829],[34,42023,42025],{"className":42024},[834,835,836,837],[34,42026,3000],{"className":42027},[78,79,837],", and ",[34,42030,42032,42050],{"className":42031,"translate":38},[37],[34,42033,42035],{"className":42034},[42],[44,42036,42037],{"xmlns":46},[48,42038,42039,42047],{},[51,42040,42041],{},[755,42042,42043,42045],{},[54,42044,18036],{},[54,42046,3000],{},[58,42048,42049],{"encoding":60},"e^n",[34,42051,42053],{"className":42052,"ariaHidden":65},[64],[34,42054,42056,42059],{"className":42055},[69],[34,42057],{"className":42058,"style":3061},[73],[34,42060,42062,42065],{"className":42061},[78],[34,42063,18036],{"className":42064},[78,79],[34,42066,42068],{"className":42067},[810],[34,42069,42071],{"className":42070},[814],[34,42072,42074],{"className":42073},[818],[34,42075,42077],{"className":42076,"style":3061},[822],[34,42078,42079,42082],{"style":825},[34,42080],{"className":42081,"style":830},[829],[34,42083,42085],{"className":42084},[834,835,836,837],[34,42086,3000],{"className":42087},[78,79,837],[316,42089,42090],{},[16,42091,42092,42093,42121,42122,42155,42156,144],{},"As ",[34,42094,42096,42109],{"className":42095,"translate":38},[37],[34,42097,42099],{"className":42098},[42],[44,42100,42101],{"xmlns":46},[48,42102,42103,42107],{},[51,42104,42105],{},[54,42106,3000],{},[58,42108,3000],{"encoding":60},[34,42110,42112],{"className":42111,"ariaHidden":65},[64],[34,42113,42115,42118],{"className":42114},[69],[34,42116],{"className":42117,"style":108},[73],[34,42119,3000],{"className":42120},[78,79]," approaches infinity, ",[34,42123,42125,42140],{"className":42124,"translate":38},[37],[34,42126,42128],{"className":42127},[42],[44,42129,42130],{"xmlns":46},[48,42131,42132,42138],{},[51,42133,42134,42136],{},[54,42135,3000],{},[176,42137,34434],{"stretchy":178},[58,42139,41953],{"encoding":60},[34,42141,42143],{"className":42142,"ariaHidden":65},[64],[34,42144,42146,42149,42152],{"className":42145},[69],[34,42147],{"className":42148,"style":2184},[73],[34,42150,3000],{"className":42151},[78,79],[34,42153,34434],{"className":42154},[258]," is approximately ",[34,42157,42159,42197],{"className":42158,"translate":38},[37],[34,42160,42162],{"className":42161},[42],[44,42163,42164],{"xmlns":46},[48,42165,42166,42194],{},[51,42167,42168,42178],{},[19376,42169,42170],{},[51,42171,42172,42174,42176],{},[347,42173,424],{},[54,42175,12569],{},[54,42177,3000],{},[755,42179,42180,42192],{},[51,42181,42182,42184,42190],{},[176,42183,179],{"fence":65},[12543,42185,42186,42188],{},[54,42187,3000],{},[54,42189,18036],{},[176,42191,189],{"fence":65},[54,42193,3000],{},[58,42195,42196],{"encoding":60},"\\sqrt{2 \\pi n} \\left(\\frac{n}{e}\\right)^n",[34,42198,42200],{"className":42199,"ariaHidden":65},[64],[34,42201,42203,42207,42268,42271],{"className":42202},[69],[34,42204],{"className":42205,"style":42206},[73],"height:1.2572em;vertical-align:-0.35em;",[34,42208,42210],{"className":42209},[78,19464],[34,42211,42213,42259],{"className":42212},[814,2523],[34,42214,42216,42256],{"className":42215},[818],[34,42217,42220,42239],{"className":42218,"style":42219},[822],"height:0.9072em;",[34,42221,42223,42226],{"className":42222,"style":23024},[10795],[34,42224],{"className":42225,"style":12709},[829],[34,42227,42230,42233,42236],{"className":42228,"style":42229},[78],"padding-left:0.833em;",[34,42231,424],{"className":42232},[78],[34,42234,12569],{"className":42235,"style":143},[78,79],[34,42237,3000],{"className":42238},[78,79],[34,42240,42242,42245],{"style":42241},"top:-2.8672em;",[34,42243],{"className":42244,"style":12709},[829],[34,42246,42249],{"className":42247,"style":42248},[10803],"min-width:0.853em;height:1.08em;",[10806,42250,42253],{"xmlns":10808,"width":10809,"height":42251,"viewBox":42252,"preserveAspectRatio":19563},"1.08em","0 0 400000 1080",[10814,42254],{"d":42255},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[34,42257,2547],{"className":42258},[2546],[34,42260,42262],{"className":42261},[818],[34,42263,42266],{"className":42264,"style":42265},[822],"height:0.1328em;",[34,42267],{},[34,42269],{"className":42270,"style":3462},[237],[34,42272,42274,42357],{"className":42273},[4177],[34,42275,42277,42283,42351],{"className":42276},[4177],[34,42278,42280],{"className":42279,"style":4182},[230,4181],[34,42281,179],{"className":42282},[4186,9740],[34,42284,42286,42289,42348],{"className":42285},[78],[34,42287],{"className":42288},[230,12689],[34,42290,42292],{"className":42291},[12543],[34,42293,42295,42340],{"className":42294},[814,2523],[34,42296,42298,42337],{"className":42297},[818],[34,42299,42301,42315,42323],{"className":42300,"style":35762},[822],[34,42302,42303,42306],{"style":17906},[34,42304],{"className":42305,"style":12709},[829],[34,42307,42309],{"className":42308},[834,835,836,837],[34,42310,42312],{"className":42311},[78,837],[34,42313,18036],{"className":42314},[78,79,837],[34,42316,42317,42320],{"style":12897},[34,42318],{"className":42319,"style":12709},[829],[34,42321],{"className":42322,"style":12904},[12836],[34,42324,42325,42328],{"style":17929},[34,42326],{"className":42327,"style":12709},[829],[34,42329,42331],{"className":42330},[834,835,836,837],[34,42332,42334],{"className":42333},[78,837],[34,42335,3000],{"className":42336},[78,79,837],[34,42338,2547],{"className":42339},[2546],[34,42341,42343],{"className":42342},[818],[34,42344,42346],{"className":42345,"style":17973},[822],[34,42347],{},[34,42349],{"className":42350},[258,12689],[34,42352,42354],{"className":42353,"style":4182},[258,4181],[34,42355,189],{"className":42356},[4186,9740],[34,42358,42360],{"className":42359},[810],[34,42361,42363],{"className":42362},[814],[34,42364,42366],{"className":42365},[818],[34,42367,42370],{"className":42368,"style":42369},[822],"height:0.9043em;",[34,42371,42373,42376],{"style":42372},"top:-3.3029em;margin-right:0.05em;",[34,42374],{"className":42375,"style":830},[829],[34,42377,42379],{"className":42378},[834,835,836,837],[34,42380,3000],{"className":42381},[78,79,837],[16,42383,42384],{},"First, introduce a limit:",[34,42386,42388],{"className":42387,"translate":38},[159],[34,42389,42391,42455],{"className":42390,"translate":38},[37],[34,42392,42394],{"className":42393},[42],[44,42395,42396],{"xmlns":46,"display":168},[48,42397,42398,42452],{},[51,42399,42400,42416,42444,42446],{},[15798,42401,42402,42408],{},[51,42403,42404,42406],{},[54,42405,37329],{},[176,42407,6321],{},[51,42409,42410,42412,42414],{},[54,42411,3000],{},[176,42413,3003],{},[54,42415,13314],{"mathvariant":2296},[755,42417,42418,42438],{},[51,42419,42420,42422,42436],{},[176,42421,179],{"fence":65},[12543,42423,42424,42430],{},[51,42425,42426,42428],{},[54,42427,3000],{},[176,42429,34434],{"stretchy":178},[755,42431,42432,42434],{},[54,42433,3000],{},[54,42435,3000],{},[176,42437,189],{"fence":65},[12543,42439,42440,42442],{},[347,42441,2213],{},[54,42443,3000],{},[176,42445,192],{},[12543,42447,42448,42450],{},[347,42449,2213],{},[54,42451,18036],{},[58,42453,42454],{"encoding":60},"\\lim_{n \\rightarrow \\infty} \\left( \\frac{n!}{n^n} \\right)^{\\frac{1}{n}} = \\frac{1}{e}",[34,42456,42458,42736],{"className":42457,"ariaHidden":65},[64],[34,42459,42461,42465,42520,42523,42727,42730,42733],{"className":42460},[69],[34,42462],{"className":42463,"style":42464},[73],"height:2.744em;vertical-align:-0.95em;",[34,42466,42468],{"className":42467},[3387,3388],[34,42469,42471,42511],{"className":42470},[814,2523],[34,42472,42474,42508],{"className":42473},[818],[34,42475,42477,42498],{"className":42476,"style":2184},[822],[34,42478,42480,42483],{"style":42479},"top:-2.4em;margin-left:0em;",[34,42481],{"className":42482,"style":12709},[829],[34,42484,42486],{"className":42485},[834,835,836,837],[34,42487,42489,42492,42495],{"className":42488},[78,837],[34,42490,3000],{"className":42491},[78,79,837],[34,42493,3003],{"className":42494},[266,837],[34,42496,13314],{"className":42497},[78,837],[34,42499,42500,42503],{"style":23024},[34,42501],{"className":42502,"style":12709},[829],[34,42504,42505],{},[34,42506,37329],{"className":42507},[3387],[34,42509,2547],{"className":42510},[2546],[34,42512,42514],{"className":42513},[818],[34,42515,42518],{"className":42516,"style":42517},[822],"height:0.7em;",[34,42519],{},[34,42521],{"className":42522,"style":3462},[237],[34,42524,42526,42633],{"className":42525},[4177],[34,42527,42529,42535,42627],{"className":42528},[4177],[34,42530,42532],{"className":42531,"style":4182},[230,4181],[34,42533,179],{"className":42534},[4186,836],[34,42536,42538,42541,42624],{"className":42537},[78],[34,42539],{"className":42540},[230,12689],[34,42542,42544],{"className":42543},[12543],[34,42545,42547,42616],{"className":42546},[814,2523],[34,42548,42550,42613],{"className":42549},[818],[34,42551,42553,42591,42599],{"className":42552,"style":15668},[822],[34,42554,42555,42558],{"style":14155},[34,42556],{"className":42557,"style":12709},[829],[34,42559,42561],{"className":42560},[78],[34,42562,42564,42567],{"className":42563},[78],[34,42565,3000],{"className":42566},[78,79],[34,42568,42570],{"className":42569},[810],[34,42571,42573],{"className":42572},[814],[34,42574,42576],{"className":42575},[818],[34,42577,42580],{"className":42578,"style":42579},[822],"height:0.5904em;",[34,42581,42582,42585],{"style":12743},[34,42583],{"className":42584,"style":830},[829],[34,42586,42588],{"className":42587},[834,835,836,837],[34,42589,3000],{"className":42590},[78,79,837],[34,42592,42593,42596],{"style":12897},[34,42594],{"className":42595,"style":12709},[829],[34,42597],{"className":42598,"style":12904},[12836],[34,42600,42601,42604],{"style":12907},[34,42602],{"className":42603,"style":12709},[829],[34,42605,42607,42610],{"className":42606},[78],[34,42608,3000],{"className":42609},[78,79],[34,42611,34434],{"className":42612},[258],[34,42614,2547],{"className":42615},[2546],[34,42617,42619],{"className":42618},[818],[34,42620,42622],{"className":42621,"style":14669},[822],[34,42623],{},[34,42625],{"className":42626},[258,12689],[34,42628,42630],{"className":42629,"style":4182},[258,4181],[34,42631,189],{"className":42632},[4186,836],[34,42634,42636],{"className":42635},[810],[34,42637,42639],{"className":42638},[814],[34,42640,42642],{"className":42641},[818],[34,42643,42646],{"className":42644,"style":42645},[822],"height:1.7939em;",[34,42647,42649,42652],{"style":42648},"top:-4.2029em;margin-right:0.05em;",[34,42650],{"className":42651,"style":12709},[829],[34,42653,42655],{"className":42654},[834,835,836,837],[34,42656,42658],{"className":42657},[78,837],[34,42659,42661,42664,42724],{"className":42660},[78,837],[34,42662],{"className":42663},[230,12689,834,10776,12798],[34,42665,42667],{"className":42666},[12543],[34,42668,42670,42716],{"className":42669},[814,2523],[34,42671,42673,42713],{"className":42672},[818],[34,42674,42677,42691,42699],{"className":42675,"style":42676},[822],"height:0.8443em;",[34,42678,42679,42682],{"style":12814},[34,42680],{"className":42681,"style":12709},[829],[34,42683,42685],{"className":42684},[834,10776,9740,837],[34,42686,42688],{"className":42687},[78,837],[34,42689,3000],{"className":42690},[78,79,837],[34,42692,42693,42696],{"style":12829},[34,42694],{"className":42695,"style":12709},[829],[34,42697],{"className":42698,"style":12837},[12836,837],[34,42700,42701,42704],{"style":12840},[34,42702],{"className":42703,"style":12709},[829],[34,42705,42707],{"className":42706},[834,10776,9740,837],[34,42708,42710],{"className":42709},[78,837],[34,42711,2213],{"className":42712},[78,837],[34,42714,2547],{"className":42715},[2546],[34,42717,42719],{"className":42718},[818],[34,42720,42722],{"className":42721,"style":12862},[822],[34,42723],{},[34,42725],{"className":42726},[258,12689,834,10776,12798],[34,42728],{"className":42729,"style":262},[237],[34,42731,192],{"className":42732},[266],[34,42734],{"className":42735,"style":262},[237],[34,42737,42739,42743],{"className":42738},[69],[34,42740],{"className":42741,"style":42742},[73],"height:2.0074em;vertical-align:-0.686em;",[34,42744,42746,42749,42802],{"className":42745},[78],[34,42747],{"className":42748},[230,12689],[34,42750,42752],{"className":42751},[12543],[34,42753,42755,42794],{"className":42754},[814,2523],[34,42756,42758,42791],{"className":42757},[818],[34,42759,42761,42772,42780],{"className":42760,"style":16800},[822],[34,42762,42763,42766],{"style":14155},[34,42764],{"className":42765,"style":12709},[829],[34,42767,42769],{"className":42768},[78],[34,42770,18036],{"className":42771},[78,79],[34,42773,42774,42777],{"style":12897},[34,42775],{"className":42776,"style":12709},[829],[34,42778],{"className":42779,"style":12904},[12836],[34,42781,42782,42785],{"style":12907},[34,42783],{"className":42784,"style":12709},[829],[34,42786,42788],{"className":42787},[78],[34,42789,2213],{"className":42790},[78],[34,42792,2547],{"className":42793},[2546],[34,42795,42797],{"className":42796},[818],[34,42798,42800],{"className":42799,"style":14669},[822],[34,42801],{},[34,42803],{"className":42804},[258,12689],[16,42806,42807],{},"Proof:",[34,42809,42811],{"className":42810,"translate":38},[159],[34,42812,42814,42925],{"className":42813,"translate":38},[37],[34,42815,42817],{"className":42816},[42],[44,42818,42819],{"xmlns":46,"display":168},[48,42820,42821,42922],{},[51,42822,42823,42839,42867,42869,42885],{},[15798,42824,42825,42831],{},[51,42826,42827,42829],{},[54,42828,37329],{},[176,42830,6321],{},[51,42832,42833,42835,42837],{},[54,42834,3000],{},[176,42836,3003],{},[54,42838,13314],{"mathvariant":2296},[755,42840,42841,42861],{},[51,42842,42843,42845,42859],{},[176,42844,179],{"fence":65},[12543,42846,42847,42853],{},[51,42848,42849,42851],{},[54,42850,3000],{},[176,42852,34434],{"stretchy":178},[755,42854,42855,42857],{},[54,42856,3000],{},[54,42858,3000],{},[176,42860,189],{"fence":65},[12543,42862,42863,42865],{},[347,42864,2213],{},[54,42866,3000],{},[176,42868,192],{},[15798,42870,42871,42877],{},[51,42872,42873,42875],{},[54,42874,37329],{},[176,42876,6321],{},[51,42878,42879,42881,42883],{},[54,42880,3000],{},[176,42882,3003],{},[54,42884,13314],{"mathvariant":2296},[755,42886,42887,42889],{},[54,42888,18036],{},[51,42890,42891,42897,42900,42902],{},[12543,42892,42893,42895],{},[347,42894,2213],{},[54,42896,3000],{},[54,42898,42899],{},"ln",[176,42901,6321],{},[51,42903,42904,42906,42920],{},[176,42905,179],{"fence":65},[12543,42907,42908,42914],{},[51,42909,42910,42912],{},[54,42911,3000],{},[176,42913,34434],{"stretchy":178},[755,42915,42916,42918],{},[54,42917,3000],{},[54,42919,3000],{},[176,42921,189],{"fence":65},[58,42923,42924],{"encoding":60},"\\lim_{n \\rightarrow \\infty} \\left( \\frac{n!}{n^n} \\right)^{\\frac{1}{n}} = \\lim_{n \\rightarrow \\infty} e^{\\frac{1}{n} \\ln \\left( \\frac{n!}{n^n} \\right)}",[34,42926,42928,43199],{"className":42927,"ariaHidden":65},[64],[34,42929,42931,42934,42987,42990,43190,43193,43196],{"className":42930},[69],[34,42932],{"className":42933,"style":42464},[73],[34,42935,42937],{"className":42936},[3387,3388],[34,42938,42940,42979],{"className":42939},[814,2523],[34,42941,42943,42976],{"className":42942},[818],[34,42944,42946,42966],{"className":42945,"style":2184},[822],[34,42947,42948,42951],{"style":42479},[34,42949],{"className":42950,"style":12709},[829],[34,42952,42954],{"className":42953},[834,835,836,837],[34,42955,42957,42960,42963],{"className":42956},[78,837],[34,42958,3000],{"className":42959},[78,79,837],[34,42961,3003],{"className":42962},[266,837],[34,42964,13314],{"className":42965},[78,837],[34,42967,42968,42971],{"style":23024},[34,42969],{"className":42970,"style":12709},[829],[34,42972,42973],{},[34,42974,37329],{"className":42975},[3387],[34,42977,2547],{"className":42978},[2546],[34,42980,42982],{"className":42981},[818],[34,42983,42985],{"className":42984,"style":42517},[822],[34,42986],{},[34,42988],{"className":42989,"style":3462},[237],[34,42991,42993,43099],{"className":42992},[4177],[34,42994,42996,43002,43093],{"className":42995},[4177],[34,42997,42999],{"className":42998,"style":4182},[230,4181],[34,43000,179],{"className":43001},[4186,836],[34,43003,43005,43008,43090],{"className":43004},[78],[34,43006],{"className":43007},[230,12689],[34,43009,43011],{"className":43010},[12543],[34,43012,43014,43082],{"className":43013},[814,2523],[34,43015,43017,43079],{"className":43016},[818],[34,43018,43020,43057,43065],{"className":43019,"style":15668},[822],[34,43021,43022,43025],{"style":14155},[34,43023],{"className":43024,"style":12709},[829],[34,43026,43028],{"className":43027},[78],[34,43029,43031,43034],{"className":43030},[78],[34,43032,3000],{"className":43033},[78,79],[34,43035,43037],{"className":43036},[810],[34,43038,43040],{"className":43039},[814],[34,43041,43043],{"className":43042},[818],[34,43044,43046],{"className":43045,"style":42579},[822],[34,43047,43048,43051],{"style":12743},[34,43049],{"className":43050,"style":830},[829],[34,43052,43054],{"className":43053},[834,835,836,837],[34,43055,3000],{"className":43056},[78,79,837],[34,43058,43059,43062],{"style":12897},[34,43060],{"className":43061,"style":12709},[829],[34,43063],{"className":43064,"style":12904},[12836],[34,43066,43067,43070],{"style":12907},[34,43068],{"className":43069,"style":12709},[829],[34,43071,43073,43076],{"className":43072},[78],[34,43074,3000],{"className":43075},[78,79],[34,43077,34434],{"className":43078},[258],[34,43080,2547],{"className":43081},[2546],[34,43083,43085],{"className":43084},[818],[34,43086,43088],{"className":43087,"style":14669},[822],[34,43089],{},[34,43091],{"className":43092},[258,12689],[34,43094,43096],{"className":43095,"style":4182},[258,4181],[34,43097,189],{"className":43098},[4186,836],[34,43100,43102],{"className":43101},[810],[34,43103,43105],{"className":43104},[814],[34,43106,43108],{"className":43107},[818],[34,43109,43111],{"className":43110,"style":42645},[822],[34,43112,43113,43116],{"style":42648},[34,43114],{"className":43115,"style":12709},[829],[34,43117,43119],{"className":43118},[834,835,836,837],[34,43120,43122],{"className":43121},[78,837],[34,43123,43125,43128,43187],{"className":43124},[78,837],[34,43126],{"className":43127},[230,12689,834,10776,12798],[34,43129,43131],{"className":43130},[12543],[34,43132,43134,43179],{"className":43133},[814,2523],[34,43135,43137,43176],{"className":43136},[818],[34,43138,43140,43154,43162],{"className":43139,"style":42676},[822],[34,43141,43142,43145],{"style":12814},[34,43143],{"className":43144,"style":12709},[829],[34,43146,43148],{"className":43147},[834,10776,9740,837],[34,43149,43151],{"className":43150},[78,837],[34,43152,3000],{"className":43153},[78,79,837],[34,43155,43156,43159],{"style":12829},[34,43157],{"className":43158,"style":12709},[829],[34,43160],{"className":43161,"style":12837},[12836,837],[34,43163,43164,43167],{"style":12840},[34,43165],{"className":43166,"style":12709},[829],[34,43168,43170],{"className":43169},[834,10776,9740,837],[34,43171,43173],{"className":43172},[78,837],[34,43174,2213],{"className":43175},[78,837],[34,43177,2547],{"className":43178},[2546],[34,43180,43182],{"className":43181},[818],[34,43183,43185],{"className":43184,"style":12862},[822],[34,43186],{},[34,43188],{"className":43189},[258,12689,834,10776,12798],[34,43191],{"className":43192,"style":262},[237],[34,43194,192],{"className":43195},[266],[34,43197],{"className":43198,"style":262},[237],[34,43200,43202,43206,43259,43262],{"className":43201},[69],[34,43203],{"className":43204,"style":43205},[73],"height:1.8105em;vertical-align:-0.7em;",[34,43207,43209],{"className":43208},[3387,3388],[34,43210,43212,43251],{"className":43211},[814,2523],[34,43213,43215,43248],{"className":43214},[818],[34,43216,43218,43238],{"className":43217,"style":2184},[822],[34,43219,43220,43223],{"style":42479},[34,43221],{"className":43222,"style":12709},[829],[34,43224,43226],{"className":43225},[834,835,836,837],[34,43227,43229,43232,43235],{"className":43228},[78,837],[34,43230,3000],{"className":43231},[78,79,837],[34,43233,3003],{"className":43234},[266,837],[34,43236,13314],{"className":43237},[78,837],[34,43239,43240,43243],{"style":23024},[34,43241],{"className":43242,"style":12709},[829],[34,43244,43245],{},[34,43246,37329],{"className":43247},[3387],[34,43249,2547],{"className":43250},[2546],[34,43252,43254],{"className":43253},[818],[34,43255,43257],{"className":43256,"style":42517},[822],[34,43258],{},[34,43260],{"className":43261,"style":3462},[237],[34,43263,43265,43268],{"className":43264},[78],[34,43266,18036],{"className":43267},[78,79],[34,43269,43271],{"className":43270},[810],[34,43272,43274],{"className":43273},[814],[34,43275,43277],{"className":43276},[818],[34,43278,43281],{"className":43279,"style":43280},[822],"height:1.1105em;",[34,43282,43284,43287],{"style":43283},"top:-3.413em;margin-right:0.05em;",[34,43285],{"className":43286,"style":12709},[829],[34,43288,43290],{"className":43289},[834,835,836,837],[34,43291,43293,43361,43364,43373],{"className":43292},[78,837],[34,43294,43296,43299,43358],{"className":43295},[78,837],[34,43297],{"className":43298},[230,12689,834,10776,12798],[34,43300,43302],{"className":43301},[12543],[34,43303,43305,43350],{"className":43304},[814,2523],[34,43306,43308,43347],{"className":43307},[818],[34,43309,43311,43325,43333],{"className":43310,"style":42676},[822],[34,43312,43313,43316],{"style":12814},[34,43314],{"className":43315,"style":12709},[829],[34,43317,43319],{"className":43318},[834,10776,9740,837],[34,43320,43322],{"className":43321},[78,837],[34,43323,3000],{"className":43324},[78,79,837],[34,43326,43327,43330],{"style":12829},[34,43328],{"className":43329,"style":12709},[829],[34,43331],{"className":43332,"style":12837},[12836,837],[34,43334,43335,43338],{"style":12840},[34,43336],{"className":43337,"style":12709},[829],[34,43339,43341],{"className":43340},[834,10776,9740,837],[34,43342,43344],{"className":43343},[78,837],[34,43345,2213],{"className":43346},[78,837],[34,43348,2547],{"className":43349},[2546],[34,43351,43353],{"className":43352},[818],[34,43354,43356],{"className":43355,"style":12862},[822],[34,43357],{},[34,43359],{"className":43360},[258,12689,834,10776,12798],[34,43362],{"className":43363,"style":16971},[237,837],[34,43365,43367,43370],{"className":43366},[3387,837],[34,43368,7102],{"className":43369},[837],[34,43371,3000],{"className":43372},[837],[34,43374,43376,43383,43479],{"className":43375},[4177,837],[34,43377,43380],{"className":43378,"style":43379},[230,834,10776,12798,837,4181],"top:0.075em;",[34,43381,179],{"className":43382},[837],[34,43384,43386,43389,43476],{"className":43385},[78,837],[34,43387],{"className":43388},[230,12689,834,10776,12798],[34,43390,43392],{"className":43391},[12543],[34,43393,43395,43468],{"className":43394},[814,2523],[34,43396,43398,43465],{"className":43397},[818],[34,43399,43401,43440,43448],{"className":43400,"style":12811},[822],[34,43402,43403,43406],{"style":12814},[34,43404],{"className":43405,"style":12709},[829],[34,43407,43409],{"className":43408},[834,10776,9740,837],[34,43410,43412],{"className":43411},[78,837],[34,43413,43415,43418],{"className":43414},[78,837],[34,43416,3000],{"className":43417},[78,79,837],[34,43419,43421],{"className":43420},[810],[34,43422,43424],{"className":43423},[814],[34,43425,43427],{"className":43426},[818],[34,43428,43431],{"className":43429,"style":43430},[822],"height:0.7246em;",[34,43432,43434,43437],{"style":43433},"top:-2.794em;margin-right:0.1em;",[34,43435],{"className":43436,"style":10772},[829],[34,43438,3000],{"className":43439},[78,79,837],[34,43441,43442,43445],{"style":12829},[34,43443],{"className":43444,"style":12709},[829],[34,43446],{"className":43447,"style":12837},[12836,837],[34,43449,43450,43453],{"style":12840},[34,43451],{"className":43452,"style":12709},[829],[34,43454,43456],{"className":43455},[834,10776,9740,837],[34,43457,43459,43462],{"className":43458},[78,837],[34,43460,3000],{"className":43461},[78,79,837],[34,43463,34434],{"className":43464},[258,837],[34,43466,2547],{"className":43467},[2546],[34,43469,43471],{"className":43470},[818],[34,43472,43474],{"className":43473,"style":12862},[822],[34,43475],{},[34,43477],{"className":43478},[258,12689,834,10776,12798],[34,43480,43482],{"className":43481,"style":43379},[258,834,10776,12798,837,4181],[34,43483,189],{"className":43484},[837],[16,43486,43487],{},"which is equivalent to proving:",[34,43489,43491],{"className":43490,"translate":38},[159],[34,43492,43494,43558],{"className":43493,"translate":38},[37],[34,43495,43497],{"className":43496},[42],[44,43498,43499],{"xmlns":46,"display":168},[48,43500,43501,43555],{},[51,43502,43503,43519,43525,43527,43529,43549,43551,43553],{},[15798,43504,43505,43511],{},[51,43506,43507,43509],{},[54,43508,37329],{},[176,43510,6321],{},[51,43512,43513,43515,43517],{},[54,43514,3000],{},[176,43516,3003],{},[54,43518,13314],{"mathvariant":2296},[12543,43520,43521,43523],{},[347,43522,2213],{},[54,43524,3000],{},[54,43526,42899],{},[176,43528,6321],{},[51,43530,43531,43533,43547],{},[176,43532,179],{"fence":65},[12543,43534,43535,43541],{},[51,43536,43537,43539],{},[54,43538,3000],{},[176,43540,34434],{"stretchy":178},[755,43542,43543,43545],{},[54,43544,3000],{},[54,43546,3000],{},[176,43548,189],{"fence":65},[176,43550,192],{},[176,43552,6463],{},[347,43554,2213],{},[58,43556,43557],{"encoding":60},"\\lim_{n \\rightarrow \\infty} \\frac{1}{n} \\ln \\left( \\frac{n!}{n^n} \\right) = -1",[34,43559,43561,43809],{"className":43560,"ariaHidden":65},[64],[34,43562,43564,43567,43620,43623,43685,43688,43691,43694,43800,43803,43806],{"className":43563},[69],[34,43565],{"className":43566,"style":14693},[73],[34,43568,43570],{"className":43569},[3387,3388],[34,43571,43573,43612],{"className":43572},[814,2523],[34,43574,43576,43609],{"className":43575},[818],[34,43577,43579,43599],{"className":43578,"style":2184},[822],[34,43580,43581,43584],{"style":42479},[34,43582],{"className":43583,"style":12709},[829],[34,43585,43587],{"className":43586},[834,835,836,837],[34,43588,43590,43593,43596],{"className":43589},[78,837],[34,43591,3000],{"className":43592},[78,79,837],[34,43594,3003],{"className":43595},[266,837],[34,43597,13314],{"className":43598},[78,837],[34,43600,43601,43604],{"style":23024},[34,43602],{"className":43603,"style":12709},[829],[34,43605,43606],{},[34,43607,37329],{"className":43608},[3387],[34,43610,2547],{"className":43611},[2546],[34,43613,43615],{"className":43614},[818],[34,43616,43618],{"className":43617,"style":42517},[822],[34,43619],{},[34,43621],{"className":43622,"style":3462},[237],[34,43624,43626,43629,43682],{"className":43625},[78],[34,43627],{"className":43628},[230,12689],[34,43630,43632],{"className":43631},[12543],[34,43633,43635,43674],{"className":43634},[814,2523],[34,43636,43638,43671],{"className":43637},[818],[34,43639,43641,43652,43660],{"className":43640,"style":16800},[822],[34,43642,43643,43646],{"style":14155},[34,43644],{"className":43645,"style":12709},[829],[34,43647,43649],{"className":43648},[78],[34,43650,3000],{"className":43651},[78,79],[34,43653,43654,43657],{"style":12897},[34,43655],{"className":43656,"style":12709},[829],[34,43658],{"className":43659,"style":12904},[12836],[34,43661,43662,43665],{"style":12907},[34,43663],{"className":43664,"style":12709},[829],[34,43666,43668],{"className":43667},[78],[34,43669,2213],{"className":43670},[78],[34,43672,2547],{"className":43673},[2546],[34,43675,43677],{"className":43676},[818],[34,43678,43680],{"className":43679,"style":14669},[822],[34,43681],{},[34,43683],{"className":43684},[258,12689],[34,43686],{"className":43687,"style":3462},[237],[34,43689,42899],{"className":43690},[3387],[34,43692],{"className":43693,"style":3462},[237],[34,43695,43697,43703,43794],{"className":43696},[4177],[34,43698,43700],{"className":43699,"style":4182},[230,4181],[34,43701,179],{"className":43702},[4186,836],[34,43704,43706,43709,43791],{"className":43705},[78],[34,43707],{"className":43708},[230,12689],[34,43710,43712],{"className":43711},[12543],[34,43713,43715,43783],{"className":43714},[814,2523],[34,43716,43718,43780],{"className":43717},[818],[34,43719,43721,43758,43766],{"className":43720,"style":15668},[822],[34,43722,43723,43726],{"style":14155},[34,43724],{"className":43725,"style":12709},[829],[34,43727,43729],{"className":43728},[78],[34,43730,43732,43735],{"className":43731},[78],[34,43733,3000],{"className":43734},[78,79],[34,43736,43738],{"className":43737},[810],[34,43739,43741],{"className":43740},[814],[34,43742,43744],{"className":43743},[818],[34,43745,43747],{"className":43746,"style":42579},[822],[34,43748,43749,43752],{"style":12743},[34,43750],{"className":43751,"style":830},[829],[34,43753,43755],{"className":43754},[834,835,836,837],[34,43756,3000],{"className":43757},[78,79,837],[34,43759,43760,43763],{"style":12897},[34,43761],{"className":43762,"style":12709},[829],[34,43764],{"className":43765,"style":12904},[12836],[34,43767,43768,43771],{"style":12907},[34,43769],{"className":43770,"style":12709},[829],[34,43772,43774,43777],{"className":43773},[78],[34,43775,3000],{"className":43776},[78,79],[34,43778,34434],{"className":43779},[258],[34,43781,2547],{"className":43782},[2546],[34,43784,43786],{"className":43785},[818],[34,43787,43789],{"className":43788,"style":14669},[822],[34,43790],{},[34,43792],{"className":43793},[258,12689],[34,43795,43797],{"className":43796,"style":4182},[258,4181],[34,43798,189],{"className":43799},[4186,836],[34,43801],{"className":43802,"style":262},[237],[34,43804,192],{"className":43805},[266],[34,43807],{"className":43808,"style":262},[237],[34,43810,43812,43815,43818],{"className":43811},[69],[34,43813],{"className":43814,"style":511},[73],[34,43816,6463],{"className":43817},[78],[34,43819,2213],{"className":43820},[78],[16,43822,43823],{},"We 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\\rightarrow \\infty} \\frac{1}{n} \\ln \\left( \\frac{n!}{n^n} \\right) \\\\ = \\lim_{n \\rightarrow \\infty} \\frac{1}{n} \\left( \\ln \\frac{1}{n} + \\ln \\frac{2}{n} + \\cdots + \\ln \\frac{n}{n} \\right) \\\\ = \\int_{0}^{1} \\ln x \\, dx = (x \\ln x - x) \\Big|_{0}^{1} = 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\\sim \\frac{1}{e^n}",[34,44970,44972,45078],{"className":44971,"ariaHidden":65},[64],[34,44973,44975,44978,45069,45072,45075],{"className":44974},[69],[34,44976],{"className":44977,"style":15649},[73],[34,44979,44981,44984,45066],{"className":44980},[78],[34,44982],{"className":44983},[230,12689],[34,44985,44987],{"className":44986},[12543],[34,44988,44990,45058],{"className":44989},[814,2523],[34,44991,44993,45055],{"className":44992},[818],[34,44994,44996,45033,45041],{"className":44995,"style":15668},[822],[34,44997,44998,45001],{"style":14155},[34,44999],{"className":45000,"style":12709},[829],[34,45002,45004],{"className":45003},[78],[34,45005,45007,45010],{"className":45006},[78],[34,45008,3000],{"className":45009},[78,79],[34,45011,45013],{"className":45012},[810],[34,45014,45016],{"className":45015},[814],[34,45017,45019],{"className":45018},[818],[34,45020,45022],{"className":45021,"style":42579},[822],[34,45023,45024,45027],{"style":12743},[34,45025],{"className":45026,"style":830},[829],[34,45028,45030],{"className":45029},[834,835,836,837],[34,45031,3000],{"className":45032},[78,79,837],[34,45034,45035,45038],{"style":12897},[34,45036],{"className":45037,"style":12709},[829],[34,45039],{"className":45040,"style":12904},[12836],[34,45042,45043,45046],{"style":12907},[34,45044],{"className":45045,"style":12709},[829],[34,45047,45049,45052],{"className":45048},[78],[34,45050,3000],{"className":45051},[78,79],[34,45053,34434],{"className":45054},[258],[34,45056,2547],{"className":45057},[2546],[34,45059,45061],{"className":45060},[818],[34,45062,45064],{"className":45063,"style":14669},[822],[34,45065],{},[34,45067],{"className":45068},[258,12689],[34,45070],{"className":45071,"style":262},[237],[34,45073,14925],{"className":45074},[266],[34,45076],{"className":45077,"style":262},[237],[34,45079,45081,45084],{"className":45080},[69],[34,45082],{"className":45083,"style":42742},[73],[34,45085,45087,45090,45169],{"className":45086},[78],[34,45088],{"className":45089},[230,12689],[34,45091,45093],{"className":45092},[12543],[34,45094,45096,45161],{"className":45095},[814,2523],[34,45097,45099,45158],{"className":45098},[818],[34,45100,45102,45139,45147],{"className":45101,"style":16800},[822],[34,45103,45104,45107],{"style":14155},[34,45105],{"className":45106,"style":12709},[829],[34,45108,45110],{"className":45109},[78],[34,45111,45113,45116],{"className":45112},[78],[34,45114,18036],{"className":45115},[78,79],[34,45117,45119],{"className":45118},[810],[34,45120,45122],{"className":45121},[814],[34,45123,45125],{"className":45124},[818],[34,45126,45128],{"className":45127,"style":42579},[822],[34,45129,45130,45133],{"style":12743},[34,45131],{"className":45132,"style":830},[829],[34,45134,45136],{"className":45135},[834,835,836,837],[34,45137,3000],{"className":45138},[78,79,837],[34,45140,45141,45144],{"style":12897},[34,45142],{"className":45143,"style":12709},[829],[34,45145],{"className":45146,"style":12904},[12836],[34,45148,45149,45152],{"style":12907},[34,45150],{"className":45151,"style":12709},[829],[34,45153,45155],{"className":45154},[78],[34,45156,2213],{"className":45157},[78],[34,45159,2547],{"className":45160},[2546],[34,45162,45164],{"className":45163},[818],[34,45165,45167],{"className":45166,"style":14669},[822],[34,45168],{},[34,45170],{"className":45171},[258,12689],[16,45173,45174],{},"We would like to verify:",[34,45176,45178],{"className":45177,"translate":38},[159],[34,45179,45181,45233],{"className":45180,"translate":38},[37],[34,45182,45184],{"className":45183},[42],[44,45185,45186],{"xmlns":46,"display":168},[48,45187,45188,45230],{},[51,45189,45190,45206,45220,45226,45228],{},[15798,45191,45192,45198],{},[51,45193,45194,45196],{},[54,45195,37329],{},[176,45197,6321],{},[51,45199,45200,45202,45204],{},[54,45201,3000],{},[176,45203,3003],{},[54,45205,13314],{"mathvariant":2296},[12543,45207,45208,45214],{},[51,45209,45210,45212],{},[54,45211,3000],{},[176,45213,34434],{"stretchy":178},[755,45215,45216,45218],{},[54,45217,3000],{},[54,45219,3000],{},[755,45221,45222,45224],{},[54,45223,18036],{},[54,45225,3000],{},[176,45227,192],{},[347,45229,2213],{},[58,45231,45232],{"encoding":60},"\\lim_{n \\rightarrow \\infty} \\frac{n!}{n^n} e^n = 1",[34,45234,45236,45428],{"className":45235,"ariaHidden":65},[64],[34,45237,45239,45243,45296,45299,45390,45419,45422,45425],{"className":45238},[69],[34,45240],{"className":45241,"style":45242},[73],"height:2.0714em;vertical-align:-0.7em;",[34,45244,45246],{"className":45245},[3387,3388],[34,45247,45249,45288],{"className":45248},[814,2523],[34,45250,45252,45285],{"className":45251},[818],[34,45253,45255,45275],{"className":45254,"style":2184},[822],[34,45256,45257,45260],{"style":42479},[34,45258],{"className":45259,"style":12709},[829],[34,45261,45263],{"className":45262},[834,835,836,837],[34,45264,45266,45269,45272],{"className":45265},[78,837],[34,45267,3000],{"className":45268},[78,79,837],[34,45270,3003],{"className":45271},[266,837],[34,45273,13314],{"className":45274},[78,837],[34,45276,45277,45280],{"style":23024},[34,45278],{"className":45279,"style":12709},[829],[34,45281,45282],{},[34,45283,37329],{"className":45284},[3387],[34,45286,2547],{"className":45287},[2546],[34,45289,45291],{"className":45290},[818],[34,45292,45294],{"className":45293,"style":42517},[822],[34,45295],{},[34,45297],{"className":45298,"style":3462},[237],[34,45300,45302,45305,45387],{"className":45301},[78],[34,45303],{"className":45304},[230,12689],[34,45306,45308],{"className":45307},[12543],[34,45309,45311,45379],{"className":45310},[814,2523],[34,45312,45314,45376],{"className":45313},[818],[34,45315,45317,45354,45362],{"className":45316,"style":15668},[822],[34,45318,45319,45322],{"style":14155},[34,45320],{"className":45321,"style":12709},[829],[34,45323,45325],{"className":45324},[78],[34,45326,45328,45331],{"className":45327},[78],[34,45329,3000],{"className":45330},[78,79],[34,45332,45334],{"className":45333},[810],[34,45335,45337],{"className":45336},[814],[34,45338,45340],{"className":45339},[818],[34,45341,45343],{"className":45342,"style":42579},[822],[34,45344,45345,45348],{"style":12743},[34,45346],{"className":45347,"style":830},[829],[34,45349,45351],{"className":45350},[834,835,836,837],[34,45352,3000],{"className":45353},[78,79,837],[34,45355,45356,45359],{"style":12897},[34,45357],{"className":45358,"style":12709},[829],[34,45360],{"className":45361,"style":12904},[12836],[34,45363,45364,45367],{"style":12907},[34,45365],{"className":45366,"style":12709},[829],[34,45368,45370,45373],{"className":45369},[78],[34,45371,3000],{"className":45372},[78,79],[34,45374,34434],{"className":45375},[258],[34,45377,2547],{"className":45378},[2546],[34,45380,45382],{"className":45381},[818],[34,45383,45385],{"className":45384,"style":14669},[822],[34,45386],{},[34,45388],{"className":45389},[258,12689],[34,45391,45393,45396],{"className":45392},[78],[34,45394,18036],{"className":45395},[78,79],[34,45397,45399],{"className":45398},[810],[34,45400,45402],{"className":45401},[814],[34,45403,45405],{"className":45404},[818],[34,45406,45408],{"className":45407,"style":19071},[822],[34,45409,45410,45413],{"style":4710},[34,45411],{"className":45412,"style":830},[829],[34,45414,45416],{"className":45415},[834,835,836,837],[34,45417,3000],{"className":45418},[78,79,837],[34,45420],{"className":45421,"style":262},[237],[34,45423,192],{"className":45424},[266],[34,45426],{"className":45427,"style":262},[237],[34,45429,45431,45434],{"className":45430},[69],[34,45432],{"className":45433,"style":391},[73],[34,45435,2213],{"className":45436},[78],[16,45438,45439],{},"Let’s test this.",[16,45441,45442],{},[20,45443,45444],{},"Using the ratio test:",[34,45446,45448],{"className":45447,"translate":38},[159],[34,45449,45451,45559],{"className":45450,"translate":38},[37],[34,45452,45454],{"className":45453},[42],[44,45455,45456],{"xmlns":46,"display":168},[48,45457,45458,45556],{},[51,45459,45460,45466,45468,45482,45488,45490,45492,45504,45506,45544],{},[2468,45461,45462,45464],{},[54,45463,11468],{},[54,45465,3000],{},[176,45467,192],{},[12543,45469,45470,45476],{},[51,45471,45472,45474],{},[54,45473,3000],{},[176,45475,34434],{"stretchy":178},[755,45477,45478,45480],{},[54,45479,3000],{},[54,45481,3000],{},[755,45483,45484,45486],{},[54,45485,18036],{},[54,45487,3000],{},[176,45489,5228],{"separator":65},[237,45491],{"width":1197},[2468,45493,45494,45496],{},[54,45495,11468],{},[51,45497,45498,45500,45502],{},[54,45499,3000],{},[176,45501,184],{},[347,45503,2213],{},[176,45505,192],{},[12543,45507,45508,45522],{},[51,45509,45510,45512,45514,45516,45518,45520],{},[176,45511,179],{"stretchy":178},[54,45513,3000],{},[176,45515,184],{},[347,45517,2213],{},[176,45519,189],{"stretchy":178},[176,45521,34434],{"stretchy":178},[51,45523,45524,45526,45528,45530,45532],{},[176,45525,179],{"stretchy":178},[54,45527,3000],{},[176,45529,184],{},[347,45531,2213],{},[755,45533,45534,45536],{},[176,45535,189],{"stretchy":178},[51,45537,45538,45540,45542],{},[54,45539,3000],{},[176,45541,184],{},[347,45543,2213],{},[755,45545,45546,45548],{},[54,45547,18036],{},[51,45549,45550,45552,45554],{},[54,45551,3000],{},[176,45553,184],{},[347,45555,2213],{},[58,45557,45558],{"encoding":60},"a_n = \\frac{n!}{n^n} e^n, \\quad a_{n+1} = \\frac{(n+1)!}{(n+1)^{n+1}} e^{n+1}",[34,45560,45562,45617,45810],{"className":45561,"ariaHidden":65},[64],[34,45563,45565,45568,45608,45611,45614],{"className":45564},[69],[34,45566],{"className":45567,"style":2625},[73],[34,45569,45571,45574],{"className":45570},[78],[34,45572,11468],{"className":45573},[78,79],[34,45575,45577],{"className":45576},[810],[34,45578,45580,45600],{"className":45579},[814,2523],[34,45581,45583,45597],{"className":45582},[818],[34,45584,45586],{"className":45585,"style":5870},[822],[34,45587,45588,45591],{"style":2533},[34,45589],{"className":45590,"style":830},[829],[34,45592,45594],{"className":45593},[834,835,836,837],[34,45595,3000],{"className":45596},[78,79,837],[34,45598,2547],{"className":45599},[2546],[34,45601,45603],{"className":45602},[818],[34,45604,45606],{"className":45605,"style":2554},[822],[34,45607],{},[34,45609],{"className":45610,"style":262},[237],[34,45612,192],{"className":45613},[266],[34,45615],{"className":45616,"style":262},[237],[34,45618,45620,45623,45714,45743,45746,45749,45752,45801,45804,45807],{"className":45619},[69],[34,45621],{"className":45622,"style":15649},[73],[34,45624,45626,45629,45711],{"className":45625},[78],[34,45627],{"className":45628},[230,12689],[34,45630,45632],{"className":45631},[12543],[34,45633,45635,45703],{"className":45634},[814,2523],[34,45636,45638,45700],{"className":45637},[818],[34,45639,45641,45678,45686],{"className":45640,"style":15668},[822],[34,45642,45643,45646],{"style":14155},[34,45644],{"className":45645,"style":12709},[829],[34,45647,45649],{"className":45648},[78],[34,45650,45652,45655],{"className":45651},[78],[34,45653,3000],{"className":45654},[78,79],[34,45656,45658],{"className":45657},[810],[34,45659,45661],{"className":45660},[814],[34,45662,45664],{"className":45663},[818],[34,45665,45667],{"className":45666,"style":42579},[822],[34,45668,45669,45672],{"style":12743},[34,45670],{"className":45671,"style":830},[829],[34,45673,45675],{"className":45674},[834,835,836,837],[34,45676,3000],{"className":45677},[78,79,837],[34,45679,45680,45683],{"style":12897},[34,45681],{"className":45682,"style":12709},[829],[34,45684],{"className":45685,"style":12904},[12836],[34,45687,45688,45691],{"style":12907},[34,45689],{"className":45690,"style":12709},[829],[34,45692,45694,45697],{"className":45693},[78],[34,45695,3000],{"className":45696},[78,79],[34,45698,34434],{"className":45699},[258],[34,45701,2547],{"className":45702},[2546],[34,45704,45706],{"className":45705},[818],[34,45707,45709],{"className":45708,"style":14669},[822],[34,45710],{},[34,45712],{"className":45713},[258,12689],[34,45715,45717,45720],{"className":45716},[78],[34,45718,18036],{"className":45719},[78,79],[34,45721,45723],{"className":45722},[810],[34,45724,45726],{"className":45725},[814],[34,45727,45729],{"className":45728},[818],[34,45730,45732],{"className":45731,"style":19071},[822],[34,45733,45734,45737],{"style":4710},[34,45735],{"className":45736,"style":830},[829],[34,45738,45740],{"className":45739},[834,835,836,837],[34,45741,3000],{"className":45742},[78,79,837],[34,45744,5228],{"className":45745},[5317],[34,45747],{"className":45748,"style":1297},[237],[34,45750],{"className":45751,"style":3462},[237],[34,45753,45755,45758],{"className":45754},[78],[34,45756,11468],{"className":45757},[78,79],[34,45759,45761],{"className":45760},[810],[34,45762,45764,45793],{"className":45763},[814,2523],[34,45765,45767,45790],{"className":45766},[818],[34,45768,45770],{"className":45769,"style":2530},[822],[34,45771,45772,45775],{"style":2533},[34,45773],{"className":45774,"style":830},[829],[34,45776,45778],{"className":45777},[834,835,836,837],[34,45779,45781,45784,45787],{"className":45780},[78,837],[34,45782,3000],{"className":45783},[78,79,837],[34,45785,184],{"className":45786},[242,837],[34,45788,2213],{"className":45789},[78,837],[34,45791,2547],{"className":45792},[2546],[34,45794,45796],{"className":45795},[818],[34,45797,45799],{"className":45798,"style":6830},[822],[34,45800],{},[34,45802],{"className":45803,"style":262},[237],[34,45805,192],{"className":45806},[266],[34,45808],{"className":45809,"style":262},[237],[34,45811,45813,45816,45949],{"className":45812},[69],[34,45814],{"className":45815,"style":23317},[73],[34,45817,45819,45822,45946],{"className":45818},[78],[34,45820],{"className":45821},[230,12689],[34,45823,45825],{"className":45824},[12543],[34,45826,45828,45938],{"className":45827},[814,2523],[34,45829,45831,45935],{"className":45830},[818],[34,45832,45834,45898,45906],{"className":45833,"style":14583},[822],[34,45835,45836,45839],{"style":14155},[34,45837],{"className":45838,"style":12709},[829],[34,45840,45842,45845,45848,45851,45854,45857,45860],{"className":45841},[78],[34,45843,179],{"className":45844},[230],[34,45846,3000],{"className":45847},[78,79],[34,45849],{"className":45850,"style":238},[237],[34,45852,184],{"className":45853},[242],[34,45855],{"className":45856,"style":238},[237],[34,45858,2213],{"className":45859},[78],[34,45861,45863,45866],{"className":45862},[258],[34,45864,189],{"className":45865},[258],[34,45867,45869],{"className":45868},[810],[34,45870,45872],{"className":45871},[814],[34,45873,45875],{"className":45874},[818],[34,45876,45878],{"className":45877,"style":15259},[822],[34,45879,45880,45883],{"style":12743},[34,45881],{"className":45882,"style":830},[829],[34,45884,45886],{"className":45885},[834,835,836,837],[34,45887,45889,45892,45895],{"className":45888},[78,837],[34,45890,3000],{"className":45891},[78,79,837],[34,45893,184],{"className":45894},[242,837],[34,45896,2213],{"className":45897},[78,837],[34,45899,45900,45903],{"style":12897},[34,45901],{"className":45902,"style":12709},[829],[34,45904],{"className":45905,"style":12904},[12836],[34,45907,45908,45911],{"style":12907},[34,45909],{"className":45910,"style":12709},[829],[34,45912,45914,45917,45920,45923,45926,45929,45932],{"className":45913},[78],[34,45915,179],{"className":45916},[230],[34,45918,3000],{"className":45919},[78,79],[34,45921],{"className":45922,"style":238},[237],[34,45924,184],{"className":45925},[242],[34,45927],{"className":45928,"style":238},[237],[34,45930,2213],{"className":45931},[78],[34,45933,39708],{"className":45934},[258],[34,45936,2547],{"className":45937},[2546],[34,45939,45941],{"className":45940},[818],[34,45942,45944],{"className":45943,"style":17481},[822],[34,45945],{},[34,45947],{"className":45948},[258,12689],[34,45950,45952,45955],{"className":45951},[78],[34,45953,18036],{"className":45954},[78,79],[34,45956,45958],{"className":45957},[810],[34,45959,45961],{"className":45960},[814],[34,45962,45964],{"className":45963},[818],[34,45965,45967],{"className":45966,"style":4707},[822],[34,45968,45969,45972],{"style":4710},[34,45970],{"className":45971,"style":830},[829],[34,45973,45975],{"className":45974},[834,835,836,837],[34,45976,45978,45981,45984],{"className":45977},[78,837],[34,45979,3000],{"className":45980},[78,79,837],[34,45982,184],{"className":45983},[242,837],[34,45985,2213],{"className":45986},[78,837],[34,45988,45990],{"className":45989,"translate":38},[159],[34,45991,45993,46071],{"className":45992,"translate":38},[37],[34,45994,45996],{"className":45995},[42],[44,45997,45998],{"xmlns":46,"display":168},[48,45999,46000,46068],{},[51,46001,46002,46018,46038,46040,46064,46066],{},[15798,46003,46004,46010],{},[51,46005,46006,46008],{},[54,46007,37329],{},[176,46009,6321],{},[51,46011,46012,46014,46016],{},[54,46013,3000],{},[176,46015,3003],{},[54,46017,13314],{"mathvariant":2296},[12543,46019,46020,46032],{},[2468,46021,46022,46024],{},[54,46023,11468],{},[51,46025,46026,46028,46030],{},[54,46027,3000],{},[176,46029,184],{},[347,46031,2213],{},[2468,46033,46034,46036],{},[54,46035,11468],{},[54,46037,3000],{},[176,46039,192],{},[12543,46041,46042,46044],{},[54,46043,18036],{},[755,46045,46046,46062],{},[51,46047,46048,46050,46052,46054,46060],{},[176,46049,179],{"fence":65},[347,46051,2213],{},[176,46053,184],{},[12543,46055,46056,46058],{},[347,46057,2213],{},[54,46059,3000],{},[176,46061,189],{"fence":65},[54,46063,3000],{},[176,46065,13189],{},[347,46067,2213],{},[58,46069,46070],{"encoding":60},"\\lim_{n \\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^n} > 1",[34,46072,46074,46292,46491],{"className":46073,"ariaHidden":65},[64],[34,46075,46077,46081,46134,46137,46283,46286,46289],{"className":46076},[69],[34,46078],{"className":46079,"style":46080},[73],"height:1.9436em;vertical-align:-0.836em;",[34,46082,46084],{"className":46083},[3387,3388],[34,46085,46087,46126],{"className":46086},[814,2523],[34,46088,46090,46123],{"className":46089},[818],[34,46091,46093,46113],{"className":46092,"style":2184},[822],[34,46094,46095,46098],{"style":42479},[34,46096],{"className":46097,"style":12709},[829],[34,46099,46101],{"className":46100},[834,835,836,837],[34,46102,46104,46107,46110],{"className":46103},[78,837],[34,46105,3000],{"className":46106},[78,79,837],[34,46108,3003],{"className":46109},[266,837],[34,46111,13314],{"className":46112},[78,837],[34,46114,46115,46118],{"style":23024},[34,46116],{"className":46117,"style":12709},[829],[34,46119,46120],{},[34,46121,37329],{"className":46122},[3387],[34,46124,2547],{"className":46125},[2546],[34,46127,46129],{"className":46128},[818],[34,46130,46132],{"className":46131,"style":42517},[822],[34,46133],{},[34,46135],{"className":46136,"style":3462},[237],[34,46138,46140,46143,46280],{"className":46139},[78],[34,46141],{"className":46142},[230,12689],[34,46144,46146],{"className":46145},[12543],[34,46147,46149,46271],{"className":46148},[814,2523],[34,46150,46152,46268],{"className":46151},[818],[34,46153,46155,46203,46211],{"className":46154,"style":26758},[822],[34,46156,46157,46160],{"style":14155},[34,46158],{"className":46159,"style":12709},[829],[34,46161,46163],{"className":46162},[78],[34,46164,46166,46169],{"className":46165},[78],[34,46167,11468],{"className":46168},[78,79],[34,46170,46172],{"className":46171},[810],[34,46173,46175,46195],{"className":46174},[814,2523],[34,46176,46178,46192],{"className":46177},[818],[34,46179,46181],{"className":46180,"style":5870},[822],[34,46182,46183,46186],{"style":2533},[34,46184],{"className":46185,"style":830},[829],[34,46187,46189],{"className":46188},[834,835,836,837],[34,46190,3000],{"className":46191},[78,79,837],[34,46193,2547],{"className":46194},[2546],[34,46196,46198],{"className":46197},[818],[34,46199,46201],{"className":46200,"style":2554},[822],[34,46202],{},[34,46204,46205,46208],{"style":12897},[34,46206],{"className":46207,"style":12709},[829],[34,46209],{"className":46210,"style":12904},[12836],[34,46212,46213,46216],{"style":12907},[34,46214],{"className":46215,"style":12709},[829],[34,46217,46219],{"className":46218},[78],[34,46220,46222,46225],{"className":46221},[78],[34,46223,11468],{"className":46224},[78,79],[34,46226,46228],{"className":46227},[810],[34,46229,46231,46260],{"className":46230},[814,2523],[34,46232,46234,46257],{"className":46233},[818],[34,46235,46237],{"className":46236,"style":2530},[822],[34,46238,46239,46242],{"style":2533},[34,46240],{"className":46241,"style":830},[829],[34,46243,46245],{"className":46244},[834,835,836,837],[34,46246,46248,46251,46254],{"className":46247},[78,837],[34,46249,3000],{"className":46250},[78,79,837],[34,46252,184],{"className":46253},[242,837],[34,46255,2213],{"className":46256},[78,837],[34,46258,2547],{"className":46259},[2546],[34,46261,46263],{"className":46262},[818],[34,46264,46266],{"className":46265,"style":6830},[822],[34,46267],{},[34,46269,2547],{"className":46270},[2546],[34,46272,46274],{"className":46273},[818],[34,46275,46278],{"className":46276,"style":46277},[822],"height:0.836em;",[34,46279],{},[34,46281],{"className":46282},[258,12689],[34,46284],{"className":46285,"style":262},[237],[34,46287,192],{"className":46288},[266],[34,46290],{"className":46291,"style":262},[237],[34,46293,46295,46299,46482,46485,46488],{"className":46294},[69],[34,46296],{"className":46297,"style":46298},[73],"height:2.2519em;vertical-align:-1.1443em;",[34,46300,46302,46305,46479],{"className":46301},[78],[34,46303],{"className":46304},[230,12689],[34,46306,46308],{"className":46307},[12543],[34,46309,46311,46470],{"className":46310},[814,2523],[34,46312,46314,46467],{"className":46313},[818],[34,46315,46317,46448,46456],{"className":46316,"style":26758},[822],[34,46318,46320,46323],{"style":46319},"top:-2.2057em;",[34,46321],{"className":46322,"style":12709},[829],[34,46324,46326],{"className":46325},[78],[34,46327,46329,46425],{"className":46328},[4177],[34,46330,46332,46338,46341,46344,46347,46350,46419],{"className":46331},[4177],[34,46333,46335],{"className":46334,"style":4182},[230,4181],[34,46336,179],{"className":46337},[4186,9740],[34,46339,2213],{"className":46340},[78],[34,46342],{"className":46343,"style":238},[237],[34,46345,184],{"className":46346},[242],[34,46348],{"className":46349,"style":238},[237],[34,46351,46353,46356,46416],{"className":46352},[78],[34,46354],{"className":46355},[230,12689],[34,46357,46359],{"className":46358},[12543],[34,46360,46362,46408],{"className":46361},[814,2523],[34,46363,46365,46405],{"className":46364},[818],[34,46366,46369,46383,46391],{"className":46367,"style":46368},[822],"height:0.8451em;",[34,46370,46371,46374],{"style":17906},[34,46372],{"className":46373,"style":12709},[829],[34,46375,46377],{"className":46376},[834,835,836,837],[34,46378,46380],{"className":46379},[78,837],[34,46381,3000],{"className":46382},[78,79,837],[34,46384,46385,46388],{"style":12897},[34,46386],{"className":46387,"style":12709},[829],[34,46389],{"className":46390,"style":12904},[12836],[34,46392,46393,46396],{"style":17929},[34,46394],{"className":46395,"style":12709},[829],[34,46397,46399],{"className":46398},[834,835,836,837],[34,46400,46402],{"className":46401},[78,837],[34,46403,2213],{"className":46404},[78,837],[34,46406,2547],{"className":46407},[2546],[34,46409,46411],{"className":46410},[818],[34,46412,46414],{"className":46413,"style":17973},[822],[34,46415],{},[34,46417],{"className":46418},[258,12689],[34,46420,46422],{"className":46421,"style":4182},[258,4181],[34,46423,189],{"className":46424},[4186,9740],[34,46426,46428],{"className":46427},[810],[34,46429,46431],{"className":46430},[814],[34,46432,46434],{"className":46433},[818],[34,46435,46437],{"className":46436,"style":42369},[822],[34,46438,46439,46442],{"style":42372},[34,46440],{"className":46441,"style":830},[829],[34,46443,46445],{"className":46444},[834,835,836,837],[34,46446,3000],{"className":46447},[78,79,837],[34,46449,46450,46453],{"style":12897},[34,46451],{"className":46452,"style":12709},[829],[34,46454],{"className":46455,"style":12904},[12836],[34,46457,46458,46461],{"style":12907},[34,46459],{"className":46460,"style":12709},[829],[34,46462,46464],{"className":46463},[78],[34,46465,18036],{"className":46466},[78,79],[34,46468,2547],{"className":46469},[2546],[34,46471,46473],{"className":46472},[818],[34,46474,46477],{"className":46475,"style":46476},[822],"height:1.1443em;",[34,46478],{},[34,46480],{"className":46481},[258,12689],[34,46483],{"className":46484,"style":262},[237],[34,46486,13189],{"className":46487},[266],[34,46489],{"className":46490,"style":262},[237],[34,46492,46494,46497],{"className":46493},[69],[34,46495],{"className":46496,"style":391},[73],[34,46498,2213],{"className":46499},[78],[16,46501,46502,46503,46531,46532,46560,46561,46631],{},"Although the limit of the expression is ",[34,46504,46506,46519],{"className":46505,"translate":38},[37],[34,46507,46509],{"className":46508},[42],[44,46510,46511],{"xmlns":46},[48,46512,46513,46517],{},[51,46514,46515],{},[54,46516,18036],{},[58,46518,18036],{"encoding":60},[34,46520,46522],{"className":46521,"ariaHidden":65},[64],[34,46523,46525,46528],{"className":46524},[69],[34,46526],{"className":46527,"style":108},[73],[34,46529,18036],{"className":46530},[78,79],", it is always approaching ",[34,46533,46535,46548],{"className":46534,"translate":38},[37],[34,46536,46538],{"className":46537},[42],[44,46539,46540],{"xmlns":46},[48,46541,46542,46546],{},[51,46543,46544],{},[54,46545,18036],{},[58,46547,18036],{"encoding":60},[34,46549,46551],{"className":46550,"ariaHidden":65},[64],[34,46552,46554,46557],{"className":46553},[69],[34,46555],{"className":46556,"style":108},[73],[34,46558,18036],{"className":46559},[78,79]," from below, so it is still greater than 1, indicating that ",[34,46562,46564,46582],{"className":46563,"translate":38},[37],[34,46565,46567],{"className":46566},[42],[44,46568,46569],{"xmlns":46},[48,46570,46571,46579],{},[51,46572,46573],{},[2468,46574,46575,46577],{},[54,46576,11468],{},[54,46578,3000],{},[58,46580,46581],{"encoding":60},"a_n",[34,46583,46585],{"className":46584,"ariaHidden":65},[64],[34,46586,46588,46591],{"className":46587},[69],[34,46589],{"className":46590,"style":2625},[73],[34,46592,46594,46597],{"className":46593},[78],[34,46595,11468],{"className":46596},[78,79],[34,46598,46600],{"className":46599},[810],[34,46601,46603,46623],{"className":46602},[814,2523],[34,46604,46606,46620],{"className":46605},[818],[34,46607,46609],{"className":46608,"style":5870},[822],[34,46610,46611,46614],{"style":2533},[34,46612],{"className":46613,"style":830},[829],[34,46615,46617],{"className":46616},[834,835,836,837],[34,46618,3000],{"className":46619},[78,79,837],[34,46621,2547],{"className":46622},[2546],[34,46624,46626],{"className":46625},[818],[34,46627,46629],{"className":46628,"style":2554},[822],[34,46630],{}," is divergent.",[16,46633,46634],{},"Mathematicians have attempted various approaches and found:",[34,46636,46638],{"className":46637,"translate":38},[159],[34,46639,46641,46691],{"className":46640,"translate":38},[37],[34,46642,46644],{"className":46643},[42],[44,46645,46646],{"xmlns":46,"display":168},[48,46647,46648,46688],{},[51,46649,46650,46656,46658,46682],{},[2468,46651,46652,46654],{},[54,46653,11468],{},[54,46655,3000],{},[176,46657,192],{},[12543,46659,46660,46666],{},[51,46661,46662,46664],{},[54,46663,3000],{},[176,46665,34434],{"stretchy":178},[755,46667,46668,46670],{},[54,46669,3000],{},[51,46671,46672,46674,46676],{},[54,46673,3000],{},[176,46675,184],{},[12543,46677,46678,46680],{},[347,46679,2213],{},[347,46681,424],{},[755,46683,46684,46686],{},[54,46685,18036],{},[54,46687,3000],{},[58,46689,46690],{"encoding":60},"a_n = \\frac{n!}{n^{n + \\frac{1}{2}}} e^n",[34,46692,46694,46749],{"className":46693,"ariaHidden":65},[64],[34,46695,46697,46700,46740,46743,46746],{"className":46696},[69],[34,46698],{"className":46699,"style":2625},[73],[34,46701,46703,46706],{"className":46702},[78],[34,46704,11468],{"className":46705},[78,79],[34,46707,46709],{"className":46708},[810],[34,46710,46712,46732],{"className":46711},[814,2523],[34,46713,46715,46729],{"className":46714},[818],[34,46716,46718],{"className":46717,"style":5870},[822],[34,46719,46720,46723],{"style":2533},[34,46721],{"className":46722,"style":830},[829],[34,46724,46726],{"className":46725},[834,835,836,837],[34,46727,3000],{"className":46728},[78,79,837],[34,46730,2547],{"className":46731},[2546],[34,46733,46735],{"className":46734},[818],[34,46736,46738],{"className":46737,"style":2554},[822],[34,46739],{},[34,46741],{"className":46742,"style":262},[237],[34,46744,192],{"className":46745},[266],[34,46747],{"className":46748,"style":262},[237],[34,46750,46752,46756,46925],{"className":46751},[69],[34,46753],{"className":46754,"style":46755},[73],"height:2.201em;vertical-align:-0.8296em;",[34,46757,46759,46762,46922],{"className":46758},[78],[34,46760],{"className":46761},[230,12689],[34,46763,46765],{"className":46764},[12543],[34,46766,46768,46913],{"className":46767},[814,2523],[34,46769,46771,46910],{"className":46770},[818],[34,46772,46774,46888,46896],{"className":46773,"style":15668},[822],[34,46775,46777,46780],{"style":46776},"top:-2.1704em;",[34,46778],{"className":46779,"style":12709},[829],[34,46781,46783],{"className":46782},[78],[34,46784,46786,46789],{"className":46785},[78],[34,46787,3000],{"className":46788},[78,79],[34,46790,46792],{"className":46791},[810],[34,46793,46795],{"className":46794},[814],[34,46796,46798],{"className":46797},[818],[34,46799,46802],{"className":46800,"style":46801},[822],"height:0.9396em;",[34,46803,46805,46808],{"style":46804},"top:-3.3486em;margin-right:0.05em;",[34,46806],{"className":46807,"style":12709},[829],[34,46809,46811],{"className":46810},[834,835,836,837],[34,46812,46814,46817,46820],{"className":46813},[78,837],[34,46815,3000],{"className":46816},[78,79,837],[34,46818,184],{"className":46819},[242,837],[34,46821,46823,46826,46885],{"className":46822},[78,837],[34,46824],{"className":46825},[230,12689,834,10776,12798],[34,46827,46829],{"className":46828},[12543],[34,46830,46832,46877],{"className":46831},[814,2523],[34,46833,46835,46874],{"className":46834},[818],[34,46836,46838,46852,46860],{"className":46837,"style":42676},[822],[34,46839,46840,46843],{"style":12814},[34,46841],{"className":46842,"style":12709},[829],[34,46844,46846],{"className":46845},[834,10776,9740,837],[34,46847,46849],{"className":46848},[78,837],[34,46850,424],{"className":46851},[78,837],[34,46853,46854,46857],{"style":12829},[34,46855],{"className":46856,"style":12709},[829],[34,46858],{"className":46859,"style":12837},[12836,837],[34,46861,46862,46865],{"style":12840},[34,46863],{"className":46864,"style":12709},[829],[34,46866,46868],{"className":46867},[834,10776,9740,837],[34,46869,46871],{"className":46870},[78,837],[34,46872,2213],{"className":46873},[78,837],[34,46875,2547],{"className":46876},[2546],[34,46878,46880],{"className":46879},[818],[34,46881,46883],{"className":46882,"style":12862},[822],[34,46884],{},[34,46886],{"className":46887},[258,12689,834,10776,12798],[34,46889,46890,46893],{"style":12897},[34,46891],{"className":46892,"style":12709},[829],[34,46894],{"className":46895,"style":12904},[12836],[34,46897,46898,46901],{"style":12907},[34,46899],{"className":46900,"style":12709},[829],[34,46902,46904,46907],{"className":46903},[78],[34,46905,3000],{"className":46906},[78,79],[34,46908,34434],{"className":46909},[258],[34,46911,2547],{"className":46912},[2546],[34,46914,46916],{"className":46915},[818],[34,46917,46920],{"className":46918,"style":46919},[822],"height:0.8296em;",[34,46921],{},[34,46923],{"className":46924},[258,12689],[34,46926,46928,46931],{"className":46927},[78],[34,46929,18036],{"className":46930},[78,79],[34,46932,46934],{"className":46933},[810],[34,46935,46937],{"className":46936},[814],[34,46938,46940],{"className":46939},[818],[34,46941,46943],{"className":46942,"style":19071},[822],[34,46944,46945,46948],{"style":4710},[34,46946],{"className":46947,"style":830},[829],[34,46949,46951],{"className":46950},[834,835,836,837],[34,46952,3000],{"className":46953},[78,79,837],[16,46955,46956,46957,47026,47027,47030],{},"To prove that ",[34,46958,46960,46977],{"className":46959,"translate":38},[37],[34,46961,46963],{"className":46962},[42],[44,46964,46965],{"xmlns":46},[48,46966,46967,46975],{},[51,46968,46969],{},[2468,46970,46971,46973],{},[54,46972,11468],{},[54,46974,3000],{},[58,46976,46581],{"encoding":60},[34,46978,46980],{"className":46979,"ariaHidden":65},[64],[34,46981,46983,46986],{"className":46982},[69],[34,46984],{"className":46985,"style":2625},[73],[34,46987,46989,46992],{"className":46988},[78],[34,46990,11468],{"className":46991},[78,79],[34,46993,46995],{"className":46994},[810],[34,46996,46998,47018],{"className":46997},[814,2523],[34,46999,47001,47015],{"className":47000},[818],[34,47002,47004],{"className":47003,"style":5870},[822],[34,47005,47006,47009],{"style":2533},[34,47007],{"className":47008,"style":830},[829],[34,47010,47012],{"className":47011},[834,835,836,837],[34,47013,3000],{"className":47014},[78,79,837],[34,47016,2547],{"className":47017},[2546],[34,47019,47021],{"className":47020},[818],[34,47022,47024],{"className":47023,"style":2554},[822],[34,47025],{}," converges, use the ",[20,47028,47029],{},"ratio test"," 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\\rightarrow \\infty} \\frac{a_{n+1}}{a_n} = \\frac{e}{\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}}}",[34,47819,47821,48037],{"className":47820,"ariaHidden":65},[64],[34,47822,47824,47827,47880,47883,48028,48031,48034],{"className":47823},[69],[34,47825],{"className":47826,"style":46080},[73],[34,47828,47830],{"className":47829},[3387,3388],[34,47831,47833,47872],{"className":47832},[814,2523],[34,47834,47836,47869],{"className":47835},[818],[34,47837,47839,47859],{"className":47838,"style":2184},[822],[34,47840,47841,47844],{"style":42479},[34,47842],{"className":47843,"style":12709},[829],[34,47845,47847],{"className":47846},[834,835,836,837],[34,47848,47850,47853,47856],{"className":47849},[78,837],[34,47851,3000],{"className":47852},[78,79,837],[34,47854,3003],{"className":47855},[266,837],[34,47857,13314],{"className":47858},[78,837],[34,47860,47861,47864],{"style":23024},[34,47862],{"className":47863,"style":12709},[829],[34,47865,47866],{},[34,47867,37329],{"className":47868},[3387],[34,47870,2547],{"className":47871},[2546],[34,47873,47875],{"className":47874},[818],[34,47876,47878],{"className":47877,"style":42517},[822],[34,47879],{},[34,47881],{"className":47882,"style":3462},[237],[34,47884,47886,47889,48025],{"className":47885},[78],[34,47887],{"className":47888},[230,12689],[34,47890,47892],{"className":47891},[12543],[34,47893,47895,48017],{"className":47894},[814,2523],[34,47896,47898,48014],{"className":47897},[818],[34,47899,47901,47949,47957],{"className":47900,"style":26758},[822],[34,47902,47903,47906],{"style":14155},[34,47904],{"className":47905,"style":12709},[829],[34,47907,47909],{"className":47908},[78],[34,47910,47912,47915],{"className":47911},[78],[34,47913,11468],{"className":47914},[78,79],[34,47916,47918],{"className":47917},[810],[34,47919,47921,47941],{"className":47920},[814,2523],[34,47922,47924,47938],{"className":47923},[818],[34,47925,47927],{"className":47926,"style":5870},[822],[34,47928,47929,47932],{"style":2533},[34,47930],{"className":47931,"style":830},[829],[34,47933,47935],{"className":47934},[834,835,836,837],[34,47936,3000],{"className":47937},[78,79,837],[34,47939,2547],{"className":47940},[2546],[34,47942,47944],{"className":47943},[818],[34,47945,47947],{"className":47946,"style":2554},[822],[34,47948],{},[34,47950,47951,47954],{"style":12897},[34,47952],{"className":47953,"style":12709},[829],[34,47955],{"className":47956,"style":12904},[12836],[34,47958,47959,47962],{"style":12907},[34,47960],{"className":47961,"style":12709},[829],[34,47963,47965],{"className":47964},[78],[34,47966,47968,47971],{"className":47967},[78],[34,47969,11468],{"className":47970},[78,79],[34,47972,47974],{"className":47973},[810],[34,47975,47977,48006],{"className":47976},[814,2523],[34,47978,47980,48003],{"className":47979},[818],[34,47981,47983],{"className":47982,"style":2530},[822],[34,47984,47985,47988],{"style":2533},[34,47986],{"className":47987,"style":830},[829],[34,47989,47991],{"className":47990},[834,835,836,837],[34,47992,47994,47997,48000],{"className":47993},[78,837],[34,47995,3000],{"className":47996},[78,79,837],[34,47998,184],{"className":47999},[242,837],[34,48001,2213],{"className":48002},[78,837],[34,48004,2547],{"className":48005},[2546],[34,48007,48009],{"className":48008},[818],[34,48010,48012],{"className":48011,"style":6830},[822],[34,48013],{},[34,48015,2547],{"className":48016},[2546],[34,48018,48020],{"className":48019},[818],[34,48021,48023],{"className":48022,"style":46277},[822],[34,48024],{},[34,48026],{"className":48027},[258,12689],[34,48029],{"className":48030,"style":262},[237],[34,48032,192],{"className":48033},[266],[34,48035],{"className":48036,"style":262},[237],[34,48038,48040,48044],{"className":48039},[69],[34,48041],{"className":48042,"style":48043},[73],"height:2.5415em;vertical-align:-1.4339em;",[34,48045,48047,48050,48301],{"className":48046},[78],[34,48048],{"className":48049},[230,12689],[34,48051,48053],{"className":48052},[12543],[34,48054,48056,48292],{"className":48055},[814,2523],[34,48057,48059,48289],{"className":48058},[818],[34,48060,48062,48268,48277],{"className":48061,"style":26758},[822],[34,48063,48065,48069],{"style":48064},"top:-2.11em;",[34,48066],{"className":48067,"style":48068},[829],"height:3.1939em;",[34,48070,48072],{"className":48071},[78],[34,48073,48075,48170],{"className":48074},[4177],[34,48076,48078,48084,48087,48090,48093,48096,48164],{"className":48077},[4177],[34,48079,48081],{"className":48080,"style":4182},[230,4181],[34,48082,179],{"className":48083},[4186,9740],[34,48085,2213],{"className":48086},[78],[34,48088],{"className":48089,"style":238},[237],[34,48091,184],{"className":48092},[242],[34,48094],{"className":48095,"style":238},[237],[34,48097,48099,48102,48161],{"className":48098},[78],[34,48100],{"className":48101},[230,12689],[34,48103,48105],{"className":48104},[12543],[34,48106,48108,48153],{"className":48107},[814,2523],[34,48109,48111,48150],{"className":48110},[818],[34,48112,48114,48128,48136],{"className":48113,"style":46368},[822],[34,48115,48116,48119],{"style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this with:",[34,48308,48310],{"className":48309,"translate":38},[159],[34,48311,48313,48397],{"className":48312,"translate":38},[37],[34,48314,48316],{"className":48315},[42],[44,48317,48318],{"xmlns":46,"display":168},[48,48319,48320,48394],{},[51,48321,48322,48352,48354],{},[755,48323,48324,48340],{},[51,48325,48326,48328,48330,48332,48338],{},[176,48327,179],{"fence":65},[347,48329,2213],{},[176,48331,184],{},[12543,48333,48334,48336],{},[347,48335,2213],{},[54,48337,3000],{},[176,48339,189],{"fence":65},[51,48341,48342,48344,48346],{},[54,48343,3000],{},[176,48345,184],{},[12543,48347,48348,48350],{},[347,48349,2213],{},[347,48351,424],{},[176,48353,192],{},[755,48355,48356,48358],{},[54,48357,18036],{},[51,48359,48360,48362,48364,48366,48372,48374,48376,48378],{},[176,48361,179],{"stretchy":178},[54,48363,3000],{},[176,48365,184],{},[12543,48367,48368,48370],{},[347,48369,2213],{},[347,48371,424],{},[176,48373,189],{"stretchy":178},[54,48375,42899],{},[176,48377,6321],{},[51,48379,48380,48382,48384,48386,48392],{},[176,48381,179],{"fence":65},[347,48383,2213],{},[176,48385,184],{},[12543,48387,48388,48390],{},[347,48389,2213],{},[54,48391,3000],{},[176,48393,189],{"fence":65},[58,48395,48396],{"encoding":60},"\\left(1 + \\frac{1}{n}\\right)^{n + \\frac{1}{2}} = e^{(n + \\frac{1}{2}) \\ln \\left(1 + \\frac{1}{n}\\right)}",[34,48398,48400,48604],{"className":48399,"ariaHidden":65},[64],[34,48401,48403,48406,48595,48598,48601],{"className":48402},[69],[34,48404],{"className":48405,"style":42464},[73],[34,48407,48409,48498],{"className":48408},[4177],[34,48410,48412,48418,48421,48424,48427,48430,48492],{"className":48411},[4177],[34,48413,48415],{"className":48414,"style":4182},[230,4181],[34,48416,179],{"className":48417},[4186,836],[34,48419,2213],{"className":48420},[78],[34,48422],{"className":48423,"style":238},[237],[34,48425,184],{"className":48426},[242],[34,48428],{"className":48429,"style":238},[237],[34,48431,48433,48436,48489],{"className":48432},[78],[34,48434],{"className":48435},[230,12689],[34,48437,48439],{"className":48438},[12543],[34,48440,48442,48481],{"className":48441},[814,2523],[34,48443,48445,48478],{"className":48444},[818],[34,48446,48448,48459,48467],{"className":48447,"style":16800},[822],[34,48449,48450,48453],{"style":14155},[34,48451],{"className":48452,"style":12709},[829],[34,48454,48456],{"className":48455},[78],[34,48457,3000],{"className":48458},[78,79],[34,48460,48461,48464],{"style":12897},[34,48462],{"className":48463,"style":12709},[829],[34,48465],{"className":48466,"style":12904},[12836],[34,48468,48469,48472],{"style":12907},[34,48470],{"className":48471,"style":12709},[829],[34,48473,48475],{"className":48474},[78],[34,48476,2213],{"className":48477},[78],[34,48479,2547],{"className":48480},[2546],[34,48482,48484],{"className":48483},[818],[34,48485,48487],{"className":48486,"style":14669},[822],[34,48488],{},[34,48490],{"className":48491},[258,12689],[34,48493,48495],{"className":48494,"style":4182},[258,4181],[34,48496,189],{"className":48497},[4186,836],[34,48499,48501],{"className":48500},[810],[34,48502,48504],{"className":48503},[814],[34,48505,48507],{"className":48506},[818],[34,48508,48510],{"className":48509,"style":42645},[822],[34,48511,48512,48515],{"style":42648},[34,48513],{"className":48514,"style":12709},[829],[34,48516,48518],{"className":48517},[834,835,836,837],[34,48519,48521,48524,48527],{"className":48520},[78,837],[34,48522,3000],{"className":48523},[78,79,837],[34,48525,184],{"className":48526},[242,837],[34,48528,48530,48533,48592],{"className":48529},[78,837],[34,48531],{"className":48532},[230,12689,834,10776,12798],[34,48534,48536],{"className":48535},[12543],[34,48537,48539,48584],{"className":48538},[814,2523],[34,48540,48542,48581],{"className":48541},[818],[34,48543,48545,48559,48567],{"className":48544,"style":42676},[822],[34,48546,48547,48550],{"style":12814},[34,48548],{"className":48549,"style":12709},[829],[34,48551,48553],{"className":48552},[834,10776,9740,837],[34,48554,48556],{"className":48555},[78,837],[34,48557,424],{"className":48558},[78,837],[34,48560,48561,48564],{"style":12829},[34,48562],{"className":48563,"style":12709},[829],[34,48565],{"className":48566,"style":12837},[12836,837],[34,48568,48569,48572],{"style":12840},[34,48570],{"className":48571,"style":12709},[829],[34,48573,48575],{"className":48574},[834,10776,9740,837],[34,48576,48578],{"className":48577},[78,837],[34,48579,2213],{"className":48580},[78,837],[34,48582,2547],{"className":48583},[2546],[34,48585,48587],{"className":48586},[818],[34,48588,48590],{"className":48589,"style":12862},[822],[34,48591],{},[34,48593],{"className":48594},[258,12689,834,10776,12798],[34,48596],{"className":48597,"style":262},[237],[34,48599,192],{"className":48600},[266],[34,48602],{"className":48603,"style":262},[237],[34,48605,48607,48610],{"className":48606},[69],[34,48608],{"className":48609,"style":43280},[73],[34,48611,48613,48616],{"className":48612},[78],[34,48614,18036],{"className":48615},[78,79],[34,48617,48619],{"className":48618},[810],[34,48620,48622],{"className":48621},[814],[34,48623,48625],{"className":48624},[818],[34,48626,48628],{"className":48627,"style":43280},[822],[34,48629,48630,48633],{"style":43283},[34,48631],{"className":48632,"style":12709},[829],[34,48634,48636],{"className":48635},[834,835,836,837],[34,48637,48639,48642,48645,48648,48716,48719,48722,48731],{"className":48638},[78,837],[34,48640,179],{"className":48641},[230,837],[34,48643,3000],{"className":48644},[78,79,837],[34,48646,184],{"className":48647},[242,837],[34,48649,48651,48654,48713],{"className":48650},[78,837],[34,48652],{"className":48653},[230,12689,834,10776,12798],[34,48655,48657],{"className":48656},[12543],[34,48658,48660,48705],{"className":48659},[814,2523],[34,48661,48663,48702],{"className":48662},[818],[34,48664,48666,48680,48688],{"className":48665,"style":42676},[822],[34,48667,48668,48671],{"style":12814},[34,48669],{"className":48670,"style":12709},[829],[34,48672,48674],{"className":48673},[834,10776,9740,837],[34,48675,48677],{"className":48676},[78,837],[34,48678,424],{"className":48679},[78,837],[34,48681,48682,48685],{"style":12829},[34,48683],{"className":48684,"style":12709},[829],[34,48686],{"className":48687,"style":12837},[12836,837],[34,4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is monotonically decreasing and ",[34,51555,51557,51579],{"className":51556,"translate":38},[37],[34,51558,51560],{"className":51559},[42],[44,51561,51562],{"xmlns":46},[48,51563,51564,51576],{},[51,51565,51566,51572,51574],{},[2468,51567,51568,51570],{},[54,51569,11468],{},[54,51571,3000],{},[176,51573,13189],{},[347,51575,2287],{},[58,51577,51578],{"encoding":60},"a_n > 0",[34,51580,51582,51637],{"className":51581,"ariaHidden":65},[64],[34,51583,51585,51588,51628,51631,51634],{"className":51584},[69],[34,51586],{"className":51587,"style":22845},[73],[34,51589,51591,51594],{"className":51590},[78],[34,51592,11468],{"className":51593},[78,79],[34,51595,51597],{"className":51596},[810],[34,51598,51600,51620],{"className":51599},[814,2523],[34,51601,51603,51617],{"className":51602},[818],[34,51604,51606],{"className":51605,"style":5870},[822],[34,51607,51608,51611],{"style":2533},[34,51609],{"className":51610,"style":830},[829],[34,51612,51614],{"className":51613},[834,835,836,837],[34,51615,3000],{"className":51616},[78,79,837],[34,51618,2547],{"className":51619},[2546],[34,51621,51623],{"className":51622},[818],[34,51624,51626],{"className":51625,"style":2554},[822],[34,51627],{},[34,51629],{"className":51630,"style":262},[237],[34,51632,13189],{"className":51633},[266],[34,51635],{"className":51636,"style":262},[237],[34,51638,51640,51643],{"className":51639},[69],[34,51641],{"className":51642,"style":391},[73],[34,51644,2287],{"className":51645},[78],", so it must converge, i.e., it has a limit.",[16,51648,51649],{},"Let:",[34,51651,51653],{"className":51652,"translate":38},[159],[34,51654,51656,51718],{"className":51655,"translate":38},[37],[34,51657,51659],{"className":51658},[42],[44,51660,51661],{"xmlns":46,"display":168},[48,51662,51663,51715],{},[51,51664,51665,51681,51705,51711,51713],{},[15798,51666,51667,51673],{},[51,51668,51669,51671],{},[54,51670,37329],{},[176,51672,6321],{},[51,51674,51675,51677,51679],{},[54,51676,3000],{},[176,51678,3003],{},[54,51680,13314],{"mathvariant":2296},[12543,51682,51683,51689],{},[51,51684,51685,51687],{},[54,51686,3000],{},[176,51688,34434],{"stretchy":178},[755,51690,51691,51693],{},[54,51692,3000],{},[51,51694,51695,51697,51699],{},[54,51696,3000],{},[176,51698,184],{},[12543,51700,51701,51703],{},[347,51702,2213],{},[347,51704,424],{},[755,51706,51707,51709],{},[54,51708,18036],{},[54,51710,3000],{},[176,51712,192],{},[54,51714,11468],{},[58,51716,51717],{"encoding":60},"\\lim_{n \\rightarrow \\infty} \\frac{n!}{n^{n + \\frac{1}{2}}} e^n = a",[34,51719,51721,51986],{"className":51720,"ariaHidden":65},[64],[34,51722,51724,51727,51780,51783,51948,51977,51980,51983],{"className":51723},[69],[34,51725],{"className":51726,"style":46755},[73],[34,51728,51730],{"className":51729},[3387,3388],[34,51731,51733,51772],{"className":51732},[814,2523],[34,51734,51736,51769],{"className":51735},[818],[34,51737,51739,51759],{"className":51738,"style":2184},[822],[34,51740,51741,51744],{"style":42479},[34,51742],{"className":51743,"style":12709},[829],[34,51745,51747],{"className":51746},[834,835,836,837],[34,51748,51750,51753,51756],{"className":51749},[78,837],[34,51751,3000],{"className":51752},[78,79,837],[34,51754,3003],{"className":51755},[266,837],[34,51757,13314],{"className":51758},[78,837],[34,51760,51761,51764],{"style":23024},[34,51762],{"className":51763,"style":12709},[829],[34,51765,51766],{},[34,51767,37329],{"className":51768},[3387],[34,51770,2547],{"className":51771},[2546],[34,51773,51775],{"className":51774},[818],[34,51776,51778],{"className":51777,"style":42517},[822],[34,51779],{},[34,51781],{"className":51782,"style":3462},[237],[34,51784,51786,51789,51945],{"className":51785},[78],[34,51787],{"className":51788},[230,12689],[34,51790,51792],{"className":51791},[12543],[34,51793,51795,51937],{"className":51794},[814,2523],[34,51796,51798,51934],{"className":51797},[818],[34,51799,51801,51912,51920],{"className":51800,"style":15668},[822],[34,51802,51803,51806],{"style":46776},[34,51804],{"className":51805,"style":12709},[829],[34,51807,51809],{"className":51808},[78],[34,51810,51812,51815],{"className":51811},[78],[34,51813,3000],{"className":51814},[78,79],[34,51816,51818],{"className":51817},[810],[34,51819,51821],{"className":51820},[814],[34,51822,51824],{"className":51823},[818],[34,51825,51827],{"className":51826,"style":46801},[822],[34,51828,51829,51832],{"style":46804},[34,51830],{"className":51831,"style":12709},[829],[34,51833,51835],{"className":51834},[834,835,836,837],[34,51836,51838,51841,51844],{"className":51837},[78,837],[34,51839,3000],{"className":51840},[78,79,837],[34,51842,184],{"className":51843},[242,837],[34,51845,51847,51850,51909],{"className":51846},[78,837],[34,51848],{"className":51849},[230,12689,834,10776,12798],[34,51851,51853],{"className":51852},[12543],[34,51854,51856,51901],{"className":51855},[814,2523],[34,51857,51859,51898],{"className":51858},[818],[34,51860,51862,51876,51884],{"className":51861,"style":42676},[822],[34,51863,51864,51867],{"style":12814},[34,51865],{"className":51866,"style":12709},[829],[34,51868,51870],{"className":51869},[834,10776,9740,837],[34,51871,51873],{"className":51872},[78,837],[34,51874,424],{"className":51875},[78,837],[34,51877,51878,51881],{"style":12829},[34,51879],{"className":51880,"style":12709},[829],[34,51882],{"className":51883,"style":12837},[12836,837],[34,51885,51886,51889],{"style":12840},[34,51887],{"className":51888,"style":12709},[829],[34,51890,51892],{"className":51891},[834,10776,9740,837],[34,51893,51895],{"className":51894},[78,837],[34,51896,2213],{"className":51897},[78,837],[34,51899,2547],{"className":51900},[2546],[34,51902,51904],{"className":51903},[818],[34,51905,51907],{"className":51906,"style":12862},[822],[34,51908],{},[34,51910],{"className":51911},[258,12689,834,10776,12798],[34,51913,51914,51917],{"style":12897},[34,51915],{"className":51916,"style":12709},[829],[34,51918],{"className":51919,"style":12904},[12836],[34,51921,51922,51925],{"style":12907},[34,51923],{"className":51924,"style":12709},[829],[34,51926,51928,51931],{"className":51927},[78],[34,51929,3000],{"className":51930},[78,79],[34,51932,34434],{"className":51933},[258],[34,51935,2547],{"className":51936},[2546],[34,51938,51940],{"className":51939},[818],[34,51941,51943],{"className":51942,"style":46919},[822],[34,51944],{},[34,51946],{"className":51947},[258,12689],[34,51949,51951,51954],{"className":51950},[78],[34,51952,18036],{"className":51953},[78,79],[34,51955,51957],{"className":51956},[810],[34,51958,51960],{"className":51959},[814],[34,51961,51963],{"className":51962},[818],[34,51964,51966],{"className":51965,"style":19071},[822],[34,51967,51968,51971],{"style":4710},[34,51969],{"className":51970,"style":830},[829],[34,51972,51974],{"className":51973},[834,835,836,837],[34,51975,3000],{"className":51976},[78,79,837],[34,51978],{"className":51979,"style":262},[237],[34,51981,192],{"className":51982},[266],[34,51984],{"className":51985,"style":262},[237],[34,51987,51989,51992],{"className":51988},[69],[34,51990],{"className":51991,"style":108},[73],[34,51993,11468],{"className":51994},[78,79],[16,51996,51997],{},"Then, we need to find this limit.",[16,51999,52000],{},"From the earlier Wallis formula:",[34,52002,52004],{"className":52003,"translate":38},[159],[34,52005,52007,52096],{"className":52006,"translate":38},[37],[34,52008,52010],{"className":52009},[42],[44,52011,52012],{"xmlns":46,"display":168},[48,52013,52014,52094],{},[51,52015,52016,52032,52086,52088],{},[15798,52017,52018,52024],{},[51,52019,52020,52022],{},[54,52021,37329],{},[176,52023,6321],{},[51,52025,52026,52028,52030],{},[54,52027,3324],{},[176,52029,3003],{},[54,52031,13314],{"mathvariant":2296},[12543,52033,52034,52052],{},[51,52035,52036,52038,52040,52042,52044,52046],{},[176,52037,179],{"stretchy":178},[347,52039,424],{},[54,52041,3324],{},[176,52043,189],{"stretchy":178},[176,52045,34434],{"stretchy":178},[755,52047,52048,52050],{},[176,52049,34434],{"stretchy":178},[347,52051,424],{},[51,52053,52054,52056,52058,52060,52062,52064,52066,52068,52070,52072,52074,52076,52078,52080,52082,52084],{},[176,52055,179],{"stretchy":178},[347,52057,424],{},[54,52059,3324],{},[176,52061,184],{},[347,52063,2213],{},[176,52065,189],{"stretchy":178},[176,52067,34434],{"stretchy":178},[176,52069,34434],{"stretchy":178},[176,52071,179],{"stretchy":178},[347,52073,424],{},[54,52075,3324],{},[176,52077,6463],{},[347,52079,2213],{},[176,52081,189],{"stretchy":178},[176,52083,34434],{"stretchy":178},[176,52085,34434],{"stretchy":178},[176,52087,192],{},[12543,52089,52090,52092],{},[54,52091,12569],{},[347,52093,424],{},[58,52095,41635],{"encoding":60},[34,52097,52099,52315],{"className":52098,"ariaHidden":65},[64],[34,52100,52102,52105,52158,52161,52306,52309,52312],{"className":52101},[69],[34,52103],{"className":52104,"style":39605},[73],[34,52106,52108],{"className":52107},[3387,3388],[34,52109,52111,52150],{"className":52110},[814,2523],[34,52112,52114,52147],{"className":52113},[818],[34,52115,52117,52137],{"className":52116,"style":2184},[822],[34,52118,52119,52122],{"style":37648},[34,52120],{"className":52121,"style":12709},[829],[34,52123,52125],{"className":52124},[834,835,836,837],[34,52126,52128,52131,52134],{"className":52127},[78,837],[34,52129,3324],{"className":52130,"style":3446},[78,79,837],[34,52132,3003],{"className":52133},[266,837],[34,52135,13314],{"className":52136},[78,837],[34,52138,52139,52142],{"style":23024},[34,52140],{"className":52141,"style":12709},[829],[34,52143,52144],{},[34,52145,37329],{"className":52146},[3387],[34,52148,2547],{"className":52149},[2546],[34,52151,52153],{"className":52152},[818],[34,52154,52156],{"className":52155,"style":37686},[822],[34,52157],{},[34,52159],{"className":52160,"style":3462},[237],[34,52162,52164,52167,52303],{"className":52163},[78],[34,52165],{"className":52166},[230,12689],[34,52168,52170],{"className":52169},[12543],[34,52171,52173,52295],{"className":52172},[814,2523],[34,52174,52176,52292],{"className":52175},[818],[34,52177,52179,52235,52243],{"className":52178,"style":15228},[822],[34,52180,52181,52184],{"style":14155},[34,52182],{"className":52183,"style":12709},[829],[34,52185,52187,52190,52193,52196,52199,52202,52205,52208,52211,52214,52217,52220,52223,52226,52229,52232],{"className":52186},[78],[34,52188,179],{"className":52189},[230],[34,52191,424],{"className":52192},[78],[34,52194,3324],{"className":52195,"style":3446},[78,79],[34,52197],{"className":52198,"style":238},[237],[34,52200,184],{"className":52201},[242],[34,52203],{"className":52204,"style":238},[237],[34,52206,2213],{"className":52207},[78],[34,52209,34895],{"className":52210},[258],[34,52212,179],{"className":52213},[230],[34,52215,424],{"className":52216},[78],[34,52218,3324],{"className":52219,"style":3446},[78,79],[34,52221],{"className":52222,"style":238},[237],[34,52224,6463],{"className":52225},[242],[34,52227],{"className":52228,"style":238},[237],[34,52230,2213],{"className":52231},[78],[34,52233,34895],{"className":52234},[258],[34,52236,52237,52240],{"style":12897},[34,52238],{"className":52239,"style":12709},[829],[34,52241],{"className":52242,"style":12904},[12836],[34,52244,52245,52248],{"style":12907},[34,52246],{"className":52247,"style":12709},[829],[34,52249,52251,52254,52257,52260,52263],{"className":52250},[78],[34,52252,179],{"className":52253},[230],[34,52255,424],{"className":52256},[78],[34,52258,3324],{"className":52259,"style":3446},[78,79],[34,52261,39708],{"className":52262},[258],[34,52264,52266,52269],{"className":52265},[258],[34,52267,34434],{"className":52268},[258],[34,52270,52272],{"className":52271},[810],[34,52273,52275],{"className":52274},[814],[34,52276,52278],{"className":52277},[818],[34,52279,52281],{"className":52280,"style":800},[822],[34,52282,52283,52286],{"style":825},[34,52284],{"className":52285,"style":830},[829],[34,52287,52289],{"className":52288},[834,835,836,837],[34,52290,424],{"className":52291},[78,837],[34,52293,2547],{"className":52294},[2546],[34,52296,52298],{"className":52297},[818],[34,52299,52301],{"className":52300,"style":17481},[822],[34,52302],{},[34,52304],{"className":52305},[258,12689],[34,52307],{"className":52308,"style":262},[237],[34,52310,192],{"className":52311},[266],[34,52313],{"className":52314,"style":262},[237],[34,52316,52318,52321],{"className":52317},[69],[34,52319],{"className":52320,"style":26739},[73],[34,52322,52324,52327,52380],{"className":52323},[78],[34,52325],{"className":52326},[230,12689],[34,52328,52330],{"className":52329},[12543],[34,52331,52333,52372],{"className":52332},[814,2523],[34,52334,52336,52369],{"className":52335},[818],[34,52337,52339,52350,52358],{"className":52338,"style":26758},[822],[34,52340,52341,52344],{"style":14155},[34,52342],{"className":52343,"style":12709},[829],[34,52345,52347],{"className":52346},[78],[34,52348,424],{"className":52349},[78],[34,52351,52352,52355],{"style":12897},[34,52353],{"className":52354,"style":12709},[829],[34,52356],{"className":52357,"style":12904},[12836],[34,52359,52360,52363],{"style":12907},[34,52361],{"className":52362,"style":12709},[829],[34,52364,52366],{"className":52365},[78],[34,52367,12569],{"className":52368,"style":143},[78,79],[34,52370,2547],{"className":52371},[2546],[34,52373,52375],{"className":52374},[818],[34,52376,52378],{"className":52377,"style":14669},[822],[34,52379],{},[34,52381],{"className":52382},[258,12689],[34,52384,52386],{"className":52385,"translate":38},[159],[34,52387,52389,52447],{"className":52388,"translate":38},[37],[34,52390,52392],{"className":52391},[42],[44,52393,52394],{"xmlns":46,"display":168},[48,52395,52396,52444],{},[51,52397,52398,52400,52402,52404,52406,52408,52410,52412,52414,52416,52418,52420,52422,52424,52426,52428,52430,52432,52434,52440,52442],{},[176,52399,179],{"stretchy":178},[347,52401,424],{},[54,52403,3000],{},[176,52405,189],{"stretchy":178},[176,52407,34434],{"stretchy":178},[176,52409,34434],{"stretchy":178},[176,52411,192],{},[347,52413,424],{},[54,52415,3000],{},[176,52417,179],{"stretchy":178},[347,52419,424],{},[54,52421,3000],{},[176,52423,6463],{},[347,52425,424],{},[176,52427,189],{"stretchy":178},[176,52429,6706],{},[347,52431,424],{},[176,52433,192],{},[755,52435,52436,52438],{},[347,52437,424],{},[54,52439,3000],{},[54,52441,3000],{},[176,52443,34434],{"stretchy":178},[58,52445,52446],{"encoding":60},"(2n)!! = 2n (2n-2) \\cdots 2 = 2^n n!",[34,52448,52450,52477,52507,52540],{"className":52449,"ariaHidden":65},[64],[34,52451,52453,52456,52459,52462,52465,52468,52471,52474],{"className":52452},[69],[34,52454],{"className":52455,"style":223},[73],[34,52457,179],{"className":52458},[230],[34,52460,424],{"className":52461},[78],[34,52463,3000],{"className":52464},[78,79],[34,52466,34895],{"className":52467},[258],[34,52469],{"className":52470,"style":262},[237],[34,52472,192],{"className":52473},[266],[34,52475],{"className":52476,"style":262},[237],[34,52478,52480,52483,52486,52489,52492,52495,52498,52501,52504],{"className":52479},[69],[34,52481],{"className":52482,"style":223},[73],[34,52484,424],{"className":52485},[78],[34,52487,3000],{"className":52488},[78,79],[34,52490,179],{"className":52491},[230],[34,52493,424],{"className":52494},[78],[34,52496,3000],{"className":52497},[78,79],[34,52499],{"className":52500,"style":238},[237],[34,52502,6463],{"className":52503},[242],[34,52505],{"className":52506,"style":238},[237],[34,52508,52510,52513,52516,52519,52522,52525,52528,52531,52534,52537],{"className":52509},[69],[34,52511],{"className":52512,"style":223},[73],[34,52514,424],{"className":52515},[78],[34,52517,189],{"className":52518},[258],[34,52520],{"className":52521,"style":3462},[237],[34,52523,6706],{"className":52524},[4177],[34,52526],{"className":52527,"style":3462},[237],[34,52529,424],{"className":52530},[78],[34,52532],{"className":52533,"style":262},[237],[34,52535,192],{"className":52536},[266],[34,52538],{"className":52539,"style":262},[237],[34,52541,52543,52546,52575,52578],{"className":52542},[69],[34,52544],{"className":52545,"style":19071},[73],[34,52547,52549,52552],{"className":52548},[78],[34,52550,424],{"className":52551},[78],[34,52553,52555],{"className":52554},[810],[34,52556,52558],{"className":52557},[814],[34,52559,52561],{"className":52560},[818],[34,52562,52564],{"className":52563,"style":19071},[822],[34,52565,52566,52569],{"style":4710},[34,52567],{"className":52568,"style":830},[829],[34,52570,52572],{"className":52571},[834,835,836,837],[34,52573,3000],{"className":52574},[78,79,837],[34,52576,3000],{"className":52577},[78,79],[34,52579,34434],{"className":52580},[258],[34,52582,52584],{"className":52583,"translate":38},[159],[34,52585,52587,52673],{"className":52586,"translate":38},[37],[34,52588,52590],{"className":52589},[42],[44,52591,52592],{"xmlns":46,"display":168},[48,52593,52594,52670],{},[51,52595,52596,52598,52600,52602,52604,52606,52608,52610,52612,52614,52642,52644],{},[176,52597,179],{"stretchy":178},[347,52599,424],{},[54,52601,3000],{},[176,52603,6463],{},[347,52605,2213],{},[176,52607,189],{"stretchy":178},[176,52609,34434],{"stretchy":178},[176,52611,34434],{"stretchy":178},[176,52613,192],{},[12543,52615,52616,52628],{},[51,52617,52618,52620,52622,52624,52626],{},[176,52619,179],{"stretchy":178},[347,52621,424],{},[54,52623,3000],{},[176,52625,189],{"stretchy":178},[176,52627,34434],{"stretchy":178},[51,52629,52630,52632,52634,52636,52638,52640],{},[176,52631,179],{"stretchy":178},[347,52633,424],{},[54,52635,3000],{},[176,52637,189],{"stretchy":178},[176,52639,34434],{"stretchy":178},[176,52641,34434],{"stretchy":178},[176,52643,192],{},[12543,52645,52646,52658],{},[51,52647,52648,52650,52652,52654,52656],{},[176,52649,179],{"stretchy":178},[347,52651,424],{},[54,52653,3000],{},[176,52655,189],{"stretchy":178},[176,52657,34434],{"stretchy":178},[51,52659,52660,52666,52668],{},[755,52661,52662,52664],{},[347,52663,424],{},[54,52665,3000],{},[54,52667,3000],{},[176,52669,34434],{"stretchy":178},[58,52671,52672],{"encoding":60},"(2n-1)!! = \\frac{(2n)!}{(2n)!!} = \\frac{(2n)!}{2^n n!}",[34,52674,52676,52700,52721,52816],{"className":52675,"ariaHidden":65},[64],[34,52677,52679,52682,52685,52688,52691,52694,52697],{"className":52678},[69],[34,52680],{"className":52681,"style":223},[73],[34,52683,179],{"className":52684},[230],[34,52686,424],{"className":52687},[78],[34,52689,3000],{"className":52690},[78,79],[34,52692],{"className":52693,"style":238},[237],[34,52695,6463],{"className":52696},[242],[34,52698],{"className":52699,"style":238},[237],[34,52701,52703,52706,52709,52712,52715,52718],{"className":52702},[69],[34,52704],{"className":52705,"style":223},[73],[34,52707,2213],{"className":52708},[78],[34,52710,34895],{"className":52711},[258],[34,52713],{"className":52714,"style":262},[237],[34,52716,192],{"className":52717},[266],[34,52719],{"className":52720,"style":262},[237],[34,52722,52724,52727,52807,52810,52813],{"className":52723},[69],[34,52725],{"className":52726,"style":23317},[73],[34,52728,52730,52733,52804],{"className":52729},[78],[34,52731],{"className":52732},[230,12689],[34,52734,52736],{"className":52735},[12543],[34,52737,52739,52796],{"className":52738},[814,2523],[34,52740,52742,52793],{"className":52741},[818],[34,52743,52745,52765,52773],{"className":52744,"style":14583},[822],[34,52746,52747,52750],{"style":14155},[34,52748],{"className":52749,"style":12709},[829],[34,52751,52753,52756,52759,52762],{"className":52752},[78],[34,52754,179],{"className":52755},[230],[34,52757,424],{"className":52758},[78],[34,52760,3000],{"className":52761},[78,79],[34,52763,34895],{"className":52764},[258],[34,52766,52767,52770],{"style":12897},[34,52768],{"className":52769,"style":12709},[829],[34,52771],{"className":52772,"style":12904},[12836],[34,52774,52775,52778],{"style":12907},[34,52776],{"className":52777,"style":12709},[829],[34,52779,52781,52784,52787,52790],{"className":52780},[78],[34,52782,179],{"className":52783},[230],[34,52785,424],{"className":52786},[78],[34,52788,3000],{"className":52789},[78,79],[34,52791,39708],{"className":52792},[258],[34,52794,2547],{"className":52795},[2546],[34,52797,52799],{"className":52798},[818],[34,52800,52802],{"className":52801,"style":17481},[822],[34,52803],{},[34,52805],{"className":52806},[258,12689],[34,52808],{"className":52809,"style":262},[237],[34,52811,192],{"className":52812},[266],[34,52814],{"className":52815,"style":262},[237],[34,52817,52819,52822],{"className":52818},[69],[34,52820],{"className":52821,"style":14564},[73],[34,52823,52825,52828,52922],{"className":52824},[78],[34,52826],{"className":52827},[230,12689],[34,52829,52831],{"className":52830},[12543],[34,52832,52834,52914],{"className":52833},[814,2523],[34,52835,52837,52911],{"className":52836},[818],[34,52838,52840,52883,52891],{"className":52839,"style":14583},[822],[34,52841,52842,52845],{"style":14155},[34,52843],{"className":52844,"style":12709},[829],[34,52846,52848,52877,52880],{"className":52847},[78],[34,52849,52851,52854],{"className":52850},[78],[34,52852,424],{"className":52853},[78],[34,52855,52857],{"className":52856},[810],[34,52858,52860],{"className":52859},[814],[34,52861,52863],{"className":52862},[818],[34,52864,52866],{"className":52865,"style":42579},[822],[34,52867,52868,52871],{"style":12743},[34,52869],{"className":52870,"style":830},[829],[34,52872,52874],{"className":52873},[834,835,836,837],[34,52875,3000],{"className":52876},[78,79,837],[34,52878,3000],{"className":52879},[78,79],[34,52881,34434],{"className":52882},[258],[34,52884,52885,52888],{"style":12897},[34,52886],{"className":52887,"style":12709},[829],[34,52889],{"className":52890,"style":12904},[12836],[34,52892,52893,52896],{"style":12907},[34,52894],{"className":52895,"style":12709},[829],[34,52897,52899,52902,52905,52908],{"className":52898},[78],[34,52900,179],{"className":52901},[230],[34,52903,424],{"className":52904},[78],[34,52906,3000],{"className":52907},[78,79],[34,52909,39708],{"className":52910},[258],[34,52912,2547],{"className":52913},[2546],[34,52915,52917],{"className":52916},[818],[34,52918,52920],{"className":52919,"style":14669},[822],[34,52921],{},[34,52923],{"className":52924},[258,12689],[34,52926,52928],{"className":52927,"translate":38},[159],[34,52929,52931,53043],{"className":52930,"translate":38},[37],[34,52932,52934],{"className":52933},[42],[44,52935,52936],{"xmlns":46,"display":168},[48,52937,52938,53040],{},[51,52939,52940,52942,52944,52946,52948,52950,52952,52954,52956,52958,52994,52996],{},[176,52941,179],{"stretchy":178},[347,52943,424],{},[54,52945,3000],{},[176,52947,184],{},[347,52949,2213],{},[176,52951,189],{"stretchy":178},[176,52953,34434],{"stretchy":178},[176,52955,34434],{"stretchy":178},[176,52957,192],{},[12543,52959,52960,52976],{},[51,52961,52962,52964,52966,52968,52970,52972,52974],{},[176,52963,179],{"stretchy":178},[347,52965,424],{},[54,52967,3000],{},[176,52969,184],{},[347,52971,424],{},[176,52973,189],{"stretchy":178},[176,52975,34434],{"stretchy":178},[51,52977,52978,52980,52982,52984,52986,52988,52990,52992],{},[176,52979,179],{"stretchy":178},[347,52981,424],{},[54,52983,3000],{},[176,52985,184],{},[347,52987,424],{},[176,52989,189],{"stretchy":178},[176,52991,34434],{"stretchy":178},[176,52993,34434],{"stretchy":178},[176,52995,192],{},[12543,52997,52998,53014],{},[51,52999,53000,53002,53004,53006,53008,53010,53012],{},[176,53001,179],{"stretchy":178},[347,53003,424],{},[54,53005,3000],{},[176,53007,184],{},[347,53009,424],{},[176,53011,189],{"stretchy":178},[176,53013,34434],{"stretchy":178},[51,53015,53016,53028,53030,53032,53034,53036,53038],{},[755,53017,53018,53020],{},[347,53019,424],{},[51,53021,53022,53024,53026],{},[54,53023,3000],{},[176,53025,184],{},[347,53027,2213],{},[176,53029,179],{"stretchy":178},[54,53031,3000],{},[176,53033,184],{},[347,53035,2213],{},[176,53037,189],{"stretchy":178},[176,53039,34434],{"stretchy":178},[58,53041,53042],{"encoding":60},"(2n+1)!! = \\frac{(2n+2)!}{(2n+2)!!} = \\frac{(2n+2)!}{2^{n+1} (n+1)!}",[34,53044,53046,53070,53091,53210],{"className":53045,"ariaHidden":65},[64],[34,53047,53049,53052,53055,53058,53061,53064,53067],{"className":53048},[69],[34,53050],{"className":53051,"style":223},[73],[34,53053,179],{"className":53054},[230],[34,53056,424],{"className":53057},[78],[34,53059,3000],{"className":53060},[78,79],[34,53062],{"className":53063,"style":238},[237],[34,53065,184],{"className":53066},[242],[34,53068],{"className":53069,"style":238},[237],[34,53071,53073,53076,53079,53082,53085,53088],{"className":53072},[69],[34,53074],{"className":53075,"style":223},[73],[34,53077,2213],{"className":53078},[78],[34,53080,34895],{"className":53081},[258],[34,53083],{"className":53084,"style":262},[237],[34,53086,192],{"className":53087},[266],[34,53089],{"className":53090,"style":262},[237],[34,53092,53094,53097,53201,53204,53207],{"className":53093},[69],[34,53095],{"className":53096,"style":23317},[73],[34,53098,53100,53103,53198],{"className":53099},[78],[34,53101],{"className":53102},[230,12689],[34,53104,53106],{"className":53105},[12543],[34,53107,53109,53190],{"className":53108},[814,2523],[34,53110,53112,53187],{"className":53111},[818],[34,53113,53115,53147,53155],{"className":53114,"style":14583},[822],[34,53116,53117,53120],{"style":14155},[34,53118],{"className":53119,"style":12709},[829],[34,53121,53123,53126,53129,53132,53135,53138,53141,53144],{"className":53122},[78],[34,53124,179],{"className":53125},[230],[34,53127,424],{"className":53128},[78],[34,53130,3000],{"className":53131},[78,79],[34,53133],{"className":53134,"style":238},[237],[34,53136,184],{"className":53137},[242],[34,53139],{"className":53140,"style":238},[237],[34,53142,424],{"className":53143},[78],[34,53145,34895],{"className":53146},[258],[34,53148,53149,53152],{"style":12897},[34,53150],{"className":53151,"style":12709},[829],[34,53153],{"className":53154,"style":12904},[12836],[34,53156,53157,53160],{"style":12907},[34,53158],{"className":53159,"style":12709},[829],[34,53161,53163,53166,53169,53172,53175,53178,53181,53184],{"className":53162},[78],[34,53164,179],{"className":53165},[230],[34,53167,424],{"className":53168},[78],[34,53170,3000],{"className":53171},[78,79],[34,53173],{"className":53174,"style":238},[237],[34,53176,184],{"className":53177},[242],[34,53179],{"className":53180,"style":238},[237],[34,53182,424],{"className":53183},[78],[34,53185,39708],{"className":53186},[258],[34,53188,2547],{"className":53189},[2546],[34,53191,53193],{"className":53192},[818],[34,53194,53196],{"className":53195,"style":17481},[822],[34,53197],{},[34,53199],{"className":53200},[258,12689],[34,53202],{"className":53203,"style":262},[237],[34,53205,192],{"className":53206},[266],[34,53208],{"className":53209,"style":262},[237],[34,53211,53213,53216],{"className":53212},[69],[34,53214],{"className":53215,"style":23317},[73],[34,53217,53219,53222,53352],{"className":53218},[78],[34,53220],{"className":53221},[230,12689],[34,53223,53225],{"className":53224},[12543],[34,53226,53228,53344],{"className":53227},[814,2523],[34,53229,53231,53341],{"className":53230},[818],[34,53232,53234,53301,53309],{"className":53233,"style":14583},[822],[34,53235,53236,53239],{"style":14155},[34,53237],{"className":53238,"style":12709},[829],[34,53240,53242,53280,53283,53286,53289,53292,53295,53298],{"className":53241},[78],[34,53243,53245,53248],{"className":53244},[78],[34,53246,424],{"className":53247},[78],[34,53249,53251],{"className":53250},[810],[34,53252,53254],{"className":53253},[814],[34,53255,53257],{"className":53256},[818],[34,53258,53260],{"className":53259,"style":15259},[822],[34,53261,53262,53265],{"style":12743},[34,53263],{"className":53264,"style":830},[829],[34,53266,53268],{"className":53267},[834,835,836,837],[34,53269,53271,53274,53277],{"className":53270},[78,837],[34,53272,3000],{"className":53273},[78,79,837],[34,53275,184],{"className":53276},[242,837],[34,53278,2213],{"className":53279},[78,837],[34,53281,179],{"className":53282},[230],[34,53284,3000],{"className":53285},[78,79],[34,53287],{"className":53288,"style":238},[237],[34,53290,184],{"className":53291},[242],[34,53293],{"className":53294,"style":238},[237],[34,53296,2213],{"className":53297},[78],[34,53299,39708],{"className":53300},[258],[34,53302,53303,53306],{"style":12897},[34,53304],{"className":53305,"style":12709},[829],[34,53307],{"className":53308,"style":12904},[12836],[34,53310,53311,53314],{"style":12907},[34,53312],{"className":53313,"style":12709},[829],[34,53315,53317,53320,53323,53326,53329,53332,53335,53338],{"className":53316},[78],[34,53318,179],{"className":53319},[230],[34,53321,424],{"className":53322},[78],[34,53324,3000],{"className":53325},[78,79],[34,53327],{"className":53328,"style":238},[237],[34,53330,184],{"className":53331},[242],[34,53333],{"className":53334,"style":238},[237],[34,53336,424],{"className":53337},[78],[34,53339,39708],{"className":53340},[258],[34,53342,2547],{"className":53343},[2546],[34,53345,53347],{"className":53346},[818],[34,53348,53350],{"className":53349,"style":17481},[822],[34,53351],{},[34,53353],{"className":53354},[258,12689],[16,53356,53357],{},"Thus, Wallis' formula becomes:",[34,53359,53361],{"className":53360,"translate":38},[159],[34,53362,53364,53461],{"className":53363,"translate":38},[37],[34,53365,53367],{"className":53366},[42],[44,53368,53369],{"xmlns":46,"display":168},[48,53370,53371,53458],{},[51,53372,53373,53389,53450,53452],{},[15798,53374,53375,53381],{},[51,53376,53377,53379],{},[54,53378,37329],{},[176,53380,6321],{},[51,53382,53383,53385,53387],{},[54,53384,3000],{},[176,53386,3003],{},[54,53388,13314],{"mathvariant":2296},[12543,53390,53391,53424],{},[51,53392,53393,53400,53402,53404,53406,53412,53414,53416,53418,53420,53422],{},[755,53394,53395,53398],{},[347,53396,53397],{},"16",[54,53399,3000],{},[176,53401,179],{"stretchy":178},[54,53403,3000],{},[176,53405,34434],{"stretchy":178},[755,53407,53408,53410],{},[176,53409,189],{"stretchy":178},[347,53411,990],{},[176,53413,179],{"stretchy":178},[347,53415,424],{},[54,53417,3000],{},[176,53419,184],{},[347,53421,424],{},[176,53423,189],{"stretchy":178},[51,53425,53426,53428,53430,53432,53434,53436,53438,53440,53442,53444,53446,53448],{},[176,53427,179],{"stretchy":178},[347,53429,424],{},[54,53431,3000],{},[176,53433,189],{"stretchy":178},[176,53435,34434],{"stretchy":178},[176,53437,179],{"stretchy":178},[347,53439,424],{},[54,53441,3000],{},[176,53443,184],{},[347,53445,424],{},[176,53447,189],{"stretchy":178},[176,53449,34434],{"stretchy":178},[176,53451,192],{},[12543,53453,53454,53456],{},[54,53455,12569],{},[347,53457,424],{},[58,53459,53460],{"encoding":60},"\\lim_{n \\rightarrow \\infty} \\frac{16^n (n!)^4 (2n+2)}{(2n)!(2n+2)!} = \\frac{\\pi}{2}",[34,53462,53464,53722],{"className":53463,"ariaHidden":65},[64],[34,53465,53467,53470,53523,53526,53713,53716,53719],{"className":53466},[69],[34,53468],{"className":53469,"style":39605},[73],[34,53471,53473],{"className":53472},[3387,3388],[34,53474,53476,53515],{"className":53475},[814,2523],[34,53477,53479,53512],{"className":53478},[818],[34,53480,53482,53502],{"className":53481,"style":2184},[822],[34,53483,53484,53487],{"style":42479},[34,53485],{"className":53486,"style":12709},[829],[34,53488,53490],{"className":53489},[834,835,836,837],[34,53491,53493,53496,53499],{"className":53492},[78,837],[34,53494,3000],{"className":53495},[78,79,837],[34,53497,3003],{"className":53498},[266,837],[34,53500,13314],{"className":53501},[78,837],[34,53503,53504,53507],{"style":23024},[34,53505],{"className":53506,"style":12709},[829],[34,53508,53509],{},[34,53510,37329],{"className":53511},[3387],[34,53513,2547],{"className":53514},[2546],[34,53516,53518],{"className":53517},[818],[34,53519,53521],{"className":53520,"style":42517},[822],[34,53522],{},[34,53524],{"className":53525,"style":3462},[237],[34,53527,53529,53532,53710],{"className":53528},[78],[34,53530],{"className":53531},[230,12689],[34,53533,53535],{"className":53534},[12543],[34,53536,53538,53702],{"className":53537},[814,2523],[34,53539,53541,53699],{"className":53540},[818],[34,53542,53544,53588,53596],{"className":53543,"style":15228},[822],[34,53545,53546,53549],{"style":14155},[34,53547],{"className":53548,"style":12709},[829],[34,53550,53552,53555,53558,53561,53564,53567,53570,53573,53576,53579,53582,53585],{"className":53551},[78],[34,53553,179],{"className":53554},[230],[34,53556,424],{"className":53557},[78],[34,53559,3000],{"className":53560},[78,79],[34,53562,39708],{"className":53563},[258],[34,53565,179],{"className":53566},[230],[34,53568,424],{"className":53569},[78],[34,53571,3000],{"className":53572},[78,79],[34,53574],{"className":53575,"style":238},[237],[34,53577,184],{"className":53578},[242],[34,53580],{"className":53581,"style":238},[237],[34,53583,424],{"className":53584},[78],[34,53586,39708],{"className":53587},[258],[34,53589,53590,53593],{"style":12897},[34,53591],{"className":53592,"style":12709},[829],[34,53594],{"className":53595,"style":12904},[12836],[34,53597,53598,53601],{"style":12907},[34,53599],{"className":53600,"style":12709},[829],[34,53602,53604,53607,53637,53640,53643,53646,53675,53678,53681,53684,53687,53690,53693,53696],{"className":53603},[78],[34,53605,2213],{"className":53606},[78],[34,53608,53610,53614],{"className":53609},[78],[34,53611,53613],{"className":53612},[78],"6",[34,53615,53617],{"className":53616},[810],[34,53618,53620],{"className":53619},[814],[34,53621,53623],{"className":53622},[818],[34,53624,53626],{"className":53625,"style":3061},[822],[34,53627,53628,53631],{"style":825},[34,53629],{"className":53630,"style":830},[829],[34,53632,53634],{"className":53633},[834,835,836,837],[34,53635,3000],{"className":53636},[78,79,837],[34,53638,179],{"className":53639},[230],[34,53641,3000],{"className":53642},[78,79],[34,53644,34434],{"className":53645},[258],[34,53647,53649,53652],{"className":53648},[258],[34,53650,189],{"className":53651},[258],[34,53653,53655],{"className":53654},[810],[34,53656,53658],{"className":53657},[814],[34,53659,53661],{"className":53660},[818],[34,53662,53664],{"className":53663,"style":800},[822],[34,53665,53666,53669],{"style":825},[34,53667],{"className":53668,"style":830},[829],[34,53670,53672],{"className":53671},[834,835,836,837],[34,53673,990],{"className":53674},[78,837],[34,53676,179],{"className":53677},[230],[34,53679,424],{"className":53680},[78],[34,53682,3000],{"className":53683},[78,79],[34,53685],{"className":53686,"style":238},[237],[34,53688,184],{"className":53689},[242],[34,53691],{"className":53692,"style":238},[237],[34,53694,424],{"className":53695},[78],[34,53697,189],{"className":53698},[258],[34,53700,2547],{"className":53701},[2546],[34,53703,53705],{"className":53704},[818],[34,53706,53708],{"className":53707,"style":17481},[822],[34,53709],{},[34,53711],{"className":53712},[258,12689],[34,53714],{"className":53715,"style":262},[237],[34,53717,192],{"className":53718},[266],[34,53720],{"className":53721,"style":262},[237],[34,53723,53725,53728],{"className":53724},[69],[34,53726],{"className":53727,"style":26739},[73],[34,53729,53731,53734,53787],{"className":53730},[78],[34,53732],{"className":53733},[230,12689],[34,53735,53737],{"className":53736},[12543],[34,53738,53740,53779],{"className":53739},[814,2523],[34,53741,53743,53776],{"className":53742},[818],[34,53744,53746,53757,53765],{"className":53745,"style":26758},[822],[34,53747,53748,53751],{"style":14155},[34,53749],{"className":53750,"style":12709},[829],[34,53752,53754],{"className":53753},[78],[34,53755,424],{"className":53756},[78],[34,53758,53759,53762],{"style":12897},[34,53760],{"className":53761,"style":12709},[829],[34,53763],{"className":53764,"style":12904},[12836],[34,53766,53767,53770],{"style":12907},[34,53768],{"className":53769,"style":12709},[829],[34,53771,53773],{"className":53772},[78],[34,53774,12569],{"className":53775,"style":143},[78,79],[34,53777,2547],{"className":53778},[2546],[34,53780,53782],{"className":53781},[818],[34,53783,53785],{"className":53784,"style":14669},[822],[34,53786],{},[34,53788],{"className":53789},[258,12689],[16,53791,53792],{},"Using:",[34,53794,53796],{"className":53795,"translate":38},[159],[34,53797,53799,53845],{"className":53798,"translate":38},[37],[34,53800,53802],{"className":53801},[42],[44,53803,53804],{"xmlns":46,"display":168},[48,53805,53806,53842],{},[51,53807,53808,53810,53812,53814,53816,53826],{},[54,53809,3000],{},[176,53811,34434],{"stretchy":178},[176,53813,14925],{},[54,53815,11468],{},[755,53817,53818,53820],{},[54,53819,18036],{},[51,53821,53822,53824],{},[176,53823,6463],{},[54,53825,3000],{},[755,53827,53828,53830],{},[54,53829,3000],{},[51,53831,53832,53834,53836],{},[54,53833,3000],{},[176,53835,184],{},[12543,53837,53838,53840],{},[347,53839,2213],{},[347,53841,424],{},[58,53843,53844],{"encoding":60},"n! \\sim a e^{-n} n^{n + \\frac{1}{2}}",[34,53846,53848,53869],{"className":53847,"ariaHidden":65},[64],[34,53849,53851,53854,53857,53860,53863,53866],{"className":53850},[69],[34,53852],{"className":53853,"style":2184},[73],[34,53855,3000],{"className":53856},[78,79],[34,53858,34434],{"className":53859},[258],[34,53861],{"className":53862,"style":262},[237],[34,53864,14925],{"className":53865},[266],[34,53867],{"className":53868,"style":262},[237],[34,53870,53872,53876,53879,53915],{"className":53871},[69],[34,53873],{"className":53874,"style":53875},[73],"height:1.004em;",[34,53877,11468],{"className":53878},[78,79],[34,53880,53882,53885],{"className":53881},[78],[34,53883,18036],{"className":53884},[78,79],[34,53886,53888],{"className":53887},[810],[34,53889,53891],{"className":53890},[814],[34,53892,53894],{"className":53893},[818],[34,53895,53898],{"className":53896,"style":53897},[822],"height:0.8213em;",[34,53899,53900,53903],{"style":4710},[34,53901],{"className":53902,"style":830},[829],[34,53904,53906],{"className":53905},[834,835,836,837],[34,53907,53909,53912],{"className":53908},[78,837],[34,53910,6463],{"className":53911},[78,837],[34,53913,3000],{"className":53914},[78,79,837],[34,53916,53918,53921],{"className":53917},[78],[34,53919,3000],{"className":53920},[78,79],[34,53922,53924],{"className":53923},[810],[34,53925,53927],{"className":53926},[814],[34,53928,53930],{"className":53929},[818],[34,53931,53933],{"className":53932,"style":53875},[822],[34,53934,53935,53938],{"style":43283},[34,53936],{"className":53937,"style":12709},[829],[34,53939,53941],{"className":53940},[834,835,836,837],[34,53942,53944,53947,53950],{"className":53943},[78,837],[34,53945,3000],{"className":53946},[78,79,837],[34,53948,184],{"className":53949},[242,837],[34,53951,53953,53956,54015],{"className":53952},[78,837],[34,53954],{"className":53955},[230,12689,834,10776,12798],[34,53957,53959],{"className":53958},[12543],[34,53960,53962,54007],{"className":53961},[814,2523],[34,53963,53965,54004],{"className":53964},[818],[34,53966,53968,53982,53990],{"className":53967,"style":42676},[822],[34,53969,53970,53973],{"style":12814},[34,53971],{"className":53972,"style":12709},[829],[34,53974,53976],{"className":53975},[834,10776,9740,837],[34,53977,53979],{"className":53978},[78,837],[34,53980,424],{"className":53981},[78,837],[34,53983,53984,53987],{"style":12829},[34,53985],{"className":53986,"style":12709},[829],[34,53988],{"className":53989,"style":12837},[12836,837],[34,53991,53992,53995],{"style":12840},[34,53993],{"className":53994,"style":12709},[829],[34,53996,53998],{"className":53997},[834,10776,9740,837],[34,53999,54001],{"className":54000},[78,837],[34,54002,2213],{"className":54003},[78,837],[34,54005,2547],{"className":54006},[2546],[34,54008,54010],{"className":54009},[818],[34,54011,54013],{"className":54012,"style":12862},[822],[34,54014],{},[34,54016],{"className":54017},[258,12689,834,10776,12798],[16,54019,54020],{},"Substitute into:",[34,54022,54024],{"className":54023,"translate":38},[159],[34,54025,54027,54117],{"className":54026,"translate":38},[37],[34,54028,54030],{"className":54029},[42],[44,54031,54032],{"xmlns":46,"display":168},[48,54033,54034,54114],{},[51,54035,54036,54052,54106,54108],{},[15798,54037,54038,54044],{},[51,54039,54040,54042],{},[54,54041,37329],{},[176,54043,6321],{},[51,54045,54046,54048,54050],{},[54,54047,3000],{},[176,54049,3003],{},[54,54051,13314],{"mathvariant":2296},[12543,54053,54054,54068],{},[51,54055,54056,54062],{},[755,54057,54058,54060],{},[54,54059,11468],{},[347,54061,424],{},[755,54063,54064,54066],{},[54,54065,18036],{},[347,54067,424],{},[51,54069,54070,54072],{},[347,54071,990],{},[755,54073,54074,54092],{},[51,54075,54076,54078,54090],{},[176,54077,179],{"fence":65},[12543,54079,54080,54088],{},[51,54081,54082,54084,54086],{},[54,54083,3000],{},[176,54085,184],{},[347,54087,2213],{},[54,54089,3000],{},[176,54091,189],{"fence":65},[51,54093,54094,54096,54098,54100],{},[347,54095,424],{},[54,54097,3000],{},[176,54099,184],{},[12543,54101,54102,54104],{},[347,54103,429],{},[347,54105,424],{},[176,54107,192],{},[12543,54109,54110,54112],{},[54,54111,12569],{},[347,54113,424],{},[58,54115,54116],{"encoding":60},"\\lim_{n \\rightarrow \\infty} \\frac{a^2 e^2}{4 \\left(\\frac{n+1}{n}\\right)^{2n + \\frac{3}{2}}} = \\frac{\\pi}{2}",[34,54118,54120,54504],{"className":54119,"ariaHidden":65},[64],[34,54121,54123,54127,54180,54183,54495,54498,54501],{"className":54122},[69],[34,54124],{"className":54125,"style":54126},[73],"height:2.925em;vertical-align:-1.4339em;",[34,54128,54130],{"className":54129},[3387,3388],[34,54131,54133,54172],{"className":54132},[814,2523],[34,54134,54136,54169],{"className":54135},[818],[34,54137,54139,54159],{"className":54138,"style":2184},[822],[34,54140,54141,54144],{"style":42479},[34,54142],{"className":54143,"style":12709},[829],[34,54145,54147],{"className":54146},[834,835,836,837],[34,54148,54150,54153,54156],{"className":54149},[78,837],[34,54151,3000],{"className":54152},[78,79,837],[34,54154,3003],{"className":54155},[266,837],[34,54157,13314],{"className":54158},[78,837],[34,54160,54161,54164],{"style":23024},[34,54162],{"className":54163,"style":12709},[829],[34,54165,54166],{},[34,54167,37329],{"className":54168},[3387],[34,54170,2547],{"className":54171},[2546],[34,54173,54175],{"className":54174},[818],[34,54176,54178],{"className":54177,"style":42517},[822],[34,54179],{},[34,54181],{"className":54182,"style":3462},[237],[34,54184,54186,54189,54492],{"className":54185},[78],[34,54187],{"className":54188},[230,12689],[34,54190,54192],{"className":54191},[12543],[34,54193,54195,54484],{"className":54194},[814,2523],[34,54196,54198,54481],{"className":54197},[818],[34,54199,54201,54407,54415],{"className":54200,"style":15228},[822],[34,54202,54203,54206],{"style":48064},[34,54204],{"className":54205,"style":48068},[829],[34,54207,54209,54212,54215],{"className":54208},[78],[34,54210,990],{"className":54211},[78],[34,54213],{"className":54214,"style":3462},[237],[34,54216,54218,54307],{"className":54217},[4177],[34,54219,54221,54227,54301],{"className":54220},[4177],[34,54222,54224],{"className":54223,"style":4182},[230,4181],[34,54225,179],{"className":54226},[4186,9740],[34,54228,54230,54233,54298],{"className":54229},[78],[34,54231],{"className":54232},[230,12689],[34,54234,54236],{"className":54235},[12543],[34,54237,54239,54290],{"className":54238},[814,2523],[34,54240,54242,54287],{"className":54241},[818],[34,54243,54245,54259,54267],{"className":54244,"style":46368},[822],[34,54246,54247,54250],{"style":17906},[34,54248],{"className":54249,"style":12709},[829],[34,54251,54253],{"className":54252},[834,835,836,837],[34,54254,54256],{"className":54255},[78,837],[34,54257,3000],{"className":54258},[78,79,837],[34,54260,54261,54264],{"style":12897},[34,54262],{"className":54263,"style":12709},[829],[34,54265],{"className":54266,"style":12904},[12836],[34,54268,54269,54272],{"style":17929},[34,54270],{"className":54271,"style":12709},[829],[34,54273,54275],{"className":54274},[834,835,836,837],[34,54276,54278,54281,54284],{"className":54277},[78,837],[34,54279,3000],{"className":54280},[78,79,837],[34,54282,184],{"className":54283},[242,837],[34,54285,2213],{"className":54286},[78,837],[34,54288,2547],{"className":54289},[2546],[34,54291,54293],{"className":54292},[818],[34,54294,54296],{"className":54295,"style":17973},[822],[34,54297],{},[34,54299],{"className":54300},[258,12689],[34,54302,54304],{"className":54303,"style":4182},[258,4181],[34,54305,189],{"className":54306},[4186,9740],[34,54308,54310],{"className":54309},[810],[34,54311,54313],{"className":54312},[814],[34,54314,54316],{"className":54315},[818],[34,54317,54319],{"className":54318,"style":48182},[822],[34,54320,54321,54324],{"style":44875},[34,54322],{"className":54323,"style":12709},[829],[34,54325,54327],{"className":54326},[834,835,836,837],[34,54328,54330,54333,54336,54339],{"className":54329},[78,837],[34,54331,424],{"className":54332},[78,837],[34,54334,3000],{"className":54335},[78,79,837],[34,54337,184],{"className":54338},[242,837],[34,54340,54342,54345,54404],{"className":54341},[78,837],[34,54343],{"className":54344},[230,12689,834,10776,12798],[34,54346,54348],{"className":54347},[12543],[34,54349,54351,54396],{"className":54350},[814,2523],[34,54352,54354,54393],{"className":54353},[818],[34,54355,54357,54371,54379],{"className":54356,"style":42676},[822],[34,54358,54359,54362],{"style":12814},[34,54360],{"className":54361,"style":12709},[829],[34,54363,54365],{"className":54364},[834,10776,9740,837],[34,54366,54368],{"className":54367},[78,837],[34,54369,424],{"className":54370},[78,837],[34,54372,54373,54376],{"style":12829},[34,54374],{"className":54375,"style":12709},[829],[34,54377],{"className":54378,"style":12837},[12836,837],[34,54380,54381,54384],{"style":12840},[34,54382],{"className":54383,"style":12709},[829],[34,54385,54387],{"className":54386},[834,10776,9740,837],[34,54388,54390],{"className":54389},[78,837],[34,54391,429],{"className":54392},[78,837],[34,54394,2547],{"className":54395},[2546],[34,54397,54399],{"className":54398},[818],[34,54400,54402],{"className":54401,"style":12862},[822],[34,54403],{},[34,54405],{"className":54406},[258,12689,834,10776,12798],[34,54408,54409,54412],{"style":48270},[34,54410],{"className":54411,"style":48068},[829],[34,54413],{"className":54414,"style":12904},[12836],[34,54416,54417,54420],{"style":48279},[34,54418],{"className":54419,"style":48068},[829],[34,54421,54423,54452],{"className":54422},[78],[34,54424,54426,54429],{"className":54425},[78],[34,54427,11468],{"className":54428},[78,79],[34,54430,54432],{"className":54431},[810],[34,54433,54435],{"className":54434},[814],[34,54436,54438],{"className":54437},[818],[34,54439,54441],{"className":54440,"style":800},[822],[34,54442,54443,54446],{"style":825},[34,54444],{"className":54445,"style":830},[829],[34,54447,54449],{"className":54448},[834,835,836,837],[34,54450,424],{"className":54451},[78,837],[34,54453,54455,54458],{"className":54454},[78],[34,54456,18036],{"className":54457},[78,79],[34,54459,54461],{"className":54460},[810],[34,54462,54464],{"className":54463},[814],[34,54465,54467],{"className":54466},[818],[34,54468,54470],{"className":54469,"style":800},[822],[34,54471,54472,54475],{"style":825},[34,54473],{"className":54474,"style":830},[829],[34,54476,54478],{"className":54477},[834,835,836,837],[34,54479,424],{"className":54480},[78,837],[34,54482,2547],{"className":54483},[2546],[34,54485,54487],{"className":54486},[818],[34,54488,54490],{"className":54489,"style":48298},[822],[34,54491],{},[34,54493],{"className":54494},[258,12689],[34,54496],{"className":54497,"style":262},[237],[34,54499,192],{"className":54500},[266],[34,54502],{"className":54503,"style":262},[237],[34,54505,54507,54510],{"className":54506},[69],[34,54508],{"className":54509,"style":26739},[73],[34,54511,54513,54516,54569],{"className":54512},[78],[34,54514],{"className":54515},[230,12689],[34,54517,54519],{"className":54518},[12543],[34,54520,54522,54561],{"className":54521},[814,2523],[34,54523,54525,54558],{"className":54524},[818],[34,54526,54528,54539,54547],{"className":54527,"style":26758},[822],[34,54529,54530,54533],{"style":14155},[34,54531],{"className":54532,"style":12709},[829],[34,54534,54536],{"className":54535},[78],[34,54537,424],{"className":54538},[78],[34,54540,54541,54544],{"style":12897},[34,54542],{"className":54543,"style":12709},[829],[34,54545],{"className":54546,"style":12904},[12836],[34,54548,54549,54552],{"style":12907},[34,54550],{"className":54551,"style":12709},[829],[34,54553,54555],{"className":54554},[78],[34,54556,12569],{"className":54557,"style":143},[78,79],[34,54559,2547],{"className":54560},[2546],[34,54562,54564],{"className":54563},[818],[34,54565,54567],{"className":54566,"style":14669},[822],[34,54568],{},[34,54570],{"className":54571},[258,12689],[16,54573,54574],{},"Solve for:",[34,54576,54578],{"className":54577,"translate":38},[159],[34,54579,54581,54605],{"className":54580,"translate":38},[37],[34,54582,54584],{"className":54583},[42],[44,54585,54586],{"xmlns":46,"display":168},[48,54587,54588,54602],{},[51,54589,54590,54592,54594],{},[54,54591,11468],{},[176,54593,192],{},[19376,54595,54596],{},[51,54597,54598,54600],{},[347,54599,424],{},[54,54601,12569],{},[58,54603,54604],{"encoding":60},"a = \\sqrt{2 \\pi}",[34,54606,54608,54626],{"className":54607,"ariaHidden":65},[64],[34,54609,54611,54614,54617,54620,54623],{"className":54610},[69],[34,54612],{"className":54613,"style":108},[73],[34,54615,11468],{"className":54616},[78,79],[34,54618],{"className":54619,"style":262},[237],[34,54621,192],{"className":54622},[266],[34,54624],{"className":54625,"style":262},[237],[34,54627,54629,54633],{"className":54628},[69],[34,54630],{"className":54631,"style":54632},[73],"height:1.04em;vertical-align:-0.0839em;",[34,54634,54636],{"className":54635},[78,19464],[34,54637,54639,54677],{"className":54638},[814,2523],[34,54640,54642,54674],{"className":54641},[818],[34,54643,54646,54661],{"className":54644,"style":54645},[822],"height:0.9561em;",[34,54647,54649,54652],{"className":54648,"style":23024},[10795],[34,54650],{"className":54651,"style":12709},[829],[34,54653,54655,54658],{"className":54654,"style":42229},[78],[34,54656,424],{"className":54657},[78],[34,54659,12569],{"className":54660,"style":143},[78,79],[34,54662,54664,54667],{"style":54663},"top:-2.9161em;",[34,54665],{"className":54666,"style":12709},[829],[34,54668,54670],{"className":54669,"style":42248},[10803],[10806,54671,54672],{"xmlns":10808,"width":10809,"height":42251,"viewBox":42252,"preserveAspectRatio":19563},[10814,54673],{"d":42255},[34,54675,2547],{"className":54676},[2546],[34,54678,54680],{"className":54679},[818],[34,54681,54684],{"className":54682,"style":54683},[822],"height:0.0839em;",[34,54685],{},[16,54687,54688,54689,54717],{},"Thus, as ",[34,54690,54692,54705],{"className":54691,"translate":38},[37],[34,54693,54695],{"className":54694},[42],[44,54696,54697],{"xmlns":46},[48,54698,54699,54703],{},[51,54700,54701],{},[54,54702,3000],{},[58,54704,3000],{"encoding":60},[34,54706,54708],{"className":54707,"ariaHidden":65},[64],[34,54709,54711,54714],{"className":54710},[69],[34,54712],{"className":54713,"style":108},[73],[34,54715,3000],{"className":54716},[78,79]," approaches infinity:",[34,54719,54721],{"className":54720,"translate":38},[159],[34,54722,54724,54776],{"className":54723,"translate":38},[37],[34,54725,54727],{"className":54726},[42],[44,54728,54729],{"xmlns":46,"display":168},[48,54730,54731,54773],{},[51,54732,54733,54735,54737,54739,54747,54757],{},[54,54734,3000],{},[176,54736,34434],{"stretchy":178},[176,54738,14925],{},[19376,54740,54741],{},[51,54742,54743,54745],{},[347,54744,424],{},[54,54746,12569],{},[755,54748,54749,54751],{},[54,54750,18036],{},[51,54752,54753,54755],{},[176,54754,6463],{},[54,54756,3000],{},[755,54758,54759,54761],{},[54,54760,3000],{},[51,54762,54763,54765,54767],{},[54,54764,3000],{},[176,54766,184],{},[12543,54768,54769,54771],{},[347,54770,2213],{},[347,54772,424],{},[58,54774,54775],{"encoding":60},"n! \\sim \\sqrt{2 \\pi} e^{-n} n^{n + \\frac{1}{2}}",[34,54777,54779,54800],{"className":54778,"ariaHidden":65},[64],[34,54780,54782,54785,54788,54791,54794,54797],{"className":54781},[69],[34,54783],{"className":54784,"style":2184},[73],[34,54786,3000],{"className":54787},[78,79],[34,54789,34434],{"className":54790},[258],[34,54792],{"className":54793,"style":262},[237],[34,54795,14925],{"className":54796},[266],[34,54798],{"className":54799,"style":262},[237],[34,54801,54803,54807,54857,54892],{"className":54802},[69],[34,54804],{"className":54805,"style":54806},[73],"height:1.0879em;vertical-align:-0.0839em;",[34,54808,54810],{"className":54809},[78,19464],[34,54811,54813,54849],{"className":54812},[814,2523],[34,54814,54816,54846],{"className":54815},[818],[34,54817,54819,54834],{"className":54818,"style":54645},[822],[34,54820,54822,54825],{"className":54821,"style":23024},[10795],[34,54823],{"className":54824,"style":12709},[829],[34,54826,54828,54831],{"className":54827,"style":42229},[78],[34,54829,424],{"className":54830},[78],[34,54832,12569],{"className":54833,"style":143},[78,79],[34,54835,54836,54839],{"style":54663},[34,54837],{"className":54838,"style":12709},[829],[34,54840,54842],{"className":54841,"style":42248},[10803],[10806,54843,54844],{"xmlns":10808,"width":10809,"height":42251,"viewBox":42252,"preserveAspectRatio":19563},[10814,54845],{"d":42255},[34,54847,2547],{"className":54848},[2546],[34,54850,54852],{"className":54851},[818],[34,54853,54855],{"className":54854,"style":54683},[822],[34,54856],{},[34,54858,54860,54863],{"className":54859},[78],[34,54861,18036],{"className":54862},[78,79],[34,54864,54866],{"className":54865},[810],[34,54867,54869],{"className":54868},[814],[34,54870,54872],{"className":54871},[818],[34,54873,54875],{"className":54874,"style":53897},[822],[34,54876,54877,54880],{"style":4710},[34,54878],{"className":54879,"style":830},[829],[34,54881,54883],{"className":54882},[834,835,836,837],[34,54884,54886,54889],{"className":54885},[78,837],[34,54887,6463],{"className":54888},[78,837],[34,54890,3000],{"className":54891},[78,79,837],[34,54893,54895,54898],{"className":54894},[78],[34,54896,3000],{"className":54897},[78,79],[34,54899,54901],{"className":54900},[810],[34,54902,54904],{"className":54903},[814],[34,54905,54907],{"className":54906},[818],[34,54908,54910],{"className":54909,"style":53875},[822],[34,54911,54912,54915],{"style":43283},[34,54913],{"className":54914,"style":12709},[829],[34,54916,54918],{"className":54917},[834,835,836,837],[34,54919,54921,54924,54927],{"className":54920},[78,837],[34,54922,3000],{"className":54923},[78,79,837],[34,54925,184],{"className":54926},[242,837],[34,54928,54930,54933,54992],{"className":54929},[78,837],[34,54931],{"className":54932},[230,12689,834,10776,12798],[34,54934,54936],{"className":54935},[12543],[34,54937,54939,54984],{"className":54938},[814,2523],[34,54940,54942,54981],{"className":54941},[818],[34,54943,54945,54959,54967],{"className":54944,"style":42676},[822],[34,54946,54947,54950],{"style":12814},[34,54948],{"className":54949,"style":12709},[829],[34,54951,54953],{"className":54952},[834,10776,9740,837],[34,54954,54956],{"className":54955},[78,837],[34,54957,424],{"className":54958},[78,837],[34,54960,54961,54964],{"style":12829},[34,54962],{"className":54963,"style":12709},[829],[34,54965],{"className":54966,"style":12837},[12836,837],[34,54968,54969,54972],{"style":12840},[34,54970],{"className":54971,"style":12709},[829],[34,54973,54975],{"className":54974},[834,10776,9740,837],[34,54976,54978],{"className":54977},[78,837],[34,54979,2213],{"className":54980},[78,837],[34,54982,2547],{"className":54983},[2546],[34,54985,54987],{"className":54986},[818],[34,54988,54990],{"className":54989,"style":12862},[822],[34,54991],{},[34,54993],{"className":54994},[258,12689,834,10776,12798],[16,54996,54997],{},"or equivalently:",[34,54999,55001],{"className":55000,"translate":38},[159],[34,55002,55004,55048],{"className":55003,"translate":38},[37],[34,55005,55007],{"className":55006},[42],[44,55008,55009],{"xmlns":46,"display":168},[48,55010,55011,55045],{},[51,55012,55013,55015,55017,55019,55029],{},[54,55014,3000],{},[176,55016,34434],{"stretchy":178},[176,55018,14925],{},[19376,55020,55021],{},[51,55022,55023,55025,55027],{},[347,55024,424],{},[54,55026,12569],{},[54,55028,3000],{},[755,55030,55031,55043],{},[51,55032,55033,55035,55041],{},[176,55034,179],{"fence":65},[12543,55036,55037,55039],{},[54,55038,3000],{},[54,55040,18036],{},[176,55042,189],{"fence":65},[54,55044,3000],{},[58,55046,55047],{"encoding":60},"n! \\sim \\sqrt{2 \\pi n} 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formula finds a relationship of n!,nⁿ,eⁿ while n trends to infinity.",[11695],"btl6ousWJKFfaDjpk8J08ikbFDtBilVoN5eqz0FmTv0",1786880524993]