[{"data":1,"prerenderedAt":26379},["ShallowReactive",2],{"lesson:\u002Fdeep-learning\u002Foptimization\u002Fsecond-order-and-approximate-methods":3,"course-wordcounts":20603,"ref-card-index":21515,"nav:deep-learning":26197,"tikz:8d5cd4e8319babc7459cb8039aecb4097f1aba31b5d34e737fcf59e5c2d66473":26371,"tikz:96d1763be284bab4a09203750480d18dafdfcd6b8becb9f18cdeefd3a054e291":26372,"tikz:8afa14a91a39bc0661176a03286d5ff9783acf58518d74622ef31943de1e84f5":26373,"tikz:a8162f396adc7234bf26a23bfb0ee4230653ac39f0c6a7f77f1ec144050a0e30":26374,"tikz:83cca56c424b2db323f5fb3ca68d89bdea67d712bcbd477359ef7cd205e468bc":26375,"tikz:29df4f79bc50e39b6c5dfdadf89bc2c66ae6736f93be34f049afd1dbdcf013ce":26376,"tikz:459393d9d992ed0270d9e66f8c9b22ccf5950f5a4cebc050f02d94e2573ab031":26377,"tikz:ecb097e5a997f4fb92411b09e494c18a871afaf39781e83a42d521404592a578":26378},{"id":4,"title":5,"blurb":6,"body":7,"brief":20572,"category":20573,"description":20574,"draft":20575,"extension":20576,"meta":20577,"module":20579,"navigation":20580,"path":20581,"practice":20582,"rawbody":20583,"readingTime":20584,"seo":20589,"sources":20590,"status":20598,"stem":20599,"summary":20600,"topics":20601,"__hash__":20602},"course\u002F06.deep-learning\u002F03.optimization\u002F05.second-order-and-approximate-methods.md","Second-Order & Approximate Methods","",{"type":8,"value":9,"toc":20551},"minimark",[10,270,275,379,782,839,1246,1253,1506,1510,1708,1852,1855,2362,2690,3131,4344,4696,5209,5369,5968,5972,5975,6014,6163,6166,6169,6840,6907,6911,7115,7333,7529,8052,8055,8285,8289,8480,8751,8773,8777,8807,9279,9633,9636,9712,9739,9743,9874,10223,10726,11647,11927,11930,12441,13424,13833,14441,14445,14741,14998,15270,15357,15815,15870,15874,15916,16026,16422,16489,16670,16792,17059,18244,18263,18353,18645,18924,18927,18941,18945,19040,19043,19047,19050,19342,19416,19420,19423,19470,19473,19477,20225,20547],[11,12,13,14,19,20,24,25,29,30,100,101,162,163,167,168,171,172,175,176,254,255,259,260],"p",{},"Every optimizer so far (",[15,16,18],"a",{"href":17},"\u002Fdeep-learning\u002Foptimization\u002Fgradient-descent-and-sgd","SGD",",\n",[15,21,23],{"href":22},"\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods","momentum, and the adaptive methods",")\nis ",[26,27,28],"strong",{},"first-order",": it sees only the gradient ",[31,32,35],"span",{"className":33},[34],"katex",[31,36,40,69],{"className":37,"ariaHidden":39},[38],"katex-html","true",[31,41,44,49,56,61,66],{"className":42},[43],"base",[31,45],{"className":46,"style":48},[47],"strut","height:0.625em;vertical-align:-0.1944em;",[31,50,55],{"className":51,"style":54},[52,53],"mord","mathnormal","margin-right:0.0359em;","g",[31,57],{"className":58,"style":60},[59],"mspace","margin-right:0.2778em;",[31,62,65],{"className":63},[64],"mrel","=",[31,67],{"className":68,"style":60},[59],[31,70,72,76,80,85,90,95],{"className":71},[43],[31,73],{"className":74,"style":75},[47],"height:1em;vertical-align:-0.25em;",[31,77,79],{"className":78},[52],"∇",[31,81,84],{"className":82},[52,83],"mathcal","L",[31,86,89],{"className":87},[88],"mopen","(",[31,91,94],{"className":92,"style":93},[52,53],"margin-right:0.0278em;","θ",[31,96,99],{"className":97},[98],"mclose",")"," and steps\n",[31,102,104],{"className":103},[34],[31,105,107,127,149],{"className":106,"ariaHidden":39},[38],[31,108,110,114,117,120,124],{"className":109},[43],[31,111],{"className":112,"style":113},[47],"height:0.6944em;",[31,115,94],{"className":116,"style":93},[52,53],[31,118],{"className":119,"style":60},[59],[31,121,123],{"className":122},[64],"←",[31,125],{"className":126,"style":60},[59],[31,128,130,134,137,141,146],{"className":129},[43],[31,131],{"className":132,"style":133},[47],"height:0.7778em;vertical-align:-0.0833em;",[31,135,94],{"className":136,"style":93},[52,53],[31,138],{"className":139,"style":140},[59],"margin-right:0.2222em;",[31,142,145],{"className":143},[144],"mbin","−",[31,147],{"className":148,"style":140},[59],[31,150,152,155,159],{"className":151},[43],[31,153],{"className":154,"style":48},[47],[31,156,158],{"className":157,"style":54},[52,53],"η",[31,160,55],{"className":161,"style":54},[52,53],". The gradient is a flat plane tangent to the loss; it\ngives the downhill ",[164,165,166],"em",{},"direction"," but nothing about how the surface ",[164,169,170],{},"curves",". A\n",[26,173,174],{},"second-order"," method also reads the curvature (the Hessian ",[31,177,179],{"className":178},[34],[31,180,182,203],{"className":181,"ariaHidden":39},[38],[31,183,185,189,194,197,200],{"className":184},[43],[31,186],{"className":187,"style":188},[47],"height:0.6833em;",[31,190,193],{"className":191,"style":192},[52,53],"margin-right:0.0813em;","H",[31,195],{"className":196,"style":60},[59],[31,198,65],{"className":199},[64],[31,201],{"className":202,"style":60},[59],[31,204,206,210,251],{"className":205},[43],[31,207],{"className":208,"style":209},[47],"height:0.8141em;",[31,211,213,216],{"className":212},[52],[31,214,79],{"className":215},[52],[31,217,220],{"className":218},[219],"msupsub",[31,221,224],{"className":222},[223],"vlist-t",[31,225,228],{"className":226},[227],"vlist-r",[31,229,232],{"className":230,"style":209},[231],"vlist",[31,233,235,240],{"style":234},"top:-3.063em;margin-right:0.05em;",[31,236],{"className":237,"style":239},[238],"pstrut","height:2.7em;",[31,241,247],{"className":242},[243,244,245,246],"sizing","reset-size6","size3","mtight",[31,248,250],{"className":249},[52,246],"2",[31,252,84],{"className":253},[52,83],")\nand uses it to choose both direction and step length at once. On the\n",[15,256,258],{"href":257},"\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape","ill-conditioned"," ravines that\nmake first-order descent crawl, that extra information is decisive. The problem is\nits cost, and most of this lesson is about approximations that capture a fraction\nof it cheaply.",[261,262,263],"sup",{},[15,264,269],{"href":265,"ariaDescribedBy":266,"dataFootnoteRef":6,"id":268},"#user-content-fn-gf-secondorder",[267],"footnote-label","user-content-fnref-gf-secondorder","1",[271,272,274],"h2",{"id":273},"newtons-method","Newton's method",[11,276,277,278,293,294,378],{},"Expand the loss to second order around the current iterate ",[31,279,281],{"className":280},[34],[31,282,284],{"className":283,"ariaHidden":39},[38],[31,285,287,290],{"className":286},[43],[31,288],{"className":289,"style":113},[47],[31,291,94],{"className":292,"style":93},[52,53],". Writing the step\nas ",[31,295,297],{"className":296},[34],[31,298,300,319,369],{"className":299,"ariaHidden":39},[38],[31,301,303,306,310,313,316],{"className":302},[43],[31,304],{"className":305,"style":188},[47],[31,307,309],{"className":308},[52],"Δ",[31,311],{"className":312,"style":60},[59],[31,314,65],{"className":315},[64],[31,317],{"className":318,"style":60},[59],[31,320,322,326,360,363,366],{"className":321},[43],[31,323],{"className":324,"style":325},[47],"height:0.8352em;vertical-align:-0.0833em;",[31,327,329,332],{"className":328},[52],[31,330,94],{"className":331,"style":93},[52,53],[31,333,335],{"className":334},[219],[31,336,338],{"className":337},[223],[31,339,341],{"className":340},[227],[31,342,345],{"className":343,"style":344},[231],"height:0.7519em;",[31,346,347,350],{"style":234},[31,348],{"className":349,"style":239},[238],[31,351,353],{"className":352},[243,244,245,246],[31,354,356],{"className":355},[52,246],[31,357,359],{"className":358},[52,246],"′",[31,361],{"className":362,"style":140},[59],[31,364,145],{"className":365},[144],[31,367],{"className":368,"style":140},[59],[31,370,372,375],{"className":371},[43],[31,373],{"className":374,"style":113},[47],[31,376,94],{"className":377,"style":93},[52,53],", Taylor's theorem gives the local quadratic model",[31,380,383],{"className":381},[382],"katex-display",[31,384,386],{"className":385},[34],[31,387,389,414,442,475,532,686,729],{"className":388,"ariaHidden":39},[38],[31,390,392,395,398,401,404,407,411],{"className":391},[43],[31,393],{"className":394,"style":75},[47],[31,396,84],{"className":397},[52,83],[31,399,89],{"className":400},[88],[31,402,94],{"className":403,"style":93},[52,53],[31,405],{"className":406,"style":140},[59],[31,408,410],{"className":409},[144],"+",[31,412],{"className":413,"style":140},[59],[31,415,417,420,423,426,429,432,436,439],{"className":416},[43],[31,418],{"className":419,"style":75},[47],[31,421,309],{"className":422},[52],[31,424,99],{"className":425},[98],[31,427],{"className":428,"style":60},[59],[31,430],{"className":431,"style":60},[59],[31,433,435],{"className":434},[64],"≈",[31,437],{"className":438,"style":60},[59],[31,440],{"className":441,"style":60},[59],[31,443,445,448,451,454,457,460,463,466,469,472],{"className":444},[43],[31,446],{"className":447,"style":75},[47],[31,449,84],{"className":450},[52,83],[31,452,89],{"className":453},[88],[31,455,94],{"className":456,"style":93},[52,53],[31,458,99],{"className":459},[98],[31,461],{"className":462,"style":60},[59],[31,464],{"className":465,"style":140},[59],[31,467,410],{"className":468},[144],[31,470],{"className":471,"style":60},[59],[31,473],{"className":474,"style":140},[59],[31,476,478,482,514,517,520,523,526,529],{"className":477},[43],[31,479],{"className":480,"style":481},[47],"height:1.0935em;vertical-align:-0.1944em;",[31,483,485,488],{"className":484},[52],[31,486,55],{"className":487,"style":54},[52,53],[31,489,491],{"className":490},[219],[31,492,494],{"className":493},[223],[31,495,497],{"className":496},[227],[31,498,501],{"className":499,"style":500},[231],"height:0.8991em;",[31,502,504,507],{"style":503},"top:-3.113em;margin-right:0.05em;",[31,505],{"className":506,"style":239},[238],[31,508,510],{"className":509},[243,244,245,246],[31,511,513],{"className":512},[52,246],"⊤",[31,515,309],{"className":516},[52],[31,518],{"className":519,"style":60},[59],[31,521],{"className":522,"style":140},[59],[31,524,410],{"className":525},[144],[31,527],{"className":528,"style":60},[59],[31,530],{"className":531,"style":140},[59],[31,533,535,539,620,624,653,656,659,662,667,671,674,677,680,683],{"className":534},[43],[31,536],{"className":537,"style":538},[47],"height:1.2441em;vertical-align:-0.345em;",[31,540,542,546,617],{"className":541},[52],[31,543],{"className":544},[88,545],"nulldelimiter",[31,547,550],{"className":548},[549],"mfrac",[31,551,554,608],{"className":552},[223,553],"vlist-t2",[31,555,557,603],{"className":556},[227],[31,558,561,577,588],{"className":559,"style":560},[231],"height:0.8451em;",[31,562,564,568],{"style":563},"top:-2.655em;",[31,565],{"className":566,"style":567},[238],"height:3em;",[31,569,571],{"className":570},[243,244,245,246],[31,572,574],{"className":573},[52,246],[31,575,250],{"className":576},[52,246],[31,578,580,583],{"style":579},"top:-3.23em;",[31,581],{"className":582,"style":567},[238],[31,584],{"className":585,"style":587},[586],"frac-line","border-bottom-width:0.04em;",[31,589,591,594],{"style":590},"top:-3.394em;",[31,592],{"className":593,"style":567},[238],[31,595,597],{"className":596},[243,244,245,246],[31,598,600],{"className":599},[52,246],[31,601,269],{"className":602},[52,246],[31,604,607],{"className":605},[606],"vlist-s","​",[31,609,611],{"className":610},[227],[31,612,615],{"className":613,"style":614},[231],"height:0.345em;",[31,616],{},[31,618],{"className":619},[98,545],[31,621],{"className":622,"style":623},[59],"margin-right:0.1667em;",[31,625,627,630],{"className":626},[52],[31,628,309],{"className":629},[52],[31,631,633],{"className":632},[219],[31,634,636],{"className":635},[223],[31,637,639],{"className":638},[227],[31,640,642],{"className":641,"style":500},[231],[31,643,644,647],{"style":503},[31,645],{"className":646,"style":239},[238],[31,648,650],{"className":649},[243,244,245,246],[31,651,513],{"className":652},[52,246],[31,654,193],{"className":655,"style":192},[52,53],[31,657],{"className":658,"style":623},[59],[31,660,309],{"className":661},[52],[31,663,666],{"className":664},[665],"mpunct",",",[31,668],{"className":669,"style":670},[59],"margin-right:2em;",[31,672],{"className":673,"style":623},[59],[31,675,55],{"className":676,"style":54},[52,53],[31,678],{"className":679,"style":60},[59],[31,681,65],{"className":682},[64],[31,684],{"className":685,"style":60},[59],[31,687,689,692,695,698,701,704,707,710,714,717,720,723,726],{"className":688},[43],[31,690],{"className":691,"style":75},[47],[31,693,79],{"className":694},[52],[31,696,84],{"className":697},[52,83],[31,699,89],{"className":700},[88],[31,702,94],{"className":703,"style":93},[52,53],[31,705,99],{"className":706},[98],[31,708,666],{"className":709},[665],[31,711],{"className":712,"style":713},[59],"margin-right:1em;",[31,715],{"className":716,"style":623},[59],[31,718,193],{"className":719,"style":192},[52,53],[31,721],{"className":722,"style":60},[59],[31,724,65],{"className":725},[64],[31,727],{"className":728,"style":60},[59],[31,730,732,736,766,769,772,775,778],{"className":731},[43],[31,733],{"className":734,"style":735},[47],"height:1.1141em;vertical-align:-0.25em;",[31,737,739,742],{"className":738},[52],[31,740,79],{"className":741},[52],[31,743,745],{"className":744},[219],[31,746,748],{"className":747},[223],[31,749,751],{"className":750},[227],[31,752,755],{"className":753,"style":754},[231],"height:0.8641em;",[31,756,757,760],{"style":503},[31,758],{"className":759,"style":239},[238],[31,761,763],{"className":762},[243,244,245,246],[31,764,250],{"className":765},[52,246],[31,767,84],{"className":768},[52,83],[31,770,89],{"className":771},[88],[31,773,94],{"className":774,"style":93},[52,53],[31,776,99],{"className":777},[98],[31,779,781],{"className":780},[52],".",[11,783,784,785,800,801,838],{},"This is a paraboloid in ",[31,786,788],{"className":787},[34],[31,789,791],{"className":790,"ariaHidden":39},[38],[31,792,794,797],{"className":793},[43],[31,795],{"className":796,"style":188},[47],[31,798,309],{"className":799},[52],". If ",[31,802,804],{"className":803},[34],[31,805,807,827],{"className":806,"ariaHidden":39},[38],[31,808,810,814,817,820,824],{"className":809},[43],[31,811],{"className":812,"style":813},[47],"height:0.7224em;vertical-align:-0.0391em;",[31,815,193],{"className":816,"style":192},[52,53],[31,818],{"className":819,"style":60},[59],[31,821,823],{"className":822},[64],"≻",[31,825],{"className":826,"style":60},[59],[31,828,830,834],{"className":829},[43],[31,831],{"className":832,"style":833},[47],"height:0.6444em;",[31,835,837],{"className":836},[52],"0"," it has a unique minimum, found by\nsetting the gradient of the right-hand side to zero:",[31,840,842],{"className":841},[382],[31,843,845],{"className":844},[34],[31,846,848,1080,1099,1120,1146,1192],{"className":847,"ariaHidden":39},[38],[31,849,851,855,898,902,905,1071,1074,1077],{"className":850},[43],[31,852],{"className":853,"style":854},[47],"height:1.2491em;vertical-align:-0.35em;",[31,856,858,861],{"className":857},[52],[31,859,79],{"className":860},[52],[31,862,864],{"className":863},[219],[31,865,867,889],{"className":866},[223,553],[31,868,870,886],{"className":869},[227],[31,871,874],{"className":872,"style":873},[231],"height:0.3283em;",[31,875,877,880],{"style":876},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[31,878],{"className":879,"style":239},[238],[31,881,883],{"className":882},[243,244,245,246],[31,884,309],{"className":885},[52,246],[31,887,607],{"className":888},[606],[31,890,892],{"className":891},[227],[31,893,896],{"className":894,"style":895},[231],"height:0.15em;",[31,897],{},[31,899],{"className":900,"style":901},[59],"margin-right:-0.1667em;",[31,903],{"className":904,"style":623},[59],[31,906,909,920,949,952,955,958,961,1029,1058,1061,1064],{"className":907},[908],"minner",[31,910,914],{"className":911,"style":913},[88,912],"delimcenter","top:0em;",[31,915,919],{"className":916},[917,918],"delimsizing","size1","[",[31,921,923,926],{"className":922},[52],[31,924,55],{"className":925,"style":54},[52,53],[31,927,929],{"className":928},[219],[31,930,932],{"className":931},[223],[31,933,935],{"className":934},[227],[31,936,938],{"className":937,"style":500},[231],[31,939,940,943],{"style":503},[31,941],{"className":942,"style":239},[238],[31,944,946],{"className":945},[243,244,245,246],[31,947,513],{"className":948},[52,246],[31,950,309],{"className":951},[52],[31,953],{"className":954,"style":140},[59],[31,956,410],{"className":957},[144],[31,959],{"className":960,"style":140},[59],[31,962,964,967,1026],{"className":963},[52],[31,965],{"className":966},[88,545],[31,968,970],{"className":969},[549],[31,971,973,1018],{"className":972},[223,553],[31,974,976,1015],{"className":975},[227],[31,977,979,993,1001],{"className":978,"style":560},[231],[31,980,981,984],{"style":563},[31,982],{"className":983,"style":567},[238],[31,985,987],{"className":986},[243,244,245,246],[31,988,990],{"className":989},[52,246],[31,991,250],{"className":992},[52,246],[31,994,995,998],{"style":579},[31,996],{"className":997,"style":567},[238],[31,999],{"className":1000,"style":587},[586],[31,1002,1003,1006],{"style":590},[31,1004],{"className":1005,"style":567},[238],[31,1007,1009],{"className":1008},[243,244,245,246],[31,1010,1012],{"className":1011},[52,246],[31,1013,269],{"className":1014},[52,246],[31,1016,607],{"className":1017},[606],[31,1019,1021],{"className":1020},[227],[31,1022,1024],{"className":1023,"style":614},[231],[31,1025],{},[31,1027],{"className":1028},[98,545],[31,1030,1032,1035],{"className":1031},[52],[31,1033,309],{"className":1034},[52],[31,1036,1038],{"className":1037},[219],[31,1039,1041],{"className":1040},[223],[31,1042,1044],{"className":1043},[227],[31,1045,1047],{"className":1046,"style":500},[231],[31,1048,1049,1052],{"style":503},[31,1050],{"className":1051,"style":239},[238],[31,1053,1055],{"className":1054},[243,244,245,246],[31,1056,513],{"className":1057},[52,246],[31,1059,193],{"className":1060,"style":192},[52,53],[31,1062,309],{"className":1063},[52],[31,1065,1067],{"className":1066,"style":913},[98,912],[31,1068,1070],{"className":1069},[917,918],"]",[31,1072],{"className":1073,"style":60},[59],[31,1075,65],{"className":1076},[64],[31,1078],{"className":1079,"style":60},[59],[31,1081,1083,1087,1090,1093,1096],{"className":1082},[43],[31,1084],{"className":1085,"style":1086},[47],"height:0.7778em;vertical-align:-0.1944em;",[31,1088,55],{"className":1089,"style":54},[52,53],[31,1091],{"className":1092,"style":140},[59],[31,1094,410],{"className":1095},[144],[31,1097],{"className":1098,"style":140},[59],[31,1100,1102,1105,1108,1111,1114,1117],{"className":1101},[43],[31,1103],{"className":1104,"style":188},[47],[31,1106,193],{"className":1107,"style":192},[52,53],[31,1109,309],{"className":1110},[52],[31,1112],{"className":1113,"style":60},[59],[31,1115,65],{"className":1116},[64],[31,1118],{"className":1119,"style":60},[59],[31,1121,1123,1127,1130,1133,1136,1140,1143],{"className":1122},[43],[31,1124],{"className":1125,"style":1126},[47],"height:0.6684em;vertical-align:-0.024em;",[31,1128,837],{"className":1129},[52],[31,1131],{"className":1132,"style":60},[59],[31,1134],{"className":1135,"style":60},[59],[31,1137,1139],{"className":1138},[64],"⟹",[31,1141],{"className":1142,"style":60},[59],[31,1144],{"className":1145,"style":60},[59],[31,1147,1149,1153,1183,1186,1189],{"className":1148},[43],[31,1150],{"className":1151,"style":1152},[47],"height:0.7387em;",[31,1154,1156,1159],{"className":1155},[52],[31,1157,309],{"className":1158},[52],[31,1160,1162],{"className":1161},[219],[31,1163,1165],{"className":1164},[223],[31,1166,1168],{"className":1167},[227],[31,1169,1171],{"className":1170,"style":1152},[231],[31,1172,1173,1176],{"style":503},[31,1174],{"className":1175,"style":239},[238],[31,1177,1179],{"className":1178},[243,244,245,246],[31,1180,1182],{"className":1181},[144,246],"⋆",[31,1184],{"className":1185,"style":60},[59],[31,1187,65],{"className":1188},[64],[31,1190],{"className":1191,"style":60},[59],[31,1193,1195,1199,1202,1205,1240,1243],{"className":1194},[43],[31,1196],{"className":1197,"style":1198},[47],"height:1.0585em;vertical-align:-0.1944em;",[31,1200,145],{"className":1201},[52],[31,1203],{"className":1204,"style":623},[59],[31,1206,1208,1211],{"className":1207},[52],[31,1209,193],{"className":1210,"style":192},[52,53],[31,1212,1214],{"className":1213},[219],[31,1215,1217],{"className":1216},[223],[31,1218,1220],{"className":1219},[227],[31,1221,1223],{"className":1222,"style":754},[231],[31,1224,1225,1228],{"style":503},[31,1226],{"className":1227,"style":239},[238],[31,1229,1231],{"className":1230},[243,244,245,246],[31,1232,1234,1237],{"className":1233},[52,246],[31,1235,145],{"className":1236},[52,246],[31,1238,269],{"className":1239},[52,246],[31,1241,55],{"className":1242,"style":54},[52,53],[31,1244,781],{"className":1245},[52],[11,1247,1248,1249,1252],{},"The minimizer of the quadratic is one step away, and that step is the ",[26,1250,1251],{},"Newton step",".\nTaking it as the update gives Newton's method.",[11,1254,1255,1256,1329,1330,1446,1447,1450,1451,1505],{},"The model being minimized is a bowl, not a plane. First-order descent fits the loss with\nits tangent plane ",[31,1257,1259],{"className":1258},[34],[31,1260,1262,1289],{"className":1261,"ariaHidden":39},[38],[31,1263,1265,1268,1271,1274,1277,1280,1283,1286],{"className":1264},[43],[31,1266],{"className":1267,"style":75},[47],[31,1269,84],{"className":1270},[52,83],[31,1272,89],{"className":1273},[88],[31,1275,94],{"className":1276,"style":93},[52,53],[31,1278,99],{"className":1279},[98],[31,1281],{"className":1282,"style":140},[59],[31,1284,410],{"className":1285},[144],[31,1287],{"className":1288,"style":140},[59],[31,1290,1292,1296,1326],{"className":1291},[43],[31,1293],{"className":1294,"style":1295},[47],"height:1.0435em;vertical-align:-0.1944em;",[31,1297,1299,1302],{"className":1298},[52],[31,1300,55],{"className":1301,"style":54},[52,53],[31,1303,1305],{"className":1304},[219],[31,1306,1308],{"className":1307},[223],[31,1309,1311],{"className":1310},[227],[31,1312,1315],{"className":1313,"style":1314},[231],"height:0.8491em;",[31,1316,1317,1320],{"style":234},[31,1318],{"className":1319,"style":239},[238],[31,1321,1323],{"className":1322},[243,244,245,246],[31,1324,513],{"className":1325},[52,246],[31,1327,309],{"className":1328},[52]," and can only report a direction of\ndecrease; the plane has no bottom, so the step length has to come from a separately tuned\nlearning rate. The second-order model adds the quadratic term ",[31,1331,1333],{"className":1332},[34],[31,1334,1336],{"className":1335,"ariaHidden":39},[38],[31,1337,1339,1343,1411,1440,1443],{"className":1338},[43],[31,1340],{"className":1341,"style":1342},[47],"height:1.1941em;vertical-align:-0.345em;",[31,1344,1346,1349,1408],{"className":1345},[52],[31,1347],{"className":1348},[88,545],[31,1350,1352],{"className":1351},[549],[31,1353,1355,1400],{"className":1354},[223,553],[31,1356,1358,1397],{"className":1357},[227],[31,1359,1361,1375,1383],{"className":1360,"style":560},[231],[31,1362,1363,1366],{"style":563},[31,1364],{"className":1365,"style":567},[238],[31,1367,1369],{"className":1368},[243,244,245,246],[31,1370,1372],{"className":1371},[52,246],[31,1373,250],{"className":1374},[52,246],[31,1376,1377,1380],{"style":579},[31,1378],{"className":1379,"style":567},[238],[31,1381],{"className":1382,"style":587},[586],[31,1384,1385,1388],{"style":590},[31,1386],{"className":1387,"style":567},[238],[31,1389,1391],{"className":1390},[243,244,245,246],[31,1392,1394],{"className":1393},[52,246],[31,1395,269],{"className":1396},[52,246],[31,1398,607],{"className":1399},[606],[31,1401,1403],{"className":1402},[227],[31,1404,1406],{"className":1405,"style":614},[231],[31,1407],{},[31,1409],{"className":1410},[98,545],[31,1412,1414,1417],{"className":1413},[52],[31,1415,309],{"className":1416},[52],[31,1418,1420],{"className":1419},[219],[31,1421,1423],{"className":1422},[223],[31,1424,1426],{"className":1425},[227],[31,1427,1429],{"className":1428,"style":1314},[231],[31,1430,1431,1434],{"style":234},[31,1432],{"className":1433,"style":239},[238],[31,1435,1437],{"className":1436},[243,244,245,246],[31,1438,513],{"className":1439},[52,246],[31,1441,193],{"className":1442,"style":192},[52,53],[31,1444,309],{"className":1445},[52],",\nwhich closes the surface into a paraboloid with a definite minimum. Minimizing that bowl\nfixes direction ",[164,1448,1449],{},"and"," length together, and ",[31,1452,1454],{"className":1453},[34],[31,1455,1457],{"className":1456,"ariaHidden":39},[38],[31,1458,1460,1464,1467,1502],{"className":1459},[43],[31,1461],{"className":1462,"style":1463},[47],"height:1.0085em;vertical-align:-0.1944em;",[31,1465,145],{"className":1466},[52],[31,1468,1470,1473],{"className":1469},[52],[31,1471,193],{"className":1472,"style":192},[52,53],[31,1474,1476],{"className":1475},[219],[31,1477,1479],{"className":1478},[223],[31,1480,1482],{"className":1481},[227],[31,1483,1485],{"className":1484,"style":209},[231],[31,1486,1487,1490],{"style":234},[31,1488],{"className":1489,"style":239},[238],[31,1491,1493],{"className":1492},[243,244,245,246],[31,1494,1496,1499],{"className":1495},[52,246],[31,1497,145],{"className":1498},[52,246],[31,1500,269],{"className":1501},[52,246],[31,1503,55],{"className":1504,"style":54},[52,53]," is the vector from the current point\nto its floor.",[1507,1508],"tikz-figure",{"hash":1509},"8d5cd4e8319babc7459cb8039aecb4097f1aba31b5d34e737fcf59e5c2d66473",[1511,1512,1514],"callout",{"type":1513},"definition",[11,1515,1516,1519,1520,1621,1622,1637,1638,1653,1654,1669,1670,1673,1674,1707],{},[26,1517,1518],{},"Definition (Newton's method)."," The second-order update\n",[31,1521,1523],{"className":1522},[34],[31,1524,1526,1550,1574],{"className":1525,"ariaHidden":39},[38],[31,1527,1529,1532,1535,1538,1541,1544,1547],{"className":1528},[43],[31,1530],{"className":1531,"style":113},[47],[31,1533,94],{"className":1534,"style":93},[52,53],[31,1536],{"className":1537,"style":60},[59],[31,1539],{"className":1540,"style":60},[59],[31,1542,123],{"className":1543},[64],[31,1545],{"className":1546,"style":60},[59],[31,1548],{"className":1549,"style":60},[59],[31,1551,1553,1556,1559,1562,1565,1568,1571],{"className":1552},[43],[31,1554],{"className":1555,"style":133},[47],[31,1557,94],{"className":1558,"style":93},[52,53],[31,1560],{"className":1561,"style":60},[59],[31,1563],{"className":1564,"style":140},[59],[31,1566,145],{"className":1567},[144],[31,1569],{"className":1570,"style":60},[59],[31,1572],{"className":1573,"style":140},[59],[31,1575,1577,1580,1615,1618],{"className":1576},[43],[31,1578],{"className":1579,"style":1463},[47],[31,1581,1583,1586],{"className":1582},[52],[31,1584,193],{"className":1585,"style":192},[52,53],[31,1587,1589],{"className":1588},[219],[31,1590,1592],{"className":1591},[223],[31,1593,1595],{"className":1594},[227],[31,1596,1598],{"className":1597,"style":209},[231],[31,1599,1600,1603],{"style":234},[31,1601],{"className":1602,"style":239},[238],[31,1604,1606],{"className":1605},[243,244,245,246],[31,1607,1609,1612],{"className":1608},[52,246],[31,1610,145],{"className":1611},[52,246],[31,1613,269],{"className":1614},[52,246],[31,1616,55],{"className":1617,"style":54},[52,53],[31,1619,666],{"className":1620},[665],"\nwhich moves directly to the minimum of the second-order Taylor model of ",[31,1623,1625],{"className":1624},[34],[31,1626,1628],{"className":1627,"ariaHidden":39},[38],[31,1629,1631,1634],{"className":1630},[43],[31,1632],{"className":1633,"style":188},[47],[31,1635,84],{"className":1636},[52,83],"\nat ",[31,1639,1641],{"className":1640},[34],[31,1642,1644],{"className":1643,"ariaHidden":39},[38],[31,1645,1647,1650],{"className":1646},[43],[31,1648],{"className":1649,"style":113},[47],[31,1651,94],{"className":1652,"style":93},[52,53],". When ",[31,1655,1657],{"className":1656},[34],[31,1658,1660],{"className":1659,"ariaHidden":39},[38],[31,1661,1663,1666],{"className":1662},[43],[31,1664],{"className":1665,"style":188},[47],[31,1667,84],{"className":1668},[52,83]," is ",[164,1671,1672],{},"exactly"," quadratic and ",[31,1675,1677],{"className":1676},[34],[31,1678,1680,1698],{"className":1679,"ariaHidden":39},[38],[31,1681,1683,1686,1689,1692,1695],{"className":1682},[43],[31,1684],{"className":1685,"style":813},[47],[31,1687,193],{"className":1688,"style":192},[52,53],[31,1690],{"className":1691,"style":60},[59],[31,1693,823],{"className":1694},[64],[31,1696],{"className":1697,"style":60},[59],[31,1699,1701,1704],{"className":1700},[43],[31,1702],{"className":1703,"style":833},[47],[31,1705,837],{"className":1706},[52],", it reaches the\ntrue minimum in a single step from any starting point.",[11,1709,1710,1711,1729,1730,1745,1746,1749,1750,1797,1798,1851],{},"The geometry explains the advantage. Gradient descent steps along ",[31,1712,1714],{"className":1713},[34],[31,1715,1717],{"className":1716,"ariaHidden":39},[38],[31,1718,1720,1723,1726],{"className":1719},[43],[31,1721],{"className":1722,"style":1086},[47],[31,1724,145],{"className":1725},[52],[31,1727,55],{"className":1728,"style":54},[52,53],", perpendicular to\nthe contours of ",[31,1731,1733],{"className":1732},[34],[31,1734,1736],{"className":1735,"ariaHidden":39},[38],[31,1737,1739,1742],{"className":1738},[43],[31,1740],{"className":1741,"style":188},[47],[31,1743,84],{"className":1744},[52,83]," ",[164,1747,1748],{},"as drawn in raw coordinates","; on a stretched bowl that\ndirection does not point at the minimum. Newton multiplies by ",[31,1751,1753],{"className":1752},[34],[31,1754,1756],{"className":1755,"ariaHidden":39},[38],[31,1757,1759,1762],{"className":1758},[43],[31,1760],{"className":1761,"style":209},[47],[31,1763,1765,1768],{"className":1764},[52],[31,1766,193],{"className":1767,"style":192},[52,53],[31,1769,1771],{"className":1770},[219],[31,1772,1774],{"className":1773},[223],[31,1775,1777],{"className":1776},[227],[31,1778,1780],{"className":1779,"style":209},[231],[31,1781,1782,1785],{"style":234},[31,1783],{"className":1784,"style":239},[238],[31,1786,1788],{"className":1787},[243,244,245,246],[31,1789,1791,1794],{"className":1790},[52,246],[31,1792,145],{"className":1793},[52,246],[31,1795,269],{"className":1796},[52,246],", which rescales\neach direction by its curvature (stretching the gently curved axes and shrinking the\nsteep ones), so that in the rescaled coordinates the bowl is round and ",[31,1799,1801],{"className":1800},[34],[31,1802,1804],{"className":1803,"ariaHidden":39},[38],[31,1805,1807,1810,1813,1848],{"className":1806},[43],[31,1808],{"className":1809,"style":1463},[47],[31,1811,145],{"className":1812},[52],[31,1814,1816,1819],{"className":1815},[52],[31,1817,193],{"className":1818,"style":192},[52,53],[31,1820,1822],{"className":1821},[219],[31,1823,1825],{"className":1824},[223],[31,1826,1828],{"className":1827},[227],[31,1829,1831],{"className":1830,"style":209},[231],[31,1832,1833,1836],{"style":234},[31,1834],{"className":1835,"style":239},[238],[31,1837,1839],{"className":1838},[243,244,245,246],[31,1840,1842,1845],{"className":1841},[52,246],[31,1843,145],{"className":1844},[52,246],[31,1846,269],{"className":1847},[52,246],[31,1849,55],{"className":1850,"style":54},[52,53]," aims\nstraight at the bottom.",[1507,1853],{"hash":1854},"96d1763be284bab4a09203750480d18dafdfcd6b8becb9f18cdeefd3a054e291",[11,1856,1857,1858,1873,1874,1745,1877,2020,2021,2028,2029,2274,2275,2361],{},"Because the rescaling cancels the eigenvalue spread of ",[31,1859,1861],{"className":1860},[34],[31,1862,1864],{"className":1863,"ariaHidden":39},[38],[31,1865,1867,1870],{"className":1866},[43],[31,1868],{"className":1869,"style":188},[47],[31,1871,193],{"className":1872,"style":192},[52,53],", Newton's method is\n",[26,1875,1876],{},"invariant to the condition number",[31,1878,1880],{"className":1879},[34],[31,1881,1883,1911],{"className":1882,"ariaHidden":39},[38],[31,1884,1886,1889,1893,1896,1899,1902,1905,1908],{"className":1885},[43],[31,1887],{"className":1888,"style":75},[47],[31,1890,1892],{"className":1891},[52,53],"κ",[31,1894,89],{"className":1895},[88],[31,1897,193],{"className":1898,"style":192},[52,53],[31,1900,99],{"className":1901},[98],[31,1903],{"className":1904,"style":60},[59],[31,1906,65],{"className":1907},[64],[31,1909],{"className":1910,"style":60},[59],[31,1912,1914,1917,1968,1972],{"className":1913},[43],[31,1915],{"className":1916,"style":75},[47],[31,1918,1920,1924],{"className":1919},[52],[31,1921,1923],{"className":1922},[52,53],"λ",[31,1925,1927],{"className":1926},[219],[31,1928,1930,1960],{"className":1929},[223,553],[31,1931,1933,1957],{"className":1932},[227],[31,1934,1937],{"className":1935,"style":1936},[231],"height:0.1514em;",[31,1938,1939,1942],{"style":876},[31,1940],{"className":1941,"style":239},[238],[31,1943,1945],{"className":1944},[243,244,245,246],[31,1946,1948],{"className":1947},[52,246],[31,1949,1952],{"className":1950},[1951,246],"mop",[31,1953,1956],{"className":1954},[52,1955,246],"mathrm","max",[31,1958,607],{"className":1959},[606],[31,1961,1963],{"className":1962},[227],[31,1964,1966],{"className":1965,"style":895},[231],[31,1967],{},[31,1969,1971],{"className":1970},[52],"\u002F",[31,1973,1975,1978],{"className":1974},[52],[31,1976,1923],{"className":1977},[52,53],[31,1979,1981],{"className":1980},[219],[31,1982,1984,2012],{"className":1983},[223,553],[31,1985,1987,2009],{"className":1986},[227],[31,1988,1991],{"className":1989,"style":1990},[231],"height:0.3175em;",[31,1992,1993,1996],{"style":876},[31,1994],{"className":1995,"style":239},[238],[31,1997,1999],{"className":1998},[243,244,245,246],[31,2000,2002],{"className":2001},[52,246],[31,2003,2005],{"className":2004},[1951,246],[31,2006,2008],{"className":2007},[52,1955,246],"min",[31,2010,607],{"className":2011},[606],[31,2013,2015],{"className":2014},[227],[31,2016,2018],{"className":2017,"style":895},[231],[31,2019],{},": a\nbowl stretched a thousand-to-one is solved as fast as a round one. This removes\nexactly the problem that momentum and adaptive\nrates only mitigate.",[261,2022,2023],{},[15,2024,250],{"href":2025,"ariaDescribedBy":2026,"dataFootnoteRef":6,"id":2027},"#user-content-fn-gf-hessian",[267],"user-content-fnref-gf-hessian"," To see the one-step claim algebraically, take an exact quadratic\n",[31,2030,2032],{"className":2031},[34],[31,2033,2035,2062,2152,2236],{"className":2034,"ariaHidden":39},[38],[31,2036,2038,2041,2044,2047,2050,2053,2056,2059],{"className":2037},[43],[31,2039],{"className":2040,"style":75},[47],[31,2042,84],{"className":2043},[52,83],[31,2045,89],{"className":2046},[88],[31,2048,94],{"className":2049,"style":93},[52,53],[31,2051,99],{"className":2052},[98],[31,2054],{"className":2055,"style":60},[59],[31,2057,65],{"className":2058},[64],[31,2060],{"className":2061,"style":60},[59],[31,2063,2065,2069,2137,2140,2143,2146,2149],{"className":2064},[43],[31,2066],{"className":2067,"style":2068},[47],"height:1.1901em;vertical-align:-0.345em;",[31,2070,2072,2075,2134],{"className":2071},[52],[31,2073],{"className":2074},[88,545],[31,2076,2078],{"className":2077},[549],[31,2079,2081,2126],{"className":2080},[223,553],[31,2082,2084,2123],{"className":2083},[227],[31,2085,2087,2101,2109],{"className":2086,"style":560},[231],[31,2088,2089,2092],{"style":563},[31,2090],{"className":2091,"style":567},[238],[31,2093,2095],{"className":2094},[243,244,245,246],[31,2096,2098],{"className":2097},[52,246],[31,2099,250],{"className":2100},[52,246],[31,2102,2103,2106],{"style":579},[31,2104],{"className":2105,"style":567},[238],[31,2107],{"className":2108,"style":587},[586],[31,2110,2111,2114],{"style":590},[31,2112],{"className":2113,"style":567},[238],[31,2115,2117],{"className":2116},[243,244,245,246],[31,2118,2120],{"className":2119},[52,246],[31,2121,269],{"className":2122},[52,246],[31,2124,607],{"className":2125},[606],[31,2127,2129],{"className":2128},[227],[31,2130,2132],{"className":2131,"style":614},[231],[31,2133],{},[31,2135],{"className":2136},[98,545],[31,2138,89],{"className":2139},[88],[31,2141,94],{"className":2142,"style":93},[52,53],[31,2144],{"className":2145,"style":140},[59],[31,2147,145],{"className":2148},[144],[31,2150],{"className":2151,"style":140},[59],[31,2153,2155,2159,2189,2218,2221,2224,2227,2230,2233],{"className":2154},[43],[31,2156],{"className":2157,"style":2158},[47],"height:1.0991em;vertical-align:-0.25em;",[31,2160,2162,2165],{"className":2161},[52],[31,2163,94],{"className":2164,"style":93},[52,53],[31,2166,2168],{"className":2167},[219],[31,2169,2171],{"className":2170},[223],[31,2172,2174],{"className":2173},[227],[31,2175,2178],{"className":2176,"style":2177},[231],"height:0.6887em;",[31,2179,2180,2183],{"style":234},[31,2181],{"className":2182,"style":239},[238],[31,2184,2186],{"className":2185},[243,244,245,246],[31,2187,1182],{"className":2188},[144,246],[31,2190,2192,2195],{"className":2191},[98],[31,2193,99],{"className":2194},[98],[31,2196,2198],{"className":2197},[219],[31,2199,2201],{"className":2200},[223],[31,2202,2204],{"className":2203},[227],[31,2205,2207],{"className":2206,"style":1314},[231],[31,2208,2209,2212],{"style":234},[31,2210],{"className":2211,"style":239},[238],[31,2213,2215],{"className":2214},[243,244,245,246],[31,2216,513],{"className":2217},[52,246],[31,2219,193],{"className":2220,"style":192},[52,53],[31,2222,89],{"className":2223},[88],[31,2225,94],{"className":2226,"style":93},[52,53],[31,2228],{"className":2229,"style":140},[59],[31,2231,145],{"className":2232},[144],[31,2234],{"className":2235,"style":140},[59],[31,2237,2239,2242,2271],{"className":2238},[43],[31,2240],{"className":2241,"style":75},[47],[31,2243,2245,2248],{"className":2244},[52],[31,2246,94],{"className":2247,"style":93},[52,53],[31,2249,2251],{"className":2250},[219],[31,2252,2254],{"className":2253},[223],[31,2255,2257],{"className":2256},[227],[31,2258,2260],{"className":2259,"style":2177},[231],[31,2261,2262,2265],{"style":234},[31,2263],{"className":2264,"style":239},[238],[31,2266,2268],{"className":2267},[243,244,245,246],[31,2269,1182],{"className":2270},[144,246],[31,2272,99],{"className":2273},[98],",\nfor which ",[31,2276,2278],{"className":2277},[34],[31,2279,2281,2299,2323],{"className":2280,"ariaHidden":39},[38],[31,2282,2284,2287,2290,2293,2296],{"className":2283},[43],[31,2285],{"className":2286,"style":48},[47],[31,2288,55],{"className":2289,"style":54},[52,53],[31,2291],{"className":2292,"style":60},[59],[31,2294,65],{"className":2295},[64],[31,2297],{"className":2298,"style":60},[59],[31,2300,2302,2305,2308,2311,2314,2317,2320],{"className":2301},[43],[31,2303],{"className":2304,"style":75},[47],[31,2306,193],{"className":2307,"style":192},[52,53],[31,2309,89],{"className":2310},[88],[31,2312,94],{"className":2313,"style":93},[52,53],[31,2315],{"className":2316,"style":140},[59],[31,2318,145],{"className":2319},[144],[31,2321],{"className":2322,"style":140},[59],[31,2324,2326,2329,2358],{"className":2325},[43],[31,2327],{"className":2328,"style":75},[47],[31,2330,2332,2335],{"className":2331},[52],[31,2333,94],{"className":2334,"style":93},[52,53],[31,2336,2338],{"className":2337},[219],[31,2339,2341],{"className":2340},[223],[31,2342,2344],{"className":2343},[227],[31,2345,2347],{"className":2346,"style":2177},[231],[31,2348,2349,2352],{"style":234},[31,2350],{"className":2351,"style":239},[238],[31,2353,2355],{"className":2354},[243,244,245,246],[31,2356,1182],{"className":2357},[144,246],[31,2359,99],{"className":2360},[98],":",[31,2363,2365],{"className":2364},[382],[31,2366,2368],{"className":2367},[34],[31,2369,2371,2389,2442,2460,2519,2566,2584,2605,2652],{"className":2370,"ariaHidden":39},[38],[31,2372,2374,2377,2380,2383,2386],{"className":2373},[43],[31,2375],{"className":2376,"style":133},[47],[31,2378,94],{"className":2379,"style":93},[52,53],[31,2381],{"className":2382,"style":140},[59],[31,2384,145],{"className":2385},[144],[31,2387],{"className":2388,"style":140},[59],[31,2390,2392,2395,2430,2433,2436,2439],{"className":2391},[43],[31,2393],{"className":2394,"style":1198},[47],[31,2396,2398,2401],{"className":2397},[52],[31,2399,193],{"className":2400,"style":192},[52,53],[31,2402,2404],{"className":2403},[219],[31,2405,2407],{"className":2406},[223],[31,2408,2410],{"className":2409},[227],[31,2411,2413],{"className":2412,"style":754},[231],[31,2414,2415,2418],{"style":503},[31,2416],{"className":2417,"style":239},[238],[31,2419,2421],{"className":2420},[243,244,245,246],[31,2422,2424,2427],{"className":2423},[52,246],[31,2425,145],{"className":2426},[52,246],[31,2428,269],{"className":2429},[52,246],[31,2431,55],{"className":2432,"style":54},[52,53],[31,2434],{"className":2435,"style":60},[59],[31,2437,65],{"className":2438},[64],[31,2440],{"className":2441,"style":60},[59],[31,2443,2445,2448,2451,2454,2457],{"className":2444},[43],[31,2446],{"className":2447,"style":133},[47],[31,2449,94],{"className":2450,"style":93},[52,53],[31,2452],{"className":2453,"style":140},[59],[31,2455,145],{"className":2456},[144],[31,2458],{"className":2459,"style":140},[59],[31,2461,2463,2466,2501,2504,2507,2510,2513,2516],{"className":2462},[43],[31,2464],{"className":2465,"style":735},[47],[31,2467,2469,2472],{"className":2468},[52],[31,2470,193],{"className":2471,"style":192},[52,53],[31,2473,2475],{"className":2474},[219],[31,2476,2478],{"className":2477},[223],[31,2479,2481],{"className":2480},[227],[31,2482,2484],{"className":2483,"style":754},[231],[31,2485,2486,2489],{"style":503},[31,2487],{"className":2488,"style":239},[238],[31,2490,2492],{"className":2491},[243,244,245,246],[31,2493,2495,2498],{"className":2494},[52,246],[31,2496,145],{"className":2497},[52,246],[31,2499,269],{"className":2500},[52,246],[31,2502,193],{"className":2503,"style":192},[52,53],[31,2505,89],{"className":2506},[88],[31,2508,94],{"className":2509,"style":93},[52,53],[31,2511],{"className":2512,"style":140},[59],[31,2514,145],{"className":2515},[144],[31,2517],{"className":2518,"style":140},[59],[31,2520,2522,2525,2554,2557,2560,2563],{"className":2521},[43],[31,2523],{"className":2524,"style":75},[47],[31,2526,2528,2531],{"className":2527},[52],[31,2529,94],{"className":2530,"style":93},[52,53],[31,2532,2534],{"className":2533},[219],[31,2535,2537],{"className":2536},[223],[31,2538,2540],{"className":2539},[227],[31,2541,2543],{"className":2542,"style":1152},[231],[31,2544,2545,2548],{"style":503},[31,2546],{"className":2547,"style":239},[238],[31,2549,2551],{"className":2550},[243,244,245,246],[31,2552,1182],{"className":2553},[144,246],[31,2555,99],{"className":2556},[98],[31,2558],{"className":2559,"style":60},[59],[31,2561,65],{"className":2562},[64],[31,2564],{"className":2565,"style":60},[59],[31,2567,2569,2572,2575,2578,2581],{"className":2568},[43],[31,2570],{"className":2571,"style":133},[47],[31,2573,94],{"className":2574,"style":93},[52,53],[31,2576],{"className":2577,"style":140},[59],[31,2579,145],{"className":2580},[144],[31,2582],{"className":2583,"style":140},[59],[31,2585,2587,2590,2593,2596,2599,2602],{"className":2586},[43],[31,2588],{"className":2589,"style":75},[47],[31,2591,89],{"className":2592},[88],[31,2594,94],{"className":2595,"style":93},[52,53],[31,2597],{"className":2598,"style":140},[59],[31,2600,145],{"className":2601},[144],[31,2603],{"className":2604,"style":140},[59],[31,2606,2608,2611,2640,2643,2646,2649],{"className":2607},[43],[31,2609],{"className":2610,"style":75},[47],[31,2612,2614,2617],{"className":2613},[52],[31,2615,94],{"className":2616,"style":93},[52,53],[31,2618,2620],{"className":2619},[219],[31,2621,2623],{"className":2622},[223],[31,2624,2626],{"className":2625},[227],[31,2627,2629],{"className":2628,"style":1152},[231],[31,2630,2631,2634],{"style":503},[31,2632],{"className":2633,"style":239},[238],[31,2635,2637],{"className":2636},[243,244,245,246],[31,2638,1182],{"className":2639},[144,246],[31,2641,99],{"className":2642},[98],[31,2644],{"className":2645,"style":60},[59],[31,2647,65],{"className":2648},[64],[31,2650],{"className":2651,"style":60},[59],[31,2653,2655,2658,2687],{"className":2654},[43],[31,2656],{"className":2657,"style":1152},[47],[31,2659,2661,2664],{"className":2660},[52],[31,2662,94],{"className":2663,"style":93},[52,53],[31,2665,2667],{"className":2666},[219],[31,2668,2670],{"className":2669},[223],[31,2671,2673],{"className":2672},[227],[31,2674,2676],{"className":2675,"style":1152},[231],[31,2677,2678,2681],{"style":503},[31,2679],{"className":2680,"style":239},[238],[31,2682,2684],{"className":2683},[243,244,245,246],[31,2685,1182],{"className":2686},[144,246],[31,2688,781],{"className":2689},[52],[1511,2691,2693],{"type":2692},"theorem",[11,2694,2695,2698,2699,2714,2715,2783,2784,2825,2826,2867,2868,2871,2872,3130],{},[26,2696,2697],{},"Theorem (Local quadratic convergence)."," If ",[31,2700,2702],{"className":2701},[34],[31,2703,2705],{"className":2704,"ariaHidden":39},[38],[31,2706,2708,2711],{"className":2707},[43],[31,2709],{"className":2710,"style":188},[47],[31,2712,84],{"className":2713},[52,83]," is twice continuously\ndifferentiable, ",[31,2716,2718],{"className":2717},[34],[31,2719,2721,2774],{"className":2720,"ariaHidden":39},[38],[31,2722,2724,2727,2730,2733,2762,2765,2768,2771],{"className":2723},[43],[31,2725],{"className":2726,"style":75},[47],[31,2728,193],{"className":2729,"style":192},[52,53],[31,2731,89],{"className":2732},[88],[31,2734,2736,2739],{"className":2735},[52],[31,2737,94],{"className":2738,"style":93},[52,53],[31,2740,2742],{"className":2741},[219],[31,2743,2745],{"className":2744},[223],[31,2746,2748],{"className":2747},[227],[31,2749,2751],{"className":2750,"style":2177},[231],[31,2752,2753,2756],{"style":234},[31,2754],{"className":2755,"style":239},[238],[31,2757,2759],{"className":2758},[243,244,245,246],[31,2760,1182],{"className":2761},[144,246],[31,2763,99],{"className":2764},[98],[31,2766],{"className":2767,"style":60},[59],[31,2769,823],{"className":2770},[64],[31,2772],{"className":2773,"style":60},[59],[31,2775,2777,2780],{"className":2776},[43],[31,2778],{"className":2779,"style":833},[47],[31,2781,837],{"className":2782},[52]," at a minimum ",[31,2785,2787],{"className":2786},[34],[31,2788,2790],{"className":2789,"ariaHidden":39},[38],[31,2791,2793,2796],{"className":2792},[43],[31,2794],{"className":2795,"style":113},[47],[31,2797,2799,2802],{"className":2798},[52],[31,2800,94],{"className":2801,"style":93},[52,53],[31,2803,2805],{"className":2804},[219],[31,2806,2808],{"className":2807},[223],[31,2809,2811],{"className":2810},[227],[31,2812,2814],{"className":2813,"style":2177},[231],[31,2815,2816,2819],{"style":234},[31,2817],{"className":2818,"style":239},[238],[31,2820,2822],{"className":2821},[243,244,245,246],[31,2823,1182],{"className":2824},[144,246],", and the iterate\nstarts close enough to ",[31,2827,2829],{"className":2828},[34],[31,2830,2832],{"className":2831,"ariaHidden":39},[38],[31,2833,2835,2838],{"className":2834},[43],[31,2836],{"className":2837,"style":113},[47],[31,2839,2841,2844],{"className":2840},[52],[31,2842,94],{"className":2843,"style":93},[52,53],[31,2845,2847],{"className":2846},[219],[31,2848,2850],{"className":2849},[223],[31,2851,2853],{"className":2852},[227],[31,2854,2856],{"className":2855,"style":2177},[231],[31,2857,2858,2861],{"style":234},[31,2859],{"className":2860,"style":239},[238],[31,2862,2864],{"className":2863},[243,244,245,246],[31,2865,1182],{"className":2866},[144,246],", Newton's method converges ",[164,2869,2870],{},"quadratically",":\n",[31,2873,2875],{"className":2874},[34],[31,2876,2878,2951,2999,3065],{"className":2877,"ariaHidden":39},[38],[31,2879,2881,2884,2888,2942,2945,2948],{"className":2880},[43],[31,2882],{"className":2883,"style":75},[47],[31,2885,2887],{"className":2886},[88],"∥",[31,2889,2891,2894],{"className":2890},[52],[31,2892,94],{"className":2893,"style":93},[52,53],[31,2895,2897],{"className":2896},[219],[31,2898,2900,2933],{"className":2899},[223,553],[31,2901,2903,2930],{"className":2902},[227],[31,2904,2907],{"className":2905,"style":2906},[231],"height:0.3361em;",[31,2908,2910,2913],{"style":2909},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[31,2911],{"className":2912,"style":239},[238],[31,2914,2916],{"className":2915},[243,244,245,246],[31,2917,2919,2924,2927],{"className":2918},[52,246],[31,2920,2923],{"className":2921,"style":2922},[52,53,246],"margin-right:0.0315em;","k",[31,2925,410],{"className":2926},[144,246],[31,2928,269],{"className":2929},[52,246],[31,2931,607],{"className":2932},[606],[31,2934,2936],{"className":2935},[227],[31,2937,2940],{"className":2938,"style":2939},[231],"height:0.2083em;",[31,2941],{},[31,2943],{"className":2944,"style":140},[59],[31,2946,145],{"className":2947},[144],[31,2949],{"className":2950,"style":140},[59],[31,2952,2954,2957,2986,2989,2992,2996],{"className":2953},[43],[31,2955],{"className":2956,"style":75},[47],[31,2958,2960,2963],{"className":2959},[52],[31,2961,94],{"className":2962,"style":93},[52,53],[31,2964,2966],{"className":2965},[219],[31,2967,2969],{"className":2968},[223],[31,2970,2972],{"className":2971},[227],[31,2973,2975],{"className":2974,"style":2177},[231],[31,2976,2977,2980],{"style":234},[31,2978],{"className":2979,"style":239},[238],[31,2981,2983],{"className":2982},[243,244,245,246],[31,2984,1182],{"className":2985},[144,246],[31,2987,2887],{"className":2988},[98],[31,2990],{"className":2991,"style":60},[59],[31,2993,2995],{"className":2994},[64],"≤",[31,2997],{"className":2998,"style":60},[59],[31,3000,3002,3005,3010,3013,3016,3056,3059,3062],{"className":3001},[43],[31,3003],{"className":3004,"style":75},[47],[31,3006,3009],{"className":3007,"style":3008},[52,53],"margin-right:0.0715em;","C",[31,3011],{"className":3012,"style":623},[59],[31,3014,2887],{"className":3015},[88],[31,3017,3019,3022],{"className":3018},[52],[31,3020,94],{"className":3021,"style":93},[52,53],[31,3023,3025],{"className":3024},[219],[31,3026,3028,3048],{"className":3027},[223,553],[31,3029,3031,3045],{"className":3030},[227],[31,3032,3034],{"className":3033,"style":2906},[231],[31,3035,3036,3039],{"style":2909},[31,3037],{"className":3038,"style":239},[238],[31,3040,3042],{"className":3041},[243,244,245,246],[31,3043,2923],{"className":3044,"style":2922},[52,53,246],[31,3046,607],{"className":3047},[606],[31,3049,3051],{"className":3050},[227],[31,3052,3054],{"className":3053,"style":895},[231],[31,3055],{},[31,3057],{"className":3058,"style":140},[59],[31,3060,145],{"className":3061},[144],[31,3063],{"className":3064,"style":140},[59],[31,3066,3068,3072,3101],{"className":3067},[43],[31,3069],{"className":3070,"style":3071},[47],"height:1.0641em;vertical-align:-0.25em;",[31,3073,3075,3078],{"className":3074},[52],[31,3076,94],{"className":3077,"style":93},[52,53],[31,3079,3081],{"className":3080},[219],[31,3082,3084],{"className":3083},[223],[31,3085,3087],{"className":3086},[227],[31,3088,3090],{"className":3089,"style":2177},[231],[31,3091,3092,3095],{"style":234},[31,3093],{"className":3094,"style":239},[238],[31,3096,3098],{"className":3097},[243,244,245,246],[31,3099,1182],{"className":3100},[144,246],[31,3102,3104,3107],{"className":3103},[98],[31,3105,2887],{"className":3106},[98],[31,3108,3110],{"className":3109},[219],[31,3111,3113],{"className":3112},[223],[31,3114,3116],{"className":3115},[227],[31,3117,3119],{"className":3118,"style":209},[231],[31,3120,3121,3124],{"style":234},[31,3122],{"className":3123,"style":239},[238],[31,3125,3127],{"className":3126},[243,244,245,246],[31,3128,250],{"className":3129},[52,246],".\nThe number of correct digits roughly doubles each step.",[1511,3132,3134],{"type":3133},"proof",[11,3135,3136,3139,3140,3403,3404,3419,3420,3461,3462,3805,3806,4029,4030,4077,4078,4320,4321],{},[26,3137,3138],{},"Proof sketch."," The Newton update is ",[31,3141,3143],{"className":3142},[34],[31,3144,3146,3211,3267],{"className":3145,"ariaHidden":39},[38],[31,3147,3149,3153,3202,3205,3208],{"className":3148},[43],[31,3150],{"className":3151,"style":3152},[47],"height:0.9028em;vertical-align:-0.2083em;",[31,3154,3156,3159],{"className":3155},[52],[31,3157,94],{"className":3158,"style":93},[52,53],[31,3160,3162],{"className":3161},[219],[31,3163,3165,3194],{"className":3164},[223,553],[31,3166,3168,3191],{"className":3167},[227],[31,3169,3171],{"className":3170,"style":2906},[231],[31,3172,3173,3176],{"style":2909},[31,3174],{"className":3175,"style":239},[238],[31,3177,3179],{"className":3178},[243,244,245,246],[31,3180,3182,3185,3188],{"className":3181},[52,246],[31,3183,2923],{"className":3184,"style":2922},[52,53,246],[31,3186,410],{"className":3187},[144,246],[31,3189,269],{"className":3190},[52,246],[31,3192,607],{"className":3193},[606],[31,3195,3197],{"className":3196},[227],[31,3198,3200],{"className":3199,"style":2939},[231],[31,3201],{},[31,3203],{"className":3204,"style":60},[59],[31,3206,65],{"className":3207},[64],[31,3209],{"className":3210,"style":60},[59],[31,3212,3214,3218,3258,3261,3264],{"className":3213},[43],[31,3215],{"className":3216,"style":3217},[47],"height:0.8444em;vertical-align:-0.15em;",[31,3219,3221,3224],{"className":3220},[52],[31,3222,94],{"className":3223,"style":93},[52,53],[31,3225,3227],{"className":3226},[219],[31,3228,3230,3250],{"className":3229},[223,553],[31,3231,3233,3247],{"className":3232},[227],[31,3234,3236],{"className":3235,"style":2906},[231],[31,3237,3238,3241],{"style":2909},[31,3239],{"className":3240,"style":239},[238],[31,3242,3244],{"className":3243},[243,244,245,246],[31,3245,2923],{"className":3246,"style":2922},[52,53,246],[31,3248,607],{"className":3249},[606],[31,3251,3253],{"className":3252},[227],[31,3254,3256],{"className":3255,"style":895},[231],[31,3257],{},[31,3259],{"className":3260,"style":140},[59],[31,3262,145],{"className":3263},[144],[31,3265],{"className":3266,"style":140},[59],[31,3268,3270,3273,3276,3279,3319,3354,3357,3360,3400],{"className":3269},[43],[31,3271],{"className":3272,"style":3071},[47],[31,3274,193],{"className":3275,"style":192},[52,53],[31,3277,89],{"className":3278},[88],[31,3280,3282,3285],{"className":3281},[52],[31,3283,94],{"className":3284,"style":93},[52,53],[31,3286,3288],{"className":3287},[219],[31,3289,3291,3311],{"className":3290},[223,553],[31,3292,3294,3308],{"className":3293},[227],[31,3295,3297],{"className":3296,"style":2906},[231],[31,3298,3299,3302],{"style":2909},[31,3300],{"className":3301,"style":239},[238],[31,3303,3305],{"className":3304},[243,244,245,246],[31,3306,2923],{"className":3307,"style":2922},[52,53,246],[31,3309,607],{"className":3310},[606],[31,3312,3314],{"className":3313},[227],[31,3315,3317],{"className":3316,"style":895},[231],[31,3318],{},[31,3320,3322,3325],{"className":3321},[98],[31,3323,99],{"className":3324},[98],[31,3326,3328],{"className":3327},[219],[31,3329,3331],{"className":3330},[223],[31,3332,3334],{"className":3333},[227],[31,3335,3337],{"className":3336,"style":209},[231],[31,3338,3339,3342],{"style":234},[31,3340],{"className":3341,"style":239},[238],[31,3343,3345],{"className":3344},[243,244,245,246],[31,3346,3348,3351],{"className":3347},[52,246],[31,3349,145],{"className":3350},[52,246],[31,3352,269],{"className":3353},[52,246],[31,3355,55],{"className":3356,"style":54},[52,53],[31,3358,89],{"className":3359},[88],[31,3361,3363,3366],{"className":3362},[52],[31,3364,94],{"className":3365,"style":93},[52,53],[31,3367,3369],{"className":3368},[219],[31,3370,3372,3392],{"className":3371},[223,553],[31,3373,3375,3389],{"className":3374},[227],[31,3376,3378],{"className":3377,"style":2906},[231],[31,3379,3380,3383],{"style":2909},[31,3381],{"className":3382,"style":239},[238],[31,3384,3386],{"className":3385},[243,244,245,246],[31,3387,2923],{"className":3388,"style":2922},[52,53,246],[31,3390,607],{"className":3391},[606],[31,3393,3395],{"className":3394},[227],[31,3396,3398],{"className":3397,"style":895},[231],[31,3399],{},[31,3401,99],{"className":3402},[98],".\nTaylor-expand ",[31,3405,3407],{"className":3406},[34],[31,3408,3410],{"className":3409,"ariaHidden":39},[38],[31,3411,3413,3416],{"className":3412},[43],[31,3414],{"className":3415,"style":48},[47],[31,3417,55],{"className":3418,"style":54},[52,53]," about ",[31,3421,3423],{"className":3422},[34],[31,3424,3426],{"className":3425,"ariaHidden":39},[38],[31,3427,3429,3432],{"className":3428},[43],[31,3430],{"className":3431,"style":113},[47],[31,3433,3435,3438],{"className":3434},[52],[31,3436,94],{"className":3437,"style":93},[52,53],[31,3439,3441],{"className":3440},[219],[31,3442,3444],{"className":3443},[223],[31,3445,3447],{"className":3446},[227],[31,3448,3450],{"className":3449,"style":2177},[231],[31,3451,3452,3455],{"style":234},[31,3453],{"className":3454,"style":239},[238],[31,3456,3458],{"className":3457},[243,244,245,246],[31,3459,1182],{"className":3460},[144,246],": ",[31,3463,3465],{"className":3464},[34],[31,3466,3468,3532,3628,3675,3738],{"className":3467,"ariaHidden":39},[38],[31,3469,3471,3474,3477,3480,3520,3523,3526,3529],{"className":3470},[43],[31,3472],{"className":3473,"style":75},[47],[31,3475,55],{"className":3476,"style":54},[52,53],[31,3478,89],{"className":3479},[88],[31,3481,3483,3486],{"className":3482},[52],[31,3484,94],{"className":3485,"style":93},[52,53],[31,3487,3489],{"className":3488},[219],[31,3490,3492,3512],{"className":3491},[223,553],[31,3493,3495,3509],{"className":3494},[227],[31,3496,3498],{"className":3497,"style":2906},[231],[31,3499,3500,3503],{"style":2909},[31,3501],{"className":3502,"style":239},[238],[31,3504,3506],{"className":3505},[243,244,245,246],[31,3507,2923],{"className":3508,"style":2922},[52,53,246],[31,3510,607],{"className":3511},[606],[31,3513,3515],{"className":3514},[227],[31,3516,3518],{"className":3517,"style":895},[231],[31,3519],{},[31,3521,99],{"className":3522},[98],[31,3524],{"className":3525,"style":60},[59],[31,3527,65],{"className":3528},[64],[31,3530],{"className":3531,"style":60},[59],[31,3533,3535,3538,3541,3544,3573,3576,3579,3619,3622,3625],{"className":3534},[43],[31,3536],{"className":3537,"style":75},[47],[31,3539,193],{"className":3540,"style":192},[52,53],[31,3542,89],{"className":3543},[88],[31,3545,3547,3550],{"className":3546},[52],[31,3548,94],{"className":3549,"style":93},[52,53],[31,3551,3553],{"className":3552},[219],[31,3554,3556],{"className":3555},[223],[31,3557,3559],{"className":3558},[227],[31,3560,3562],{"className":3561,"style":2177},[231],[31,3563,3564,3567],{"style":234},[31,3565],{"className":3566,"style":239},[238],[31,3568,3570],{"className":3569},[243,244,245,246],[31,3571,1182],{"className":3572},[144,246],[31,3574,99],{"className":3575},[98],[31,3577,89],{"className":3578},[88],[31,3580,3582,3585],{"className":3581},[52],[31,3583,94],{"className":3584,"style":93},[52,53],[31,3586,3588],{"className":3587},[219],[31,3589,3591,3611],{"className":3590},[223,553],[31,3592,3594,3608],{"className":3593},[227],[31,3595,3597],{"className":3596,"style":2906},[231],[31,3598,3599,3602],{"style":2909},[31,3600],{"className":3601,"style":239},[238],[31,3603,3605],{"className":3604},[243,244,245,246],[31,3606,2923],{"className":3607,"style":2922},[52,53,246],[31,3609,607],{"className":3610},[606],[31,3612,3614],{"className":3613},[227],[31,3615,3617],{"className":3616,"style":895},[231],[31,3618],{},[31,3620],{"className":3621,"style":140},[59],[31,3623,145],{"className":3624},[144],[31,3626],{"className":3627,"style":140},[59],[31,3629,3631,3634,3663,3666,3669,3672],{"className":3630},[43],[31,3632],{"className":3633,"style":75},[47],[31,3635,3637,3640],{"className":3636},[52],[31,3638,94],{"className":3639,"style":93},[52,53],[31,3641,3643],{"className":3642},[219],[31,3644,3646],{"className":3645},[223],[31,3647,3649],{"className":3648},[227],[31,3650,3652],{"className":3651,"style":2177},[231],[31,3653,3654,3657],{"style":234},[31,3655],{"className":3656,"style":239},[238],[31,3658,3660],{"className":3659},[243,244,245,246],[31,3661,1182],{"className":3662},[144,246],[31,3664,99],{"className":3665},[98],[31,3667],{"className":3668,"style":140},[59],[31,3670,410],{"className":3671},[144],[31,3673],{"className":3674,"style":140},[59],[31,3676,3678,3681,3685,3689,3729,3732,3735],{"className":3677},[43],[31,3679],{"className":3680,"style":75},[47],[31,3682,3684],{"className":3683,"style":93},[52,53],"O",[31,3686,3688],{"className":3687},[88],"(∥",[31,3690,3692,3695],{"className":3691},[52],[31,3693,94],{"className":3694,"style":93},[52,53],[31,3696,3698],{"className":3697},[219],[31,3699,3701,3721],{"className":3700},[223,553],[31,3702,3704,3718],{"className":3703},[227],[31,3705,3707],{"className":3706,"style":2906},[231],[31,3708,3709,3712],{"style":2909},[31,3710],{"className":3711,"style":239},[238],[31,3713,3715],{"className":3714},[243,244,245,246],[31,3716,2923],{"className":3717,"style":2922},[52,53,246],[31,3719,607],{"className":3720},[606],[31,3722,3724],{"className":3723},[227],[31,3725,3727],{"className":3726,"style":895},[231],[31,3728],{},[31,3730],{"className":3731,"style":140},[59],[31,3733,145],{"className":3734},[144],[31,3736],{"className":3737,"style":140},[59],[31,3739,3741,3744,3773,3802],{"className":3740},[43],[31,3742],{"className":3743,"style":3071},[47],[31,3745,3747,3750],{"className":3746},[52],[31,3748,94],{"className":3749,"style":93},[52,53],[31,3751,3753],{"className":3752},[219],[31,3754,3756],{"className":3755},[223],[31,3757,3759],{"className":3758},[227],[31,3760,3762],{"className":3761,"style":2177},[231],[31,3763,3764,3767],{"style":234},[31,3765],{"className":3766,"style":239},[238],[31,3768,3770],{"className":3769},[243,244,245,246],[31,3771,1182],{"className":3772},[144,246],[31,3774,3776,3779],{"className":3775},[98],[31,3777,2887],{"className":3778},[98],[31,3780,3782],{"className":3781},[219],[31,3783,3785],{"className":3784},[223],[31,3786,3788],{"className":3787},[227],[31,3789,3791],{"className":3790,"style":209},[231],[31,3792,3793,3796],{"style":234},[31,3794],{"className":3795,"style":239},[238],[31,3797,3799],{"className":3798},[243,244,245,246],[31,3800,250],{"className":3801},[52,246],[31,3803,99],{"className":3804},[98],",\nand ",[31,3807,3809],{"className":3808},[34],[31,3810,3812,3876,3929,3990],{"className":3811,"ariaHidden":39},[38],[31,3813,3815,3818,3821,3824,3864,3867,3870,3873],{"className":3814},[43],[31,3816],{"className":3817,"style":75},[47],[31,3819,193],{"className":3820,"style":192},[52,53],[31,3822,89],{"className":3823},[88],[31,3825,3827,3830],{"className":3826},[52],[31,3828,94],{"className":3829,"style":93},[52,53],[31,3831,3833],{"className":3832},[219],[31,3834,3836,3856],{"className":3835},[223,553],[31,3837,3839,3853],{"className":3838},[227],[31,3840,3842],{"className":3841,"style":2906},[231],[31,3843,3844,3847],{"style":2909},[31,3845],{"className":3846,"style":239},[238],[31,3848,3850],{"className":3849},[243,244,245,246],[31,3851,2923],{"className":3852,"style":2922},[52,53,246],[31,3854,607],{"className":3855},[606],[31,3857,3859],{"className":3858},[227],[31,3860,3862],{"className":3861,"style":895},[231],[31,3863],{},[31,3865,99],{"className":3866},[98],[31,3868],{"className":3869,"style":60},[59],[31,3871,65],{"className":3872},[64],[31,3874],{"className":3875,"style":60},[59],[31,3877,3879,3882,3885,3888,3917,3920,3923,3926],{"className":3878},[43],[31,3880],{"className":3881,"style":75},[47],[31,3883,193],{"className":3884,"style":192},[52,53],[31,3886,89],{"className":3887},[88],[31,3889,3891,3894],{"className":3890},[52],[31,3892,94],{"className":3893,"style":93},[52,53],[31,3895,3897],{"className":3896},[219],[31,3898,3900],{"className":3899},[223],[31,3901,3903],{"className":3902},[227],[31,3904,3906],{"className":3905,"style":2177},[231],[31,3907,3908,3911],{"style":234},[31,3909],{"className":3910,"style":239},[238],[31,3912,3914],{"className":3913},[243,244,245,246],[31,3915,1182],{"className":3916},[144,246],[31,3918,99],{"className":3919},[98],[31,3921],{"className":3922,"style":140},[59],[31,3924,410],{"className":3925},[144],[31,3927],{"className":3928,"style":140},[59],[31,3930,3932,3935,3938,3941,3981,3984,3987],{"className":3931},[43],[31,3933],{"className":3934,"style":75},[47],[31,3936,3684],{"className":3937,"style":93},[52,53],[31,3939,3688],{"className":3940},[88],[31,3942,3944,3947],{"className":3943},[52],[31,3945,94],{"className":3946,"style":93},[52,53],[31,3948,3950],{"className":3949},[219],[31,3951,3953,3973],{"className":3952},[223,553],[31,3954,3956,3970],{"className":3955},[227],[31,3957,3959],{"className":3958,"style":2906},[231],[31,3960,3961,3964],{"style":2909},[31,3962],{"className":3963,"style":239},[238],[31,3965,3967],{"className":3966},[243,244,245,246],[31,3968,2923],{"className":3969,"style":2922},[52,53,246],[31,3971,607],{"className":3972},[606],[31,3974,3976],{"className":3975},[227],[31,3977,3979],{"className":3978,"style":895},[231],[31,3980],{},[31,3982],{"className":3983,"style":140},[59],[31,3985,145],{"className":3986},[144],[31,3988],{"className":3989,"style":140},[59],[31,3991,3993,3996,4025],{"className":3992},[43],[31,3994],{"className":3995,"style":75},[47],[31,3997,3999,4002],{"className":3998},[52],[31,4000,94],{"className":4001,"style":93},[52,53],[31,4003,4005],{"className":4004},[219],[31,4006,4008],{"className":4007},[223],[31,4009,4011],{"className":4010},[227],[31,4012,4014],{"className":4013,"style":2177},[231],[31,4015,4016,4019],{"style":234},[31,4017],{"className":4018,"style":239},[238],[31,4020,4022],{"className":4021},[243,244,245,246],[31,4023,1182],{"className":4024},[144,246],[31,4026,4028],{"className":4027},[98],"∥)",". Substituting,\nthe linear error term cancels exactly — that is what ",[31,4031,4033],{"className":4032},[34],[31,4034,4036],{"className":4035,"ariaHidden":39},[38],[31,4037,4039,4042],{"className":4038},[43],[31,4040],{"className":4041,"style":209},[47],[31,4043,4045,4048],{"className":4044},[52],[31,4046,193],{"className":4047,"style":192},[52,53],[31,4049,4051],{"className":4050},[219],[31,4052,4054],{"className":4053},[223],[31,4055,4057],{"className":4056},[227],[31,4058,4060],{"className":4059,"style":209},[231],[31,4061,4062,4065],{"style":234},[31,4063],{"className":4064,"style":239},[238],[31,4066,4068],{"className":4067},[243,244,245,246],[31,4069,4071,4074],{"className":4070},[52,246],[31,4072,145],{"className":4073},[52,246],[31,4075,269],{"className":4076},[52,246]," buys — leaving only the\nquadratic remainder, so ",[31,4079,4081],{"className":4080},[34],[31,4082,4084,4148,4192,4253],{"className":4083,"ariaHidden":39},[38],[31,4085,4087,4090,4139,4142,4145],{"className":4086},[43],[31,4088],{"className":4089,"style":3152},[47],[31,4091,4093,4096],{"className":4092},[52],[31,4094,94],{"className":4095,"style":93},[52,53],[31,4097,4099],{"className":4098},[219],[31,4100,4102,4131],{"className":4101},[223,553],[31,4103,4105,4128],{"className":4104},[227],[31,4106,4108],{"className":4107,"style":2906},[231],[31,4109,4110,4113],{"style":2909},[31,4111],{"className":4112,"style":239},[238],[31,4114,4116],{"className":4115},[243,244,245,246],[31,4117,4119,4122,4125],{"className":4118},[52,246],[31,4120,2923],{"className":4121,"style":2922},[52,53,246],[31,4123,410],{"className":4124},[144,246],[31,4126,269],{"className":4127},[52,246],[31,4129,607],{"className":4130},[606],[31,4132,4134],{"className":4133},[227],[31,4135,4137],{"className":4136,"style":2939},[231],[31,4138],{},[31,4140],{"className":4141,"style":140},[59],[31,4143,145],{"className":4144},[144],[31,4146],{"className":4147,"style":140},[59],[31,4149,4151,4154,4183,4186,4189],{"className":4150},[43],[31,4152],{"className":4153,"style":113},[47],[31,4155,4157,4160],{"className":4156},[52],[31,4158,94],{"className":4159,"style":93},[52,53],[31,4161,4163],{"className":4162},[219],[31,4164,4166],{"className":4165},[223],[31,4167,4169],{"className":4168},[227],[31,4170,4172],{"className":4171,"style":2177},[231],[31,4173,4174,4177],{"style":234},[31,4175],{"className":4176,"style":239},[238],[31,4178,4180],{"className":4179},[243,244,245,246],[31,4181,1182],{"className":4182},[144,246],[31,4184],{"className":4185,"style":60},[59],[31,4187,65],{"className":4188},[64],[31,4190],{"className":4191,"style":60},[59],[31,4193,4195,4198,4201,4204,4244,4247,4250],{"className":4194},[43],[31,4196],{"className":4197,"style":75},[47],[31,4199,3684],{"className":4200,"style":93},[52,53],[31,4202,3688],{"className":4203},[88],[31,4205,4207,4210],{"className":4206},[52],[31,4208,94],{"className":4209,"style":93},[52,53],[31,4211,4213],{"className":4212},[219],[31,4214,4216,4236],{"className":4215},[223,553],[31,4217,4219,4233],{"className":4218},[227],[31,4220,4222],{"className":4221,"style":2906},[231],[31,4223,4224,4227],{"style":2909},[31,4225],{"className":4226,"style":239},[238],[31,4228,4230],{"className":4229},[243,244,245,246],[31,4231,2923],{"className":4232,"style":2922},[52,53,246],[31,4234,607],{"className":4235},[606],[31,4237,4239],{"className":4238},[227],[31,4240,4242],{"className":4241,"style":895},[231],[31,4243],{},[31,4245],{"className":4246,"style":140},[59],[31,4248,145],{"className":4249},[144],[31,4251],{"className":4252,"style":140},[59],[31,4254,4256,4259,4288,4317],{"className":4255},[43],[31,4257],{"className":4258,"style":3071},[47],[31,4260,4262,4265],{"className":4261},[52],[31,4263,94],{"className":4264,"style":93},[52,53],[31,4266,4268],{"className":4267},[219],[31,4269,4271],{"className":4270},[223],[31,4272,4274],{"className":4273},[227],[31,4275,4277],{"className":4276,"style":2177},[231],[31,4278,4279,4282],{"style":234},[31,4280],{"className":4281,"style":239},[238],[31,4283,4285],{"className":4284},[243,244,245,246],[31,4286,1182],{"className":4287},[144,246],[31,4289,4291,4294],{"className":4290},[98],[31,4292,2887],{"className":4293},[98],[31,4295,4297],{"className":4296},[219],[31,4298,4300],{"className":4299},[223],[31,4301,4303],{"className":4302},[227],[31,4304,4306],{"className":4305,"style":209},[231],[31,4307,4308,4311],{"style":234},[31,4309],{"className":4310,"style":239},[238],[31,4312,4314],{"className":4313},[243,244,245,246],[31,4315,250],{"className":4316},[52,246],[31,4318,99],{"className":4319},[98],". ",[31,4322,4324],{"className":4323},[34],[31,4325,4327],{"className":4326,"ariaHidden":39},[38],[31,4328,4330,4334],{"className":4329},[43],[31,4331],{"className":4332,"style":4333},[47],"height:0.675em;",[31,4335,4339],{"className":4336},[4337,4338],"enclosing","qed",[31,4340,4343],{"className":4341},[52,4342],"amsrm","□",[11,4345,4346,4347,4350,4351,4390,4391,4406,4407,4443,4444,4552,4553,4695],{},"The invariance runs deeper than the condition number. Newton's method is ",[26,4348,4349],{},"affine\ninvariant",": reparametrize by any invertible linear map ",[31,4352,4354],{"className":4353},[34],[31,4355,4357,4377],{"className":4356,"ariaHidden":39},[38],[31,4358,4360,4364,4368,4371,4374],{"className":4359},[43],[31,4361],{"className":4362,"style":4363},[47],"height:0.8889em;vertical-align:-0.1944em;",[31,4365,4367],{"className":4366},[52,53],"ϕ",[31,4369],{"className":4370,"style":60},[59],[31,4372,65],{"className":4373},[64],[31,4375],{"className":4376,"style":60},[59],[31,4378,4380,4383,4387],{"className":4379},[43],[31,4381],{"className":4382,"style":113},[47],[31,4384,4386],{"className":4385},[52,53],"A",[31,4388,94],{"className":4389,"style":93},[52,53],", and the\niterates it produces are the exact images under ",[31,4392,4394],{"className":4393},[34],[31,4395,4397],{"className":4396,"ariaHidden":39},[38],[31,4398,4400,4403],{"className":4399},[43],[31,4401],{"className":4402,"style":188},[47],[31,4404,4386],{"className":4405},[52,53]," of the iterates on the original\nproblem. A quick check: under ",[31,4408,4410],{"className":4409},[34],[31,4411,4413,4431],{"className":4412,"ariaHidden":39},[38],[31,4414,4416,4419,4422,4425,4428],{"className":4415},[43],[31,4417],{"className":4418,"style":4363},[47],[31,4420,4367],{"className":4421},[52,53],[31,4423],{"className":4424,"style":60},[59],[31,4426,65],{"className":4427},[64],[31,4429],{"className":4430,"style":60},[59],[31,4432,4434,4437,4440],{"className":4433},[43],[31,4435],{"className":4436,"style":113},[47],[31,4438,4386],{"className":4439},[52,53],[31,4441,94],{"className":4442,"style":93},[52,53]," the gradient transforms as\n",[31,4445,4447],{"className":4446},[34],[31,4448,4450,4508],{"className":4449,"ariaHidden":39},[38],[31,4451,4453,4457,4499,4502,4505],{"className":4452},[43],[31,4454],{"className":4455,"style":4456},[47],"height:0.7167em;vertical-align:-0.2861em;",[31,4458,4460,4463],{"className":4459},[52],[31,4461,55],{"className":4462,"style":54},[52,53],[31,4464,4466],{"className":4465},[219],[31,4467,4469,4490],{"className":4468},[223,553],[31,4470,4472,4487],{"className":4471},[227],[31,4473,4475],{"className":4474,"style":2906},[231],[31,4476,4478,4481],{"style":4477},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[31,4479],{"className":4480,"style":239},[238],[31,4482,4484],{"className":4483},[243,244,245,246],[31,4485,4367],{"className":4486},[52,53,246],[31,4488,607],{"className":4489},[606],[31,4491,4493],{"className":4492},[227],[31,4494,4497],{"className":4495,"style":4496},[231],"height:0.2861em;",[31,4498],{},[31,4500],{"className":4501,"style":60},[59],[31,4503,65],{"className":4504},[64],[31,4506],{"className":4507,"style":60},[59],[31,4509,4511,4514,4549],{"className":4510},[43],[31,4512],{"className":4513,"style":1295},[47],[31,4515,4517,4520],{"className":4516},[52],[31,4518,4386],{"className":4519},[52,53],[31,4521,4523],{"className":4522},[219],[31,4524,4526],{"className":4525},[223],[31,4527,4529],{"className":4528},[227],[31,4530,4532],{"className":4531,"style":1314},[231],[31,4533,4534,4537],{"style":234},[31,4535],{"className":4536,"style":239},[238],[31,4538,4540],{"className":4539},[243,244,245,246],[31,4541,4543,4546],{"className":4542},[52,246],[31,4544,145],{"className":4545},[52,246],[31,4547,513],{"className":4548},[52,246],[31,4550,55],{"className":4551,"style":54},[52,53]," and the Hessian as ",[31,4554,4556],{"className":4555},[34],[31,4557,4559,4616],{"className":4558,"ariaHidden":39},[38],[31,4560,4562,4566,4607,4610,4613],{"className":4561},[43],[31,4563],{"className":4564,"style":4565},[47],"height:0.9694em;vertical-align:-0.2861em;",[31,4567,4569,4572],{"className":4568},[52],[31,4570,193],{"className":4571,"style":192},[52,53],[31,4573,4575],{"className":4574},[219],[31,4576,4578,4599],{"className":4577},[223,553],[31,4579,4581,4596],{"className":4580},[227],[31,4582,4584],{"className":4583,"style":2906},[231],[31,4585,4587,4590],{"style":4586},"top:-2.55em;margin-left:-0.0813em;margin-right:0.05em;",[31,4588],{"className":4589,"style":239},[238],[31,4591,4593],{"className":4592},[243,244,245,246],[31,4594,4367],{"className":4595},[52,53,246],[31,4597,607],{"className":4598},[606],[31,4600,4602],{"className":4601},[227],[31,4603,4605],{"className":4604,"style":4496},[231],[31,4606],{},[31,4608],{"className":4609,"style":60},[59],[31,4611,65],{"className":4612},[64],[31,4614],{"className":4615,"style":60},[59],[31,4617,4619,4622,4657,4660],{"className":4618},[43],[31,4620],{"className":4621,"style":1314},[47],[31,4623,4625,4628],{"className":4624},[52],[31,4626,4386],{"className":4627},[52,53],[31,4629,4631],{"className":4630},[219],[31,4632,4634],{"className":4633},[223],[31,4635,4637],{"className":4636},[227],[31,4638,4640],{"className":4639,"style":1314},[231],[31,4641,4642,4645],{"style":234},[31,4643],{"className":4644,"style":239},[238],[31,4646,4648],{"className":4647},[243,244,245,246],[31,4649,4651,4654],{"className":4650},[52,246],[31,4652,145],{"className":4653},[52,246],[31,4655,513],{"className":4656},[52,246],[31,4658,193],{"className":4659,"style":192},[52,53],[31,4661,4663,4666],{"className":4662},[52],[31,4664,4386],{"className":4665},[52,53],[31,4667,4669],{"className":4668},[219],[31,4670,4672],{"className":4671},[223],[31,4673,4675],{"className":4674},[227],[31,4676,4678],{"className":4677,"style":209},[231],[31,4679,4680,4683],{"style":234},[31,4681],{"className":4682,"style":239},[238],[31,4684,4686],{"className":4685},[243,244,245,246],[31,4687,4689,4692],{"className":4688},[52,246],[31,4690,145],{"className":4691},[52,246],[31,4693,269],{"className":4694},[52,246],", so the Newton\ndirection is",[31,4697,4699],{"className":4698},[382],[31,4700,4702],{"className":4701},[34],[31,4703,4705,4823,5008,5134],{"className":4704,"ariaHidden":39},[38],[31,4706,4708,4712,4715,4774,4814,4817,4820],{"className":4707},[43],[31,4709],{"className":4710,"style":4711},[47],"height:1.2917em;vertical-align:-0.4275em;",[31,4713,145],{"className":4714},[52],[31,4716,4718,4721],{"className":4717},[52],[31,4719,193],{"className":4720,"style":192},[52,53],[31,4722,4724],{"className":4723},[219],[31,4725,4727,4765],{"className":4726},[223,553],[31,4728,4730,4762],{"className":4729},[227],[31,4731,4733,4745],{"className":4732,"style":754},[231],[31,4734,4736,4739],{"style":4735},"top:-2.4086em;margin-left:-0.0813em;margin-right:0.05em;",[31,4737],{"className":4738,"style":239},[238],[31,4740,4742],{"className":4741},[243,244,245,246],[31,4743,4367],{"className":4744},[52,53,246],[31,4746,4747,4750],{"style":503},[31,4748],{"className":4749,"style":239},[238],[31,4751,4753],{"className":4752},[243,244,245,246],[31,4754,4756,4759],{"className":4755},[52,246],[31,4757,145],{"className":4758},[52,246],[31,4760,269],{"className":4761},[52,246],[31,4763,607],{"className":4764},[606],[31,4766,4768],{"className":4767},[227],[31,4769,4772],{"className":4770,"style":4771},[231],"height:0.4275em;",[31,4773],{},[31,4775,4777,4780],{"className":4776},[52],[31,4778,55],{"className":4779,"style":54},[52,53],[31,4781,4783],{"className":4782},[219],[31,4784,4786,4806],{"className":4785},[223,553],[31,4787,4789,4803],{"className":4788},[227],[31,4790,4792],{"className":4791,"style":2906},[231],[31,4793,4794,4797],{"style":4477},[31,4795],{"className":4796,"style":239},[238],[31,4798,4800],{"className":4799},[243,244,245,246],[31,4801,4367],{"className":4802},[52,53,246],[31,4804,607],{"className":4805},[606],[31,4807,4809],{"className":4808},[227],[31,4810,4812],{"className":4811,"style":4496},[231],[31,4813],{},[31,4815],{"className":4816,"style":60},[59],[31,4818,65],{"className":4819},[64],[31,4821],{"className":4822,"style":60},[59],[31,4824,4826,4830,4833,4836,4958,4961,4996,4999,5002,5005],{"className":4825},[43],[31,4827],{"className":4828,"style":4829},[47],"height:1.4531em;vertical-align:-0.35em;",[31,4831,145],{"className":4832},[52],[31,4834],{"className":4835,"style":623},[59],[31,4837,4839,4927],{"className":4838},[908],[31,4840,4842,4848,4883,4886,4921],{"className":4841},[908],[31,4843,4845],{"className":4844,"style":913},[88,912],[31,4846,89],{"className":4847},[917,918],[31,4849,4851,4854],{"className":4850},[52],[31,4852,4386],{"className":4853},[52,53],[31,4855,4857],{"className":4856},[219],[31,4858,4860],{"className":4859},[223],[31,4861,4863],{"className":4862},[227],[31,4864,4866],{"className":4865,"style":500},[231],[31,4867,4868,4871],{"style":503},[31,4869],{"className":4870,"style":239},[238],[31,4872,4874],{"className":4873},[243,244,245,246],[31,4875,4877,4880],{"className":4876},[52,246],[31,4878,145],{"className":4879},[52,246],[31,4881,513],{"className":4882},[52,246],[31,4884,193],{"className":4885,"style":192},[52,53],[31,4887,4889,4892],{"className":4888},[52],[31,4890,4386],{"className":4891},[52,53],[31,4893,4895],{"className":4894},[219],[31,4896,4898],{"className":4897},[223],[31,4899,4901],{"className":4900},[227],[31,4902,4904],{"className":4903,"style":754},[231],[31,4905,4906,4909],{"style":503},[31,4907],{"className":4908,"style":239},[238],[31,4910,4912],{"className":4911},[243,244,245,246],[31,4913,4915,4918],{"className":4914},[52,246],[31,4916,145],{"className":4917},[52,246],[31,4919,269],{"className":4920},[52,246],[31,4922,4924],{"className":4923,"style":913},[98,912],[31,4925,99],{"className":4926},[917,918],[31,4928,4930],{"className":4929},[219],[31,4931,4933],{"className":4932},[223],[31,4934,4936],{"className":4935},[227],[31,4937,4940],{"className":4938,"style":4939},[231],"height:1.1031em;",[31,4941,4943,4946],{"style":4942},"top:-3.352em;margin-right:0.05em;",[31,4944],{"className":4945,"style":239},[238],[31,4947,4949],{"className":4948},[243,244,245,246],[31,4950,4952,4955],{"className":4951},[52,246],[31,4953,145],{"className":4954},[52,246],[31,4956,269],{"className":4957},[52,246],[31,4959],{"className":4960,"style":623},[59],[31,4962,4964,4967],{"className":4963},[52],[31,4965,4386],{"className":4966},[52,53],[31,4968,4970],{"className":4969},[219],[31,4971,4973],{"className":4972},[223],[31,4974,4976],{"className":4975},[227],[31,4977,4979],{"className":4978,"style":500},[231],[31,4980,4981,4984],{"style":503},[31,4982],{"className":4983,"style":239},[238],[31,4985,4987],{"className":4986},[243,244,245,246],[31,4988,4990,4993],{"className":4989},[52,246],[31,4991,145],{"className":4992},[52,246],[31,4994,513],{"className":4995},[52,246],[31,4997,55],{"className":4998,"style":54},[52,53],[31,5000],{"className":5001,"style":60},[59],[31,5003,65],{"className":5004},[64],[31,5006],{"className":5007,"style":60},[59],[31,5009,5011,5014,5017,5020,5023,5058,5087,5122,5125,5128,5131],{"className":5010},[43],[31,5012],{"className":5013,"style":481},[47],[31,5015,145],{"className":5016},[52],[31,5018,4386],{"className":5019},[52,53],[31,5021],{"className":5022,"style":623},[59],[31,5024,5026,5029],{"className":5025},[52],[31,5027,193],{"className":5028,"style":192},[52,53],[31,5030,5032],{"className":5031},[219],[31,5033,5035],{"className":5034},[223],[31,5036,5038],{"className":5037},[227],[31,5039,5041],{"className":5040,"style":754},[231],[31,5042,5043,5046],{"style":503},[31,5044],{"className":5045,"style":239},[238],[31,5047,5049],{"className":5048},[243,244,245,246],[31,5050,5052,5055],{"className":5051},[52,246],[31,5053,145],{"className":5054},[52,246],[31,5056,269],{"className":5057},[52,246],[31,5059,5061,5064],{"className":5060},[52],[31,5062,4386],{"className":5063},[52,53],[31,5065,5067],{"className":5066},[219],[31,5068,5070],{"className":5069},[223],[31,5071,5073],{"className":5072},[227],[31,5074,5076],{"className":5075,"style":500},[231],[31,5077,5078,5081],{"style":503},[31,5079],{"className":5080,"style":239},[238],[31,5082,5084],{"className":5083},[243,244,245,246],[31,5085,513],{"className":5086},[52,246],[31,5088,5090,5093],{"className":5089},[52],[31,5091,4386],{"className":5092},[52,53],[31,5094,5096],{"className":5095},[219],[31,5097,5099],{"className":5098},[223],[31,5100,5102],{"className":5101},[227],[31,5103,5105],{"className":5104,"style":500},[231],[31,5106,5107,5110],{"style":503},[31,5108],{"className":5109,"style":239},[238],[31,5111,5113],{"className":5112},[243,244,245,246],[31,5114,5116,5119],{"className":5115},[52,246],[31,5117,145],{"className":5118},[52,246],[31,5120,513],{"className":5121},[52,246],[31,5123,55],{"className":5124,"style":54},[52,53],[31,5126],{"className":5127,"style":60},[59],[31,5129,65],{"className":5130},[64],[31,5132],{"className":5133,"style":60},[59],[31,5135,5137,5141,5144,5147,5203,5206],{"className":5136},[43],[31,5138],{"className":5139,"style":5140},[47],"height:1.2141em;vertical-align:-0.35em;",[31,5142,4386],{"className":5143},[52,53],[31,5145],{"className":5146,"style":623},[59],[31,5148,5150,5156,5159,5194,5197],{"className":5149},[908],[31,5151,5153],{"className":5152,"style":913},[88,912],[31,5154,89],{"className":5155},[917,918],[31,5157,145],{"className":5158},[52],[31,5160,5162,5165],{"className":5161},[52],[31,5163,193],{"className":5164,"style":192},[52,53],[31,5166,5168],{"className":5167},[219],[31,5169,5171],{"className":5170},[223],[31,5172,5174],{"className":5173},[227],[31,5175,5177],{"className":5176,"style":754},[231],[31,5178,5179,5182],{"style":503},[31,5180],{"className":5181,"style":239},[238],[31,5183,5185],{"className":5184},[243,244,245,246],[31,5186,5188,5191],{"className":5187},[52,246],[31,5189,145],{"className":5190},[52,246],[31,5192,269],{"className":5193},[52,246],[31,5195,55],{"className":5196,"style":54},[52,53],[31,5198,5200],{"className":5199,"style":913},[98,912],[31,5201,99],{"className":5202},[917,918],[31,5204],{"className":5205,"style":623},[59],[31,5207,666],{"className":5208},[665],[11,5210,5211,5212,5227,5228,5340,5341,5368],{},"the original step carried through ",[31,5213,5215],{"className":5214},[34],[31,5216,5218],{"className":5217,"ariaHidden":39},[38],[31,5219,5221,5224],{"className":5220},[43],[31,5222],{"className":5223,"style":188},[47],[31,5225,4386],{"className":5226},[52,53],". Gradient descent has no such property — the\ndirection ",[31,5229,5231],{"className":5230},[34],[31,5232,5234,5293],{"className":5233,"ariaHidden":39},[38],[31,5235,5237,5241,5244,5284,5287,5290],{"className":5236},[43],[31,5238],{"className":5239,"style":5240},[47],"height:0.8694em;vertical-align:-0.2861em;",[31,5242,145],{"className":5243},[52],[31,5245,5247,5250],{"className":5246},[52],[31,5248,55],{"className":5249,"style":54},[52,53],[31,5251,5253],{"className":5252},[219],[31,5254,5256,5276],{"className":5255},[223,553],[31,5257,5259,5273],{"className":5258},[227],[31,5260,5262],{"className":5261,"style":2906},[231],[31,5263,5264,5267],{"style":4477},[31,5265],{"className":5266,"style":239},[238],[31,5268,5270],{"className":5269},[243,244,245,246],[31,5271,4367],{"className":5272},[52,53,246],[31,5274,607],{"className":5275},[606],[31,5277,5279],{"className":5278},[227],[31,5280,5282],{"className":5281,"style":4496},[231],[31,5283],{},[31,5285],{"className":5286,"style":60},[59],[31,5288,65],{"className":5289},[64],[31,5291],{"className":5292,"style":60},[59],[31,5294,5296,5299,5302,5337],{"className":5295},[43],[31,5297],{"className":5298,"style":1295},[47],[31,5300,145],{"className":5301},[52],[31,5303,5305,5308],{"className":5304},[52],[31,5306,4386],{"className":5307},[52,53],[31,5309,5311],{"className":5310},[219],[31,5312,5314],{"className":5313},[223],[31,5315,5317],{"className":5316},[227],[31,5318,5320],{"className":5319,"style":1314},[231],[31,5321,5322,5325],{"style":234},[31,5323],{"className":5324,"style":239},[238],[31,5326,5328],{"className":5327},[243,244,245,246],[31,5329,5331,5334],{"className":5330},[52,246],[31,5332,145],{"className":5333},[52,246],[31,5335,513],{"className":5336},[52,246],[31,5338,55],{"className":5339,"style":54},[52,53]," does not equal ",[31,5342,5344],{"className":5343},[34],[31,5345,5347],{"className":5346,"ariaHidden":39},[38],[31,5348,5350,5353,5356,5359,5362,5365],{"className":5349},[43],[31,5351],{"className":5352,"style":75},[47],[31,5354,4386],{"className":5355},[52,53],[31,5357,89],{"className":5358},[88],[31,5360,145],{"className":5361},[52],[31,5363,55],{"className":5364,"style":54},[52,53],[31,5366,99],{"className":5367},[98],", so a bad choice of units alone\ncan make first-order descent crawl. Newton reads geometry, not coordinates, which is why\nthe same step handles a bowl stretched a thousand-to-one and a round one alike.",[11,5370,5371,5372,5387,5388,5391,5392,5529,5530,5569,5570,5585,5586,5637,5638,5688,5689,5752,5753,5768,5769,5817,5818,5866,5867,1971,5917,5967],{},"The cost is high. Forming ",[31,5373,5375],{"className":5374},[34],[31,5376,5378],{"className":5377,"ariaHidden":39},[38],[31,5379,5381,5384],{"className":5380},[43],[31,5382],{"className":5383,"style":188},[47],[31,5385,193],{"className":5386,"style":192},[52,53]," needs the second derivative with respect\nto every ",[164,5389,5390],{},"pair"," of parameters — ",[31,5393,5395],{"className":5394},[34],[31,5396,5398,5490],{"className":5397,"ariaHidden":39},[38],[31,5399,5401,5405,5481,5484,5487],{"className":5400},[43],[31,5402],{"className":5403,"style":5404},[47],"height:1.2801em;vertical-align:-0.35em;",[31,5406,5408,5414,5475],{"className":5407},[52],[31,5409,5411],{"className":5410,"style":913},[88,912],[31,5412,89],{"className":5413},[917,918],[31,5415,5417],{"className":5416},[549],[31,5418,5420,5467],{"className":5419},[223,553],[31,5421,5423,5464],{"className":5422},[227],[31,5424,5427,5442],{"className":5425,"style":5426},[231],"height:0.9301em;",[31,5428,5430,5433],{"style":5429},"top:-2.355em;",[31,5431],{"className":5432,"style":239},[238],[31,5434,5436],{"className":5435},[243,244,245,246],[31,5437,5439],{"className":5438},[52,246],[31,5440,250],{"className":5441},[52,246],[31,5443,5445,5448],{"style":5444},"top:-3.144em;",[31,5446],{"className":5447,"style":239},[238],[31,5449,5451],{"className":5450},[243,244,245,246],[31,5452,5454,5458,5461],{"className":5453},[52,246],[31,5455,5457],{"className":5456},[52,53,246],"d",[31,5459,410],{"className":5460},[144,246],[31,5462,269],{"className":5463},[52,246],[31,5465,607],{"className":5466},[606],[31,5468,5470],{"className":5469},[227],[31,5471,5473],{"className":5472,"style":614},[231],[31,5474],{},[31,5476,5478],{"className":5477,"style":913},[98,912],[31,5479,99],{"className":5480},[917,918],[31,5482],{"className":5483,"style":60},[59],[31,5485,435],{"className":5486},[64],[31,5488],{"className":5489,"style":60},[59],[31,5491,5493,5496,5525],{"className":5492},[43],[31,5494],{"className":5495,"style":3071},[47],[31,5497,5499,5502],{"className":5498},[52],[31,5500,5457],{"className":5501},[52,53],[31,5503,5505],{"className":5504},[219],[31,5506,5508],{"className":5507},[223],[31,5509,5511],{"className":5510},[227],[31,5512,5514],{"className":5513,"style":209},[231],[31,5515,5516,5519],{"style":234},[31,5517],{"className":5518,"style":239},[238],[31,5520,5522],{"className":5521},[243,244,245,246],[31,5523,250],{"className":5524},[52,246],[31,5526,5528],{"className":5527},[52],"\u002F2"," distinct entries — and\nsolving the linear system ",[31,5531,5533],{"className":5532},[34],[31,5534,5536,5557],{"className":5535,"ariaHidden":39},[38],[31,5537,5539,5542,5545,5548,5551,5554],{"className":5538},[43],[31,5540],{"className":5541,"style":188},[47],[31,5543,193],{"className":5544,"style":192},[52,53],[31,5546,309],{"className":5547},[52],[31,5549],{"className":5550,"style":60},[59],[31,5552,65],{"className":5553},[64],[31,5555],{"className":5556,"style":60},[59],[31,5558,5560,5563,5566],{"className":5559},[43],[31,5561],{"className":5562,"style":1086},[47],[31,5564,145],{"className":5565},[52],[31,5567,55],{"className":5568,"style":54},[52,53]," (whether by inverting ",[31,5571,5573],{"className":5572},[34],[31,5574,5576],{"className":5575,"ariaHidden":39},[38],[31,5577,5579,5582],{"className":5578},[43],[31,5580],{"className":5581,"style":188},[47],[31,5583,193],{"className":5584,"style":192},[52,53]," or by Cholesky\nfactorization) costs ",[31,5587,5589],{"className":5588},[34],[31,5590,5592],{"className":5591,"ariaHidden":39},[38],[31,5593,5595,5598,5601,5604,5634],{"className":5594},[43],[31,5596],{"className":5597,"style":3071},[47],[31,5599,3684],{"className":5600,"style":93},[52,53],[31,5602,89],{"className":5603},[88],[31,5605,5607,5610],{"className":5606},[52],[31,5608,5457],{"className":5609},[52,53],[31,5611,5613],{"className":5612},[219],[31,5614,5616],{"className":5615},[223],[31,5617,5619],{"className":5618},[227],[31,5620,5622],{"className":5621,"style":209},[231],[31,5623,5624,5627],{"style":234},[31,5625],{"className":5626,"style":239},[238],[31,5628,5630],{"className":5629},[243,244,245,246],[31,5631,5633],{"className":5632},[52,246],"3",[31,5635,99],{"className":5636},[98]," arithmetic and ",[31,5639,5641],{"className":5640},[34],[31,5642,5644],{"className":5643,"ariaHidden":39},[38],[31,5645,5647,5650,5653,5656,5685],{"className":5646},[43],[31,5648],{"className":5649,"style":3071},[47],[31,5651,3684],{"className":5652,"style":93},[52,53],[31,5654,89],{"className":5655},[88],[31,5657,5659,5662],{"className":5658},[52],[31,5660,5457],{"className":5661},[52,53],[31,5663,5665],{"className":5664},[219],[31,5666,5668],{"className":5667},[223],[31,5669,5671],{"className":5670},[227],[31,5672,5674],{"className":5673,"style":209},[231],[31,5675,5676,5679],{"style":234},[31,5677],{"className":5678,"style":239},[238],[31,5680,5682],{"className":5681},[243,244,245,246],[31,5683,250],{"className":5684},[52,246],[31,5686,99],{"className":5687},[98]," storage. For a ",[31,5690,5692],{"className":5691},[34],[31,5693,5695,5713],{"className":5694,"ariaHidden":39},[38],[31,5696,5698,5701,5704,5707,5710],{"className":5697},[43],[31,5699],{"className":5700,"style":113},[47],[31,5702,5457],{"className":5703},[52,53],[31,5705],{"className":5706,"style":60},[59],[31,5708,65],{"className":5709},[64],[31,5711],{"className":5712,"style":60},[59],[31,5714,5716,5719,5722],{"className":5715},[43],[31,5717],{"className":5718,"style":209},[47],[31,5720,269],{"className":5721},[52],[31,5723,5725,5728],{"className":5724},[52],[31,5726,837],{"className":5727},[52],[31,5729,5731],{"className":5730},[219],[31,5732,5734],{"className":5733},[223],[31,5735,5737],{"className":5736},[227],[31,5738,5740],{"className":5739,"style":209},[231],[31,5741,5742,5745],{"style":234},[31,5743],{"className":5744,"style":239},[238],[31,5746,5748],{"className":5747},[243,244,245,246],[31,5749,5751],{"className":5750},[52,246],"7"," network,\n",[31,5754,5756],{"className":5755},[34],[31,5757,5759],{"className":5758,"ariaHidden":39},[38],[31,5760,5762,5765],{"className":5761},[43],[31,5763],{"className":5764,"style":188},[47],[31,5766,193],{"className":5767,"style":192},[52,53]," alone is ",[31,5770,5772],{"className":5771},[34],[31,5773,5775],{"className":5774,"ariaHidden":39},[38],[31,5776,5778,5781,5784],{"className":5777},[43],[31,5779],{"className":5780,"style":209},[47],[31,5782,269],{"className":5783},[52],[31,5785,5787,5790],{"className":5786},[52],[31,5788,837],{"className":5789},[52],[31,5791,5793],{"className":5792},[219],[31,5794,5796],{"className":5795},[223],[31,5797,5799],{"className":5798},[227],[31,5800,5802],{"className":5801,"style":209},[231],[31,5803,5804,5807],{"style":234},[31,5805],{"className":5806,"style":239},[238],[31,5808,5810],{"className":5809},[243,244,245,246],[31,5811,5813],{"className":5812},[52,246],[31,5814,5816],{"className":5815},[52,246],"14"," numbers; the factorization is ",[31,5819,5821],{"className":5820},[34],[31,5822,5824],{"className":5823,"ariaHidden":39},[38],[31,5825,5827,5830,5833],{"className":5826},[43],[31,5828],{"className":5829,"style":209},[47],[31,5831,269],{"className":5832},[52],[31,5834,5836,5839],{"className":5835},[52],[31,5837,837],{"className":5838},[52],[31,5840,5842],{"className":5841},[219],[31,5843,5845],{"className":5844},[223],[31,5846,5848],{"className":5847},[227],[31,5849,5851],{"className":5850,"style":209},[231],[31,5852,5853,5856],{"style":234},[31,5854],{"className":5855,"style":239},[238],[31,5857,5859],{"className":5858},[243,244,245,246],[31,5860,5862],{"className":5861},[52,246],[31,5863,5865],{"className":5864},[52,246],"21"," operations, per step. The\nentire remainder of the lesson is a catalog of ways to keep the curvature information\nwhile retreating from that ",[31,5868,5870],{"className":5869},[34],[31,5871,5873],{"className":5872,"ariaHidden":39},[38],[31,5874,5876,5879,5882,5885,5914],{"className":5875},[43],[31,5877],{"className":5878,"style":3071},[47],[31,5880,3684],{"className":5881,"style":93},[52,53],[31,5883,89],{"className":5884},[88],[31,5886,5888,5891],{"className":5887},[52],[31,5889,5457],{"className":5890},[52,53],[31,5892,5894],{"className":5893},[219],[31,5895,5897],{"className":5896},[223],[31,5898,5900],{"className":5899},[227],[31,5901,5903],{"className":5902,"style":209},[231],[31,5904,5905,5908],{"style":234},[31,5906],{"className":5907,"style":239},[238],[31,5909,5911],{"className":5910},[243,244,245,246],[31,5912,250],{"className":5913},[52,246],[31,5915,99],{"className":5916},[98],[31,5918,5920],{"className":5919},[34],[31,5921,5923],{"className":5922,"ariaHidden":39},[38],[31,5924,5926,5929,5932,5935,5964],{"className":5925},[43],[31,5927],{"className":5928,"style":3071},[47],[31,5930,3684],{"className":5931,"style":93},[52,53],[31,5933,89],{"className":5934},[88],[31,5936,5938,5941],{"className":5937},[52],[31,5939,5457],{"className":5940},[52,53],[31,5942,5944],{"className":5943},[219],[31,5945,5947],{"className":5946},[223],[31,5948,5950],{"className":5949},[227],[31,5951,5953],{"className":5952,"style":209},[231],[31,5954,5955,5958],{"style":234},[31,5956],{"className":5957,"style":239},[238],[31,5959,5961],{"className":5960},[243,244,245,246],[31,5962,5633],{"className":5963},[52,246],[31,5965,99],{"className":5966},[98]," cost.",[271,5969,5971],{"id":5970},"why-second-order-methods-are-rare-in-deep-learning","Why second-order methods are rare in deep learning",[11,5973,5974],{},"Newton's method works well on small, smooth, deterministic problems. In deep\nlearning all three properties fail at once, and each failure alone rules the method out.",[5976,5977,5979,5980],"h3",{"id":5978},"the-hessian-is-dd","The Hessian is ",[31,5981,5983],{"className":5982},[34],[31,5984,5986,6005],{"className":5985,"ariaHidden":39},[38],[31,5987,5989,5992,5995,5998,6002],{"className":5988},[43],[31,5990],{"className":5991,"style":133},[47],[31,5993,5457],{"className":5994},[52,53],[31,5996],{"className":5997,"style":140},[59],[31,5999,6001],{"className":6000},[144],"×",[31,6003],{"className":6004,"style":140},[59],[31,6006,6008,6011],{"className":6007},[43],[31,6009],{"className":6010,"style":113},[47],[31,6012,5457],{"className":6013},[52,53],[11,6015,6016,6017,6032,6033,6066,6067,6069,6070,6111,6112,6162],{},"The first wall is sheer size. The Hessian of a network with ",[31,6018,6020],{"className":6019},[34],[31,6021,6023],{"className":6022,"ariaHidden":39},[38],[31,6024,6026,6029],{"className":6025},[43],[31,6027],{"className":6028,"style":113},[47],[31,6030,5457],{"className":6031},[52,53]," parameters is a\n",[31,6034,6036],{"className":6035},[34],[31,6037,6039,6057],{"className":6038,"ariaHidden":39},[38],[31,6040,6042,6045,6048,6051,6054],{"className":6041},[43],[31,6043],{"className":6044,"style":133},[47],[31,6046,5457],{"className":6047},[52,53],[31,6049],{"className":6050,"style":140},[59],[31,6052,6001],{"className":6053},[144],[31,6055],{"className":6056,"style":140},[59],[31,6058,6060,6063],{"className":6059},[43],[31,6061],{"className":6062,"style":113},[47],[31,6064,5457],{"className":6065},[52,53]," symmetric matrix (one entry per ",[164,6068,5390],{}," of parameters), so its storage\ngrows as ",[31,6071,6073],{"className":6072},[34],[31,6074,6076],{"className":6075,"ariaHidden":39},[38],[31,6077,6079,6082],{"className":6078},[43],[31,6080],{"className":6081,"style":209},[47],[31,6083,6085,6088],{"className":6084},[52],[31,6086,5457],{"className":6087},[52,53],[31,6089,6091],{"className":6090},[219],[31,6092,6094],{"className":6093},[223],[31,6095,6097],{"className":6096},[227],[31,6098,6100],{"className":6099,"style":209},[231],[31,6101,6102,6105],{"style":234},[31,6103],{"className":6104,"style":239},[238],[31,6106,6108],{"className":6107},[243,244,245,246],[31,6109,250],{"className":6110},[52,246]," and forming its inverse costs ",[31,6113,6115],{"className":6114},[34],[31,6116,6118],{"className":6117,"ariaHidden":39},[38],[31,6119,6121,6124,6127,6130,6159],{"className":6120},[43],[31,6122],{"className":6123,"style":3071},[47],[31,6125,3684],{"className":6126,"style":93},[52,53],[31,6128,89],{"className":6129},[88],[31,6131,6133,6136],{"className":6132},[52],[31,6134,5457],{"className":6135},[52,53],[31,6137,6139],{"className":6138},[219],[31,6140,6142],{"className":6141},[223],[31,6143,6145],{"className":6144},[227],[31,6146,6148],{"className":6147,"style":209},[231],[31,6149,6150,6153],{"style":234},[31,6151],{"className":6152,"style":239},[238],[31,6154,6156],{"className":6155},[243,244,245,246],[31,6157,5633],{"className":6158},[52,246],[31,6160,99],{"className":6161},[98]," by Gaussian elimination.",[1507,6164],{"hash":6165},"8afa14a91a39bc0661176a03286d5ff9783acf58518d74622ef31943de1e84f5",[11,6167,6168],{},"The numbers are far from feasible. The table quantifies the gap: a gradient\nthat fits in memory pairs with a Hessian that cannot be stored on any hardware.",[6170,6171,6172,6261],"table",{},[6173,6174,6175],"thead",{},[6176,6177,6178,6197,6200,6203],"tr",{},[6179,6180,6181,6196],"th",{},[31,6182,6184],{"className":6183},[34],[31,6185,6187],{"className":6186,"ariaHidden":39},[38],[31,6188,6190,6193],{"className":6189},[43],[31,6191],{"className":6192,"style":113},[47],[31,6194,5457],{"className":6195},[52,53]," (parameters)",[6179,6198,6199],{},"gradient entries",[6179,6201,6202],{},"Hessian entries",[6179,6204,6205,6206],{},"inversion cost ",[31,6207,6209],{"className":6208},[34],[31,6210,6212,6226],{"className":6211,"ariaHidden":39},[38],[31,6213,6215,6219,6223],{"className":6214},[43],[31,6216],{"className":6217,"style":6218},[47],"height:0.3669em;",[31,6220,6222],{"className":6221},[64],"∼",[31,6224],{"className":6225,"style":60},[59],[31,6227,6229,6232],{"className":6228},[43],[31,6230],{"className":6231,"style":209},[47],[31,6233,6235,6238],{"className":6234},[52],[31,6236,5457],{"className":6237},[52,53],[31,6239,6241],{"className":6240},[219],[31,6242,6244],{"className":6243},[223],[31,6245,6247],{"className":6246},[227],[31,6248,6250],{"className":6249,"style":209},[231],[31,6251,6252,6255],{"style":234},[31,6253],{"className":6254,"style":239},[238],[31,6256,6258],{"className":6257},[243,244,245,246],[31,6259,5633],{"className":6260},[52,246],[6262,6263,6264,6453,6647],"tbody",{},[6176,6265,6266,6313,6359,6406],{},[6267,6268,6269],"td",{},[31,6270,6272],{"className":6271},[34],[31,6273,6275],{"className":6274,"ariaHidden":39},[38],[31,6276,6278,6281,6284],{"className":6277},[43],[31,6279],{"className":6280,"style":209},[47],[31,6282,269],{"className":6283},[52],[31,6285,6287,6290],{"className":6286},[52],[31,6288,837],{"className":6289},[52],[31,6291,6293],{"className":6292},[219],[31,6294,6296],{"className":6295},[223],[31,6297,6299],{"className":6298},[227],[31,6300,6302],{"className":6301,"style":209},[231],[31,6303,6304,6307],{"style":234},[31,6305],{"className":6306,"style":239},[238],[31,6308,6310],{"className":6309},[243,244,245,246],[31,6311,5633],{"className":6312},[52,246],[6267,6314,6315],{},[31,6316,6318],{"className":6317},[34],[31,6319,6321],{"className":6320,"ariaHidden":39},[38],[31,6322,6324,6327,6330],{"className":6323},[43],[31,6325],{"className":6326,"style":209},[47],[31,6328,269],{"className":6329},[52],[31,6331,6333,6336],{"className":6332},[52],[31,6334,837],{"className":6335},[52],[31,6337,6339],{"className":6338},[219],[31,6340,6342],{"className":6341},[223],[31,6343,6345],{"className":6344},[227],[31,6346,6348],{"className":6347,"style":209},[231],[31,6349,6350,6353],{"style":234},[31,6351],{"className":6352,"style":239},[238],[31,6354,6356],{"className":6355},[243,244,245,246],[31,6357,5633],{"className":6358},[52,246],[6267,6360,6361],{},[31,6362,6364],{"className":6363},[34],[31,6365,6367],{"className":6366,"ariaHidden":39},[38],[31,6368,6370,6373,6376],{"className":6369},[43],[31,6371],{"className":6372,"style":209},[47],[31,6374,269],{"className":6375},[52],[31,6377,6379,6382],{"className":6378},[52],[31,6380,837],{"className":6381},[52],[31,6383,6385],{"className":6384},[219],[31,6386,6388],{"className":6387},[223],[31,6389,6391],{"className":6390},[227],[31,6392,6394],{"className":6393,"style":209},[231],[31,6395,6396,6399],{"style":234},[31,6397],{"className":6398,"style":239},[238],[31,6400,6402],{"className":6401},[243,244,245,246],[31,6403,6405],{"className":6404},[52,246],"6",[6267,6407,6408],{},[31,6409,6411],{"className":6410},[34],[31,6412,6414],{"className":6413,"ariaHidden":39},[38],[31,6415,6417,6420,6423],{"className":6416},[43],[31,6418],{"className":6419,"style":209},[47],[31,6421,269],{"className":6422},[52],[31,6424,6426,6429],{"className":6425},[52],[31,6427,837],{"className":6428},[52],[31,6430,6432],{"className":6431},[219],[31,6433,6435],{"className":6434},[223],[31,6436,6438],{"className":6437},[227],[31,6439,6441],{"className":6440,"style":209},[231],[31,6442,6443,6446],{"style":234},[31,6444],{"className":6445,"style":239},[238],[31,6447,6449],{"className":6448},[243,244,245,246],[31,6450,6452],{"className":6451},[52,246],"9",[6176,6454,6455,6501,6547,6597],{},[6267,6456,6457],{},[31,6458,6460],{"className":6459},[34],[31,6461,6463],{"className":6462,"ariaHidden":39},[38],[31,6464,6466,6469,6472],{"className":6465},[43],[31,6467],{"className":6468,"style":209},[47],[31,6470,269],{"className":6471},[52],[31,6473,6475,6478],{"className":6474},[52],[31,6476,837],{"className":6477},[52],[31,6479,6481],{"className":6480},[219],[31,6482,6484],{"className":6483},[223],[31,6485,6487],{"className":6486},[227],[31,6488,6490],{"className":6489,"style":209},[231],[31,6491,6492,6495],{"style":234},[31,6493],{"className":6494,"style":239},[238],[31,6496,6498],{"className":6497},[243,244,245,246],[31,6499,6405],{"className":6500},[52,246],[6267,6502,6503],{},[31,6504,6506],{"className":6505},[34],[31,6507,6509],{"className":6508,"ariaHidden":39},[38],[31,6510,6512,6515,6518],{"className":6511},[43],[31,6513],{"className":6514,"style":209},[47],[31,6516,269],{"className":6517},[52],[31,6519,6521,6524],{"className":6520},[52],[31,6522,837],{"className":6523},[52],[31,6525,6527],{"className":6526},[219],[31,6528,6530],{"className":6529},[223],[31,6531,6533],{"className":6532},[227],[31,6534,6536],{"className":6535,"style":209},[231],[31,6537,6538,6541],{"style":234},[31,6539],{"className":6540,"style":239},[238],[31,6542,6544],{"className":6543},[243,244,245,246],[31,6545,6405],{"className":6546},[52,246],[6267,6548,6549],{},[31,6550,6552],{"className":6551},[34],[31,6553,6555],{"className":6554,"ariaHidden":39},[38],[31,6556,6558,6561,6564],{"className":6557},[43],[31,6559],{"className":6560,"style":209},[47],[31,6562,269],{"className":6563},[52],[31,6565,6567,6570],{"className":6566},[52],[31,6568,837],{"className":6569},[52],[31,6571,6573],{"className":6572},[219],[31,6574,6576],{"className":6575},[223],[31,6577,6579],{"className":6578},[227],[31,6580,6582],{"className":6581,"style":209},[231],[31,6583,6584,6587],{"style":234},[31,6585],{"className":6586,"style":239},[238],[31,6588,6590],{"className":6589},[243,244,245,246],[31,6591,6593],{"className":6592},[52,246],[31,6594,6596],{"className":6595},[52,246],"12",[6267,6598,6599],{},[31,6600,6602],{"className":6601},[34],[31,6603,6605],{"className":6604,"ariaHidden":39},[38],[31,6606,6608,6611,6614],{"className":6607},[43],[31,6609],{"className":6610,"style":209},[47],[31,6612,269],{"className":6613},[52],[31,6615,6617,6620],{"className":6616},[52],[31,6618,837],{"className":6619},[52],[31,6621,6623],{"className":6622},[219],[31,6624,6626],{"className":6625},[223],[31,6627,6629],{"className":6628},[227],[31,6630,6632],{"className":6631,"style":209},[231],[31,6633,6634,6637],{"style":234},[31,6635],{"className":6636,"style":239},[238],[31,6638,6640],{"className":6639},[243,244,245,246],[31,6641,6643],{"className":6642},[52,246],[31,6644,6646],{"className":6645},[52,246],"18",[6176,6648,6649,6695,6741,6790],{},[6267,6650,6651],{},[31,6652,6654],{"className":6653},[34],[31,6655,6657],{"className":6656,"ariaHidden":39},[38],[31,6658,6660,6663,6666],{"className":6659},[43],[31,6661],{"className":6662,"style":209},[47],[31,6664,269],{"className":6665},[52],[31,6667,6669,6672],{"className":6668},[52],[31,6670,837],{"className":6671},[52],[31,6673,6675],{"className":6674},[219],[31,6676,6678],{"className":6677},[223],[31,6679,6681],{"className":6680},[227],[31,6682,6684],{"className":6683,"style":209},[231],[31,6685,6686,6689],{"style":234},[31,6687],{"className":6688,"style":239},[238],[31,6690,6692],{"className":6691},[243,244,245,246],[31,6693,6452],{"className":6694},[52,246],[6267,6696,6697],{},[31,6698,6700],{"className":6699},[34],[31,6701,6703],{"className":6702,"ariaHidden":39},[38],[31,6704,6706,6709,6712],{"className":6705},[43],[31,6707],{"className":6708,"style":209},[47],[31,6710,269],{"className":6711},[52],[31,6713,6715,6718],{"className":6714},[52],[31,6716,837],{"className":6717},[52],[31,6719,6721],{"className":6720},[219],[31,6722,6724],{"className":6723},[223],[31,6725,6727],{"className":6726},[227],[31,6728,6730],{"className":6729,"style":209},[231],[31,6731,6732,6735],{"style":234},[31,6733],{"className":6734,"style":239},[238],[31,6736,6738],{"className":6737},[243,244,245,246],[31,6739,6452],{"className":6740},[52,246],[6267,6742,6743],{},[31,6744,6746],{"className":6745},[34],[31,6747,6749],{"className":6748,"ariaHidden":39},[38],[31,6750,6752,6755,6758],{"className":6751},[43],[31,6753],{"className":6754,"style":209},[47],[31,6756,269],{"className":6757},[52],[31,6759,6761,6764],{"className":6760},[52],[31,6762,837],{"className":6763},[52],[31,6765,6767],{"className":6766},[219],[31,6768,6770],{"className":6769},[223],[31,6771,6773],{"className":6772},[227],[31,6774,6776],{"className":6775,"style":209},[231],[31,6777,6778,6781],{"style":234},[31,6779],{"className":6780,"style":239},[238],[31,6782,6784],{"className":6783},[243,244,245,246],[31,6785,6787],{"className":6786},[52,246],[31,6788,6646],{"className":6789},[52,246],[6267,6791,6792],{},[31,6793,6795],{"className":6794},[34],[31,6796,6798],{"className":6797,"ariaHidden":39},[38],[31,6799,6801,6804,6807],{"className":6800},[43],[31,6802],{"className":6803,"style":209},[47],[31,6805,269],{"className":6806},[52],[31,6808,6810,6813],{"className":6809},[52],[31,6811,837],{"className":6812},[52],[31,6814,6816],{"className":6815},[219],[31,6817,6819],{"className":6818},[223],[31,6820,6822],{"className":6821},[227],[31,6823,6825],{"className":6824,"style":209},[231],[31,6826,6827,6830],{"style":234},[31,6828],{"className":6829,"style":239},[238],[31,6831,6833],{"className":6832},[243,244,245,246],[31,6834,6836],{"className":6835},[52,246],[31,6837,6839],{"className":6838},[52,246],"27",[11,6841,6842,6843,6858,6859,6903,6904,781],{},"Even storing ",[31,6844,6846],{"className":6845},[34],[31,6847,6849],{"className":6848,"ariaHidden":39},[38],[31,6850,6852,6855],{"className":6851},[43],[31,6853],{"className":6854,"style":188},[47],[31,6856,193],{"className":6857,"style":192},[52,53]," once for a modest ",[31,6860,6862],{"className":6861},[34],[31,6863,6865],{"className":6864,"ariaHidden":39},[38],[31,6866,6868,6871,6874],{"className":6867},[43],[31,6869],{"className":6870,"style":209},[47],[31,6872,269],{"className":6873},[52],[31,6875,6877,6880],{"className":6876},[52],[31,6878,837],{"className":6879},[52],[31,6881,6883],{"className":6882},[219],[31,6884,6886],{"className":6885},[223],[31,6887,6889],{"className":6888},[227],[31,6890,6892],{"className":6891,"style":209},[231],[31,6893,6894,6897],{"style":234},[31,6895],{"className":6896,"style":239},[238],[31,6898,6900],{"className":6899},[243,244,245,246],[31,6901,6405],{"className":6902},[52,246],"-parameter network needs a terabyte at\nsingle precision; inverting it is hopeless. And the optimizer must do this ",[164,6905,6906],{},"every step",[5976,6908,6910],{"id":6909},"newton-is-attracted-to-saddle-points","Newton is attracted to saddle points",[11,6912,6913,6914,6917,6918,6921,6922,6989,6990,7007,7008,7061,7062,7114],{},"The second wall is qualitative, and worse: where it does run, Newton's method walks\n",[164,6915,6916],{},"toward"," the very saddle points that\n",[15,6919,6920],{"href":257},"dominate high-dimensional loss surfaces",".\nDiagonalize ",[31,6923,6925],{"className":6924},[34],[31,6926,6928,6946],{"className":6927,"ariaHidden":39},[38],[31,6929,6931,6934,6937,6940,6943],{"className":6930},[43],[31,6932],{"className":6933,"style":188},[47],[31,6935,193],{"className":6936,"style":192},[52,53],[31,6938],{"className":6939,"style":60},[59],[31,6941,65],{"className":6942},[64],[31,6944],{"className":6945,"style":60},[59],[31,6947,6949,6952,6956,6960],{"className":6948},[43],[31,6950],{"className":6951,"style":1295},[47],[31,6953,6955],{"className":6954},[52,53],"Q",[31,6957,6959],{"className":6958},[52],"Λ",[31,6961,6963,6966],{"className":6962},[52],[31,6964,6955],{"className":6965},[52,53],[31,6967,6969],{"className":6968},[219],[31,6970,6972],{"className":6971},[223],[31,6973,6975],{"className":6974},[227],[31,6976,6978],{"className":6977,"style":1314},[231],[31,6979,6980,6983],{"style":234},[31,6981],{"className":6982,"style":239},[238],[31,6984,6986],{"className":6985},[243,244,245,246],[31,6987,513],{"className":6988},[52,246]," and resolve the Newton step into the eigenbasis.\nAlong eigendirection ",[31,6991,6993],{"className":6992},[34],[31,6994,6996],{"className":6995,"ariaHidden":39},[38],[31,6997,6999,7003],{"className":6998},[43],[31,7000],{"className":7001,"style":7002},[47],"height:0.6595em;",[31,7004,7006],{"className":7005},[52,53],"i",", with eigenvalue ",[31,7009,7011],{"className":7010},[34],[31,7012,7014],{"className":7013,"ariaHidden":39},[38],[31,7015,7017,7020],{"className":7016},[43],[31,7018],{"className":7019,"style":3217},[47],[31,7021,7023,7026],{"className":7022},[52],[31,7024,1923],{"className":7025},[52,53],[31,7027,7029],{"className":7028},[219],[31,7030,7032,7053],{"className":7031},[223,553],[31,7033,7035,7050],{"className":7034},[227],[31,7036,7039],{"className":7037,"style":7038},[231],"height:0.3117em;",[31,7040,7041,7044],{"style":876},[31,7042],{"className":7043,"style":239},[238],[31,7045,7047],{"className":7046},[243,244,245,246],[31,7048,7006],{"className":7049},[52,53,246],[31,7051,607],{"className":7052},[606],[31,7054,7056],{"className":7055},[227],[31,7057,7059],{"className":7058,"style":895},[231],[31,7060],{}," and gradient component ",[31,7063,7065],{"className":7064},[34],[31,7066,7068],{"className":7067,"ariaHidden":39},[38],[31,7069,7071,7074],{"className":7070},[43],[31,7072],{"className":7073,"style":48},[47],[31,7075,7077,7080],{"className":7076},[52],[31,7078,55],{"className":7079,"style":54},[52,53],[31,7081,7083],{"className":7082},[219],[31,7084,7086,7106],{"className":7085},[223,553],[31,7087,7089,7103],{"className":7088},[227],[31,7090,7092],{"className":7091,"style":7038},[231],[31,7093,7094,7097],{"style":4477},[31,7095],{"className":7096,"style":239},[238],[31,7098,7100],{"className":7099},[243,244,245,246],[31,7101,7006],{"className":7102},[52,53,246],[31,7104,607],{"className":7105},[606],[31,7107,7109],{"className":7108},[227],[31,7110,7112],{"className":7111,"style":895},[231],[31,7113],{},", the\nstep is",[31,7116,7118],{"className":7117},[382],[31,7119,7121],{"className":7120},[34],[31,7122,7124,7180],{"className":7123,"ariaHidden":39},[38],[31,7125,7127,7131,7171,7174,7177],{"className":7126},[43],[31,7128],{"className":7129,"style":7130},[47],"height:0.8333em;vertical-align:-0.15em;",[31,7132,7134,7137],{"className":7133},[52],[31,7135,309],{"className":7136},[52],[31,7138,7140],{"className":7139},[219],[31,7141,7143,7163],{"className":7142},[223,553],[31,7144,7146,7160],{"className":7145},[227],[31,7147,7149],{"className":7148,"style":7038},[231],[31,7150,7151,7154],{"style":876},[31,7152],{"className":7153,"style":239},[238],[31,7155,7157],{"className":7156},[243,244,245,246],[31,7158,7006],{"className":7159},[52,53,246],[31,7161,607],{"className":7162},[606],[31,7164,7166],{"className":7165},[227],[31,7167,7169],{"className":7168,"style":895},[231],[31,7170],{},[31,7172],{"className":7173,"style":60},[59],[31,7175,65],{"className":7176},[64],[31,7178],{"className":7179,"style":60},[59],[31,7181,7183,7187,7190,7330],{"className":7182},[43],[31,7184],{"className":7185,"style":7186},[47],"height:1.9436em;vertical-align:-0.836em;",[31,7188,145],{"className":7189},[52],[31,7191,7193,7196,7327],{"className":7192},[52],[31,7194],{"className":7195},[88,545],[31,7197,7199],{"className":7198},[549],[31,7200,7202,7318],{"className":7201},[223,553],[31,7203,7205,7315],{"className":7204},[227],[31,7206,7209,7258,7266],{"className":7207,"style":7208},[231],"height:1.1076em;",[31,7210,7212,7215],{"style":7211},"top:-2.314em;",[31,7213],{"className":7214,"style":567},[238],[31,7216,7218],{"className":7217},[52],[31,7219,7221,7224],{"className":7220},[52],[31,7222,1923],{"className":7223},[52,53],[31,7225,7227],{"className":7226},[219],[31,7228,7230,7250],{"className":7229},[223,553],[31,7231,7233,7247],{"className":7232},[227],[31,7234,7236],{"className":7235,"style":7038},[231],[31,7237,7238,7241],{"style":876},[31,7239],{"className":7240,"style":239},[238],[31,7242,7244],{"className":7243},[243,244,245,246],[31,7245,7006],{"className":7246},[52,53,246],[31,7248,607],{"className":7249},[606],[31,7251,7253],{"className":7252},[227],[31,7254,7256],{"className":7255,"style":895},[231],[31,7257],{},[31,7259,7260,7263],{"style":579},[31,7261],{"className":7262,"style":567},[238],[31,7264],{"className":7265,"style":587},[586],[31,7267,7269,7272],{"style":7268},"top:-3.677em;",[31,7270],{"className":7271,"style":567},[238],[31,7273,7275],{"className":7274},[52],[31,7276,7278,7281],{"className":7277},[52],[31,7279,55],{"className":7280,"style":54},[52,53],[31,7282,7284],{"className":7283},[219],[31,7285,7287,7307],{"className":7286},[223,553],[31,7288,7290,7304],{"className":7289},[227],[31,7291,7293],{"className":7292,"style":7038},[231],[31,7294,7295,7298],{"style":4477},[31,7296],{"className":7297,"style":239},[238],[31,7299,7301],{"className":7300},[243,244,245,246],[31,7302,7006],{"className":7303},[52,53,246],[31,7305,607],{"className":7306},[606],[31,7308,7310],{"className":7309},[227],[31,7311,7313],{"className":7312,"style":895},[231],[31,7314],{},[31,7316,607],{"className":7317},[606],[31,7319,7321],{"className":7320},[227],[31,7322,7325],{"className":7323,"style":7324},[231],"height:0.836em;",[31,7326],{},[31,7328],{"className":7329},[98,545],[31,7331,781],{"className":7332},[52],[11,7334,7335,7336,7391,7392,7395,7396,7411,7412,7464,7465,7468,7469,7472,7473,7506,7507,781,7522],{},"Here is the defect. The first-order step ",[31,7337,7339],{"className":7338},[34],[31,7340,7342],{"className":7341,"ariaHidden":39},[38],[31,7343,7345,7348,7351],{"className":7344},[43],[31,7346],{"className":7347,"style":1086},[47],[31,7349,145],{"className":7350},[52],[31,7352,7354,7357],{"className":7353},[52],[31,7355,55],{"className":7356,"style":54},[52,53],[31,7358,7360],{"className":7359},[219],[31,7361,7363,7383],{"className":7362},[223,553],[31,7364,7366,7380],{"className":7365},[227],[31,7367,7369],{"className":7368,"style":7038},[231],[31,7370,7371,7374],{"style":4477},[31,7372],{"className":7373,"style":239},[238],[31,7375,7377],{"className":7376},[243,244,245,246],[31,7378,7006],{"className":7379},[52,53,246],[31,7381,607],{"className":7382},[606],[31,7384,7386],{"className":7385},[227],[31,7387,7389],{"className":7388,"style":895},[231],[31,7390],{}," always moves ",[164,7393,7394],{},"downhill"," along axis ",[31,7397,7399],{"className":7398},[34],[31,7400,7402],{"className":7401,"ariaHidden":39},[38],[31,7403,7405,7408],{"className":7404},[43],[31,7406],{"className":7407,"style":7002},[47],[31,7409,7006],{"className":7410},[52,53],".\nThe Newton step divides by ",[31,7413,7415],{"className":7414},[34],[31,7416,7418],{"className":7417,"ariaHidden":39},[38],[31,7419,7421,7424],{"className":7420},[43],[31,7422],{"className":7423,"style":3217},[47],[31,7425,7427,7430],{"className":7426},[52],[31,7428,1923],{"className":7429},[52,53],[31,7431,7433],{"className":7432},[219],[31,7434,7436,7456],{"className":7435},[223,553],[31,7437,7439,7453],{"className":7438},[227],[31,7440,7442],{"className":7441,"style":7038},[231],[31,7443,7444,7447],{"style":876},[31,7445],{"className":7446,"style":239},[238],[31,7448,7450],{"className":7449},[243,244,245,246],[31,7451,7006],{"className":7452},[52,53,246],[31,7454,607],{"className":7455},[606],[31,7457,7459],{"className":7458},[227],[31,7460,7462],{"className":7461,"style":895},[231],[31,7463],{},", and dividing by a ",[26,7466,7467],{},"negative"," eigenvalue\n",[164,7470,7471],{},"flips the sign"," — turning a downhill move into an uphill one, straight toward the\ncritical point. Newton does not distinguish whether a critical point is a minimum,\na maximum, or a saddle; it converges to all of them, because it solves ",[31,7474,7476],{"className":7475},[34],[31,7477,7479,7497],{"className":7478,"ariaHidden":39},[38],[31,7480,7482,7485,7488,7491,7494],{"className":7481},[43],[31,7483],{"className":7484,"style":48},[47],[31,7486,55],{"className":7487,"style":54},[52,53],[31,7489],{"className":7490,"style":60},[59],[31,7492,65],{"className":7493},[64],[31,7495],{"className":7496,"style":60},[59],[31,7498,7500,7503],{"className":7499},[43],[31,7501],{"className":7502,"style":833},[47],[31,7504,837],{"className":7505},[52],"\nrather than minimizing ",[31,7508,7510],{"className":7509},[34],[31,7511,7513],{"className":7512,"ariaHidden":39},[38],[31,7514,7516,7519],{"className":7515},[43],[31,7517],{"className":7518,"style":188},[47],[31,7520,84],{"className":7521},[52,83],[261,7523,7524],{},[15,7525,5633],{"href":7526,"ariaDescribedBy":7527,"dataFootnoteRef":6,"id":7528},"#user-content-fn-gf-saddlefree",[267],"user-content-fnref-gf-saddlefree",[6170,7530,7531,7754],{},[6173,7532,7533],{},[6176,7534,7535,7538,7592,7650,7751],{},[6179,7536,7537],{},"Eigendirection",[6179,7539,7540],{},[31,7541,7543],{"className":7542},[34],[31,7544,7546],{"className":7545,"ariaHidden":39},[38],[31,7547,7549,7552],{"className":7548},[43],[31,7550],{"className":7551,"style":3217},[47],[31,7553,7555,7558],{"className":7554},[52],[31,7556,1923],{"className":7557},[52,53],[31,7559,7561],{"className":7560},[219],[31,7562,7564,7584],{"className":7563},[223,553],[31,7565,7567,7581],{"className":7566},[227],[31,7568,7570],{"className":7569,"style":7038},[231],[31,7571,7572,7575],{"style":876},[31,7573],{"className":7574,"style":239},[238],[31,7576,7578],{"className":7577},[243,244,245,246],[31,7579,7006],{"className":7580},[52,53,246],[31,7582,607],{"className":7583},[606],[31,7585,7587],{"className":7586},[227],[31,7588,7590],{"className":7589,"style":895},[231],[31,7591],{},[6179,7593,7594,7595],{},"first-order step ",[31,7596,7598],{"className":7597},[34],[31,7599,7601],{"className":7600,"ariaHidden":39},[38],[31,7602,7604,7607,7610],{"className":7603},[43],[31,7605],{"className":7606,"style":1086},[47],[31,7608,145],{"className":7609},[52],[31,7611,7613,7616],{"className":7612},[52],[31,7614,55],{"className":7615,"style":54},[52,53],[31,7617,7619],{"className":7618},[219],[31,7620,7622,7642],{"className":7621},[223,553],[31,7623,7625,7639],{"className":7624},[227],[31,7626,7628],{"className":7627,"style":7038},[231],[31,7629,7630,7633],{"style":4477},[31,7631],{"className":7632,"style":239},[238],[31,7634,7636],{"className":7635},[243,244,245,246],[31,7637,7006],{"className":7638},[52,53,246],[31,7640,607],{"className":7641},[606],[31,7643,7645],{"className":7644},[227],[31,7646,7648],{"className":7647,"style":895},[231],[31,7649],{},[6179,7651,7652,7653],{},"Newton step ",[31,7654,7656],{"className":7655},[34],[31,7657,7659],{"className":7658,"ariaHidden":39},[38],[31,7660,7662,7665,7668,7708,7711],{"className":7661},[43],[31,7663],{"className":7664,"style":75},[47],[31,7666,145],{"className":7667},[52],[31,7669,7671,7674],{"className":7670},[52],[31,7672,55],{"className":7673,"style":54},[52,53],[31,7675,7677],{"className":7676},[219],[31,7678,7680,7700],{"className":7679},[223,553],[31,7681,7683,7697],{"className":7682},[227],[31,7684,7686],{"className":7685,"style":7038},[231],[31,7687,7688,7691],{"style":4477},[31,7689],{"className":7690,"style":239},[238],[31,7692,7694],{"className":7693},[243,244,245,246],[31,7695,7006],{"className":7696},[52,53,246],[31,7698,607],{"className":7699},[606],[31,7701,7703],{"className":7702},[227],[31,7704,7706],{"className":7705,"style":895},[231],[31,7707],{},[31,7709,1971],{"className":7710},[52],[31,7712,7714,7717],{"className":7713},[52],[31,7715,1923],{"className":7716},[52,53],[31,7718,7720],{"className":7719},[219],[31,7721,7723,7743],{"className":7722},[223,553],[31,7724,7726,7740],{"className":7725},[227],[31,7727,7729],{"className":7728,"style":7038},[231],[31,7730,7731,7734],{"style":876},[31,7732],{"className":7733,"style":239},[238],[31,7735,7737],{"className":7736},[243,244,245,246],[31,7738,7006],{"className":7739},[52,53,246],[31,7741,607],{"className":7742},[606],[31,7744,7746],{"className":7745},[227],[31,7747,7749],{"className":7748,"style":895},[231],[31,7750],{},[6179,7752,7753],{},"effect",[6262,7755,7756,7842,7931],{},[6176,7757,7758,7761,7834,7836,7839],{},[6267,7759,7760],{},"up-curving",[6267,7762,7763],{},[31,7764,7766],{"className":7765},[34],[31,7767,7769,7825],{"className":7768,"ariaHidden":39},[38],[31,7770,7772,7775,7815,7818,7822],{"className":7771},[43],[31,7773],{"className":7774,"style":3217},[47],[31,7776,7778,7781],{"className":7777},[52],[31,7779,1923],{"className":7780},[52,53],[31,7782,7784],{"className":7783},[219],[31,7785,7787,7807],{"className":7786},[223,553],[31,7788,7790,7804],{"className":7789},[227],[31,7791,7793],{"className":7792,"style":7038},[231],[31,7794,7795,7798],{"style":876},[31,7796],{"className":7797,"style":239},[238],[31,7799,7801],{"className":7800},[243,244,245,246],[31,7802,7006],{"className":7803},[52,53,246],[31,7805,607],{"className":7806},[606],[31,7808,7810],{"className":7809},[227],[31,7811,7813],{"className":7812,"style":895},[231],[31,7814],{},[31,7816],{"className":7817,"style":60},[59],[31,7819,7821],{"className":7820},[64],">",[31,7823],{"className":7824,"style":60},[59],[31,7826,7828,7831],{"className":7827},[43],[31,7829],{"className":7830,"style":833},[47],[31,7832,837],{"className":7833},[52],[6267,7835,7394],{},[6267,7837,7838],{},"downhill, rescaled",[6267,7840,7841],{},"toward minimum (correct)",[6176,7843,7844,7847,7920,7922,7928],{},[6267,7845,7846],{},"down-curving",[6267,7848,7849],{},[31,7850,7852],{"className":7851},[34],[31,7853,7855,7911],{"className":7854,"ariaHidden":39},[38],[31,7856,7858,7861,7901,7904,7908],{"className":7857},[43],[31,7859],{"className":7860,"style":3217},[47],[31,7862,7864,7867],{"className":7863},[52],[31,7865,1923],{"className":7866},[52,53],[31,7868,7870],{"className":7869},[219],[31,7871,7873,7893],{"className":7872},[223,553],[31,7874,7876,7890],{"className":7875},[227],[31,7877,7879],{"className":7878,"style":7038},[231],[31,7880,7881,7884],{"style":876},[31,7882],{"className":7883,"style":239},[238],[31,7885,7887],{"className":7886},[243,244,245,246],[31,7888,7006],{"className":7889},[52,53,246],[31,7891,607],{"className":7892},[606],[31,7894,7896],{"className":7895},[227],[31,7897,7899],{"className":7898,"style":895},[231],[31,7900],{},[31,7902],{"className":7903,"style":60},[59],[31,7905,7907],{"className":7906},[64],"\u003C",[31,7909],{"className":7910,"style":60},[59],[31,7912,7914,7917],{"className":7913},[43],[31,7915],{"className":7916,"style":833},[47],[31,7918,837],{"className":7919},[52],[6267,7921,7394],{},[6267,7923,7924,7927],{},[164,7925,7926],{},"uphill"," (sign flips)",[6267,7929,7930],{},"toward saddle (wrong)",[6176,7932,7933,7936,8008,8011,8049],{},[6267,7934,7935],{},"flat",[6267,7937,7938],{},[31,7939,7941],{"className":7940},[34],[31,7942,7944,7999],{"className":7943,"ariaHidden":39},[38],[31,7945,7947,7950,7990,7993,7996],{"className":7946},[43],[31,7948],{"className":7949,"style":3217},[47],[31,7951,7953,7956],{"className":7952},[52],[31,7954,1923],{"className":7955},[52,53],[31,7957,7959],{"className":7958},[219],[31,7960,7962,7982],{"className":7961},[223,553],[31,7963,7965,7979],{"className":7964},[227],[31,7966,7968],{"className":7967,"style":7038},[231],[31,7969,7970,7973],{"style":876},[31,7971],{"className":7972,"style":239},[238],[31,7974,7976],{"className":7975},[243,244,245,246],[31,7977,7006],{"className":7978},[52,53,246],[31,7980,607],{"className":7981},[606],[31,7983,7985],{"className":7984},[227],[31,7986,7988],{"className":7987,"style":895},[231],[31,7989],{},[31,7991],{"className":7992,"style":60},[59],[31,7994,435],{"className":7995},[64],[31,7997],{"className":7998,"style":60},[59],[31,8000,8002,8005],{"className":8001},[43],[31,8003],{"className":8004,"style":833},[47],[31,8006,837],{"className":8007},[52],[6267,8009,8010],{},"small",[6267,8012,8013,8014,99],{},"blows up (",[31,8015,8017],{"className":8016},[34],[31,8018,8020,8040],{"className":8019,"ariaHidden":39},[38],[31,8021,8023,8027,8031,8034,8037],{"className":8022},[43],[31,8024],{"className":8025,"style":8026},[47],"height:0.6667em;vertical-align:-0.0833em;",[31,8028,8030],{"className":8029},[52],"÷",[31,8032],{"className":8033,"style":60},[59],[31,8035,435],{"className":8036},[64],[31,8038],{"className":8039,"style":60},[59],[31,8041,8043,8046],{"className":8042},[43],[31,8044],{"className":8045,"style":833},[47],[31,8047,837],{"className":8048},[52],[6267,8050,8051],{},"catastrophic step",[1507,8053],{"hash":8054},"a8162f396adc7234bf26a23bfb0ee4230653ac39f0c6a7f77f1ec144050a0e30",[11,8056,8057,8058,8061,8062,8109,8110,8164,8165,8168,8169,8249,8250,8269,8270,781],{},"Since high-loss critical points are overwhelmingly saddles, an unmodified Newton step\nis not merely useless but actively harmful. The fix is to force the curvature positive:\n",[26,8059,8060],{},"saddle-free Newton"," replaces ",[31,8063,8065],{"className":8064},[34],[31,8066,8068],{"className":8067,"ariaHidden":39},[38],[31,8069,8071,8074],{"className":8070},[43],[31,8072],{"className":8073,"style":209},[47],[31,8075,8077,8080],{"className":8076},[52],[31,8078,193],{"className":8079,"style":192},[52,53],[31,8081,8083],{"className":8082},[219],[31,8084,8086],{"className":8085},[223],[31,8087,8089],{"className":8088},[227],[31,8090,8092],{"className":8091,"style":209},[231],[31,8093,8094,8097],{"style":234},[31,8095],{"className":8096,"style":239},[238],[31,8098,8100],{"className":8099},[243,244,245,246],[31,8101,8103,8106],{"className":8102},[52,246],[31,8104,145],{"className":8105},[52,246],[31,8107,269],{"className":8108},[52,246]," with ",[31,8111,8113],{"className":8112},[34],[31,8114,8116],{"className":8115,"ariaHidden":39},[38],[31,8117,8119,8122,8126,8129],{"className":8118},[43],[31,8120],{"className":8121,"style":3071},[47],[31,8123,8125],{"className":8124},[52],"∣",[31,8127,193],{"className":8128,"style":192},[52,53],[31,8130,8132,8135],{"className":8131},[52],[31,8133,8125],{"className":8134},[52],[31,8136,8138],{"className":8137},[219],[31,8139,8141],{"className":8140},[223],[31,8142,8144],{"className":8143},[227],[31,8145,8147],{"className":8146,"style":209},[231],[31,8148,8149,8152],{"style":234},[31,8150],{"className":8151,"style":239},[238],[31,8153,8155],{"className":8154},[243,244,245,246],[31,8156,8158,8161],{"className":8157},[52,246],[31,8159,145],{"className":8160},[52,246],[31,8162,269],{"className":8163},[52,246]," (take the absolute values of\nthe eigenvalues), and classic ",[26,8166,8167],{},"damping"," (Levenberg–Marquardt) solves\n",[31,8170,8172],{"className":8171},[34],[31,8173,8175,8196],{"className":8174,"ariaHidden":39},[38],[31,8176,8178,8181,8184,8187,8190,8193],{"className":8177},[43],[31,8179],{"className":8180,"style":75},[47],[31,8182,89],{"className":8183},[88],[31,8185,193],{"className":8186,"style":192},[52,53],[31,8188],{"className":8189,"style":140},[59],[31,8191,410],{"className":8192},[144],[31,8194],{"className":8195,"style":140},[59],[31,8197,8199,8202,8206,8211,8246],{"className":8198},[43],[31,8200],{"className":8201,"style":3071},[47],[31,8203,8205],{"className":8204},[52,53],"μ",[31,8207,8210],{"className":8208,"style":8209},[52,53],"margin-right:0.0785em;","I",[31,8212,8214,8217],{"className":8213},[98],[31,8215,99],{"className":8216},[98],[31,8218,8220],{"className":8219},[219],[31,8221,8223],{"className":8222},[223],[31,8224,8226],{"className":8225},[227],[31,8227,8229],{"className":8228,"style":209},[231],[31,8230,8231,8234],{"style":234},[31,8232],{"className":8233,"style":239},[238],[31,8235,8237],{"className":8236},[243,244,245,246],[31,8238,8240,8243],{"className":8239},[52,246],[31,8241,145],{"className":8242},[52,246],[31,8244,269],{"className":8245},[52,246],[31,8247,55],{"className":8248,"style":54},[52,53],", adding ",[31,8251,8253],{"className":8252},[34],[31,8254,8256],{"className":8255,"ariaHidden":39},[38],[31,8257,8259,8263,8266],{"className":8258},[43],[31,8260],{"className":8261,"style":8262},[47],"height:0.8778em;vertical-align:-0.1944em;",[31,8264,8205],{"className":8265},[52,53],[31,8267,8210],{"className":8268,"style":8209},[52,53]," until the matrix is positive definite. Both restore\ndescent, but both still need ",[31,8271,8273],{"className":8272},[34],[31,8274,8276],{"className":8275,"ariaHidden":39},[38],[31,8277,8279,8282],{"className":8278},[43],[31,8280],{"className":8281,"style":188},[47],[31,8283,193],{"className":8284,"style":192},[52,53],[5976,8286,8288],{"id":8287},"minibatch-noise-corrupts-the-estimate","Minibatch noise corrupts the estimate",[11,8290,8291,8292,8307,8308,8323,8324,8327,8328,8380,8381,8479],{},"The third wall is statistical. Deep nets are trained on minibatches, so both ",[31,8293,8295],{"className":8294},[34],[31,8296,8298],{"className":8297,"ariaHidden":39},[38],[31,8299,8301,8304],{"className":8300},[43],[31,8302],{"className":8303,"style":48},[47],[31,8305,55],{"className":8306,"style":54},[52,53]," and\n",[31,8309,8311],{"className":8310},[34],[31,8312,8314],{"className":8313,"ariaHidden":39},[38],[31,8315,8317,8320],{"className":8316},[43],[31,8318],{"className":8319,"style":188},[47],[31,8321,193],{"className":8322,"style":192},[52,53]," are noisy estimates from a sample. First-order steps tolerate this (noise averages\nout over many steps), but the Newton step ",[26,8325,8326],{},"divides by"," the Hessian estimate, and a\nratio amplifies error in the denominator. A small eigenvalue ",[31,8329,8331],{"className":8330},[34],[31,8332,8334],{"className":8333,"ariaHidden":39},[38],[31,8335,8337,8340],{"className":8336},[43],[31,8338],{"className":8339,"style":3217},[47],[31,8341,8343,8346],{"className":8342},[52],[31,8344,1923],{"className":8345},[52,53],[31,8347,8349],{"className":8348},[219],[31,8350,8352,8372],{"className":8351},[223,553],[31,8353,8355,8369],{"className":8354},[227],[31,8356,8358],{"className":8357,"style":7038},[231],[31,8359,8360,8363],{"style":876},[31,8361],{"className":8362,"style":239},[238],[31,8364,8366],{"className":8365},[243,244,245,246],[31,8367,7006],{"className":8368},[52,53,246],[31,8370,607],{"className":8371},[606],[31,8373,8375],{"className":8374},[227],[31,8376,8378],{"className":8377,"style":895},[231],[31,8379],{}," estimated with\neven modest relative error produces a huge swing in ",[31,8382,8384],{"className":8383},[34],[31,8385,8387],{"className":8386,"ariaHidden":39},[38],[31,8388,8390,8393,8396,8436,8439],{"className":8389},[43],[31,8391],{"className":8392,"style":75},[47],[31,8394,145],{"className":8395},[52],[31,8397,8399,8402],{"className":8398},[52],[31,8400,55],{"className":8401,"style":54},[52,53],[31,8403,8405],{"className":8404},[219],[31,8406,8408,8428],{"className":8407},[223,553],[31,8409,8411,8425],{"className":8410},[227],[31,8412,8414],{"className":8413,"style":7038},[231],[31,8415,8416,8419],{"style":4477},[31,8417],{"className":8418,"style":239},[238],[31,8420,8422],{"className":8421},[243,244,245,246],[31,8423,7006],{"className":8424},[52,53,246],[31,8426,607],{"className":8427},[606],[31,8429,8431],{"className":8430},[227],[31,8432,8434],{"className":8433,"style":895},[231],[31,8435],{},[31,8437,1971],{"className":8438},[52],[31,8440,8442,8445],{"className":8441},[52],[31,8443,1923],{"className":8444},[52,53],[31,8446,8448],{"className":8447},[219],[31,8449,8451,8471],{"className":8450},[223,553],[31,8452,8454,8468],{"className":8453},[227],[31,8455,8457],{"className":8456,"style":7038},[231],[31,8458,8459,8462],{"style":876},[31,8460],{"className":8461,"style":239},[238],[31,8463,8465],{"className":8464},[243,244,245,246],[31,8466,7006],{"className":8467},[52,53,246],[31,8469,607],{"className":8470},[606],[31,8472,8474],{"className":8473},[227],[31,8475,8477],{"className":8476,"style":895},[231],[31,8478],{},". The very curvature\ninformation Newton depends on is the part the noise corrupts most.",[6170,8481,8482,8495],{},[6173,8483,8484],{},[6176,8485,8486,8489,8492],{},[6179,8487,8488],{},"Assumption Newton needs",[6179,8490,8491],{},"Reality in deep learning",[6179,8493,8494],{},"Consequence",[6262,8496,8497,8646,8705,8740],{},[6176,8498,8499,8517,8628],{},[6267,8500,8501,8516],{},[31,8502,8504],{"className":8503},[34],[31,8505,8507],{"className":8506,"ariaHidden":39},[38],[31,8508,8510,8513],{"className":8509},[43],[31,8511],{"className":8512,"style":188},[47],[31,8514,193],{"className":8515,"style":192},[52,53]," small and storable",[6267,8518,8519,8534,8535,8579,8580,1669,8595],{},[31,8520,8522],{"className":8521},[34],[31,8523,8525],{"className":8524,"ariaHidden":39},[38],[31,8526,8528,8531],{"className":8527},[43],[31,8529],{"className":8530,"style":113},[47],[31,8532,5457],{"className":8533},[52,53]," up to ",[31,8536,8538],{"className":8537},[34],[31,8539,8541],{"className":8540,"ariaHidden":39},[38],[31,8542,8544,8547,8550],{"className":8543},[43],[31,8545],{"className":8546,"style":209},[47],[31,8548,269],{"className":8549},[52],[31,8551,8553,8556],{"className":8552},[52],[31,8554,837],{"className":8555},[52],[31,8557,8559],{"className":8558},[219],[31,8560,8562],{"className":8561},[223],[31,8563,8565],{"className":8564},[227],[31,8566,8568],{"className":8567,"style":209},[231],[31,8569,8570,8573],{"style":234},[31,8571],{"className":8572,"style":239},[238],[31,8574,8576],{"className":8575},[243,244,245,246],[31,8577,6452],{"className":8578},[52,246],", ",[31,8581,8583],{"className":8582},[34],[31,8584,8586],{"className":8585,"ariaHidden":39},[38],[31,8587,8589,8592],{"className":8588},[43],[31,8590],{"className":8591,"style":188},[47],[31,8593,193],{"className":8594,"style":192},[52,53],[31,8596,8598],{"className":8597},[34],[31,8599,8601,8619],{"className":8600,"ariaHidden":39},[38],[31,8602,8604,8607,8610,8613,8616],{"className":8603},[43],[31,8605],{"className":8606,"style":133},[47],[31,8608,5457],{"className":8609},[52,53],[31,8611],{"className":8612,"style":140},[59],[31,8614,6001],{"className":8615},[144],[31,8617],{"className":8618,"style":140},[59],[31,8620,8622,8625],{"className":8621},[43],[31,8623],{"className":8624,"style":113},[47],[31,8626,5457],{"className":8627},[52,53],[6267,8629,8630,8631],{},"cannot form or invert ",[31,8632,8634],{"className":8633},[34],[31,8635,8637],{"className":8636,"ariaHidden":39},[38],[31,8638,8640,8643],{"className":8639},[43],[31,8641],{"className":8642,"style":188},[47],[31,8644,193],{"className":8645,"style":192},[52,53],[6176,8647,8648,8684,8702],{},[6267,8649,8650,8683],{},[31,8651,8653],{"className":8652},[34],[31,8654,8656,8674],{"className":8655,"ariaHidden":39},[38],[31,8657,8659,8662,8665,8668,8671],{"className":8658},[43],[31,8660],{"className":8661,"style":813},[47],[31,8663,193],{"className":8664,"style":192},[52,53],[31,8666],{"className":8667,"style":60},[59],[31,8669,823],{"className":8670},[64],[31,8672],{"className":8673,"style":60},[59],[31,8675,8677,8680],{"className":8676},[43],[31,8678],{"className":8679,"style":833},[47],[31,8681,837],{"className":8682},[52]," (a bowl)",[6267,8685,8686,8687],{},"mostly saddles, mixed-sign ",[31,8688,8690],{"className":8689},[34],[31,8691,8693],{"className":8692,"ariaHidden":39},[38],[31,8694,8696,8699],{"className":8695},[43],[31,8697],{"className":8698,"style":188},[47],[31,8700,193],{"className":8701,"style":192},[52,53],[6267,8703,8704],{},"attracted to saddles",[6176,8706,8707,8734,8737],{},[6267,8708,8709,8733],{},[31,8710,8712],{"className":8711},[34],[31,8713,8715],{"className":8714,"ariaHidden":39},[38],[31,8716,8718,8721,8724,8727,8730],{"className":8717},[43],[31,8719],{"className":8720,"style":8262},[47],[31,8722,55],{"className":8723,"style":54},[52,53],[31,8725,666],{"className":8726},[665],[31,8728],{"className":8729,"style":623},[59],[31,8731,193],{"className":8732,"style":192},[52,53]," exact",[6267,8735,8736],{},"minibatch estimates, noisy",[6267,8738,8739],{},"division amplifies the noise",[6176,8741,8742,8745,8748],{},[6267,8743,8744],{},"problem deterministic",[6267,8746,8747],{},"stochastic objective",[6267,8749,8750],{},"step direction unstable",[11,8752,8753,8754,8757,8758,781],{},"The rest of the lesson is the response: methods that capture ",[164,8755,8756],{},"some"," curvature without\never forming, storing, or inverting ",[31,8759,8761],{"className":8760},[34],[31,8762,8764],{"className":8763,"ariaHidden":39},[38],[31,8765,8767,8770],{"className":8766},[43],[31,8768],{"className":8769,"style":188},[47],[31,8771,193],{"className":8772,"style":192},[52,53],[271,8774,8776],{"id":8775},"conjugate-gradients","Conjugate gradients",[11,8778,8779,8780,8783,8784,8786,8787,8790,8791,8806],{},"Steepest descent on a quadratic wastes effort because successive steps are orthogonal,\nand on a stretched bowl orthogonal steps zig-zag, each step partly ",[164,8781,8782],{},"undoing"," the\nprogress of the last along the previous direction. ",[26,8785,8776],{}," (CG) fixes\nthis by choosing search directions that are ",[164,8788,8789],{},"conjugate"," with respect to ",[31,8792,8794],{"className":8793},[34],[31,8795,8797],{"className":8796,"ariaHidden":39},[38],[31,8798,8800,8803],{"className":8799},[43],[31,8801],{"className":8802,"style":188},[47],[31,8804,193],{"className":8805,"style":192},[52,53],", so that\nminimizing along a new direction never disturbs the minimization already achieved along\nthe old ones.",[1511,8808,8809],{"type":1513},[11,8810,8811,8814,8815,8916,8917,8919,8920,8935,8936,9063,9064,9138,9139,9191,9192,9244,9245,9262,9263,9278],{},[26,8812,8813],{},"Definition (Conjugate directions)."," Two directions ",[31,8816,8818],{"className":8817},[34],[31,8819,8821],{"className":8820,"ariaHidden":39},[38],[31,8822,8824,8828,8868,8871,8874],{"className":8823},[43],[31,8825],{"className":8826,"style":8827},[47],"height:0.9805em;vertical-align:-0.2861em;",[31,8829,8831,8834],{"className":8830},[52],[31,8832,5457],{"className":8833},[52,53],[31,8835,8837],{"className":8836},[219],[31,8838,8840,8860],{"className":8839},[223,553],[31,8841,8843,8857],{"className":8842},[227],[31,8844,8846],{"className":8845,"style":7038},[231],[31,8847,8848,8851],{"style":876},[31,8849],{"className":8850,"style":239},[238],[31,8852,8854],{"className":8853},[243,244,245,246],[31,8855,7006],{"className":8856},[52,53,246],[31,8858,607],{"className":8859},[606],[31,8861,8863],{"className":8862},[227],[31,8864,8866],{"className":8865,"style":895},[231],[31,8867],{},[31,8869,666],{"className":8870},[665],[31,8872],{"className":8873,"style":623},[59],[31,8875,8877,8880],{"className":8876},[52],[31,8878,5457],{"className":8879},[52,53],[31,8881,8883],{"className":8882},[219],[31,8884,8886,8908],{"className":8885},[223,553],[31,8887,8889,8905],{"className":8888},[227],[31,8890,8892],{"className":8891,"style":7038},[231],[31,8893,8894,8897],{"style":876},[31,8895],{"className":8896,"style":239},[238],[31,8898,8900],{"className":8899},[243,244,245,246],[31,8901,8904],{"className":8902,"style":8903},[52,53,246],"margin-right:0.0572em;","j",[31,8906,607],{"className":8907},[606],[31,8909,8911],{"className":8910},[227],[31,8912,8914],{"className":8913,"style":4496},[231],[31,8915],{}," are ",[26,8918,8789],{},"\nwith respect to ",[31,8921,8923],{"className":8922},[34],[31,8924,8926],{"className":8925,"ariaHidden":39},[38],[31,8927,8929,8932],{"className":8928},[43],[31,8930],{"className":8931,"style":188},[47],[31,8933,193],{"className":8934,"style":192},[52,53]," if ",[31,8937,8939],{"className":8938},[34],[31,8940,8942,9054],{"className":8941,"ariaHidden":39},[38],[31,8943,8945,8949,9002,9005,9045,9048,9051],{"className":8944},[43],[31,8946],{"className":8947,"style":8948},[47],"height:1.1352em;vertical-align:-0.2861em;",[31,8950,8952,8955],{"className":8951},[52],[31,8953,5457],{"className":8954},[52,53],[31,8956,8958],{"className":8957},[219],[31,8959,8961,8993],{"className":8960},[223,553],[31,8962,8964,8990],{"className":8963},[227],[31,8965,8967,8979],{"className":8966,"style":1314},[231],[31,8968,8970,8973],{"style":8969},"top:-2.4413em;margin-left:0em;margin-right:0.05em;",[31,8971],{"className":8972,"style":239},[238],[31,8974,8976],{"className":8975},[243,244,245,246],[31,8977,7006],{"className":8978},[52,53,246],[31,8980,8981,8984],{"style":234},[31,8982],{"className":8983,"style":239},[238],[31,8985,8987],{"className":8986},[243,244,245,246],[31,8988,513],{"className":8989},[52,246],[31,8991,607],{"className":8992},[606],[31,8994,8996],{"className":8995},[227],[31,8997,9000],{"className":8998,"style":8999},[231],"height:0.2587em;",[31,9001],{},[31,9003,193],{"className":9004,"style":192},[52,53],[31,9006,9008,9011],{"className":9007},[52],[31,9009,5457],{"className":9010},[52,53],[31,9012,9014],{"className":9013},[219],[31,9015,9017,9037],{"className":9016},[223,553],[31,9018,9020,9034],{"className":9019},[227],[31,9021,9023],{"className":9022,"style":7038},[231],[31,9024,9025,9028],{"style":876},[31,9026],{"className":9027,"style":239},[238],[31,9029,9031],{"className":9030},[243,244,245,246],[31,9032,8904],{"className":9033,"style":8903},[52,53,246],[31,9035,607],{"className":9036},[606],[31,9038,9040],{"className":9039},[227],[31,9041,9043],{"className":9042,"style":4496},[231],[31,9044],{},[31,9046],{"className":9047,"style":60},[59],[31,9049,65],{"className":9050},[64],[31,9052],{"className":9053,"style":60},[59],[31,9055,9057,9060],{"className":9056},[43],[31,9058],{"className":9059,"style":833},[47],[31,9061,837],{"className":9062},[52]," for ",[31,9065,9067],{"className":9066},[34],[31,9068,9070,9128],{"className":9069,"ariaHidden":39},[38],[31,9071,9073,9076,9079,9082,9125],{"className":9072},[43],[31,9074],{"className":9075,"style":4363},[47],[31,9077,7006],{"className":9078},[52,53],[31,9080],{"className":9081,"style":60},[59],[31,9083,9085,9118,9122],{"className":9084},[64],[31,9086,9088],{"className":9087},[64],[31,9089,9092],{"className":9090},[52,9091],"vbox",[31,9093,9096],{"className":9094},[9095],"thinbox",[31,9097,9100,9103,9114],{"className":9098},[9099],"rlap",[31,9101],{"className":9102,"style":4363},[47],[31,9104,9107],{"className":9105},[9106],"inner",[31,9108,9110],{"className":9109},[52],[31,9111,9113],{"className":9112},[64],"",[31,9115],{"className":9116},[9117],"fix",[31,9119],{"className":9120},[59,9121],"nobreak",[31,9123,65],{"className":9124},[64],[31,9126],{"className":9127,"style":60},[59],[31,9129,9131,9135],{"className":9130},[43],[31,9132],{"className":9133,"style":9134},[47],"height:0.854em;vertical-align:-0.1944em;",[31,9136,8904],{"className":9137,"style":8903},[52,53],". Minimizing a quadratic\nexactly along ",[31,9140,9142],{"className":9141},[34],[31,9143,9145],{"className":9144,"ariaHidden":39},[38],[31,9146,9148,9151],{"className":9147},[43],[31,9149],{"className":9150,"style":8827},[47],[31,9152,9154,9157],{"className":9153},[52],[31,9155,5457],{"className":9156},[52,53],[31,9158,9160],{"className":9159},[219],[31,9161,9163,9183],{"className":9162},[223,553],[31,9164,9166,9180],{"className":9165},[227],[31,9167,9169],{"className":9168,"style":7038},[231],[31,9170,9171,9174],{"style":876},[31,9172],{"className":9173,"style":239},[238],[31,9175,9177],{"className":9176},[243,244,245,246],[31,9178,8904],{"className":9179,"style":8903},[52,53,246],[31,9181,607],{"className":9182},[606],[31,9184,9186],{"className":9185},[227],[31,9187,9189],{"className":9188,"style":4496},[231],[31,9190],{}," leaves its minimum along every earlier conjugate ",[31,9193,9195],{"className":9194},[34],[31,9196,9198],{"className":9197,"ariaHidden":39},[38],[31,9199,9201,9204],{"className":9200},[43],[31,9202],{"className":9203,"style":3217},[47],[31,9205,9207,9210],{"className":9206},[52],[31,9208,5457],{"className":9209},[52,53],[31,9211,9213],{"className":9212},[219],[31,9214,9216,9236],{"className":9215},[223,553],[31,9217,9219,9233],{"className":9218},[227],[31,9220,9222],{"className":9221,"style":7038},[231],[31,9223,9224,9227],{"style":876},[31,9225],{"className":9226,"style":239},[238],[31,9228,9230],{"className":9229},[243,244,245,246],[31,9231,7006],{"className":9232},[52,53,246],[31,9234,607],{"className":9235},[606],[31,9237,9239],{"className":9238},[227],[31,9240,9242],{"className":9241,"style":895},[231],[31,9243],{}," undisturbed,\nso ",[31,9246,9248],{"className":9247},[34],[31,9249,9251],{"className":9250,"ariaHidden":39},[38],[31,9252,9254,9258],{"className":9253},[43],[31,9255],{"className":9256,"style":9257},[47],"height:0.4306em;",[31,9259,9261],{"className":9260},[52,53],"n"," conjugate line-searches reach the exact minimum of an ",[31,9264,9266],{"className":9265},[34],[31,9267,9269],{"className":9268,"ariaHidden":39},[38],[31,9270,9272,9275],{"className":9271},[43],[31,9273],{"className":9274,"style":9257},[47],[31,9276,9261],{"className":9277},[52,53],"-dimensional quadratic\n— with no zig-zag and no Hessian inverse.",[11,9280,9281,9282,9339,9340,9616,9617,9632],{},"The new direction reuses the previous one, bent just enough to be conjugate to it. The\ncombining coefficient ",[31,9283,9285],{"className":9284},[34],[31,9286,9288],{"className":9287,"ariaHidden":39},[38],[31,9289,9291,9294],{"className":9290},[43],[31,9292],{"className":9293,"style":4363},[47],[31,9295,9297,9302],{"className":9296},[52],[31,9298,9301],{"className":9299,"style":9300},[52,53],"margin-right:0.0528em;","β",[31,9303,9305],{"className":9304},[219],[31,9306,9308,9331],{"className":9307},[223,553],[31,9309,9311,9328],{"className":9310},[227],[31,9312,9315],{"className":9313,"style":9314},[231],"height:0.2806em;",[31,9316,9318,9321],{"style":9317},"top:-2.55em;margin-left:-0.0528em;margin-right:0.05em;",[31,9319],{"className":9320,"style":239},[238],[31,9322,9324],{"className":9323},[243,244,245,246],[31,9325,9327],{"className":9326},[52,53,246],"t",[31,9329,607],{"className":9330},[606],[31,9332,9334],{"className":9333},[227],[31,9335,9337],{"className":9336,"style":895},[231],[31,9338],{}," (the Fletcher–Reeves form ",[31,9341,9343],{"className":9342},[34],[31,9344,9346,9401],{"className":9345,"ariaHidden":39},[38],[31,9347,9349,9352,9392,9395,9398],{"className":9348},[43],[31,9350],{"className":9351,"style":4363},[47],[31,9353,9355,9358],{"className":9354},[52],[31,9356,9301],{"className":9357,"style":9300},[52,53],[31,9359,9361],{"className":9360},[219],[31,9362,9364,9384],{"className":9363},[223,553],[31,9365,9367,9381],{"className":9366},[227],[31,9368,9370],{"className":9369,"style":9314},[231],[31,9371,9372,9375],{"style":9317},[31,9373],{"className":9374,"style":239},[238],[31,9376,9378],{"className":9377},[243,244,245,246],[31,9379,9327],{"className":9380},[52,53,246],[31,9382,607],{"className":9383},[606],[31,9385,9387],{"className":9386},[227],[31,9388,9390],{"className":9389,"style":895},[231],[31,9391],{},[31,9393],{"className":9394,"style":60},[59],[31,9396,65],{"className":9397},[64],[31,9399],{"className":9400,"style":60},[59],[31,9402,9404,9408,9461,9501,9504,9566],{"className":9403},[43],[31,9405],{"className":9406,"style":9407},[47],"height:1.1555em;vertical-align:-0.3064em;",[31,9409,9411,9414],{"className":9410},[52],[31,9412,55],{"className":9413,"style":54},[52,53],[31,9415,9417],{"className":9416},[219],[31,9418,9420,9452],{"className":9419},[223,553],[31,9421,9423,9449],{"className":9422},[227],[31,9424,9426,9438],{"className":9425,"style":1314},[231],[31,9427,9429,9432],{"style":9428},"top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;",[31,9430],{"className":9431,"style":239},[238],[31,9433,9435],{"className":9434},[243,244,245,246],[31,9436,9327],{"className":9437},[52,53,246],[31,9439,9440,9443],{"style":234},[31,9441],{"className":9442,"style":239},[238],[31,9444,9446],{"className":9445},[243,244,245,246],[31,9447,513],{"className":9448},[52,246],[31,9450,607],{"className":9451},[606],[31,9453,9455],{"className":9454},[227],[31,9456,9459],{"className":9457,"style":9458},[231],"height:0.247em;",[31,9460],{},[31,9462,9464,9467],{"className":9463},[52],[31,9465,55],{"className":9466,"style":54},[52,53],[31,9468,9470],{"className":9469},[219],[31,9471,9473,9493],{"className":9472},[223,553],[31,9474,9476,9490],{"className":9475},[227],[31,9477,9479],{"className":9478,"style":9314},[231],[31,9480,9481,9484],{"style":4477},[31,9482],{"className":9483,"style":239},[238],[31,9485,9487],{"className":9486},[243,244,245,246],[31,9488,9327],{"className":9489},[52,53,246],[31,9491,607],{"className":9492},[606],[31,9494,9496],{"className":9495},[227],[31,9497,9499],{"className":9498,"style":895},[231],[31,9500],{},[31,9502,1971],{"className":9503},[52],[31,9505,9507,9510],{"className":9506},[52],[31,9508,55],{"className":9509,"style":54},[52,53],[31,9511,9513],{"className":9512},[219],[31,9514,9516,9557],{"className":9515},[223,553],[31,9517,9519,9554],{"className":9518},[227],[31,9520,9522,9543],{"className":9521,"style":1314},[231],[31,9523,9525,9528],{"style":9524},"top:-2.4519em;margin-left:-0.0359em;margin-right:0.05em;",[31,9526],{"className":9527,"style":239},[238],[31,9529,9531],{"className":9530},[243,244,245,246],[31,9532,9534,9537,9540],{"className":9533},[52,246],[31,9535,9327],{"className":9536},[52,53,246],[31,9538,145],{"className":9539},[144,246],[31,9541,269],{"className":9542},[52,246],[31,9544,9545,9548],{"style":234},[31,9546],{"className":9547,"style":239},[238],[31,9549,9551],{"className":9550},[243,244,245,246],[31,9552,513],{"className":9553},[52,246],[31,9555,607],{"className":9556},[606],[31,9558,9560],{"className":9559},[227],[31,9561,9564],{"className":9562,"style":9563},[231],"height:0.3064em;",[31,9565],{},[31,9567,9569,9572],{"className":9568},[52],[31,9570,55],{"className":9571,"style":54},[52,53],[31,9573,9575],{"className":9574},[219],[31,9576,9578,9608],{"className":9577},[223,553],[31,9579,9581,9605],{"className":9580},[227],[31,9582,9585],{"className":9583,"style":9584},[231],"height:0.3011em;",[31,9586,9587,9590],{"style":4477},[31,9588],{"className":9589,"style":239},[238],[31,9591,9593],{"className":9592},[243,244,245,246],[31,9594,9596,9599,9602],{"className":9595},[52,246],[31,9597,9327],{"className":9598},[52,53,246],[31,9600,145],{"className":9601},[144,246],[31,9603,269],{"className":9604},[52,246],[31,9606,607],{"className":9607},[606],[31,9609,9611],{"className":9610},[227],[31,9612,9614],{"className":9613,"style":2939},[231],[31,9615],{},") is computed from gradients alone, so CG never touches ",[31,9618,9620],{"className":9619},[34],[31,9621,9623],{"className":9622,"ariaHidden":39},[38],[31,9624,9626,9629],{"className":9625},[43],[31,9627],{"className":9628,"style":188},[47],[31,9630,193],{"className":9631,"style":192},[52,53],"\nexplicitly. Each step needs one gradient and one line search.",[1507,9634],{"hash":9635},"83cca56c424b2db323f5fb3ca68d89bdea67d712bcbd477359ef7cd205e468bc",[9637,9638,9642],"pre",{"className":9639,"code":9640,"language":9641,"meta":6,"style":6},"language-algorithm shiki shiki-themes Vesper Light - Orange Boost (Quick Open Adjusted) vesper","caption: $\\textsc{ConjugateGradients}(\\mathcal{L}, \\theta_0)$ — minimize without forming $H$\n$g_0 \\gets \\nabla\\mathcal{L}(\\theta_0)$\n$d_0 \\gets -g_0$ \u002F\u002F first direction is steepest descent\nfor $t \\gets 0, 1, 2, \\dots$ do\n  $a_t \\gets \\arg\\min_a \\mathcal{L}(\\theta_t + a\\,d_t)$ \u002F\u002F line search along $d_t$\n  $\\theta_{t+1} \\gets \\theta_t + a_t\\,d_t$\n  $g_{t+1} \\gets \\nabla\\mathcal{L}(\\theta_{t+1})$\n  if $\\norm{g_{t+1}} \\le \\epsilon$ then \u002F\u002F converged\n    return $\\theta_{t+1}$\n  $b_t \\gets (g_{t+1}^\\top g_{t+1}) \u002F (g_t^\\top g_t)$ \u002F\u002F Fletcher-Reeves coefficient\n  $d_{t+1} \\gets -g_{t+1} + b_t\\,d_t$ \u002F\u002F bend to be conjugate to $d_t$\n","algorithm",[9643,9644,9645,9652,9658,9664,9670,9676,9682,9688,9694,9700,9706],"code",{"__ignoreMap":6},[31,9646,9649],{"class":9647,"line":9648},"line",1,[31,9650,9651],{},"caption: $\\textsc{ConjugateGradients}(\\mathcal{L}, \\theta_0)$ — minimize without forming $H$\n",[31,9653,9655],{"class":9647,"line":9654},2,[31,9656,9657],{},"$g_0 \\gets \\nabla\\mathcal{L}(\\theta_0)$\n",[31,9659,9661],{"class":9647,"line":9660},3,[31,9662,9663],{},"$d_0 \\gets -g_0$ \u002F\u002F first direction is steepest descent\n",[31,9665,9667],{"class":9647,"line":9666},4,[31,9668,9669],{},"for $t \\gets 0, 1, 2, \\dots$ do\n",[31,9671,9673],{"class":9647,"line":9672},5,[31,9674,9675],{},"  $a_t \\gets \\arg\\min_a \\mathcal{L}(\\theta_t + a\\,d_t)$ \u002F\u002F line search along $d_t$\n",[31,9677,9679],{"class":9647,"line":9678},6,[31,9680,9681],{},"  $\\theta_{t+1} \\gets \\theta_t + a_t\\,d_t$\n",[31,9683,9685],{"class":9647,"line":9684},7,[31,9686,9687],{},"  $g_{t+1} \\gets \\nabla\\mathcal{L}(\\theta_{t+1})$\n",[31,9689,9691],{"class":9647,"line":9690},8,[31,9692,9693],{},"  if $\\norm{g_{t+1}} \\le \\epsilon$ then \u002F\u002F converged\n",[31,9695,9697],{"class":9647,"line":9696},9,[31,9698,9699],{},"    return $\\theta_{t+1}$\n",[31,9701,9703],{"class":9647,"line":9702},10,[31,9704,9705],{},"  $b_t \\gets (g_{t+1}^\\top g_{t+1}) \u002F (g_t^\\top g_t)$ \u002F\u002F Fletcher-Reeves coefficient\n",[31,9707,9709],{"class":9647,"line":9708},11,[31,9710,9711],{},"  $d_{t+1} \\gets -g_{t+1} + b_t\\,d_t$ \u002F\u002F bend to be conjugate to $d_t$\n",[11,9713,9714,9715,9730,9731],{},"On a strictly quadratic objective CG terminates in at most ",[31,9716,9718],{"className":9717},[34],[31,9719,9721],{"className":9720,"ariaHidden":39},[38],[31,9722,9724,9727],{"className":9723},[43],[31,9725],{"className":9726,"style":113},[47],[31,9728,5457],{"className":9729},[52,53]," steps; on a general\nnonlinear loss it is run with periodic restarts and remains a strong batch optimizer,\nthough minibatch noise in the line search limits its use in pure deep learning.",[261,9732,9733],{},[15,9734,9738],{"href":9735,"ariaDescribedBy":9736,"dataFootnoteRef":6,"id":9737},"#user-content-fn-gf-cg",[267],"user-content-fnref-gf-cg","4",[271,9740,9742],{"id":9741},"quasi-newton-bfgs-and-l-bfgs","Quasi-Newton: BFGS and L-BFGS",[11,9744,9745,9746,9793,9794,9797,9798,9801,9802,9869,9870,9873],{},"Newton needs ",[31,9747,9749],{"className":9748},[34],[31,9750,9752],{"className":9751,"ariaHidden":39},[38],[31,9753,9755,9758],{"className":9754},[43],[31,9756],{"className":9757,"style":209},[47],[31,9759,9761,9764],{"className":9760},[52],[31,9762,193],{"className":9763,"style":192},[52,53],[31,9765,9767],{"className":9766},[219],[31,9768,9770],{"className":9769},[223],[31,9771,9773],{"className":9772},[227],[31,9774,9776],{"className":9775,"style":209},[231],[31,9777,9778,9781],{"style":234},[31,9779],{"className":9780,"style":239},[238],[31,9782,9784],{"className":9783},[243,244,245,246],[31,9785,9787,9790],{"className":9786},[52,246],[31,9788,145],{"className":9789},[52,246],[31,9791,269],{"className":9792},[52,246]," but the cost is prohibitive. ",[26,9795,9796],{},"Quasi-Newton"," methods ",[164,9799,9800],{},"learn"," an\napproximation ",[31,9803,9805],{"className":9804},[34],[31,9806,9808,9828],{"className":9807,"ariaHidden":39},[38],[31,9809,9811,9814,9819,9822,9825],{"className":9810},[43],[31,9812],{"className":9813,"style":188},[47],[31,9815,9818],{"className":9816,"style":9817},[52,53],"margin-right:0.0502em;","B",[31,9820],{"className":9821,"style":60},[59],[31,9823,435],{"className":9824},[64],[31,9826],{"className":9827,"style":60},[59],[31,9829,9831,9834],{"className":9830},[43],[31,9832],{"className":9833,"style":209},[47],[31,9835,9837,9840],{"className":9836},[52],[31,9838,193],{"className":9839,"style":192},[52,53],[31,9841,9843],{"className":9842},[219],[31,9844,9846],{"className":9845},[223],[31,9847,9849],{"className":9848},[227],[31,9850,9852],{"className":9851,"style":209},[231],[31,9853,9854,9857],{"style":234},[31,9855],{"className":9856,"style":239},[238],[31,9858,9860],{"className":9859},[243,244,245,246],[31,9861,9863,9866],{"className":9862},[52,246],[31,9864,145],{"className":9865},[52,246],[31,9867,269],{"className":9868},[52,246]," as they go, reading curvature off the change in the\ngradient. The ",[26,9871,9872],{},"secant condition"," anchors the construction: over a step, the gradient's\nchange encodes the curvature along that step. Writing",[31,9875,9877],{"className":9876},[382],[31,9878,9880],{"className":9879},[34],[31,9881,9883,9940,10004,10109,10174],{"className":9882,"ariaHidden":39},[38],[31,9884,9886,9890,9931,9934,9937],{"className":9885},[43],[31,9887],{"className":9888,"style":9889},[47],"height:0.5806em;vertical-align:-0.15em;",[31,9891,9893,9897],{"className":9892},[52],[31,9894,9896],{"className":9895},[52,53],"s",[31,9898,9900],{"className":9899},[219],[31,9901,9903,9923],{"className":9902},[223,553],[31,9904,9906,9920],{"className":9905},[227],[31,9907,9909],{"className":9908,"style":9314},[231],[31,9910,9911,9914],{"style":876},[31,9912],{"className":9913,"style":239},[238],[31,9915,9917],{"className":9916},[243,244,245,246],[31,9918,9327],{"className":9919},[52,53,246],[31,9921,607],{"className":9922},[606],[31,9924,9926],{"className":9925},[227],[31,9927,9929],{"className":9928,"style":895},[231],[31,9930],{},[31,9932],{"className":9933,"style":60},[59],[31,9935,65],{"className":9936},[64],[31,9938],{"className":9939,"style":60},[59],[31,9941,9943,9946,9995,9998,10001],{"className":9942},[43],[31,9944],{"className":9945,"style":3152},[47],[31,9947,9949,9952],{"className":9948},[52],[31,9950,94],{"className":9951,"style":93},[52,53],[31,9953,9955],{"className":9954},[219],[31,9956,9958,9987],{"className":9957},[223,553],[31,9959,9961,9984],{"className":9960},[227],[31,9962,9964],{"className":9963,"style":9584},[231],[31,9965,9966,9969],{"style":2909},[31,9967],{"className":9968,"style":239},[238],[31,9970,9972],{"className":9971},[243,244,245,246],[31,9973,9975,9978,9981],{"className":9974},[52,246],[31,9976,9327],{"className":9977},[52,53,246],[31,9979,410],{"className":9980},[144,246],[31,9982,269],{"className":9983},[52,246],[31,9985,607],{"className":9986},[606],[31,9988,9990],{"className":9989},[227],[31,9991,9993],{"className":9992,"style":2939},[231],[31,9994],{},[31,9996],{"className":9997,"style":140},[59],[31,9999,145],{"className":10000},[144],[31,10002],{"className":10003,"style":140},[59],[31,10005,10007,10010,10050,10053,10056,10059,10100,10103,10106],{"className":10006},[43],[31,10008],{"className":10009,"style":4363},[47],[31,10011,10013,10016],{"className":10012},[52],[31,10014,94],{"className":10015,"style":93},[52,53],[31,10017,10019],{"className":10018},[219],[31,10020,10022,10042],{"className":10021},[223,553],[31,10023,10025,10039],{"className":10024},[227],[31,10026,10028],{"className":10027,"style":9314},[231],[31,10029,10030,10033],{"style":2909},[31,10031],{"className":10032,"style":239},[238],[31,10034,10036],{"className":10035},[243,244,245,246],[31,10037,9327],{"className":10038},[52,53,246],[31,10040,607],{"className":10041},[606],[31,10043,10045],{"className":10044},[227],[31,10046,10048],{"className":10047,"style":895},[231],[31,10049],{},[31,10051,666],{"className":10052},[665],[31,10054],{"className":10055,"style":670},[59],[31,10057],{"className":10058,"style":623},[59],[31,10060,10062,10066],{"className":10061},[52],[31,10063,10065],{"className":10064,"style":54},[52,53],"y",[31,10067,10069],{"className":10068},[219],[31,10070,10072,10092],{"className":10071},[223,553],[31,10073,10075,10089],{"className":10074},[227],[31,10076,10078],{"className":10077,"style":9314},[231],[31,10079,10080,10083],{"style":4477},[31,10081],{"className":10082,"style":239},[238],[31,10084,10086],{"className":10085},[243,244,245,246],[31,10087,9327],{"className":10088},[52,53,246],[31,10090,607],{"className":10091},[606],[31,10093,10095],{"className":10094},[227],[31,10096,10098],{"className":10097,"style":895},[231],[31,10099],{},[31,10101],{"className":10102,"style":60},[59],[31,10104,65],{"className":10105},[64],[31,10107],{"className":10108,"style":60},[59],[31,10110,10112,10116,10165,10168,10171],{"className":10111},[43],[31,10113],{"className":10114,"style":10115},[47],"height:0.7917em;vertical-align:-0.2083em;",[31,10117,10119,10122],{"className":10118},[52],[31,10120,55],{"className":10121,"style":54},[52,53],[31,10123,10125],{"className":10124},[219],[31,10126,10128,10157],{"className":10127},[223,553],[31,10129,10131,10154],{"className":10130},[227],[31,10132,10134],{"className":10133,"style":9584},[231],[31,10135,10136,10139],{"style":4477},[31,10137],{"className":10138,"style":239},[238],[31,10140,10142],{"className":10141},[243,244,245,246],[31,10143,10145,10148,10151],{"className":10144},[52,246],[31,10146,9327],{"className":10147},[52,53,246],[31,10149,410],{"className":10150},[144,246],[31,10152,269],{"className":10153},[52,246],[31,10155,607],{"className":10156},[606],[31,10158,10160],{"className":10159},[227],[31,10161,10163],{"className":10162,"style":2939},[231],[31,10164],{},[31,10166],{"className":10167,"style":140},[59],[31,10169,145],{"className":10170},[144],[31,10172],{"className":10173,"style":140},[59],[31,10175,10177,10180,10220],{"className":10176},[43],[31,10178],{"className":10179,"style":48},[47],[31,10181,10183,10186],{"className":10182},[52],[31,10184,55],{"className":10185,"style":54},[52,53],[31,10187,10189],{"className":10188},[219],[31,10190,10192,10212],{"className":10191},[223,553],[31,10193,10195,10209],{"className":10194},[227],[31,10196,10198],{"className":10197,"style":9314},[231],[31,10199,10200,10203],{"style":4477},[31,10201],{"className":10202,"style":239},[238],[31,10204,10206],{"className":10205},[243,244,245,246],[31,10207,9327],{"className":10208},[52,53,246],[31,10210,607],{"className":10211},[606],[31,10213,10215],{"className":10214},[227],[31,10216,10218],{"className":10217,"style":895},[231],[31,10219],{},[31,10221,666],{"className":10222},[665],[11,10224,10225,10226,10241,10242,10356,10357,10404,10405,4320,10547,10550,10551,10604,10605,10709,10710,10725],{},"a first-order expansion of ",[31,10227,10229],{"className":10228},[34],[31,10230,10232],{"className":10231,"ariaHidden":39},[38],[31,10233,10235,10238],{"className":10234},[43],[31,10236],{"className":10237,"style":48},[47],[31,10239,55],{"className":10240,"style":54},[52,53]," gives ",[31,10243,10245],{"className":10244},[34],[31,10246,10248,10304],{"className":10247,"ariaHidden":39},[38],[31,10249,10251,10255,10295,10298,10301],{"className":10250},[43],[31,10252],{"className":10253,"style":10254},[47],"height:0.6776em;vertical-align:-0.1944em;",[31,10256,10258,10261],{"className":10257},[52],[31,10259,10065],{"className":10260,"style":54},[52,53],[31,10262,10264],{"className":10263},[219],[31,10265,10267,10287],{"className":10266},[223,553],[31,10268,10270,10284],{"className":10269},[227],[31,10271,10273],{"className":10272,"style":9314},[231],[31,10274,10275,10278],{"style":4477},[31,10276],{"className":10277,"style":239},[238],[31,10279,10281],{"className":10280},[243,244,245,246],[31,10282,9327],{"className":10283},[52,53,246],[31,10285,607],{"className":10286},[606],[31,10288,10290],{"className":10289},[227],[31,10291,10293],{"className":10292,"style":895},[231],[31,10294],{},[31,10296],{"className":10297,"style":60},[59],[31,10299,435],{"className":10300},[64],[31,10302],{"className":10303,"style":60},[59],[31,10305,10307,10310,10313,10316],{"className":10306},[43],[31,10308],{"className":10309,"style":7130},[47],[31,10311,193],{"className":10312,"style":192},[52,53],[31,10314],{"className":10315,"style":623},[59],[31,10317,10319,10322],{"className":10318},[52],[31,10320,9896],{"className":10321},[52,53],[31,10323,10325],{"className":10324},[219],[31,10326,10328,10348],{"className":10327},[223,553],[31,10329,10331,10345],{"className":10330},[227],[31,10332,10334],{"className":10333,"style":9314},[231],[31,10335,10336,10339],{"style":876},[31,10337],{"className":10338,"style":239},[238],[31,10340,10342],{"className":10341},[243,244,245,246],[31,10343,9327],{"className":10344},[52,53,246],[31,10346,607],{"className":10347},[606],[31,10349,10351],{"className":10350},[227],[31,10352,10354],{"className":10353,"style":895},[231],[31,10355],{},", so ",[31,10358,10360],{"className":10359},[34],[31,10361,10363],{"className":10362,"ariaHidden":39},[38],[31,10364,10366,10369],{"className":10365},[43],[31,10367],{"className":10368,"style":209},[47],[31,10370,10372,10375],{"className":10371},[52],[31,10373,193],{"className":10374,"style":192},[52,53],[31,10376,10378],{"className":10377},[219],[31,10379,10381],{"className":10380},[223],[31,10382,10384],{"className":10383},[227],[31,10385,10387],{"className":10386,"style":209},[231],[31,10388,10389,10392],{"style":234},[31,10390],{"className":10391,"style":239},[238],[31,10393,10395],{"className":10394},[243,244,245,246],[31,10396,10398,10401],{"className":10397},[52,246],[31,10399,145],{"className":10400},[52,246],[31,10402,269],{"className":10403},[52,246]," must satisfy\n",[31,10406,10408],{"className":10407},[34],[31,10409,10411,10501],{"className":10410,"ariaHidden":39},[38],[31,10412,10414,10417,10452,10492,10495,10498],{"className":10413},[43],[31,10415],{"className":10416,"style":1463},[47],[31,10418,10420,10423],{"className":10419},[52],[31,10421,193],{"className":10422,"style":192},[52,53],[31,10424,10426],{"className":10425},[219],[31,10427,10429],{"className":10428},[223],[31,10430,10432],{"className":10431},[227],[31,10433,10435],{"className":10434,"style":209},[231],[31,10436,10437,10440],{"style":234},[31,10438],{"className":10439,"style":239},[238],[31,10441,10443],{"className":10442},[243,244,245,246],[31,10444,10446,10449],{"className":10445},[52,246],[31,10447,145],{"className":10448},[52,246],[31,10450,269],{"className":10451},[52,246],[31,10453,10455,10458],{"className":10454},[52],[31,10456,10065],{"className":10457,"style":54},[52,53],[31,10459,10461],{"className":10460},[219],[31,10462,10464,10484],{"className":10463},[223,553],[31,10465,10467,10481],{"className":10466},[227],[31,10468,10470],{"className":10469,"style":9314},[231],[31,10471,10472,10475],{"style":4477},[31,10473],{"className":10474,"style":239},[238],[31,10476,10478],{"className":10477},[243,244,245,246],[31,10479,9327],{"className":10480},[52,53,246],[31,10482,607],{"className":10483},[606],[31,10485,10487],{"className":10486},[227],[31,10488,10490],{"className":10489,"style":895},[231],[31,10491],{},[31,10493],{"className":10494,"style":60},[59],[31,10496,435],{"className":10497},[64],[31,10499],{"className":10500,"style":60},[59],[31,10502,10504,10507],{"className":10503},[43],[31,10505],{"className":10506,"style":9889},[47],[31,10508,10510,10513],{"className":10509},[52],[31,10511,9896],{"className":10512},[52,53],[31,10514,10516],{"className":10515},[219],[31,10517,10519,10539],{"className":10518},[223,553],[31,10520,10522,10536],{"className":10521},[227],[31,10523,10525],{"className":10524,"style":9314},[231],[31,10526,10527,10530],{"style":876},[31,10528],{"className":10529,"style":239},[238],[31,10531,10533],{"className":10532},[243,244,245,246],[31,10534,9327],{"className":10535},[52,53,246],[31,10537,607],{"className":10538},[606],[31,10540,10542],{"className":10541},[227],[31,10543,10545],{"className":10544,"style":895},[231],[31,10546],{},[26,10548,10549],{},"BFGS"," updates ",[31,10552,10554],{"className":10553},[34],[31,10555,10557],{"className":10556,"ariaHidden":39},[38],[31,10558,10560,10563],{"className":10559},[43],[31,10561],{"className":10562,"style":7130},[47],[31,10564,10566,10569],{"className":10565},[52],[31,10567,9818],{"className":10568,"style":9817},[52,53],[31,10570,10572],{"className":10571},[219],[31,10573,10575,10596],{"className":10574},[223,553],[31,10576,10578,10593],{"className":10577},[227],[31,10579,10581],{"className":10580,"style":9314},[231],[31,10582,10584,10587],{"style":10583},"top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;",[31,10585],{"className":10586,"style":239},[238],[31,10588,10590],{"className":10589},[243,244,245,246],[31,10591,9327],{"className":10592},[52,53,246],[31,10594,607],{"className":10595},[606],[31,10597,10599],{"className":10598},[227],[31,10600,10602],{"className":10601,"style":895},[231],[31,10603],{}," at each step so it reproduces this\nrelation on the latest ",[31,10606,10608],{"className":10607},[34],[31,10609,10611],{"className":10610,"ariaHidden":39},[38],[31,10612,10614,10617,10620,10660,10663,10666,10706],{"className":10613},[43],[31,10615],{"className":10616,"style":75},[47],[31,10618,89],{"className":10619},[88],[31,10621,10623,10626],{"className":10622},[52],[31,10624,9896],{"className":10625},[52,53],[31,10627,10629],{"className":10628},[219],[31,10630,10632,10652],{"className":10631},[223,553],[31,10633,10635,10649],{"className":10634},[227],[31,10636,10638],{"className":10637,"style":9314},[231],[31,10639,10640,10643],{"style":876},[31,10641],{"className":10642,"style":239},[238],[31,10644,10646],{"className":10645},[243,244,245,246],[31,10647,9327],{"className":10648},[52,53,246],[31,10650,607],{"className":10651},[606],[31,10653,10655],{"className":10654},[227],[31,10656,10658],{"className":10657,"style":895},[231],[31,10659],{},[31,10661,666],{"className":10662},[665],[31,10664],{"className":10665,"style":623},[59],[31,10667,10669,10672],{"className":10668},[52],[31,10670,10065],{"className":10671,"style":54},[52,53],[31,10673,10675],{"className":10674},[219],[31,10676,10678,10698],{"className":10677},[223,553],[31,10679,10681,10695],{"className":10680},[227],[31,10682,10684],{"className":10683,"style":9314},[231],[31,10685,10686,10689],{"style":4477},[31,10687],{"className":10688,"style":239},[238],[31,10690,10692],{"className":10691},[243,244,245,246],[31,10693,9327],{"className":10694},[52,53,246],[31,10696,607],{"className":10697},[606],[31,10699,10701],{"className":10700},[227],[31,10702,10704],{"className":10703,"style":895},[231],[31,10705],{},[31,10707,99],{"className":10708},[98]," pair while changing ",[31,10711,10713],{"className":10712},[34],[31,10714,10716],{"className":10715,"ariaHidden":39},[38],[31,10717,10719,10722],{"className":10718},[43],[31,10720],{"className":10721,"style":188},[47],[31,10723,9818],{"className":10724,"style":9817},[52,53]," as little as possible\n(a rank-two correction):",[31,10727,10729],{"className":10728},[382],[31,10730,10732],{"className":10731},[34],[31,10733,10735,10800,11400],{"className":10734,"ariaHidden":39},[38],[31,10736,10738,10742,10791,10794,10797],{"className":10737},[43],[31,10739],{"className":10740,"style":10741},[47],"height:0.8917em;vertical-align:-0.2083em;",[31,10743,10745,10748],{"className":10744},[52],[31,10746,9818],{"className":10747,"style":9817},[52,53],[31,10749,10751],{"className":10750},[219],[31,10752,10754,10783],{"className":10753},[223,553],[31,10755,10757,10780],{"className":10756},[227],[31,10758,10760],{"className":10759,"style":9584},[231],[31,10761,10762,10765],{"style":10583},[31,10763],{"className":10764,"style":239},[238],[31,10766,10768],{"className":10767},[243,244,245,246],[31,10769,10771,10774,10777],{"className":10770},[52,246],[31,10772,9327],{"className":10773},[52,53,246],[31,10775,410],{"className":10776},[144,246],[31,10778,269],{"className":10779},[52,246],[31,10781,607],{"className":10782},[606],[31,10784,10786],{"className":10785},[227],[31,10787,10789],{"className":10788,"style":2939},[231],[31,10790],{},[31,10792],{"className":10793,"style":60},[59],[31,10795,65],{"className":10796},[64],[31,10798],{"className":10799,"style":60},[59],[31,10801,10803,10807,11079,11082,11122,11125,11391,11394,11397],{"className":10802},[43],[31,10804],{"className":10805,"style":10806},[47],"height:2.4928em;vertical-align:-0.9667em;",[31,10808,10810,10816,10819,10822,10825,10828,11073],{"className":10809},[908],[31,10811,10813],{"className":10812,"style":913},[88,912],[31,10814,89],{"className":10815},[917,245],[31,10817,8210],{"className":10818,"style":8209},[52,53],[31,10820],{"className":10821,"style":140},[59],[31,10823,145],{"className":10824},[144],[31,10826],{"className":10827,"style":140},[59],[31,10829,10831,10834,11070],{"className":10830},[52],[31,10832],{"className":10833},[88,545],[31,10835,10837],{"className":10836},[549],[31,10838,10840,11061],{"className":10839},[223,553],[31,10841,10843,11058],{"className":10842},[227],[31,10844,10847,10951,10959],{"className":10845,"style":10846},[231],"height:1.5261em;",[31,10848,10850,10853],{"style":10849},"top:-2.2791em;",[31,10851],{"className":10852,"style":567},[238],[31,10854,10856,10911],{"className":10855},[52],[31,10857,10859,10862],{"className":10858},[52],[31,10860,10065],{"className":10861,"style":54},[52,53],[31,10863,10865],{"className":10864},[219],[31,10866,10868,10902],{"className":10867},[223,553],[31,10869,10871,10899],{"className":10870},[227],[31,10872,10875,10887],{"className":10873,"style":10874},[231],"height:0.8309em;",[31,10876,10878,10881],{"style":10877},"top:-2.4542em;margin-left:-0.0359em;margin-right:0.05em;",[31,10879],{"className":10880,"style":239},[238],[31,10882,10884],{"className":10883},[243,244,245,246],[31,10885,9327],{"className":10886},[52,53,246],[31,10888,10890,10893],{"style":10889},"top:-3.0448em;margin-right:0.05em;",[31,10891],{"className":10892,"style":239},[238],[31,10894,10896],{"className":10895},[243,244,245,246],[31,10897,513],{"className":10898},[52,246],[31,10900,607],{"className":10901},[606],[31,10903,10905],{"className":10904},[227],[31,10906,10909],{"className":10907,"style":10908},[231],"height:0.2458em;",[31,10910],{},[31,10912,10914,10917],{"className":10913},[52],[31,10915,9896],{"className":10916},[52,53],[31,10918,10920],{"className":10919},[219],[31,10921,10923,10943],{"className":10922},[223,553],[31,10924,10926,10940],{"className":10925},[227],[31,10927,10929],{"className":10928,"style":9314},[231],[31,10930,10931,10934],{"style":876},[31,10932],{"className":10933,"style":239},[238],[31,10935,10937],{"className":10936},[243,244,245,246],[31,10938,9327],{"className":10939},[52,53,246],[31,10941,607],{"className":10942},[606],[31,10944,10946],{"className":10945},[227],[31,10947,10949],{"className":10948,"style":895},[231],[31,10950],{},[31,10952,10953,10956],{"style":579},[31,10954],{"className":10955,"style":567},[238],[31,10957],{"className":10958,"style":587},[586],[31,10960,10961,10964],{"style":7268},[31,10962],{"className":10963,"style":567},[238],[31,10965,10967,11007],{"className":10966},[52],[31,10968,10970,10973],{"className":10969},[52],[31,10971,9896],{"className":10972},[52,53],[31,10974,10976],{"className":10975},[219],[31,10977,10979,10999],{"className":10978},[223,553],[31,10980,10982,10996],{"className":10981},[227],[31,10983,10985],{"className":10984,"style":9314},[231],[31,10986,10987,10990],{"style":876},[31,10988],{"className":10989,"style":239},[238],[31,10991,10993],{"className":10992},[243,244,245,246],[31,10994,9327],{"className":10995},[52,53,246],[31,10997,607],{"className":10998},[606],[31,11000,11002],{"className":11001},[227],[31,11003,11005],{"className":11004,"style":895},[231],[31,11006],{},[31,11008,11010,11013],{"className":11009},[52],[31,11011,10065],{"className":11012,"style":54},[52,53],[31,11014,11016],{"className":11015},[219],[31,11017,11019,11050],{"className":11018},[223,553],[31,11020,11022,11047],{"className":11021},[227],[31,11023,11025,11036],{"className":11024,"style":1314},[231],[31,11026,11027,11030],{"style":9428},[31,11028],{"className":11029,"style":239},[238],[31,11031,11033],{"className":11032},[243,244,245,246],[31,11034,9327],{"className":11035},[52,53,246],[31,11037,11038,11041],{"style":234},[31,11039],{"className":11040,"style":239},[238],[31,11042,11044],{"className":11043},[243,244,245,246],[31,11045,513],{"className":11046},[52,246],[31,11048,607],{"className":11049},[606],[31,11051,11053],{"className":11052},[227],[31,11054,11056],{"className":11055,"style":9458},[231],[31,11057],{},[31,11059,607],{"className":11060},[606],[31,11062,11064],{"className":11063},[227],[31,11065,11068],{"className":11066,"style":11067},[231],"height:0.9667em;",[31,11069],{},[31,11071],{"className":11072},[98,545],[31,11074,11076],{"className":11075,"style":913},[98,912],[31,11077,99],{"className":11078},[917,245],[31,11080],{"className":11081,"style":623},[59],[31,11083,11085,11088],{"className":11084},[52],[31,11086,9818],{"className":11087,"style":9817},[52,53],[31,11089,11091],{"className":11090},[219],[31,11092,11094,11114],{"className":11093},[223,553],[31,11095,11097,11111],{"className":11096},[227],[31,11098,11100],{"className":11099,"style":9314},[231],[31,11101,11102,11105],{"style":10583},[31,11103],{"className":11104,"style":239},[238],[31,11106,11108],{"className":11107},[243,244,245,246],[31,11109,9327],{"className":11110},[52,53,246],[31,11112,607],{"className":11113},[606],[31,11115,11117],{"className":11116},[227],[31,11118,11120],{"className":11119,"style":895},[231],[31,11121],{},[31,11123],{"className":11124,"style":623},[59],[31,11126,11128,11134,11137,11140,11143,11146,11385],{"className":11127},[908],[31,11129,11131],{"className":11130,"style":913},[88,912],[31,11132,89],{"className":11133},[917,245],[31,11135,8210],{"className":11136,"style":8209},[52,53],[31,11138],{"className":11139,"style":140},[59],[31,11141,145],{"className":11142},[144],[31,11144],{"className":11145,"style":140},[59],[31,11147,11149,11152,11382],{"className":11148},[52],[31,11150],{"className":11151},[88,545],[31,11153,11155],{"className":11154},[549],[31,11156,11158,11374],{"className":11157},[223,553],[31,11159,11161,11371],{"className":11160},[227],[31,11162,11164,11263,11271],{"className":11163,"style":10846},[231],[31,11165,11166,11169],{"style":10849},[31,11167],{"className":11168,"style":567},[238],[31,11170,11172,11223],{"className":11171},[52],[31,11173,11175,11178],{"className":11174},[52],[31,11176,10065],{"className":11177,"style":54},[52,53],[31,11179,11181],{"className":11180},[219],[31,11182,11184,11215],{"className":11183},[223,553],[31,11185,11187,11212],{"className":11186},[227],[31,11188,11190,11201],{"className":11189,"style":10874},[231],[31,11191,11192,11195],{"style":10877},[31,11193],{"className":11194,"style":239},[238],[31,11196,11198],{"className":11197},[243,244,245,246],[31,11199,9327],{"className":11200},[52,53,246],[31,11202,11203,11206],{"style":10889},[31,11204],{"className":11205,"style":239},[238],[31,11207,11209],{"className":11208},[243,244,245,246],[31,11210,513],{"className":11211},[52,246],[31,11213,607],{"className":11214},[606],[31,11216,11218],{"className":11217},[227],[31,11219,11221],{"className":11220,"style":10908},[231],[31,11222],{},[31,11224,11226,11229],{"className":11225},[52],[31,11227,9896],{"className":11228},[52,53],[31,11230,11232],{"className":11231},[219],[31,11233,11235,11255],{"className":11234},[223,553],[31,11236,11238,11252],{"className":11237},[227],[31,11239,11241],{"className":11240,"style":9314},[231],[31,11242,11243,11246],{"style":876},[31,11244],{"className":11245,"style":239},[238],[31,11247,11249],{"className":11248},[243,244,245,246],[31,11250,9327],{"className":11251},[52,53,246],[31,11253,607],{"className":11254},[606],[31,11256,11258],{"className":11257},[227],[31,11259,11261],{"className":11260,"style":895},[231],[31,11262],{},[31,11264,11265,11268],{"style":579},[31,11266],{"className":11267,"style":567},[238],[31,11269],{"className":11270,"style":587},[586],[31,11272,11273,11276],{"style":7268},[31,11274],{"className":11275,"style":567},[238],[31,11277,11279,11319],{"className":11278},[52],[31,11280,11282,11285],{"className":11281},[52],[31,11283,10065],{"className":11284,"style":54},[52,53],[31,11286,11288],{"className":11287},[219],[31,11289,11291,11311],{"className":11290},[223,553],[31,11292,11294,11308],{"className":11293},[227],[31,11295,11297],{"className":11296,"style":9314},[231],[31,11298,11299,11302],{"style":4477},[31,11300],{"className":11301,"style":239},[238],[31,11303,11305],{"className":11304},[243,244,245,246],[31,11306,9327],{"className":11307},[52,53,246],[31,11309,607],{"className":11310},[606],[31,11312,11314],{"className":11313},[227],[31,11315,11317],{"className":11316,"style":895},[231],[31,11318],{},[31,11320,11322,11325],{"className":11321},[52],[31,11323,9896],{"className":11324},[52,53],[31,11326,11328],{"className":11327},[219],[31,11329,11331,11363],{"className":11330},[223,553],[31,11332,11334,11360],{"className":11333},[227],[31,11335,11337,11349],{"className":11336,"style":1314},[231],[31,11338,11340,11343],{"style":11339},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[31,11341],{"className":11342,"style":239},[238],[31,11344,11346],{"className":11345},[243,244,245,246],[31,11347,9327],{"className":11348},[52,53,246],[31,11350,11351,11354],{"style":234},[31,11352],{"className":11353,"style":239},[238],[31,11355,11357],{"className":11356},[243,244,245,246],[31,11358,513],{"className":11359},[52,246],[31,11361,607],{"className":11362},[606],[31,11364,11366],{"className":11365},[227],[31,11367,11369],{"className":11368,"style":9458},[231],[31,11370],{},[31,11372,607],{"className":11373},[606],[31,11375,11377],{"className":11376},[227],[31,11378,11380],{"className":11379,"style":11067},[231],[31,11381],{},[31,11383],{"className":11384},[98,545],[31,11386,11388],{"className":11387,"style":913},[98,912],[31,11389,99],{"className":11390},[917,245],[31,11392],{"className":11393,"style":140},[59],[31,11395,410],{"className":11396},[144],[31,11398],{"className":11399,"style":140},[59],[31,11401,11403,11406,11644],{"className":11402},[43],[31,11404],{"className":11405,"style":10806},[47],[31,11407,11409,11412,11641],{"className":11408},[52],[31,11410],{"className":11411},[88,545],[31,11413,11415],{"className":11414},[549],[31,11416,11418,11633],{"className":11417},[223,553],[31,11419,11421,11630],{"className":11420},[227],[31,11422,11424,11523,11531],{"className":11423,"style":10846},[231],[31,11425,11426,11429],{"style":10849},[31,11427],{"className":11428,"style":567},[238],[31,11430,11432,11483],{"className":11431},[52],[31,11433,11435,11438],{"className":11434},[52],[31,11436,10065],{"className":11437,"style":54},[52,53],[31,11439,11441],{"className":11440},[219],[31,11442,11444,11475],{"className":11443},[223,553],[31,11445,11447,11472],{"className":11446},[227],[31,11448,11450,11461],{"className":11449,"style":10874},[231],[31,11451,11452,11455],{"style":10877},[31,11453],{"className":11454,"style":239},[238],[31,11456,11458],{"className":11457},[243,244,245,246],[31,11459,9327],{"className":11460},[52,53,246],[31,11462,11463,11466],{"style":10889},[31,11464],{"className":11465,"style":239},[238],[31,11467,11469],{"className":11468},[243,244,245,246],[31,11470,513],{"className":11471},[52,246],[31,11473,607],{"className":11474},[606],[31,11476,11478],{"className":11477},[227],[31,11479,11481],{"className":11480,"style":10908},[231],[31,11482],{},[31,11484,11486,11489],{"className":11485},[52],[31,11487,9896],{"className":11488},[52,53],[31,11490,11492],{"className":11491},[219],[31,11493,11495,11515],{"className":11494},[223,553],[31,11496,11498,11512],{"className":11497},[227],[31,11499,11501],{"className":11500,"style":9314},[231],[31,11502,11503,11506],{"style":876},[31,11504],{"className":11505,"style":239},[238],[31,11507,11509],{"className":11508},[243,244,245,246],[31,11510,9327],{"className":11511},[52,53,246],[31,11513,607],{"className":11514},[606],[31,11516,11518],{"className":11517},[227],[31,11519,11521],{"className":11520,"style":895},[231],[31,11522],{},[31,11524,11525,11528],{"style":579},[31,11526],{"className":11527,"style":567},[238],[31,11529],{"className":11530,"style":587},[586],[31,11532,11533,11536],{"style":7268},[31,11534],{"className":11535,"style":567},[238],[31,11537,11539,11579],{"className":11538},[52],[31,11540,11542,11545],{"className":11541},[52],[31,11543,9896],{"className":11544},[52,53],[31,11546,11548],{"className":11547},[219],[31,11549,11551,11571],{"className":11550},[223,553],[31,11552,11554,11568],{"className":11553},[227],[31,11555,11557],{"className":11556,"style":9314},[231],[31,11558,11559,11562],{"style":876},[31,11560],{"className":11561,"style":239},[238],[31,11563,11565],{"className":11564},[243,244,245,246],[31,11566,9327],{"className":11567},[52,53,246],[31,11569,607],{"className":11570},[606],[31,11572,11574],{"className":11573},[227],[31,11575,11577],{"className":11576,"style":895},[231],[31,11578],{},[31,11580,11582,11585],{"className":11581},[52],[31,11583,9896],{"className":11584},[52,53],[31,11586,11588],{"className":11587},[219],[31,11589,11591,11622],{"className":11590},[223,553],[31,11592,11594,11619],{"className":11593},[227],[31,11595,11597,11608],{"className":11596,"style":1314},[231],[31,11598,11599,11602],{"style":11339},[31,11600],{"className":11601,"style":239},[238],[31,11603,11605],{"className":11604},[243,244,245,246],[31,11606,9327],{"className":11607},[52,53,246],[31,11609,11610,11613],{"style":234},[31,11611],{"className":11612,"style":239},[238],[31,11614,11616],{"className":11615},[243,244,245,246],[31,11617,513],{"className":11618},[52,246],[31,11620,607],{"className":11621},[606],[31,11623,11625],{"className":11624},[227],[31,11626,11628],{"className":11627,"style":9458},[231],[31,11629],{},[31,11631,607],{"className":11632},[606],[31,11634,11636],{"className":11635},[227],[31,11637,11639],{"className":11638,"style":11067},[231],[31,11640],{},[31,11642],{"className":11643},[98,545],[31,11645,781],{"className":11646},[52],[11,11648,11649,11650,11910,11911,11926],{},"The step is then ",[31,11651,11653],{"className":11652},[34],[31,11654,11656,11720,11775],{"className":11655,"ariaHidden":39},[38],[31,11657,11659,11662,11711,11714,11717],{"className":11658},[43],[31,11660],{"className":11661,"style":3152},[47],[31,11663,11665,11668],{"className":11664},[52],[31,11666,94],{"className":11667,"style":93},[52,53],[31,11669,11671],{"className":11670},[219],[31,11672,11674,11703],{"className":11673},[223,553],[31,11675,11677,11700],{"className":11676},[227],[31,11678,11680],{"className":11679,"style":9584},[231],[31,11681,11682,11685],{"style":2909},[31,11683],{"className":11684,"style":239},[238],[31,11686,11688],{"className":11687},[243,244,245,246],[31,11689,11691,11694,11697],{"className":11690},[52,246],[31,11692,9327],{"className":11693},[52,53,246],[31,11695,410],{"className":11696},[144,246],[31,11698,269],{"className":11699},[52,246],[31,11701,607],{"className":11702},[606],[31,11704,11706],{"className":11705},[227],[31,11707,11709],{"className":11708,"style":2939},[231],[31,11710],{},[31,11712],{"className":11713,"style":60},[59],[31,11715,65],{"className":11716},[64],[31,11718],{"className":11719,"style":60},[59],[31,11721,11723,11726,11766,11769,11772],{"className":11722},[43],[31,11724],{"className":11725,"style":3217},[47],[31,11727,11729,11732],{"className":11728},[52],[31,11730,94],{"className":11731,"style":93},[52,53],[31,11733,11735],{"className":11734},[219],[31,11736,11738,11758],{"className":11737},[223,553],[31,11739,11741,11755],{"className":11740},[227],[31,11742,11744],{"className":11743,"style":9314},[231],[31,11745,11746,11749],{"style":2909},[31,11747],{"className":11748,"style":239},[238],[31,11750,11752],{"className":11751},[243,244,245,246],[31,11753,9327],{"className":11754},[52,53,246],[31,11756,607],{"className":11757},[606],[31,11759,11761],{"className":11760},[227],[31,11762,11764],{"className":11763,"style":895},[231],[31,11765],{},[31,11767],{"className":11768,"style":140},[59],[31,11770,145],{"className":11771},[144],[31,11773],{"className":11774,"style":140},[59],[31,11776,11778,11781,11824,11827,11867,11870],{"className":11777},[43],[31,11779],{"className":11780,"style":8262},[47],[31,11782,11784,11789],{"className":11783},[52],[31,11785,11788],{"className":11786,"style":11787},[52,53],"margin-right:0.0037em;","α",[31,11790,11792],{"className":11791},[219],[31,11793,11795,11816],{"className":11794},[223,553],[31,11796,11798,11813],{"className":11797},[227],[31,11799,11801],{"className":11800,"style":9314},[231],[31,11802,11804,11807],{"style":11803},"top:-2.55em;margin-left:-0.0037em;margin-right:0.05em;",[31,11805],{"className":11806,"style":239},[238],[31,11808,11810],{"className":11809},[243,244,245,246],[31,11811,9327],{"className":11812},[52,53,246],[31,11814,607],{"className":11815},[606],[31,11817,11819],{"className":11818},[227],[31,11820,11822],{"className":11821,"style":895},[231],[31,11823],{},[31,11825],{"className":11826,"style":623},[59],[31,11828,11830,11833],{"className":11829},[52],[31,11831,9818],{"className":11832,"style":9817},[52,53],[31,11834,11836],{"className":11835},[219],[31,11837,11839,11859],{"className":11838},[223,553],[31,11840,11842,11856],{"className":11841},[227],[31,11843,11845],{"className":11844,"style":9314},[231],[31,11846,11847,11850],{"style":10583},[31,11848],{"className":11849,"style":239},[238],[31,11851,11853],{"className":11852},[243,244,245,246],[31,11854,9327],{"className":11855},[52,53,246],[31,11857,607],{"className":11858},[606],[31,11860,11862],{"className":11861},[227],[31,11863,11865],{"className":11864,"style":895},[231],[31,11866],{},[31,11868],{"className":11869,"style":623},[59],[31,11871,11873,11876],{"className":11872},[52],[31,11874,55],{"className":11875,"style":54},[52,53],[31,11877,11879],{"className":11878},[219],[31,11880,11882,11902],{"className":11881},[223,553],[31,11883,11885,11899],{"className":11884},[227],[31,11886,11888],{"className":11887,"style":9314},[231],[31,11889,11890,11893],{"style":4477},[31,11891],{"className":11892,"style":239},[238],[31,11894,11896],{"className":11895},[243,244,245,246],[31,11897,9327],{"className":11898},[52,53,246],[31,11900,607],{"className":11901},[606],[31,11903,11905],{"className":11904},[227],[31,11906,11908],{"className":11907,"style":895},[231],[31,11909],{},", a Newton-like move\nthat never forms or inverts ",[31,11912,11914],{"className":11913},[34],[31,11915,11917],{"className":11916,"ariaHidden":39},[38],[31,11918,11920,11923],{"className":11919},[43],[31,11921],{"className":11922,"style":188},[47],[31,11924,193],{"className":11925,"style":192},[52,53],", only matrix–vector products and a line search.",[1507,11928],{"hash":11929},"29df4f79bc50e39b6c5dfdadf89bc2c66ae6736f93be34f049afd1dbdcf013ce",[11,11931,11932,11933,12089,12090,12151,12152,12274,12275,12278,12279,12331,12332,12335,12336,12440],{},"The rank-two form is engineered to satisfy three demands at once: it reproduces the\nsecant relation ",[31,11934,11936],{"className":11935},[34],[31,11937,11939,12043],{"className":11938,"ariaHidden":39},[38],[31,11940,11942,11945,11994,12034,12037,12040],{"className":11941},[43],[31,11943],{"className":11944,"style":10741},[47],[31,11946,11948,11951],{"className":11947},[52],[31,11949,9818],{"className":11950,"style":9817},[52,53],[31,11952,11954],{"className":11953},[219],[31,11955,11957,11986],{"className":11956},[223,553],[31,11958,11960,11983],{"className":11959},[227],[31,11961,11963],{"className":11962,"style":9584},[231],[31,11964,11965,11968],{"style":10583},[31,11966],{"className":11967,"style":239},[238],[31,11969,11971],{"className":11970},[243,244,245,246],[31,11972,11974,11977,11980],{"className":11973},[52,246],[31,11975,9327],{"className":11976},[52,53,246],[31,11978,410],{"className":11979},[144,246],[31,11981,269],{"className":11982},[52,246],[31,11984,607],{"className":11985},[606],[31,11987,11989],{"className":11988},[227],[31,11990,11992],{"className":11991,"style":2939},[231],[31,11993],{},[31,11995,11997,12000],{"className":11996},[52],[31,11998,10065],{"className":11999,"style":54},[52,53],[31,12001,12003],{"className":12002},[219],[31,12004,12006,12026],{"className":12005},[223,553],[31,12007,12009,12023],{"className":12008},[227],[31,12010,12012],{"className":12011,"style":9314},[231],[31,12013,12014,12017],{"style":4477},[31,12015],{"className":12016,"style":239},[238],[31,12018,12020],{"className":12019},[243,244,245,246],[31,12021,9327],{"className":12022},[52,53,246],[31,12024,607],{"className":12025},[606],[31,12027,12029],{"className":12028},[227],[31,12030,12032],{"className":12031,"style":895},[231],[31,12033],{},[31,12035],{"className":12036,"style":60},[59],[31,12038,65],{"className":12039},[64],[31,12041],{"className":12042,"style":60},[59],[31,12044,12046,12049],{"className":12045},[43],[31,12047],{"className":12048,"style":9889},[47],[31,12050,12052,12055],{"className":12051},[52],[31,12053,9896],{"className":12054},[52,53],[31,12056,12058],{"className":12057},[219],[31,12059,12061,12081],{"className":12060},[223,553],[31,12062,12064,12078],{"className":12063},[227],[31,12065,12067],{"className":12066,"style":9314},[231],[31,12068,12069,12072],{"style":876},[31,12070],{"className":12071,"style":239},[238],[31,12073,12075],{"className":12074},[243,244,245,246],[31,12076,9327],{"className":12077},[52,53,246],[31,12079,607],{"className":12080},[606],[31,12082,12084],{"className":12083},[227],[31,12085,12087],{"className":12086,"style":895},[231],[31,12088],{},", it keeps ",[31,12091,12093],{"className":12092},[34],[31,12094,12096],{"className":12095,"ariaHidden":39},[38],[31,12097,12099,12102],{"className":12098},[43],[31,12100],{"className":12101,"style":10741},[47],[31,12103,12105,12108],{"className":12104},[52],[31,12106,9818],{"className":12107,"style":9817},[52,53],[31,12109,12111],{"className":12110},[219],[31,12112,12114,12143],{"className":12113},[223,553],[31,12115,12117,12140],{"className":12116},[227],[31,12118,12120],{"className":12119,"style":9584},[231],[31,12121,12122,12125],{"style":10583},[31,12123],{"className":12124,"style":239},[238],[31,12126,12128],{"className":12127},[243,244,245,246],[31,12129,12131,12134,12137],{"className":12130},[52,246],[31,12132,9327],{"className":12133},[52,53,246],[31,12135,410],{"className":12136},[144,246],[31,12138,269],{"className":12139},[52,246],[31,12141,607],{"className":12142},[606],[31,12144,12146],{"className":12145},[227],[31,12147,12149],{"className":12148,"style":2939},[231],[31,12150],{}," symmetric and positive definite\nwhenever ",[31,12153,12155],{"className":12154},[34],[31,12156,12158,12265],{"className":12157,"ariaHidden":39},[38],[31,12159,12161,12165,12216,12256,12259,12262],{"className":12160},[43],[31,12162],{"className":12163,"style":12164},[47],"height:1.0961em;vertical-align:-0.247em;",[31,12166,12168,12171],{"className":12167},[52],[31,12169,10065],{"className":12170,"style":54},[52,53],[31,12172,12174],{"className":12173},[219],[31,12175,12177,12208],{"className":12176},[223,553],[31,12178,12180,12205],{"className":12179},[227],[31,12181,12183,12194],{"className":12182,"style":1314},[231],[31,12184,12185,12188],{"style":9428},[31,12186],{"className":12187,"style":239},[238],[31,12189,12191],{"className":12190},[243,244,245,246],[31,12192,9327],{"className":12193},[52,53,246],[31,12195,12196,12199],{"style":234},[31,12197],{"className":12198,"style":239},[238],[31,12200,12202],{"className":12201},[243,244,245,246],[31,12203,513],{"className":12204},[52,246],[31,12206,607],{"className":12207},[606],[31,12209,12211],{"className":12210},[227],[31,12212,12214],{"className":12213,"style":9458},[231],[31,12215],{},[31,12217,12219,12222],{"className":12218},[52],[31,12220,9896],{"className":12221},[52,53],[31,12223,12225],{"className":12224},[219],[31,12226,12228,12248],{"className":12227},[223,553],[31,12229,12231,12245],{"className":12230},[227],[31,12232,12234],{"className":12233,"style":9314},[231],[31,12235,12236,12239],{"style":876},[31,12237],{"className":12238,"style":239},[238],[31,12240,12242],{"className":12241},[243,244,245,246],[31,12243,9327],{"className":12244},[52,53,246],[31,12246,607],{"className":12247},[606],[31,12249,12251],{"className":12250},[227],[31,12252,12254],{"className":12253,"style":895},[231],[31,12255],{},[31,12257],{"className":12258,"style":60},[59],[31,12260,7821],{"className":12261},[64],[31,12263],{"className":12264,"style":60},[59],[31,12266,12268,12271],{"className":12267},[43],[31,12269],{"className":12270,"style":833},[47],[31,12272,837],{"className":12273},[52]," (the ",[26,12276,12277],{},"curvature condition",", guaranteed by a proper line\nsearch), and among all matrices meeting the secant relation it is the closest to ",[31,12280,12282],{"className":12281},[34],[31,12283,12285],{"className":12284,"ariaHidden":39},[38],[31,12286,12288,12291],{"className":12287},[43],[31,12289],{"className":12290,"style":7130},[47],[31,12292,12294,12297],{"className":12293},[52],[31,12295,9818],{"className":12296,"style":9817},[52,53],[31,12298,12300],{"className":12299},[219],[31,12301,12303,12323],{"className":12302},[223,553],[31,12304,12306,12320],{"className":12305},[227],[31,12307,12309],{"className":12308,"style":9314},[231],[31,12310,12311,12314],{"style":10583},[31,12312],{"className":12313,"style":239},[238],[31,12315,12317],{"className":12316},[243,244,245,246],[31,12318,9327],{"className":12319},[52,53,246],[31,12321,607],{"className":12322},[606],[31,12324,12326],{"className":12325},[227],[31,12327,12329],{"className":12328,"style":895},[231],[31,12330],{}," in\na weighted Frobenius norm. Positive definiteness matters: it forces every step to be a\ndescent direction, so unlike raw Newton, BFGS cannot sign-flip toward a saddle. The\ncurvature is ",[164,12333,12334],{},"learned",", one ",[31,12337,12339],{"className":12338},[34],[31,12340,12342],{"className":12341,"ariaHidden":39},[38],[31,12343,12345,12348,12351,12391,12394,12397,12437],{"className":12344},[43],[31,12346],{"className":12347,"style":75},[47],[31,12349,89],{"className":12350},[88],[31,12352,12354,12357],{"className":12353},[52],[31,12355,9896],{"className":12356},[52,53],[31,12358,12360],{"className":12359},[219],[31,12361,12363,12383],{"className":12362},[223,553],[31,12364,12366,12380],{"className":12365},[227],[31,12367,12369],{"className":12368,"style":9314},[231],[31,12370,12371,12374],{"style":876},[31,12372],{"className":12373,"style":239},[238],[31,12375,12377],{"className":12376},[243,244,245,246],[31,12378,9327],{"className":12379},[52,53,246],[31,12381,607],{"className":12382},[606],[31,12384,12386],{"className":12385},[227],[31,12387,12389],{"className":12388,"style":895},[231],[31,12390],{},[31,12392,666],{"className":12393},[665],[31,12395],{"className":12396,"style":623},[59],[31,12398,12400,12403],{"className":12399},[52],[31,12401,10065],{"className":12402,"style":54},[52,53],[31,12404,12406],{"className":12405},[219],[31,12407,12409,12429],{"className":12408},[223,553],[31,12410,12412,12426],{"className":12411},[227],[31,12413,12415],{"className":12414,"style":9314},[231],[31,12416,12417,12420],{"style":4477},[31,12418],{"className":12419,"style":239},[238],[31,12421,12423],{"className":12422},[243,244,245,246],[31,12424,9327],{"className":12425},[52,53,246],[31,12427,607],{"className":12428},[606],[31,12430,12432],{"className":12431},[227],[31,12433,12435],{"className":12434,"style":895},[231],[31,12436],{},[31,12438,99],{"className":12439},[98]," pair at a time, and it accumulates the effect of\nmany past pairs into a single matrix.",[11,12442,12443,12444,12459,12460,4320,12493,12496,12497,12512,12513,12529,12530,12634,12635,12669,12670,12686,12687,12779,12780,12832,12833,13099,13100,13147,13148,13166,13167,13259,13260,13287,13288,13338,13339,12669,13389,13416,13417],{},"The remaining problem is memory: ",[31,12445,12447],{"className":12446},[34],[31,12448,12450],{"className":12449,"ariaHidden":39},[38],[31,12451,12453,12456],{"className":12452},[43],[31,12454],{"className":12455,"style":188},[47],[31,12457,9818],{"className":12458,"style":9817},[52,53]," is still ",[31,12461,12463],{"className":12462},[34],[31,12464,12466,12484],{"className":12465,"ariaHidden":39},[38],[31,12467,12469,12472,12475,12478,12481],{"className":12468},[43],[31,12470],{"className":12471,"style":133},[47],[31,12473,5457],{"className":12474},[52,53],[31,12476],{"className":12477,"style":140},[59],[31,12479,6001],{"className":12480},[144],[31,12482],{"className":12483,"style":140},[59],[31,12485,12487,12490],{"className":12486},[43],[31,12488],{"className":12489,"style":113},[47],[31,12491,5457],{"className":12492},[52,53],[26,12494,12495],{},"L-BFGS"," (limited-memory\nBFGS) never stores ",[31,12498,12500],{"className":12499},[34],[31,12501,12503],{"className":12502,"ariaHidden":39},[38],[31,12504,12506,12509],{"className":12505},[43],[31,12507],{"className":12508,"style":188},[47],[31,12510,9818],{"className":12511,"style":9817},[52,53]," at all. It keeps only the last ",[31,12514,12516],{"className":12515},[34],[31,12517,12519],{"className":12518,"ariaHidden":39},[38],[31,12520,12522,12525],{"className":12521},[43],[31,12523],{"className":12524,"style":9257},[47],[31,12526,12528],{"className":12527},[52,53],"m"," pairs ",[31,12531,12533],{"className":12532},[34],[31,12534,12536],{"className":12535,"ariaHidden":39},[38],[31,12537,12539,12542,12545,12585,12588,12591,12631],{"className":12538},[43],[31,12540],{"className":12541,"style":75},[47],[31,12543,89],{"className":12544},[88],[31,12546,12548,12551],{"className":12547},[52],[31,12549,9896],{"className":12550},[52,53],[31,12552,12554],{"className":12553},[219],[31,12555,12557,12577],{"className":12556},[223,553],[31,12558,12560,12574],{"className":12559},[227],[31,12561,12563],{"className":12562,"style":9314},[231],[31,12564,12565,12568],{"style":876},[31,12566],{"className":12567,"style":239},[238],[31,12569,12571],{"className":12570},[243,244,245,246],[31,12572,9327],{"className":12573},[52,53,246],[31,12575,607],{"className":12576},[606],[31,12578,12580],{"className":12579},[227],[31,12581,12583],{"className":12582,"style":895},[231],[31,12584],{},[31,12586,666],{"className":12587},[665],[31,12589],{"className":12590,"style":623},[59],[31,12592,12594,12597],{"className":12593},[52],[31,12595,10065],{"className":12596,"style":54},[52,53],[31,12598,12600],{"className":12599},[219],[31,12601,12603,12623],{"className":12602},[223,553],[31,12604,12606,12620],{"className":12605},[227],[31,12607,12609],{"className":12608,"style":9314},[231],[31,12610,12611,12614],{"style":4477},[31,12612],{"className":12613,"style":239},[238],[31,12615,12617],{"className":12616},[243,244,245,246],[31,12618,9327],{"className":12619},[52,53,246],[31,12621,607],{"className":12622},[606],[31,12624,12626],{"className":12625},[227],[31,12627,12629],{"className":12628,"style":895},[231],[31,12630],{},[31,12632,99],{"className":12633},[98]," (typically\n",[31,12636,12638],{"className":12637},[34],[31,12639,12641,12659],{"className":12640,"ariaHidden":39},[38],[31,12642,12644,12647,12650,12653,12656],{"className":12643},[43],[31,12645],{"className":12646,"style":9257},[47],[31,12648,12528],{"className":12649},[52,53],[31,12651],{"className":12652,"style":60},[59],[31,12654,65],{"className":12655},[64],[31,12657],{"className":12658,"style":60},[59],[31,12660,12662,12665],{"className":12661},[43],[31,12663],{"className":12664,"style":833},[47],[31,12666,12668],{"className":12667},[52],"5"," to ",[31,12671,12673],{"className":12672},[34],[31,12674,12676],{"className":12675,"ariaHidden":39},[38],[31,12677,12679,12682],{"className":12678},[43],[31,12680],{"className":12681,"style":833},[47],[31,12683,12685],{"className":12684},[52],"20",") and reconstructs the action ",[31,12688,12690],{"className":12689},[34],[31,12691,12693],{"className":12692,"ariaHidden":39},[38],[31,12694,12696,12699,12739],{"className":12695},[43],[31,12697],{"className":12698,"style":8262},[47],[31,12700,12702,12705],{"className":12701},[52],[31,12703,9818],{"className":12704,"style":9817},[52,53],[31,12706,12708],{"className":12707},[219],[31,12709,12711,12731],{"className":12710},[223,553],[31,12712,12714,12728],{"className":12713},[227],[31,12715,12717],{"className":12716,"style":9314},[231],[31,12718,12719,12722],{"style":10583},[31,12720],{"className":12721,"style":239},[238],[31,12723,12725],{"className":12724},[243,244,245,246],[31,12726,9327],{"className":12727},[52,53,246],[31,12729,607],{"className":12730},[606],[31,12732,12734],{"className":12733},[227],[31,12735,12737],{"className":12736,"style":895},[231],[31,12738],{},[31,12740,12742,12745],{"className":12741},[52],[31,12743,55],{"className":12744,"style":54},[52,53],[31,12746,12748],{"className":12747},[219],[31,12749,12751,12771],{"className":12750},[223,553],[31,12752,12754,12768],{"className":12753},[227],[31,12755,12757],{"className":12756,"style":9314},[231],[31,12758,12759,12762],{"style":4477},[31,12760],{"className":12761,"style":239},[238],[31,12763,12765],{"className":12764},[243,244,245,246],[31,12766,9327],{"className":12767},[52,53,246],[31,12769,607],{"className":12770},[606],[31,12772,12774],{"className":12773},[227],[31,12775,12777],{"className":12776,"style":895},[231],[31,12778],{}," on the fly by a two-loop\nrecursion over those vectors: a backward sweep peels off the recent curvature\ncorrections into scalars ",[31,12781,12783],{"className":12782},[34],[31,12784,12786],{"className":12785,"ariaHidden":39},[38],[31,12787,12789,12792],{"className":12788},[43],[31,12790],{"className":12791,"style":9889},[47],[31,12793,12795,12798],{"className":12794},[52],[31,12796,11788],{"className":12797,"style":11787},[52,53],[31,12799,12801],{"className":12800},[219],[31,12802,12804,12824],{"className":12803},[223,553],[31,12805,12807,12821],{"className":12806},[227],[31,12808,12810],{"className":12809,"style":7038},[231],[31,12811,12812,12815],{"style":11803},[31,12813],{"className":12814,"style":239},[238],[31,12816,12818],{"className":12817},[243,244,245,246],[31,12819,7006],{"className":12820},[52,53,246],[31,12822,607],{"className":12823},[606],[31,12825,12827],{"className":12826},[227],[31,12828,12830],{"className":12829,"style":895},[231],[31,12831],{},", a scaling by ",[31,12834,12836],{"className":12835},[34],[31,12837,12839,12859],{"className":12838,"ariaHidden":39},[38],[31,12840,12842,12845,12850,12853,12856],{"className":12841},[43],[31,12843],{"className":12844,"style":48},[47],[31,12846,12849],{"className":12847,"style":12848},[52,53],"margin-right:0.0556em;","γ",[31,12851],{"className":12852,"style":60},[59],[31,12854,65],{"className":12855},[64],[31,12857],{"className":12858,"style":60},[59],[31,12860,12862,12865,12868,12929,12978,12981,12984,12987,13047,13096],{"className":12861},[43],[31,12863],{"className":12864,"style":9407},[47],[31,12866,89],{"className":12867},[88],[31,12869,12871,12874],{"className":12870},[52],[31,12872,9896],{"className":12873},[52,53],[31,12875,12877],{"className":12876},[219],[31,12878,12880,12921],{"className":12879},[223,553],[31,12881,12883,12918],{"className":12882},[227],[31,12884,12886,12907],{"className":12885,"style":1314},[231],[31,12887,12889,12892],{"style":12888},"top:-2.4519em;margin-left:0em;margin-right:0.05em;",[31,12890],{"className":12891,"style":239},[238],[31,12893,12895],{"className":12894},[243,244,245,246],[31,12896,12898,12901,12904],{"className":12897},[52,246],[31,12899,9327],{"className":12900},[52,53,246],[31,12902,145],{"className":12903},[144,246],[31,12905,269],{"className":12906},[52,246],[31,12908,12909,12912],{"style":234},[31,12910],{"className":12911,"style":239},[238],[31,12913,12915],{"className":12914},[243,244,245,246],[31,12916,513],{"className":12917},[52,246],[31,12919,607],{"className":12920},[606],[31,12922,12924],{"className":12923},[227],[31,12925,12927],{"className":12926,"style":9563},[231],[31,12928],{},[31,12930,12932,12935],{"className":12931},[52],[31,12933,10065],{"className":12934,"style":54},[52,53],[31,12936,12938],{"className":12937},[219],[31,12939,12941,12970],{"className":12940},[223,553],[31,12942,12944,12967],{"className":12943},[227],[31,12945,12947],{"className":12946,"style":9584},[231],[31,12948,12949,12952],{"style":4477},[31,12950],{"className":12951,"style":239},[238],[31,12953,12955],{"className":12954},[243,244,245,246],[31,12956,12958,12961,12964],{"className":12957},[52,246],[31,12959,9327],{"className":12960},[52,53,246],[31,12962,145],{"className":12963},[144,246],[31,12965,269],{"className":12966},[52,246],[31,12968,607],{"className":12969},[606],[31,12971,12973],{"className":12972},[227],[31,12974,12976],{"className":12975,"style":2939},[231],[31,12977],{},[31,12979,99],{"className":12980},[98],[31,12982,1971],{"className":12983},[52],[31,12985,89],{"className":12986},[88],[31,12988,12990,12993],{"className":12989},[52],[31,12991,10065],{"className":12992,"style":54},[52,53],[31,12994,12996],{"className":12995},[219],[31,12997,12999,13039],{"className":12998},[223,553],[31,13000,13002,13036],{"className":13001},[227],[31,13003,13005,13025],{"className":13004,"style":1314},[231],[31,13006,13007,13010],{"style":9524},[31,13008],{"className":13009,"style":239},[238],[31,13011,13013],{"className":13012},[243,244,245,246],[31,13014,13016,13019,13022],{"className":13015},[52,246],[31,13017,9327],{"className":13018},[52,53,246],[31,13020,145],{"className":13021},[144,246],[31,13023,269],{"className":13024},[52,246],[31,13026,13027,13030],{"style":234},[31,13028],{"className":13029,"style":239},[238],[31,13031,13033],{"className":13032},[243,244,245,246],[31,13034,513],{"className":13035},[52,246],[31,13037,607],{"className":13038},[606],[31,13040,13042],{"className":13041},[227],[31,13043,13045],{"className":13044,"style":9563},[231],[31,13046],{},[31,13048,13050,13053],{"className":13049},[52],[31,13051,10065],{"className":13052,"style":54},[52,53],[31,13054,13056],{"className":13055},[219],[31,13057,13059,13088],{"className":13058},[223,553],[31,13060,13062,13085],{"className":13061},[227],[31,13063,13065],{"className":13064,"style":9584},[231],[31,13066,13067,13070],{"style":4477},[31,13068],{"className":13069,"style":239},[238],[31,13071,13073],{"className":13072},[243,244,245,246],[31,13074,13076,13079,13082],{"className":13075},[52,246],[31,13077,9327],{"className":13078},[52,53,246],[31,13080,145],{"className":13081},[144,246],[31,13083,269],{"className":13084},[52,246],[31,13086,607],{"className":13087},[606],[31,13089,13091],{"className":13090},[227],[31,13092,13094],{"className":13093,"style":2939},[231],[31,13095],{},[31,13097,99],{"className":13098},[98]," approximates the bulk of ",[31,13101,13103],{"className":13102},[34],[31,13104,13106],{"className":13105,"ariaHidden":39},[38],[31,13107,13109,13112],{"className":13108},[43],[31,13110],{"className":13111,"style":209},[47],[31,13113,13115,13118],{"className":13114},[52],[31,13116,193],{"className":13117,"style":192},[52,53],[31,13119,13121],{"className":13120},[219],[31,13122,13124],{"className":13123},[223],[31,13125,13127],{"className":13126},[227],[31,13128,13130],{"className":13129,"style":209},[231],[31,13131,13132,13135],{"style":234},[31,13133],{"className":13134,"style":239},[238],[31,13136,13138],{"className":13137},[243,244,245,246],[31,13139,13141,13144],{"className":13140},[52,246],[31,13142,145],{"className":13143},[52,246],[31,13145,269],{"className":13146},[52,246],", and a forward sweep folds the\ncorrections back in. Each sweep is ",[31,13149,13151],{"className":13150},[34],[31,13152,13154],{"className":13153,"ariaHidden":39},[38],[31,13155,13157,13160,13163],{"className":13156},[43],[31,13158],{"className":13159,"style":833},[47],[31,13161,250],{"className":13162},[52],[31,13164,12528],{"className":13165},[52,53]," inner products and vector updates, so one\nmatrix–vector action ",[31,13168,13170],{"className":13169},[34],[31,13171,13173],{"className":13172,"ariaHidden":39},[38],[31,13174,13176,13179,13219],{"className":13175},[43],[31,13177],{"className":13178,"style":8262},[47],[31,13180,13182,13185],{"className":13181},[52],[31,13183,9818],{"className":13184,"style":9817},[52,53],[31,13186,13188],{"className":13187},[219],[31,13189,13191,13211],{"className":13190},[223,553],[31,13192,13194,13208],{"className":13193},[227],[31,13195,13197],{"className":13196,"style":9314},[231],[31,13198,13199,13202],{"style":10583},[31,13200],{"className":13201,"style":239},[238],[31,13203,13205],{"className":13204},[243,244,245,246],[31,13206,9327],{"className":13207},[52,53,246],[31,13209,607],{"className":13210},[606],[31,13212,13214],{"className":13213},[227],[31,13215,13217],{"className":13216,"style":895},[231],[31,13218],{},[31,13220,13222,13225],{"className":13221},[52],[31,13223,55],{"className":13224,"style":54},[52,53],[31,13226,13228],{"className":13227},[219],[31,13229,13231,13251],{"className":13230},[223,553],[31,13232,13234,13248],{"className":13233},[227],[31,13235,13237],{"className":13236,"style":9314},[231],[31,13238,13239,13242],{"style":4477},[31,13240],{"className":13241,"style":239},[238],[31,13243,13245],{"className":13244},[243,244,245,246],[31,13246,9327],{"className":13247},[52,53,246],[31,13249,607],{"className":13250},[606],[31,13252,13254],{"className":13253},[227],[31,13255,13257],{"className":13256,"style":895},[231],[31,13258],{}," costs ",[31,13261,13263],{"className":13262},[34],[31,13264,13266],{"className":13265,"ariaHidden":39},[38],[31,13267,13269,13272,13275,13278,13281,13284],{"className":13268},[43],[31,13270],{"className":13271,"style":75},[47],[31,13273,3684],{"className":13274,"style":93},[52,53],[31,13276,89],{"className":13277},[88],[31,13279,12528],{"className":13280},[52,53],[31,13282,5457],{"className":13283},[52,53],[31,13285,99],{"className":13286},[98]," rather than ",[31,13289,13291],{"className":13290},[34],[31,13292,13294],{"className":13293,"ariaHidden":39},[38],[31,13295,13297,13300,13303,13306,13335],{"className":13296},[43],[31,13298],{"className":13299,"style":3071},[47],[31,13301,3684],{"className":13302,"style":93},[52,53],[31,13304,89],{"className":13305},[88],[31,13307,13309,13312],{"className":13308},[52],[31,13310,5457],{"className":13311},[52,53],[31,13313,13315],{"className":13314},[219],[31,13316,13318],{"className":13317},[223],[31,13319,13321],{"className":13320},[227],[31,13322,13324],{"className":13323,"style":209},[231],[31,13325,13326,13329],{"style":234},[31,13327],{"className":13328,"style":239},[238],[31,13330,13332],{"className":13331},[243,244,245,246],[31,13333,250],{"className":13334},[52,246],[31,13336,99],{"className":13337},[98],". Memory drops from\n",[31,13340,13342],{"className":13341},[34],[31,13343,13345],{"className":13344,"ariaHidden":39},[38],[31,13346,13348,13351,13354,13357,13386],{"className":13347},[43],[31,13349],{"className":13350,"style":3071},[47],[31,13352,3684],{"className":13353,"style":93},[52,53],[31,13355,89],{"className":13356},[88],[31,13358,13360,13363],{"className":13359},[52],[31,13361,5457],{"className":13362},[52,53],[31,13364,13366],{"className":13365},[219],[31,13367,13369],{"className":13368},[223],[31,13370,13372],{"className":13371},[227],[31,13373,13375],{"className":13374,"style":209},[231],[31,13376,13377,13380],{"style":234},[31,13378],{"className":13379,"style":239},[238],[31,13381,13383],{"className":13382},[243,244,245,246],[31,13384,250],{"className":13385},[52,246],[31,13387,99],{"className":13388},[98],[31,13390,13392],{"className":13391},[34],[31,13393,13395],{"className":13394,"ariaHidden":39},[38],[31,13396,13398,13401,13404,13407,13410,13413],{"className":13397},[43],[31,13399],{"className":13400,"style":75},[47],[31,13402,3684],{"className":13403,"style":93},[52,53],[31,13405,89],{"className":13406},[88],[31,13408,12528],{"className":13409},[52,53],[31,13411,5457],{"className":13412},[52,53],[31,13414,99],{"className":13415},[98],", the same order as the gradient. That is what makes a quasi-Newton\nmethod usable at deep-learning scale (in the full-batch regime).",[261,13418,13419],{},[15,13420,12668],{"href":13421,"ariaDescribedBy":13422,"dataFootnoteRef":6,"id":13423},"#user-content-fn-gf-bfgs",[267],"user-content-fnref-gf-bfgs",[11,13425,13426,13427,13531,13532,13703,13704,13707,13708,13760,13761,13816,13817,13832],{},"The reason L-BFGS is still uncommon on deep nets is the same noise wall Newton hit, now\nin a subtler place. The secant pair ",[31,13428,13430],{"className":13429},[34],[31,13431,13433],{"className":13432,"ariaHidden":39},[38],[31,13434,13436,13439,13442,13482,13485,13488,13528],{"className":13435},[43],[31,13437],{"className":13438,"style":75},[47],[31,13440,89],{"className":13441},[88],[31,13443,13445,13448],{"className":13444},[52],[31,13446,9896],{"className":13447},[52,53],[31,13449,13451],{"className":13450},[219],[31,13452,13454,13474],{"className":13453},[223,553],[31,13455,13457,13471],{"className":13456},[227],[31,13458,13460],{"className":13459,"style":9314},[231],[31,13461,13462,13465],{"style":876},[31,13463],{"className":13464,"style":239},[238],[31,13466,13468],{"className":13467},[243,244,245,246],[31,13469,9327],{"className":13470},[52,53,246],[31,13472,607],{"className":13473},[606],[31,13475,13477],{"className":13476},[227],[31,13478,13480],{"className":13479,"style":895},[231],[31,13481],{},[31,13483,666],{"className":13484},[665],[31,13486],{"className":13487,"style":623},[59],[31,13489,13491,13494],{"className":13490},[52],[31,13492,10065],{"className":13493,"style":54},[52,53],[31,13495,13497],{"className":13496},[219],[31,13498,13500,13520],{"className":13499},[223,553],[31,13501,13503,13517],{"className":13502},[227],[31,13504,13506],{"className":13505,"style":9314},[231],[31,13507,13508,13511],{"style":4477},[31,13509],{"className":13510,"style":239},[238],[31,13512,13514],{"className":13513},[243,244,245,246],[31,13515,9327],{"className":13516},[52,53,246],[31,13518,607],{"className":13519},[606],[31,13521,13523],{"className":13522},[227],[31,13524,13526],{"className":13525,"style":895},[231],[31,13527],{},[31,13529,99],{"className":13530},[98]," is built from the difference\n",[31,13533,13535],{"className":13534},[34],[31,13536,13538,13593,13657],{"className":13537,"ariaHidden":39},[38],[31,13539,13541,13544,13584,13587,13590],{"className":13540},[43],[31,13542],{"className":13543,"style":48},[47],[31,13545,13547,13550],{"className":13546},[52],[31,13548,10065],{"className":13549,"style":54},[52,53],[31,13551,13553],{"className":13552},[219],[31,13554,13556,13576],{"className":13555},[223,553],[31,13557,13559,13573],{"className":13558},[227],[31,13560,13562],{"className":13561,"style":9314},[231],[31,13563,13564,13567],{"style":4477},[31,13565],{"className":13566,"style":239},[238],[31,13568,13570],{"className":13569},[243,244,245,246],[31,13571,9327],{"className":13572},[52,53,246],[31,13574,607],{"className":13575},[606],[31,13577,13579],{"className":13578},[227],[31,13580,13582],{"className":13581,"style":895},[231],[31,13583],{},[31,13585],{"className":13586,"style":60},[59],[31,13588,65],{"className":13589},[64],[31,13591],{"className":13592,"style":60},[59],[31,13594,13596,13599,13648,13651,13654],{"className":13595},[43],[31,13597],{"className":13598,"style":10115},[47],[31,13600,13602,13605],{"className":13601},[52],[31,13603,55],{"className":13604,"style":54},[52,53],[31,13606,13608],{"className":13607},[219],[31,13609,13611,13640],{"className":13610},[223,553],[31,13612,13614,13637],{"className":13613},[227],[31,13615,13617],{"className":13616,"style":9584},[231],[31,13618,13619,13622],{"style":4477},[31,13620],{"className":13621,"style":239},[238],[31,13623,13625],{"className":13624},[243,244,245,246],[31,13626,13628,13631,13634],{"className":13627},[52,246],[31,13629,9327],{"className":13630},[52,53,246],[31,13632,410],{"className":13633},[144,246],[31,13635,269],{"className":13636},[52,246],[31,13638,607],{"className":13639},[606],[31,13641,13643],{"className":13642},[227],[31,13644,13646],{"className":13645,"style":2939},[231],[31,13647],{},[31,13649],{"className":13650,"style":140},[59],[31,13652,145],{"className":13653},[144],[31,13655],{"className":13656,"style":140},[59],[31,13658,13660,13663],{"className":13659},[43],[31,13661],{"className":13662,"style":48},[47],[31,13664,13666,13669],{"className":13665},[52],[31,13667,55],{"className":13668,"style":54},[52,53],[31,13670,13672],{"className":13671},[219],[31,13673,13675,13695],{"className":13674},[223,553],[31,13676,13678,13692],{"className":13677},[227],[31,13679,13681],{"className":13680,"style":9314},[231],[31,13682,13683,13686],{"style":4477},[31,13684],{"className":13685,"style":239},[238],[31,13687,13689],{"className":13688},[243,244,245,246],[31,13690,9327],{"className":13691},[52,53,246],[31,13693,607],{"className":13694},[606],[31,13696,13698],{"className":13697},[227],[31,13699,13701],{"className":13700,"style":895},[231],[31,13702],{}," of two gradients. Under minibatching those gradients are computed\non ",[164,13705,13706],{},"different"," samples, so ",[31,13709,13711],{"className":13710},[34],[31,13712,13714],{"className":13713,"ariaHidden":39},[38],[31,13715,13717,13720],{"className":13716},[43],[31,13718],{"className":13719,"style":48},[47],[31,13721,13723,13726],{"className":13722},[52],[31,13724,10065],{"className":13725,"style":54},[52,53],[31,13727,13729],{"className":13728},[219],[31,13730,13732,13752],{"className":13731},[223,553],[31,13733,13735,13749],{"className":13734},[227],[31,13736,13738],{"className":13737,"style":9314},[231],[31,13739,13740,13743],{"style":4477},[31,13741],{"className":13742,"style":239},[238],[31,13744,13746],{"className":13745},[243,244,245,246],[31,13747,9327],{"className":13748},[52,53,246],[31,13750,607],{"className":13751},[606],[31,13753,13755],{"className":13754},[227],[31,13756,13758],{"className":13757,"style":895},[231],[31,13759],{}," mixes the true curvature signal ",[31,13762,13764],{"className":13763},[34],[31,13765,13767],{"className":13766,"ariaHidden":39},[38],[31,13768,13770,13773,13776],{"className":13769},[43],[31,13771],{"className":13772,"style":7130},[47],[31,13774,193],{"className":13775,"style":192},[52,53],[31,13777,13779,13782],{"className":13778},[52],[31,13780,9896],{"className":13781},[52,53],[31,13783,13785],{"className":13784},[219],[31,13786,13788,13808],{"className":13787},[223,553],[31,13789,13791,13805],{"className":13790},[227],[31,13792,13794],{"className":13793,"style":9314},[231],[31,13795,13796,13799],{"style":876},[31,13797],{"className":13798,"style":239},[238],[31,13800,13802],{"className":13801},[243,244,245,246],[31,13803,9327],{"className":13804},[52,53,246],[31,13806,607],{"className":13807},[606],[31,13809,13811],{"className":13810},[227],[31,13812,13814],{"className":13813,"style":895},[231],[31,13815],{}," with the\ndifference of two independent noise terms. That noise does not average away inside the\nBFGS update the way it does across SGD steps — it corrupts the very curvature estimate the\nmethod accumulates, and a stale or noisy ",[31,13818,13820],{"className":13819},[34],[31,13821,13823],{"className":13822,"ariaHidden":39},[38],[31,13824,13826,13829],{"className":13825},[43],[31,13827],{"className":13828,"style":188},[47],[31,13830,9818],{"className":13831,"style":9817},[52,53]," can point the step in a wrong direction. L-BFGS\nworks best with large or full batches, where the gradients are clean\nenough; on noisy minibatch gradients the first-order methods are preferable.",[6170,13834,13835,13854],{},[6173,13836,13837],{},[6176,13838,13839,13842,13845,13848,13851],{},[6179,13840,13841],{},"Method",[6179,13843,13844],{},"Hessian use",[6179,13846,13847],{},"Memory",[6179,13849,13850],{},"Per-step cost",[6179,13852,13853],{},"Reaches quadratic min in",[6262,13855,13856,14002,14112,14330],{},[6176,13857,13858,13861,13880,13932,13984],{},[6267,13859,13860],{},"Newton",[6267,13862,13863,13864,13879],{},"exact ",[31,13865,13867],{"className":13866},[34],[31,13868,13870],{"className":13869,"ariaHidden":39},[38],[31,13871,13873,13876],{"className":13872},[43],[31,13874],{"className":13875,"style":188},[47],[31,13877,193],{"className":13878,"style":192},[52,53],", inverts it",[6267,13881,13882],{},[31,13883,13885],{"className":13884},[34],[31,13886,13888],{"className":13887,"ariaHidden":39},[38],[31,13889,13891,13894,13897,13900,13929],{"className":13890},[43],[31,13892],{"className":13893,"style":3071},[47],[31,13895,3684],{"className":13896,"style":93},[52,53],[31,13898,89],{"className":13899},[88],[31,13901,13903,13906],{"className":13902},[52],[31,13904,5457],{"className":13905},[52,53],[31,13907,13909],{"className":13908},[219],[31,13910,13912],{"className":13911},[223],[31,13913,13915],{"className":13914},[227],[31,13916,13918],{"className":13917,"style":209},[231],[31,13919,13920,13923],{"style":234},[31,13921],{"className":13922,"style":239},[238],[31,13924,13926],{"className":13925},[243,244,245,246],[31,13927,250],{"className":13928},[52,246],[31,13930,99],{"className":13931},[98],[6267,13933,13934],{},[31,13935,13937],{"className":13936},[34],[31,13938,13940],{"className":13939,"ariaHidden":39},[38],[31,13941,13943,13946,13949,13952,13981],{"className":13942},[43],[31,13944],{"className":13945,"style":3071},[47],[31,13947,3684],{"className":13948,"style":93},[52,53],[31,13950,89],{"className":13951},[88],[31,13953,13955,13958],{"className":13954},[52],[31,13956,5457],{"className":13957},[52,53],[31,13959,13961],{"className":13960},[219],[31,13962,13964],{"className":13963},[223],[31,13965,13967],{"className":13966},[227],[31,13968,13970],{"className":13969,"style":209},[231],[31,13971,13972,13975],{"style":234},[31,13973],{"className":13974,"style":239},[238],[31,13976,13978],{"className":13977},[243,244,245,246],[31,13979,5633],{"className":13980},[52,246],[31,13982,99],{"className":13983},[98],[6267,13985,13986,14001],{},[31,13987,13989],{"className":13988},[34],[31,13990,13992],{"className":13991,"ariaHidden":39},[38],[31,13993,13995,13998],{"className":13994},[43],[31,13996],{"className":13997,"style":833},[47],[31,13999,269],{"className":14000},[52]," step",[6176,14003,14004,14006,14028,14054,14081],{},[6267,14005,8776],{},[6267,14007,14008,14009,14027],{},"implicit (via ",[31,14010,14012],{"className":14011},[34],[31,14013,14015],{"className":14014,"ariaHidden":39},[38],[31,14016,14018,14021,14024],{"className":14017},[43],[31,14019],{"className":14020,"style":113},[47],[31,14022,193],{"className":14023,"style":192},[52,53],[31,14025,5457],{"className":14026},[52,53]," products)",[6267,14029,14030],{},[31,14031,14033],{"className":14032},[34],[31,14034,14036],{"className":14035,"ariaHidden":39},[38],[31,14037,14039,14042,14045,14048,14051],{"className":14038},[43],[31,14040],{"className":14041,"style":75},[47],[31,14043,3684],{"className":14044,"style":93},[52,53],[31,14046,89],{"className":14047},[88],[31,14049,5457],{"className":14050},[52,53],[31,14052,99],{"className":14053},[98],[6267,14055,14056,14080],{},[31,14057,14059],{"className":14058},[34],[31,14060,14062],{"className":14061,"ariaHidden":39},[38],[31,14063,14065,14068,14071,14074,14077],{"className":14064},[43],[31,14066],{"className":14067,"style":75},[47],[31,14069,3684],{"className":14070,"style":93},[52,53],[31,14072,89],{"className":14073},[88],[31,14075,5457],{"className":14076},[52,53],[31,14078,99],{"className":14079},[98]," + line search",[6267,14082,14083,14111],{},[31,14084,14086],{"className":14085},[34],[31,14087,14089,14102],{"className":14088,"ariaHidden":39},[38],[31,14090,14092,14096,14099],{"className":14091},[43],[31,14093],{"className":14094,"style":14095},[47],"height:0.7719em;vertical-align:-0.136em;",[31,14097,2995],{"className":14098},[64],[31,14100],{"className":14101,"style":60},[59],[31,14103,14105,14108],{"className":14104},[43],[31,14106],{"className":14107,"style":113},[47],[31,14109,5457],{"className":14110},[52,53]," steps",[6176,14113,14114,14116,14197,14249,14301],{},[6267,14115,10549],{},[6267,14117,14118,14119,14166,14167],{},"approximates ",[31,14120,14122],{"className":14121},[34],[31,14123,14125],{"className":14124,"ariaHidden":39},[38],[31,14126,14128,14131],{"className":14127},[43],[31,14129],{"className":14130,"style":209},[47],[31,14132,14134,14137],{"className":14133},[52],[31,14135,193],{"className":14136,"style":192},[52,53],[31,14138,14140],{"className":14139},[219],[31,14141,14143],{"className":14142},[223],[31,14144,14146],{"className":14145},[227],[31,14147,14149],{"className":14148,"style":209},[231],[31,14150,14151,14154],{"style":234},[31,14152],{"className":14153,"style":239},[238],[31,14155,14157],{"className":14156},[243,244,245,246],[31,14158,14160,14163],{"className":14159},[52,246],[31,14161,145],{"className":14162},[52,246],[31,14164,269],{"className":14165},[52,246]," from ",[31,14168,14170],{"className":14169},[34],[31,14171,14173],{"className":14172,"ariaHidden":39},[38],[31,14174,14176,14179,14182,14185,14188,14191,14194],{"className":14175},[43],[31,14177],{"className":14178,"style":75},[47],[31,14180,89],{"className":14181},[88],[31,14183,9896],{"className":14184},[52,53],[31,14186,666],{"className":14187},[665],[31,14189],{"className":14190,"style":623},[59],[31,14192,10065],{"className":14193,"style":54},[52,53],[31,14195,99],{"className":14196},[98],[6267,14198,14199],{},[31,14200,14202],{"className":14201},[34],[31,14203,14205],{"className":14204,"ariaHidden":39},[38],[31,14206,14208,14211,14214,14217,14246],{"className":14207},[43],[31,14209],{"className":14210,"style":3071},[47],[31,14212,3684],{"className":14213,"style":93},[52,53],[31,14215,89],{"className":14216},[88],[31,14218,14220,14223],{"className":14219},[52],[31,14221,5457],{"className":14222},[52,53],[31,14224,14226],{"className":14225},[219],[31,14227,14229],{"className":14228},[223],[31,14230,14232],{"className":14231},[227],[31,14233,14235],{"className":14234,"style":209},[231],[31,14236,14237,14240],{"style":234},[31,14238],{"className":14239,"style":239},[238],[31,14241,14243],{"className":14242},[243,244,245,246],[31,14244,250],{"className":14245},[52,246],[31,14247,99],{"className":14248},[98],[6267,14250,14251],{},[31,14252,14254],{"className":14253},[34],[31,14255,14257],{"className":14256,"ariaHidden":39},[38],[31,14258,14260,14263,14266,14269,14298],{"className":14259},[43],[31,14261],{"className":14262,"style":3071},[47],[31,14264,3684],{"className":14265,"style":93},[52,53],[31,14267,89],{"className":14268},[88],[31,14270,14272,14275],{"className":14271},[52],[31,14273,5457],{"className":14274},[52,53],[31,14276,14278],{"className":14277},[219],[31,14279,14281],{"className":14280},[223],[31,14282,14284],{"className":14283},[227],[31,14285,14287],{"className":14286,"style":209},[231],[31,14288,14289,14292],{"style":234},[31,14290],{"className":14291,"style":239},[238],[31,14293,14295],{"className":14294},[243,244,245,246],[31,14296,250],{"className":14297},[52,246],[31,14299,99],{"className":14300},[98],[6267,14302,14303,14111],{},[31,14304,14306],{"className":14305},[34],[31,14307,14309,14321],{"className":14308,"ariaHidden":39},[38],[31,14310,14312,14315,14318],{"className":14311},[43],[31,14313],{"className":14314,"style":14095},[47],[31,14316,2995],{"className":14317},[64],[31,14319],{"className":14320,"style":60},[59],[31,14322,14324,14327],{"className":14323},[43],[31,14325],{"className":14326,"style":113},[47],[31,14328,5457],{"className":14329},[52,53],[6176,14331,14332,14334,14353,14382,14411],{},[6267,14333,12495],{},[6267,14335,14336,14337,14352],{},"last ",[31,14338,14340],{"className":14339},[34],[31,14341,14343],{"className":14342,"ariaHidden":39},[38],[31,14344,14346,14349],{"className":14345},[43],[31,14347],{"className":14348,"style":9257},[47],[31,14350,12528],{"className":14351},[52,53]," curvature pairs",[6267,14354,14355],{},[31,14356,14358],{"className":14357},[34],[31,14359,14361],{"className":14360,"ariaHidden":39},[38],[31,14362,14364,14367,14370,14373,14376,14379],{"className":14363},[43],[31,14365],{"className":14366,"style":75},[47],[31,14368,3684],{"className":14369,"style":93},[52,53],[31,14371,89],{"className":14372},[88],[31,14374,12528],{"className":14375},[52,53],[31,14377,5457],{"className":14378},[52,53],[31,14380,99],{"className":14381},[98],[6267,14383,14384],{},[31,14385,14387],{"className":14386},[34],[31,14388,14390],{"className":14389,"ariaHidden":39},[38],[31,14391,14393,14396,14399,14402,14405,14408],{"className":14392},[43],[31,14394],{"className":14395,"style":75},[47],[31,14397,3684],{"className":14398,"style":93},[52,53],[31,14400,89],{"className":14401},[88],[31,14403,12528],{"className":14404},[52,53],[31,14406,5457],{"className":14407},[52,53],[31,14409,99],{"className":14410},[98],[6267,14412,14413,14414,14111],{},"superlinear, ",[31,14415,14417],{"className":14416},[34],[31,14418,14420,14432],{"className":14419,"ariaHidden":39},[38],[31,14421,14423,14426,14429],{"className":14422},[43],[31,14424],{"className":14425,"style":14095},[47],[31,14427,2995],{"className":14428},[64],[31,14430],{"className":14431,"style":60},[59],[31,14433,14435,14438],{"className":14434},[43],[31,14436],{"className":14437,"style":113},[47],[31,14439,5457],{"className":14440},[52,53],[271,14442,14444],{"id":14443},"gauss-newton-and-the-fisher-matrix","Gauss-Newton and the Fisher matrix",[11,14446,14447,14448,14463,14464,14691,14692,14740],{},"Before the natural gradient there is a simpler positive-semidefinite stand-in for ",[31,14449,14451],{"className":14450},[34],[31,14452,14454],{"className":14453,"ariaHidden":39},[38],[31,14455,14457,14460],{"className":14456},[43],[31,14458],{"className":14459,"style":188},[47],[31,14461,193],{"className":14462,"style":192},[52,53],".\nFor a loss that is a sum of squared residuals ",[31,14465,14467],{"className":14466},[34],[31,14468,14470,14488],{"className":14469,"ariaHidden":39},[38],[31,14471,14473,14476,14479,14482,14485],{"className":14472},[43],[31,14474],{"className":14475,"style":188},[47],[31,14477,84],{"className":14478},[52,83],[31,14480],{"className":14481,"style":60},[59],[31,14483,65],{"className":14484},[64],[31,14486],{"className":14487,"style":60},[59],[31,14489,14491,14494,14562,14565,14612,14615,14656,14659,14662],{"className":14490},[43],[31,14492],{"className":14493,"style":2068},[47],[31,14495,14497,14500,14559],{"className":14496},[52],[31,14498],{"className":14499},[88,545],[31,14501,14503],{"className":14502},[549],[31,14504,14506,14551],{"className":14505},[223,553],[31,14507,14509,14548],{"className":14508},[227],[31,14510,14512,14526,14534],{"className":14511,"style":560},[231],[31,14513,14514,14517],{"style":563},[31,14515],{"className":14516,"style":567},[238],[31,14518,14520],{"className":14519},[243,244,245,246],[31,14521,14523],{"className":14522},[52,246],[31,14524,250],{"className":14525},[52,246],[31,14527,14528,14531],{"style":579},[31,14529],{"className":14530,"style":567},[238],[31,14532],{"className":14533,"style":587},[586],[31,14535,14536,14539],{"style":590},[31,14537],{"className":14538,"style":567},[238],[31,14540,14542],{"className":14541},[243,244,245,246],[31,14543,14545],{"className":14544},[52,246],[31,14546,269],{"className":14547},[52,246],[31,14549,607],{"className":14550},[606],[31,14552,14554],{"className":14553},[227],[31,14555,14557],{"className":14556,"style":614},[231],[31,14558],{},[31,14560],{"className":14561},[98,545],[31,14563],{"className":14564,"style":623},[59],[31,14566,14568,14575],{"className":14567},[1951],[31,14569,14574],{"className":14570,"style":14573},[1951,14571,14572],"op-symbol","small-op","position:relative;top:0em;","∑",[31,14576,14578],{"className":14577},[219],[31,14579,14581,14603],{"className":14580},[223,553],[31,14582,14584,14600],{"className":14583},[227],[31,14585,14588],{"className":14586,"style":14587},[231],"height:0.162em;",[31,14589,14591,14594],{"style":14590},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[31,14592],{"className":14593,"style":239},[238],[31,14595,14597],{"className":14596},[243,244,245,246],[31,14598,7006],{"className":14599},[52,53,246],[31,14601,607],{"className":14602},[606],[31,14604,14606],{"className":14605},[227],[31,14607,14610],{"className":14608,"style":14609},[231],"height:0.2997em;",[31,14611],{},[31,14613],{"className":14614,"style":623},[59],[31,14616,14618,14622],{"className":14617},[52],[31,14619,14621],{"className":14620,"style":93},[52,53],"r",[31,14623,14625],{"className":14624},[219],[31,14626,14628,14648],{"className":14627},[223,553],[31,14629,14631,14645],{"className":14630},[227],[31,14632,14634],{"className":14633,"style":7038},[231],[31,14635,14636,14639],{"style":2909},[31,14637],{"className":14638,"style":239},[238],[31,14640,14642],{"className":14641},[243,244,245,246],[31,14643,7006],{"className":14644},[52,53,246],[31,14646,607],{"className":14647},[606],[31,14649,14651],{"className":14650},[227],[31,14652,14654],{"className":14653,"style":895},[231],[31,14655],{},[31,14657,89],{"className":14658},[88],[31,14660,94],{"className":14661,"style":93},[52,53],[31,14663,14665,14668],{"className":14664},[98],[31,14666,99],{"className":14667},[98],[31,14669,14671],{"className":14670},[219],[31,14672,14674],{"className":14673},[223],[31,14675,14677],{"className":14676},[227],[31,14678,14680],{"className":14679,"style":209},[231],[31,14681,14682,14685],{"style":234},[31,14683],{"className":14684,"style":239},[238],[31,14686,14688],{"className":14687},[243,244,245,246],[31,14689,250],{"className":14690},[52,246],",\nwrite the Jacobian of the residual vector as ",[31,14693,14695],{"className":14694},[34],[31,14696,14698,14718],{"className":14697,"ariaHidden":39},[38],[31,14699,14701,14704,14709,14712,14715],{"className":14700},[43],[31,14702],{"className":14703,"style":188},[47],[31,14705,14708],{"className":14706,"style":14707},[52,53],"margin-right:0.0962em;","J",[31,14710],{"className":14711,"style":60},[59],[31,14713,65],{"className":14714},[64],[31,14716],{"className":14717,"style":60},[59],[31,14719,14721,14724,14728,14731,14734,14737],{"className":14720},[43],[31,14722],{"className":14723,"style":75},[47],[31,14725,14727],{"className":14726,"style":12848},[52],"∂",[31,14729,14621],{"className":14730,"style":93},[52,53],[31,14732,1971],{"className":14733},[52],[31,14735,14727],{"className":14736,"style":12848},[52],[31,14738,94],{"className":14739,"style":93},[52,53],". The exact\nHessian is",[31,14742,14744],{"className":14743},[382],[31,14745,14747],{"className":14746},[34],[31,14748,14750,14768,14822],{"className":14749,"ariaHidden":39},[38],[31,14751,14753,14756,14759,14762,14765],{"className":14752},[43],[31,14754],{"className":14755,"style":188},[47],[31,14757,193],{"className":14758,"style":192},[52,53],[31,14760],{"className":14761,"style":60},[59],[31,14763,65],{"className":14764},[64],[31,14766],{"className":14767,"style":60},[59],[31,14769,14771,14775,14804,14807,14810,14813,14816,14819],{"className":14770},[43],[31,14772],{"className":14773,"style":14774},[47],"height:0.9824em;vertical-align:-0.0833em;",[31,14776,14778,14781],{"className":14777},[52],[31,14779,14708],{"className":14780,"style":14707},[52,53],[31,14782,14784],{"className":14783},[219],[31,14785,14787],{"className":14786},[223],[31,14788,14790],{"className":14789},[227],[31,14791,14793],{"className":14792,"style":500},[231],[31,14794,14795,14798],{"style":503},[31,14796],{"className":14797,"style":239},[238],[31,14799,14801],{"className":14800},[243,244,245,246],[31,14802,513],{"className":14803},[52,246],[31,14805,14708],{"className":14806,"style":14707},[52,53],[31,14808],{"className":14809,"style":60},[59],[31,14811],{"className":14812,"style":140},[59],[31,14814,410],{"className":14815},[144],[31,14817],{"className":14818,"style":60},[59],[31,14820],{"className":14821,"style":140},[59],[31,14823,14825,14829,14880,14883,14923,14926,14955,14995],{"className":14824},[43],[31,14826],{"className":14827,"style":14828},[47],"height:2.3277em;vertical-align:-1.2777em;",[31,14830,14833],{"className":14831},[1951,14832],"op-limits",[31,14834,14836,14871],{"className":14835},[223,553],[31,14837,14839,14868],{"className":14838},[227],[31,14840,14843,14856],{"className":14841,"style":14842},[231],"height:1.05em;",[31,14844,14846,14850],{"style":14845},"top:-1.8723em;margin-left:0em;",[31,14847],{"className":14848,"style":14849},[238],"height:3.05em;",[31,14851,14853],{"className":14852},[243,244,245,246],[31,14854,7006],{"className":14855},[52,53,246],[31,14857,14859,14862],{"style":14858},"top:-3.05em;",[31,14860],{"className":14861,"style":14849},[238],[31,14863,14864],{},[31,14865,14574],{"className":14866},[1951,14571,14867],"large-op",[31,14869,607],{"className":14870},[606],[31,14872,14874],{"className":14873},[227],[31,14875,14878],{"className":14876,"style":14877},[231],"height:1.2777em;",[31,14879],{},[31,14881],{"className":14882,"style":623},[59],[31,14884,14886,14889],{"className":14885},[52],[31,14887,14621],{"className":14888,"style":93},[52,53],[31,14890,14892],{"className":14891},[219],[31,14893,14895,14915],{"className":14894},[223,553],[31,14896,14898,14912],{"className":14897},[227],[31,14899,14901],{"className":14900,"style":7038},[231],[31,14902,14903,14906],{"style":2909},[31,14904],{"className":14905,"style":239},[238],[31,14907,14909],{"className":14908},[243,244,245,246],[31,14910,7006],{"className":14911},[52,53,246],[31,14913,607],{"className":14914},[606],[31,14916,14918],{"className":14917},[227],[31,14919,14921],{"className":14920,"style":895},[231],[31,14922],{},[31,14924],{"className":14925,"style":623},[59],[31,14927,14929,14932],{"className":14928},[52],[31,14930,79],{"className":14931},[52],[31,14933,14935],{"className":14934},[219],[31,14936,14938],{"className":14937},[223],[31,14939,14941],{"className":14940},[227],[31,14942,14944],{"className":14943,"style":754},[231],[31,14945,14946,14949],{"style":503},[31,14947],{"className":14948,"style":239},[238],[31,14950,14952],{"className":14951},[243,244,245,246],[31,14953,250],{"className":14954},[52,246],[31,14956,14958,14961],{"className":14957},[52],[31,14959,14621],{"className":14960,"style":93},[52,53],[31,14962,14964],{"className":14963},[219],[31,14965,14967,14987],{"className":14966},[223,553],[31,14968,14970,14984],{"className":14969},[227],[31,14971,14973],{"className":14972,"style":7038},[231],[31,14974,14975,14978],{"style":2909},[31,14976],{"className":14977,"style":239},[238],[31,14979,14981],{"className":14980},[243,244,245,246],[31,14982,7006],{"className":14983},[52,53,246],[31,14985,607],{"className":14986},[606],[31,14988,14990],{"className":14989},[227],[31,14991,14993],{"className":14992,"style":895},[231],[31,14994],{},[31,14996,781],{"className":14997},[52],[11,14999,15000,15001,15004,15005,15068,15069,15151,15152,15204,15205,15269],{},"The ",[26,15002,15003],{},"Gauss-Newton"," approximation drops the second term and keeps only ",[31,15006,15008],{"className":15007},[34],[31,15009,15011,15030],{"className":15010,"ariaHidden":39},[38],[31,15012,15014,15017,15021,15024,15027],{"className":15013},[43],[31,15015],{"className":15016,"style":188},[47],[31,15018,15020],{"className":15019},[52,53],"G",[31,15022],{"className":15023,"style":60},[59],[31,15025,65],{"className":15026},[64],[31,15028],{"className":15029,"style":60},[59],[31,15031,15033,15036,15065],{"className":15032},[43],[31,15034],{"className":15035,"style":1314},[47],[31,15037,15039,15042],{"className":15038},[52],[31,15040,14708],{"className":15041,"style":14707},[52,53],[31,15043,15045],{"className":15044},[219],[31,15046,15048],{"className":15047},[223],[31,15049,15051],{"className":15050},[227],[31,15052,15054],{"className":15053,"style":1314},[231],[31,15055,15056,15059],{"style":234},[31,15057],{"className":15058,"style":239},[238],[31,15060,15062],{"className":15061},[243,244,245,246],[31,15063,513],{"className":15064},[52,246],[31,15066,14708],{"className":15067,"style":14707},[52,53],".\nThat term is the part carrying the residual curvature ",[31,15070,15072],{"className":15071},[34],[31,15073,15075],{"className":15074,"ariaHidden":39},[38],[31,15076,15078,15082,15111],{"className":15077},[43],[31,15079],{"className":15080,"style":15081},[47],"height:0.9641em;vertical-align:-0.15em;",[31,15083,15085,15088],{"className":15084},[52],[31,15086,79],{"className":15087},[52],[31,15089,15091],{"className":15090},[219],[31,15092,15094],{"className":15093},[223],[31,15095,15097],{"className":15096},[227],[31,15098,15100],{"className":15099,"style":209},[231],[31,15101,15102,15105],{"style":234},[31,15103],{"className":15104,"style":239},[238],[31,15106,15108],{"className":15107},[243,244,245,246],[31,15109,250],{"className":15110},[52,246],[31,15112,15114,15117],{"className":15113},[52],[31,15115,14621],{"className":15116,"style":93},[52,53],[31,15118,15120],{"className":15119},[219],[31,15121,15123,15143],{"className":15122},[223,553],[31,15124,15126,15140],{"className":15125},[227],[31,15127,15129],{"className":15128,"style":7038},[231],[31,15130,15131,15134],{"style":2909},[31,15132],{"className":15133,"style":239},[238],[31,15135,15137],{"className":15136},[243,244,245,246],[31,15138,7006],{"className":15139},[52,53,246],[31,15141,607],{"className":15142},[606],[31,15144,15146],{"className":15145},[227],[31,15147,15149],{"className":15148,"style":895},[231],[31,15150],{},"; near a good fit the\nresiduals ",[31,15153,15155],{"className":15154},[34],[31,15156,15158],{"className":15157,"ariaHidden":39},[38],[31,15159,15161,15164],{"className":15160},[43],[31,15162],{"className":15163,"style":9889},[47],[31,15165,15167,15170],{"className":15166},[52],[31,15168,14621],{"className":15169,"style":93},[52,53],[31,15171,15173],{"className":15172},[219],[31,15174,15176,15196],{"className":15175},[223,553],[31,15177,15179,15193],{"className":15178},[227],[31,15180,15182],{"className":15181,"style":7038},[231],[31,15183,15184,15187],{"style":2909},[31,15185],{"className":15186,"style":239},[238],[31,15188,15190],{"className":15189},[243,244,245,246],[31,15191,7006],{"className":15192},[52,53,246],[31,15194,607],{"className":15195},[606],[31,15197,15199],{"className":15198},[227],[31,15200,15202],{"className":15201,"style":895},[231],[31,15203],{}," are small, so dropping it costs little, and what remains is an outer\nproduct ",[31,15206,15208],{"className":15207},[34],[31,15209,15211,15260],{"className":15210,"ariaHidden":39},[38],[31,15212,15214,15218,15247,15250,15253,15257],{"className":15213},[43],[31,15215],{"className":15216,"style":15217},[47],"height:0.9851em;vertical-align:-0.136em;",[31,15219,15221,15224],{"className":15220},[52],[31,15222,14708],{"className":15223,"style":14707},[52,53],[31,15225,15227],{"className":15226},[219],[31,15228,15230],{"className":15229},[223],[31,15231,15233],{"className":15232},[227],[31,15234,15236],{"className":15235,"style":1314},[231],[31,15237,15238,15241],{"style":234},[31,15239],{"className":15240,"style":239},[238],[31,15242,15244],{"className":15243},[243,244,245,246],[31,15245,513],{"className":15246},[52,246],[31,15248,14708],{"className":15249,"style":14707},[52,53],[31,15251],{"className":15252,"style":60},[59],[31,15254,15256],{"className":15255},[64],"⪰",[31,15258],{"className":15259,"style":60},[59],[31,15261,15263,15266],{"className":15262},[43],[31,15264],{"className":15265,"style":833},[47],[31,15267,837],{"className":15268},[52]," that can never be indefinite. Gauss-Newton gets the same\nimmunity to saddles that damping enforces by hand, and it needs only first\nderivatives of the residuals.",[11,15271,15000,15272,15275,15276,15356],{},[26,15273,15274],{},"Fisher information matrix"," generalizes this to probabilistic models. When the loss\nis the negative log-likelihood of the model's output distribution ",[31,15277,15279],{"className":15278},[34],[31,15280,15282,15343],{"className":15281,"ariaHidden":39},[38],[31,15283,15285,15288,15328,15331,15334,15337,15340],{"className":15284},[43],[31,15286],{"className":15287,"style":75},[47],[31,15289,15291,15294],{"className":15290},[52],[31,15292,11],{"className":15293},[52,53],[31,15295,15297],{"className":15296},[219],[31,15298,15300,15320],{"className":15299},[223,553],[31,15301,15303,15317],{"className":15302},[227],[31,15304,15306],{"className":15305,"style":2906},[231],[31,15307,15308,15311],{"style":876},[31,15309],{"className":15310,"style":239},[238],[31,15312,15314],{"className":15313},[243,244,245,246],[31,15315,94],{"className":15316,"style":93},[52,53,246],[31,15318,607],{"className":15319},[606],[31,15321,15323],{"className":15322},[227],[31,15324,15326],{"className":15325,"style":895},[231],[31,15327],{},[31,15329,89],{"className":15330},[88],[31,15332,10065],{"className":15333,"style":54},[52,53],[31,15335],{"className":15336,"style":60},[59],[31,15338,8125],{"className":15339},[64],[31,15341],{"className":15342,"style":60},[59],[31,15344,15346,15349,15353],{"className":15345},[43],[31,15347],{"className":15348,"style":75},[47],[31,15350,15352],{"className":15351},[52,53],"x",[31,15354,99],{"className":15355},[98],", the\nFisher is the expected outer product of the score,",[31,15358,15360],{"className":15359},[382],[31,15361,15363],{"className":15362},[34],[31,15364,15366,15386],{"className":15365,"ariaHidden":39},[38],[31,15367,15369,15372,15377,15380,15383],{"className":15368},[43],[31,15370],{"className":15371,"style":188},[47],[31,15373,15376],{"className":15374,"style":15375},[52,53],"margin-right:0.1389em;","F",[31,15378],{"className":15379,"style":60},[59],[31,15381,65],{"className":15382},[64],[31,15384],{"className":15385,"style":60},[59],[31,15387,15389,15392,15437,15440,15531,15534,15537,15809,15812],{"className":15388},[43],[31,15390],{"className":15391,"style":854},[47],[31,15393,15395,15400],{"className":15394},[52],[31,15396,15399],{"className":15397},[52,15398],"mathbb","E",[31,15401,15403],{"className":15402},[219],[31,15404,15406,15429],{"className":15405},[223,553],[31,15407,15409,15426],{"className":15408},[227],[31,15410,15412],{"className":15411,"style":1936},[231],[31,15413,15414,15417],{"style":876},[31,15415],{"className":15416,"style":239},[238],[31,15418,15420],{"className":15419},[243,244,245,246],[31,15421,15423],{"className":15422},[52,246],[31,15424,15352],{"className":15425},[52,53,246],[31,15427,607],{"className":15428},[606],[31,15430,15432],{"className":15431},[227],[31,15433,15435],{"className":15434,"style":895},[231],[31,15436],{},[31,15438],{"className":15439,"style":623},[59],[31,15441,15443,15446],{"className":15442},[52],[31,15444,15399],{"className":15445},[52,15398],[31,15447,15449],{"className":15448},[219],[31,15450,15452,15523],{"className":15451},[223,553],[31,15453,15455,15520],{"className":15454},[227],[31,15456,15458],{"className":15457,"style":1936},[231],[31,15459,15460,15463],{"style":876},[31,15461],{"className":15462,"style":239},[238],[31,15464,15466],{"className":15465},[243,244,245,246],[31,15467,15469,15472,15475],{"className":15468},[52,246],[31,15470,10065],{"className":15471,"style":54},[52,53,246],[31,15473,6222],{"className":15474},[64,246],[31,15476,15478,15481],{"className":15477},[52,246],[31,15479,11],{"className":15480},[52,53,246],[31,15482,15484],{"className":15483},[219],[31,15485,15487,15511],{"className":15486},[223,553],[31,15488,15490,15508],{"className":15489},[227],[31,15491,15494],{"className":15492,"style":15493},[231],"height:0.3448em;",[31,15495,15497,15501],{"style":15496},"top:-2.3488em;margin-left:0em;margin-right:0.0714em;",[31,15498],{"className":15499,"style":15500},[238],"height:2.5em;",[31,15502,15505],{"className":15503},[243,15504,918,246],"reset-size3",[31,15506,94],{"className":15507,"style":93},[52,53,246],[31,15509,607],{"className":15510},[606],[31,15512,15514],{"className":15513},[227],[31,15515,15518],{"className":15516,"style":15517},[231],"height:0.1512em;",[31,15519],{},[31,15521,607],{"className":15522},[606],[31,15524,15526],{"className":15525},[227],[31,15527,15529],{"className":15528,"style":4496},[231],[31,15530],{},[31,15532],{"className":15533,"style":901},[59],[31,15535],{"className":15536,"style":623},[59],[31,15538,15540,15546,15586,15589,15597,15600,15640,15643,15646,15649,15652,15655,15658,15661,15664,15704,15707,15713,15716,15756,15759,15762,15765,15768,15771,15774,15803],{"className":15539},[908],[31,15541,15543],{"className":15542,"style":913},[88,912],[31,15544,919],{"className":15545},[917,918],[31,15547,15549,15552],{"className":15548},[52],[31,15550,79],{"className":15551},[52],[31,15553,15555],{"className":15554},[219],[31,15556,15558,15578],{"className":15557},[223,553],[31,15559,15561,15575],{"className":15560},[227],[31,15562,15564],{"className":15563,"style":2906},[231],[31,15565,15566,15569],{"style":876},[31,15567],{"className":15568,"style":239},[238],[31,15570,15572],{"className":15571},[243,244,245,246],[31,15573,94],{"className":15574,"style":93},[52,53,246],[31,15576,607],{"className":15577},[606],[31,15579,15581],{"className":15580},[227],[31,15582,15584],{"className":15583,"style":895},[231],[31,15585],{},[31,15587],{"className":15588,"style":623},[59],[31,15590,15592],{"className":15591},[1951],[31,15593,15596],{"className":15594,"style":15595},[52,1955],"margin-right:0.0139em;","log",[31,15598],{"className":15599,"style":623},[59],[31,15601,15603,15606],{"className":15602},[52],[31,15604,11],{"className":15605},[52,53],[31,15607,15609],{"className":15608},[219],[31,15610,15612,15632],{"className":15611},[223,553],[31,15613,15615,15629],{"className":15614},[227],[31,15616,15618],{"className":15617,"style":2906},[231],[31,15619,15620,15623],{"style":876},[31,15621],{"className":15622,"style":239},[238],[31,15624,15626],{"className":15625},[243,244,245,246],[31,15627,94],{"className":15628,"style":93},[52,53,246],[31,15630,607],{"className":15631},[606],[31,15633,15635],{"className":15634},[227],[31,15636,15638],{"className":15637,"style":895},[231],[31,15639],{},[31,15641,89],{"className":15642},[88],[31,15644,10065],{"className":15645,"style":54},[52,53],[31,15647],{"className":15648,"style":60},[59],[31,15650,8125],{"className":15651},[64],[31,15653],{"className":15654,"style":60},[59],[31,15656,15352],{"className":15657},[52,53],[31,15659,99],{"className":15660},[98],[31,15662],{"className":15663,"style":623},[59],[31,15665,15667,15670],{"className":15666},[52],[31,15668,79],{"className":15669},[52],[31,15671,15673],{"className":15672},[219],[31,15674,15676,15696],{"className":15675},[223,553],[31,15677,15679,15693],{"className":15678},[227],[31,15680,15682],{"className":15681,"style":2906},[231],[31,15683,15684,15687],{"style":876},[31,15685],{"className":15686,"style":239},[238],[31,15688,15690],{"className":15689},[243,244,245,246],[31,15691,94],{"className":15692,"style":93},[52,53,246],[31,15694,607],{"className":15695},[606],[31,15697,15699],{"className":15698},[227],[31,15700,15702],{"className":15701,"style":895},[231],[31,15703],{},[31,15705],{"className":15706,"style":623},[59],[31,15708,15710],{"className":15709},[1951],[31,15711,15596],{"className":15712,"style":15595},[52,1955],[31,15714],{"className":15715,"style":623},[59],[31,15717,15719,15722],{"className":15718},[52],[31,15720,11],{"className":15721},[52,53],[31,15723,15725],{"className":15724},[219],[31,15726,15728,15748],{"className":15727},[223,553],[31,15729,15731,15745],{"className":15730},[227],[31,15732,15734],{"className":15733,"style":2906},[231],[31,15735,15736,15739],{"style":876},[31,15737],{"className":15738,"style":239},[238],[31,15740,15742],{"className":15741},[243,244,245,246],[31,15743,94],{"className":15744,"style":93},[52,53,246],[31,15746,607],{"className":15747},[606],[31,15749,15751],{"className":15750},[227],[31,15752,15754],{"className":15753,"style":895},[231],[31,15755],{},[31,15757,89],{"className":15758},[88],[31,15760,10065],{"className":15761,"style":54},[52,53],[31,15763],{"className":15764,"style":60},[59],[31,15766,8125],{"className":15767},[64],[31,15769],{"className":15770,"style":60},[59],[31,15772,15352],{"className":15773},[52,53],[31,15775,15777,15780],{"className":15776},[98],[31,15778,99],{"className":15779},[98],[31,15781,15783],{"className":15782},[219],[31,15784,15786],{"className":15785},[223],[31,15787,15789],{"className":15788},[227],[31,15790,15792],{"className":15791,"style":500},[231],[31,15793,15794,15797],{"style":503},[31,15795],{"className":15796,"style":239},[238],[31,15798,15800],{"className":15799},[243,244,245,246],[31,15801,513],{"className":15802},[52,246],[31,15804,15806],{"className":15805,"style":913},[98,912],[31,15807,1070],{"className":15808},[917,918],[31,15810],{"className":15811,"style":623},[59],[31,15813,781],{"className":15814},[52],[11,15816,15817,15818,15833,15834,15837,15838,15853,15854,15869],{},"Two facts make ",[31,15819,15821],{"className":15820},[34],[31,15822,15824],{"className":15823,"ariaHidden":39},[38],[31,15825,15827,15830],{"className":15826},[43],[31,15828],{"className":15829,"style":188},[47],[31,15831,15376],{"className":15832,"style":15375},[52,53]," attractive. It equals the expected Hessian of the negative\nlog-likelihood (so it ",[164,15835,15836],{},"is"," curvature, not a crude substitute), and being an expectation\nof outer products it is positive semidefinite by construction — no sign-flips toward\nsaddles. For the exponential-family output layers used in practice, ",[31,15839,15841],{"className":15840},[34],[31,15842,15844],{"className":15843,"ariaHidden":39},[38],[31,15845,15847,15850],{"className":15846},[43],[31,15848],{"className":15849,"style":188},[47],[31,15851,15376],{"className":15852,"style":15375},[52,53]," coincides with\nthe Gauss-Newton matrix ",[31,15855,15857],{"className":15856},[34],[31,15858,15860],{"className":15859,"ariaHidden":39},[38],[31,15861,15863,15866],{"className":15862},[43],[31,15864],{"className":15865,"style":188},[47],[31,15867,15020],{"className":15868},[52,53],", tying the two views together.",[271,15871,15873],{"id":15872},"natural-gradient-k-fac-and-hessian-free","Natural gradient, K-FAC, and Hessian-free",[11,15875,15000,15876,15879,15880,15895,15896,15911,15912,15915],{},[26,15877,15878],{},"natural gradient"," preconditions by ",[31,15881,15883],{"className":15882},[34],[31,15884,15886],{"className":15885,"ariaHidden":39},[38],[31,15887,15889,15892],{"className":15888},[43],[31,15890],{"className":15891,"style":188},[47],[31,15893,15376],{"className":15894,"style":15375},[52,53]," instead of ",[31,15897,15899],{"className":15898},[34],[31,15900,15902],{"className":15901,"ariaHidden":39},[38],[31,15903,15905,15908],{"className":15904},[43],[31,15906],{"className":15907,"style":188},[47],[31,15909,193],{"className":15910,"style":192},[52,53],", measuring distance in the\nspace of the model's output ",[164,15913,15914],{},"distributions"," rather than its raw parameters:",[31,15917,15919],{"className":15918},[382],[31,15920,15922],{"className":15921},[34],[31,15923,15925,15949,15973],{"className":15924,"ariaHidden":39},[38],[31,15926,15928,15931,15934,15937,15940,15943,15946],{"className":15927},[43],[31,15929],{"className":15930,"style":113},[47],[31,15932,94],{"className":15933,"style":93},[52,53],[31,15935],{"className":15936,"style":60},[59],[31,15938],{"className":15939,"style":60},[59],[31,15941,123],{"className":15942},[64],[31,15944],{"className":15945,"style":60},[59],[31,15947],{"className":15948,"style":60},[59],[31,15950,15952,15955,15958,15961,15964,15967,15970],{"className":15951},[43],[31,15953],{"className":15954,"style":133},[47],[31,15956,94],{"className":15957,"style":93},[52,53],[31,15959],{"className":15960,"style":60},[59],[31,15962],{"className":15963,"style":140},[59],[31,15965,145],{"className":15966},[144],[31,15968],{"className":15969,"style":60},[59],[31,15971],{"className":15972,"style":140},[59],[31,15974,15976,15979,15982,15985,16020,16023],{"className":15975},[43],[31,15977],{"className":15978,"style":1198},[47],[31,15980,158],{"className":15981,"style":54},[52,53],[31,15983],{"className":15984,"style":623},[59],[31,15986,15988,15991],{"className":15987},[52],[31,15989,15376],{"className":15990,"style":15375},[52,53],[31,15992,15994],{"className":15993},[219],[31,15995,15997],{"className":15996},[223],[31,15998,16000],{"className":15999},[227],[31,16001,16003],{"className":16002,"style":754},[231],[31,16004,16005,16008],{"style":503},[31,16006],{"className":16007,"style":239},[238],[31,16009,16011],{"className":16010},[243,244,245,246],[31,16012,16014,16017],{"className":16013},[52,246],[31,16015,145],{"className":16016},[52,246],[31,16018,269],{"className":16019},[52,246],[31,16021,55],{"className":16022,"style":54},[52,53],[31,16024,781],{"className":16025},[52],[11,16027,16028,16029,16032,16033,16268,16269,16284,16285,16300,16301,16351,16352,16367,16368,16421],{},"The motivation is invariance. Ordinary gradient descent moves fastest in the direction of\nsteepest descent measured by Euclidean distance in ",[164,16030,16031],{},"parameter"," space, but a parameter is\nan arbitrary label — reparametrize the network and the same distance means something\ndifferent. The natural gradient instead takes the steepest-descent direction under the\nKullback–Leibler divergence between output distributions, a quantity that does not depend\non how the parameters are named. To second order that divergence is\n",[31,16034,16036],{"className":16035},[34],[31,16037,16039,16159],{"className":16038,"ariaHidden":39},[38],[31,16040,16042,16045,16052,16055,16095,16098,16147,16150,16153,16156],{"className":16041},[43],[31,16043],{"className":16044,"style":75},[47],[31,16046,16048],{"className":16047},[52],[31,16049,16051],{"className":16050},[52,1955],"KL",[31,16053,89],{"className":16054},[88],[31,16056,16058,16061],{"className":16057},[52],[31,16059,11],{"className":16060},[52,53],[31,16062,16064],{"className":16063},[219],[31,16065,16067,16087],{"className":16066},[223,553],[31,16068,16070,16084],{"className":16069},[227],[31,16071,16073],{"className":16072,"style":2906},[231],[31,16074,16075,16078],{"style":876},[31,16076],{"className":16077,"style":239},[238],[31,16079,16081],{"className":16080},[243,244,245,246],[31,16082,94],{"className":16083,"style":93},[52,53,246],[31,16085,607],{"className":16086},[606],[31,16088,16090],{"className":16089},[227],[31,16091,16093],{"className":16092,"style":895},[231],[31,16094],{},[31,16096,2887],{"className":16097},[52],[31,16099,16101,16104],{"className":16100},[52],[31,16102,11],{"className":16103},[52,53],[31,16105,16107],{"className":16106},[219],[31,16108,16110,16139],{"className":16109},[223,553],[31,16111,16113,16136],{"className":16112},[227],[31,16114,16116],{"className":16115,"style":2906},[231],[31,16117,16118,16121],{"style":876},[31,16119],{"className":16120,"style":239},[238],[31,16122,16124],{"className":16123},[243,244,245,246],[31,16125,16127,16130,16133],{"className":16126},[52,246],[31,16128,94],{"className":16129,"style":93},[52,53,246],[31,16131,410],{"className":16132},[144,246],[31,16134,309],{"className":16135},[52,246],[31,16137,607],{"className":16138},[606],[31,16140,16142],{"className":16141},[227],[31,16143,16145],{"className":16144,"style":2939},[231],[31,16146],{},[31,16148,99],{"className":16149},[98],[31,16151],{"className":16152,"style":60},[59],[31,16154,435],{"className":16155},[64],[31,16157],{"className":16158,"style":60},[59],[31,16160,16162,16165,16233,16262,16265],{"className":16161},[43],[31,16163],{"className":16164,"style":1342},[47],[31,16166,16168,16171,16230],{"className":16167},[52],[31,16169],{"className":16170},[88,545],[31,16172,16174],{"className":16173},[549],[31,16175,16177,16222],{"className":16176},[223,553],[31,16178,16180,16219],{"className":16179},[227],[31,16181,16183,16197,16205],{"className":16182,"style":560},[231],[31,16184,16185,16188],{"style":563},[31,16186],{"className":16187,"style":567},[238],[31,16189,16191],{"className":16190},[243,244,245,246],[31,16192,16194],{"className":16193},[52,246],[31,16195,250],{"className":16196},[52,246],[31,16198,16199,16202],{"style":579},[31,16200],{"className":16201,"style":567},[238],[31,16203],{"className":16204,"style":587},[586],[31,16206,16207,16210],{"style":590},[31,16208],{"className":16209,"style":567},[238],[31,16211,16213],{"className":16212},[243,244,245,246],[31,16214,16216],{"className":16215},[52,246],[31,16217,269],{"className":16218},[52,246],[31,16220,607],{"className":16221},[606],[31,16223,16225],{"className":16224},[227],[31,16226,16228],{"className":16227,"style":614},[231],[31,16229],{},[31,16231],{"className":16232},[98,545],[31,16234,16236,16239],{"className":16235},[52],[31,16237,309],{"className":16238},[52],[31,16240,16242],{"className":16241},[219],[31,16243,16245],{"className":16244},[223],[31,16246,16248],{"className":16247},[227],[31,16249,16251],{"className":16250,"style":1314},[231],[31,16252,16253,16256],{"style":234},[31,16254],{"className":16255,"style":239},[238],[31,16257,16259],{"className":16258},[243,244,245,246],[31,16260,513],{"className":16261},[52,246],[31,16263,15376],{"className":16264,"style":15375},[52,53],[31,16266,309],{"className":16267},[52],", so\n",[31,16270,16272],{"className":16271},[34],[31,16273,16275],{"className":16274,"ariaHidden":39},[38],[31,16276,16278,16281],{"className":16277},[43],[31,16279],{"className":16280,"style":188},[47],[31,16282,15376],{"className":16283,"style":15375},[52,53]," plays exactly the role ",[31,16286,16288],{"className":16287},[34],[31,16289,16291],{"className":16290,"ariaHidden":39},[38],[31,16292,16294,16297],{"className":16293},[43],[31,16295],{"className":16296,"style":188},[47],[31,16298,193],{"className":16299,"style":192},[52,53]," played in the Taylor bowl, and ",[31,16302,16304],{"className":16303},[34],[31,16305,16307],{"className":16306,"ariaHidden":39},[38],[31,16308,16310,16313,16348],{"className":16309},[43],[31,16311],{"className":16312,"style":1463},[47],[31,16314,16316,16319],{"className":16315},[52],[31,16317,15376],{"className":16318,"style":15375},[52,53],[31,16320,16322],{"className":16321},[219],[31,16323,16325],{"className":16324},[223],[31,16326,16328],{"className":16327},[227],[31,16329,16331],{"className":16330,"style":209},[231],[31,16332,16333,16336],{"style":234},[31,16334],{"className":16335,"style":239},[238],[31,16337,16339],{"className":16338},[243,244,245,246],[31,16340,16342,16345],{"className":16341},[52,246],[31,16343,145],{"className":16344},[52,246],[31,16346,269],{"className":16347},[52,246],[31,16349,55],{"className":16350,"style":54},[52,53]," is the step that\ndescends the loss per unit of distributional change. Because ",[31,16353,16355],{"className":16354},[34],[31,16356,16358],{"className":16357,"ariaHidden":39},[38],[31,16359,16361,16364],{"className":16360},[43],[31,16362],{"className":16363,"style":188},[47],[31,16365,15376],{"className":16366,"style":15375},[52,53]," is positive semidefinite,\nthat step never climbs toward a saddle the way ",[31,16369,16371],{"className":16370},[34],[31,16372,16374],{"className":16373,"ariaHidden":39},[38],[31,16375,16377,16380,16383,16418],{"className":16376},[43],[31,16378],{"className":16379,"style":1463},[47],[31,16381,145],{"className":16382},[52],[31,16384,16386,16389],{"className":16385},[52],[31,16387,193],{"className":16388,"style":192},[52,53],[31,16390,16392],{"className":16391},[219],[31,16393,16395],{"className":16394},[223],[31,16396,16398],{"className":16397},[227],[31,16399,16401],{"className":16400,"style":209},[231],[31,16402,16403,16406],{"style":234},[31,16404],{"className":16405,"style":239},[238],[31,16407,16409],{"className":16408},[243,244,245,246],[31,16410,16412,16415],{"className":16411},[52,246],[31,16413,145],{"className":16414},[52,246],[31,16416,269],{"className":16417},[52,246],[31,16419,55],{"className":16420,"style":54},[52,53]," can.",[11,16423,16424,12459,16439,16472,16473,16488],{},[31,16425,16427],{"className":16426},[34],[31,16428,16430],{"className":16429,"ariaHidden":39},[38],[31,16431,16433,16436],{"className":16432},[43],[31,16434],{"className":16435,"style":188},[47],[31,16437,15376],{"className":16438,"style":15375},[52,53],[31,16440,16442],{"className":16441},[34],[31,16443,16445,16463],{"className":16444,"ariaHidden":39},[38],[31,16446,16448,16451,16454,16457,16460],{"className":16447},[43],[31,16449],{"className":16450,"style":133},[47],[31,16452,5457],{"className":16453},[52,53],[31,16455],{"className":16456,"style":140},[59],[31,16458,6001],{"className":16459},[144],[31,16461],{"className":16462,"style":140},[59],[31,16464,16466,16469],{"className":16465},[43],[31,16467],{"className":16468,"style":113},[47],[31,16470,5457],{"className":16471},[52,53],", so like ",[31,16474,16476],{"className":16475},[34],[31,16477,16479],{"className":16478,"ariaHidden":39},[38],[31,16480,16482,16485],{"className":16481},[43],[31,16483],{"className":16484,"style":188},[47],[31,16486,193],{"className":16487,"style":192},[52,53]," it is used only through approximations.",[6170,16490,16491,16503],{},[6173,16492,16493],{},[6176,16494,16495,16497,16500],{},[6179,16496,13841],{},[6179,16498,16499],{},"Curvature matrix",[6179,16501,16502],{},"Trick that makes it affordable",[6262,16504,16505,16558,16619],{},[6176,16506,16507,16510,16528],{},[6267,16508,16509],{},"Natural gradient",[6267,16511,16512,16513],{},"Fisher information ",[31,16514,16516],{"className":16515},[34],[31,16517,16519],{"className":16518,"ariaHidden":39},[38],[31,16520,16522,16525],{"className":16521},[43],[31,16523],{"className":16524,"style":188},[47],[31,16526,15376],{"className":16527,"style":15375},[52,53],[6267,16529,16530,16531],{},"curvature in output-distribution space, always ",[31,16532,16534],{"className":16533},[34],[31,16535,16537,16549],{"className":16536,"ariaHidden":39},[38],[31,16538,16540,16543,16546],{"className":16539},[43],[31,16541],{"className":16542,"style":14095},[47],[31,16544,15256],{"className":16545},[64],[31,16547],{"className":16548,"style":60},[59],[31,16550,16552,16555],{"className":16551},[43],[31,16553],{"className":16554,"style":833},[47],[31,16556,837],{"className":16557},[52],[6176,16559,16560,16563,16581],{},[6267,16561,16562],{},"K-FAC",[6267,16564,16565,16566],{},"block-diagonal ",[31,16567,16569],{"className":16568},[34],[31,16570,16572],{"className":16571,"ariaHidden":39},[38],[31,16573,16575,16578],{"className":16574},[43],[31,16576],{"className":16577,"style":188},[47],[31,16579,15376],{"className":16580,"style":15375},[52,53],[6267,16582,16583,16584],{},"factor each layer's block as a Kronecker product ",[31,16585,16587],{"className":16586},[34],[31,16588,16590,16610],{"className":16589,"ariaHidden":39},[38],[31,16591,16593,16597,16600,16603,16607],{"className":16592},[43],[31,16594],{"className":16595,"style":16596},[47],"height:0.7667em;vertical-align:-0.0833em;",[31,16598,4386],{"className":16599},[52,53],[31,16601],{"className":16602,"style":140},[59],[31,16604,16606],{"className":16605},[144],"⊗",[31,16608],{"className":16609,"style":140},[59],[31,16611,16613,16616],{"className":16612},[43],[31,16614],{"className":16615,"style":188},[47],[31,16617,15020],{"className":16618},[52,53],[6176,16620,16621,16624,16643],{},[6267,16622,16623],{},"Hessian-free",[6267,16625,16626,16627,16642],{},"Hessian ",[31,16628,16630],{"className":16629},[34],[31,16631,16633],{"className":16632,"ariaHidden":39},[38],[31,16634,16636,16639],{"className":16635},[43],[31,16637],{"className":16638,"style":188},[47],[31,16640,193],{"className":16641,"style":192},[52,53]," (implicit)",[6267,16644,16645,16646,16665,16666],{},"inner CG using ",[31,16647,16649],{"className":16648},[34],[31,16650,16652],{"className":16651,"ariaHidden":39},[38],[31,16653,16655,16658,16661],{"className":16654},[43],[31,16656],{"className":16657,"style":188},[47],[31,16659,193],{"className":16660,"style":192},[52,53],[31,16662,16664],{"className":16663,"style":54},[52,53],"v"," products via ",[15,16667,16669],{"href":16668},"\u002Fdeep-learning\u002Fneural-networks\u002Fbackpropagation","autodiff",[1511,16671,16672],{"type":1513},[11,16673,16674,16677,16678,16717,16718,16722,16723,16741,16742,16757,16758,16791],{},[26,16675,16676],{},"Definition (Hessian-free optimization)."," A truncated-Newton method that solves\n",[31,16679,16681],{"className":16680},[34],[31,16682,16684,16705],{"className":16683,"ariaHidden":39},[38],[31,16685,16687,16690,16693,16696,16699,16702],{"className":16686},[43],[31,16688],{"className":16689,"style":188},[47],[31,16691,193],{"className":16692,"style":192},[52,53],[31,16694,309],{"className":16695},[52],[31,16697],{"className":16698,"style":60},[59],[31,16700,65],{"className":16701},[64],[31,16703],{"className":16704,"style":60},[59],[31,16706,16708,16711,16714],{"className":16707},[43],[31,16709],{"className":16710,"style":1086},[47],[31,16712,145],{"className":16713},[52],[31,16715,55],{"className":16716,"style":54},[52,53]," approximately by an inner ",[15,16719,16721],{"href":16720},"#conjugate-gradients","conjugate-gradient","\nloop, supplying each Hessian–vector product ",[31,16724,16726],{"className":16725},[34],[31,16727,16729],{"className":16728,"ariaHidden":39},[38],[31,16730,16732,16735,16738],{"className":16731},[43],[31,16733],{"className":16734,"style":188},[47],[31,16736,193],{"className":16737,"style":192},[52,53],[31,16739,16664],{"className":16740,"style":54},[52,53]," through a single extra\nbackpropagation pass — never forming ",[31,16743,16745],{"className":16744},[34],[31,16746,16748],{"className":16747,"ariaHidden":39},[38],[31,16749,16751,16754],{"className":16750},[43],[31,16752],{"className":16753,"style":188},[47],[31,16755,193],{"className":16756,"style":192},[52,53],". Curvature is used; the ",[31,16759,16761],{"className":16760},[34],[31,16762,16764,16782],{"className":16763,"ariaHidden":39},[38],[31,16765,16767,16770,16773,16776,16779],{"className":16766},[43],[31,16768],{"className":16769,"style":133},[47],[31,16771,5457],{"className":16772},[52,53],[31,16774],{"className":16775,"style":140},[59],[31,16777,6001],{"className":16778},[144],[31,16780],{"className":16781,"style":140},[59],[31,16783,16785,16788],{"className":16784},[43],[31,16786],{"className":16787,"style":113},[47],[31,16789,5457],{"className":16790},[52,53]," matrix\nis never materialized.",[1511,16793,16795],{"type":16794},"remark",[11,16796,16797,16800,16801,16834,16835,17007,17008,17058],{},[26,16798,16799],{},"Remark (K-FAC)."," Kronecker-Factored Approximate Curvature approximates each layer's\nFisher block as a Kronecker product ",[31,16802,16804],{"className":16803},[34],[31,16805,16807,16825],{"className":16806,"ariaHidden":39},[38],[31,16808,16810,16813,16816,16819,16822],{"className":16809},[43],[31,16811],{"className":16812,"style":16596},[47],[31,16814,4386],{"className":16815},[52,53],[31,16817],{"className":16818,"style":140},[59],[31,16820,16606],{"className":16821},[144],[31,16823],{"className":16824,"style":140},[59],[31,16826,16828,16831],{"className":16827},[43],[31,16829],{"className":16830,"style":188},[47],[31,16832,15020],{"className":16833},[52,53]," of two small factors — one from the\nlayer's inputs, one from its output gradients. Inverting a Kronecker product is just\ninverting the two small factors, ",[31,16836,16838],{"className":16837},[34],[31,16839,16841,16862,16915,16966],{"className":16840,"ariaHidden":39},[38],[31,16842,16844,16847,16850,16853,16856,16859],{"className":16843},[43],[31,16845],{"className":16846,"style":75},[47],[31,16848,89],{"className":16849},[88],[31,16851,4386],{"className":16852},[52,53],[31,16854],{"className":16855,"style":140},[59],[31,16857,16606],{"className":16858},[144],[31,16860],{"className":16861,"style":140},[59],[31,16863,16865,16868,16871,16906,16909,16912],{"className":16864},[43],[31,16866],{"className":16867,"style":3071},[47],[31,16869,15020],{"className":16870},[52,53],[31,16872,16874,16877],{"className":16873},[98],[31,16875,99],{"className":16876},[98],[31,16878,16880],{"className":16879},[219],[31,16881,16883],{"className":16882},[223],[31,16884,16886],{"className":16885},[227],[31,16887,16889],{"className":16888,"style":209},[231],[31,16890,16891,16894],{"style":234},[31,16892],{"className":16893,"style":239},[238],[31,16895,16897],{"className":16896},[243,244,245,246],[31,16898,16900,16903],{"className":16899},[52,246],[31,16901,145],{"className":16902},[52,246],[31,16904,269],{"className":16905},[52,246],[31,16907],{"className":16908,"style":60},[59],[31,16910,65],{"className":16911},[64],[31,16913],{"className":16914,"style":60},[59],[31,16916,16918,16922,16957,16960,16963],{"className":16917},[43],[31,16919],{"className":16920,"style":16921},[47],"height:0.8974em;vertical-align:-0.0833em;",[31,16923,16925,16928],{"className":16924},[52],[31,16926,4386],{"className":16927},[52,53],[31,16929,16931],{"className":16930},[219],[31,16932,16934],{"className":16933},[223],[31,16935,16937],{"className":16936},[227],[31,16938,16940],{"className":16939,"style":209},[231],[31,16941,16942,16945],{"style":234},[31,16943],{"className":16944,"style":239},[238],[31,16946,16948],{"className":16947},[243,244,245,246],[31,16949,16951,16954],{"className":16950},[52,246],[31,16952,145],{"className":16953},[52,246],[31,16955,269],{"className":16956},[52,246],[31,16958],{"className":16959,"style":140},[59],[31,16961,16606],{"className":16962},[144],[31,16964],{"className":16965,"style":140},[59],[31,16967,16969,16972],{"className":16968},[43],[31,16970],{"className":16971,"style":209},[47],[31,16973,16975,16978],{"className":16974},[52],[31,16976,15020],{"className":16977},[52,53],[31,16979,16981],{"className":16980},[219],[31,16982,16984],{"className":16983},[223],[31,16985,16987],{"className":16986},[227],[31,16988,16990],{"className":16989,"style":209},[231],[31,16991,16992,16995],{"style":234},[31,16993],{"className":16994,"style":239},[238],[31,16996,16998],{"className":16997},[243,244,245,246],[31,16999,17001,17004],{"className":17000},[52,246],[31,17002,145],{"className":17003},[52,246],[31,17005,269],{"className":17006},[52,246],",\ncollapsing a per-layer ",[31,17009,17011],{"className":17010},[34],[31,17012,17014],{"className":17013,"ariaHidden":39},[38],[31,17015,17017,17020,17023,17026,17055],{"className":17016},[43],[31,17018],{"className":17019,"style":3071},[47],[31,17021,3684],{"className":17022,"style":93},[52,53],[31,17024,89],{"className":17025},[88],[31,17027,17029,17032],{"className":17028},[52],[31,17030,5457],{"className":17031},[52,53],[31,17033,17035],{"className":17034},[219],[31,17036,17038],{"className":17037},[223],[31,17039,17041],{"className":17040},[227],[31,17042,17044],{"className":17043,"style":209},[231],[31,17045,17046,17049],{"style":234},[31,17047],{"className":17048,"style":239},[238],[31,17050,17052],{"className":17051},[243,244,245,246],[31,17053,5633],{"className":17054},[52,246],[31,17056,99],{"className":17057},[98]," inverse to two cheap ones.",[11,17060,17061,17062,17122,17123,17182,17183,17343,17344,17451,17452,17485,17486,17501,17502,17622,17623,17501,17638,17757,17758,17816,17817,17875,17876,18029,18030,18243],{},"For a layer with ",[31,17063,17065],{"className":17064},[34],[31,17066,17068],{"className":17067,"ariaHidden":39},[38],[31,17069,17071,17074],{"className":17070},[43],[31,17072],{"className":17073,"style":9889},[47],[31,17075,17077,17080],{"className":17076},[52],[31,17078,9261],{"className":17079},[52,53],[31,17081,17083],{"className":17082},[219],[31,17084,17086,17114],{"className":17085},[223,553],[31,17087,17089,17111],{"className":17088},[227],[31,17090,17092],{"className":17091,"style":1990},[231],[31,17093,17094,17097],{"style":876},[31,17095],{"className":17096,"style":239},[238],[31,17098,17100],{"className":17099},[243,244,245,246],[31,17101,17103],{"className":17102},[52,246],[31,17104,17107],{"className":17105},[52,17106,246],"text",[31,17108,17110],{"className":17109},[52,246],"in",[31,17112,607],{"className":17113},[606],[31,17115,17117],{"className":17116},[227],[31,17118,17120],{"className":17119,"style":895},[231],[31,17121],{}," inputs and ",[31,17124,17126],{"className":17125},[34],[31,17127,17129],{"className":17128,"ariaHidden":39},[38],[31,17130,17132,17135],{"className":17131},[43],[31,17133],{"className":17134,"style":9889},[47],[31,17136,17138,17141],{"className":17137},[52],[31,17139,9261],{"className":17140},[52,53],[31,17142,17144],{"className":17143},[219],[31,17145,17147,17174],{"className":17146},[223,553],[31,17148,17150,17171],{"className":17149},[227],[31,17151,17153],{"className":17152,"style":9314},[231],[31,17154,17155,17158],{"style":876},[31,17156],{"className":17157,"style":239},[238],[31,17159,17161],{"className":17160},[243,244,245,246],[31,17162,17164],{"className":17163},[52,246],[31,17165,17167],{"className":17166},[52,17106,246],[31,17168,17170],{"className":17169},[52,246],"out",[31,17172,607],{"className":17173},[606],[31,17175,17177],{"className":17176},[227],[31,17178,17180],{"className":17179,"style":895},[231],[31,17181],{}," outputs, its weight has\n",[31,17184,17186],{"className":17185},[34],[31,17187,17189,17245],{"className":17188,"ariaHidden":39},[38],[31,17190,17192,17195,17236,17239,17242],{"className":17191},[43],[31,17193],{"className":17194,"style":3217},[47],[31,17196,17198,17201],{"className":17197},[52],[31,17199,5457],{"className":17200},[52,53],[31,17202,17204],{"className":17203},[219],[31,17205,17207,17228],{"className":17206},[223,553],[31,17208,17210,17225],{"className":17209},[227],[31,17211,17213],{"className":17212,"style":2906},[231],[31,17214,17215,17218],{"style":876},[31,17216],{"className":17217,"style":239},[238],[31,17219,17221],{"className":17220},[243,244,245,246],[31,17222,17224],{"className":17223},[52,246],"ℓ",[31,17226,607],{"className":17227},[606],[31,17229,17231],{"className":17230},[227],[31,17232,17234],{"className":17233,"style":895},[231],[31,17235],{},[31,17237],{"className":17238,"style":60},[59],[31,17240,65],{"className":17241},[64],[31,17243],{"className":17244,"style":60},[59],[31,17246,17248,17251,17297],{"className":17247},[43],[31,17249],{"className":17250,"style":9889},[47],[31,17252,17254,17257],{"className":17253},[52],[31,17255,9261],{"className":17256},[52,53],[31,17258,17260],{"className":17259},[219],[31,17261,17263,17289],{"className":17262},[223,553],[31,17264,17266,17286],{"className":17265},[227],[31,17267,17269],{"className":17268,"style":1990},[231],[31,17270,17271,17274],{"style":876},[31,17272],{"className":17273,"style":239},[238],[31,17275,17277],{"className":17276},[243,244,245,246],[31,17278,17280],{"className":17279},[52,246],[31,17281,17283],{"className":17282},[52,17106,246],[31,17284,17110],{"className":17285},[52,246],[31,17287,607],{"className":17288},[606],[31,17290,17292],{"className":17291},[227],[31,17293,17295],{"className":17294,"style":895},[231],[31,17296],{},[31,17298,17300,17303],{"className":17299},[52],[31,17301,9261],{"className":17302},[52,53],[31,17304,17306],{"className":17305},[219],[31,17307,17309,17335],{"className":17308},[223,553],[31,17310,17312,17332],{"className":17311},[227],[31,17313,17315],{"className":17314,"style":9314},[231],[31,17316,17317,17320],{"style":876},[31,17318],{"className":17319,"style":239},[238],[31,17321,17323],{"className":17322},[243,244,245,246],[31,17324,17326],{"className":17325},[52,246],[31,17327,17329],{"className":17328},[52,17106,246],[31,17330,17170],{"className":17331},[52,246],[31,17333,607],{"className":17334},[606],[31,17336,17338],{"className":17337},[227],[31,17339,17341],{"className":17340,"style":895},[231],[31,17342],{}," parameters and its Fisher block is\n",[31,17345,17347],{"className":17346},[34],[31,17348,17350,17405],{"className":17349,"ariaHidden":39},[38],[31,17351,17353,17356,17396,17399,17402],{"className":17352},[43],[31,17354],{"className":17355,"style":3217},[47],[31,17357,17359,17362],{"className":17358},[52],[31,17360,5457],{"className":17361},[52,53],[31,17363,17365],{"className":17364},[219],[31,17366,17368,17388],{"className":17367},[223,553],[31,17369,17371,17385],{"className":17370},[227],[31,17372,17374],{"className":17373,"style":2906},[231],[31,17375,17376,17379],{"style":876},[31,17377],{"className":17378,"style":239},[238],[31,17380,17382],{"className":17381},[243,244,245,246],[31,17383,17224],{"className":17384},[52,246],[31,17386,607],{"className":17387},[606],[31,17389,17391],{"className":17390},[227],[31,17392,17394],{"className":17393,"style":895},[231],[31,17395],{},[31,17397],{"className":17398,"style":140},[59],[31,17400,6001],{"className":17401},[144],[31,17403],{"className":17404,"style":140},[59],[31,17406,17408,17411],{"className":17407},[43],[31,17409],{"className":17410,"style":3217},[47],[31,17412,17414,17417],{"className":17413},[52],[31,17415,5457],{"className":17416},[52,53],[31,17418,17420],{"className":17419},[219],[31,17421,17423,17443],{"className":17422},[223,553],[31,17424,17426,17440],{"className":17425},[227],[31,17427,17429],{"className":17428,"style":2906},[231],[31,17430,17431,17434],{"style":876},[31,17432],{"className":17433,"style":239},[238],[31,17435,17437],{"className":17436},[243,244,245,246],[31,17438,17224],{"className":17439},[52,246],[31,17441,607],{"className":17442},[606],[31,17444,17446],{"className":17445},[227],[31,17447,17449],{"className":17448,"style":895},[231],[31,17450],{},". K-FAC models that block as ",[31,17453,17455],{"className":17454},[34],[31,17456,17458,17476],{"className":17457,"ariaHidden":39},[38],[31,17459,17461,17464,17467,17470,17473],{"className":17460},[43],[31,17462],{"className":17463,"style":16596},[47],[31,17465,4386],{"className":17466},[52,53],[31,17468],{"className":17469,"style":140},[59],[31,17471,16606],{"className":17472},[144],[31,17474],{"className":17475,"style":140},[59],[31,17477,17479,17482],{"className":17478},[43],[31,17480],{"className":17481,"style":188},[47],[31,17483,15020],{"className":17484},[52,53],", where ",[31,17487,17489],{"className":17488},[34],[31,17490,17492],{"className":17491,"ariaHidden":39},[38],[31,17493,17495,17498],{"className":17494},[43],[31,17496],{"className":17497,"style":188},[47],[31,17499,4386],{"className":17500},[52,53]," is the\n",[31,17503,17505],{"className":17504},[34],[31,17506,17508,17570],{"className":17507,"ariaHidden":39},[38],[31,17509,17511,17515,17561,17564,17567],{"className":17510},[43],[31,17512],{"className":17513,"style":17514},[47],"height:0.7333em;vertical-align:-0.15em;",[31,17516,17518,17521],{"className":17517},[52],[31,17519,9261],{"className":17520},[52,53],[31,17522,17524],{"className":17523},[219],[31,17525,17527,17553],{"className":17526},[223,553],[31,17528,17530,17550],{"className":17529},[227],[31,17531,17533],{"className":17532,"style":1990},[231],[31,17534,17535,17538],{"style":876},[31,17536],{"className":17537,"style":239},[238],[31,17539,17541],{"className":17540},[243,244,245,246],[31,17542,17544],{"className":17543},[52,246],[31,17545,17547],{"className":17546},[52,17106,246],[31,17548,17110],{"className":17549},[52,246],[31,17551,607],{"className":17552},[606],[31,17554,17556],{"className":17555},[227],[31,17557,17559],{"className":17558,"style":895},[231],[31,17560],{},[31,17562],{"className":17563,"style":140},[59],[31,17565,6001],{"className":17566},[144],[31,17568],{"className":17569,"style":140},[59],[31,17571,17573,17576],{"className":17572},[43],[31,17574],{"className":17575,"style":9889},[47],[31,17577,17579,17582],{"className":17578},[52],[31,17580,9261],{"className":17581},[52,53],[31,17583,17585],{"className":17584},[219],[31,17586,17588,17614],{"className":17587},[223,553],[31,17589,17591,17611],{"className":17590},[227],[31,17592,17594],{"className":17593,"style":1990},[231],[31,17595,17596,17599],{"style":876},[31,17597],{"className":17598,"style":239},[238],[31,17600,17602],{"className":17601},[243,244,245,246],[31,17603,17605],{"className":17604},[52,246],[31,17606,17608],{"className":17607},[52,17106,246],[31,17609,17110],{"className":17610},[52,246],[31,17612,607],{"className":17613},[606],[31,17615,17617],{"className":17616},[227],[31,17618,17620],{"className":17619,"style":895},[231],[31,17621],{}," covariance of the layer's inputs and ",[31,17624,17626],{"className":17625},[34],[31,17627,17629],{"className":17628,"ariaHidden":39},[38],[31,17630,17632,17635],{"className":17631},[43],[31,17633],{"className":17634,"style":188},[47],[31,17636,15020],{"className":17637},[52,53],[31,17639,17641],{"className":17640},[34],[31,17642,17644,17705],{"className":17643,"ariaHidden":39},[38],[31,17645,17647,17650,17696,17699,17702],{"className":17646},[43],[31,17648],{"className":17649,"style":17514},[47],[31,17651,17653,17656],{"className":17652},[52],[31,17654,9261],{"className":17655},[52,53],[31,17657,17659],{"className":17658},[219],[31,17660,17662,17688],{"className":17661},[223,553],[31,17663,17665,17685],{"className":17664},[227],[31,17666,17668],{"className":17667,"style":9314},[231],[31,17669,17670,17673],{"style":876},[31,17671],{"className":17672,"style":239},[238],[31,17674,17676],{"className":17675},[243,244,245,246],[31,17677,17679],{"className":17678},[52,246],[31,17680,17682],{"className":17681},[52,17106,246],[31,17683,17170],{"className":17684},[52,246],[31,17686,607],{"className":17687},[606],[31,17689,17691],{"className":17690},[227],[31,17692,17694],{"className":17693,"style":895},[231],[31,17695],{},[31,17697],{"className":17698,"style":140},[59],[31,17700,6001],{"className":17701},[144],[31,17703],{"className":17704,"style":140},[59],[31,17706,17708,17711],{"className":17707},[43],[31,17709],{"className":17710,"style":9889},[47],[31,17712,17714,17717],{"className":17713},[52],[31,17715,9261],{"className":17716},[52,53],[31,17718,17720],{"className":17719},[219],[31,17721,17723,17749],{"className":17722},[223,553],[31,17724,17726,17746],{"className":17725},[227],[31,17727,17729],{"className":17728,"style":9314},[231],[31,17730,17731,17734],{"style":876},[31,17732],{"className":17733,"style":239},[238],[31,17735,17737],{"className":17736},[243,244,245,246],[31,17738,17740],{"className":17739},[52,246],[31,17741,17743],{"className":17742},[52,17106,246],[31,17744,17170],{"className":17745},[52,246],[31,17747,607],{"className":17748},[606],[31,17750,17752],{"className":17751},[227],[31,17753,17755],{"className":17754,"style":895},[231],[31,17756],{}," covariance of its output gradients. Storing and\ninverting two factors of size ",[31,17759,17761],{"className":17760},[34],[31,17762,17764],{"className":17763,"ariaHidden":39},[38],[31,17765,17767,17770],{"className":17766},[43],[31,17768],{"className":17769,"style":9889},[47],[31,17771,17773,17776],{"className":17772},[52],[31,17774,9261],{"className":17775},[52,53],[31,17777,17779],{"className":17778},[219],[31,17780,17782,17808],{"className":17781},[223,553],[31,17783,17785,17805],{"className":17784},[227],[31,17786,17788],{"className":17787,"style":1990},[231],[31,17789,17790,17793],{"style":876},[31,17791],{"className":17792,"style":239},[238],[31,17794,17796],{"className":17795},[243,244,245,246],[31,17797,17799],{"className":17798},[52,246],[31,17800,17802],{"className":17801},[52,17106,246],[31,17803,17110],{"className":17804},[52,246],[31,17806,607],{"className":17807},[606],[31,17809,17811],{"className":17810},[227],[31,17812,17814],{"className":17813,"style":895},[231],[31,17815],{}," and ",[31,17818,17820],{"className":17819},[34],[31,17821,17823],{"className":17822,"ariaHidden":39},[38],[31,17824,17826,17829],{"className":17825},[43],[31,17827],{"className":17828,"style":9889},[47],[31,17830,17832,17835],{"className":17831},[52],[31,17833,9261],{"className":17834},[52,53],[31,17836,17838],{"className":17837},[219],[31,17839,17841,17867],{"className":17840},[223,553],[31,17842,17844,17864],{"className":17843},[227],[31,17845,17847],{"className":17846,"style":9314},[231],[31,17848,17849,17852],{"style":876},[31,17850],{"className":17851,"style":239},[238],[31,17853,17855],{"className":17854},[243,244,245,246],[31,17856,17858],{"className":17857},[52,246],[31,17859,17861],{"className":17860},[52,17106,246],[31,17862,17170],{"className":17863},[52,246],[31,17865,607],{"className":17866},[606],[31,17868,17870],{"className":17869},[227],[31,17871,17873],{"className":17872,"style":895},[231],[31,17874],{}," costs\n",[31,17877,17879],{"className":17878},[34],[31,17880,17882,17963],{"className":17881,"ariaHidden":39},[38],[31,17883,17885,17889,17892,17895,17954,17957,17960],{"className":17884},[43],[31,17886],{"className":17887,"style":17888},[47],"height:1.0786em;vertical-align:-0.2645em;",[31,17890,3684],{"className":17891,"style":93},[52,53],[31,17893,89],{"className":17894},[88],[31,17896,17898,17901],{"className":17897},[52],[31,17899,9261],{"className":17900},[52,53],[31,17902,17904],{"className":17903},[219],[31,17905,17907,17945],{"className":17906},[223,553],[31,17908,17910,17942],{"className":17909},[227],[31,17911,17913,17931],{"className":17912,"style":209},[231],[31,17914,17916,17919],{"style":17915},"top:-2.4355em;margin-left:0em;margin-right:0.05em;",[31,17917],{"className":17918,"style":239},[238],[31,17920,17922],{"className":17921},[243,244,245,246],[31,17923,17925],{"className":17924},[52,246],[31,17926,17928],{"className":17927},[52,17106,246],[31,17929,17110],{"className":17930},[52,246],[31,17932,17933,17936],{"style":234},[31,17934],{"className":17935,"style":239},[238],[31,17937,17939],{"className":17938},[243,244,245,246],[31,17940,5633],{"className":17941},[52,246],[31,17943,607],{"className":17944},[606],[31,17946,17948],{"className":17947},[227],[31,17949,17952],{"className":17950,"style":17951},[231],"height:0.2645em;",[31,17953],{},[31,17955],{"className":17956,"style":140},[59],[31,17958,410],{"className":17959},[144],[31,17961],{"className":17962,"style":140},[59],[31,17964,17966,17969,18026],{"className":17965},[43],[31,17967],{"className":17968,"style":3071},[47],[31,17970,17972,17975],{"className":17971},[52],[31,17973,9261],{"className":17974},[52,53],[31,17976,17978],{"className":17977},[219],[31,17979,17981,18018],{"className":17980},[223,553],[31,17982,17984,18015],{"className":17983},[227],[31,17985,17987,18004],{"className":17986,"style":209},[231],[31,17988,17989,17992],{"style":11339},[31,17990],{"className":17991,"style":239},[238],[31,17993,17995],{"className":17994},[243,244,245,246],[31,17996,17998],{"className":17997},[52,246],[31,17999,18001],{"className":18000},[52,17106,246],[31,18002,17170],{"className":18003},[52,246],[31,18005,18006,18009],{"style":234},[31,18007],{"className":18008,"style":239},[238],[31,18010,18012],{"className":18011},[243,244,245,246],[31,18013,5633],{"className":18014},[52,246],[31,18016,607],{"className":18017},[606],[31,18019,18021],{"className":18020},[227],[31,18022,18024],{"className":18023,"style":9458},[231],[31,18025],{},[31,18027,99],{"className":18028},[98]," instead of the ",[31,18031,18033],{"className":18032},[34],[31,18034,18036,18114],{"className":18035,"ariaHidden":39},[38],[31,18037,18039,18043,18046,18049,18102,18105,18108,18111],{"className":18038},[43],[31,18040],{"className":18041,"style":18042},[47],"height:1.0972em;vertical-align:-0.2831em;",[31,18044,3684],{"className":18045,"style":93},[52,53],[31,18047,89],{"className":18048},[88],[31,18050,18052,18055],{"className":18051},[52],[31,18053,5457],{"className":18054},[52,53],[31,18056,18058],{"className":18057},[219],[31,18059,18061,18093],{"className":18060},[223,553],[31,18062,18064,18090],{"className":18063},[227],[31,18065,18067,18079],{"className":18066,"style":209},[231],[31,18068,18070,18073],{"style":18069},"top:-2.4169em;margin-left:0em;margin-right:0.05em;",[31,18071],{"className":18072,"style":239},[238],[31,18074,18076],{"className":18075},[243,244,245,246],[31,18077,17224],{"className":18078},[52,246],[31,18080,18081,18084],{"style":234},[31,18082],{"className":18083,"style":239},[238],[31,18085,18087],{"className":18086},[243,244,245,246],[31,18088,5633],{"className":18089},[52,246],[31,18091,607],{"className":18092},[606],[31,18094,18096],{"className":18095},[227],[31,18097,18100],{"className":18098,"style":18099},[231],"height:0.2831em;",[31,18101],{},[31,18103,99],{"className":18104},[98],[31,18106],{"className":18107,"style":60},[59],[31,18109,65],{"className":18110},[64],[31,18112],{"className":18113,"style":60},[59],[31,18115,18117,18120,18123,18126,18183,18240],{"className":18116},[43],[31,18118],{"className":18119,"style":17888},[47],[31,18121,3684],{"className":18122,"style":93},[52,53],[31,18124,89],{"className":18125},[88],[31,18127,18129,18132],{"className":18128},[52],[31,18130,9261],{"className":18131},[52,53],[31,18133,18135],{"className":18134},[219],[31,18136,18138,18175],{"className":18137},[223,553],[31,18139,18141,18172],{"className":18140},[227],[31,18142,18144,18161],{"className":18143,"style":209},[231],[31,18145,18146,18149],{"style":17915},[31,18147],{"className":18148,"style":239},[238],[31,18150,18152],{"className":18151},[243,244,245,246],[31,18153,18155],{"className":18154},[52,246],[31,18156,18158],{"className":18157},[52,17106,246],[31,18159,17110],{"className":18160},[52,246],[31,18162,18163,18166],{"style":234},[31,18164],{"className":18165,"style":239},[238],[31,18167,18169],{"className":18168},[243,244,245,246],[31,18170,5633],{"className":18171},[52,246],[31,18173,607],{"className":18174},[606],[31,18176,18178],{"className":18177},[227],[31,18179,18181],{"className":18180,"style":17951},[231],[31,18182],{},[31,18184,18186,18189],{"className":18185},[52],[31,18187,9261],{"className":18188},[52,53],[31,18190,18192],{"className":18191},[219],[31,18193,18195,18232],{"className":18194},[223,553],[31,18196,18198,18229],{"className":18197},[227],[31,18199,18201,18218],{"className":18200,"style":209},[231],[31,18202,18203,18206],{"style":11339},[31,18204],{"className":18205,"style":239},[238],[31,18207,18209],{"className":18208},[243,244,245,246],[31,18210,18212],{"className":18211},[52,246],[31,18213,18215],{"className":18214},[52,17106,246],[31,18216,17170],{"className":18217},[52,246],[31,18219,18220,18223],{"style":234},[31,18221],{"className":18222,"style":239},[238],[31,18224,18226],{"className":18225},[243,244,245,246],[31,18227,5633],{"className":18228},[52,246],[31,18230,607],{"className":18231},[606],[31,18233,18235],{"className":18234},[227],[31,18236,18238],{"className":18237,"style":9458},[231],[31,18239],{},[31,18241,99],{"className":18242},[98]," of the full block — a difference of many orders of\nmagnitude for a wide layer. The Kronecker structure is an assumption (it holds exactly\nonly if inputs and output-gradients are statistically independent), but it captures\nbetween-parameter curvature that a diagonal preconditioner throws away.",[5976,18245,18247,18248],{"id":18246},"hessian-vector-products-without-forming-h","Hessian-vector products without forming ",[31,18249,18251],{"className":18250},[34],[31,18252,18254],{"className":18253,"ariaHidden":39},[38],[31,18255,18257,18260],{"className":18256},[43],[31,18258],{"className":18259,"style":188},[47],[31,18261,193],{"className":18262,"style":192},[52,53],[11,18264,18265,18266,18299,18300,18318,18319,18337,18338,666],{},"The trick under Hessian-free optimization is that the whole ",[31,18267,18269],{"className":18268},[34],[31,18270,18272,18290],{"className":18271,"ariaHidden":39},[38],[31,18273,18275,18278,18281,18284,18287],{"className":18274},[43],[31,18276],{"className":18277,"style":133},[47],[31,18279,5457],{"className":18280},[52,53],[31,18282],{"className":18283,"style":140},[59],[31,18285,6001],{"className":18286},[144],[31,18288],{"className":18289,"style":140},[59],[31,18291,18293,18296],{"className":18292},[43],[31,18294],{"className":18295,"style":113},[47],[31,18297,5457],{"className":18298},[52,53]," matrix is never\nneeded — only its action ",[31,18301,18303],{"className":18302},[34],[31,18304,18306],{"className":18305,"ariaHidden":39},[38],[31,18307,18309,18312,18315],{"className":18308},[43],[31,18310],{"className":18311,"style":188},[47],[31,18313,193],{"className":18314,"style":192},[52,53],[31,18316,16664],{"className":18317,"style":54},[52,53]," on chosen vectors, which inner CG consumes one at a time.\nAnd ",[31,18320,18322],{"className":18321},[34],[31,18323,18325],{"className":18324,"ariaHidden":39},[38],[31,18326,18328,18331,18334],{"className":18327},[43],[31,18329],{"className":18330,"style":188},[47],[31,18332,193],{"className":18333,"style":192},[52,53],[31,18335,16664],{"className":18336,"style":54},[52,53]," is itself a gradient. For any fixed ",[31,18339,18341],{"className":18340},[34],[31,18342,18344],{"className":18343,"ariaHidden":39},[38],[31,18345,18347,18350],{"className":18346},[43],[31,18348],{"className":18349,"style":9257},[47],[31,18351,16664],{"className":18352,"style":54},[52,53],[31,18354,18356],{"className":18355},[382],[31,18357,18359],{"className":18358},[34],[31,18360,18362,18383,18497],{"className":18361,"ariaHidden":39},[38],[31,18363,18365,18368,18371,18374,18377,18380],{"className":18364},[43],[31,18366],{"className":18367,"style":188},[47],[31,18369,193],{"className":18370,"style":192},[52,53],[31,18372,16664],{"className":18373,"style":54},[52,53],[31,18375],{"className":18376,"style":60},[59],[31,18378,65],{"className":18379},[64],[31,18381],{"className":18382,"style":60},[59],[31,18384,18386,18389,18429,18432,18488,18491,18494],{"className":18385},[43],[31,18387],{"className":18388,"style":854},[47],[31,18390,18392,18395],{"className":18391},[52],[31,18393,79],{"className":18394},[52],[31,18396,18398],{"className":18397},[219],[31,18399,18401,18421],{"className":18400},[223,553],[31,18402,18404,18418],{"className":18403},[227],[31,18405,18407],{"className":18406,"style":2906},[231],[31,18408,18409,18412],{"style":876},[31,18410],{"className":18411,"style":239},[238],[31,18413,18415],{"className":18414},[243,244,245,246],[31,18416,94],{"className":18417,"style":93},[52,53,246],[31,18419,607],{"className":18420},[606],[31,18422,18424],{"className":18423},[227],[31,18425,18427],{"className":18426,"style":895},[231],[31,18428],{},[31,18430],{"className":18431,"style":623},[59],[31,18433,18435,18441,18444,18447,18450,18479,18482],{"className":18434},[908],[31,18436,18438],{"className":18437,"style":913},[88,912],[31,18439,89],{"className":18440},[917,918],[31,18442,55],{"className":18443,"style":54},[52,53],[31,18445,89],{"className":18446},[88],[31,18448,94],{"className":18449,"style":93},[52,53],[31,18451,18453,18456],{"className":18452},[98],[31,18454,99],{"className":18455},[98],[31,18457,18459],{"className":18458},[219],[31,18460,18462],{"className":18461},[223],[31,18463,18465],{"className":18464},[227],[31,18466,18468],{"className":18467,"style":500},[231],[31,18469,18470,18473],{"style":503},[31,18471],{"className":18472,"style":239},[238],[31,18474,18476],{"className":18475},[243,244,245,246],[31,18477,513],{"className":18478},[52,246],[31,18480,16664],{"className":18481,"style":54},[52,53],[31,18483,18485],{"className":18484,"style":913},[98,912],[31,18486,99],{"className":18487},[917,918],[31,18489],{"className":18490,"style":60},[59],[31,18492,65],{"className":18493},[64],[31,18495],{"className":18496,"style":60},[59],[31,18498,18500,18503,18543,18546,18639,18642],{"className":18499},[43],[31,18501],{"className":18502,"style":854},[47],[31,18504,18506,18509],{"className":18505},[52],[31,18507,79],{"className":18508},[52],[31,18510,18512],{"className":18511},[219],[31,18513,18515,18535],{"className":18514},[223,553],[31,18516,18518,18532],{"className":18517},[227],[31,18519,18521],{"className":18520,"style":2906},[231],[31,18522,18523,18526],{"style":876},[31,18524],{"className":18525,"style":239},[238],[31,18527,18529],{"className":18528},[243,244,245,246],[31,18530,94],{"className":18531,"style":93},[52,53,246],[31,18533,607],{"className":18534},[606],[31,18536,18538],{"className":18537},[227],[31,18539,18541],{"className":18540,"style":895},[231],[31,18542],{},[31,18544],{"className":18545,"style":623},[59],[31,18547,18549,18555,18558,18598,18601,18630,18633],{"className":18548},[908],[31,18550,18552],{"className":18551,"style":913},[88,912],[31,18553,89],{"className":18554},[917,918],[31,18556,89],{"className":18557},[88],[31,18559,18561,18564],{"className":18560},[52],[31,18562,79],{"className":18563},[52],[31,18565,18567],{"className":18566},[219],[31,18568,18570,18590],{"className":18569},[223,553],[31,18571,18573,18587],{"className":18572},[227],[31,18574,18576],{"className":18575,"style":2906},[231],[31,18577,18578,18581],{"style":876},[31,18579],{"className":18580,"style":239},[238],[31,18582,18584],{"className":18583},[243,244,245,246],[31,18585,94],{"className":18586,"style":93},[52,53,246],[31,18588,607],{"className":18589},[606],[31,18591,18593],{"className":18592},[227],[31,18594,18596],{"className":18595,"style":895},[231],[31,18597],{},[31,18599,84],{"className":18600},[52,83],[31,18602,18604,18607],{"className":18603},[98],[31,18605,99],{"className":18606},[98],[31,18608,18610],{"className":18609},[219],[31,18611,18613],{"className":18612},[223],[31,18614,18616],{"className":18615},[227],[31,18617,18619],{"className":18618,"style":500},[231],[31,18620,18621,18624],{"style":503},[31,18622],{"className":18623,"style":239},[238],[31,18625,18627],{"className":18626},[243,244,245,246],[31,18628,513],{"className":18629},[52,246],[31,18631,16664],{"className":18632,"style":54},[52,53],[31,18634,18636],{"className":18635,"style":913},[98,912],[31,18637,99],{"className":18638},[917,918],[31,18640],{"className":18641,"style":623},[59],[31,18643,666],{"className":18644},[665],[11,18646,18647,18648,18692,18693,18708,18709,18724,18725,18798,18799,4320,18814,18817,18818,18833,18834,18878,18879,18882,18883,18898,18899,18923],{},"because differentiating ",[31,18649,18651],{"className":18650},[34],[31,18652,18654],{"className":18653,"ariaHidden":39},[38],[31,18655,18657,18660,18689],{"className":18656},[43],[31,18658],{"className":18659,"style":1295},[47],[31,18661,18663,18666],{"className":18662},[52],[31,18664,55],{"className":18665,"style":54},[52,53],[31,18667,18669],{"className":18668},[219],[31,18670,18672],{"className":18671},[223],[31,18673,18675],{"className":18674},[227],[31,18676,18678],{"className":18677,"style":1314},[231],[31,18679,18680,18683],{"style":234},[31,18681],{"className":18682,"style":239},[238],[31,18684,18686],{"className":18685},[243,244,245,246],[31,18687,513],{"className":18688},[52,246],[31,18690,16664],{"className":18691,"style":54},[52,53]," (a scalar, with ",[31,18694,18696],{"className":18695},[34],[31,18697,18699],{"className":18698,"ariaHidden":39},[38],[31,18700,18702,18705],{"className":18701},[43],[31,18703],{"className":18704,"style":9257},[47],[31,18706,16664],{"className":18707,"style":54},[52,53]," held constant) with respect to\n",[31,18710,18712],{"className":18711},[34],[31,18713,18715],{"className":18714,"ariaHidden":39},[38],[31,18716,18718,18721],{"className":18717},[43],[31,18719],{"className":18720,"style":113},[47],[31,18722,94],{"className":18723,"style":93},[52,53]," contracts the second derivative ",[31,18726,18728],{"className":18727},[34],[31,18729,18731,18789],{"className":18730,"ariaHidden":39},[38],[31,18732,18734,18737,18777,18780,18783,18786],{"className":18733},[43],[31,18735],{"className":18736,"style":8262},[47],[31,18738,18740,18743],{"className":18739},[52],[31,18741,79],{"className":18742},[52],[31,18744,18746],{"className":18745},[219],[31,18747,18749,18769],{"className":18748},[223,553],[31,18750,18752,18766],{"className":18751},[227],[31,18753,18755],{"className":18754,"style":2906},[231],[31,18756,18757,18760],{"style":876},[31,18758],{"className":18759,"style":239},[238],[31,18761,18763],{"className":18762},[243,244,245,246],[31,18764,94],{"className":18765,"style":93},[52,53,246],[31,18767,607],{"className":18768},[606],[31,18770,18772],{"className":18771},[227],[31,18773,18775],{"className":18774,"style":895},[231],[31,18776],{},[31,18778,55],{"className":18779,"style":54},[52,53],[31,18781],{"className":18782,"style":60},[59],[31,18784,65],{"className":18785},[64],[31,18787],{"className":18788,"style":60},[59],[31,18790,18792,18795],{"className":18791},[43],[31,18793],{"className":18794,"style":188},[47],[31,18796,193],{"className":18797,"style":192},[52,53]," against ",[31,18800,18802],{"className":18801},[34],[31,18803,18805],{"className":18804,"ariaHidden":39},[38],[31,18806,18808,18811],{"className":18807},[43],[31,18809],{"className":18810,"style":9257},[47],[31,18812,16664],{"className":18813,"style":54},[52,53],[26,18815,18816],{},"Pearlmutter's\ntrick"," computes this at the cost of one extra backward pass: run the ordinary backward\npass to get ",[31,18819,18821],{"className":18820},[34],[31,18822,18824],{"className":18823,"ariaHidden":39},[38],[31,18825,18827,18830],{"className":18826},[43],[31,18828],{"className":18829,"style":48},[47],[31,18831,55],{"className":18832,"style":54},[52,53],", take the scalar ",[31,18835,18837],{"className":18836},[34],[31,18838,18840],{"className":18839,"ariaHidden":39},[38],[31,18841,18843,18846,18875],{"className":18842},[43],[31,18844],{"className":18845,"style":1295},[47],[31,18847,18849,18852],{"className":18848},[52],[31,18850,55],{"className":18851,"style":54},[52,53],[31,18853,18855],{"className":18854},[219],[31,18856,18858],{"className":18857},[223],[31,18859,18861],{"className":18860},[227],[31,18862,18864],{"className":18863,"style":1314},[231],[31,18865,18866,18869],{"style":234},[31,18867],{"className":18868,"style":239},[238],[31,18870,18872],{"className":18871},[243,244,245,246],[31,18873,513],{"className":18874},[52,246],[31,18876,16664],{"className":18877,"style":54},[52,53],", and backpropagate ",[164,18880,18881],{},"that",". No entry of ",[31,18884,18886],{"className":18885},[34],[31,18887,18889],{"className":18888,"ariaHidden":39},[38],[31,18890,18892,18895],{"className":18891},[43],[31,18893],{"className":18894,"style":188},[47],[31,18896,193],{"className":18897,"style":192},[52,53]," is\never materialized, storage stays ",[31,18900,18902],{"className":18901},[34],[31,18903,18905],{"className":18904,"ariaHidden":39},[38],[31,18906,18908,18911,18914,18917,18920],{"className":18907},[43],[31,18909],{"className":18910,"style":75},[47],[31,18912,3684],{"className":18913,"style":93},[52,53],[31,18915,89],{"className":18916},[88],[31,18918,5457],{"className":18919},[52,53],[31,18921,99],{"className":18922},[98],", and each CG iteration inside the inner solve\ncosts one forward–backward pair.",[1507,18925],{"hash":18926},"459393d9d992ed0270d9e66f8c9b22ccf5950f5a4cebc050f02d94e2573ab031",[11,18928,18929,18930,18933,18934],{},"Curvature is ",[164,18931,18932],{},"accessible"," — just never as a stored matrix.",[261,18935,18936],{},[15,18937,6405],{"href":18938,"ariaDescribedBy":18939,"dataFootnoteRef":6,"id":18940},"#user-content-fn-gf-natural",[267],"user-content-fnref-gf-natural",[271,18942,18944],{"id":18943},"a-ladder-of-approximations","A ladder of approximations",[11,18946,18947,18948,18972,18973,18988,18989,19039],{},"Every method here occupies one rung of a single ladder that trades curvature accuracy\nagainst cost per step. At the bottom is plain gradient descent, which uses no curvature\nand costs ",[31,18949,18951],{"className":18950},[34],[31,18952,18954],{"className":18953,"ariaHidden":39},[38],[31,18955,18957,18960,18963,18966,18969],{"className":18956},[43],[31,18958],{"className":18959,"style":75},[47],[31,18961,3684],{"className":18962,"style":93},[52,53],[31,18964,89],{"className":18965},[88],[31,18967,5457],{"className":18968},[52,53],[31,18970,99],{"className":18971},[98],". Each rung up costs more and captures more of ",[31,18974,18976],{"className":18975},[34],[31,18977,18979],{"className":18978,"ariaHidden":39},[38],[31,18980,18982,18985],{"className":18981},[43],[31,18983],{"className":18984,"style":188},[47],[31,18986,193],{"className":18987,"style":192},[52,53],": a diagonal estimate\n(one number per parameter, which is what Adam does), then a per-layer Kronecker block\n(K-FAC), then a low-rank running estimate from curvature pairs (L-BFGS), and at the top\nthe full Hessian solve of exact Newton at ",[31,18990,18992],{"className":18991},[34],[31,18993,18995],{"className":18994,"ariaHidden":39},[38],[31,18996,18998,19001,19004,19007,19036],{"className":18997},[43],[31,18999],{"className":19000,"style":3071},[47],[31,19002,3684],{"className":19003,"style":93},[52,53],[31,19005,89],{"className":19006},[88],[31,19008,19010,19013],{"className":19009},[52],[31,19011,5457],{"className":19012},[52,53],[31,19014,19016],{"className":19015},[219],[31,19017,19019],{"className":19018},[223],[31,19020,19022],{"className":19021},[227],[31,19023,19025],{"className":19024,"style":209},[231],[31,19026,19027,19030],{"style":234},[31,19028],{"className":19029,"style":239},[238],[31,19031,19033],{"className":19032},[243,244,245,246],[31,19034,5633],{"className":19035},[52,246],[31,19037,99],{"className":19038},[98],". Deep learning almost always sits on\nthe lowest two rungs, because the higher ones cannot tolerate minibatch noise and their\nper-step cost is not worthwhile when the gradient itself is only an estimate.",[1507,19041],{"hash":19042},"ecb097e5a997f4fb92411b09e494c18a871afaf39781e83a42d521404592a578",[271,19044,19046],{"id":19045},"first-order-versus-second-order","First-order versus second-order",[11,19048,19049],{},"The whole chapter resolves into one comparison. First-order methods are cheap, noise-\ntolerant, and scale to billions of parameters but suffer under ill-conditioning;\nsecond-order methods fix conditioning and converge in far fewer steps but cost too much\nper step and break under saddles and noise. Deep learning sits in the top row, borrowing\npieces of the bottom row (momentum, adaptive rates, occasionally L-BFGS) when\nit can.",[6170,19051,19052,19064],{},[6173,19053,19054],{},[6176,19055,19056,19058,19061],{},[6179,19057],{},[6179,19059,19060],{},"First-order (SGD, Adam)",[6179,19062,19063],{},"Second-order (Newton, quasi-Newton)",[6262,19065,19066,19122,19284,19295,19306,19317,19331],{},[6176,19067,19068,19071,19089],{},[6267,19069,19070],{},"Information used",[6267,19072,19073,19074],{},"gradient ",[31,19075,19077],{"className":19076},[34],[31,19078,19080],{"className":19079,"ariaHidden":39},[38],[31,19081,19083,19086],{"className":19082},[43],[31,19084],{"className":19085,"style":48},[47],[31,19087,55],{"className":19088,"style":54},[52,53],[6267,19090,19073,19091,19106,19107],{},[31,19092,19094],{"className":19093},[34],[31,19095,19097],{"className":19096,"ariaHidden":39},[38],[31,19098,19100,19103],{"className":19099},[43],[31,19101],{"className":19102,"style":48},[47],[31,19104,55],{"className":19105,"style":54},[52,53]," + curvature ",[31,19108,19110],{"className":19109},[34],[31,19111,19113],{"className":19112,"ariaHidden":39},[38],[31,19114,19116,19119],{"className":19115},[43],[31,19117],{"className":19118,"style":188},[47],[31,19120,193],{"className":19121,"style":192},[52,53],[6176,19123,19124,19126,19152],{},[6267,19125,13850],{},[6267,19127,19128],{},[31,19129,19131],{"className":19130},[34],[31,19132,19134],{"className":19133,"ariaHidden":39},[38],[31,19135,19137,19140,19143,19146,19149],{"className":19136},[43],[31,19138],{"className":19139,"style":75},[47],[31,19141,3684],{"className":19142,"style":93},[52,53],[31,19144,89],{"className":19145},[88],[31,19147,5457],{"className":19148},[52,53],[31,19150,99],{"className":19151},[98],[6267,19153,19154,19204,19205,19255,19256,19283],{},[31,19155,19157],{"className":19156},[34],[31,19158,19160],{"className":19159,"ariaHidden":39},[38],[31,19161,19163,19166,19169,19172,19201],{"className":19162},[43],[31,19164],{"className":19165,"style":3071},[47],[31,19167,3684],{"className":19168,"style":93},[52,53],[31,19170,89],{"className":19171},[88],[31,19173,19175,19178],{"className":19174},[52],[31,19176,5457],{"className":19177},[52,53],[31,19179,19181],{"className":19180},[219],[31,19182,19184],{"className":19183},[223],[31,19185,19187],{"className":19186},[227],[31,19188,19190],{"className":19189,"style":209},[231],[31,19191,19192,19195],{"style":234},[31,19193],{"className":19194,"style":239},[238],[31,19196,19198],{"className":19197},[243,244,245,246],[31,19199,250],{"className":19200},[52,246],[31,19202,99],{"className":19203},[98],"–",[31,19206,19208],{"className":19207},[34],[31,19209,19211],{"className":19210,"ariaHidden":39},[38],[31,19212,19214,19217,19220,19223,19252],{"className":19213},[43],[31,19215],{"className":19216,"style":3071},[47],[31,19218,3684],{"className":19219,"style":93},[52,53],[31,19221,89],{"className":19222},[88],[31,19224,19226,19229],{"className":19225},[52],[31,19227,5457],{"className":19228},[52,53],[31,19230,19232],{"className":19231},[219],[31,19233,19235],{"className":19234},[223],[31,19236,19238],{"className":19237},[227],[31,19239,19241],{"className":19240,"style":209},[231],[31,19242,19243,19246],{"style":234},[31,19244],{"className":19245,"style":239},[238],[31,19247,19249],{"className":19248},[243,244,245,246],[31,19250,5633],{"className":19251},[52,246],[31,19253,99],{"className":19254},[98]," (exact), ",[31,19257,19259],{"className":19258},[34],[31,19260,19262],{"className":19261,"ariaHidden":39},[38],[31,19263,19265,19268,19271,19274,19277,19280],{"className":19264},[43],[31,19266],{"className":19267,"style":75},[47],[31,19269,3684],{"className":19270,"style":93},[52,53],[31,19272,89],{"className":19273},[88],[31,19275,12528],{"className":19276},[52,53],[31,19278,5457],{"className":19279},[52,53],[31,19281,99],{"className":19282},[98]," (L-BFGS)",[6176,19285,19286,19289,19292],{},[6267,19287,19288],{},"Steps to converge",[6267,19290,19291],{},"many",[6267,19293,19294],{},"few (quadratic \u002F superlinear)",[6176,19296,19297,19300,19303],{},[6267,19298,19299],{},"Robust to conditioning",[6267,19301,19302],{},"no (needs momentum \u002F adaptive rates)",[6267,19304,19305],{},"yes (rescales by curvature)",[6176,19307,19308,19311,19314],{},[6267,19309,19310],{},"Robust to minibatch noise",[6267,19312,19313],{},"yes (noise averages out)",[6267,19315,19316],{},"no (division amplifies it)",[6176,19318,19319,19322,19325],{},[6267,19320,19321],{},"Saddle behavior",[6267,19323,19324],{},"escapes (with noise \u002F momentum)",[6267,19326,19327,19330],{},[164,19328,19329],{},"attracted"," unless modified",[6176,19332,19333,19336,19339],{},[6267,19334,19335],{},"Deep-learning suitability",[6267,19337,19338],{},"the default",[6267,19340,19341],{},"rare; full-batch or approximate only",[1511,19343,19344],{"type":16794},[11,19345,19346,1745,19349,19352,19353,19356,19357,19381,19382,19415],{},[26,19347,19348],{},"Remark (Why Adam, not Newton).",[15,19350,19351],{"href":22},"Adam and the adaptive methods","\ncan be read as the cheapest possible second-order approximation: they precondition by\na ",[164,19354,19355],{},"diagonal"," estimate of curvature (one scale per parameter, ",[31,19358,19360],{"className":19359},[34],[31,19361,19363],{"className":19362,"ariaHidden":39},[38],[31,19364,19366,19369,19372,19375,19378],{"className":19365},[43],[31,19367],{"className":19368,"style":75},[47],[31,19370,3684],{"className":19371,"style":93},[52,53],[31,19373,89],{"className":19374},[88],[31,19376,5457],{"className":19377},[52,53],[31,19379,99],{"className":19380},[98]," memory) instead\nof the full ",[31,19383,19385],{"className":19384},[34],[31,19386,19388,19406],{"className":19387,"ariaHidden":39},[38],[31,19389,19391,19394,19397,19400,19403],{"className":19390},[43],[31,19392],{"className":19393,"style":133},[47],[31,19395,5457],{"className":19396},[52,53],[31,19398],{"className":19399,"style":140},[59],[31,19401,6001],{"className":19402},[144],[31,19404],{"className":19405,"style":140},[59],[31,19407,19409,19412],{"className":19408},[43],[31,19410],{"className":19411,"style":113},[47],[31,19413,5457],{"className":19414},[52,53]," Hessian. They keep the conditioning benefit of second-order\nmethods at first-order cost, and that trade is why they, not\nNewton, dominate deep learning.",[271,19417,19419],{"id":19418},"modern-second-order-optimizers","Modern second-order optimizers",[11,19421,19422],{},"Goodfellow's §8.6 stops at the classical toolkit; the past decade pushed\napproximate second-order optimization back toward large-scale training.",[19424,19425,19426,19440,19454],"ul",{},[19427,19428,19429,19432,19433],"li",{},[26,19430,19431],{},"K-FAC in full."," Martens and Grosse's paper is the source of the\nKronecker-factored Fisher approximation above, and the first to make natural\ngradient practical on real networks.",[261,19434,19435],{},[15,19436,5751],{"href":19437,"ariaDescribedBy":19438,"dataFootnoteRef":6,"id":19439},"#user-content-fn-kfac",[267],"user-content-fnref-kfac",[19427,19441,19442,19445,19446],{},[26,19443,19444],{},"Shampoo."," Gupta, Koren, and Singer keep a preconditioner per tensor\ndimension (Kronecker-factored full-matrix AdaGrad); a distributed version has\ntrained production language models faster than Adam.",[261,19447,19448],{},[15,19449,19453],{"href":19450,"ariaDescribedBy":19451,"dataFootnoteRef":6,"id":19452},"#user-content-fn-shampoo",[267],"user-content-fnref-shampoo","8",[19427,19455,19456,19459,19460,19462,19463],{},[26,19457,19458],{},"Sophia."," Liu et al. use a clipped ",[164,19461,19355],{}," Hessian estimate to stay\nsaddle-safe and roughly halve language-model pre-training steps — the ladder's\ndiagonal rung, but reading true curvature rather than Adam's gradient second\nmoment.",[261,19464,19465],{},[15,19466,6452],{"href":19467,"ariaDescribedBy":19468,"dataFootnoteRef":6,"id":19469},"#user-content-fn-sophia",[267],"user-content-fnref-sophia",[11,19471,19472],{},"The trend runs up the ladder: as models grow, a little real curvature increasingly\nbecomes worth its cost, and the saddle and noise walls are handled by the same\npositive-definite surrogates and clipping this lesson derived.",[271,19474,19476],{"id":19475},"takeaways","Takeaways",[19424,19478,19479,19573,19744,19793,20076,20193],{},[19427,19480,19481,1745,19483,19569,19570,19572],{},[26,19482,274],{},[31,19484,19486],{"className":19485},[34],[31,19487,19489,19507,19525],{"className":19488,"ariaHidden":39},[38],[31,19490,19492,19495,19498,19501,19504],{"className":19491},[43],[31,19493],{"className":19494,"style":113},[47],[31,19496,94],{"className":19497,"style":93},[52,53],[31,19499],{"className":19500,"style":60},[59],[31,19502,123],{"className":19503},[64],[31,19505],{"className":19506,"style":60},[59],[31,19508,19510,19513,19516,19519,19522],{"className":19509},[43],[31,19511],{"className":19512,"style":133},[47],[31,19514,94],{"className":19515,"style":93},[52,53],[31,19517],{"className":19518,"style":140},[59],[31,19520,145],{"className":19521},[144],[31,19523],{"className":19524,"style":140},[59],[31,19526,19528,19531,19566],{"className":19527},[43],[31,19529],{"className":19530,"style":1463},[47],[31,19532,19534,19537],{"className":19533},[52],[31,19535,193],{"className":19536,"style":192},[52,53],[31,19538,19540],{"className":19539},[219],[31,19541,19543],{"className":19542},[223],[31,19544,19546],{"className":19545},[227],[31,19547,19549],{"className":19548,"style":209},[231],[31,19550,19551,19554],{"style":234},[31,19552],{"className":19553,"style":239},[238],[31,19555,19557],{"className":19556},[243,244,245,246],[31,19558,19560,19563],{"className":19559},[52,246],[31,19561,145],{"className":19562},[52,246],[31,19564,269],{"className":19565},[52,246],[31,19567,55],{"className":19568,"style":54},[52,53]," jumps to the minimum of the local\nquadratic in one step and is ",[26,19571,1876],{}," — it rescales each\ndirection by its curvature, curing the ill-conditioning that makes first-order descent\nzig-zag.",[19427,19574,19575,19576,19611,19612,19666,19667,19720,19721,19736,19737,19739,19740,19743],{},"Three walls keep it out of deep learning: the Hessian is ",[26,19577,19578],{},[31,19579,19581],{"className":19580},[34],[31,19582,19584,19602],{"className":19583,"ariaHidden":39},[38],[31,19585,19587,19590,19593,19596,19599],{"className":19586},[43],[31,19588],{"className":19589,"style":133},[47],[31,19591,5457],{"className":19592},[52,53],[31,19594],{"className":19595,"style":140},[59],[31,19597,6001],{"className":19598},[144],[31,19600],{"className":19601,"style":140},[59],[31,19603,19605,19608],{"className":19604},[43],[31,19606],{"className":19607,"style":113},[47],[31,19609,5457],{"className":19610},[52,53]," (storage\n",[31,19613,19615],{"className":19614},[34],[31,19616,19618,19631],{"className":19617,"ariaHidden":39},[38],[31,19619,19621,19624,19628],{"className":19620},[43],[31,19622],{"className":19623,"style":9257},[47],[31,19625,19627],{"className":19626},[64],"∝",[31,19629],{"className":19630,"style":60},[59],[31,19632,19634,19637],{"className":19633},[43],[31,19635],{"className":19636,"style":209},[47],[31,19638,19640,19643],{"className":19639},[52],[31,19641,5457],{"className":19642},[52,53],[31,19644,19646],{"className":19645},[219],[31,19647,19649],{"className":19648},[223],[31,19650,19652],{"className":19651},[227],[31,19653,19655],{"className":19654,"style":209},[231],[31,19656,19657,19660],{"style":234},[31,19658],{"className":19659,"style":239},[238],[31,19661,19663],{"className":19662},[243,244,245,246],[31,19664,250],{"className":19665},[52,246],", inversion ",[31,19668,19670],{"className":19669},[34],[31,19671,19673,19685],{"className":19672,"ariaHidden":39},[38],[31,19674,19676,19679,19682],{"className":19675},[43],[31,19677],{"className":19678,"style":9257},[47],[31,19680,19627],{"className":19681},[64],[31,19683],{"className":19684,"style":60},[59],[31,19686,19688,19691],{"className":19687},[43],[31,19689],{"className":19690,"style":209},[47],[31,19692,19694,19697],{"className":19693},[52],[31,19695,5457],{"className":19696},[52,53],[31,19698,19700],{"className":19699},[219],[31,19701,19703],{"className":19702},[223],[31,19704,19706],{"className":19705},[227],[31,19707,19709],{"className":19708,"style":209},[231],[31,19710,19711,19714],{"style":234},[31,19712],{"className":19713,"style":239},[238],[31,19715,19717],{"className":19716},[243,244,245,246],[31,19718,5633],{"className":19719},[52,246],", infeasible for ",[31,19722,19724],{"className":19723},[34],[31,19725,19727],{"className":19726,"ariaHidden":39},[38],[31,19728,19730,19733],{"className":19729},[43],[31,19731],{"className":19732,"style":113},[47],[31,19734,5457],{"className":19735},[52,53]," up to billions); Newton is\n",[26,19738,8704],{}," because dividing by a negative eigenvalue flips a downhill\nstep uphill; and ",[26,19741,19742],{},"minibatch noise"," in the denominator is amplified by the division.",[19427,19745,19746,19748,19749,19764,19765,19792],{},[26,19747,8776],{}," steps in ",[31,19750,19752],{"className":19751},[34],[31,19753,19755],{"className":19754,"ariaHidden":39},[38],[31,19756,19758,19761],{"className":19757},[43],[31,19759],{"className":19760,"style":188},[47],[31,19762,193],{"className":19763,"style":192},[52,53],"-conjugate directions to eliminate the zig-zag of\nsteepest descent and reach a quadratic's minimum in ",[31,19766,19768],{"className":19767},[34],[31,19769,19771,19783],{"className":19770,"ariaHidden":39},[38],[31,19772,19774,19777,19780],{"className":19773},[43],[31,19775],{"className":19776,"style":14095},[47],[31,19778,2995],{"className":19779},[64],[31,19781],{"className":19782,"style":60},[59],[31,19784,19786,19789],{"className":19785},[43],[31,19787],{"className":19788,"style":113},[47],[31,19790,5457],{"className":19791},[52,53]," steps using gradients\nonly — no Hessian inverse.",[19427,19794,19795,19797,19798,19863,19864,19974,19975,19977,19978,20028,20029,20031,20032,20047,20048,20075],{},[26,19796,9796],{}," methods build ",[31,19799,19801],{"className":19800},[34],[31,19802,19804,19822],{"className":19803,"ariaHidden":39},[38],[31,19805,19807,19810,19813,19816,19819],{"className":19806},[43],[31,19808],{"className":19809,"style":188},[47],[31,19811,9818],{"className":19812,"style":9817},[52,53],[31,19814],{"className":19815,"style":60},[59],[31,19817,435],{"className":19818},[64],[31,19820],{"className":19821,"style":60},[59],[31,19823,19825,19828],{"className":19824},[43],[31,19826],{"className":19827,"style":209},[47],[31,19829,19831,19834],{"className":19830},[52],[31,19832,193],{"className":19833,"style":192},[52,53],[31,19835,19837],{"className":19836},[219],[31,19838,19840],{"className":19839},[223],[31,19841,19843],{"className":19842},[227],[31,19844,19846],{"className":19845,"style":209},[231],[31,19847,19848,19851],{"style":234},[31,19849],{"className":19850,"style":239},[238],[31,19852,19854],{"className":19853},[243,244,245,246],[31,19855,19857,19860],{"className":19856},[52,246],[31,19858,145],{"className":19859},[52,246],[31,19861,269],{"className":19862},[52,246]," from gradient differences via the\nsecant condition ",[31,19865,19867],{"className":19866},[34],[31,19868,19870,19925],{"className":19869,"ariaHidden":39},[38],[31,19871,19873,19876,19916,19919,19922],{"className":19872},[43],[31,19874],{"className":19875,"style":10254},[47],[31,19877,19879,19882],{"className":19878},[52],[31,19880,10065],{"className":19881,"style":54},[52,53],[31,19883,19885],{"className":19884},[219],[31,19886,19888,19908],{"className":19887},[223,553],[31,19889,19891,19905],{"className":19890},[227],[31,19892,19894],{"className":19893,"style":9314},[231],[31,19895,19896,19899],{"style":4477},[31,19897],{"className":19898,"style":239},[238],[31,19900,19902],{"className":19901},[243,244,245,246],[31,19903,9327],{"className":19904},[52,53,246],[31,19906,607],{"className":19907},[606],[31,19909,19911],{"className":19910},[227],[31,19912,19914],{"className":19913,"style":895},[231],[31,19915],{},[31,19917],{"className":19918,"style":60},[59],[31,19920,435],{"className":19921},[64],[31,19923],{"className":19924,"style":60},[59],[31,19926,19928,19931,19934],{"className":19927},[43],[31,19929],{"className":19930,"style":7130},[47],[31,19932,193],{"className":19933,"style":192},[52,53],[31,19935,19937,19940],{"className":19936},[52],[31,19938,9896],{"className":19939},[52,53],[31,19941,19943],{"className":19942},[219],[31,19944,19946,19966],{"className":19945},[223,553],[31,19947,19949,19963],{"className":19948},[227],[31,19950,19952],{"className":19951,"style":9314},[231],[31,19953,19954,19957],{"style":876},[31,19955],{"className":19956,"style":239},[238],[31,19958,19960],{"className":19959},[243,244,245,246],[31,19961,9327],{"className":19962},[52,53,246],[31,19964,607],{"className":19965},[606],[31,19967,19969],{"className":19968},[227],[31,19970,19972],{"className":19971,"style":895},[231],[31,19973],{},"; ",[26,19976,10549],{}," stores the full ",[31,19979,19981],{"className":19980},[34],[31,19982,19984],{"className":19983,"ariaHidden":39},[38],[31,19985,19987,19990,19993,19996,20025],{"className":19986},[43],[31,19988],{"className":19989,"style":3071},[47],[31,19991,3684],{"className":19992,"style":93},[52,53],[31,19994,89],{"className":19995},[88],[31,19997,19999,20002],{"className":19998},[52],[31,20000,5457],{"className":20001},[52,53],[31,20003,20005],{"className":20004},[219],[31,20006,20008],{"className":20007},[223],[31,20009,20011],{"className":20010},[227],[31,20012,20014],{"className":20013,"style":209},[231],[31,20015,20016,20019],{"style":234},[31,20017],{"className":20018,"style":239},[238],[31,20020,20022],{"className":20021},[243,244,245,246],[31,20023,250],{"className":20024},[52,246],[31,20026,99],{"className":20027},[98]," matrix,\n",[26,20030,12495],{}," keeps only the last ",[31,20033,20035],{"className":20034},[34],[31,20036,20038],{"className":20037,"ariaHidden":39},[38],[31,20039,20041,20044],{"className":20040},[43],[31,20042],{"className":20043,"style":9257},[47],[31,20045,12528],{"className":20046},[52,53]," curvature pairs at ",[31,20049,20051],{"className":20050},[34],[31,20052,20054],{"className":20053,"ariaHidden":39},[38],[31,20055,20057,20060,20063,20066,20069,20072],{"className":20056},[43],[31,20058],{"className":20059,"style":75},[47],[31,20061,3684],{"className":20062,"style":93},[52,53],[31,20064,89],{"className":20065},[88],[31,20067,12528],{"className":20068},[52,53],[31,20070,5457],{"className":20071},[52,53],[31,20073,99],{"className":20074},[98]," memory.",[19427,20077,15000,20078,20080,20081,20096,20097,20099,20100,20115,20116,20118,20119,20158,20159,20177,20178,781],{},[26,20079,15878],{}," preconditions by the always-positive Fisher matrix ",[31,20082,20084],{"className":20083},[34],[31,20085,20087],{"className":20086,"ariaHidden":39},[38],[31,20088,20090,20093],{"className":20089},[43],[31,20091],{"className":20092,"style":188},[47],[31,20094,15376],{"className":20095,"style":15375},[52,53],";\n",[26,20098,16562],{}," factors ",[31,20101,20103],{"className":20102},[34],[31,20104,20106],{"className":20105,"ariaHidden":39},[38],[31,20107,20109,20112],{"className":20108},[43],[31,20110],{"className":20111,"style":188},[47],[31,20113,15376],{"className":20114,"style":15375},[52,53]," per layer as a Kronecker product; ",[26,20117,16623],{}," optimization\nsolves ",[31,20120,20122],{"className":20121},[34],[31,20123,20125,20146],{"className":20124,"ariaHidden":39},[38],[31,20126,20128,20131,20134,20137,20140,20143],{"className":20127},[43],[31,20129],{"className":20130,"style":188},[47],[31,20132,193],{"className":20133,"style":192},[52,53],[31,20135,309],{"className":20136},[52],[31,20138],{"className":20139,"style":60},[59],[31,20141,65],{"className":20142},[64],[31,20144],{"className":20145,"style":60},[59],[31,20147,20149,20152,20155],{"className":20148},[43],[31,20150],{"className":20151,"style":1086},[47],[31,20153,145],{"className":20154},[52],[31,20156,55],{"className":20157,"style":54},[52,53]," by inner CG using ",[31,20160,20162],{"className":20161},[34],[31,20163,20165],{"className":20164,"ariaHidden":39},[38],[31,20166,20168,20171,20174],{"className":20167},[43],[31,20169],{"className":20170,"style":188},[47],[31,20172,193],{"className":20173,"style":192},[52,53],[31,20175,16664],{"className":20176,"style":54},[52,53]," products from one extra backward pass —\ncurvature without ever storing ",[31,20179,20181],{"className":20180},[34],[31,20182,20184],{"className":20183,"ariaHidden":39},[38],[31,20185,20187,20190],{"className":20186},[43],[31,20188],{"className":20189,"style":188},[47],[31,20191,193],{"className":20192,"style":192},[52,53],[19427,20194,20195,20196,20199,20200,20224],{},"Deep learning stays first-order: ",[26,20197,20198],{},"Adam"," is the diagonal, ",[31,20201,20203],{"className":20202},[34],[31,20204,20206],{"className":20205,"ariaHidden":39},[38],[31,20207,20209,20212,20215,20218,20221],{"className":20208},[43],[31,20210],{"className":20211,"style":75},[47],[31,20213,3684],{"className":20214,"style":93},[52,53],[31,20216,89],{"className":20217},[88],[31,20219,5457],{"className":20220},[52,53],[31,20222,99],{"className":20223},[98]," approximation to\nsecond-order preconditioning, keeping the conditioning benefit at first-order cost.",[20226,20227,20230,20235],"section",{"className":20228,"dataFootnotes":6},[20229],"footnotes",[271,20231,20234],{"className":20232,"id":267},[20233],"sr-only","Footnotes",[20236,20237,20238,20255,20322,20335,20364,20470,20502,20517,20532],"ol",{},[19427,20239,20241,8579,20244,20247,20248],{"id":20240},"user-content-fn-gf-secondorder",[26,20242,20243],{},"Goodfellow",[164,20245,20246],{},"Deep Learning",", §8.6 — Approximate Second-Order Methods: why curvature converges in fewer steps and the price that keeps exact second-order methods out of deep learning. ",[15,20249,20254],{"href":20250,"ariaLabel":20251,"className":20252,"dataFootnoteBackref":6},"#user-content-fnref-gf-secondorder","Back to reference 1",[20253],"data-footnote-backref","↩",[19427,20256,20258,8579,20260,20262,20263,20316,20317],{"id":20257},"user-content-fn-gf-hessian",[26,20259,20243],{},[164,20261,20246],{},", §4.3.1 — Beyond the Gradient: Jacobian and Hessian Matrices: the second-order Taylor model, the Newton step ",[31,20264,20266],{"className":20265},[34],[31,20267,20269],{"className":20268,"ariaHidden":39},[38],[31,20270,20272,20275,20278,20313],{"className":20271},[43],[31,20273],{"className":20274,"style":1463},[47],[31,20276,145],{"className":20277},[52],[31,20279,20281,20284],{"className":20280},[52],[31,20282,193],{"className":20283,"style":192},[52,53],[31,20285,20287],{"className":20286},[219],[31,20288,20290],{"className":20289},[223],[31,20291,20293],{"className":20292},[227],[31,20294,20296],{"className":20295,"style":209},[231],[31,20297,20298,20301],{"style":234},[31,20299],{"className":20300,"style":239},[238],[31,20302,20304],{"className":20303},[243,244,245,246],[31,20305,20307,20310],{"className":20306},[52,246],[31,20308,145],{"className":20309},[52,246],[31,20311,269],{"className":20312},[52,246],[31,20314,55],{"className":20315,"style":54},[52,53],", and condition-number invariance. ",[15,20318,20254],{"href":20319,"ariaLabel":20320,"className":20321,"dataFootnoteBackref":6},"#user-content-fnref-gf-hessian","Back to reference 2",[20253],[19427,20323,20325,8579,20327,20329,20330],{"id":20324},"user-content-fn-gf-saddlefree",[26,20326,20243],{},[164,20328,20246],{},", §8.2.3, §8.6 — Newton's attraction to saddle points (dividing by a negative eigenvalue) and the saddle-free \u002F damped fixes that force positive curvature. ",[15,20331,20254],{"href":20332,"ariaLabel":20333,"className":20334,"dataFootnoteBackref":6},"#user-content-fnref-gf-saddlefree","Back to reference 3",[20253],[19427,20336,20338,8579,20340,20342,20343,20358,20359],{"id":20337},"user-content-fn-gf-cg",[26,20339,20243],{},[164,20341,20246],{},", §8.6 — Conjugate Gradients: ",[31,20344,20346],{"className":20345},[34],[31,20347,20349],{"className":20348,"ariaHidden":39},[38],[31,20350,20352,20355],{"className":20351},[43],[31,20353],{"className":20354,"style":188},[47],[31,20356,193],{"className":20357,"style":192},[52,53],"-conjugate search directions that eliminate the zig-zag of steepest descent without forming the Hessian. ",[15,20360,20254],{"href":20361,"ariaLabel":20362,"className":20363,"dataFootnoteBackref":6},"#user-content-fnref-gf-cg","Back to reference 4",[20253],[19427,20365,20367,8579,20369,20371,20372,20437,20438,4320,20465],{"id":20366},"user-content-fn-gf-bfgs",[26,20368,20243],{},[164,20370,20246],{},", §8.6 — BFGS and L-BFGS: learning ",[31,20373,20375],{"className":20374},[34],[31,20376,20378,20396],{"className":20377,"ariaHidden":39},[38],[31,20379,20381,20384,20387,20390,20393],{"className":20380},[43],[31,20382],{"className":20383,"style":188},[47],[31,20385,9818],{"className":20386,"style":9817},[52,53],[31,20388],{"className":20389,"style":60},[59],[31,20391,435],{"className":20392},[64],[31,20394],{"className":20395,"style":60},[59],[31,20397,20399,20402],{"className":20398},[43],[31,20400],{"className":20401,"style":209},[47],[31,20403,20405,20408],{"className":20404},[52],[31,20406,193],{"className":20407,"style":192},[52,53],[31,20409,20411],{"className":20410},[219],[31,20412,20414],{"className":20413},[223],[31,20415,20417],{"className":20416},[227],[31,20418,20420],{"className":20419,"style":209},[231],[31,20421,20422,20425],{"style":234},[31,20423],{"className":20424,"style":239},[238],[31,20426,20428],{"className":20427},[243,244,245,246],[31,20429,20431,20434],{"className":20430},[52,246],[31,20432,145],{"className":20433},[52,246],[31,20435,269],{"className":20436},[52,246]," from the secant condition, and the limited-memory recursion that drops storage to ",[31,20439,20441],{"className":20440},[34],[31,20442,20444],{"className":20443,"ariaHidden":39},[38],[31,20445,20447,20450,20453,20456,20459,20462],{"className":20446},[43],[31,20448],{"className":20449,"style":75},[47],[31,20451,3684],{"className":20452,"style":93},[52,53],[31,20454,89],{"className":20455},[88],[31,20457,12528],{"className":20458},[52,53],[31,20460,5457],{"className":20461},[52,53],[31,20463,99],{"className":20464},[98],[15,20466,20254],{"href":20467,"ariaLabel":20468,"className":20469,"dataFootnoteBackref":6},"#user-content-fnref-gf-bfgs","Back to reference 5",[20253],[19427,20471,20473,8579,20475,20477,20478,20496,20497],{"id":20472},"user-content-fn-gf-natural",[26,20474,20243],{},[164,20476,20246],{},", §8.6, §12.1 — natural gradient via the Fisher information matrix, K-FAC's Kronecker factorization, and Hessian-free optimization through ",[31,20479,20481],{"className":20480},[34],[31,20482,20484],{"className":20483,"ariaHidden":39},[38],[31,20485,20487,20490,20493],{"className":20486},[43],[31,20488],{"className":20489,"style":188},[47],[31,20491,193],{"className":20492,"style":192},[52,53],[31,20494,16664],{"className":20495,"style":54},[52,53]," products. ",[15,20498,20254],{"href":20499,"ariaLabel":20500,"className":20501,"dataFootnoteBackref":6},"#user-content-fnref-gf-natural","Back to reference 6",[20253],[19427,20503,20505,8579,20508,20511,20512],{"id":20504},"user-content-fn-kfac",[26,20506,20507],{},"Martens & Grosse",[164,20509,20510],{},"Optimizing Neural Networks with Kronecker-Factored Approximate Curvature",", ICML 2015 — the K-FAC approximation of the Fisher block as a Kronecker product of two small factors. ",[15,20513,20254],{"href":20514,"ariaLabel":20515,"className":20516,"dataFootnoteBackref":6},"#user-content-fnref-kfac","Back to reference 7",[20253],[19427,20518,20520,8579,20523,20526,20527],{"id":20519},"user-content-fn-shampoo",[26,20521,20522],{},"Gupta, Koren & Singer",[164,20524,20525],{},"Shampoo: Preconditioned Stochastic Tensor Optimization",", ICML 2018 — per-dimension Kronecker-factored preconditioning, later scaled to production language models. ",[15,20528,20254],{"href":20529,"ariaLabel":20530,"className":20531,"dataFootnoteBackref":6},"#user-content-fnref-shampoo","Back to reference 8",[20253],[19427,20533,20535,8579,20538,20541,20542],{"id":20534},"user-content-fn-sophia",[26,20536,20537],{},"Liu et al.",[164,20539,20540],{},"Sophia: A Scalable Stochastic Second-Order Optimizer for Language Model Pre-training",", ICLR 2024 — a clipped diagonal-Hessian preconditioner that roughly halves pre-training steps. ",[15,20543,20254],{"href":20544,"ariaLabel":20545,"className":20546,"dataFootnoteBackref":6},"#user-content-fnref-sophia","Back to reference 9",[20253],[20548,20549,20550],"style",{},"html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark-mode .shiki span {color: var(--shiki-dark-mode);background: var(--shiki-dark-mode-bg);font-style: var(--shiki-dark-mode-font-style);font-weight: var(--shiki-dark-mode-font-weight);text-decoration: var(--shiki-dark-mode-text-decoration);}html.dark-mode .shiki span {color: var(--shiki-dark-mode);background: var(--shiki-dark-mode-bg);font-style: var(--shiki-dark-mode-font-style);font-weight: var(--shiki-dark-mode-font-weight);text-decoration: var(--shiki-dark-mode-text-decoration);}",{"title":6,"searchDepth":9654,"depth":9654,"links":20552},[20553,20554,20560,20561,20562,20563,20567,20568,20569,20570,20571],{"id":273,"depth":9654,"text":274},{"id":5970,"depth":9654,"text":5971,"children":20555},[20556,20558,20559],{"id":5978,"depth":9660,"text":20557},"The Hessian is d×d",{"id":6909,"depth":9660,"text":6910},{"id":8287,"depth":9660,"text":8288},{"id":8775,"depth":9654,"text":8776},{"id":9741,"depth":9654,"text":9742},{"id":14443,"depth":9654,"text":14444},{"id":15872,"depth":9654,"text":15873,"children":20564},[20565],{"id":18246,"depth":9660,"text":20566},"Hessian-vector products without forming H",{"id":18943,"depth":9654,"text":18944},{"id":19045,"depth":9654,"text":19046},{"id":19418,"depth":9654,"text":19419},{"id":19475,"depth":9654,"text":19476},{"id":267,"depth":9654,"text":20234},[],"computer-science","Every optimizer so far (SGD,\nmomentum, and the adaptive methods)\nis first-order: it sees only the gradient g=∇L(θ) and steps\nθ←θ−ηg. The gradient is a flat plane tangent to the loss; it\ngives the downhill direction but nothing about how the surface curves. A\nsecond-order method also reads the curvature (the Hessian H=∇2L)\nand uses it to choose both direction and step length at once. On the\nill-conditioned ravines that\nmake first-order descent crawl, that extra information is decisive. The problem is\nits cost, and most of this lesson is about approximations that capture a fraction\nof it cheaply.1",false,"md",{"moduleNumber":9660,"lessonNumber":9672,"order":20578},305,"Optimization",true,"\u002Fdeep-learning\u002Foptimization\u002Fsecond-order-and-approximate-methods",[],"---\ntitle: Second-Order & Approximate Methods\nmodule: Optimization\nmoduleNumber: 3\nlessonNumber: 5\norder: 305\nsummary: >\n  Newton's method reads the curvature of the loss off its Hessian and jumps to the\n  minimum of the local quadratic in a single step, rescaling away the\n  ill-conditioning that slows first-order descent. We derive it, then explain the\n  three obstacles that keep it out of deep learning: a $d \\times d$ Hessian for $d$ in\n  the billions, an attraction to saddle points, and minibatch noise. The alternative is\n  approximation (conjugate gradients, BFGS and L-BFGS, the natural gradient and\n  Hessian-free methods), each buying some of Newton's curvature information without\n  ever forming or inverting $H$.\ntopics: [Optimization]\nsources:\n  - book: Goodfellow\n    ref: \"§8.6 — Approximate Second-Order Methods\"\n  - book: Goodfellow\n    ref: \"§4.3.1 — Beyond the Gradient: Jacobian and Hessian Matrices; §8.2 Challenges in Optimization\"\n  - book: Chollet\n    ref: \"§2.4 — Gradient-Based Optimization\"\n---\n\nEvery optimizer so far ([SGD](\u002Fdeep-learning\u002Foptimization\u002Fgradient-descent-and-sgd),\n[momentum, and the adaptive methods](\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods))\nis **first-order**: it sees only the gradient $g = \\nabla\\mathcal{L}(\\theta)$ and steps\n$\\theta \\gets \\theta - \\eta g$. The gradient is a flat plane tangent to the loss; it\ngives the downhill _direction_ but nothing about how the surface _curves_. A\n**second-order** method also reads the curvature (the Hessian $H = \\nabla^2\\mathcal{L}$)\nand uses it to choose both direction and step length at once. On the\n[ill-conditioned](\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape) ravines that\nmake first-order descent crawl, that extra information is decisive. The problem is\nits cost, and most of this lesson is about approximations that capture a fraction\nof it cheaply.[^gf-secondorder]\n\n## Newton's method\n\nExpand the loss to second order around the current iterate $\\theta$. Writing the step\nas $\\Delta = \\theta' - \\theta$, Taylor's theorem gives the local quadratic model\n\n$$\n\\mathcal{L}(\\theta + \\Delta) \\;\\approx\\; \\mathcal{L}(\\theta) \\;+\\; g^\\top \\Delta \\;+\\;\n\\tfrac12\\, \\Delta^\\top H\\, \\Delta,\n\\qquad g = \\nabla\\mathcal{L}(\\theta),\\quad H = \\nabla^2\\mathcal{L}(\\theta).\n$$\n\nThis is a paraboloid in $\\Delta$. If $H \\succ 0$ it has a unique minimum, found by\nsetting the gradient of the right-hand side to zero:\n\n$$\n\\nabla_\\Delta\\!\\brackets{ g^\\top\\Delta + \\tfrac12 \\Delta^\\top H \\Delta }\n= g + H\\Delta = 0\n\\;\\Longrightarrow\\;\n\\Delta^\\star = -\\,H^{-1} g.\n$$\n\nThe minimizer of the quadratic is one step away, and that step is the **Newton step**.\nTaking it as the update gives Newton's method.\n\nThe model being minimized is a bowl, not a plane. First-order descent fits the loss with\nits tangent plane $\\mathcal{L}(\\theta) + g^\\top\\Delta$ and can only report a direction of\ndecrease; the plane has no bottom, so the step length has to come from a separately tuned\nlearning rate. The second-order model adds the quadratic term $\\tfrac12\\Delta^\\top H\\Delta$,\nwhich closes the surface into a paraboloid with a definite minimum. Minimizing that bowl\nfixes direction _and_ length together, and $-H^{-1}g$ is the vector from the current point\nto its floor.\n\n$$\n% caption: First order fits a tangent plane (no bottom, needs a learning rate); second\n% order fits a bowl $\\mathcal{L}+g^\\top\\Delta+\\tfrac12\\Delta^\\top H\\Delta$ whose minimum\n% $\\Delta^\\star=-H^{-1}g$ sets both the direction and the length of the step.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{green}{HTML}{1F9D4D}\n  \\definecolor{red}{HTML}{C0392B}\n  % the true loss curve (a bowl)\n  \\draw[acc, very thick] plot[domain=-2.6:3.4, samples=120]\n    (\\x, {0.35*(\\x-0.4)*(\\x-0.4) + 0.25});\n  \\node[acc, anchor=west, font=\\footnotesize] at (2.7,3.15) {\\texttt{true loss}};\n  % current iterate on the left flank\n  \\coordinate (c) at (-2.0,2.06);\n  \\fill[black] (c) circle (2.4pt);\n  \\node[anchor=east, font=\\footnotesize] at (-2.2,2.06) {\\texttt{here}};\n  % first-order tangent plane (a line): slope of the bowl at c is 0.7*(-2.4) = -1.68\n  \\draw[red, thick] (-3.0,3.74) -- (0.55,-2.15);\n  \\node[red, anchor=east, font=\\footnotesize] at (-2.75,3.55) {\\texttt{tangent plane}};\n  \\node[red, anchor=west, font=\\footnotesize] at (0.65,-2.1) {\\texttt{no bottom}};\n  % second-order model: a wider quadratic tangent at c, with its own minimum\n  \\draw[green, thick] plot[domain=-3.2:1.2, samples=80]\n    (\\x, {0.55*(\\x+0.85)*(\\x+0.85) + 0.42});\n  \\node[green, anchor=north east, font=\\footnotesize] at (-2.7,2.55) {\\texttt{quadratic model}};\n  % minimum of the second-order model\n  \\fill[green] (-0.85,0.42) circle (2.2pt);\n  \\draw[green, densely dashed] (-0.85,0.42) -- (-1.9,-0.55);\n  \\node[green, anchor=east, font=\\footnotesize] at (-1.9,-0.55) {\\texttt{model min}};\n  % the Newton step from here to model min\n  \\draw[black, ->, thick] (-2.0,1.55) -- (-0.9,1.55);\n  \\node[anchor=south, font=\\footnotesize] at (-1.45,1.55) {\\texttt{step}};\n\\end{tikzpicture}\n$$\n\n> **Definition (Newton's method).** The second-order update\n> $$\\theta \\;\\gets\\; \\theta \\;-\\; H^{-1} g,$$\n> which moves directly to the minimum of the second-order Taylor model of $\\mathcal{L}$\n> at $\\theta$. When $\\mathcal{L}$ is _exactly_ quadratic and $H \\succ 0$, it reaches the\n> true minimum in a single step from any starting point.\n\nThe geometry explains the advantage. Gradient descent steps along $-g$, perpendicular to\nthe contours of $\\mathcal{L}$ _as drawn in raw coordinates_; on a stretched bowl that\ndirection does not point at the minimum. Newton multiplies by $H^{-1}$, which rescales\neach direction by its curvature (stretching the gently curved axes and shrinking the\nsteep ones), so that in the rescaled coordinates the bowl is round and $-H^{-1}g$ aims\nstraight at the bottom.\n\n$$\n% caption: On an ill-conditioned bowl, gradient descent (red) zig-zags across the\n% contours while the Newton step (green) rescales by curvature and lands in one move.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{green}{HTML}{1F9D4D}\n  \\definecolor{red}{HTML}{C0392B}\n  % elongated elliptical contours (ill-conditioned bowl)\n  \\foreach \\r in {0.5,1.0,1.5,2.0,2.5}\n    \\draw[black] (0,0) ellipse ({\\r*2.0} and {\\r*0.72});\n  % minimum\n  \\fill[green] (0,0) circle (2.6pt);\n  \\node[green, anchor=north] at (1.5,-1.9) {\\texttt{minimum}};\n  % start point on the long axis, high up\n  \\coordinate (s) at (-3.7,1.35);\n  \\fill[black] (s) circle (2.4pt);\n  \\node[anchor=east, font=\\footnotesize] at (-3.85,1.35) {\\texttt{start}};\n  % gradient descent: zig-zag across the narrow valley\n  \\draw[red, very thick, ->] (s) -- (-2.45,-0.55);\n  \\draw[red, very thick, ->] (-2.45,-0.55) -- (-1.55,0.78);\n  \\draw[red, very thick, ->] (-1.55,0.78) -- (-0.95,-0.42);\n  \\draw[red, very thick, ->] (-0.95,-0.42) -- (-0.5,0.32);\n  \\node[red, anchor=north, font=\\footnotesize] at (-1.5,-2.0) {\\texttt{gradient descent zig-zags}};\n  % newton: straight to the minimum\n  \\draw[green, very thick, ->] (s) -- (0,0);\n  \\node[green, anchor=south, font=\\footnotesize] at (-1.5,2.0) {\\texttt{Newton: one step}};\n\\end{tikzpicture}\n$$\n\nBecause the rescaling cancels the eigenvalue spread of $H$, Newton's method is\n**invariant to the condition number** $\\kappa(H) = \\lambda_{\\max}\u002F\\lambda_{\\min}$: a\nbowl stretched a thousand-to-one is solved as fast as a round one. This removes\nexactly the problem that momentum and adaptive\nrates only mitigate.[^gf-hessian] To see the one-step claim algebraically, take an exact quadratic\n$\\mathcal{L}(\\theta) = \\tfrac12(\\theta - \\theta^\\star)^\\top H (\\theta - \\theta^\\star)$,\nfor which $g = H(\\theta - \\theta^\\star)$:\n\n$$\n\\theta - H^{-1} g\n= \\theta - H^{-1} H (\\theta - \\theta^\\star)\n= \\theta - (\\theta - \\theta^\\star)\n= \\theta^\\star.\n$$\n\n> **Theorem (Local quadratic convergence).** If $\\mathcal{L}$ is twice continuously\n> differentiable, $H(\\theta^\\star) \\succ 0$ at a minimum $\\theta^\\star$, and the iterate\n> starts close enough to $\\theta^\\star$, Newton's method converges _quadratically_:\n> $\\norm{\\theta_{k+1} - \\theta^\\star} \\le C\\,\\norm{\\theta_k - \\theta^\\star}^2$.\n> The number of correct digits roughly doubles each step.\n\n> **Proof sketch.** The Newton update is $\\theta_{k+1} = \\theta_k - H(\\theta_k)^{-1} g(\\theta_k)$.\n> Taylor-expand $g$ about $\\theta^\\star$: $g(\\theta_k) = H(\\theta^\\star)(\\theta_k - \\theta^\\star) + O(\\norm{\\theta_k - \\theta^\\star}^2)$,\n> and $H(\\theta_k) = H(\\theta^\\star) + O(\\norm{\\theta_k - \\theta^\\star})$. Substituting,\n> the linear error term cancels exactly — that is what $H^{-1}$ buys — leaving only the\n> quadratic remainder, so $\\theta_{k+1} - \\theta^\\star = O(\\norm{\\theta_k - \\theta^\\star}^2)$. $\\qed$\n\nThe invariance runs deeper than the condition number. Newton's method is **affine\ninvariant**: reparametrize by any invertible linear map $\\phi = A\\theta$, and the\niterates it produces are the exact images under $A$ of the iterates on the original\nproblem. A quick check: under $\\phi = A\\theta$ the gradient transforms as\n$g_\\phi = A^{-\\top}g$ and the Hessian as $H_\\phi = A^{-\\top} H A^{-1}$, so the Newton\ndirection is\n\n$$\n-H_\\phi^{-1} g_\\phi\n= -\\parens{A^{-\\top} H A^{-1}}^{-1} A^{-\\top} g\n= -A\\,H^{-1} A^\\top A^{-\\top} g\n= A\\parens{-H^{-1} g},\n$$\n\nthe original step carried through $A$. Gradient descent has no such property — the\ndirection $-g_\\phi = -A^{-\\top}g$ does not equal $A(-g)$, so a bad choice of units alone\ncan make first-order descent crawl. Newton reads geometry, not coordinates, which is why\nthe same step handles a bowl stretched a thousand-to-one and a round one alike.\n\nThe cost is high. Forming $H$ needs the second derivative with respect\nto every _pair_ of parameters — $\\binom{d+1}{2} \\approx d^2\u002F2$ distinct entries — and\nsolving the linear system $H\\Delta = -g$ (whether by inverting $H$ or by Cholesky\nfactorization) costs $O(d^3)$ arithmetic and $O(d^2)$ storage. For a $d = 10^7$ network,\n$H$ alone is $10^{14}$ numbers; the factorization is $10^{21}$ operations, per step. The\nentire remainder of the lesson is a catalog of ways to keep the curvature information\nwhile retreating from that $O(d^2)$\u002F$O(d^3)$ cost.\n\n## Why second-order methods are rare in deep learning\n\nNewton's method works well on small, smooth, deterministic problems. In deep\nlearning all three properties fail at once, and each failure alone rules the method out.\n\n### The Hessian is $d \\times d$\n\nThe first wall is sheer size. The Hessian of a network with $d$ parameters is a\n$d \\times d$ symmetric matrix (one entry per _pair_ of parameters), so its storage\ngrows as $d^2$ and forming its inverse costs $O(d^3)$ by Gaussian elimination.\n\n$$\n% caption: The Hessian grows quadratically in $d$ while the gradient grows only\n% linearly, so at $d=10^9$ it already holds $10^{18}$ entries.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{acclo}{HTML}{A7B5FB}\n  % small gradient column (linear in d)\n  \\node[font=\\footnotesize, anchor=south] at (0.35,2.5) {\\texttt{gradient} $g$};\n  \\draw[acc, thick, fill=acclo!40] (0.05,0) rectangle (0.65,2.3);\n  \\node[font=\\scriptsize, anchor=north] at (0.35,-0.1) {$d$};\n  % three growing Hessian squares (quadratic in d)\n  \\node[font=\\footnotesize, anchor=south] at (2.5,2.5) {\\texttt{small} $d$};\n  \\draw[acc, thick, fill=acclo!30] (2.0,1.3) rectangle (3.0,2.3);\n  \\node[font=\\footnotesize, anchor=south] at (5.2,2.5) {\\texttt{medium }$d$};\n  \\draw[acc, thick, fill=acclo!30] (4.2,0.5) rectangle (6.2,2.5);\n  \\node[font=\\footnotesize, anchor=south] at (9.1,2.85) {\\texttt{large} $d$};\n  \\draw[acc, thick, fill=acclo!30] (7.4,-0.6) rectangle (10.8,2.8);\n  \\node[font=\\footnotesize, anchor=center, text=acc] at (9.1,1.1) {$d$ \\texttt{by} $d$};\n  % the explosion annotation\n  \\node[font=\\footnotesize, text=black, anchor=north] at (5.4,-0.95) {\\texttt{storage grows as} $d^2$, \\texttt{inversion as} $d^3$};\n\\end{tikzpicture}\n$$\n\nThe numbers are far from feasible. The table quantifies the gap: a gradient\nthat fits in memory pairs with a Hessian that cannot be stored on any hardware.\n\n| $d$ (parameters) | gradient entries | Hessian entries | inversion cost $\\sim d^3$ |\n| --- | --- | --- | --- |\n| $10^3$ | $10^3$ | $10^6$ | $10^9$ |\n| $10^6$ | $10^6$ | $10^{12}$ | $10^{18}$ |\n| $10^9$ | $10^9$ | $10^{18}$ | $10^{27}$ |\n\nEven storing $H$ once for a modest $10^6$-parameter network needs a terabyte at\nsingle precision; inverting it is hopeless. And the optimizer must do this _every step_.\n\n### Newton is attracted to saddle points\n\nThe second wall is qualitative, and worse: where it does run, Newton's method walks\n_toward_ the very saddle points that\n[dominate high-dimensional loss surfaces](\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape).\nDiagonalize $H = Q\\Lambda Q^\\top$ and resolve the Newton step into the eigenbasis.\nAlong eigendirection $i$, with eigenvalue $\\lambda_i$ and gradient component $g_i$, the\nstep is\n\n$$\n\\Delta_i = -\\frac{g_i}{\\lambda_i}.\n$$\n\nHere is the defect. The first-order step $-g_i$ always moves _downhill_ along axis $i$.\nThe Newton step divides by $\\lambda_i$, and dividing by a **negative** eigenvalue\n_flips the sign_ — turning a downhill move into an uphill one, straight toward the\ncritical point. Newton does not distinguish whether a critical point is a minimum,\na maximum, or a saddle; it converges to all of them, because it solves $g = 0$\nrather than minimizing $\\mathcal{L}$.[^gf-saddlefree]\n\n| Eigendirection | $\\lambda_i$ | first-order step $-g_i$ | Newton step $-g_i\u002F\\lambda_i$ | effect |\n| --- | --- | --- | --- | --- |\n| up-curving | $\\lambda_i > 0$ | downhill | downhill, rescaled | toward minimum (correct) |\n| down-curving | $\\lambda_i \u003C 0$ | downhill | _uphill_ (sign flips) | toward saddle (wrong) |\n| flat | $\\lambda_i \\approx 0$ | small | blows up ($\\div \\approx 0$) | catastrophic step |\n\n$$\n% caption: Near a saddle, the negative-curvature axis flips the sign of its Newton\n% component, so the Newton step (red) climbs back toward the saddle.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{green}{HTML}{1F9D4D}\n  \\definecolor{red}{HTML}{C0392B}\n  % hyperbolic contours of a saddle u1^2 - u2^2\n  \\foreach \\c in {0.4,1.1,2.0} {\n    \\draw[black] plot[domain=-2.0:2.0, samples=50] ({sqrt(\\c + \\x*\\x)}, {\\x});\n    \\draw[black] plot[domain=-2.0:2.0, samples=50] ({-sqrt(\\c + \\x*\\x)}, {\\x});\n    \\draw[black] plot[domain=-2.0:2.0, samples=50] ({\\x}, {sqrt(\\c + \\x*\\x)});\n    \\draw[black] plot[domain=-2.0:2.0, samples=50] ({\\x}, {-sqrt(\\c + \\x*\\x)});\n  }\n  % the saddle\n  \\fill[black] (0,0) circle (2.6pt);\n  \\node[anchor=south west, font=\\footnotesize] at (0.12,0.08) {\\texttt{saddle}};\n  % current iterate, below the saddle on the escape (down-curving) axis\n  \\coordinate (p) at (0,-2.2);\n  \\fill[acc] (p) circle (2.4pt);\n  \\node[acc, anchor=west, font=\\footnotesize] at (0.15,-2.2) {\\texttt{here}};\n  % gradient step: continues away from the saddle (downhill, correct)\n  \\draw[green, very thick, ->] (p) -- (0,-3.3);\n  \\node[green, anchor=west, font=\\footnotesize] at (0.15,-3.1) {\\texttt{gradient leaves}};\n  % newton step: sign-flipped, climbs back toward the saddle\n  \\draw[red, very thick, ->] (p) -- (0,-0.55);\n  \\node[red, anchor=center, font=\\footnotesize] at (0,2.45) {\\texttt{Newton climbs in}};\n\\end{tikzpicture}\n$$\n\nSince high-loss critical points are overwhelmingly saddles, an unmodified Newton step\nis not merely useless but actively harmful. The fix is to force the curvature positive:\n**saddle-free Newton** replaces $H^{-1}$ with $|H|^{-1}$ (take the absolute values of\nthe eigenvalues), and classic **damping** (Levenberg–Marquardt) solves\n$(H + \\mu I)^{-1} g$, adding $\\mu I$ until the matrix is positive definite. Both restore\ndescent, but both still need $H$.\n\n### Minibatch noise corrupts the estimate\n\nThe third wall is statistical. Deep nets are trained on minibatches, so both $g$ and\n$H$ are noisy estimates from a sample. First-order steps tolerate this (noise averages\nout over many steps), but the Newton step **divides by** the Hessian estimate, and a\nratio amplifies error in the denominator. A small eigenvalue $\\lambda_i$ estimated with\neven modest relative error produces a huge swing in $-g_i\u002F\\lambda_i$. The very curvature\ninformation Newton depends on is the part the noise corrupts most.\n\n| Assumption Newton needs | Reality in deep learning | Consequence |\n| --- | --- | --- |\n| $H$ small and storable | $d$ up to $10^9$, $H$ is $d \\times d$ | cannot form or invert $H$ |\n| $H \\succ 0$ (a bowl) | mostly saddles, mixed-sign $H$ | attracted to saddles |\n| $g, H$ exact | minibatch estimates, noisy | division amplifies the noise |\n| problem deterministic | stochastic objective | step direction unstable |\n\nThe rest of the lesson is the response: methods that capture _some_ curvature without\never forming, storing, or inverting $H$.\n\n## Conjugate gradients\n\nSteepest descent on a quadratic wastes effort because successive steps are orthogonal,\nand on a stretched bowl orthogonal steps zig-zag, each step partly _undoing_ the\nprogress of the last along the previous direction. **Conjugate gradients** (CG) fixes\nthis by choosing search directions that are _conjugate_ with respect to $H$, so that\nminimizing along a new direction never disturbs the minimization already achieved along\nthe old ones.\n\n> **Definition (Conjugate directions).** Two directions $d_i, d_j$ are **conjugate**\n> with respect to $H$ if $d_i^\\top H d_j = 0$ for $i \\ne j$. Minimizing a quadratic\n> exactly along $d_j$ leaves its minimum along every earlier conjugate $d_i$ undisturbed,\n> so $n$ conjugate line-searches reach the exact minimum of an $n$-dimensional quadratic\n> — with no zig-zag and no Hessian inverse.\n\nThe new direction reuses the previous one, bent just enough to be conjugate to it. The\ncombining coefficient $\\beta_t$ (the Fletcher–Reeves form $\\beta_t = g_t^\\top g_t \u002F\ng_{t-1}^\\top g_{t-1}$) is computed from gradients alone, so CG never touches $H$\nexplicitly. Each step needs one gradient and one line search.\n\n$$\n% caption: Steepest descent (red) takes orthogonal steps that zig-zag across the\n% stretched valley; conjugate gradients (green) bends each direction to be\n% $H$-conjugate to the last and reaches the minimum without retracing.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{green}{HTML}{1F9D4D}\n  \\definecolor{red}{HTML}{C0392B}\n  % elongated contours\n  \\foreach \\r in {0.5,1.0,1.5,2.0,2.6}\n    \\draw[black] (0,0) ellipse ({\\r*1.95} and {\\r*0.7});\n  \\fill[acc] (0,0) circle (2.4pt);\n  \\node[acc, anchor=north] at (1.6,-1.9) {\\texttt{minimum}};\n  % start\n  \\coordinate (s) at (-3.6,1.45);\n  \\fill[black] (s) circle (2.2pt);\n  \\node[anchor=south, font=\\footnotesize] at (-3.6,1.6) {\\texttt{start}};\n  % steepest descent: many orthogonal zig-zag steps\n  \\draw[red, very thick, ->] (s) -- (-2.3,-0.55);\n  \\draw[red, very thick, ->] (-2.3,-0.55) -- (-1.45,0.72);\n  \\draw[red, very thick, ->] (-1.45,0.72) -- (-0.85,-0.4);\n  \\draw[red, very thick, ->] (-0.85,-0.4) -- (-0.42,0.3);\n  \\node[red, anchor=south, font=\\footnotesize] at (-1.6,2.05) {\\texttt{steepest descent}};\n  % conjugate gradients: two steps to the minimum\n  \\draw[green, very thick, ->] (s) -- (-1.6,-1.35);\n  \\draw[green, very thick, ->] (-1.6,-1.35) -- (0,0);\n  \\node[green, anchor=north, font=\\footnotesize] at (-1.6,-2.05) {\\texttt{conjugate gradients (2 steps)}};\n\\end{tikzpicture}\n$$\n\n```algorithm\ncaption: $\\textsc{ConjugateGradients}(\\mathcal{L}, \\theta_0)$ — minimize without forming $H$\n$g_0 \\gets \\nabla\\mathcal{L}(\\theta_0)$\n$d_0 \\gets -g_0$ \u002F\u002F first direction is steepest descent\nfor $t \\gets 0, 1, 2, \\dots$ do\n  $a_t \\gets \\arg\\min_a \\mathcal{L}(\\theta_t + a\\,d_t)$ \u002F\u002F line search along $d_t$\n  $\\theta_{t+1} \\gets \\theta_t + a_t\\,d_t$\n  $g_{t+1} \\gets \\nabla\\mathcal{L}(\\theta_{t+1})$\n  if $\\norm{g_{t+1}} \\le \\epsilon$ then \u002F\u002F converged\n    return $\\theta_{t+1}$\n  $b_t \\gets (g_{t+1}^\\top g_{t+1}) \u002F (g_t^\\top g_t)$ \u002F\u002F Fletcher-Reeves coefficient\n  $d_{t+1} \\gets -g_{t+1} + b_t\\,d_t$ \u002F\u002F bend to be conjugate to $d_t$\n```\n\nOn a strictly quadratic objective CG terminates in at most $d$ steps; on a general\nnonlinear loss it is run with periodic restarts and remains a strong batch optimizer,\nthough minibatch noise in the line search limits its use in pure deep learning.[^gf-cg]\n\n## Quasi-Newton: BFGS and L-BFGS\n\nNewton needs $H^{-1}$ but the cost is prohibitive. **Quasi-Newton** methods _learn_ an\napproximation $B \\approx H^{-1}$ as they go, reading curvature off the change in the\ngradient. The **secant condition** anchors the construction: over a step, the gradient's\nchange encodes the curvature along that step. Writing\n\n$$\ns_t = \\theta_{t+1} - \\theta_t,\n\\qquad\ny_t = g_{t+1} - g_t,\n$$\n\na first-order expansion of $g$ gives $y_t \\approx H\\, s_t$, so $H^{-1}$ must satisfy\n$H^{-1} y_t \\approx s_t$. **BFGS** updates $B_t$ at each step so it reproduces this\nrelation on the latest $(s_t, y_t)$ pair while changing $B$ as little as possible\n(a rank-two correction):\n\n$$\nB_{t+1}\n= \\parens{I - \\frac{s_t y_t^\\top}{y_t^\\top s_t}} B_t\n  \\parens{I - \\frac{y_t s_t^\\top}{y_t^\\top s_t}}\n  + \\frac{s_t s_t^\\top}{y_t^\\top s_t}.\n$$\n\nThe step is then $\\theta_{t+1} = \\theta_t - \\alpha_t\\, B_t\\, g_t$, a Newton-like move\nthat never forms or inverts $H$, only matrix–vector products and a line search.\n\n$$\n% caption: BFGS reads curvature off successive gradients: the secant relation\n% $y_t \\approx H s_t$ feeds a rank-two update of the inverse-Hessian estimate.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{green}{HTML}{1F9D4D}\n  \\definecolor{red}{HTML}{C0392B}\n  % a 1D loss curve (a bowl) with two iterates\n  \\draw[->, thick] (-0.3,0) -- (8.4,0) node[right, font=\\footnotesize] {\\texttt{parameter}};\n  \\draw[->, thick] (0,-0.3) -- (0,3.6) node[above, font=\\footnotesize] {\\texttt{loss}};\n  \\draw[acc, very thick] plot[domain=0.5:7.8, samples=120]\n    (\\x, {0.45 + 0.28*(\\x-5.3)*(\\x-5.3)});\n  % iterate t at left, steep tangent\n  \\fill[black] (1.7,3.06) circle (2.4pt);\n  \\node[anchor=north west, font=\\footnotesize] at (1.85,2.95) {\\texttt{here at} $t$};\n  \\draw[red, thick] (0.7,3.62) -- (2.7,1.5);\n  \\node[red, anchor=south, font=\\footnotesize] at (1.7,3.7) {\\texttt{steep gradient}};\n  % iterate t+1 at right of center, gentle tangent\n  \\fill[black] (4.0,0.94) circle (2.4pt);\n  \\node[anchor=south, font=\\footnotesize] at (3.5,1.85) {\\texttt{here at} $t{+}1$};\n  \\draw[green, thick] (3.0,1.66) -- (5.0,0.5);\n  \\node[green, anchor=center, font=\\footnotesize] at (5.5,1.35) {\\texttt{gentler gradient}};\n  % the step s_t between them\n  \\draw[black, \u003C->, thick] (1.7,-0.35) -- (4.0,-0.35);\n  \\node[black, anchor=north, font=\\footnotesize] at (2.85,-0.4) {\\texttt{step} $s$ \\texttt{(change in weight)}};\n  % annotation: change in slope = curvature\n  \\node[anchor=west, font=\\footnotesize, text=black] at (5.2,2.7) {\\texttt{slope change gives curvature}};\n\\end{tikzpicture}\n$$\n\nThe rank-two form is engineered to satisfy three demands at once: it reproduces the\nsecant relation $B_{t+1} y_t = s_t$, it keeps $B_{t+1}$ symmetric and positive definite\nwhenever $y_t^\\top s_t > 0$ (the **curvature condition**, guaranteed by a proper line\nsearch), and among all matrices meeting the secant relation it is the closest to $B_t$ in\na weighted Frobenius norm. Positive definiteness matters: it forces every step to be a\ndescent direction, so unlike raw Newton, BFGS cannot sign-flip toward a saddle. The\ncurvature is _learned_, one $(s_t, y_t)$ pair at a time, and it accumulates the effect of\nmany past pairs into a single matrix.\n\nThe remaining problem is memory: $B$ is still $d \\times d$. **L-BFGS** (limited-memory\nBFGS) never stores $B$ at all. It keeps only the last $m$ pairs $(s_t, y_t)$ (typically\n$m = 5$ to $20$) and reconstructs the action $B_t g_t$ on the fly by a two-loop\nrecursion over those vectors: a backward sweep peels off the recent curvature\ncorrections into scalars $\\alpha_i$, a scaling by $\\gamma = (s_{t-1}^\\top y_{t-1})\u002F\n(y_{t-1}^\\top y_{t-1})$ approximates the bulk of $H^{-1}$, and a forward sweep folds the\ncorrections back in. Each sweep is $2m$ inner products and vector updates, so one\nmatrix–vector action $B_t g_t$ costs $O(md)$ rather than $O(d^2)$. Memory drops from\n$O(d^2)$ to $O(md)$, the same order as the gradient. That is what makes a quasi-Newton\nmethod usable at deep-learning scale (in the full-batch regime).[^gf-bfgs]\n\nThe reason L-BFGS is still uncommon on deep nets is the same noise wall Newton hit, now\nin a subtler place. The secant pair $(s_t, y_t)$ is built from the difference\n$y_t = g_{t+1} - g_t$ of two gradients. Under minibatching those gradients are computed\non _different_ samples, so $y_t$ mixes the true curvature signal $H s_t$ with the\ndifference of two independent noise terms. That noise does not average away inside the\nBFGS update the way it does across SGD steps — it corrupts the very curvature estimate the\nmethod accumulates, and a stale or noisy $B$ can point the step in a wrong direction. L-BFGS\nworks best with large or full batches, where the gradients are clean\nenough; on noisy minibatch gradients the first-order methods are preferable.\n\n| Method | Hessian use | Memory | Per-step cost | Reaches quadratic min in |\n| --- | --- | --- | --- | --- |\n| Newton | exact $H$, inverts it | $O(d^2)$ | $O(d^3)$ | $1$ step |\n| Conjugate gradients | implicit (via $Hd$ products) | $O(d)$ | $O(d)$ + line search | $\\le d$ steps |\n| BFGS | approximates $H^{-1}$ from $(s, y)$ | $O(d^2)$ | $O(d^2)$ | $\\le d$ steps |\n| L-BFGS | last $m$ curvature pairs | $O(md)$ | $O(md)$ | superlinear, $\\le d$ steps |\n\n## Gauss-Newton and the Fisher matrix\n\nBefore the natural gradient there is a simpler positive-semidefinite stand-in for $H$.\nFor a loss that is a sum of squared residuals $\\mathcal{L} = \\tfrac12\\sum_i r_i(\\theta)^2$,\nwrite the Jacobian of the residual vector as $J = \\partial r\u002F\\partial\\theta$. The exact\nHessian is\n\n$$\nH = J^\\top J \\;+\\; \\sum_i r_i\\, \\nabla^2 r_i.\n$$\n\nThe **Gauss-Newton** approximation drops the second term and keeps only $G = J^\\top J$.\nThat term is the part carrying the residual curvature $\\nabla^2 r_i$; near a good fit the\nresiduals $r_i$ are small, so dropping it costs little, and what remains is an outer\nproduct $J^\\top J \\succeq 0$ that can never be indefinite. Gauss-Newton gets the same\nimmunity to saddles that damping enforces by hand, and it needs only first\nderivatives of the residuals.\n\nThe **Fisher information matrix** generalizes this to probabilistic models. When the loss\nis the negative log-likelihood of the model's output distribution $p_\\theta(y\\mid x)$, the\nFisher is the expected outer product of the score,\n\n$$\nF = \\mathbb{E}_{x}\\,\\mathbb{E}_{y\\sim p_\\theta}\\!\\brackets{ \\nabla_\\theta \\log p_\\theta(y\\mid x)\\,\n\\nabla_\\theta \\log p_\\theta(y\\mid x)^\\top }.\n$$\n\nTwo facts make $F$ attractive. It equals the expected Hessian of the negative\nlog-likelihood (so it _is_ curvature, not a crude substitute), and being an expectation\nof outer products it is positive semidefinite by construction — no sign-flips toward\nsaddles. For the exponential-family output layers used in practice, $F$ coincides with\nthe Gauss-Newton matrix $G$, tying the two views together.\n\n## Natural gradient, K-FAC, and Hessian-free\n\nThe **natural gradient** preconditions by $F$ instead of $H$, measuring distance in the\nspace of the model's output _distributions_ rather than its raw parameters:\n\n$$\n\\theta \\;\\gets\\; \\theta \\;-\\; \\eta\\, F^{-1} g.\n$$\n\nThe motivation is invariance. Ordinary gradient descent moves fastest in the direction of\nsteepest descent measured by Euclidean distance in _parameter_ space, but a parameter is\nan arbitrary label — reparametrize the network and the same distance means something\ndifferent. The natural gradient instead takes the steepest-descent direction under the\nKullback–Leibler divergence between output distributions, a quantity that does not depend\non how the parameters are named. To second order that divergence is\n$\\mathrm{KL}(p_\\theta \\Vert p_{\\theta+\\Delta}) \\approx \\tfrac12\\Delta^\\top F\\Delta$, so\n$F$ plays exactly the role $H$ played in the Taylor bowl, and $F^{-1}g$ is the step that\ndescends the loss per unit of distributional change. Because $F$ is positive semidefinite,\nthat step never climbs toward a saddle the way $-H^{-1}g$ can.\n\n$F$ is still $d \\times d$, so like $H$ it is used only through approximations.\n\n| Method | Curvature matrix | Trick that makes it affordable |\n| --- | --- | --- |\n| Natural gradient | Fisher information $F$ | curvature in output-distribution space, always $\\succeq 0$ |\n| K-FAC | block-diagonal $F$ | factor each layer's block as a Kronecker product $A \\otimes G$ |\n| Hessian-free | Hessian $H$ (implicit) | inner CG using $Hv$ products via [autodiff](\u002Fdeep-learning\u002Fneural-networks\u002Fbackpropagation) |\n\n> **Definition (Hessian-free optimization).** A truncated-Newton method that solves\n> $H\\Delta = -g$ approximately by an inner [conjugate-gradient](#conjugate-gradients)\n> loop, supplying each Hessian–vector product $Hv$ through a single extra\n> backpropagation pass — never forming $H$. Curvature is used; the $d \\times d$ matrix\n> is never materialized.\n\n> **Remark (K-FAC).** Kronecker-Factored Approximate Curvature approximates each layer's\n> Fisher block as a Kronecker product $A \\otimes G$ of two small factors — one from the\n> layer's inputs, one from its output gradients. Inverting a Kronecker product is just\n> inverting the two small factors, $(A \\otimes G)^{-1} = A^{-1} \\otimes G^{-1}$,\n> collapsing a per-layer $O(d^3)$ inverse to two cheap ones.\n\nFor a layer with $n_{\\text{in}}$ inputs and $n_{\\text{out}}$ outputs, its weight has\n$d_\\ell = n_{\\text{in}} n_{\\text{out}}$ parameters and its Fisher block is\n$d_\\ell \\times d_\\ell$. K-FAC models that block as $A \\otimes G$, where $A$ is the\n$n_{\\text{in}} \\times n_{\\text{in}}$ covariance of the layer's inputs and $G$ is the\n$n_{\\text{out}} \\times n_{\\text{out}}$ covariance of its output gradients. Storing and\ninverting two factors of size $n_{\\text{in}}$ and $n_{\\text{out}}$ costs\n$O(n_{\\text{in}}^3 + n_{\\text{out}}^3)$ instead of the $O(d_\\ell^3) =\nO(n_{\\text{in}}^3 n_{\\text{out}}^3)$ of the full block — a difference of many orders of\nmagnitude for a wide layer. The Kronecker structure is an assumption (it holds exactly\nonly if inputs and output-gradients are statistically independent), but it captures\nbetween-parameter curvature that a diagonal preconditioner throws away.\n\n### Hessian-vector products without forming $H$\n\nThe trick under Hessian-free optimization is that the whole $d \\times d$ matrix is never\nneeded — only its action $Hv$ on chosen vectors, which inner CG consumes one at a time.\nAnd $Hv$ is itself a gradient. For any fixed $v$,\n\n$$\nHv = \\nabla_\\theta\\parens{ g(\\theta)^\\top v } = \\nabla_\\theta\\parens{ (\\nabla_\\theta\\mathcal{L})^\\top v },\n$$\n\nbecause differentiating $g^\\top v$ (a scalar, with $v$ held constant) with respect to\n$\\theta$ contracts the second derivative $\\nabla_\\theta g = H$ against $v$. **Pearlmutter's\ntrick** computes this at the cost of one extra backward pass: run the ordinary backward\npass to get $g$, take the scalar $g^\\top v$, and backpropagate _that_. No entry of $H$ is\never materialized, storage stays $O(d)$, and each CG iteration inside the inner solve\ncosts one forward–backward pair.\n\n$$\n% caption: Pearlmutter's trick computes $Hv$ as the gradient of the scalar $g^\\top v$:\n% forward pass, backward pass for $g$, contract with $v$, backward again. The matrix\n% $H$ is never formed; storage stays $O(d)$.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize, node distance=6mm]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{green}{HTML}{1F9D4D}\n  \\tikzset{\n    bx\u002F.style={draw=acc, thick, fill=acc!8, minimum width=20mm, minimum height=8mm, align=center},\n    op\u002F.style={draw=black, thick, fill=black!4, minimum width=17mm, minimum height=8mm, align=center},\n  }\n  \\node[bx] (fwd) {\\texttt{forward}\\\\\\texttt{pass}};\n  \\node[bx, right=13mm of fwd] (bwd) {\\texttt{backward}\\\\\\texttt{pass}};\n  \\node[op, minimum width=22mm] (dot) [right=13mm of bwd] {\\texttt{dot} $g,v$};\n  \\node[bx, right=13mm of dot] (bwd2) {\\texttt{backward}\\\\\\texttt{again}};\n  \\node[op, right=13mm of bwd2] (out) {$Hv$};\n  \\draw[->, thick, acc] (fwd) -- (bwd);\n  \\draw[->, thick, acc] (bwd) -- node[above, font=\\scriptsize, black]{$g$} (dot);\n  \\draw[->, thick, acc] (dot) -- (bwd2);\n  \\draw[->, thick, green] (bwd2) -- (out);\n  % v injected into the dot product\n  \\node[op, above=9mm of dot] (v) {\\texttt{vector} $v$};\n  \\draw[->, thick, black] (v) -- (dot);\n  % annotation\n  \\node[font=\\scriptsize, black, anchor=north, text width=76mm, align=center]\n    at (dot|-fwd.south) {\\texttt{one forward pass, two backward passes; H never stored}};\n\\end{tikzpicture}\n$$\n\nCurvature is _accessible_ — just never as a stored matrix.[^gf-natural]\n\n## A ladder of approximations\n\nEvery method here occupies one rung of a single ladder that trades curvature accuracy\nagainst cost per step. At the bottom is plain gradient descent, which uses no curvature\nand costs $O(d)$. Each rung up costs more and captures more of $H$: a diagonal estimate\n(one number per parameter, which is what Adam does), then a per-layer Kronecker block\n(K-FAC), then a low-rank running estimate from curvature pairs (L-BFGS), and at the top\nthe full Hessian solve of exact Newton at $O(d^3)$. Deep learning almost always sits on\nthe lowest two rungs, because the higher ones cannot tolerate minibatch noise and their\nper-step cost is not worthwhile when the gradient itself is only an estimate.\n\n$$\n% caption: The cost\u002Faccuracy ladder. Climbing captures more of the true curvature $H$\n% (diagonal $\\to$ block $\\to$ low-rank $\\to$ full) at rising per-step cost\n% ($O(d)\\to O(d)\\to O(md)\\to O(d^3)$). Deep learning lives on the lowest rungs.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{green}{HTML}{1F9D4D}\n  % axes\n  \\draw[->, thick, black] (0,0) -- (0,6.2) node[above, font=\\footnotesize, black, anchor=south west]{\\texttt{curvature captured}};\n  \\draw[->, thick, black] (0,0) -- (10.6,0) node[below, font=\\footnotesize, black]{\\texttt{cost per step}};\n  \\tikzset{rung\u002F.style={draw=acc, thick, fill=acc!8, minimum width=40mm, minimum height=8.5mm, align=center, font=\\scriptsize}}\n  \\node[rung, anchor=west] (a) at (0.4,0.85) {\\texttt{gradient descent}};\n  \\node[rung, anchor=west] (b) at (1.9,2.05) {\\texttt{Adam (diagonal)}};\n  \\node[rung, anchor=west] (c) at (3.4,3.25) {\\texttt{K-FAC (block)}};\n  \\node[rung, anchor=west] (d) at (4.9,4.45) {\\texttt{L-BFGS (low-rank)}};\n  \\node[rung, anchor=west, fill=green!10, draw=green] (e) at (6.4,5.65) {\\texttt{full Newton}};\n  % connecting climb in the horizontal overlap band, pointing up into the next rung\n  \\draw[->, green, thick] (3.15,1.28) -- (3.15,1.62);\n  \\draw[->, green, thick] (4.65,2.48) -- (4.65,2.82);\n  \\draw[->, green, thick] (6.15,3.68) -- (6.15,4.02);\n  \\draw[->, green, thick] (7.65,4.88) -- (7.65,5.22);\n  % cost labels to the right of each rung, clear of boxes and arrows\n  \\node[font=\\scriptsize, black, anchor=west] at (a.east) {\\;$O(d)$};\n  \\node[font=\\scriptsize, black, anchor=west] at (b.east) {\\;$O(d)$};\n  \\node[font=\\scriptsize, black, anchor=west] at (c.east) {\\;$O(d)$};\n  \\node[font=\\scriptsize, black, anchor=west] at (d.east) {\\;$O(md)$};\n  \\node[font=\\scriptsize, black, anchor=west] at (e.east) {\\;$O(d^3)$};\n\\end{tikzpicture}\n$$\n\n## First-order versus second-order\n\nThe whole chapter resolves into one comparison. First-order methods are cheap, noise-\ntolerant, and scale to billions of parameters but suffer under ill-conditioning;\nsecond-order methods fix conditioning and converge in far fewer steps but cost too much\nper step and break under saddles and noise. Deep learning sits in the top row, borrowing\npieces of the bottom row (momentum, adaptive rates, occasionally L-BFGS) when\nit can.\n\n| | First-order (SGD, Adam) | Second-order (Newton, quasi-Newton) |\n| --- | --- | --- |\n| Information used | gradient $g$ | gradient $g$ + curvature $H$ |\n| Per-step cost | $O(d)$ | $O(d^2)$–$O(d^3)$ (exact), $O(md)$ (L-BFGS) |\n| Steps to converge | many | few (quadratic \u002F superlinear) |\n| Robust to conditioning | no (needs momentum \u002F adaptive rates) | yes (rescales by curvature) |\n| Robust to minibatch noise | yes (noise averages out) | no (division amplifies it) |\n| Saddle behavior | escapes (with noise \u002F momentum) | _attracted_ unless modified |\n| Deep-learning suitability | the default | rare; full-batch or approximate only |\n\n> **Remark (Why Adam, not Newton).** [Adam and the adaptive methods](\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods)\n> can be read as the cheapest possible second-order approximation: they precondition by\n> a _diagonal_ estimate of curvature (one scale per parameter, $O(d)$ memory) instead\n> of the full $d \\times d$ Hessian. They keep the conditioning benefit of second-order\n> methods at first-order cost, and that trade is why they, not\n> Newton, dominate deep learning.\n\n## Modern second-order optimizers\n\nGoodfellow's §8.6 stops at the classical toolkit; the past decade pushed\napproximate second-order optimization back toward large-scale training.\n\n- **K-FAC in full.** Martens and Grosse's paper is the source of the\n  Kronecker-factored Fisher approximation above, and the first to make natural\n  gradient practical on real networks.[^kfac]\n- **Shampoo.** Gupta, Koren, and Singer keep a preconditioner per tensor\n  dimension (Kronecker-factored full-matrix AdaGrad); a distributed version has\n  trained production language models faster than Adam.[^shampoo]\n- **Sophia.** Liu et al. use a clipped _diagonal_ Hessian estimate to stay\n  saddle-safe and roughly halve language-model pre-training steps — the ladder's\n  diagonal rung, but reading true curvature rather than Adam's gradient second\n  moment.[^sophia]\n\nThe trend runs up the ladder: as models grow, a little real curvature increasingly\nbecomes worth its cost, and the saddle and noise walls are handled by the same\npositive-definite surrogates and clipping this lesson derived.\n\n## Takeaways\n\n- **Newton's method** $\\theta \\gets \\theta - H^{-1}g$ jumps to the minimum of the local\n  quadratic in one step and is **invariant to the condition number** — it rescales each\n  direction by its curvature, curing the ill-conditioning that makes first-order descent\n  zig-zag.\n- Three walls keep it out of deep learning: the Hessian is **$d \\times d$** (storage\n  $\\propto d^2$, inversion $\\propto d^3$, infeasible for $d$ up to billions); Newton is\n  **attracted to saddles** because dividing by a negative eigenvalue flips a downhill\n  step uphill; and **minibatch noise** in the denominator is amplified by the division.\n- **Conjugate gradients** steps in $H$-conjugate directions to eliminate the zig-zag of\n  steepest descent and reach a quadratic's minimum in $\\le d$ steps using gradients\n  only — no Hessian inverse.\n- **Quasi-Newton** methods build $B \\approx H^{-1}$ from gradient differences via the\n  secant condition $y_t \\approx H s_t$; **BFGS** stores the full $O(d^2)$ matrix,\n  **L-BFGS** keeps only the last $m$ curvature pairs at $O(md)$ memory.\n- The **natural gradient** preconditions by the always-positive Fisher matrix $F$;\n  **K-FAC** factors $F$ per layer as a Kronecker product; **Hessian-free** optimization\n  solves $H\\Delta = -g$ by inner CG using $Hv$ products from one extra backward pass —\n  curvature without ever storing $H$.\n- Deep learning stays first-order: **Adam** is the diagonal, $O(d)$ approximation to\n  second-order preconditioning, keeping the conditioning benefit at first-order cost.\n\n[^gf-secondorder]: **Goodfellow**, _Deep Learning_, §8.6 — Approximate Second-Order Methods: why curvature converges in fewer steps and the price that keeps exact second-order methods out of deep learning.\n[^gf-hessian]: **Goodfellow**, _Deep Learning_, §4.3.1 — Beyond the Gradient: Jacobian and Hessian Matrices: the second-order Taylor model, the Newton step $-H^{-1}g$, and condition-number invariance.\n[^gf-saddlefree]: **Goodfellow**, _Deep Learning_, §8.2.3, §8.6 — Newton's attraction to saddle points (dividing by a negative eigenvalue) and the saddle-free \u002F damped fixes that force positive curvature.\n[^gf-cg]: **Goodfellow**, _Deep Learning_, §8.6 — Conjugate Gradients: $H$-conjugate search directions that eliminate the zig-zag of steepest descent without forming the Hessian.\n[^gf-bfgs]: **Goodfellow**, _Deep Learning_, §8.6 — BFGS and L-BFGS: learning $B \\approx H^{-1}$ from the secant condition, and the limited-memory recursion that drops storage to $O(md)$.\n[^gf-natural]: **Goodfellow**, _Deep Learning_, §8.6, §12.1 — natural gradient via the Fisher information matrix, K-FAC's Kronecker factorization, and Hessian-free optimization through $Hv$ products.\n[^kfac]: **Martens & Grosse**, _Optimizing Neural Networks with Kronecker-Factored Approximate Curvature_, ICML 2015 — the K-FAC approximation of the Fisher block as a Kronecker product of two small factors.\n[^shampoo]: **Gupta, Koren & Singer**, _Shampoo: Preconditioned Stochastic Tensor Optimization_, ICML 2018 — per-dimension Kronecker-factored preconditioning, later scaled to production language models.\n[^sophia]: **Liu et al.**, _Sophia: A Scalable Stochastic Second-Order Optimizer for Language Model Pre-training_, ICLR 2024 — a clipped diagonal-Hessian preconditioner that roughly halves pre-training steps.\n",{"text":20585,"minutes":20586,"time":20587,"words":20588},"17 min read",16.39,983400,3278,{"title":5,"description":20574},[20591,20593,20595],{"book":20243,"ref":20592},"§8.6 — Approximate Second-Order Methods",{"book":20243,"ref":20594},"§4.3.1 — Beyond the Gradient: Jacobian and Hessian Matrices; §8.2 Challenges in Optimization",{"book":20596,"ref":20597},"Chollet","§2.4 — Gradient-Based Optimization","available","06.deep-learning\u002F03.optimization\u002F05.second-order-and-approximate-methods","Newton's method reads the curvature of the loss off its Hessian and jumps to the minimum of the local quadratic in a single step, rescaling away the ill-conditioning that slows first-order descent. We derive it, then explain the three obstacles that keep it out of deep learning: a $d \\times d$ Hessian for $d$ in the billions, an attraction to saddle points, and minibatch noise. The alternative is approximation (conjugate gradients, BFGS and L-BFGS, the natural gradient and Hessian-free methods), each buying some of Newton's curvature information without ever forming or inverting $H$.\n",[20579],"GQ3pFrnVvm7FgKK9t-LpeBEcRjH7aLnCao-0gWNdPwI",{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":20604,"\u002Falgorithms\u002Ffoundations\u002Fproof-techniques":20605,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":20606,"\u002Falgorithms\u002Ffoundations\u002Fgrowth-rates-and-loop-analysis":20607,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":20608,"\u002Falgorithms\u002Ffoundations\u002Famortized-analysis":20609,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":20610,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":20611,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":20612,"\u002Falgorithms\u002Fdivide-and-conquer\u002Ffast-multiplication":20613,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":20614,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":20615,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":20616,"\u002Falgorithms\u002Fsorting\u002Fexternal-sorting":20617,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":20618,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":20619,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":20620,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":20621,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":20622,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":20623,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":20624,"\u002Falgorithms\u002Fdata-structures\u002Fspatial-data-structures":20625,"\u002Falgorithms\u002Fdata-structures\u002Fskip-lists-and-probabilistic-structures":20626,"\u002Falgorithms\u002Fdata-structures\u002Fb-trees":20627,"\u002Falgorithms\u002Fdata-structures\u002Fdata-stream-algorithms":20628,"\u002Falgorithms\u002Fdata-structures\u002Fstreaming-sketches":20629,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":20630,"\u002Falgorithms\u002Fsequences\u002Fprefix-sums":20631,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":20632,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":20633,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":20634,"\u002Falgorithms\u002Fsequences\u002Fkmp-and-z-function":20635,"\u002Falgorithms\u002Fsequences\u002Ftries":20636,"\u002Falgorithms\u002Fsequences\u002Fsuffix-arrays-and-aho-corasick":20637,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":20638,"\u002Falgorithms\u002Fgraphs\u002Fdepth-first-search":20639,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":20640,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":20641,"\u002Falgorithms\u002Fgraphs\u002Fkruskal-and-prim":20642,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":20643,"\u002Falgorithms\u002Fgraphs\u002Fall-pairs-and-negative-weights":20644,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":20645,"\u002Falgorithms\u002Fgraphs\u002Fmax-flow-min-cut":20646,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":20647,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":20648,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":20649,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":20650,"\u002Falgorithms\u002Fgraphs\u002Fbipartite-matching":20651,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":20652,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":20653,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":20654,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":20655,"\u002Falgorithms\u002Fgreedy\u002Fstable-matching":20656,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":20657,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":20658,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":20659,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":20660,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":20661,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":20662,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":20663,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":20664,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":20665,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":20666,"\u002Falgorithms\u002Fdynamic-programming\u002Fdigit-and-probability-dp":20667,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":20668,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":20669,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":20670,"\u002Falgorithms\u002Fbacktracking\u002Fgraph-backtracking":20671,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":20672,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":20673,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":20674,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":20675,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmatrix-exponentiation":20676,"\u002Falgorithms\u002Fmathematical-algorithms\u002Ffast-fourier-transform":20677,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fgradient-descent":20678,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":20679,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":20680,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":20681,"\u002Falgorithms\u002Fcomputational-geometry\u002Fpolygons-and-proximity":20682,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":20683,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":20684,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":20685,"\u002Falgorithms\u002Fintractability\u002Fapproximation-algorithms":20686,"\u002Falgorithms":20687,"\u002Fcalculus\u002Flimits-and-continuity\u002Ffunctions-and-models":20688,"\u002Fcalculus\u002Flimits-and-continuity\u002Fthe-limit-of-a-function":20689,"\u002Fcalculus\u002Flimits-and-continuity\u002Flimit-laws-and-the-precise-definition":20690,"\u002Fcalculus\u002Flimits-and-continuity\u002Fcontinuity":20691,"\u002Fcalculus\u002Fderivatives\u002Fthe-derivative-and-rates-of-change":20692,"\u002Fcalculus\u002Fderivatives\u002Fdifferentiation-rules-and-the-chain-rule":20693,"\u002Fcalculus\u002Fderivatives\u002Fimplicit-differentiation-and-related-rates":20694,"\u002Fcalculus\u002Fderivatives\u002Flinear-approximations-and-differentials":20695,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fextrema-and-the-mean-value-theorem":20696,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fhow-derivatives-shape-a-graph":20697,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fcurve-sketching-and-optimization":20698,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fnewtons-method-and-antiderivatives":20699,"\u002Fcalculus\u002Fintegrals\u002Farea-and-the-definite-integral":20700,"\u002Fcalculus\u002Fintegrals\u002Fthe-fundamental-theorem-of-calculus":20701,"\u002Fcalculus\u002Fintegrals\u002Fthe-substitution-rule":20702,"\u002Fcalculus\u002Fapplications-of-integration\u002Fareas-and-volumes":20703,"\u002Fcalculus\u002Fapplications-of-integration\u002Fwork-average-value-and-arc-length":20704,"\u002Fcalculus\u002Fapplications-of-integration\u002Fphysics-economics-and-probability":20705,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Finverse-functions-logarithms-and-exponentials":20706,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Fgrowth-decay-inverse-trig-and-hyperbolic-functions":20707,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Flhospitals-rule":20708,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fintegration-by-parts":20709,"\u002Fcalculus\u002Ftechniques-of-integration\u002Ftrigonometric-integrals-and-substitution":20710,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fpartial-fractions-and-integration-strategy":20711,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fapproximate-and-improper-integrals":20712,"\u002Fcalculus\u002Fparametric-and-polar\u002Fparametric-curves-and-their-calculus":20713,"\u002Fcalculus\u002Fparametric-and-polar\u002Fpolar-coordinates":20714,"\u002Fcalculus\u002Fparametric-and-polar\u002Fconic-sections":20715,"\u002Fcalculus\u002Fsequences-and-series\u002Fsequences":20716,"\u002Fcalculus\u002Fsequences-and-series\u002Fseries-and-the-integral-test":20717,"\u002Fcalculus\u002Fsequences-and-series\u002Fthe-convergence-tests":20718,"\u002Fcalculus\u002Fsequences-and-series\u002Fpower-series":20719,"\u002Fcalculus\u002Fsequences-and-series\u002Ftaylor-and-maclaurin-series":20720,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvectors-and-the-dot-product":20721,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fthe-cross-product-lines-and-planes":20702,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fcylinders-and-quadric-surfaces":20722,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvector-functions-and-space-curves":20723,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Farc-length-curvature-and-motion":20724,"\u002Fcalculus\u002Fpartial-derivatives\u002Ffunctions-of-several-variables":20692,"\u002Fcalculus\u002Fpartial-derivatives\u002Fpartial-derivatives":20725,"\u002Fcalculus\u002Fpartial-derivatives\u002Ftangent-planes-and-the-chain-rule":20726,"\u002Fcalculus\u002Fpartial-derivatives\u002Fdirectional-derivatives-and-the-gradient":20727,"\u002Fcalculus\u002Fpartial-derivatives\u002Foptimization-and-lagrange-multipliers":20728,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fdouble-integrals":20729,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Ftriple-integrals-and-coordinate-systems":20730,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fvector-fields-and-line-integrals":20731,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fgreens-theorem-curl-and-divergence":20732,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fsurface-integrals":20733,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fstokes-and-the-divergence-theorem":20734,"\u002Fcalculus":20735,"\u002Fmechanics\u002Ffoundations\u002Fmeasurement-and-dimensions":20736,"\u002Fmechanics\u002Ffoundations\u002Fvector-algebra":20737,"\u002Fmechanics\u002Fkinematics\u002Fone-dimensional-motion":20738,"\u002Fmechanics\u002Fkinematics\u002Fmotion-graphs":20739,"\u002Fmechanics\u002Fkinematics\u002Fprojectile-motion":20740,"\u002Fmechanics\u002Fkinematics\u002Frelative-motion":20741,"\u002Fmechanics\u002Fkinematics\u002Fcircular-motion":20742,"\u002Fmechanics\u002Fdynamics\u002Fnewtons-laws":20743,"\u002Fmechanics\u002Fdynamics\u002Ffree-body-diagrams":20744,"\u002Fmechanics\u002Fdynamics\u002Ffriction-and-curved-motion":20745,"\u002Fmechanics\u002Fdynamics\u002Fnumerical-dynamics":20746,"\u002Fmechanics\u002Fdynamics\u002Fcenter-of-mass-systems":20747,"\u002Fmechanics\u002Fenergy\u002Fwork-and-kinetic-energy":20748,"\u002Fmechanics\u002Fenergy\u002Fpotential-energy":20749,"\u002Fmechanics\u002Fenergy\u002Fmultiparticle-work":20750,"\u002Fmechanics\u002Fenergy\u002Fmass-energy-and-binding":20751,"\u002Fmechanics\u002Fenergy\u002Fphotons-and-quantization":20752,"\u002Fmechanics\u002Fmomentum\u002Fmomentum-and-collisions":20753,"\u002Fmechanics\u002Fmomentum\u002Fcenter-of-mass-collisions":20754,"\u002Fmechanics\u002Fmomentum\u002Frocket-propulsion":20755,"\u002Fmechanics\u002Frotation\u002Frotational-inertia":20756,"\u002Fmechanics\u002Frotation\u002Frotational-dynamics":20757,"\u002Fmechanics\u002Frotation\u002Frolling-motion":20758,"\u002Fmechanics\u002Frotation\u002Fangular-momentum":20759,"\u002Fmechanics\u002Frotation\u002Frolling-resistance":20760,"\u002Fmechanics\u002Frotation\u002Fgyroscopic-precession":20761,"\u002Fmechanics\u002Fgravity-and-matter\u002Fkeplerian-orbits":20762,"\u002Fmechanics\u002Fgravity-and-matter\u002Fgravitational-fields":20763,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstatic-equilibrium":20764,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-statics":20765,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-flow":20766,"\u002Fmechanics\u002Fgravity-and-matter\u002Forbital-motion":20767,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstress-and-elasticity":20768,"\u002Fmechanics\u002Foscillations-waves\u002Fdamped-oscillators":20769,"\u002Fmechanics\u002Foscillations-waves\u002Ftravelling-waves":20770,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-superposition":20771,"\u002Fmechanics\u002Foscillations-waves\u002Fstanding-waves":20772,"\u002Fmechanics\u002Foscillations-waves\u002Fsound-waves":20773,"\u002Fmechanics\u002Foscillations-waves\u002Fdoppler-effect":20774,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-packets":20775,"\u002Fmechanics\u002Foscillations-waves\u002Fbeats-and-coupling":20776,"\u002Fmechanics\u002Foscillations-waves\u002Fsimple-harmonic-motion":20777,"\u002Fmechanics\u002Foscillations-waves\u002Fpendulum-motion":20778,"\u002Fmechanics\u002Foscillations-waves\u002Fdriven-oscillators":20779,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-boundaries":20780,"\u002Fmechanics\u002Fthermodynamics\u002Fkinetic-theory-of-ideal-gases":20781,"\u002Fmechanics\u002Fthermodynamics\u002Ffirst-law-of-thermodynamics":20782,"\u002Fmechanics\u002Fthermodynamics\u002Fentropy-and-the-second-law":20783,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-processes":20784,"\u002Fmechanics\u002Fthermodynamics\u002Fphase-changes":20785,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-machines":20786,"\u002Fmechanics":20787,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcharge-and-conductors":20788,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcoulombs-law":20789,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-and-force":20790,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-maps":20791,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-dipoles":20792,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fcontinuous-charge-fields":20793,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fgauss-law-and-conductors":20794,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpoint-charge-potential":20795,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpotential-gradients-and-equipotentials":20796,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Felectrostatic-energy-and-pressure":20797,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Flaplace-boundary-problems":20798,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fcontinuous-charge-potentials":20799,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitance-fundamentals":20776,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-networks":20800,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-energy-and-force":20801,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fdielectric-polarization-and-breakdown":20802,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fcurrent-and-resistance":20772,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fkirchhoff-network-analysis":20637,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Frc-transients":20803,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-trajectories":20763,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fhall-effect":20804,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-force-on-conductors":20805,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-dipoles":20806,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmass-spectrometry":20807,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmoving-charge-fields":20808,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fbiot-savart-law":20809,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fcircular-current-loops":20810,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Famperes-law":20811,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fgauss-law-for-magnetism":20812,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmagnetic-materials":20737,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-flux":20813,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Ffaradays-law":20814,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Flenzs-law":20815,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmotional-emf":20816,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Feddy-currents":20817,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fself-inductance":20818,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-energy":20819,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Frl-circuits":20820,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-fundamentals":20755,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Freactance":20754,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Frlc-resonance":20821,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-power":20822,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Ftransformers":20823,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdisplacement-current":20824,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-waves":20825,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-momentum":20826,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdipole-radiation":20827,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fpolarization":20828,"\u002Felectricity-and-magnetism\u002Foptics\u002Freflection-and-refraction":20829,"\u002Felectricity-and-magnetism\u002Foptics\u002Fthin-lenses":20781,"\u002Felectricity-and-magnetism\u002Foptics\u002Fspherical-mirrors":20779,"\u002Felectricity-and-magnetism":20830,"\u002Flinear-algebra\u002Flinear-systems\u002Fsystems-and-echelon-forms":20831,"\u002Flinear-algebra\u002Flinear-systems\u002Fvector-and-matrix-equations":20832,"\u002Flinear-algebra\u002Flinear-systems\u002Fsolution-sets-and-applications":20833,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-independence":20834,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-transformations":20835,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-operations":20836,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-inverse-and-invertibility":20837,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fpartitioned-matrices-and-lu":20838,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fsubspaces-dimension-rank":20839,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fapplications-leontief-and-graphics":20689,"\u002Flinear-algebra\u002Fdeterminants\u002Fdeterminants-and-cofactors":20840,"\u002Flinear-algebra\u002Fdeterminants\u002Fproperties-of-determinants":20841,"\u002Flinear-algebra\u002Fdeterminants\u002Fcramer-volume-and-area":20693,"\u002Flinear-algebra\u002Fvector-spaces\u002Fvector-spaces-and-subspaces":20842,"\u002Flinear-algebra\u002Fvector-spaces\u002Fnull-and-column-spaces":20843,"\u002Flinear-algebra\u002Fvector-spaces\u002Fbases-and-independent-sets":20844,"\u002Flinear-algebra\u002Fvector-spaces\u002Fcoordinate-systems":20845,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdimension-and-rank":20846,"\u002Flinear-algebra\u002Fvector-spaces\u002Fchange-of-basis":20847,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdifference-equations-and-markov":20848,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-eigenvalues":20849,"\u002Flinear-algebra\u002Feigenvalues\u002Fthe-characteristic-equation":20850,"\u002Flinear-algebra\u002Feigenvalues\u002Fdiagonalization":20851,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-linear-transformations":20852,"\u002Flinear-algebra\u002Feigenvalues\u002Fcomplex-eigenvalues":20853,"\u002Flinear-algebra\u002Feigenvalues\u002Fdynamical-systems":20854,"\u002Flinear-algebra\u002Feigenvalues\u002Fpower-method":20855,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-length-orthogonality":20856,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Forthogonal-sets-and-projections":20857,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fgram-schmidt-and-qr":20858,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-problems":20859,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-applications":20860,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-spaces":20861,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fdiagonalizing-symmetric-matrices":20728,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fquadratic-forms":20862,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fconstrained-optimization":20863,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsingular-value-decomposition":20864,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsvd-applications-pca-imaging":20865,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-thinking-and-matrix-computation":20866,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Flu-and-cholesky":20867,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fconditioning-and-floating-point":20868,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fstability-and-error-analysis":20869,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fqr-and-numerical-least-squares":20870,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-eigenvalues-and-svd":20871,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-combinations":20872,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-independence-and-barycentric-coordinates":20873,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fconvex-combinations-and-convex-sets":20874,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fhyperplanes-and-polytopes":20875,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fcurves-and-surfaces":20876,"\u002Flinear-algebra":20877,"\u002Ftheory-of-computation":20878,"\u002Fcomputer-architecture\u002Ffoundations\u002Fbits-bytes-and-words":20879,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-representation":20880,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-arithmetic":20881,"\u002Fcomputer-architecture\u002Ffoundations\u002Ffloating-point":20882,"\u002Fcomputer-architecture\u002Ffoundations\u002Fboolean-algebra-and-bit-manipulation":20883,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fthe-machines-view":20884,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fdata-movement":20885,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farithmetic-and-logic":20886,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fcontrol-flow":20887,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fprocedures":20888,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farrays-structs-and-alignment":20889,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fmemory-layout-and-buffer-overflows":20890,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fwhat-an-isa-is":20891,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Finstruction-formats-and-operands":20892,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Faddressing-modes":20893,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fthe-y86-64-instruction-set":20894,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fy86-64-programming":20895,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Ftransistors-gates-and-boolean-functions":20896,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fcombinational-logic-and-hcl":20897,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmultiplexers-decoders-and-the-alu":20898,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmemory-elements-latches-flip-flops-and-clocking":20899,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fregister-files-and-random-access-memory":20900,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-fetch-decode-execute-cycle":20901,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-seq-stages":20902,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fcontrol-logic-and-sequencing":20903,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fassembling-seq":20904,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Ftracing-a-program":20905,"\u002Fcomputer-architecture\u002Fpipelining\u002Fpipelining-principles":20906,"\u002Fcomputer-architecture\u002Fpipelining\u002Ffrom-seq-to-pipe":20907,"\u002Fcomputer-architecture\u002Fpipelining\u002Fdata-hazards-stalling-and-forwarding":20908,"\u002Fcomputer-architecture\u002Fpipelining\u002Fcontrol-hazards-and-branch-prediction":20909,"\u002Fcomputer-architecture\u002Fpipelining\u002Fthe-complete-pipe-processor":20910,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fstorage-technologies-and-the-latency-gap":20911,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Flocality":20912,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-memories-direct-mapped":20913,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fset-associative-and-write-policies":20914,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-performance-and-cache-friendly-code":20915,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Faddress-spaces-and-translation":20916,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fpage-tables-and-page-faults":20917,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fthe-tlb-and-multi-level-page-tables":20918,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Fexceptional-control-flow":20919,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Finterrupts-and-the-kernel":20920,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fprocesses-threads-and-parallelism":20921,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fhardware-multithreading":20922,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fcache-coherence":20923,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmemory-consistency-and-synchronization":20924,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmulticore-organization":20925,"\u002Fcomputer-architecture\u002Fcapstone\u002Fthe-whole-machine":20926,"\u002Fcomputer-architecture\u002Fcapstone\u002Fassembling-a-complete-cpu":20927,"\u002Fcomputer-architecture":20878,"\u002Fdifferential-equations\u002Ffoundations\u002Fmodels-and-direction-fields":20928,"\u002Fdifferential-equations\u002Ffoundations\u002Fclassification-and-terminology":20929,"\u002Fdifferential-equations\u002Ffirst-order\u002Flinear-first-order-integrating-factors":20930,"\u002Fdifferential-equations\u002Ffirst-order\u002Fseparable-and-exact":20693,"\u002Fdifferential-equations\u002Ffirst-order\u002Fmodeling-first-order":20931,"\u002Fdifferential-equations\u002Ffirst-order\u002Fautonomous-and-population-dynamics":20692,"\u002Fdifferential-equations\u002Ffirst-order\u002Fexistence-uniqueness-euler":20699,"\u002Fdifferential-equations\u002Ffirst-order\u002Ffirst-order-difference-equations":20932,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhomogeneous-constant-coefficients":20933,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fcomplex-and-repeated-roots":20733,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fnonhomogeneous-undetermined-coefficients":20934,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fvariation-of-parameters":20935,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fmechanical-electrical-vibrations":20936,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhigher-order-linear":20937,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fpower-series-ordinary-points":20938,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fregular-singular-frobenius":20939,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fbessel-and-special-functions":20940,"\u002Fdifferential-equations\u002Flaplace\u002Flaplace-definition-ivps":20941,"\u002Fdifferential-equations\u002Flaplace\u002Fstep-impulse-convolution":20942,"\u002Fdifferential-equations\u002Fsystems\u002Fmatrices-eigenvalues-review":20943,"\u002Fdifferential-equations\u002Fsystems\u002Fconstant-coefficient-systems-phase-portraits":20944,"\u002Fdifferential-equations\u002Fsystems\u002Frepeated-eigenvalues-fundamental-matrices":20945,"\u002Fdifferential-equations\u002Fnumerical\u002Feuler-and-runge-kutta":20940,"\u002Fdifferential-equations\u002Fnumerical\u002Fmultistep-systems-stability":20946,"\u002Fdifferential-equations\u002Fnonlinear\u002Fphase-plane-autonomous-stability":20947,"\u002Fdifferential-equations\u002Fnonlinear\u002Flocally-linear-and-liapunov":20948,"\u002Fdifferential-equations\u002Fnonlinear\u002Fcompeting-species-predator-prey-limit-cycles":20949,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Ffourier-series":20950,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fheat-wave-laplace-equations":20951,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fsturm-liouville":20952,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fcalculus-of-variations":20953,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fhistorical-notes":20954,"\u002Fdifferential-equations":20955,"\u002Frelativity\u002Ffoundations\u002Fspecial-relativity-postulates":20956,"\u002Frelativity\u002Ffoundations\u002Florentz-transformation-spacetime":20957,"\u002Frelativity\u002Ffoundations\u002Ftime-dilation-length-contraction":20958,"\u002Frelativity\u002Ffoundations\u002Frelativistic-momentum-energy":20959,"\u002Frelativity\u002Ffoundations\u002Fgeneral-relativity":20838,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fminkowski-spacetime-and-the-interval":20960,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Ffour-vectors-and-index-notation":20961,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fthe-lorentz-group-and-rapidity":20962,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fdoppler-aberration-and-appearance":20963,"\u002Frelativity\u002Frelativistic-dynamics\u002Ffour-momentum-force-and-accelerated-motion":20964,"\u002Frelativity\u002Frelativistic-dynamics\u002Fparticle-decays-and-two-body-kinematics":20965,"\u002Frelativity\u002Frelativistic-dynamics\u002Fcollisions-thresholds-and-the-cm-frame":20708,"\u002Frelativity\u002Frelativistic-dynamics\u002Fmandelstam-variables-and-invariants":20966,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ffour-current-and-the-four-potential":20967,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fthe-electromagnetic-field-tensor":20968,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ftransformation-of-electric-and-magnetic-fields":20969,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fcovariant-maxwell-and-the-stress-energy-tensor":20970,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-equivalence-principle-formalized":20971,"\u002Frelativity\u002Fcurved-spacetime\u002Fmanifolds-vectors-and-the-metric":20972,"\u002Frelativity\u002Fcurved-spacetime\u002Fcovariant-derivative-and-christoffel-symbols":20973,"\u002Frelativity\u002Fcurved-spacetime\u002Fgeodesics-and-the-geodesic-equation":20974,"\u002Frelativity\u002Fcurved-spacetime\u002Fcurvature-riemann-and-geodesic-deviation":20975,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-einstein-field-equations":20930,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fthe-schwarzschild-metric":20976,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fgeodesics-and-orbits-in-schwarzschild":20977,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Flight-bending-and-null-geodesics":20978,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fperihelion-precession-of-mercury":20979,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fdeflection-of-light-and-gravitational-lensing":20980,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fgravitational-redshift-and-shapiro-delay":20859,"\u002Frelativity\u002Ftests-of-general-relativity\u002Frelativity-in-technology-gps":20981,"\u002Frelativity\u002Fblack-holes\u002Fhorizons-and-coordinate-singularities":20982,"\u002Frelativity\u002Fblack-holes\u002Frotating-and-charged-black-holes":20869,"\u002Frelativity\u002Fblack-holes\u002Fblack-hole-thermodynamics":20983,"\u002Frelativity\u002Fgravitational-waves\u002Flinearized-gravity-and-wave-solutions":20984,"\u002Frelativity\u002Fgravitational-waves\u002Fgeneration-and-the-quadrupole-formula":20985,"\u002Frelativity\u002Fgravitational-waves\u002Fdetection-ligo-and-the-first-events":20986,"\u002Frelativity\u002Fcosmological-bridge\u002Fthe-cosmological-principle-and-flrw-metric":20987,"\u002Frelativity\u002Fcosmological-bridge\u002Ffriedmann-equations-and-cosmic-dynamics":20988,"\u002Frelativity":20989,"\u002Fphysical-computing":20878,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fblackbody-radiation-and-the-planck-quantum":20990,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-photoelectric-effect-and-the-photon":20969,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fx-rays-and-the-compton-effect":20991,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-old-quantum-theory-bohr-and-sommerfeld":20992,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fde-broglie-waves-and-electron-diffraction":20993,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fwave-packets-and-the-probability-interpretation":20994,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fthe-uncertainty-principle":20995,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-schrodinger-equation-in-one-dimension":20996,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-free-particle-and-wave-packet-dynamics":20997,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fparticle-in-infinite-and-finite-square-wells":20945,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Foperators-expectation-values-and-the-harmonic-oscillator":20869,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-dirac-delta-potential":20998,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fbarrier-penetration-and-quantum-tunneling":20999,"\u002Fquantum-mechanics\u002Fformalism\u002Fhilbert-space-and-dirac-notation":21000,"\u002Fquantum-mechanics\u002Fformalism\u002Fobservables-hermitian-operators-and-eigenvalues":21001,"\u002Fquantum-mechanics\u002Fformalism\u002Fthe-postulates-and-quantum-measurement":20997,"\u002Fquantum-mechanics\u002Fformalism\u002Fposition-momentum-and-continuous-spectra":21002,"\u002Fquantum-mechanics\u002Fformalism\u002Fcommutators-and-the-generalized-uncertainty-principle":20977,"\u002Fquantum-mechanics\u002Fformalism\u002Ftime-evolution-schrodinger-and-heisenberg-pictures":20730,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fladder-operators-and-the-number-states":21003,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fcoherent-and-squeezed-states":21004,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fsymmetries-generators-and-conservation-laws":20710,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fparity-time-reversal-and-discrete-symmetries":21005,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Forbital-angular-momentum-and-spherical-harmonics":21006,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Fthe-angular-momentum-algebra":21007,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Faddition-of-angular-momenta-and-clebsch-gordan":21008,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-schrodinger-equation-in-three-dimensions":21009,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-hydrogen-atom":21010,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-isotropic-oscillator-and-hidden-symmetry":21011,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-half-pauli-matrices-and-stern-gerlach":21012,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-in-a-magnetic-field-precession-and-resonance":21013,"\u002Fquantum-mechanics\u002Fspin\u002Ftwo-level-systems-and-the-bloch-sphere":20969,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fidentical-particles-and-exchange-symmetry":21014,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fthe-pauli-principle-atoms-and-the-periodic-table":21015,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ftime-independent-perturbation-theory":21016,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ffine-structure-and-the-real-hydrogen-atom":21003,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-zeeman-and-stark-effects":20691,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-variational-method":21017,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-wkb-approximation":21018,"\u002Fquantum-mechanics":21019,"\u002Freal-analysis\u002Ffoundations\u002Fsets-logic-functions":20953,"\u002Freal-analysis\u002Ffoundations\u002Fordered-fields-completeness":21020,"\u002Freal-analysis\u002Ffoundations\u002Fabsolute-value-bounds":21021,"\u002Freal-analysis\u002Ffoundations\u002Fintervals-uncountability":20842,"\u002Freal-analysis\u002Fsequences-series\u002Fsequences-limits":21022,"\u002Freal-analysis\u002Fsequences-series\u002Flimit-laws-monotone":20728,"\u002Freal-analysis\u002Fsequences-series\u002Flimsup-bolzano-weierstrass":21023,"\u002Freal-analysis\u002Fsequences-series\u002Fcauchy-completeness":21024,"\u002Freal-analysis\u002Fsequences-series\u002Fseries-convergence":20865,"\u002Freal-analysis\u002Fsequences-series\u002Fabsolute-conditional-rearrangement":20975,"\u002Freal-analysis\u002Fmetric-spaces\u002Fmetric-spaces-norms":21025,"\u002Freal-analysis\u002Fmetric-spaces\u002Fopen-closed-sets":21026,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconvergence-completeness":21027,"\u002Freal-analysis\u002Fmetric-spaces\u002Fcompactness":21028,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconnectedness":21029,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-of-functions":21002,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuous-functions":21030,"\u002Freal-analysis\u002Fcontinuity\u002Fevt-ivt":20835,"\u002Freal-analysis\u002Fcontinuity\u002Funiform-continuity":21031,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuity-metric-spaces":21032,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-infinity-monotone":20689,"\u002Freal-analysis\u002Fdifferentiation\u002Fthe-derivative":21033,"\u002Freal-analysis\u002Fdifferentiation\u002Fmean-value-theorem":21034,"\u002Freal-analysis\u002Fdifferentiation\u002Ftaylors-theorem":20991,"\u002Freal-analysis\u002Fdifferentiation\u002Finverse-function-1d":20718,"\u002Freal-analysis\u002Friemann-integration\u002Fdarboux-integral":20869,"\u002Freal-analysis\u002Friemann-integration\u002Fintegrability-classes":21035,"\u002Freal-analysis\u002Friemann-integration\u002Fproperties-of-the-integral":21036,"\u002Freal-analysis\u002Friemann-integration\u002Ffundamental-theorem":20854,"\u002Freal-analysis\u002Friemann-integration\u002Flog-exp-improper":20979,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpointwise-uniform-convergence":21037,"\u002Freal-analysis\u002Ffunction-sequences\u002Finterchange-of-limits":21038,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpower-series-weierstrass":21039,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpicard-ode":20874,"\u002Freal-analysis\u002Fseveral-variables\u002Fdifferentiability-rn":21040,"\u002Freal-analysis\u002Fseveral-variables\u002Fgradient-chain-rule":21041,"\u002Freal-analysis\u002Fseveral-variables\u002Fhigher-derivatives-taylor-extrema":21042,"\u002Freal-analysis\u002Fseveral-variables\u002Finverse-implicit-theorems":21042,"\u002Freal-analysis\u002Fseveral-variables\u002Fmultiple-integrals":21043,"\u002Freal-analysis":21044,"\u002Fabstract-algebra\u002Ffoundations\u002Fsets-functions-relations":21045,"\u002Fabstract-algebra\u002Ffoundations\u002Fintegers-and-modular-arithmetic":21046,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fgroup-axioms-and-first-examples":21047,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fdihedral-and-symmetric-groups":21048,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fmatrix-and-quaternion-groups":21049,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fhomomorphisms-and-group-actions":21050,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fsubgroups-and-substructures":21051,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcyclic-groups":21052,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fgeneration-and-subgroup-lattices":21053,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcosets-lagrange-and-normal-subgroups":21054,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fisomorphism-theorems":21018,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcomposition-series-and-the-alternating-group":21055,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Factions-and-cayleys-theorem":21047,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fconjugation-and-the-class-equation":20964,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fsylow-theorems":21056,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fautomorphisms-and-simple-groups":21057,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fdirect-products-and-finite-abelian-groups":21058,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fsemidirect-products":21059,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fnilpotent-and-solvable-groups":21060,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fclassifying-small-groups":21061,"\u002Fabstract-algebra\u002Fring-theory\u002Frings-definitions-and-examples":21062,"\u002Fabstract-algebra\u002Fring-theory\u002Fideals-quotients-and-homomorphisms":21063,"\u002Fabstract-algebra\u002Fring-theory\u002Ffractions-and-the-chinese-remainder-theorem":21057,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Feuclidean-domains-pids-ufds":21064,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fpolynomial-rings-over-fields":21033,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fgauss-lemma-and-unique-factorization":21065,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Firreducibility-criteria-and-groebner":21066,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fintroduction-to-modules":21067,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ffree-modules-and-direct-sums":21068,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ftensor-products-and-exact-sequences":21069,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fvector-spaces-and-linear-maps":21070,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fstructure-theorem-over-pids":21071,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Frational-canonical-form":21072,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fjordan-canonical-form":21073,"\u002Fabstract-algebra\u002Ffield-theory\u002Ffield-extensions-and-algebraic-elements":21074,"\u002Fabstract-algebra\u002Ffield-theory\u002Fstraightedge-and-compass-constructions":20707,"\u002Fabstract-algebra\u002Ffield-theory\u002Fsplitting-fields-and-algebraic-closure":21075,"\u002Fabstract-algebra\u002Ffield-theory\u002Fseparable-and-cyclotomic-extensions":21076,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fthe-galois-correspondence":20949,"\u002Fabstract-algebra\u002Fgalois-theory\u002Ffinite-fields":21077,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fcyclotomic-and-abelian-extensions":21078,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fgalois-groups-of-polynomials":21006,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fsolvability-by-radicals-and-the-quintic":21078,"\u002Fabstract-algebra\u002Fcapstone\u002Fcommutative-algebra-and-algebraic-geometry":21079,"\u002Fabstract-algebra\u002Fcapstone\u002Frepresentation-and-character-theory":21080,"\u002Fabstract-algebra":21081,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fatomic-spectra-rutherford":21082,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-model-hydrogen":21083,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fx-ray-spectra-franck-hertz":21084,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-sommerfeld-old-quantum-theory":21085,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fold-quantum-theory-limits-wkb":21086,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fschrodinger-3d-hydrogen":20998,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fhydrogen-wave-functions":21087,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fradial-equation-in-full":21088,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fsymmetry-degeneracy-runge-lenz":21089,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fexpectation-values-virial":21090,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fquantum-defects-alkali-spectra":21091,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Frydberg-atoms":21092,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Frelativistic-kinetic-correction":21093,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fspin-orbit-thomas-precession":20832,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdarwin-term-fine-structure-formula":20961,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdirac-equation-hydrogen":20707,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Flamb-shift-qed":21094,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fhyperfine-structure-21cm":20691,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fnuclear-effects-isotope-shift":21095,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fperiodic-table-atomic-spectra":21096,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fcentral-field-self-consistent":20729,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fidentical-particles-hartree-fock":21023,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhelium-two-electron-atom":21097,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fls-jj-coupling-term-symbols":21098,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhund-rules-ground-terms":21099,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fzeeman-effect":21100,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fpaschen-back-intermediate":21101,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fstark-effect-polarizability":21102,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Ftime-dependent-perturbation-golden-rule":21103,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fdipole-approximation-einstein-coefficients":21104,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fselection-rules-forbidden-transitions":21105,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Flifetimes-and-line-shapes":21106,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Flaser-principles":21107,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fspectroscopy-techniques":21108,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fline-catalog-nist-asd":21109,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Flaser-cooling-doppler":21110,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fsub-doppler-trapping":21054,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fbose-einstein-condensation":21111,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Foptical-clocks-precision":21112,"\u002Fatomic-physics":21113,"\u002Fdatabases":20878,"\u002Fcategory-theory\u002Ffoundations\u002Fwhat-is-a-category":21114,"\u002Fcategory-theory\u002Ffoundations\u002Fexamples-of-categories":21115,"\u002Fcategory-theory\u002Ffoundations\u002Fspecial-morphisms":21116,"\u002Fcategory-theory\u002Ffoundations\u002Ffunctors":21045,"\u002Fcategory-theory\u002Ffoundations\u002Fnatural-transformations":21117,"\u002Fcategory-theory\u002Ffoundations\u002Fsize-and-set-theory":21118,"\u002Fcategory-theory\u002Funiversal-properties\u002Funiversal-properties":21119,"\u002Fcategory-theory\u002Funiversal-properties\u002Fproducts-and-coproducts":21120,"\u002Fcategory-theory\u002Funiversal-properties\u002Fconstructions-on-categories":21121,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Frepresentable-functors":21122,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-lemma":21123,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-consequences":21124,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits":21125,"\u002Fcategory-theory\u002Flimits-colimits\u002Fproducts-equalizers-pullbacks":21126,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcolimits":21127,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcomputing-limits":21128,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits-and-functors":21129,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions":21130,"\u002Fcategory-theory\u002Fadjunctions\u002Funits-and-counits":21131,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions-via-universal-arrows":21132,"\u002Fcategory-theory\u002Fadjunctions\u002Ffree-forgetful-adjunctions":21133,"\u002Fcategory-theory\u002Fadjoints-limits\u002Flimits-via-adjoints":21134,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fpresheaf-limits-colimits":21135,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoints-preserve-limits":21136,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoint-functor-theorem":21127,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fmonads":21137,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-eilenberg-moore":21138,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fkleisli-and-programming":21139,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-for-endofunctors":21140,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Fcartesian-closed-categories":21141,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Flambda-calculus-correspondence":21086,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Ffixed-points-and-recursion":21142,"\u002Fcategory-theory":21143,"\u002Fdeep-learning\u002Fmathematical-background\u002Flinear-algebra-for-deep-learning":21144,"\u002Fdeep-learning\u002Fmathematical-background\u002Fprobability-and-information-theory":21145,"\u002Fdeep-learning\u002Fmathematical-background\u002Fnumerical-computation":21146,"\u002Fdeep-learning\u002Fmathematical-background\u002Fcalculus":21147,"\u002Fdeep-learning\u002Ffoundations\u002Fwhat-is-deep-learning":21148,"\u002Fdeep-learning\u002Ffoundations\u002Fmachine-learning-refresher":21149,"\u002Fdeep-learning\u002Ffoundations\u002Flinear-models-and-the-perceptron":21108,"\u002Fdeep-learning\u002Fneural-networks\u002Fthe-multilayer-perceptron":21150,"\u002Fdeep-learning\u002Fneural-networks\u002Factivation-functions":21151,"\u002Fdeep-learning\u002Fneural-networks\u002Funiversal-approximation":21152,"\u002Fdeep-learning\u002Fneural-networks\u002Fbackpropagation":21153,"\u002Fdeep-learning\u002Fneural-networks\u002Floss-functions-and-output-units":21154,"\u002Fdeep-learning\u002Foptimization\u002Fgradient-descent-and-sgd":21155,"\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods":21156,"\u002Fdeep-learning\u002Foptimization\u002Finitialization":21157,"\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape":21158,"\u002Fdeep-learning\u002Foptimization\u002Fsecond-order-and-approximate-methods":20588,"\u002Fdeep-learning\u002Fregularization\u002Fregularization-overview":21159,"\u002Fdeep-learning\u002Fregularization\u002Fdropout-and-data-augmentation":21160,"\u002Fdeep-learning\u002Fregularization\u002Fearly-stopping-and-parameter-sharing":21161,"\u002Fdeep-learning\u002Fregularization\u002Fnormalization":21162,"\u002Fdeep-learning\u002Farchitectures\u002Fconvolutional-networks":21163,"\u002Fdeep-learning\u002Farchitectures\u002Fcnn-architectures":21164,"\u002Fdeep-learning\u002Farchitectures\u002Frecurrent-networks":21165,"\u002Fdeep-learning\u002Farchitectures\u002Flstm-and-gru":21166,"\u002Fdeep-learning\u002Farchitectures\u002Fattention-and-transformers":21167,"\u002Fdeep-learning\u002Farchitectures\u002Fthe-transformer-architecture":21168,"\u002Fdeep-learning\u002Farchitectures\u002Ftransformers-in-practice":21169,"\u002Fdeep-learning\u002Farchitectures\u002Fgraph-neural-networks":21170,"\u002Fdeep-learning\u002Farchitectures\u002Fstate-space-models":21171,"\u002Fdeep-learning\u002Ftheory\u002Fgeneralization-theory":21172,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-robustness":21173,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-defenses":21174,"\u002Fdeep-learning\u002Ftheory\u002Fbayesian-and-ensemble-methods":21175,"\u002Fdeep-learning\u002Ftheory\u002Fdeep-equilibrium-models":21120,"\u002Fdeep-learning\u002Fgenerative-models\u002Flinear-factor-models":21176,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoencoders":21177,"\u002Fdeep-learning\u002Fgenerative-models\u002Fvariational-autoencoders":21178,"\u002Fdeep-learning\u002Fgenerative-models\u002Fgenerative-adversarial-networks":21179,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoregressive-and-normalizing-flows":21180,"\u002Fdeep-learning\u002Fgenerative-models\u002Fenergy-based-and-boltzmann-machines":21181,"\u002Fdeep-learning\u002Fgenerative-models\u002Fdiffusion-and-score-based-models":21182,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fstructured-probabilistic-models":20672,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fmonte-carlo-and-mcmc":21183,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fapproximate-inference":21184,"\u002Fdeep-learning\u002Fpractical\u002Fpractical-methodology":20908,"\u002Fdeep-learning\u002Fpractical\u002Fhyperparameters-and-debugging":21185,"\u002Fdeep-learning\u002Fpractical\u002Frepresentation-learning":21186,"\u002Fdeep-learning\u002Fpractical\u002Ftransfer-learning":21187,"\u002Fdeep-learning\u002Fpractical\u002Fapplications":21188,"\u002Fdeep-learning\u002Fpractical\u002Fmodel-compression-and-distillation":21189,"\u002Fdeep-learning\u002Fpractical\u002Fmeta-learning-and-few-shot":21190,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models":21191,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment":21192,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fseq2seq-pretraining-and-bart":21193,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ftext-to-text-transfer-and-conditional-generation":21194,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fspeech-and-audio-models":21195,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fself-supervised-speech-and-synthesis":21196,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents":20881,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fagent-memory-retrieval-and-orchestration":21197,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmixture-of-experts":21198,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmultimodal-models":21199,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ffusion-and-vision-language-models":21200,"\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning":20918,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fmodel-free-prediction-and-control":21201,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fdeep-q-networks":21202,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic":21203,"\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback":21204,"\u002Fdeep-learning":20878,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fequilibrium-state-variables-zeroth-law":21205,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Ffirst-law-heat-and-work":21003,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fsecond-law-entropy-and-the-carnot-bound":21206,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fthermodynamic-potentials-and-maxwell-relations":21207,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fstability-response-functions-and-the-third-law":21208,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fclassical-statistics-and-equipartition":21209,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fphase-space-and-liouvilles-theorem":21210,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fensembles-and-the-equal-probability-postulate":21211,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fstatistical-entropy-boltzmann-and-gibbs":21212,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fmicrocanonical-ensemble-and-entropy":21213,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fequilibrium-conditions-temperature-pressure-chemical-potential":21050,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fideal-gas-phase-space-and-the-sackur-tetrode-entropy":21214,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Ftwo-state-systems-paramagnets-and-negative-temperature":21215,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fcanonical-ensemble-and-the-boltzmann-distribution":21216,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fpartition-function-and-the-helmholtz-free-energy":20937,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fenergy-fluctuations-and-ensemble-equivalence":20730,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fthe-einstein-solid-and-harmonic-systems":21217,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fparamagnetism-and-the-schottky-anomaly":21218,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fideal-gas-partition-function-and-the-gibbs-paradox":21219,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fequipartition-and-the-virial-theorem":20864,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fmolecular-gases-rotation-and-vibration":21220,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fgrand-canonical-ensemble-and-the-grand-partition-function":21221,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fchemical-potential-fugacity-and-number-fluctuations":20942,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fensemble-summary-and-the-thermodynamic-web":20869,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fquantum-statistics-bose-einstein-and-fermi-dirac":21222,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fderiving-the-quantum-distributions":20838,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fthe-classical-limit-and-quantum-concentration":20708,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fideal-quantum-gases-general-framework":20724,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-and-the-fermion-gas":21223,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthe-photon-gas-and-plancks-radiation-law":21224,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fblackbody-thermodynamics-and-radiation-pressure":21225,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fphonons-and-the-debye-model":20856,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-derived":21226,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthermodynamics-of-the-bose-gas-and-superfluidity":20718,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fthe-ideal-fermi-gas-at-zero-temperature":21227,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fsommerfeld-expansion-and-electrons-in-metals":21228,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":21229,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fneutron-stars-and-nuclear-matter":21230,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-cluster-expansion-and-virial-coefficients":21052,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-van-der-waals-gas-and-liquid-gas-coexistence":21231,"\u002Fstatistical-mechanics\u002Finteractions\u002Fquantum-gases-with-interactions-and-exchange":21232,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fphases-coexistence-and-classification":21233,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-ising-model-and-exact-solutions":21234,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fmean-field-theory-and-the-weiss-model":20998,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fcritical-exponents-and-landau-theory":21235,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-renormalization-group-idea":20976,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fthermodynamic-fluctuations-and-response":21236,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fbrownian-motion-and-the-langevin-equation":20837,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Flinear-response-and-the-fluctuation-dissipation-theorem":21237,"\u002Fstatistical-mechanics":21238,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fbonding-mechanisms":21239,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fmolecular-orbitals-and-h2-plus":20703,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fhydrogen-molecule-and-exchange":20963,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fvan-der-waals-forces":21240,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Frotational-vibrational-spectra":21241,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Fanharmonicity-and-rovibrational-structure":21242,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Framan-and-electronic-bands":21243,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Flasers-and-masers":21244,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fstructure-of-solids":21245,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fbravais-lattices-and-crystal-systems":20836,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Freciprocal-lattice-and-brillouin-zones":21246,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fdiffraction-and-structure-factors":21247,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonon-dispersion":21248,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonons-quantization-and-dos":21249,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fdebye-einstein-heat-capacity":20979,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fanharmonicity-and-thermal-transport":21250,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ffree-electron-gas-and-conduction":21251,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fsommerfeld-model-and-heat-capacity":21252,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ftransport-and-the-hall-effect":21253,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fscreening-and-plasmons":21254,"\u002Fcondensed-matter\u002Fband-theory\u002Fblochs-theorem-and-energy-bands":21034,"\u002Fcondensed-matter\u002Fband-theory\u002Fnearly-free-electron-model":20978,"\u002Fcondensed-matter\u002Fband-theory\u002Ftight-binding-method":21255,"\u002Fcondensed-matter\u002Fband-theory\u002Ffermi-surfaces-and-semiclassical-dynamics":21256,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fsemiconductor-bands-and-junctions":21257,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fintrinsic-and-extrinsic-semiconductors":21258,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fcarrier-transport-and-recombination":20969,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fthe-pn-junction":21259,"\u002Fcondensed-matter\u002Fsemiconductors\u002Ftransistors-and-optoelectronics":21260,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fdielectrics-and-polarization":21205,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fferroelectrics-and-piezoelectrics":21106,"\u002Fcondensed-matter\u002Fmagnetism\u002Fdiamagnetism-and-paramagnetism":20833,"\u002Fcondensed-matter\u002Fmagnetism\u002Fexchange-and-ferromagnetism":21261,"\u002Fcondensed-matter\u002Fmagnetism\u002Fantiferromagnetism-and-domains":20699,"\u002Fcondensed-matter\u002Fmagnetism\u002Fspin-waves-and-magnons":21262,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fsuperconductivity-phenomenology":21263,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Flondon-theory-and-the-meissner-effect":20844,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fginzburg-landau-theory":21264,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fbcs-theory":21099,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fjosephson-and-high-tc":21265,"\u002Fcondensed-matter\u002Fnanostructures\u002Fquantum-wells-wires-and-dots":20691,"\u002Fcondensed-matter\u002Fnanostructures\u002Finteger-quantum-hall-effect":21266,"\u002Fcondensed-matter\u002Fnanostructures\u002Ffractional-quantum-hall-and-topology":20702,"\u002Fcondensed-matter\u002Fnanostructures\u002Fgraphene-and-dirac-materials":21267,"\u002Fcondensed-matter":21019,"\u002Flogic\u002Ffoundations\u002Flogic-as-a-mathematical-model":21268,"\u002Flogic\u002Fsentential-logic\u002Fformal-languages-and-well-formed-formulas":21269,"\u002Flogic\u002Fsentential-logic\u002Ftruth-assignments-and-tautologies":21270,"\u002Flogic\u002Fsentential-logic\u002Funique-readability-and-parsing":21271,"\u002Flogic\u002Fsentential-logic\u002Finduction-and-recursion":20716,"\u002Flogic\u002Fsentential-logic\u002Fexpressive-completeness-and-normal-forms":21272,"\u002Flogic\u002Fsentential-logic\u002Fboolean-circuits":21273,"\u002Flogic\u002Fsentential-logic\u002Fcompactness-and-effectiveness":20716,"\u002Flogic\u002Ffirst-order-languages\u002Ffirst-order-languages":21274,"\u002Flogic\u002Ffirst-order-languages\u002Fstructures-truth-and-satisfaction":21129,"\u002Flogic\u002Ffirst-order-languages\u002Fdefinability-and-elementary-equivalence":21275,"\u002Flogic\u002Ffirst-order-languages\u002Fterms-substitution-and-parsing":21276,"\u002Flogic\u002Fdeductive-calculus\u002Fa-deductive-calculus":21277,"\u002Flogic\u002Fdeductive-calculus\u002Fdeduction-theorem-and-derived-rules":21275,"\u002Flogic\u002Fdeductive-calculus\u002Fsoundness":21278,"\u002Flogic\u002Fdeductive-calculus\u002Fcompleteness-and-consistency":21279,"\u002Flogic\u002Fmodels-and-theories\u002Fcompactness-and-lowenheim-skolem":21280,"\u002Flogic\u002Fmodels-and-theories\u002Ftheories-elementary-classes-and-categoricity":21281,"\u002Flogic\u002Fmodels-and-theories\u002Finterpretations-between-theories":21282,"\u002Flogic\u002Fmodels-and-theories\u002Fnonstandard-analysis":21283,"\u002Flogic\u002Farithmetic-and-definability\u002Fdefinability-in-arithmetic":21284,"\u002Flogic\u002Farithmetic-and-definability\u002Fnatural-numbers-with-successor":21285,"\u002Flogic\u002Farithmetic-and-definability\u002Fpresburger-and-reducts":21206,"\u002Flogic\u002Farithmetic-and-definability\u002Fa-subtheory-and-representability":21286,"\u002Flogic\u002Fincompleteness\u002Farithmetization-of-syntax":21279,"\u002Flogic\u002Fincompleteness\u002Fincompleteness-and-undecidability":21287,"\u002Flogic\u002Fincompleteness\u002Fsecond-incompleteness-theorem":21288,"\u002Flogic\u002Fcomputability-and-representability\u002Frecursive-functions":20922,"\u002Flogic\u002Fcomputability-and-representability\u002Frepresenting-exponentiation":21289,"\u002Flogic\u002Fsecond-order-logic\u002Fsecond-order-languages":21107,"\u002Flogic\u002Fsecond-order-logic\u002Fskolem-functions-and-many-sorted-logic":21290,"\u002Flogic\u002Fsecond-order-logic\u002Fgeneral-structures":21291,"\u002Flogic":21292,"\u002Freinforcement-learning\u002Ffoundations\u002Fwhat-is-reinforcement-learning":21293,"\u002Freinforcement-learning\u002Ffoundations\u002Fa-brief-history-of-rl":21294,"\u002Freinforcement-learning\u002Ffoundations\u002Fmulti-armed-bandits":20908,"\u002Freinforcement-learning\u002Ffoundations\u002Fbandit-exploration-algorithms":21295,"\u002Freinforcement-learning\u002Ffoundations\u002Fmarkov-decision-processes":21296,"\u002Freinforcement-learning\u002Ffoundations\u002Fvalue-functions-and-optimality":21297,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdynamic-programming":21298,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdp-async-and-gpi":21288,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods":21299,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-off-policy":21300,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftemporal-difference-learning":21301,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftd-control-sarsa-and-q-learning":21199,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping":21302,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-off-policy-methods":21303,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-and-learning":21304,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-focusing-and-decision-time":21305,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdecision-time-planning":21306,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-tree-search":20925,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-prediction":21307,"\u002Freinforcement-learning\u002Fapproximation\u002Ffeature-construction-and-nonlinear":21308,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-control":21309,"\u002Freinforcement-learning\u002Fapproximation\u002Faverage-reward-control":21310,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad":21116,"\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td":21311,"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces":21287,"\u002Freinforcement-learning\u002Fapproximation\u002Ftrue-online-and-sarsa-lambda":21312,"\u002Freinforcement-learning\u002Fapproximation\u002Fpolicy-gradient-methods":21313,"\u002Freinforcement-learning\u002Fapproximation\u002Factor-critic-and-continuous-actions":21314,"\u002Freinforcement-learning\u002Fapproximation\u002Fleast-squares-and-memory-based-methods":20936,"\u002Freinforcement-learning\u002Fapproximation\u002Fmemory-and-kernel-methods":21315,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces":21066,"\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces":21316,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks":21317,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdqn-improvements":20898,"\u002Freinforcement-learning\u002Fdeep-rl\u002Factor-critic-and-ppo":21318,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fppo-and-continuous-control":21319,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fcase-studies":21320,"\u002Freinforcement-learning\u002Fdeep-rl\u002Frl-beyond-games":21321,"\u002Freinforcement-learning\u002Fdeep-rl\u002Ffrontiers":21322,"\u002Freinforcement-learning\u002Fdeep-rl\u002Freward-design-and-open-problems":21175,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow":21323,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow-part-2":21324,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control":21325,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control-part-2":21210,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl":21326,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl-part-2":21327,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration":21328,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration-part-2":20925,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl":20992,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2":21329,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl":21330,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl-part-2":21331,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl":21332,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl-part-2":21333,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl":21334,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl-part-2":21335,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Frlhf-and-language-models":21336,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps":21337,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps-part-2":21338,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl":21339,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl-part-2":21340,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmeta-rl-and-generalization":21341,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fpsychology-of-reinforcement":21342,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Finstrumental-conditioning-and-control":21343,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-and-td-error":21344,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-in-the-brain":21345,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fanimal-learning-and-cognition":21346,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fcognitive-maps-and-planning":21347,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fneuroscience-of-reinforcement":21348,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fseveral-learning-systems":21349,"\u002Freinforcement-learning":20878,"\u002Fartificial-intelligence\u002Ffoundations\u002Fwhat-is-ai":21350,"\u002Fartificial-intelligence\u002Ffoundations\u002Ffoundations-of-ai":21351,"\u002Fartificial-intelligence\u002Ffoundations\u002Fintelligent-agents":21352,"\u002Fartificial-intelligence\u002Ffoundations\u002Fagent-architectures":21353,"\u002Fartificial-intelligence\u002Fsearch\u002Funinformed-search":21354,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-strategies-compared":21355,"\u002Fartificial-intelligence\u002Fsearch\u002Finformed-search":21356,"\u002Fartificial-intelligence\u002Fsearch\u002Fheuristic-functions":21357,"\u002Fartificial-intelligence\u002Fsearch\u002Flocal-search":21358,"\u002Fartificial-intelligence\u002Fsearch\u002Fpopulation-and-continuous-search":21359,"\u002Fartificial-intelligence\u002Fsearch\u002Fadversarial-search":21360,"\u002Fartificial-intelligence\u002Fsearch\u002Fgames-of-chance-and-imperfect-information":21361,"\u002Fartificial-intelligence\u002Fsearch\u002Fconstraint-satisfaction":21362,"\u002Fartificial-intelligence\u002Fsearch\u002Fcsp-search-and-structure":21204,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-under-uncertainty":21060,"\u002Fartificial-intelligence\u002Fsearch\u002Fbelief-state-and-online-search":21363,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-logic":21364,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-inference":21365,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic":21366,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic-in-use":21367,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Finference-and-resolution":21368,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-resolution":21186,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fclassical-planning":21369,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-graphs-and-graphplan":21370,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-in-the-real-world":21371,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-under-uncertainty":21372,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fknowledge-representation":21373,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Freasoning-systems-and-defaults":21374,"\u002Fartificial-intelligence\u002Funcertainty\u002Fprobability-and-bayes":21375,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayes-rule-and-naive-bayes":21376,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayesian-networks":21377,"\u002Fartificial-intelligence\u002Funcertainty\u002Finference-in-bayesian-networks":21378,"\u002Fartificial-intelligence\u002Funcertainty\u002Freasoning-over-time":21379,"\u002Fartificial-intelligence\u002Funcertainty\u002Ftracking-and-data-association":21380,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmaking-decisions":21119,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmarkov-decision-processes":21370,"\u002Fartificial-intelligence\u002Funcertainty\u002Fdecision-networks-and-game-theory":21381,"\u002Fartificial-intelligence\u002Funcertainty\u002Fgame-theory-and-mechanism-design":20659,"\u002Fartificial-intelligence\u002Flearning\u002Flearning-from-examples":21382,"\u002Fartificial-intelligence\u002Flearning\u002Ftheory-and-model-families":21383,"\u002Fartificial-intelligence\u002Flearning\u002Fprobabilistic-learning":21384,"\u002Fartificial-intelligence\u002Flearning\u002Fexpectation-maximization":21385,"\u002Fartificial-intelligence\u002Flearning\u002Freinforcement-learning":21386,"\u002Fartificial-intelligence\u002Flearning\u002Fgeneralization-and-policy-search":21167,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-in-learning":21387,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-based-learning-methods":21388,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fvision-and-perception":21389,"\u002Fartificial-intelligence\u002Ffrontiers\u002Freconstructing-the-3d-world":21390,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobotics":21391,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobot-planning-and-control":21392,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnatural-language-in-ai":21393,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnlp-grammar-translation-and-speech":21394,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fphilosophy-and-future":21395,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fai-ethics-and-future":21396,"\u002Fartificial-intelligence":20878,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-constituents-nuclide-chart":21135,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-size-charge-distributions":21397,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-masses-binding-energy":20695,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fsemi-empirical-mass-formula":20693,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-moments-multipoles":21228,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnuclear-force-shell-overview":21398,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fthe-deuteron":20721,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnucleon-nucleon-scattering":21031,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fmeson-theory-isospin":21399,"\u002Fnuclear-physics\u002Fnuclear-models\u002Ffermi-gas-model":21400,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fliquid-drop-collective-coordinates":21401,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fshell-model-single-particle":21091,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fcollective-model-rotations-vibrations":21402,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-law-modes":21403,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-kinetics-equilibrium":21404,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-decay-gamow-theory":21289,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-fine-structure-hindrance":21405,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fbeta-decay-energetics-neutrino":21406,"\u002Fnuclear-physics\u002Fbeta-decay\u002Ffermi-theory-beta-decay":20705,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fweak-interaction-parity-violation":20934,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fdouble-beta-decay-neutrino-mass":21407,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fgamma-multipole-radiation":21035,"\u002Fnuclear-physics\u002Fgamma-decay\u002Finternal-conversion-isomers":21408,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fangular-correlations-mossbauer":21409,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Freaction-kinematics-cross-sections":20930,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fcompound-nucleus-resonances":21018,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fdirect-reactions-optical-model":21410,"\u002Fnuclear-physics\u002Ffission\u002Ffission-barrier-dynamics":21411,"\u002Fnuclear-physics\u002Ffission\u002Fchain-reactions-reactor-physics":21412,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Ffusion-reactions-confinement":20700,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fstellar-nucleosynthesis":21075,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fbig-bang-nucleosynthesis":20937,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fcharged-particle-stopping-power":21413,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fphoton-neutron-interactions":20985,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fradiation-detectors":21049,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fdosimetry-radiation-biology":21414,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fnuclear-applications-dating-medicine":21415,"\u002Fnuclear-physics":21416,"\u002Fnatural-language-processing\u002Ffoundations\u002Fwhat-is-nlp":21417,"\u002Fnatural-language-processing\u002Ffoundations\u002Fregex-and-text-normalization":21418,"\u002Fnatural-language-processing\u002Ffoundations\u002Fminimum-edit-distance":21056,"\u002Fnatural-language-processing\u002Ffoundations\u002Fn-gram-language-models":21419,"\u002Fnatural-language-processing\u002Ffoundations\u002Fsmoothing-and-backoff":21420,"\u002Fnatural-language-processing\u002Fclassification\u002Fnaive-bayes-and-sentiment":21421,"\u002Fnatural-language-processing\u002Fclassification\u002Fevaluating-classifiers":20904,"\u002Fnatural-language-processing\u002Fclassification\u002Flogistic-regression":21422,"\u002Fnatural-language-processing\u002Fclassification\u002Fsentiment-and-affect-lexicons":21423,"\u002Fnatural-language-processing\u002Fsemantics\u002Fvector-semantics-and-embeddings":21200,"\u002Fnatural-language-processing\u002Fsemantics\u002Fstatic-word-embeddings":21424,"\u002Fnatural-language-processing\u002Fsemantics\u002Fneural-language-models":21369,"\u002Fnatural-language-processing\u002Fsequences\u002Fsequence-labeling":21425,"\u002Fnatural-language-processing\u002Fsequences\u002Fcrfs-and-neural-taggers":21426,"\u002Fnatural-language-processing\u002Fsequences\u002Frnns-and-lstms":21427,"\u002Fnatural-language-processing\u002Ftransformers\u002Ftransformers-and-attention":21428,"\u002Fnatural-language-processing\u002Ftransformers\u002Fthe-transformer-architecture":21429,"\u002Fnatural-language-processing\u002Ftransformers\u002Flarge-language-models":21430,"\u002Fnatural-language-processing\u002Ftransformers\u002Fllm-pretraining-and-scaling":21431,"\u002Fnatural-language-processing\u002Ftransformers\u002Ffine-tuning-and-prompting":20882,"\u002Fnatural-language-processing\u002Ftransformers\u002Fprompting-and-alignment":21432,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-parsing":21433,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcky-scoring-and-evaluation":21375,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdependency-parsing":21434,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fgraph-based-and-neural-dependency-parsing":21435,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fword-senses-and-wsd":21436,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fwsd-in-practice-and-induction":21437,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-roles-and-information-extraction":21438,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Frelations-events-and-templates":21439,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoreference-and-discourse":21440,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoherence-and-discourse-structure":21441,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Flogical-semantics":21299,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcompositional-semantics-and-description-logics":21442,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-parsing":21443,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fneural-semantic-parsing":21444,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Finformation-extraction":21445,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftimes-events-and-templates":21446,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdiscourse-coherence":21447,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fentity-based-and-global-coherence":21448,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-grammars":21449,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftreebanks-and-lexicalized-grammars":21450,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation":21451,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation-decoding-and-evaluation":21452,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering":21453,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering-knowledge-and-llms":21154,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-and-chatbots":21320,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-systems-and-assistants":20910,"\u002Fnatural-language-processing\u002Fapplications\u002Ftext-summarization":21454,"\u002Fnatural-language-processing\u002Fapplications\u002Fabstractive-summarization-and-evaluation":21455,"\u002Fnatural-language-processing\u002Fspeech\u002Fphonetics":21456,"\u002Fnatural-language-processing\u002Fspeech\u002Facoustic-phonetics":21457,"\u002Fnatural-language-processing\u002Fspeech\u002Fautomatic-speech-recognition":21167,"\u002Fnatural-language-processing\u002Fspeech\u002Fasr-evaluation-and-applications":21458,"\u002Fnatural-language-processing":20878,"\u002Fparticle-physics\u002Ffoundations\u002Fhistorical-overview-particle-zoo":21459,"\u002Fparticle-physics\u002Ffoundations\u002Fparticle-physics-basic-concepts":21076,"\u002Fparticle-physics\u002Ffoundations\u002Ffundamental-interactions-force-carriers":21460,"\u002Fparticle-physics\u002Funits-kinematics\u002Fnatural-units-and-scales":21220,"\u002Fparticle-physics\u002Funits-kinematics\u002Ffour-vectors-invariant-mass":21461,"\u002Fparticle-physics\u002Funits-kinematics\u002Fdecay-scattering-kinematics-mandelstam":21462,"\u002Fparticle-physics\u002Funits-kinematics\u002Fcross-sections-golden-rule":21410,"\u002Fparticle-physics\u002Fsymmetries\u002Fconservation-laws-symmetries":21463,"\u002Fparticle-physics\u002Fsymmetries\u002Fdiscrete-symmetries-cpt":21464,"\u002Fparticle-physics\u002Fsymmetries\u002Fparity-violation-weak":21026,"\u002Fparticle-physics\u002Fsymmetries\u002Fsu2-su3-flavor-symmetry":20728,"\u002Fparticle-physics\u002Fquark-model\u002Feightfold-way-su3":21465,"\u002Fparticle-physics\u002Fquark-model\u002Fmeson-spectroscopy":20980,"\u002Fparticle-physics\u002Fquark-model\u002Fbaryon-spectroscopy":21466,"\u002Fparticle-physics\u002Fquark-model\u002Fcolor-confinement-exotics":21013,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fklein-gordon-equation":21467,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fdirac-equation-spinors":21266,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fantiparticles-hole-theory":21468,"\u002Fparticle-physics\u002Fqed\u002Ffeynman-rules-qed":21469,"\u002Fparticle-physics\u002Fqed\u002Fqed-tree-processes":21235,"\u002Fparticle-physics\u002Fqed\u002Frenormalization-running-coupling":21470,"\u002Fparticle-physics\u002Fqed\u002Felectron-g-2":20699,"\u002Fparticle-physics\u002Fweak-interaction\u002Fva-structure-weak":21471,"\u002Fparticle-physics\u002Fweak-interaction\u002Fw-z-bosons-decays":21472,"\u002Fparticle-physics\u002Fweak-interaction\u002Fckm-matrix":21473,"\u002Fparticle-physics\u002Fweak-interaction\u002Fcp-violation-kaons-b-mesons":21017,"\u002Fparticle-physics\u002Fqcd\u002Fcolor-su3-gluons":21232,"\u002Fparticle-physics\u002Fqcd\u002Fasymptotic-freedom-confinement":21474,"\u002Fparticle-physics\u002Fqcd\u002Fdeep-inelastic-scattering-partons":21475,"\u002Fparticle-physics\u002Fqcd\u002Fjets-hadronization":21092,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Felectroweak-su2-u1":21476,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fspontaneous-symmetry-breaking":21137,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-mechanism":21477,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-boson-discovery":21478,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fstandard-model":21026,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-oscillations":21059,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-mass-pmns":21479,"\u002Fparticle-physics\u002Fneutrinos\u002Fdirac-majorana-experiments":21480,"\u002Fparticle-physics\u002Fexperiment\u002Faccelerators-luminosity":21481,"\u002Fparticle-physics\u002Fexperiment\u002Fdetectors-subsystems":21115,"\u002Fparticle-physics\u002Fexperiment\u002Fhow-discoveries-are-made":21482,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fbeyond-standard-model":20856,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fgrand-unified-theories":21483,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fsupersymmetry":20959,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fhierarchy-problem-naturalness":21484,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fdark-matter-candidates":21485,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fmatter-antimatter-open-questions":21282,"\u002Fparticle-physics":21486,"\u002Fastrophysics-cosmology\u002Forientation\u002Fthe-sun-and-stars":21077,"\u002Fastrophysics-cosmology\u002Forientation\u002Fstellar-death-final-states":21230,"\u002Fastrophysics-cosmology\u002Forientation\u002Fgalaxies-and-cosmology":21487,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fmagnitudes-fluxes-and-the-distance-modulus":21016,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fstellar-spectra-and-spectral-classification":21488,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Ftelescopes-and-detectors-across-the-spectrum":21076,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fthe-cosmic-distance-ladder":20988,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fblackbody-radiation-and-specific-intensity":21489,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fradiative-transfer-and-the-transfer-equation":21490,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fspectral-line-formation-and-broadening":21491,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fopacity-and-the-rosseland-mean":21492,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fhydrostatic-equilibrium-and-the-virial-theorem":21493,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equations-of-stellar-structure":21281,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equation-of-state-and-polytropes":21212,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-standard-solar-model":21494,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fthermonuclear-reaction-rates-and-the-gamow-peak":21495,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhydrogen-burning-pp-chains-and-cno":21496,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhelium-burning-and-the-triple-alpha-process":21497,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fadvanced-burning-and-neutron-capture-nucleosynthesis":21409,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fphases-of-the-interstellar-medium":21498,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fmolecular-clouds-and-gravitational-collapse":20972,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fprotostars-and-the-pre-main-sequence":21035,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-main-sequence-and-its-structure":21017,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fpost-main-sequence-low-mass-evolution":21499,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-evolution-of-massive-stars":21101,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fstellar-pulsation-and-the-instability-strip":20717,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":21500,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fcore-collapse-supernovae":21501,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fthermonuclear-supernovae-type-ia":21231,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fneutron-stars-and-pulsars":21502,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fblack-holes-schwarzschild-and-kerr":21503,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fbinary-systems-and-mass-transfer":21504,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Faccreting-compact-objects":21505,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fgravitational-waves-from-inspiraling-binaries":20698,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fmultimessenger-astronomy-and-gamma-ray-bursts":21506,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fthe-milky-way":21507,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-morphology-and-classification":20959,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-rotation-curves-and-dark-matter":21508,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Factive-galactic-nuclei-and-supermassive-black-holes":21509,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-clusters-and-large-scale-structure":21224,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-expanding-universe-and-hubbles-law":20880,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-frw-metric-and-cosmological-redshift":21510,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-friedmann-equations-and-cosmic-dynamics":21511,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fcosmological-models-and-distances":21137,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fdark-energy-and-the-accelerating-universe":21123,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fthe-thermal-history-of-the-universe":21135,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fbig-bang-nucleosynthesis":21231,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Frecombination-and-the-cosmic-microwave-background":21512,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcmb-anisotropies-and-cosmological-parameters":20697,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcosmic-inflation":20889,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fstructure-formation-and-the-growth-of-perturbations":21513,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fdark-matter-dark-energy-and-open-questions":20976,"\u002Fastrophysics-cosmology":21081,"\u002Fcolophon":21514,"\u002F":20878},4250,4808,3626,2682,4109,4786,3878,3875,3751,3415,4067,3153,3000,4042,5461,5808,3961,3749,4327,5067,4246,4655,4154,5436,2640,4003,3601,2158,4331,4189,2273,3252,4633,4964,4172,3131,5524,3160,4031,2309,4207,3226,2648,4842,5340,3307,5701,4977,4039,2615,3472,4460,3848,4075,4400,3382,3010,3602,3737,3740,3707,3922,5191,4043,3804,4542,4214,5062,2850,4361,3443,3627,4044,3766,4140,3860,4006,5199,4334,5234,3651,5509,5680,153,1375,1073,1093,1125,1146,1014,1132,876,1541,1189,1173,984,1402,1301,950,1268,1063,1107,1408,1161,925,1012,866,964,1090,1142,1085,1020,1207,973,980,728,764,1225,1329,796,929,801,878,774,1044,1488,1175,1130,890,814,870,154,4073,5140,4961,5127,4870,5382,5195,4955,5369,4501,5576,3824,4132,4289,4307,4570,3403,5084,5105,5201,5116,5341,5175,5368,5188,5211,5499,5155,4981,5125,5415,5255,5304,5130,5167,5552,5164,5094,5239,5036,5190,5004,5099,5035,5159,5088,5026,4937,5023,5264,5244,133,5114,5078,5043,5312,5170,5342,5139,5151,5049,5212,5013,5068,5079,5102,5121,5081,5029,5379,5854,5110,2139,3798,5055,5364,4984,4935,4895,4972,5289,5112,5156,4987,5031,5025,5149,5302,5042,5002,4979,4922,4960,5279,126,1877,1180,1129,907,958,1112,1300,1053,1250,1181,1241,1234,966,1050,734,1190,484,1082,926,733,761,571,607,798,804,952,977,731,784,645,771,1017,742,1004,1000,1562,1254,1288,1101,1011,1486,1061,856,992,1169,988,137,0,2037,1782,2384,2254,2123,2332,1643,1714,2089,1751,1367,1660,2511,1998,1892,1854,1791,2438,2487,1917,2375,2525,2266,1845,2275,1810,1631,2310,2166,2233,2113,2505,2347,2672,2112,2473,2592,2380,3013,2513,3256,3218,2194,2173,2205,2326,2081,3342,3152,1799,1670,1027,960,1095,1291,986,897,1209,1055,1817,1801,1593,1465,1196,1464,1201,1230,1435,1684,1461,1926,1500,1409,1284,1774,1869,162,1487,1122,1188,1001,1351,982,1005,979,1325,1046,943,1279,824,1008,989,1798,1277,1025,987,1043,1211,1074,981,939,1002,739,1139,1108,1013,1070,978,1458,1317,157,1357,1077,2355,1116,1037,1178,1637,1314,1109,1056,1702,1474,1071,1158,832,993,1404,1024,1068,1339,1106,1264,1248,913,1848,1328,1633,1224,1143,135,1378,959,1028,998,911,1527,1203,1266,1483,1165,990,938,965,1257,1418,1099,942,1352,956,1035,1398,1003,1094,1292,138,1721,1827,1449,1354,1148,1184,1285,1281,1213,1290,1271,1252,1274,1778,1591,1503,1437,1571,1584,1957,1117,1781,1648,1342,1667,1510,1965,1607,1365,1849,1259,1303,1356,1238,2208,1564,173,1671,1286,1227,1638,1529,668,1078,918,709,865,880,940,1534,1015,874,922,841,794,1194,822,1105,1658,1359,1296,1438,1921,1844,1570,1429,1324,1400,140,1787,1558,1654,1492,1747,2224,2002,2009,1323,1349,1785,1573,1722,1829,1353,1548,1552,1583,1624,1585,1245,1364,1514,1343,1397,1355,2211,1481,1770,160,2388,2293,2256,2552,2569,2478,2039,2496,2578,2814,2519,2461,2587,2492,2714,2654,3050,2447,2849,2238,2369,2061,2214,2602,2563,2186,2985,2749,3364,2038,2282,2409,2126,2573,2206,2176,2268,2182,2402,2705,2633,2414,2213,2801,3313,3410,3195,1952,2017,1509,2537,2645,2027,2415,2838,2356,1906,3184,2950,2807,2954,1683,1316,1034,1138,1763,1822,1705,1246,1701,1097,1104,1187,1032,1083,1228,916,1489,1033,1652,997,692,837,1023,888,864,1089,1231,1214,1675,1156,1075,1520,1309,139,1205,1051,735,1123,1072,915,567,768,825,1253,983,1007,762,1058,861,862,971,1208,1149,1145,1029,1084,927,810,838,857,807,936,949,2321,1622,1069,1113,1057,854,1958,1528,1618,2049,1432,1679,1796,1685,1346,1275,1476,1505,1610,2018,1599,1215,1838,1909,132,3902,2215,2240,3266,3208,3073,2454,2969,2451,1875,2728,1884,2371,2516,2842,1690,1904,2346,3146,1386,2607,1966,2668,1665,2885,1606,2577,3074,2869,2403,2433,2082,1939,1587,2460,2747,2032,2642,1619,3123,1993,2090,2339,3829,1737,2622,2340,2322,3828,4409,2305,3411,2510,4527,3030,3569,3043,2457,1946,2277,2044,2909,1693,1945,2093,2399,2115,2898,2742,2242,3895,3378,3376,2769,2223,3062,3262,2651,2949,2768,3128,2423,1977,2087,2866,3388,2830,2210,2489,2884,3945,2099,2713,3402,1692,2931,4195,3989,3206,4391,3004,3704,3494,2902,999,881,901,919,748,869,1018,1045,1049,1333,954,1092,1019,976,1771,1480,1396,953,1026,161,3533,2495,1818,3007,2595,3427,3537,2216,1895,2304,3396,1739,2073,1962,2203,1767,2666,2264,2276,2852,1807,3735,1560,4144,1669,1676,1972,2418,3291,1525,2040,2766,2337,2220,2800,3001,2078,1759,2836,1896,2026,1758,1543,1047,896,946,1060,1384,1482,815,1414,1322,1440,1240,1468,1098,1133,847,1009,1381,1052,1191,1258,1370,1712,1441,1199,957,1079,150,1262,1417,1368,1219,1136,1064,1463,1636,1059,931,1115,1736,1174,1376,1363,1411,1247,1746,1313,1299,1617,1102,1076,1495,1265,1193,1263,80,{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":21516,"\u002Falgorithms\u002Ffoundations\u002Fproof-techniques":21521,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":21525,"\u002Falgorithms\u002Ffoundations\u002Fgrowth-rates-and-loop-analysis":21529,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":21533,"\u002Falgorithms\u002Ffoundations\u002Famortized-analysis":21537,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":21541,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":21546,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":21550,"\u002Falgorithms\u002Fdivide-and-conquer\u002Ffast-multiplication":21554,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":21558,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":21563,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":21567,"\u002Falgorithms\u002Fsorting\u002Fexternal-sorting":21571,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":21575,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":21580,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":21584,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":21588,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":21592,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":21596,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":21600,"\u002Falgorithms\u002Fdata-structures\u002Fspatial-data-structures":21604,"\u002Falgorithms\u002Fdata-structures\u002Fskip-lists-and-probabilistic-structures":21608,"\u002Falgorithms\u002Fdata-structures\u002Fb-trees":21612,"\u002Falgorithms\u002Fdata-structures\u002Fdata-stream-algorithms":21616,"\u002Falgorithms\u002Fdata-structures\u002Fstreaming-sketches":21620,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":21624,"\u002Falgorithms\u002Fsequences\u002Fprefix-sums":21629,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":21633,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":21637,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":21641,"\u002Falgorithms\u002Fsequences\u002Fkmp-and-z-function":21645,"\u002Falgorithms\u002Fsequences\u002Ftries":21649,"\u002Falgorithms\u002Fsequences\u002Fsuffix-arrays-and-aho-corasick":21653,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":21657,"\u002Falgorithms\u002Fgraphs\u002Fdepth-first-search":21662,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":21666,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":21670,"\u002Falgorithms\u002Fgraphs\u002Fkruskal-and-prim":21674,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":21678,"\u002Falgorithms\u002Fgraphs\u002Fall-pairs-and-negative-weights":21682,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":21686,"\u002Falgorithms\u002Fgraphs\u002Fmax-flow-min-cut":21690,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":21694,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":21698,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":21702,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":21706,"\u002Falgorithms\u002Fgraphs\u002Fbipartite-matching":21710,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":21714,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":21719,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":21723,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":21727,"\u002Falgorithms\u002Fgreedy\u002Fstable-matching":21731,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":21735,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":21740,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":21744,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":21748,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":21752,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":21756,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":21760,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":21764,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":21768,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":21772,"\u002Falgorithms\u002Fdynamic-programming\u002Fdigit-and-probability-dp":21776,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":21780,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":21785,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":21789,"\u002Falgorithms\u002Fbacktracking\u002Fgraph-backtracking":21793,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":21797,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":21802,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":21806,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":21810,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmatrix-exponentiation":21814,"\u002Falgorithms\u002Fmathematical-algorithms\u002Ffast-fourier-transform":21818,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fgradient-descent":21822,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":21826,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":21831,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":21835,"\u002Falgorithms\u002Fcomputational-geometry\u002Fpolygons-and-proximity":21839,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":21843,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":21848,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":21852,"\u002Falgorithms\u002Fintractability\u002Fapproximation-algorithms":21856,"\u002Falgorithms":21860,"\u002Fcalculus\u002Flimits-and-continuity\u002Ffunctions-and-models":21863,"\u002Fcalculus\u002Flimits-and-continuity\u002Fthe-limit-of-a-function":21868,"\u002Fcalculus\u002Flimits-and-continuity\u002Flimit-laws-and-the-precise-definition":21872,"\u002Fcalculus\u002Flimits-and-continuity\u002Fcontinuity":21876,"\u002Fcalculus\u002Fderivatives\u002Fthe-derivative-and-rates-of-change":21880,"\u002Fcalculus\u002Fderivatives\u002Fdifferentiation-rules-and-the-chain-rule":21885,"\u002Fcalculus\u002Fderivatives\u002Fimplicit-differentiation-and-related-rates":21889,"\u002Fcalculus\u002Fderivatives\u002Flinear-approximations-and-differentials":21893,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fextrema-and-the-mean-value-theorem":21897,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fhow-derivatives-shape-a-graph":21902,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fcurve-sketching-and-optimization":21906,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fnewtons-method-and-antiderivatives":21910,"\u002Fcalculus\u002Fintegrals\u002Farea-and-the-definite-integral":21914,"\u002Fcalculus\u002Fintegrals\u002Fthe-fundamental-theorem-of-calculus":21919,"\u002Fcalculus\u002Fintegrals\u002Fthe-substitution-rule":21923,"\u002Fcalculus\u002Fapplications-of-integration\u002Fareas-and-volumes":21927,"\u002Fcalculus\u002Fapplications-of-integration\u002Fwork-average-value-and-arc-length":21932,"\u002Fcalculus\u002Fapplications-of-integration\u002Fphysics-economics-and-probability":21936,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Finverse-functions-logarithms-and-exponentials":21940,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Fgrowth-decay-inverse-trig-and-hyperbolic-functions":21945,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Flhospitals-rule":21949,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fintegration-by-parts":21953,"\u002Fcalculus\u002Ftechniques-of-integration\u002Ftrigonometric-integrals-and-substitution":21958,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fpartial-fractions-and-integration-strategy":21962,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fapproximate-and-improper-integrals":21966,"\u002Fcalculus\u002Fparametric-and-polar\u002Fparametric-curves-and-their-calculus":21970,"\u002Fcalculus\u002Fparametric-and-polar\u002Fpolar-coordinates":21975,"\u002Fcalculus\u002Fparametric-and-polar\u002Fconic-sections":21979,"\u002Fcalculus\u002Fsequences-and-series\u002Fsequences":21983,"\u002Fcalculus\u002Fsequences-and-series\u002Fseries-and-the-integral-test":21988,"\u002Fcalculus\u002Fsequences-and-series\u002Fthe-convergence-tests":21992,"\u002Fcalculus\u002Fsequences-and-series\u002Fpower-series":21996,"\u002Fcalculus\u002Fsequences-and-series\u002Ftaylor-and-maclaurin-series":22000,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvectors-and-the-dot-product":22004,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fthe-cross-product-lines-and-planes":22009,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fcylinders-and-quadric-surfaces":22013,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvector-functions-and-space-curves":22017,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Farc-length-curvature-and-motion":22021,"\u002Fcalculus\u002Fpartial-derivatives\u002Ffunctions-of-several-variables":22025,"\u002Fcalculus\u002Fpartial-derivatives\u002Fpartial-derivatives":22030,"\u002Fcalculus\u002Fpartial-derivatives\u002Ftangent-planes-and-the-chain-rule":22033,"\u002Fcalculus\u002Fpartial-derivatives\u002Fdirectional-derivatives-and-the-gradient":22037,"\u002Fcalculus\u002Fpartial-derivatives\u002Foptimization-and-lagrange-multipliers":22041,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fdouble-integrals":22045,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Ftriple-integrals-and-coordinate-systems":22050,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fvector-fields-and-line-integrals":22054,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fgreens-theorem-curl-and-divergence":22058,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fsurface-integrals":22062,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fstokes-and-the-divergence-theorem":22066,"\u002Fcalculus":22070,"\u002Fmechanics\u002Ffoundations\u002Fmeasurement-and-dimensions":22073,"\u002Fmechanics\u002Ffoundations\u002Fvector-algebra":22077,"\u002Fmechanics\u002Fkinematics\u002Fone-dimensional-motion":22081,"\u002Fmechanics\u002Fkinematics\u002Fmotion-graphs":22086,"\u002Fmechanics\u002Fkinematics\u002Fprojectile-motion":22090,"\u002Fmechanics\u002Fkinematics\u002Frelative-motion":22094,"\u002Fmechanics\u002Fkinematics\u002Fcircular-motion":22098,"\u002Fmechanics\u002Fdynamics\u002Fnewtons-laws":22102,"\u002Fmechanics\u002Fdynamics\u002Ffree-body-diagrams":22107,"\u002Fmechanics\u002Fdynamics\u002Ffriction-and-curved-motion":22111,"\u002Fmechanics\u002Fdynamics\u002Fnumerical-dynamics":22115,"\u002Fmechanics\u002Fdynamics\u002Fcenter-of-mass-systems":22119,"\u002Fmechanics\u002Fenergy\u002Fwork-and-kinetic-energy":22123,"\u002Fmechanics\u002Fenergy\u002Fpotential-energy":22128,"\u002Fmechanics\u002Fenergy\u002Fmultiparticle-work":22132,"\u002Fmechanics\u002Fenergy\u002Fmass-energy-and-binding":22136,"\u002Fmechanics\u002Fenergy\u002Fphotons-and-quantization":22140,"\u002Fmechanics\u002Fmomentum\u002Fmomentum-and-collisions":22144,"\u002Fmechanics\u002Fmomentum\u002Fcenter-of-mass-collisions":22149,"\u002Fmechanics\u002Fmomentum\u002Frocket-propulsion":22153,"\u002Fmechanics\u002Frotation\u002Frotational-inertia":22157,"\u002Fmechanics\u002Frotation\u002Frotational-dynamics":22162,"\u002Fmechanics\u002Frotation\u002Frolling-motion":22166,"\u002Fmechanics\u002Frotation\u002Fangular-momentum":22170,"\u002Fmechanics\u002Frotation\u002Frolling-resistance":22174,"\u002Fmechanics\u002Frotation\u002Fgyroscopic-precession":22178,"\u002Fmechanics\u002Fgravity-and-matter\u002Fkeplerian-orbits":22182,"\u002Fmechanics\u002Fgravity-and-matter\u002Fgravitational-fields":22187,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstatic-equilibrium":22191,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-statics":22195,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-flow":22199,"\u002Fmechanics\u002Fgravity-and-matter\u002Forbital-motion":22203,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstress-and-elasticity":22207,"\u002Fmechanics\u002Foscillations-waves\u002Fdamped-oscillators":22211,"\u002Fmechanics\u002Foscillations-waves\u002Ftravelling-waves":22216,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-superposition":22220,"\u002Fmechanics\u002Foscillations-waves\u002Fstanding-waves":22224,"\u002Fmechanics\u002Foscillations-waves\u002Fsound-waves":22228,"\u002Fmechanics\u002Foscillations-waves\u002Fdoppler-effect":22232,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-packets":22236,"\u002Fmechanics\u002Foscillations-waves\u002Fbeats-and-coupling":22240,"\u002Fmechanics\u002Foscillations-waves\u002Fsimple-harmonic-motion":22244,"\u002Fmechanics\u002Foscillations-waves\u002Fpendulum-motion":22248,"\u002Fmechanics\u002Foscillations-waves\u002Fdriven-oscillators":22252,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-boundaries":22256,"\u002Fmechanics\u002Fthermodynamics\u002Fkinetic-theory-of-ideal-gases":22260,"\u002Fmechanics\u002Fthermodynamics\u002Ffirst-law-of-thermodynamics":22265,"\u002Fmechanics\u002Fthermodynamics\u002Fentropy-and-the-second-law":22269,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-processes":22273,"\u002Fmechanics\u002Fthermodynamics\u002Fphase-changes":22277,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-machines":22281,"\u002Fmechanics":22285,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcharge-and-conductors":22288,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcoulombs-law":22293,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-and-force":22297,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-maps":22301,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-dipoles":22305,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fcontinuous-charge-fields":22309,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fgauss-law-and-conductors":22314,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpoint-charge-potential":22318,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpotential-gradients-and-equipotentials":22323,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Felectrostatic-energy-and-pressure":22327,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Flaplace-boundary-problems":22331,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fcontinuous-charge-potentials":22335,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitance-fundamentals":22339,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-networks":22344,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-energy-and-force":22348,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fdielectric-polarization-and-breakdown":22352,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fcurrent-and-resistance":22356,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fkirchhoff-network-analysis":22361,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Frc-transients":22365,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-trajectories":22369,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fhall-effect":22374,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-force-on-conductors":22378,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-dipoles":22382,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmass-spectrometry":22386,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmoving-charge-fields":22390,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fbiot-savart-law":22395,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fcircular-current-loops":22399,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Famperes-law":22403,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fgauss-law-for-magnetism":22407,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmagnetic-materials":22411,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-flux":22415,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Ffaradays-law":22420,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Flenzs-law":22424,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmotional-emf":22428,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Feddy-currents":22432,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fself-inductance":22436,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-energy":22440,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Frl-circuits":22444,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-fundamentals":22448,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Freactance":22453,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Frlc-resonance":22457,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-power":22461,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Ftransformers":22465,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdisplacement-current":22469,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-waves":22474,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-momentum":22478,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdipole-radiation":22482,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fpolarization":22486,"\u002Felectricity-and-magnetism\u002Foptics\u002Freflection-and-refraction":22490,"\u002Felectricity-and-magnetism\u002Foptics\u002Fthin-lenses":22495,"\u002Felectricity-and-magnetism\u002Foptics\u002Fspherical-mirrors":22499,"\u002Felectricity-and-magnetism":22503,"\u002Flinear-algebra\u002Flinear-systems\u002Fsystems-and-echelon-forms":22506,"\u002Flinear-algebra\u002Flinear-systems\u002Fvector-and-matrix-equations":22511,"\u002Flinear-algebra\u002Flinear-systems\u002Fsolution-sets-and-applications":22515,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-independence":22519,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-transformations":22523,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-operations":22527,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-inverse-and-invertibility":22532,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fpartitioned-matrices-and-lu":22536,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fsubspaces-dimension-rank":22540,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fapplications-leontief-and-graphics":22544,"\u002Flinear-algebra\u002Fdeterminants\u002Fdeterminants-and-cofactors":22548,"\u002Flinear-algebra\u002Fdeterminants\u002Fproperties-of-determinants":22553,"\u002Flinear-algebra\u002Fdeterminants\u002Fcramer-volume-and-area":22557,"\u002Flinear-algebra\u002Fvector-spaces\u002Fvector-spaces-and-subspaces":22561,"\u002Flinear-algebra\u002Fvector-spaces\u002Fnull-and-column-spaces":22566,"\u002Flinear-algebra\u002Fvector-spaces\u002Fbases-and-independent-sets":22570,"\u002Flinear-algebra\u002Fvector-spaces\u002Fcoordinate-systems":22574,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdimension-and-rank":22578,"\u002Flinear-algebra\u002Fvector-spaces\u002Fchange-of-basis":22582,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdifference-equations-and-markov":22586,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-eigenvalues":22590,"\u002Flinear-algebra\u002Feigenvalues\u002Fthe-characteristic-equation":22595,"\u002Flinear-algebra\u002Feigenvalues\u002Fdiagonalization":22599,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-linear-transformations":22603,"\u002Flinear-algebra\u002Feigenvalues\u002Fcomplex-eigenvalues":22607,"\u002Flinear-algebra\u002Feigenvalues\u002Fdynamical-systems":22611,"\u002Flinear-algebra\u002Feigenvalues\u002Fpower-method":22615,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-length-orthogonality":22619,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Forthogonal-sets-and-projections":22624,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fgram-schmidt-and-qr":22628,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-problems":22632,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-applications":22636,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-spaces":22640,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fdiagonalizing-symmetric-matrices":22644,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fquadratic-forms":22649,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fconstrained-optimization":22653,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsingular-value-decomposition":22657,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsvd-applications-pca-imaging":22661,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-thinking-and-matrix-computation":22665,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Flu-and-cholesky":22670,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fconditioning-and-floating-point":22674,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fstability-and-error-analysis":22678,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fqr-and-numerical-least-squares":22682,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-eigenvalues-and-svd":22686,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-combinations":22690,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-independence-and-barycentric-coordinates":22695,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fconvex-combinations-and-convex-sets":22699,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fhyperplanes-and-polytopes":22703,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fcurves-and-surfaces":22707,"\u002Flinear-algebra":22711,"\u002Ftheory-of-computation":22714,"\u002Fcomputer-architecture\u002Ffoundations\u002Fbits-bytes-and-words":22717,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-representation":22721,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-arithmetic":22725,"\u002Fcomputer-architecture\u002Ffoundations\u002Ffloating-point":22729,"\u002Fcomputer-architecture\u002Ffoundations\u002Fboolean-algebra-and-bit-manipulation":22733,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fthe-machines-view":22737,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fdata-movement":22742,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farithmetic-and-logic":22746,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fcontrol-flow":22750,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fprocedures":22754,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farrays-structs-and-alignment":22758,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fmemory-layout-and-buffer-overflows":22762,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fwhat-an-isa-is":22766,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Finstruction-formats-and-operands":22771,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Faddressing-modes":22775,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fthe-y86-64-instruction-set":22779,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fy86-64-programming":22783,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Ftransistors-gates-and-boolean-functions":22787,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fcombinational-logic-and-hcl":22792,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmultiplexers-decoders-and-the-alu":22796,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmemory-elements-latches-flip-flops-and-clocking":22800,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fregister-files-and-random-access-memory":22804,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-fetch-decode-execute-cycle":22808,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-seq-stages":22813,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fcontrol-logic-and-sequencing":22817,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fassembling-seq":22821,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Ftracing-a-program":22825,"\u002Fcomputer-architecture\u002Fpipelining\u002Fpipelining-principles":22829,"\u002Fcomputer-architecture\u002Fpipelining\u002Ffrom-seq-to-pipe":22834,"\u002Fcomputer-architecture\u002Fpipelining\u002Fdata-hazards-stalling-and-forwarding":22838,"\u002Fcomputer-architecture\u002Fpipelining\u002Fcontrol-hazards-and-branch-prediction":22842,"\u002Fcomputer-architecture\u002Fpipelining\u002Fthe-complete-pipe-processor":22846,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fstorage-technologies-and-the-latency-gap":22850,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Flocality":22855,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-memories-direct-mapped":22859,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fset-associative-and-write-policies":22863,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-performance-and-cache-friendly-code":22867,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Faddress-spaces-and-translation":22871,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fpage-tables-and-page-faults":22876,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fthe-tlb-and-multi-level-page-tables":22880,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Fexceptional-control-flow":22884,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Finterrupts-and-the-kernel":22889,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fprocesses-threads-and-parallelism":22893,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fhardware-multithreading":22898,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fcache-coherence":22902,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmemory-consistency-and-synchronization":22906,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmulticore-organization":22910,"\u002Fcomputer-architecture\u002Fcapstone\u002Fthe-whole-machine":22914,"\u002Fcomputer-architecture\u002Fcapstone\u002Fassembling-a-complete-cpu":22919,"\u002Fcomputer-architecture":22923,"\u002Fdifferential-equations\u002Ffoundations\u002Fmodels-and-direction-fields":22926,"\u002Fdifferential-equations\u002Ffoundations\u002Fclassification-and-terminology":22930,"\u002Fdifferential-equations\u002Ffirst-order\u002Flinear-first-order-integrating-factors":22934,"\u002Fdifferential-equations\u002Ffirst-order\u002Fseparable-and-exact":22939,"\u002Fdifferential-equations\u002Ffirst-order\u002Fmodeling-first-order":22943,"\u002Fdifferential-equations\u002Ffirst-order\u002Fautonomous-and-population-dynamics":22947,"\u002Fdifferential-equations\u002Ffirst-order\u002Fexistence-uniqueness-euler":22951,"\u002Fdifferential-equations\u002Ffirst-order\u002Ffirst-order-difference-equations":22955,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhomogeneous-constant-coefficients":22959,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fcomplex-and-repeated-roots":22964,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fnonhomogeneous-undetermined-coefficients":22968,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fvariation-of-parameters":22972,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fmechanical-electrical-vibrations":22976,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhigher-order-linear":22980,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fpower-series-ordinary-points":22984,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fregular-singular-frobenius":22989,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fbessel-and-special-functions":22993,"\u002Fdifferential-equations\u002Flaplace\u002Flaplace-definition-ivps":22997,"\u002Fdifferential-equations\u002Flaplace\u002Fstep-impulse-convolution":23002,"\u002Fdifferential-equations\u002Fsystems\u002Fmatrices-eigenvalues-review":23006,"\u002Fdifferential-equations\u002Fsystems\u002Fconstant-coefficient-systems-phase-portraits":23011,"\u002Fdifferential-equations\u002Fsystems\u002Frepeated-eigenvalues-fundamental-matrices":23015,"\u002Fdifferential-equations\u002Fnumerical\u002Feuler-and-runge-kutta":23019,"\u002Fdifferential-equations\u002Fnumerical\u002Fmultistep-systems-stability":23024,"\u002Fdifferential-equations\u002Fnonlinear\u002Fphase-plane-autonomous-stability":23028,"\u002Fdifferential-equations\u002Fnonlinear\u002Flocally-linear-and-liapunov":23033,"\u002Fdifferential-equations\u002Fnonlinear\u002Fcompeting-species-predator-prey-limit-cycles":23037,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Ffourier-series":23041,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fheat-wave-laplace-equations":23046,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fsturm-liouville":23050,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fcalculus-of-variations":23054,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fhistorical-notes":23059,"\u002Fdifferential-equations":23063,"\u002Frelativity\u002Ffoundations\u002Fspecial-relativity-postulates":23066,"\u002Frelativity\u002Ffoundations\u002Florentz-transformation-spacetime":23071,"\u002Frelativity\u002Ffoundations\u002Ftime-dilation-length-contraction":23075,"\u002Frelativity\u002Ffoundations\u002Frelativistic-momentum-energy":23079,"\u002Frelativity\u002Ffoundations\u002Fgeneral-relativity":23083,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fminkowski-spacetime-and-the-interval":23087,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Ffour-vectors-and-index-notation":23092,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fthe-lorentz-group-and-rapidity":23096,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fdoppler-aberration-and-appearance":23100,"\u002Frelativity\u002Frelativistic-dynamics\u002Ffour-momentum-force-and-accelerated-motion":23104,"\u002Frelativity\u002Frelativistic-dynamics\u002Fparticle-decays-and-two-body-kinematics":23109,"\u002Frelativity\u002Frelativistic-dynamics\u002Fcollisions-thresholds-and-the-cm-frame":23113,"\u002Frelativity\u002Frelativistic-dynamics\u002Fmandelstam-variables-and-invariants":23117,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ffour-current-and-the-four-potential":23121,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fthe-electromagnetic-field-tensor":23126,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ftransformation-of-electric-and-magnetic-fields":23130,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fcovariant-maxwell-and-the-stress-energy-tensor":23134,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-equivalence-principle-formalized":23138,"\u002Frelativity\u002Fcurved-spacetime\u002Fmanifolds-vectors-and-the-metric":23143,"\u002Frelativity\u002Fcurved-spacetime\u002Fcovariant-derivative-and-christoffel-symbols":23147,"\u002Frelativity\u002Fcurved-spacetime\u002Fgeodesics-and-the-geodesic-equation":23151,"\u002Frelativity\u002Fcurved-spacetime\u002Fcurvature-riemann-and-geodesic-deviation":23155,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-einstein-field-equations":23159,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fthe-schwarzschild-metric":23163,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fgeodesics-and-orbits-in-schwarzschild":23168,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Flight-bending-and-null-geodesics":23172,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fperihelion-precession-of-mercury":23176,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fdeflection-of-light-and-gravitational-lensing":23181,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fgravitational-redshift-and-shapiro-delay":23185,"\u002Frelativity\u002Ftests-of-general-relativity\u002Frelativity-in-technology-gps":23189,"\u002Frelativity\u002Fblack-holes\u002Fhorizons-and-coordinate-singularities":23193,"\u002Frelativity\u002Fblack-holes\u002Frotating-and-charged-black-holes":23198,"\u002Frelativity\u002Fblack-holes\u002Fblack-hole-thermodynamics":23202,"\u002Frelativity\u002Fgravitational-waves\u002Flinearized-gravity-and-wave-solutions":23206,"\u002Frelativity\u002Fgravitational-waves\u002Fgeneration-and-the-quadrupole-formula":23211,"\u002Frelativity\u002Fgravitational-waves\u002Fdetection-ligo-and-the-first-events":23215,"\u002Frelativity\u002Fcosmological-bridge\u002Fthe-cosmological-principle-and-flrw-metric":23219,"\u002Frelativity\u002Fcosmological-bridge\u002Ffriedmann-equations-and-cosmic-dynamics":23224,"\u002Frelativity":23228,"\u002Fphysical-computing":23231,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fblackbody-radiation-and-the-planck-quantum":23234,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-photoelectric-effect-and-the-photon":23239,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fx-rays-and-the-compton-effect":23243,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-old-quantum-theory-bohr-and-sommerfeld":23247,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fde-broglie-waves-and-electron-diffraction":23251,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fwave-packets-and-the-probability-interpretation":23256,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fthe-uncertainty-principle":23260,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-schrodinger-equation-in-one-dimension":23264,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-free-particle-and-wave-packet-dynamics":23269,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fparticle-in-infinite-and-finite-square-wells":23273,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Foperators-expectation-values-and-the-harmonic-oscillator":23277,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-dirac-delta-potential":23281,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fbarrier-penetration-and-quantum-tunneling":23285,"\u002Fquantum-mechanics\u002Fformalism\u002Fhilbert-space-and-dirac-notation":23289,"\u002Fquantum-mechanics\u002Fformalism\u002Fobservables-hermitian-operators-and-eigenvalues":23294,"\u002Fquantum-mechanics\u002Fformalism\u002Fthe-postulates-and-quantum-measurement":23298,"\u002Fquantum-mechanics\u002Fformalism\u002Fposition-momentum-and-continuous-spectra":23302,"\u002Fquantum-mechanics\u002Fformalism\u002Fcommutators-and-the-generalized-uncertainty-principle":23306,"\u002Fquantum-mechanics\u002Fformalism\u002Ftime-evolution-schrodinger-and-heisenberg-pictures":23310,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fladder-operators-and-the-number-states":23314,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fcoherent-and-squeezed-states":23319,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fsymmetries-generators-and-conservation-laws":23323,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fparity-time-reversal-and-discrete-symmetries":23327,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Forbital-angular-momentum-and-spherical-harmonics":23331,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Fthe-angular-momentum-algebra":23335,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Faddition-of-angular-momenta-and-clebsch-gordan":23339,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-schrodinger-equation-in-three-dimensions":23343,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-hydrogen-atom":23348,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-isotropic-oscillator-and-hidden-symmetry":23352,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-half-pauli-matrices-and-stern-gerlach":23356,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-in-a-magnetic-field-precession-and-resonance":23361,"\u002Fquantum-mechanics\u002Fspin\u002Ftwo-level-systems-and-the-bloch-sphere":23365,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fidentical-particles-and-exchange-symmetry":23369,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fthe-pauli-principle-atoms-and-the-periodic-table":23374,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ftime-independent-perturbation-theory":23378,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ffine-structure-and-the-real-hydrogen-atom":23383,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-zeeman-and-stark-effects":23387,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-variational-method":23391,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-wkb-approximation":23395,"\u002Fquantum-mechanics":23399,"\u002Freal-analysis\u002Ffoundations\u002Fsets-logic-functions":23402,"\u002Freal-analysis\u002Ffoundations\u002Fordered-fields-completeness":23407,"\u002Freal-analysis\u002Ffoundations\u002Fabsolute-value-bounds":23411,"\u002Freal-analysis\u002Ffoundations\u002Fintervals-uncountability":23415,"\u002Freal-analysis\u002Fsequences-series\u002Fsequences-limits":23419,"\u002Freal-analysis\u002Fsequences-series\u002Flimit-laws-monotone":23424,"\u002Freal-analysis\u002Fsequences-series\u002Flimsup-bolzano-weierstrass":23428,"\u002Freal-analysis\u002Fsequences-series\u002Fcauchy-completeness":23432,"\u002Freal-analysis\u002Fsequences-series\u002Fseries-convergence":23436,"\u002Freal-analysis\u002Fsequences-series\u002Fabsolute-conditional-rearrangement":23440,"\u002Freal-analysis\u002Fmetric-spaces\u002Fmetric-spaces-norms":23444,"\u002Freal-analysis\u002Fmetric-spaces\u002Fopen-closed-sets":23449,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconvergence-completeness":23453,"\u002Freal-analysis\u002Fmetric-spaces\u002Fcompactness":23457,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconnectedness":23461,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-of-functions":23465,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuous-functions":23469,"\u002Freal-analysis\u002Fcontinuity\u002Fevt-ivt":23473,"\u002Freal-analysis\u002Fcontinuity\u002Funiform-continuity":23477,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuity-metric-spaces":23481,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-infinity-monotone":23485,"\u002Freal-analysis\u002Fdifferentiation\u002Fthe-derivative":23489,"\u002Freal-analysis\u002Fdifferentiation\u002Fmean-value-theorem":23494,"\u002Freal-analysis\u002Fdifferentiation\u002Ftaylors-theorem":23498,"\u002Freal-analysis\u002Fdifferentiation\u002Finverse-function-1d":23502,"\u002Freal-analysis\u002Friemann-integration\u002Fdarboux-integral":23506,"\u002Freal-analysis\u002Friemann-integration\u002Fintegrability-classes":23511,"\u002Freal-analysis\u002Friemann-integration\u002Fproperties-of-the-integral":23515,"\u002Freal-analysis\u002Friemann-integration\u002Ffundamental-theorem":23519,"\u002Freal-analysis\u002Friemann-integration\u002Flog-exp-improper":23522,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpointwise-uniform-convergence":23526,"\u002Freal-analysis\u002Ffunction-sequences\u002Finterchange-of-limits":23531,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpower-series-weierstrass":23535,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpicard-ode":23539,"\u002Freal-analysis\u002Fseveral-variables\u002Fdifferentiability-rn":23543,"\u002Freal-analysis\u002Fseveral-variables\u002Fgradient-chain-rule":23548,"\u002Freal-analysis\u002Fseveral-variables\u002Fhigher-derivatives-taylor-extrema":23552,"\u002Freal-analysis\u002Fseveral-variables\u002Finverse-implicit-theorems":23556,"\u002Freal-analysis\u002Fseveral-variables\u002Fmultiple-integrals":23560,"\u002Freal-analysis":23564,"\u002Fabstract-algebra\u002Ffoundations\u002Fsets-functions-relations":23567,"\u002Fabstract-algebra\u002Ffoundations\u002Fintegers-and-modular-arithmetic":23571,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fgroup-axioms-and-first-examples":23575,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fdihedral-and-symmetric-groups":23580,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fmatrix-and-quaternion-groups":23584,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fhomomorphisms-and-group-actions":23588,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fsubgroups-and-substructures":23592,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcyclic-groups":23597,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fgeneration-and-subgroup-lattices":23601,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcosets-lagrange-and-normal-subgroups":23605,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fisomorphism-theorems":23609,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcomposition-series-and-the-alternating-group":23613,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Factions-and-cayleys-theorem":23617,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fconjugation-and-the-class-equation":23622,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fsylow-theorems":23626,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fautomorphisms-and-simple-groups":23630,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fdirect-products-and-finite-abelian-groups":23634,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fsemidirect-products":23639,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fnilpotent-and-solvable-groups":23643,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fclassifying-small-groups":23647,"\u002Fabstract-algebra\u002Fring-theory\u002Frings-definitions-and-examples":23651,"\u002Fabstract-algebra\u002Fring-theory\u002Fideals-quotients-and-homomorphisms":23656,"\u002Fabstract-algebra\u002Fring-theory\u002Ffractions-and-the-chinese-remainder-theorem":23660,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Feuclidean-domains-pids-ufds":23664,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fpolynomial-rings-over-fields":23669,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fgauss-lemma-and-unique-factorization":23673,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Firreducibility-criteria-and-groebner":23677,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fintroduction-to-modules":23681,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ffree-modules-and-direct-sums":23686,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ftensor-products-and-exact-sequences":23690,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fvector-spaces-and-linear-maps":23694,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fstructure-theorem-over-pids":23698,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Frational-canonical-form":23703,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fjordan-canonical-form":23707,"\u002Fabstract-algebra\u002Ffield-theory\u002Ffield-extensions-and-algebraic-elements":23711,"\u002Fabstract-algebra\u002Ffield-theory\u002Fstraightedge-and-compass-constructions":23716,"\u002Fabstract-algebra\u002Ffield-theory\u002Fsplitting-fields-and-algebraic-closure":23720,"\u002Fabstract-algebra\u002Ffield-theory\u002Fseparable-and-cyclotomic-extensions":23724,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fthe-galois-correspondence":23728,"\u002Fabstract-algebra\u002Fgalois-theory\u002Ffinite-fields":23733,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fcyclotomic-and-abelian-extensions":23737,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fgalois-groups-of-polynomials":23741,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fsolvability-by-radicals-and-the-quintic":23745,"\u002Fabstract-algebra\u002Fcapstone\u002Fcommutative-algebra-and-algebraic-geometry":23749,"\u002Fabstract-algebra\u002Fcapstone\u002Frepresentation-and-character-theory":23754,"\u002Fabstract-algebra":23758,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fatomic-spectra-rutherford":23761,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-model-hydrogen":23766,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fx-ray-spectra-franck-hertz":23770,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-sommerfeld-old-quantum-theory":23774,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fold-quantum-theory-limits-wkb":23778,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fschrodinger-3d-hydrogen":23782,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fhydrogen-wave-functions":23787,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fradial-equation-in-full":23791,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fsymmetry-degeneracy-runge-lenz":23795,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fexpectation-values-virial":23799,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fquantum-defects-alkali-spectra":23803,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Frydberg-atoms":23807,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Frelativistic-kinetic-correction":23811,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fspin-orbit-thomas-precession":23816,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdarwin-term-fine-structure-formula":23820,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdirac-equation-hydrogen":23824,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Flamb-shift-qed":23828,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fhyperfine-structure-21cm":23833,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fnuclear-effects-isotope-shift":23837,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fperiodic-table-atomic-spectra":23841,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fcentral-field-self-consistent":23846,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fidentical-particles-hartree-fock":23850,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhelium-two-electron-atom":23854,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fls-jj-coupling-term-symbols":23858,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhund-rules-ground-terms":23862,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fzeeman-effect":23866,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fpaschen-back-intermediate":23871,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fstark-effect-polarizability":23875,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Ftime-dependent-perturbation-golden-rule":23879,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fdipole-approximation-einstein-coefficients":23884,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fselection-rules-forbidden-transitions":23888,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Flifetimes-and-line-shapes":23892,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Flaser-principles":23896,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fspectroscopy-techniques":23901,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fline-catalog-nist-asd":23905,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Flaser-cooling-doppler":23909,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fsub-doppler-trapping":23914,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fbose-einstein-condensation":23918,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Foptical-clocks-precision":23922,"\u002Fatomic-physics":23926,"\u002Fdatabases":23929,"\u002Fcategory-theory\u002Ffoundations\u002Fwhat-is-a-category":23932,"\u002Fcategory-theory\u002Ffoundations\u002Fexamples-of-categories":23936,"\u002Fcategory-theory\u002Ffoundations\u002Fspecial-morphisms":23940,"\u002Fcategory-theory\u002Ffoundations\u002Ffunctors":23944,"\u002Fcategory-theory\u002Ffoundations\u002Fnatural-transformations":23948,"\u002Fcategory-theory\u002Ffoundations\u002Fsize-and-set-theory":23952,"\u002Fcategory-theory\u002Funiversal-properties\u002Funiversal-properties":23956,"\u002Fcategory-theory\u002Funiversal-properties\u002Fproducts-and-coproducts":23961,"\u002Fcategory-theory\u002Funiversal-properties\u002Fconstructions-on-categories":23965,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Frepresentable-functors":23969,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-lemma":23974,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-consequences":23978,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits":23982,"\u002Fcategory-theory\u002Flimits-colimits\u002Fproducts-equalizers-pullbacks":23987,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcolimits":23991,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcomputing-limits":23995,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits-and-functors":23999,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions":24003,"\u002Fcategory-theory\u002Fadjunctions\u002Funits-and-counits":24008,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions-via-universal-arrows":24012,"\u002Fcategory-theory\u002Fadjunctions\u002Ffree-forgetful-adjunctions":24016,"\u002Fcategory-theory\u002Fadjoints-limits\u002Flimits-via-adjoints":24020,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fpresheaf-limits-colimits":24025,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoints-preserve-limits":24029,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoint-functor-theorem":24033,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fmonads":24037,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-eilenberg-moore":24042,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fkleisli-and-programming":24046,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-for-endofunctors":24050,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Fcartesian-closed-categories":24054,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Flambda-calculus-correspondence":24059,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Ffixed-points-and-recursion":24063,"\u002Fcategory-theory":24067,"\u002Fdeep-learning\u002Fmathematical-background\u002Flinear-algebra-for-deep-learning":24070,"\u002Fdeep-learning\u002Fmathematical-background\u002Fprobability-and-information-theory":24074,"\u002Fdeep-learning\u002Fmathematical-background\u002Fnumerical-computation":24078,"\u002Fdeep-learning\u002Fmathematical-background\u002Fcalculus":24082,"\u002Fdeep-learning\u002Ffoundations\u002Fwhat-is-deep-learning":24085,"\u002Fdeep-learning\u002Ffoundations\u002Fmachine-learning-refresher":24089,"\u002Fdeep-learning\u002Ffoundations\u002Flinear-models-and-the-perceptron":24093,"\u002Fdeep-learning\u002Fneural-networks\u002Fthe-multilayer-perceptron":24097,"\u002Fdeep-learning\u002Fneural-networks\u002Factivation-functions":24102,"\u002Fdeep-learning\u002Fneural-networks\u002Funiversal-approximation":24106,"\u002Fdeep-learning\u002Fneural-networks\u002Fbackpropagation":24110,"\u002Fdeep-learning\u002Fneural-networks\u002Floss-functions-and-output-units":24113,"\u002Fdeep-learning\u002Foptimization\u002Fgradient-descent-and-sgd":24117,"\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods":24120,"\u002Fdeep-learning\u002Foptimization\u002Finitialization":24123,"\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape":24127,"\u002Fdeep-learning\u002Foptimization\u002Fsecond-order-and-approximate-methods":24130,"\u002Fdeep-learning\u002Fregularization\u002Fregularization-overview":24131,"\u002Fdeep-learning\u002Fregularization\u002Fdropout-and-data-augmentation":24136,"\u002Fdeep-learning\u002Fregularization\u002Fearly-stopping-and-parameter-sharing":24140,"\u002Fdeep-learning\u002Fregularization\u002Fnormalization":24144,"\u002Fdeep-learning\u002Farchitectures\u002Fconvolutional-networks":24148,"\u002Fdeep-learning\u002Farchitectures\u002Fcnn-architectures":24153,"\u002Fdeep-learning\u002Farchitectures\u002Frecurrent-networks":24157,"\u002Fdeep-learning\u002Farchitectures\u002Flstm-and-gru":24161,"\u002Fdeep-learning\u002Farchitectures\u002Fattention-and-transformers":24165,"\u002Fdeep-learning\u002Farchitectures\u002Fthe-transformer-architecture":24169,"\u002Fdeep-learning\u002Farchitectures\u002Ftransformers-in-practice":24173,"\u002Fdeep-learning\u002Farchitectures\u002Fgraph-neural-networks":24177,"\u002Fdeep-learning\u002Farchitectures\u002Fstate-space-models":24181,"\u002Fdeep-learning\u002Ftheory\u002Fgeneralization-theory":24185,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-robustness":24190,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-defenses":24194,"\u002Fdeep-learning\u002Ftheory\u002Fbayesian-and-ensemble-methods":24198,"\u002Fdeep-learning\u002Ftheory\u002Fdeep-equilibrium-models":24202,"\u002Fdeep-learning\u002Fgenerative-models\u002Flinear-factor-models":24206,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoencoders":24211,"\u002Fdeep-learning\u002Fgenerative-models\u002Fvariational-autoencoders":24215,"\u002Fdeep-learning\u002Fgenerative-models\u002Fgenerative-adversarial-networks":24219,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoregressive-and-normalizing-flows":24223,"\u002Fdeep-learning\u002Fgenerative-models\u002Fenergy-based-and-boltzmann-machines":24227,"\u002Fdeep-learning\u002Fgenerative-models\u002Fdiffusion-and-score-based-models":24231,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fstructured-probabilistic-models":24235,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fmonte-carlo-and-mcmc":24240,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fapproximate-inference":24244,"\u002Fdeep-learning\u002Fpractical\u002Fpractical-methodology":24248,"\u002Fdeep-learning\u002Fpractical\u002Fhyperparameters-and-debugging":24253,"\u002Fdeep-learning\u002Fpractical\u002Frepresentation-learning":24257,"\u002Fdeep-learning\u002Fpractical\u002Ftransfer-learning":24261,"\u002Fdeep-learning\u002Fpractical\u002Fapplications":24265,"\u002Fdeep-learning\u002Fpractical\u002Fmodel-compression-and-distillation":24269,"\u002Fdeep-learning\u002Fpractical\u002Fmeta-learning-and-few-shot":24273,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models":24277,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment":24282,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fseq2seq-pretraining-and-bart":24286,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ftext-to-text-transfer-and-conditional-generation":24290,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fspeech-and-audio-models":24294,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fself-supervised-speech-and-synthesis":24298,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents":24302,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fagent-memory-retrieval-and-orchestration":24306,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmixture-of-experts":24310,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmultimodal-models":24314,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ffusion-and-vision-language-models":24318,"\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning":24322,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fmodel-free-prediction-and-control":24327,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fdeep-q-networks":24331,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic":24335,"\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback":24339,"\u002Fdeep-learning":24343,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fequilibrium-state-variables-zeroth-law":24345,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Ffirst-law-heat-and-work":24349,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fsecond-law-entropy-and-the-carnot-bound":24353,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fthermodynamic-potentials-and-maxwell-relations":24357,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fstability-response-functions-and-the-third-law":24361,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fclassical-statistics-and-equipartition":24365,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fphase-space-and-liouvilles-theorem":24370,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fensembles-and-the-equal-probability-postulate":24374,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fstatistical-entropy-boltzmann-and-gibbs":24378,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fmicrocanonical-ensemble-and-entropy":24382,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fequilibrium-conditions-temperature-pressure-chemical-potential":24387,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fideal-gas-phase-space-and-the-sackur-tetrode-entropy":24391,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Ftwo-state-systems-paramagnets-and-negative-temperature":24395,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fcanonical-ensemble-and-the-boltzmann-distribution":24399,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fpartition-function-and-the-helmholtz-free-energy":24404,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fenergy-fluctuations-and-ensemble-equivalence":24408,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fthe-einstein-solid-and-harmonic-systems":24412,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fparamagnetism-and-the-schottky-anomaly":24416,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fideal-gas-partition-function-and-the-gibbs-paradox":24420,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fequipartition-and-the-virial-theorem":24425,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fmolecular-gases-rotation-and-vibration":24429,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fgrand-canonical-ensemble-and-the-grand-partition-function":24433,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fchemical-potential-fugacity-and-number-fluctuations":24438,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fensemble-summary-and-the-thermodynamic-web":24442,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fquantum-statistics-bose-einstein-and-fermi-dirac":24446,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fderiving-the-quantum-distributions":24451,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fthe-classical-limit-and-quantum-concentration":24455,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fideal-quantum-gases-general-framework":24459,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-and-the-fermion-gas":24463,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthe-photon-gas-and-plancks-radiation-law":24468,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fblackbody-thermodynamics-and-radiation-pressure":24472,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fphonons-and-the-debye-model":24476,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-derived":24480,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthermodynamics-of-the-bose-gas-and-superfluidity":24484,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fthe-ideal-fermi-gas-at-zero-temperature":24488,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fsommerfeld-expansion-and-electrons-in-metals":24493,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":24497,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fneutron-stars-and-nuclear-matter":24501,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-cluster-expansion-and-virial-coefficients":24505,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-van-der-waals-gas-and-liquid-gas-coexistence":24510,"\u002Fstatistical-mechanics\u002Finteractions\u002Fquantum-gases-with-interactions-and-exchange":24514,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fphases-coexistence-and-classification":24518,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-ising-model-and-exact-solutions":24523,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fmean-field-theory-and-the-weiss-model":24527,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fcritical-exponents-and-landau-theory":24531,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-renormalization-group-idea":24535,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fthermodynamic-fluctuations-and-response":24539,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fbrownian-motion-and-the-langevin-equation":24544,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Flinear-response-and-the-fluctuation-dissipation-theorem":24548,"\u002Fstatistical-mechanics":24552,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fbonding-mechanisms":24555,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fmolecular-orbitals-and-h2-plus":24560,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fhydrogen-molecule-and-exchange":24564,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fvan-der-waals-forces":24568,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Frotational-vibrational-spectra":24572,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Fanharmonicity-and-rovibrational-structure":24577,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Framan-and-electronic-bands":24581,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Flasers-and-masers":24585,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fstructure-of-solids":24589,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fbravais-lattices-and-crystal-systems":24594,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Freciprocal-lattice-and-brillouin-zones":24598,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fdiffraction-and-structure-factors":24602,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonon-dispersion":24606,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonons-quantization-and-dos":24611,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fdebye-einstein-heat-capacity":24615,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fanharmonicity-and-thermal-transport":24619,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ffree-electron-gas-and-conduction":24623,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fsommerfeld-model-and-heat-capacity":24628,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ftransport-and-the-hall-effect":24632,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fscreening-and-plasmons":24636,"\u002Fcondensed-matter\u002Fband-theory\u002Fblochs-theorem-and-energy-bands":24640,"\u002Fcondensed-matter\u002Fband-theory\u002Fnearly-free-electron-model":24645,"\u002Fcondensed-matter\u002Fband-theory\u002Ftight-binding-method":24649,"\u002Fcondensed-matter\u002Fband-theory\u002Ffermi-surfaces-and-semiclassical-dynamics":24653,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fsemiconductor-bands-and-junctions":24657,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fintrinsic-and-extrinsic-semiconductors":24662,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fcarrier-transport-and-recombination":24666,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fthe-pn-junction":24670,"\u002Fcondensed-matter\u002Fsemiconductors\u002Ftransistors-and-optoelectronics":24674,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fdielectrics-and-polarization":24678,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fferroelectrics-and-piezoelectrics":24683,"\u002Fcondensed-matter\u002Fmagnetism\u002Fdiamagnetism-and-paramagnetism":24687,"\u002Fcondensed-matter\u002Fmagnetism\u002Fexchange-and-ferromagnetism":24692,"\u002Fcondensed-matter\u002Fmagnetism\u002Fantiferromagnetism-and-domains":24696,"\u002Fcondensed-matter\u002Fmagnetism\u002Fspin-waves-and-magnons":24700,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fsuperconductivity-phenomenology":24704,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Flondon-theory-and-the-meissner-effect":24709,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fginzburg-landau-theory":24713,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fbcs-theory":24717,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fjosephson-and-high-tc":24721,"\u002Fcondensed-matter\u002Fnanostructures\u002Fquantum-wells-wires-and-dots":24725,"\u002Fcondensed-matter\u002Fnanostructures\u002Finteger-quantum-hall-effect":24730,"\u002Fcondensed-matter\u002Fnanostructures\u002Ffractional-quantum-hall-and-topology":24734,"\u002Fcondensed-matter\u002Fnanostructures\u002Fgraphene-and-dirac-materials":24738,"\u002Fcondensed-matter":24742,"\u002Flogic\u002Ffoundations\u002Flogic-as-a-mathematical-model":24745,"\u002Flogic\u002Fsentential-logic\u002Fformal-languages-and-well-formed-formulas":24749,"\u002Flogic\u002Fsentential-logic\u002Ftruth-assignments-and-tautologies":24754,"\u002Flogic\u002Fsentential-logic\u002Funique-readability-and-parsing":24758,"\u002Flogic\u002Fsentential-logic\u002Finduction-and-recursion":24762,"\u002Flogic\u002Fsentential-logic\u002Fexpressive-completeness-and-normal-forms":24766,"\u002Flogic\u002Fsentential-logic\u002Fboolean-circuits":24770,"\u002Flogic\u002Fsentential-logic\u002Fcompactness-and-effectiveness":24774,"\u002Flogic\u002Ffirst-order-languages\u002Ffirst-order-languages":24778,"\u002Flogic\u002Ffirst-order-languages\u002Fstructures-truth-and-satisfaction":24783,"\u002Flogic\u002Ffirst-order-languages\u002Fdefinability-and-elementary-equivalence":24787,"\u002Flogic\u002Ffirst-order-languages\u002Fterms-substitution-and-parsing":24791,"\u002Flogic\u002Fdeductive-calculus\u002Fa-deductive-calculus":24795,"\u002Flogic\u002Fdeductive-calculus\u002Fdeduction-theorem-and-derived-rules":24800,"\u002Flogic\u002Fdeductive-calculus\u002Fsoundness":24804,"\u002Flogic\u002Fdeductive-calculus\u002Fcompleteness-and-consistency":24808,"\u002Flogic\u002Fmodels-and-theories\u002Fcompactness-and-lowenheim-skolem":24812,"\u002Flogic\u002Fmodels-and-theories\u002Ftheories-elementary-classes-and-categoricity":24817,"\u002Flogic\u002Fmodels-and-theories\u002Finterpretations-between-theories":24821,"\u002Flogic\u002Fmodels-and-theories\u002Fnonstandard-analysis":24825,"\u002Flogic\u002Farithmetic-and-definability\u002Fdefinability-in-arithmetic":24829,"\u002Flogic\u002Farithmetic-and-definability\u002Fnatural-numbers-with-successor":24834,"\u002Flogic\u002Farithmetic-and-definability\u002Fpresburger-and-reducts":24838,"\u002Flogic\u002Farithmetic-and-definability\u002Fa-subtheory-and-representability":24842,"\u002Flogic\u002Fincompleteness\u002Farithmetization-of-syntax":24846,"\u002Flogic\u002Fincompleteness\u002Fincompleteness-and-undecidability":24851,"\u002Flogic\u002Fincompleteness\u002Fsecond-incompleteness-theorem":24855,"\u002Flogic\u002Fcomputability-and-representability\u002Frecursive-functions":24859,"\u002Flogic\u002Fcomputability-and-representability\u002Frepresenting-exponentiation":24864,"\u002Flogic\u002Fsecond-order-logic\u002Fsecond-order-languages":24868,"\u002Flogic\u002Fsecond-order-logic\u002Fskolem-functions-and-many-sorted-logic":24873,"\u002Flogic\u002Fsecond-order-logic\u002Fgeneral-structures":24877,"\u002Flogic":24881,"\u002Freinforcement-learning\u002Ffoundations\u002Fwhat-is-reinforcement-learning":24884,"\u002Freinforcement-learning\u002Ffoundations\u002Fa-brief-history-of-rl":24888,"\u002Freinforcement-learning\u002Ffoundations\u002Fmulti-armed-bandits":24892,"\u002Freinforcement-learning\u002Ffoundations\u002Fbandit-exploration-algorithms":24896,"\u002Freinforcement-learning\u002Ffoundations\u002Fmarkov-decision-processes":24900,"\u002Freinforcement-learning\u002Ffoundations\u002Fvalue-functions-and-optimality":24904,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdynamic-programming":24908,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdp-async-and-gpi":24912,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods":24916,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-off-policy":24920,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftemporal-difference-learning":24924,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftd-control-sarsa-and-q-learning":24928,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping":24932,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-off-policy-methods":24936,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-and-learning":24940,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-focusing-and-decision-time":24944,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdecision-time-planning":24948,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-tree-search":24952,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-prediction":24956,"\u002Freinforcement-learning\u002Fapproximation\u002Ffeature-construction-and-nonlinear":24961,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-control":24965,"\u002Freinforcement-learning\u002Fapproximation\u002Faverage-reward-control":24969,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad":24973,"\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td":24977,"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces":24981,"\u002Freinforcement-learning\u002Fapproximation\u002Ftrue-online-and-sarsa-lambda":24985,"\u002Freinforcement-learning\u002Fapproximation\u002Fpolicy-gradient-methods":24989,"\u002Freinforcement-learning\u002Fapproximation\u002Factor-critic-and-continuous-actions":24993,"\u002Freinforcement-learning\u002Fapproximation\u002Fleast-squares-and-memory-based-methods":24997,"\u002Freinforcement-learning\u002Fapproximation\u002Fmemory-and-kernel-methods":25001,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces":25005,"\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces":25009,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks":25013,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdqn-improvements":25017,"\u002Freinforcement-learning\u002Fdeep-rl\u002Factor-critic-and-ppo":25021,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fppo-and-continuous-control":25025,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fcase-studies":25029,"\u002Freinforcement-learning\u002Fdeep-rl\u002Frl-beyond-games":25033,"\u002Freinforcement-learning\u002Fdeep-rl\u002Ffrontiers":25037,"\u002Freinforcement-learning\u002Fdeep-rl\u002Freward-design-and-open-problems":25041,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow":25045,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow-part-2":25050,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control":25054,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control-part-2":25058,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl":25062,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl-part-2":25066,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration":25070,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration-part-2":25074,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl":25078,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2":25082,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl":25086,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl-part-2":25090,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl":25094,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl-part-2":25098,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl":25102,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl-part-2":25106,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Frlhf-and-language-models":25110,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps":25114,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps-part-2":25118,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl":25122,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl-part-2":25126,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmeta-rl-and-generalization":25130,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fpsychology-of-reinforcement":25134,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Finstrumental-conditioning-and-control":25139,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-and-td-error":25143,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-in-the-brain":25147,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fanimal-learning-and-cognition":25151,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fcognitive-maps-and-planning":25155,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fneuroscience-of-reinforcement":25159,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fseveral-learning-systems":25163,"\u002Freinforcement-learning":25167,"\u002Fartificial-intelligence\u002Ffoundations\u002Fwhat-is-ai":25169,"\u002Fartificial-intelligence\u002Ffoundations\u002Ffoundations-of-ai":25173,"\u002Fartificial-intelligence\u002Ffoundations\u002Fintelligent-agents":25177,"\u002Fartificial-intelligence\u002Ffoundations\u002Fagent-architectures":25181,"\u002Fartificial-intelligence\u002Fsearch\u002Funinformed-search":25185,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-strategies-compared":25190,"\u002Fartificial-intelligence\u002Fsearch\u002Finformed-search":25194,"\u002Fartificial-intelligence\u002Fsearch\u002Fheuristic-functions":25198,"\u002Fartificial-intelligence\u002Fsearch\u002Flocal-search":25202,"\u002Fartificial-intelligence\u002Fsearch\u002Fpopulation-and-continuous-search":25206,"\u002Fartificial-intelligence\u002Fsearch\u002Fadversarial-search":25210,"\u002Fartificial-intelligence\u002Fsearch\u002Fgames-of-chance-and-imperfect-information":25214,"\u002Fartificial-intelligence\u002Fsearch\u002Fconstraint-satisfaction":25218,"\u002Fartificial-intelligence\u002Fsearch\u002Fcsp-search-and-structure":25222,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-under-uncertainty":25226,"\u002Fartificial-intelligence\u002Fsearch\u002Fbelief-state-and-online-search":25230,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-logic":25234,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-inference":25239,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic":25243,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic-in-use":25247,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Finference-and-resolution":25251,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-resolution":25255,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fclassical-planning":25259,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-graphs-and-graphplan":25263,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-in-the-real-world":25267,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-under-uncertainty":25271,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fknowledge-representation":25275,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Freasoning-systems-and-defaults":25279,"\u002Fartificial-intelligence\u002Funcertainty\u002Fprobability-and-bayes":25283,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayes-rule-and-naive-bayes":25288,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayesian-networks":25292,"\u002Fartificial-intelligence\u002Funcertainty\u002Finference-in-bayesian-networks":25296,"\u002Fartificial-intelligence\u002Funcertainty\u002Freasoning-over-time":25300,"\u002Fartificial-intelligence\u002Funcertainty\u002Ftracking-and-data-association":25304,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmaking-decisions":25308,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmarkov-decision-processes":25312,"\u002Fartificial-intelligence\u002Funcertainty\u002Fdecision-networks-and-game-theory":25315,"\u002Fartificial-intelligence\u002Funcertainty\u002Fgame-theory-and-mechanism-design":25319,"\u002Fartificial-intelligence\u002Flearning\u002Flearning-from-examples":25323,"\u002Fartificial-intelligence\u002Flearning\u002Ftheory-and-model-families":25328,"\u002Fartificial-intelligence\u002Flearning\u002Fprobabilistic-learning":25332,"\u002Fartificial-intelligence\u002Flearning\u002Fexpectation-maximization":25336,"\u002Fartificial-intelligence\u002Flearning\u002Freinforcement-learning":25340,"\u002Fartificial-intelligence\u002Flearning\u002Fgeneralization-and-policy-search":25343,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-in-learning":25347,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-based-learning-methods":25351,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fvision-and-perception":25355,"\u002Fartificial-intelligence\u002Ffrontiers\u002Freconstructing-the-3d-world":25360,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobotics":25364,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobot-planning-and-control":25368,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnatural-language-in-ai":25372,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnlp-grammar-translation-and-speech":25376,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fphilosophy-and-future":25380,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fai-ethics-and-future":25384,"\u002Fartificial-intelligence":25388,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-constituents-nuclide-chart":25391,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-size-charge-distributions":25396,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-masses-binding-energy":25400,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fsemi-empirical-mass-formula":25404,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-moments-multipoles":25408,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnuclear-force-shell-overview":25412,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fthe-deuteron":25417,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnucleon-nucleon-scattering":25421,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fmeson-theory-isospin":25425,"\u002Fnuclear-physics\u002Fnuclear-models\u002Ffermi-gas-model":25429,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fliquid-drop-collective-coordinates":25434,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fshell-model-single-particle":25438,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fcollective-model-rotations-vibrations":25442,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-law-modes":25446,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-kinetics-equilibrium":25451,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-decay-gamow-theory":25455,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-fine-structure-hindrance":25460,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fbeta-decay-energetics-neutrino":25464,"\u002Fnuclear-physics\u002Fbeta-decay\u002Ffermi-theory-beta-decay":25469,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fweak-interaction-parity-violation":25473,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fdouble-beta-decay-neutrino-mass":25477,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fgamma-multipole-radiation":25481,"\u002Fnuclear-physics\u002Fgamma-decay\u002Finternal-conversion-isomers":25486,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fangular-correlations-mossbauer":25490,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Freaction-kinematics-cross-sections":25494,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fcompound-nucleus-resonances":25499,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fdirect-reactions-optical-model":25503,"\u002Fnuclear-physics\u002Ffission\u002Ffission-barrier-dynamics":25507,"\u002Fnuclear-physics\u002Ffission\u002Fchain-reactions-reactor-physics":25512,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Ffusion-reactions-confinement":25516,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fstellar-nucleosynthesis":25521,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fbig-bang-nucleosynthesis":25525,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fcharged-particle-stopping-power":25529,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fphoton-neutron-interactions":25534,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fradiation-detectors":25538,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fdosimetry-radiation-biology":25542,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fnuclear-applications-dating-medicine":25546,"\u002Fnuclear-physics":25550,"\u002Fnatural-language-processing\u002Ffoundations\u002Fwhat-is-nlp":25553,"\u002Fnatural-language-processing\u002Ffoundations\u002Fregex-and-text-normalization":25557,"\u002Fnatural-language-processing\u002Ffoundations\u002Fminimum-edit-distance":25561,"\u002Fnatural-language-processing\u002Ffoundations\u002Fn-gram-language-models":25565,"\u002Fnatural-language-processing\u002Ffoundations\u002Fsmoothing-and-backoff":25569,"\u002Fnatural-language-processing\u002Fclassification\u002Fnaive-bayes-and-sentiment":25573,"\u002Fnatural-language-processing\u002Fclassification\u002Fevaluating-classifiers":25578,"\u002Fnatural-language-processing\u002Fclassification\u002Flogistic-regression":25582,"\u002Fnatural-language-processing\u002Fclassification\u002Fsentiment-and-affect-lexicons":25586,"\u002Fnatural-language-processing\u002Fsemantics\u002Fvector-semantics-and-embeddings":25590,"\u002Fnatural-language-processing\u002Fsemantics\u002Fstatic-word-embeddings":25595,"\u002Fnatural-language-processing\u002Fsemantics\u002Fneural-language-models":25599,"\u002Fnatural-language-processing\u002Fsequences\u002Fsequence-labeling":25603,"\u002Fnatural-language-processing\u002Fsequences\u002Fcrfs-and-neural-taggers":25607,"\u002Fnatural-language-processing\u002Fsequences\u002Frnns-and-lstms":25611,"\u002Fnatural-language-processing\u002Ftransformers\u002Ftransformers-and-attention":25615,"\u002Fnatural-language-processing\u002Ftransformers\u002Fthe-transformer-architecture":25619,"\u002Fnatural-language-processing\u002Ftransformers\u002Flarge-language-models":25622,"\u002Fnatural-language-processing\u002Ftransformers\u002Fllm-pretraining-and-scaling":25625,"\u002Fnatural-language-processing\u002Ftransformers\u002Ffine-tuning-and-prompting":25629,"\u002Fnatural-language-processing\u002Ftransformers\u002Fprompting-and-alignment":25633,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-parsing":25637,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcky-scoring-and-evaluation":25642,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdependency-parsing":25646,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fgraph-based-and-neural-dependency-parsing":25650,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fword-senses-and-wsd":25654,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fwsd-in-practice-and-induction":25658,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-roles-and-information-extraction":25662,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Frelations-events-and-templates":25666,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoreference-and-discourse":25670,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoherence-and-discourse-structure":25674,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Flogical-semantics":25678,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcompositional-semantics-and-description-logics":25682,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-parsing":25686,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fneural-semantic-parsing":25690,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Finformation-extraction":25694,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftimes-events-and-templates":25698,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdiscourse-coherence":25702,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fentity-based-and-global-coherence":25706,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-grammars":25710,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftreebanks-and-lexicalized-grammars":25714,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation":25718,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation-decoding-and-evaluation":25722,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering":25726,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering-knowledge-and-llms":25730,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-and-chatbots":25734,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-systems-and-assistants":25738,"\u002Fnatural-language-processing\u002Fapplications\u002Ftext-summarization":25742,"\u002Fnatural-language-processing\u002Fapplications\u002Fabstractive-summarization-and-evaluation":25746,"\u002Fnatural-language-processing\u002Fspeech\u002Fphonetics":25750,"\u002Fnatural-language-processing\u002Fspeech\u002Facoustic-phonetics":25755,"\u002Fnatural-language-processing\u002Fspeech\u002Fautomatic-speech-recognition":25759,"\u002Fnatural-language-processing\u002Fspeech\u002Fasr-evaluation-and-applications":25763,"\u002Fnatural-language-processing":25767,"\u002Fparticle-physics\u002Ffoundations\u002Fhistorical-overview-particle-zoo":25770,"\u002Fparticle-physics\u002Ffoundations\u002Fparticle-physics-basic-concepts":25774,"\u002Fparticle-physics\u002Ffoundations\u002Ffundamental-interactions-force-carriers":25778,"\u002Fparticle-physics\u002Funits-kinematics\u002Fnatural-units-and-scales":25782,"\u002Fparticle-physics\u002Funits-kinematics\u002Ffour-vectors-invariant-mass":25787,"\u002Fparticle-physics\u002Funits-kinematics\u002Fdecay-scattering-kinematics-mandelstam":25791,"\u002Fparticle-physics\u002Funits-kinematics\u002Fcross-sections-golden-rule":25795,"\u002Fparticle-physics\u002Fsymmetries\u002Fconservation-laws-symmetries":25799,"\u002Fparticle-physics\u002Fsymmetries\u002Fdiscrete-symmetries-cpt":25804,"\u002Fparticle-physics\u002Fsymmetries\u002Fparity-violation-weak":25808,"\u002Fparticle-physics\u002Fsymmetries\u002Fsu2-su3-flavor-symmetry":25812,"\u002Fparticle-physics\u002Fquark-model\u002Feightfold-way-su3":25816,"\u002Fparticle-physics\u002Fquark-model\u002Fmeson-spectroscopy":25821,"\u002Fparticle-physics\u002Fquark-model\u002Fbaryon-spectroscopy":25825,"\u002Fparticle-physics\u002Fquark-model\u002Fcolor-confinement-exotics":25829,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fklein-gordon-equation":25833,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fdirac-equation-spinors":25838,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fantiparticles-hole-theory":25842,"\u002Fparticle-physics\u002Fqed\u002Ffeynman-rules-qed":25846,"\u002Fparticle-physics\u002Fqed\u002Fqed-tree-processes":25851,"\u002Fparticle-physics\u002Fqed\u002Frenormalization-running-coupling":25855,"\u002Fparticle-physics\u002Fqed\u002Felectron-g-2":25859,"\u002Fparticle-physics\u002Fweak-interaction\u002Fva-structure-weak":25863,"\u002Fparticle-physics\u002Fweak-interaction\u002Fw-z-bosons-decays":25868,"\u002Fparticle-physics\u002Fweak-interaction\u002Fckm-matrix":25872,"\u002Fparticle-physics\u002Fweak-interaction\u002Fcp-violation-kaons-b-mesons":25876,"\u002Fparticle-physics\u002Fqcd\u002Fcolor-su3-gluons":25880,"\u002Fparticle-physics\u002Fqcd\u002Fasymptotic-freedom-confinement":25885,"\u002Fparticle-physics\u002Fqcd\u002Fdeep-inelastic-scattering-partons":25889,"\u002Fparticle-physics\u002Fqcd\u002Fjets-hadronization":25893,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Felectroweak-su2-u1":25897,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fspontaneous-symmetry-breaking":25902,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-mechanism":25906,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-boson-discovery":25910,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fstandard-model":25914,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-oscillations":25918,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-mass-pmns":25923,"\u002Fparticle-physics\u002Fneutrinos\u002Fdirac-majorana-experiments":25927,"\u002Fparticle-physics\u002Fexperiment\u002Faccelerators-luminosity":25931,"\u002Fparticle-physics\u002Fexperiment\u002Fdetectors-subsystems":25936,"\u002Fparticle-physics\u002Fexperiment\u002Fhow-discoveries-are-made":25940,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fbeyond-standard-model":25944,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fgrand-unified-theories":25948,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fsupersymmetry":25952,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fhierarchy-problem-naturalness":25956,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fdark-matter-candidates":25960,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fmatter-antimatter-open-questions":25964,"\u002Fparticle-physics":25968,"\u002Fastrophysics-cosmology\u002Forientation\u002Fthe-sun-and-stars":25971,"\u002Fastrophysics-cosmology\u002Forientation\u002Fstellar-death-final-states":25976,"\u002Fastrophysics-cosmology\u002Forientation\u002Fgalaxies-and-cosmology":25980,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fmagnitudes-fluxes-and-the-distance-modulus":25984,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fstellar-spectra-and-spectral-classification":25989,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Ftelescopes-and-detectors-across-the-spectrum":25993,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fthe-cosmic-distance-ladder":25997,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fblackbody-radiation-and-specific-intensity":26001,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fradiative-transfer-and-the-transfer-equation":26006,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fspectral-line-formation-and-broadening":26010,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fopacity-and-the-rosseland-mean":26014,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fhydrostatic-equilibrium-and-the-virial-theorem":26018,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equations-of-stellar-structure":26023,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equation-of-state-and-polytropes":26027,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-standard-solar-model":26031,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fthermonuclear-reaction-rates-and-the-gamow-peak":26035,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhydrogen-burning-pp-chains-and-cno":26040,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhelium-burning-and-the-triple-alpha-process":26044,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fadvanced-burning-and-neutron-capture-nucleosynthesis":26048,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fphases-of-the-interstellar-medium":26052,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fmolecular-clouds-and-gravitational-collapse":26057,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fprotostars-and-the-pre-main-sequence":26061,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-main-sequence-and-its-structure":26065,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fpost-main-sequence-low-mass-evolution":26070,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-evolution-of-massive-stars":26074,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fstellar-pulsation-and-the-instability-strip":26078,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":26082,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fcore-collapse-supernovae":26086,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fthermonuclear-supernovae-type-ia":26090,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fneutron-stars-and-pulsars":26094,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fblack-holes-schwarzschild-and-kerr":26098,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fbinary-systems-and-mass-transfer":26102,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Faccreting-compact-objects":26107,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fgravitational-waves-from-inspiraling-binaries":26111,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fmultimessenger-astronomy-and-gamma-ray-bursts":26115,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fthe-milky-way":26119,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-morphology-and-classification":26124,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-rotation-curves-and-dark-matter":26128,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Factive-galactic-nuclei-and-supermassive-black-holes":26132,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-clusters-and-large-scale-structure":26136,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-expanding-universe-and-hubbles-law":26140,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-frw-metric-and-cosmological-redshift":26145,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-friedmann-equations-and-cosmic-dynamics":26149,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fcosmological-models-and-distances":26152,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fdark-energy-and-the-accelerating-universe":26156,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fthe-thermal-history-of-the-universe":26160,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fbig-bang-nucleosynthesis":26165,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Frecombination-and-the-cosmic-microwave-background":26169,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcmb-anisotropies-and-cosmological-parameters":26173,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcosmic-inflation":26177,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fstructure-formation-and-the-growth-of-perturbations":26181,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fdark-matter-dark-energy-and-open-questions":26185,"\u002Fastrophysics-cosmology":26189,"\u002Fcolophon":26192,"\u002F":26195},{"path":21517,"title":21518,"module":21519,"summary":21520},"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm","What Is an Algorithm?","Foundations","An algorithm is a finite, mechanical recipe that transforms inputs into outputs. We define what counts as an algorithm, how we write one down, and the three things we always ask of it: is it correct, is it fast, and can we prove it.\n",{"path":21522,"title":21523,"module":21519,"summary":21524},"\u002Falgorithms\u002Ffoundations\u002Fproof-techniques","Proof Techniques","An algorithm without a proof is a conjecture. This lesson collects the handful\nof arguments that certify the algorithms in this course — direct proof,\ncontrapositive, contradiction, ordinary and strong induction, construction, and\ndisproof by counterexample — each with a small worked\nexample and a picture. Loop invariants are a form of induction,\nrecursive correctness falls to strong induction, and the classic broken proofs\n(all horses are the same color) show where inductions go wrong.\n",{"path":21526,"title":21527,"module":21519,"summary":21528},"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis","Asymptotic Analysis","We measure an algorithm's running time as a function of its input size, then strip away machine-specific constants and lower-order terms to compare algorithms cleanly. This lesson defines the RAM model and the $O$, $\\Omega$, $\\Theta$, $o$, and $\\omega$ notations, proves the polynomial theorem, and shows how to rank growth rates with the limit test, L'Hôpital, base substitution, and the logarithm identities the arguments lean on.\n",{"path":21530,"title":21531,"module":21519,"summary":21532},"\u002Falgorithms\u002Ffoundations\u002Fgrowth-rates-and-loop-analysis","Growth Rates and Loop Analysis","With the asymptotic notations in hand, we rank the functions that actually arise in running times — from constant to factorial — proving the orderings between rungs, then read the running time of a loop nest straight off the page. Sequential blocks add, nested loops multiply, index scaling gives logarithms; a worked trace and a tour of cache-aware and galactic algorithms close the lesson.\n",{"path":21534,"title":21535,"module":21519,"summary":21536},"\u002Falgorithms\u002Ffoundations\u002Frecurrences","Recurrences and the Master Theorem","Recursive and divide-and-conquer algorithms describe their own running time with a recurrence: $T(n)$ in terms of $T$ on smaller inputs. We solve recurrences three ways — drawing the recursion tree, guessing-and-verifying by induction, and applying the Master Theorem — using merge sort as the running example, then handle unequal splits with Akra–Bazzi.\n",{"path":21538,"title":21539,"module":21519,"summary":21540},"\u002Falgorithms\u002Ffoundations\u002Famortized-analysis","Amortized Analysis","Some operations are occasionally expensive but cheap on average across any\nsequence. Amortized analysis bounds the average cost per operation over a\nworst-case sequence — not an expectation — so a rare costly step is paid for by\nthe many cheap ones around it. This lesson develops the aggregate, accounting,\nand potential methods on dynamic-array doubling, the binary counter, and a\nstack with multipop.\n",{"path":21542,"title":21543,"module":21544,"summary":21545},"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort","Divide and Conquer & Mergesort","Divide & Conquer","Divide and conquer breaks a problem into smaller copies of itself, solves\nthem recursively, and stitches the answers together. We meet the paradigm\nthrough mergesort — its merge step, its loop-invariant proof, and the\nrecursion tree that pins its cost at $\\Theta(n\\log n)$ — then count inversions\nwith the same machinery and distill the whole pattern into the master theorem.\n",{"path":21547,"title":21548,"module":21544,"summary":21549},"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort","Quicksort","Quicksort sorts in place by partitioning around a pivot and recursing on\neach side. We give Lomuto and Hoare partitioning with a correctness\ninvariant, see why a bad pivot costs $\\Theta(n^2)$ while a balanced one gives\n$\\Theta(n\\log n)$, and prove that randomizing the pivot makes the expected\ncost $\\Theta(n\\log n)$ on every input.\n",{"path":21551,"title":21552,"module":21544,"summary":21553},"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection","Linear-Time Selection","Finding the $k$-th smallest element looks like it should require sorting, but\nit does not. Quickselect adapts quicksort's partition to recurse on just one\nside, achieving expected $O(n)$. The median-of-medians algorithm guarantees a\ngood pivot with the groups-of-five trick, pushing the worst case down to a\nprovable $O(n)$.\n",{"path":21555,"title":21556,"module":21544,"summary":21557},"\u002Falgorithms\u002Fdivide-and-conquer\u002Ffast-multiplication","Fast Multiplication","Grade-school multiplication is $\\Theta(n^2)$, yet divide and conquer beats it.\nKaratsuba multiplies $n$-bit integers with three half-size products instead of\nfour, giving $\\Theta(n^{\\log_2 3})$, and Strassen multiplies matrices with\nseven block products instead of eight, giving $\\Theta(n^{\\log_2 7})$. Both\nspend cheap additions to save an expensive multiplication, and the master\ntheorem quantifies the savings.\n",{"path":21559,"title":21560,"module":21561,"summary":21562},"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort","Heaps and Heapsort","Sorting & Order Statistics","A binary heap is a tree we store flat in an array, with index arithmetic\nstanding in for pointers. We build the max-heap property bottom-up in $O(n)$\ntime, sort in place in $\\Theta(n\\log n)$ by repeatedly extracting the maximum,\nand reuse the same structure to implement a priority queue.\n",{"path":21564,"title":21565,"module":21561,"summary":21566},"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds","Lower Bounds for Comparison Sorting","Every sort we have seen runs in $\\Omega(n\\log n)$, and that is no accident.\nModeling a sort as a decision tree of comparisons, we show any such tree must\nhave $n!$ leaves, forcing height $\\ge \\log_2(n!) = \\Omega(n\\log n)$ — a bound\nno comparison sort beats in the worst case, on average, or with randomness.\n",{"path":21568,"title":21569,"module":21561,"summary":21570},"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting","Sorting in Linear Time","The $\\Omega(n\\log n)$ barrier only binds algorithms that compare. By instead\nusing keys as array indices we slip past it: counting sort runs in\n$\\Theta(n+k)$ and is stable, radix sort layers it digit by digit, and bucket\nsort averages $\\Theta(n)$ on uniform data. We see exactly when each applies.\n",{"path":21572,"title":21573,"module":21561,"summary":21574},"\u002Falgorithms\u002Fsorting\u002Fexternal-sorting","External Sorting","When the data dwarfs main memory, the cost that matters is no longer\ncomparisons but block transfers to and from disk. External merge sort sorts\nmemory-sized runs, then folds them together with a heap-driven $k$-way merge in\n$\\Theta(\\log_k(N\u002FM))$ passes. Larger fan-out cuts passes; replacement selection\nbuilds longer runs to cut them further.\n",{"path":21576,"title":21577,"module":21578,"summary":21579},"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures","Elementary Data Structures","Data Structures","Every container is built one of two ways: **contiguous** in an array, or\n**linked** through pointers. We trade cache-friendly random access against\n$O(1)$ splicing, derive the **amortized $O(1)$** append of a doubling dynamic\narray, and assemble the two ordered access disciplines — the LIFO **stack** and\nthe FIFO **queue** (with its generalization, the **deque**) — on top of both.\n",{"path":21581,"title":21582,"module":21578,"summary":21583},"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables","Hash Tables","A hash table implements the dictionary — insert, search, delete — in expected\n$O(1)$ time by scattering keys across an array with a hash function. We build\nup from direct addressing, handle collisions by chaining and by open\naddressing, analyze the load factor $\\alpha$, and see how universal hashing\nachieves its expected-time guarantee against every input.\n",{"path":21585,"title":21586,"module":21578,"summary":21587},"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees","Binary Search Trees","A binary search tree keeps keys ordered so that every operation follows a\nsingle root-to-leaf path. We state the BST property, trace search, insert,\nsuccessor, and all three delete cases on concrete trees, prove the inorder\nwalk sorts, and note the drawback — every operation costs $O(h)$, and a\ncarelessly built tree degrades to height $h = \\Theta(n)$, motivating balance.\n",{"path":21589,"title":21590,"module":21578,"summary":21591},"\u002Falgorithms\u002Fdata-structures\u002Favl-trees","AVL Trees","An AVL tree is the first balanced BST: at every node the two subtrees' heights\ndiffer by at most $1$. A Fibonacci-style minimal-node argument forces height\n$h \\le 1.44\\log_2 n = O(\\log n)$, so search, insert, and delete are all\n$O(\\log n)$. Insertion rebalances with at most one of four rotation cases\n(LL, RR, LR, RL); deletion may rotate all the way to the root.\n",{"path":21593,"title":21594,"module":21578,"summary":21595},"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees","Balanced Search Trees","An ordinary BST can degrade to height $\\Theta(n)$; balanced search trees\nguarantee $h = O(\\log n)$ by maintaining invariants and repairing them after\nevery update. We meet rotations, the local restructuring primitive, then\nred-black trees, whose color invariants force logarithmic height, and finally\nB-trees, which trade tall-and-thin for short-and-wide to win on disk.\n",{"path":21597,"title":21598,"module":21578,"summary":21599},"\u002Falgorithms\u002Fdata-structures\u002Funion-find","Disjoint Sets (Union-Find)","The disjoint-set data structure tracks a partition of elements into groups,\nanswering \"are these two in the same group?\" and merging groups on demand. A\nforest of parent pointers, sped up by union by rank and path compression,\ndrives every operation to near-constant $O(\\alpha(n))$ amortized time — the\nstructure behind connectivity queries and Kruskal's minimum spanning tree.\n",{"path":21601,"title":21602,"module":21578,"summary":21603},"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees","Fenwick & Segment Trees","A prefix-sum array answers a range sum in $O(1)$ but pays $O(n)$ per update;\na plain array updates in $O(1)$ but pays $O(n)$ per range sum. Fenwick and\nsegment trees give us _both_ in $O(\\log n)$. The Fenwick (binary indexed) tree\nis a tiny array keyed by the low bit; the segment tree is a general balanced\ntree over canonical ranges that handles any associative aggregate and, with\nlazy propagation, range updates too.\n",{"path":21605,"title":21606,"module":21578,"summary":21607},"\u002Falgorithms\u002Fdata-structures\u002Fspatial-data-structures","Spatial Data Structures","A balanced BST orders keys on a line, but points in the plane have no single\nnatural order. Quadtrees subdivide space recursively into quadrants; k-d trees\nsplit on alternating coordinates at the median. Both make range and\nnearest-neighbour queries fast by carving the plane into boxes a query can\nprune away. Range trees nest a y-tree in an x-tree for fast orthogonal range\nreporting; interval trees index intervals to answer stabbing queries.\n",{"path":21609,"title":21610,"module":21578,"summary":21611},"\u002Falgorithms\u002Fdata-structures\u002Fskip-lists-and-probabilistic-structures","Skip Lists & Probabilistic Structures","Balanced trees achieve $O(\\log n)$ with rotations and invariants; randomization\ngives the same bound far more simply. A skip list is a layered linked list whose\nexpress lanes are chosen by coin flips, giving expected $O(\\log n)$ search and\ninsert with no rebalancing. A Bloom filter trades exactness for space: a bit\narray and a few hashes answer set membership with no false negatives and a\ntunable false-positive rate, but cannot delete.\n",{"path":21613,"title":21614,"module":21578,"summary":21615},"\u002Falgorithms\u002Fdata-structures\u002Fb-trees","B-Trees","When data lives on disk, the cost that dominates is block transfers, not\ncomparisons — and a binary tree of a billion keys is thirty reads deep. A\nB-tree of minimum degree $t$ is short and wide: $t-1$ to $2t-1$ keys per node,\nall leaves at one depth, height $O(\\log_t n)$. Insertion splits a full node on\nthe way down and pushes its median up; deletion borrows or merges to keep nodes\nfull enough. High fan-out is what minimizes disk I\u002FO.\n",{"path":21617,"title":21618,"module":21578,"summary":21619},"\u002Falgorithms\u002Fdata-structures\u002Fdata-stream-algorithms","Data-Stream Algorithms","Most of this course assumes data sits in fast memory, addressable at will.\nExternal sorting relaxed that to a re-readable disk. The streaming model goes\nfurther: items arrive one at a time, are seen once, and must be discarded, with\nonly sublinear, often polylogarithmic, memory. In exchange, the answers are\napproximate and probabilistic. We set up the model, then meet reservoir\nsampling for a uniform sample of an unknown-length stream and Morris counting\nfor an approximate tally in doubly-logarithmic space.\n",{"path":21621,"title":21622,"module":21578,"summary":21623},"\u002Falgorithms\u002Fdata-structures\u002Fstreaming-sketches","Streaming Sketches","Sampling and counting kept a random subset or a single approximate tally.\nSketches go further: fixed, tiny summaries that answer questions about a\nstream's frequencies. We meet the Count–Min sketch for point frequency\nestimation, Misra–Gries for heavy hitters, and HyperLogLog for distinct\ncounts, each trading a controlled error for space that never grows with the\nstream.\n",{"path":21625,"title":21626,"module":21627,"summary":21628},"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows","Two Pointers & Sliding Windows","Sequences & Strings","A family of array idioms that collapse an obvious $O(n^2)$ scan into a single\n$O(n)$ pass by maintaining an invariant as indices move. We meet two pointers\n(converging on a sorted array, and a fast\u002Fslow pair for in-place rewriting)\nand the sliding window (fixed and variable size, amortized $O(n)$). The\ncompanion lesson on prefix sums picks up where the window's positivity\nassumption fails.\n",{"path":21630,"title":21631,"module":21627,"summary":21632},"\u002Falgorithms\u002Fsequences\u002Fprefix-sums","Prefix Sums & Difference Arrays","Prefix sums precompute the running total once so that any range-sum query is a\nsingle subtraction, $P[r{+}1]-P[l]$, in $O(1)$. A hash map of prefix\nfrequencies then counts subarrays summing to $k$ in $O(n)$ — even with negative\nentries, where the sliding window fails. The difference-array dual turns $m$\nrange-adds into $O(m+n)$, and the whole idea lifts to 2-D rectangle sums by\ninclusion–exclusion.\n",{"path":21634,"title":21635,"module":21627,"summary":21636},"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks","Monotonic Stacks & Queues","A **monotonic stack** keeps its contents sorted by popping every element that\nwould break the order before each push — turning a family of \"previous\u002Fnext\ngreater (or smaller) element\" questions into a single $O(n)$ scan. We trace\nthe next-greater-element routine push by push and prove its amortized bound,\nfuse two such scans to measure the **largest rectangle in a histogram** in\nlinear time, extend the idea to a **monotonic deque** that streams the\n**sliding-window maximum** in $O(n)$, and use asymmetric tie-breaking to\ncount **subarray minimums** without double-counting duplicates.\n",{"path":21638,"title":21639,"module":21627,"summary":21640},"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer","Binary Search on the Answer","Binary search locates the boundary of a **monotone predicate** $p(x)$ in\n$O(\\log(\\text{range}))$ probes; sorted arrays are only one instance. We first\nestablish the half-open `while (lo \u003C hi)` template for $\\textsc{lower\\_bound}$\nand $\\textsc{upper\\_bound}$, then generalize to \"binary search on the answer\":\nwhenever feasibility is monotone in a numeric parameter, we binary search the\nparameter itself, calling a feasibility check at each step.\n",{"path":21642,"title":21643,"module":21627,"summary":21644},"\u002Falgorithms\u002Fsequences\u002Fstring-matching","String Matching: Naive & Rabin–Karp","Given a text $T$ of length $n$ and a pattern $P$ of length $m$, find every\noccurrence of $P$ in $T$. The naive scan costs $O(nm)$ and re-reads text it has\nalready seen. Rabin–Karp fixes the first inefficiency with a **rolling hash**:\neach length-$m$ window is summarized by one number, updated in $O(1)$ per slide,\nverified on a hash match to kill collisions, for expected $O(n+m)$. A companion\nlesson removes the re-reading entirely with KMP and the Z-function.\n",{"path":21646,"title":21647,"module":21627,"summary":21648},"\u002Falgorithms\u002Fsequences\u002Fkmp-and-z-function","String Matching: KMP & the Z-Function","Two linear-time matchers that beat Rabin–Karp's expected bound with a\nworst-case guarantee and no randomness. KMP precomputes a **failure function**\n$\\pi$ so a mismatch slides the pattern by $q-\\pi[q-1]$ and the text pointer\nnever backs up, for $O(n+m)$. The **Z-function** computes the longest\nprefix-match at every position via the Z-box, giving the same bound from a\ndifferent angle; the two encodings of a string's self-overlap convert freely.\n",{"path":21650,"title":21651,"module":21627,"summary":21652},"\u002Falgorithms\u002Fsequences\u002Ftries","Tries & Prefix Trees","A **trie** stores a set of strings in a tree keyed by _characters_, so that\ninsert, search, delete, and prefix-test all run in $O(L)$ time — the length\nof the key, _independent of how many keys are stored_. Shared prefixes are\nstored once, which makes tries the natural structure for autocomplete,\nwildcard dictionaries, board word-search, and — over the alphabet $\\{0,1\\}$\n— the maximum-XOR-pair problem. Radix (Patricia) trees compress the chains.\n",{"path":21654,"title":21655,"module":21627,"summary":21656},"\u002Falgorithms\u002Fsequences\u002Fsuffix-arrays-and-aho-corasick","Suffix Arrays, LCP & Aho–Corasick","A **suffix array** sorts all $n$ suffixes of a string, indexing every substring\nat once; built in $O(n\\log n)$, it locates a pattern by binary search in\n$O(m\\log n)$. Its companion **LCP array** (Kasai's $O(n)$ algorithm) counts\ndistinct substrings and finds the longest repeated substring. **Aho–Corasick**\ngeneralises KMP to a whole dictionary: a trie of patterns plus failure links\nscans the text once in $O(\\text{text} + \\text{matches})$ to report every\noccurrence of every pattern. Manacher's algorithm finds all palindromic\nsubstrings in $O(n)$.\n",{"path":21658,"title":21659,"module":21660,"summary":21661},"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal","Graph Representations and Traversal","Graphs","A graph captures _relationships_ — who connects to whom. We fix the\nvocabulary, weigh the two standard representations (adjacency list versus\nmatrix), then meet the single search skeleton behind everything that follows:\nWhatever-First-Search, and its breadth-first reading, which finds shortest\npaths by number of edges in $O(V + E)$.\n",{"path":21663,"title":21664,"module":21660,"summary":21665},"\u002Falgorithms\u002Fgraphs\u002Fdepth-first-search","Depth-First Search","Swap BFS's queue for a stack and the search plunges instead of fanning out.\nDepth-first search stamps every vertex with discovery and finish times that\nnest like parentheses, classifies each edge as tree, back, forward, or cross,\nand — through the back edge — decides in one pass whether a graph has a cycle.\nThese timestamps underpin topological sort, strong\nconnectivity, and the rest of this module.\n",{"path":21667,"title":21668,"module":21660,"summary":21669},"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc","Topological Sort and Strong Connectivity","Directed acyclic graphs model dependencies: tasks that must precede other\ntasks. A _topological order_ lays such a graph out in a line so every edge\npoints forward, and depth-first finish times yield one almost for free.\nWe then ask the harder question for graphs _with_ cycles: which vertices can\nreach each other? The answer is the strongly connected components, found by a\ntwo-pass DFS.\n",{"path":21671,"title":21672,"module":21660,"summary":21673},"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees","Minimum Spanning Trees","Given a weighted network, how do we connect everything as cheaply as possible?\nThe answer is a minimum spanning tree, and one lemma — the cut property —\njustifies _every_ correct MST algorithm. We prove the cut and cycle\nproperties by exchange arguments, use them to settle uniqueness, and meet the\noldest MST algorithm, Borůvka's, whose parallel component-merging rounds fall\nstraight out of the cut rule.\n",{"path":21675,"title":21676,"module":21660,"summary":21677},"\u002Falgorithms\u002Fgraphs\u002Fkruskal-and-prim","Kruskal and Prim","The two minimum-spanning-tree algorithms you will actually implement.\nKruskal grows a forest edge by edge, cheapest first, using a union-find\nstructure to reject cycle-closing edges; Prim grows one tree outward from a\nroot with a priority queue, exactly Dijkstra rekeyed by attachment cost. Both\ntraced in full on a nine-town graph, with the edge cases, the bottleneck\nproperty, and where each one wins.\n",{"path":21679,"title":21680,"module":21660,"summary":21681},"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths","Shortest Paths","Finding the cheapest route through a weighted network is one of the most-used\nalgorithms in computing, and a single operation — _relaxation_ — underlies\nevery method. We build the primitive, prove the triangle inequality and\noptimal substructure that make it work, then meet Dijkstra's algorithm: the\ngreedy solution for non-negative weights, traced vertex by vertex, with the\ncut argument that proves each extraction is final.\n",{"path":21683,"title":21684,"module":21660,"summary":21685},"\u002Falgorithms\u002Fgraphs\u002Fall-pairs-and-negative-weights","All-Pairs and Negative Weights","Dijkstra's greedy schedule breaks the moment an edge goes negative. We give it\nup for dynamic programming: Bellman-Ford derived as a DP over edge budgets,\nwith its negative-cycle detector, and Floyd-Warshall computing the distance\nbetween _every_ pair of vertices via a DP over which vertices a path may pass\nthrough. We close with Johnson's algorithm and the arbitrage problems that\nnegative cycles encode.\n",{"path":21687,"title":21688,"module":21660,"summary":21689},"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow","Network Flow","How much can flow through a network from source to sink? We build flow\nnetworks with capacity and conservation constraints, increase a flow by\npushing along augmenting paths in the residual graph, and see how reverse\nedges let the algorithm undo earlier routing. Ford-Fulkerson and its BFS refinement\nEdmonds-Karp find a maximum flow, traced end to end on a worked network.\n",{"path":21691,"title":21692,"module":21660,"summary":21693},"\u002Falgorithms\u002Fgraphs\u002Fmax-flow-min-cut","Max-Flow Min-Cut and Applications","Why is the flow found when no augmenting path remains actually optimal? The\nanswer is a duality theorem: the maximum flow equals the minimum cut. We prove\nit, read the minimum cut off the final residual graph, then derive bipartite\nmatching and a catalog of modeling reductions from the flow\nabstraction — before touching the modern algorithms that supersede\nEdmonds-Karp.\n",{"path":21695,"title":21696,"module":21660,"summary":21697},"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points","Bridges & Articulation Points","A **bridge** is an edge whose removal disconnects the graph; an **articulation\npoint** is a vertex whose removal does. Both are single points of failure in a\nnetwork. A single depth-first search computes discovery times and **low-links**,\nand two local criteria — $low[v] > disc[u]$ for bridges, $low[v] \\ge disc[u]$\nfor cut vertices — find them all in $O(V+E)$.\n",{"path":21699,"title":21700,"module":21660,"summary":21701},"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor","Lowest Common Ancestor & Binary Lifting","Given a rooted tree, the lowest common ancestor of $u$ and $v$ is the deepest\nnode that is an ancestor of both. A naive walk answers one query in $O(h)$;\n**binary lifting** precomputes the $2^k$-th ancestor of every node in\n$O(n\\log n)$, then answers $k$-th-ancestor and LCA queries in $O(\\log n)$ each.\nWe derive both jumps, apply them to tree distance, and compare against the\nEuler-tour + RMQ and Tarjan offline alternatives.\n",{"path":21703,"title":21704,"module":21660,"summary":21705},"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat","2-SAT via Implication Graphs","A boolean formula whose every clause has exactly two literals can be solved in\n_linear_ time — even though its three-literal cousin is NP-complete. The idea\nis to read each clause as a pair of implications, build a directed graph on the\n$2n$ literals, and ask a question we already know how to answer: which literals\nshare a strongly connected component? The formula is satisfiable iff no variable\nlands in the same SCC as its own negation, and the SCCs' topological order\nyields a satisfying assignment for free.\n",{"path":21707,"title":21708,"module":21660,"summary":21709},"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours","Eulerian Tours","An **Eulerian tour** uses every _edge_ of a graph exactly once. We give the\nexact parity and balance conditions under which one exists (even degree\nfor undirected graphs, in-degree equal to out-degree for directed) and Hierholzer's\n$O(E)$ algorithm that constructs one by splicing closed sub-tours. We contrast\nthis sharply with the **Hamiltonian** problem (visit every _vertex_ once),\nwhich is NP-complete: visiting edges is easy, visiting vertices is hard.\n",{"path":21711,"title":21712,"module":21660,"summary":21713},"\u002Falgorithms\u002Fgraphs\u002Fbipartite-matching","Bipartite Matching","Pairing applicants to jobs, students to slots, files to disks: all are\n**maximum bipartite matching**. We solve it combinatorially with **augmenting\npaths** (Kuhn's algorithm, $O(VE)$), speed it up to $O(E\\sqrt V)$ with\n**Hopcroft–Karp**, and uncover the structure behind it — **König's theorem**\n(max matching equals min vertex cover) and **Hall's marriage theorem** (a\nperfect matching exists iff every set has enough neighbors).\n",{"path":21715,"title":21716,"module":21717,"summary":21718},"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method","The Greedy Method","Greedy Algorithms","A greedy algorithm builds a solution one locally-best choice at a time and\nnever looks back. We isolate the two properties that make this work — the\ngreedy-choice property and optimal substructure — prove the canonical\nactivity-selection algorithm correct with an exchange argument, watch greedy\nfail on the 0\u002F1 knapsack, and glimpse matroids as the theory\nthat says exactly when the greedy method is optimal.\n",{"path":21720,"title":21721,"module":21717,"summary":21722},"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals","Scheduling & Interval Partitioning","Three classic scheduling problems all yield to greedy algorithms — and all\nthree turn on a single design decision: which key to sort by. Interval\nscheduling sorts by **finish** time to pack the most compatible jobs;\ninterval partitioning sorts by **start** time and proves the rooms needed\nequal the maximum overlap **depth**; minimizing maximum lateness sorts by\n**deadline** and is justified by an adjacent-swap exchange argument.\n",{"path":21724,"title":21725,"module":21717,"summary":21726},"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes","Huffman Codes","Huffman coding builds a\nprovably optimal prefix-free binary code by repeatedly merging the two least\nfrequent symbols. We develop prefix-free codes as binary trees, give the\nalgorithm with a priority queue, build a Huffman tree from example\nfrequencies, prove optimality with the same greedy-choice-plus-substructure\nargument, and pin the running time at $O(n\\log n)$.\n",{"path":21728,"title":21729,"module":21717,"summary":21730},"\u002Falgorithms\u002Fgreedy\u002Fmatroids","Matroids & Exchange Arguments","The capstone of the greedy module: _why_ and _when_ a greedy algorithm is\nprovably optimal. We recap the two correctness templates — **greedy-stays-ahead**\nand the **exchange argument** — then meet the **matroid** $M=(S,\\mathcal{I})$, an\nabstraction whose **exchange property** is the structure greedy needs.\nThe matroid–greedy theorem says sorting by weight and taking what stays\nindependent yields a maximum-weight basis _if and only if_ the structure is a\nmatroid. Kruskal's MST is the canonical instance; 0\u002F1 knapsack and TSP are the\ncanonical failures.\n",{"path":21732,"title":21733,"module":21717,"summary":21734},"\u002Falgorithms\u002Fgreedy\u002Fstable-matching","Stable Matching (Gale–Shapley)","Two sides each rank the other; we want a matching with no **blocking pair** — no\ntwo participants who both prefer each other to their assigned partners. The\n**Gale–Shapley deferred-acceptance** algorithm has proposers propose in\npreference order while receivers tentatively hold the best offer so far. We prove\nit terminates in $\\O(n^2)$ proposals, returns a **perfect** matching, and that\nthe matching is **stable**. A sharper asymmetry follows: deferred acceptance is\n**proposer-optimal** and **receiver-pessimal**, the structural fact behind the\nresidency match and school-choice systems.\n",{"path":21736,"title":21737,"module":21738,"summary":21739},"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples","Principles of Dynamic Programming","Dynamic Programming","Dynamic programming is recursion with memory: when a recursive solution\nre-solves the same subproblems again and again, we solve each one once and\nstore the answer. We identify the two structural conditions that make this\nwork — overlapping subproblems and optimal substructure — contrast top-down\nmemoization with bottom-up tabulation, and distil the whole method into a\nfive-step recipe.\n",{"path":21741,"title":21742,"module":21738,"summary":21743},"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp","Sequence Alignment & LCS","Two strings can be compared by how much of one appears inside the\nother. The longest common subsequence (LCS) and edit distance are the two\nclassic measures, and they are the _same_ dynamic program with different\ncosts. We derive the LCS recurrence by examining the last characters, fill a\nworked DP table, reconstruct the subsequence, and then show edit distance as\nthe identical $\\Theta(mn)$ pattern.\n",{"path":21745,"title":21746,"module":21738,"summary":21747},"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence","Longest Increasing Subsequence","Given a sequence of numbers, how long is its longest strictly increasing\nsubsequence? A first dynamic program indexes subproblems by the element each\nsubsequence _ends at_, giving an $O(n^2)$ solution with parent-pointer\nreconstruction. A sharper idea, the patience-sorting _tails_ array searched by\nbinary search, drops the time to $O(n\\log n)$. We then fold in the\nvariants: non-decreasing, counting, Russian-doll envelopes, and bitonic.\n",{"path":21749,"title":21750,"module":21738,"summary":21751},"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack","Knapsack & Subset Problems","We start from $\\textsc{Subset-sum}$ — does some sublist hit a target $t$? — and its\ninclude\u002Fexclude recurrence over a boolean table $A(i, u)$, then bolt on values\nto get 0\u002F1 knapsack as the same machine with $\\lor$ promoted to $\\max$. We fill\nboth tables, recover the chosen items, and confront the surprise that the\n$\\Theta(nt)$ running time is only _pseudo-polynomial_ — exponential in the bit\nlength $b$, and unimprovable unless $\\mathrm{P}=\\mathrm{NP}$ since subset-sum is\n$\\textsc{NP-complete}$. The fractional variant reveals the sharp line between greedy\nand dynamic programming.\n",{"path":21753,"title":21754,"module":21738,"summary":21755},"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded","Coin Change & Unbounded Knapsack","The previous lesson let each item be taken at most once. Drop that cap — items\nmay be reused _any number of times_ — and the 0\u002F1 knapsack collapses from a\ntwo-dimensional table to a one-dimensional one, because there is no longer a\nprefix of \"already-used\" items to track. We meet **unbounded knapsack**, then\nits most famous instance, **coin change**: the minimum-coins recurrence\n$C[a] = 1 + \\min_c C[a-c]$, and the counting variant where the _order of the\nloops_ decides whether you count unordered combinations or ordered sequences —\nthe classic bug. Greed fails in general but works for canonical coin systems.\n",{"path":21757,"title":21758,"module":21738,"summary":21759},"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp","Interval DP","Many problems ask for the best way to combine a contiguous range of items, and\nthe answer is a dynamic program over subintervals $[i,j]$ that chooses a split\npoint $k$. We derive the pattern from matrix-chain multiplication —\nparenthesising a product to minimize scalar multiplications in $O(n^3)$ — distil\nit into a reusable template filled by increasing interval length, and then meet\nits sharpest variant: the \"last operation\" trick behind Burst Balloons and\ncutting a stick, where fixing the _last_ move (not the first) makes the two\nsides independent.\n",{"path":21761,"title":21762,"module":21738,"summary":21763},"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp","Dynamic Programming on Trees","When the subproblems of a dynamic program are _rooted subtrees_, a single\npost-order DFS solves the whole thing in $O(n)$: each node combines the\nalready-computed answers of its children. We meet the archetype — maximum-weight\nindependent set on a tree — then the \"path through a node\" pattern behind tree\ndiameter and maximum path sum, and finally **rerooting**, which computes a\nper-node answer for _every_ node as root in $O(n)$ with two passes.\n",{"path":21765,"title":21766,"module":21738,"summary":21767},"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp","Bitmask DP","When a subproblem depends not on an index or a prefix but on _which subset_ of\na small ground set has been used, we can encode that subset as the bits of an\ninteger and index a DP table by it. With $n \\le \\sim 20$ the $2^n$ subsets fit\nin a table, turning $\\Theta(n!)$ brute force into $O(2^n \\cdot \\text{poly}(n))$.\nWe meet the bit tricks, the Held–Karp TSP archetype, assignment by mask,\nsubset-sum partitioning, and submask enumeration with its $3^n$ bound.\n",{"path":21769,"title":21770,"module":21738,"summary":21771},"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations","DP Optimizations","A correct DP recurrence is only half the battle; its naive evaluation is often\na factor of $n$ slower than necessary. This capstone surveys five techniques,\nmonotonic-queue, the convex hull trick, divide-and-conquer optimization,\nKnuth's optimization, and SOS DP, that each exploit _structure in the\ntransition_ (a sliding window, linear costs, monotone optimal splits, the\nquadrangle inequality, or subset lattices) to shave an $O(n)$, $O(\\log n)$, or\nworse factor off the running time.\n",{"path":21773,"title":21774,"module":21738,"summary":21775},"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs","Dynamic Programming on Graphs","Many graph algorithms are dynamic programs: the subproblem is the\n_best value reachable under a restricted resource_ — intermediate vertices\nallowed, edges allowed, or a topological prefix — and edge _relaxation_ is the\nDP transition. We frame Floyd–Warshall as the archetype ($O(V^3)$ all-pairs\nshortest paths), Bellman–Ford as a DP over path length (the at-most-$K$-stops\nvariant), DAG-DP in topological order ($O(V+E)$), and Warshall's transitive\nclosure as the boolean analog.\n",{"path":21777,"title":21778,"module":21738,"summary":21779},"\u002Falgorithms\u002Fdynamic-programming\u002Fdigit-and-probability-dp","Digit & Probability DP","Two DP patterns with unusual state. _Digit DP_ counts the\nintegers in a range $[L, R]$ that satisfy a digit constraint by walking the\ndecimal places of the bound, carrying a _tight_ flag that marks when the prefix\nstill equals the bound's. _Probability\u002FExpectation DP_ replaces \"best value\" with\n\"expected value,\" using linearity of expectation to make each state an\naverage over its weighted transitions — the natural tool for expected step\ncounts and absorbing Markov chains.\n",{"path":21781,"title":21782,"module":21783,"summary":21784},"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals","Backtracking: Subsets, Permutations & Combinations","Backtracking & Search","Backtracking builds a solution one choice at a time and abandons a partial\nsolution the moment it cannot be completed, exploring a state-space tree by\ndepth-first search. We meet the universal choose\u002Fexplore\u002Fun-choose template,\nderive the canonical enumerations — subsets ($2^n$), permutations ($n!$), and\ncombinations ($\\binom{n}{k}$) — handle duplicate elements by skipping equal\nsiblings, and see how pruning turns an exponential search into a tractable one.\n",{"path":21786,"title":21787,"module":21783,"summary":21788},"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search","Constraint Search: N-Queens & Sudoku","Many hard puzzles are **constraint satisfaction problems**: assign each\nvariable a value from its domain so that every constraint holds. Backtracking\nsolves them by assigning variables one at a time and rejecting a partial\nassignment the instant a constraint breaks. We make the rejection cheap — $O(1)$\nconflict checks for N-Queens via column and diagonal sets — and prune harder\nwith **forward checking**, **MRV** ordering, and **constraint propagation**,\nwhich is what lets an exponential search actually finish.\n",{"path":21790,"title":21791,"module":21783,"summary":21792},"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound","Branch & Bound and Meet in the Middle","Plain backtracking prunes a search tree by _feasibility_; for _optimization_\nproblems we can prune far more aggressively by _value_. **Branch and bound**\nkeeps the best complete solution found so far and discards any partial solution\nwhose optimistic bound cannot beat it. **Meet in the middle** splits the\ninstance in two, enumerates each half, and recombines by binary search — turning\n$2^n$ into $O(2^{n\u002F2}\\,n)$ and pushing exact search out to $n \\approx 40$.\n",{"path":21794,"title":21795,"module":21783,"summary":21796},"\u002Falgorithms\u002Fbacktracking\u002Fgraph-backtracking","Graph Backtracking: m-Coloring & Hamiltonian Paths","Two famous graph problems have no known efficient algorithm, yet yield cleanly\nto backtracking with the right pruning. **Graph $m$-coloring** assigns one of\n$m$ colors to each vertex so no edge is monochromatic; we color vertices in turn\nand reject a color the instant a neighbor already has it. **Hamiltonian\npath\u002Fcycle** asks for a walk visiting every vertex exactly once; we extend a path\ngreedily and backtrack on dead ends. Both are NP-complete, so the worst case is\nexponential — but feasibility pruning and good vertex ordering make real\ninstances tractable, and the contrast with the easy Eulerian condition shows why.\n",{"path":21798,"title":21799,"module":21800,"summary":21801},"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics","Number Theory: GCD & Modular Arithmetic","Mathematical Algorithms","This lesson opens the mathematical-algorithms module with the bedrock of\ncomputational number theory. We prove Euclid's recurrence\n$\\gcd(a,b)=\\gcd(b,\\,a\\bmod b)$ and its $O(\\log\\min(a,b))$ running time, extend\nit to recover Bézout coefficients $x,y$ with $ax+by=\\gcd(a,b)$, and build\nmodular arithmetic on residue classes — including when a modular inverse\n$a^{-1}\\bmod m$ exists and how to compute it.\n",{"path":21803,"title":21804,"module":21800,"summary":21805},"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality","Modular Exponentiation & Primality","Computing $a^n \\bmod m$ naively costs $n$ multiplications; **repeated squaring**\ndoes it in $O(\\log n)$ by reading the bits of the exponent. We use this routine\nto state **Fermat's little theorem** (and the modular inverse it gives), then to\ntest primality — trial division, the probabilistic **Fermat** and **Miller–Rabin**\ntests, and the deterministic witness set that settles primality for every 64-bit\nnumber.\n",{"path":21807,"title":21808,"module":21800,"summary":21809},"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization","Sieves & Factorization","The previous lesson tested one number for primality; here we ask for _all_\nprimes up to $n$ at once. The **sieve of Eratosthenes** cross-cuts composites\nin $O(n\\log\\log n)$, and a **linear sieve** does it in $O(n)$ while recording\neach number's **smallest prime factor**, which then factors any $x \\le n$ in\n$O(\\log x)$. From a factorization $x = \\prod p_i^{e_i}$ the multiplicative\nfunctions $\\tau$, $\\sigma$, and Euler's totient $\\varphi$ fall out immediately.\n",{"path":21811,"title":21812,"module":21800,"summary":21813},"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics","Combinatorics & Counting","Counting is the arithmetic of finite sets. We build up from permutations\n$n!$ and combinations $\\binom{n}{k}$, prove Pascal's rule by a bijection,\nand count multisets with stars and bars. The practical core is computing\n$\\binom{n}{k}\\bmod p$ in $O(1)$ from precomputed factorials and inverse\nfactorials. We close with inclusion–exclusion and the Chinese Remainder\nTheorem, both of which lean on the modular inverse from the previous lesson.\n",{"path":21815,"title":21816,"module":21800,"summary":21817},"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmatrix-exponentiation","Matrix Exponentiation","A linear recurrence advances by a fixed linear rule, so one step is a\n**matrix–vector** product and $n$ steps are a **matrix power**. Packaging\nFibonacci, and any $k$-term recurrence, into a transition matrix lets us jump\nto the $n$-th term in $O(k^3 \\log n)$ by **exponentiation by squaring** — the\nsame doubling trick from modular exponentiation, now over matrices.\n",{"path":21819,"title":21820,"module":21800,"summary":21821},"\u002Falgorithms\u002Fmathematical-algorithms\u002Ffast-fourier-transform","Fast Fourier Transform","Multiplying two degree-$n$ polynomials by the schoolbook method costs\n$\\Theta(n^2)$. Evaluating them at the **$n$-th roots of unity** turns\nmultiplication into pointwise products, and the **Cooley–Tukey FFT** computes\nall those evaluations in $\\Theta(n\\log n)$ by splitting even and odd\ncoefficients. The inverse FFT interpolates back, giving $\\Theta(n\\log n)$\npolynomial and big-integer multiplication.\n",{"path":21823,"title":21824,"module":21800,"summary":21825},"\u002Falgorithms\u002Fmathematical-algorithms\u002Fgradient-descent","Numerical Optimization and Gradient Descent","Most of this course chases **discrete** optima over finite structures; here the\nsearch space is **continuous** and the objective $f$ is differentiable. The\n**gradient** points uphill, so stepping against it —\n$x_{t+1} = x_t - \\eta\\,\\nabla f(x_t)$ — walks downhill. **Convexity** makes every\nlocal minimum global; for convex $L$-smooth $f$ gradient descent converges at\n$O(1\u002Ft)$, and **geometrically** under strong convexity. **Newton's method** uses\nthe Hessian for local quadratic convergence, and **bisection** is the robust\nbracketing fallback for roots.\n",{"path":21827,"title":21828,"module":21829,"summary":21830},"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives","Geometric Primitives & Orientation","Computational Geometry","Computational geometry is built on a single reliable primitive — the\n**orientation test**, a sign of a cross product that tells whether three points\nturn left, right, or lie collinear. From points-as-vectors and the dot and\ncross products we derive orientation, segment intersection, the shoelace area\nformula, and point-in-polygon tests, keeping all arithmetic **exact and\ninteger** so that no floating-point rounding can corrupt a sign.\n",{"path":21832,"title":21833,"module":21829,"summary":21834},"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull","Convex Hull","The convex hull is the smallest convex polygon enclosing a point set — the\nrubber band snapped around the nails. We build it with Andrew's monotone chain,\nsorting by $(x,y)$ and sweeping a lower and upper hull while popping any\nnon-left turn via the orientation primitive, in $O(n\\log n)$. A reduction from\nsorting shows that bound is optimal, and the hull yields diameter, smallest\nenclosing rectangle, and more through rotating calipers.\n",{"path":21836,"title":21837,"module":21829,"summary":21838},"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line","Sweep-Line Algorithms","The plane-sweep paradigm turns a static $2$-D geometry problem into a dynamic\n$1$-D ordered-set problem: a vertical line sweeps left to right, stopping at an\n$x$-sorted **event queue** while a balanced-BST **status structure** tracks the\nobjects it currently crosses, ordered by $y$. We derive Bentley–Ottmann segment\nintersection in $O((n+k)\\log n)$, recover closest-pair in $O(n\\log n)$, and\nreduce skyline, rectangle-area, and overlap problems to $\\pm1$ event sweeps.\n",{"path":21840,"title":21841,"module":21829,"summary":21842},"\u002Falgorithms\u002Fcomputational-geometry\u002Fpolygons-and-proximity","Polygons & Proximity","Four classics that live on top of the orientation primitive and the convex\nhull. **Closest pair** falls to divide-and-conquer in $\\Theta(n\\log n)$, where a\npacking argument caps the cross-boundary combine at seven neighbours per point.\n**Point-in-polygon** is the ray-casting parity test or the winding-number count\nthat also handles self-intersecting boundaries, both with their edge caveats. The **shoelace formula**\ngives signed area as a sum of cross products, and **rotating calipers** walk the\nhull to read off diameter and width in $O(n)$.\n",{"path":21844,"title":21845,"module":21846,"summary":21847},"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions","P, NP, and Reductions","Intractability","Most problems we have met so far have fast algorithms. A vast and important\nfamily seemingly does not. This lesson builds the vocabulary for that\ndivide: decision problems, the class $\\mathsf{P}$ of problems we can solve\nquickly, the class $\\mathsf{NP}$ of problems whose solutions we can _check_\nquickly, and polynomial-time reductions, the tool that lets us compare the\ndifficulty of two problems without solving either.\n",{"path":21849,"title":21850,"module":21846,"summary":21851},"\u002Falgorithms\u002Fintractability\u002Fnp-completeness","NP-Completeness","Some problems in $\\mathsf{NP}$ are universally hardest: every other problem\nin $\\mathsf{NP}$ reduces to them. This lesson defines $\\mathsf{NP}$-hard and\n$\\mathsf{NP}$-complete, states the Cook–Levin theorem that anchors the\ntheory on **SAT**, walks the web of reductions that grows from it, and gives\nthe four-step recipe for proving a brand-new problem $\\mathsf{NP}$-complete.\n",{"path":21853,"title":21854,"module":21846,"summary":21855},"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness","Coping with NP-Hardness","An $\\mathsf{NP}$-hardness proof rules out an exact polynomial-time algorithm,\nnot the need for answers. This lesson surveys four practical responses to\nhardness: approximation algorithms with a provable ratio (worked through a\n2-approximation for vertex cover), heuristics and local search, exact\nexponential methods like branch and bound, and exploiting special structure\nin the instances you actually face.\n",{"path":21857,"title":21858,"module":21846,"summary":21859},"\u002Falgorithms\u002Fintractability\u002Fapproximation-algorithms","Approximation Algorithms","When a problem is $\\mathsf{NP}$-hard we can still ask for a solution\nprovably close to optimal. This lesson makes the approximation ratio\n$\\rho$ precise, separates absolute from relative guarantees, and proves the\nratios of four classic algorithms: greedy set cover ($H_n \\approx \\ln n$),\nthe MST-doubling $2$-approximation for metric TSP, load balancing, and the\nknapsack FPTAS. It closes with the hierarchy PTAS \u002F FPTAS and the limits of\ninapproximability.\n",{"path":21861,"title":21862,"module":6,"summary":6},"\u002Falgorithms","Algorithms",{"path":21864,"title":21865,"module":21866,"summary":21867},"\u002Fcalculus\u002Flimits-and-continuity\u002Ffunctions-and-models","Functions and Mathematical Models","Limits and Continuity","A function assigns exactly one output to each input and can be presented four ways: verbally, numerically, graphically, or by a formula. The elementary families — linear, polynomial, power, rational, trigonometric, exponential — model most elementary phenomena, and transformation, combination, and composition build every other function from them.\n",{"path":21869,"title":21870,"module":21866,"summary":21871},"\u002Fcalculus\u002Flimits-and-continuity\u002Fthe-limit-of-a-function","The Limit of a Function","The tangent and velocity problems both ask for a value a ratio approaches but never reaches — the limit. Its intuitive two-sided form splits into one-sided limits that must agree; a limit fails to exist when they disagree or when the function grows without bound, the latter producing a vertical asymptote.\n",{"path":21873,"title":21874,"module":21866,"summary":21875},"\u002Fcalculus\u002Flimits-and-continuity\u002Flimit-laws-and-the-precise-definition","Limit Laws and the ε–δ Definition","The Limit Laws reduce a limit to arithmetic on simpler limits, and direct substitution settles polynomials and rational functions outright. The 0\u002F0 forms that resist substitution yield to algebra or the Squeeze Theorem, and the ε–δ definition makes \"arbitrarily close\" precise as a pair of quantified inequalities.\n",{"path":21877,"title":21878,"module":21866,"summary":21879},"\u002Fcalculus\u002Flimits-and-continuity\u002Fcontinuity","Continuity","A function is continuous at a point when its limit there equals its value, so the graph has no break. Continuity fails in three geometric ways; it is closed under arithmetic and composition, so the elementary families and their combinations are continuous; and on a closed interval it forces the Intermediate Value Theorem, which locates roots.\n",{"path":21881,"title":21882,"module":21883,"summary":21884},"\u002Fcalculus\u002Fderivatives\u002Fthe-derivative-and-rates-of-change","The Derivative and Rates of Change","Derivatives","A single limit with three readings: the slope of the tangent line, the instantaneous velocity of a moving object, and the rate of change of one quantity with respect to another. Built from the difference quotient, extended from a value at one point to a function of x, and undefined exactly where a corner, jump, or vertical tangent appears.\n",{"path":21886,"title":21887,"module":21883,"summary":21888},"\u002Fcalculus\u002Fderivatives\u002Fdifferentiation-rules-and-the-chain-rule","Differentiation Rules and the Chain Rule","Computing every derivative from the limit definition is tedious. A short list of rules — power, constant multiple, sum, product, quotient — differentiates any polynomial or rational function by inspection. The trigonometric derivatives follow from one limit, and the chain rule extends everything to composite functions by multiplying rates along the composition.\n",{"path":21890,"title":21891,"module":21883,"summary":21892},"\u002Fcalculus\u002Fderivatives\u002Fimplicit-differentiation-and-related-rates","Implicit Differentiation and Related Rates","Not every curve is the graph of y = f(x). Implicit differentiation finds a slope from an equation in x and y directly, treating y as an unknown function and differentiating both sides. The same chain-rule idea drives related rates, where one measured rate of change forces another through a geometric constraint, and interprets the derivative as a rate across the sciences.\n",{"path":21894,"title":21895,"module":21883,"summary":21896},"\u002Fcalculus\u002Fderivatives\u002Flinear-approximations-and-differentials","Linear Approximations and Differentials","A differentiable curve looks like its tangent line under enough magnification, so the tangent is a usable stand-in for the function near the point of contact. The linear approximation and its linearization, written in the language of differentials dy and dx, estimate both function values and the measurement error propagated into a computed quantity.\n",{"path":21898,"title":21899,"module":21900,"summary":21901},"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fextrema-and-the-mean-value-theorem","Extrema and the Mean Value Theorem","Applications of Derivatives","Absolute and local extrema, the Extreme Value Theorem that guarantees them, and Fermat's Theorem pinning candidates to critical numbers. The Closed Interval Method turns the search for extrema into a finite checklist. Rolle's Theorem and the Mean Value Theorem then connect a function's values to its derivative, giving the tool that most of differential calculus rests on.\n",{"path":21903,"title":21904,"module":21900,"summary":21905},"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fhow-derivatives-shape-a-graph","How Derivatives Shape a Graph","The sign of the first derivative fixes where a function rises and falls, and a sign change identifies each local extremum through the First Derivative Test. The second derivative sets concavity and inflection points and gives a faster Second Derivative Test. Limits at infinity describe end behavior and the horizontal asymptotes a curve settles toward.\n",{"path":21907,"title":21908,"module":21900,"summary":21909},"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fcurve-sketching-and-optimization","Curve Sketching and Optimization","A checklist that synthesizes domain, symmetry, asymptotes, monotonicity, extrema, and concavity into a hand sketch of any function, plus the slant asymptote for rational functions whose degree exceeds the denominator's. The same extremum machinery, applied to a word problem, becomes the optimization template: model one quantity, reduce it to a function of a single variable, and find its absolute extremum.\n",{"path":21911,"title":21912,"module":21900,"summary":21913},"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fnewtons-method-and-antiderivatives","Newton's Method and Antiderivatives","Newton's method solves $f(x) = 0$ by repeatedly replacing the curve with its tangent line and jumping to the tangent's root, converging fast when it works and diverging when the derivative is small. Antiderivatives reverse differentiation: every antiderivative of a function differs from another by a constant, so the general antiderivative is a family of parallel curves, pinned to one by an initial condition.\n",{"path":21915,"title":21916,"module":21917,"summary":21918},"\u002Fcalculus\u002Fintegrals\u002Farea-and-the-definite-integral","Area and the Definite Integral","Integrals","The area under a curve is defined as a limit of sums of rectangle areas. The same limit — a Riemann sum taken as the mesh shrinks to zero — defines the definite integral, a single number measuring signed area, total distance, and every accumulated quantity built the same way. Its properties, comparison bounds, and reading as net area follow directly from the limit.\n",{"path":21920,"title":21921,"module":21917,"summary":21922},"\u002Fcalculus\u002Fintegrals\u002Fthe-fundamental-theorem-of-calculus","The Fundamental Theorem of Calculus","Differentiation and integration are inverse operations. Part 1 says the derivative of an area-accumulation function is the integrand; Part 2 says a definite integral equals the change in any antiderivative across the interval. Together they replace limits of Riemann sums with antiderivative lookups, define the indefinite integral, and give the Net Change Theorem for rates.\n",{"path":21924,"title":21925,"module":21917,"summary":21926},"\u002Fcalculus\u002Fintegrals\u002Fthe-substitution-rule","The Substitution Rule","Substitution runs the Chain Rule backward: spotting an inner function whose derivative also appears in the integrand lets the variable change to $u$ and collapse a composite integral to a simple one. The rule applies to indefinite and definite integrals, with two ways to handle the limits, and it yields the symmetry shortcuts that double even integrands and vanish odd ones.\n",{"path":21928,"title":21929,"module":21930,"summary":21931},"\u002Fcalculus\u002Fapplications-of-integration\u002Fareas-and-volumes","Areas Between Curves and Volumes","Applications of Integration","A definite integral computes any quantity that a limit of Riemann sums approximates. Applied to geometry it gives the area between two curves and the volume of a solid: by cross-sections, by disks and washers when the region is revolved, and by cylindrical shells when inverting the boundary is awkward.\n",{"path":21933,"title":21934,"module":21930,"summary":21935},"\u002Fcalculus\u002Fapplications-of-integration\u002Fwork-average-value-and-arc-length","Work, Average Value, Arc Length, and Surface Area","The work done by a force that varies with position, the average value of a function and the Mean Value Theorem it satisfies, the length of a curve, and the area of a surface swept out by revolving that curve. Each is a limit of Riemann sums, hence a definite integral.\n",{"path":21937,"title":21938,"module":21930,"summary":21939},"\u002Fcalculus\u002Fapplications-of-integration\u002Fphysics-economics-and-probability","Applications to Physics, Economics, and Probability","Definite integrals in physics, economics, and statistics: the force a fluid exerts on a submerged plate, the balance point of a plane region, the money consumers save at a market price, and the probability that a continuous random variable lands in an interval, together with its mean.\n",{"path":21941,"title":21942,"module":21943,"summary":21944},"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Finverse-functions-logarithms-and-exponentials","Inverse Functions, Logarithms, and Exponentials","Exponential, Logarithmic, and Inverse Functions","A one-to-one function has an inverse that reverses it, with a graph mirrored across y = x and a derivative given by the reciprocal-slope rule. The exponential e^x is its own derivative and the natural logarithm has derivative 1\u002Fx; logarithmic differentiation turns products, quotients, and variable powers into sums.\n",{"path":21946,"title":21947,"module":21943,"summary":21948},"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Fgrowth-decay-inverse-trig-and-hyperbolic-functions","Growth, Decay, Inverse Trigonometric, and Hyperbolic Functions","Any quantity whose rate of change is proportional to its size grows or decays exponentially, the single equation y' = ky behind populations, radioactive decay, cooling, and continuously compounded interest. The inverse trigonometric functions have algebraic derivatives, and the hyperbolic functions, built from e^x and e^{-x}, describe the hanging cable.\n",{"path":21950,"title":21951,"module":21943,"summary":21952},"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Flhospitals-rule","Indeterminate Forms and l'Hospital's Rule","When a limit produces 0\u002F0 or infinity over infinity, the value is undetermined by the forms alone. l'Hospital's Rule resolves both by replacing the ratio of functions with the ratio of their derivatives. Products, differences, and powers reduce to a quotient the rule can handle, and repeated use ranks the growth of logarithms, powers, and exponentials.\n",{"path":21954,"title":21955,"module":21956,"summary":21957},"\u002Fcalculus\u002Ftechniques-of-integration\u002Fintegration-by-parts","Integration by Parts","Techniques of Integration","The product rule for derivatives reverses into integration by parts, trading the integral of $u\\,\\d v$ for the integral of $v\\,\\d u$ whenever the second is easier. The LIATE ordering fixes which factor to differentiate. Standard cases: a polynomial against a transcendental factor, repeated parts, cyclic integrals that solve for themselves, and reduction formulas that peel an exponent down by recursion.\n",{"path":21959,"title":21960,"module":21956,"summary":21961},"\u002Fcalculus\u002Ftechniques-of-integration\u002Ftrigonometric-integrals-and-substitution","Trigonometric Integrals and Substitution","Two related techniques. Trigonometric integrals evaluate powers and products of sine, cosine, tangent, and secant by splitting off one factor and converting the rest with a Pythagorean identity, or by dropping even powers with half-angle formulas. Trigonometric substitution runs the idea in reverse: replace x by a sine, tangent, or secant to clear a radical, integrate, then read the answer back off a reference triangle.\n",{"path":21963,"title":21964,"module":21956,"summary":21965},"\u002Fcalculus\u002Ftechniques-of-integration\u002Fpartial-fractions-and-integration-strategy","Partial Fractions and Integration Strategy","Any rational function integrates in closed form: factor the denominator, split the fraction into simple pieces by partial fractions, and integrate each piece as a logarithm or an arctangent. Four denominator cases exhaust the possibilities. A four-step strategy then sorts an arbitrary integrand by its shape to the technique that fits it, and a short catalog records elementary functions whose antiderivatives are not elementary.\n",{"path":21967,"title":21968,"module":21956,"summary":21969},"\u002Fcalculus\u002Ftechniques-of-integration\u002Fapproximate-and-improper-integrals","Approximate and Improper Integrals","Two definite integrals the Fundamental Theorem cannot reach. With no antiderivative available, the Midpoint, Trapezoidal, and Simpson rules approximate the integral from sample values, each carrying a provable error bound. With an infinite interval or an integrand that blows up, the improper integral is defined as a limit that either converges or diverges; the Comparison Test settles which without evaluating it.\n",{"path":21971,"title":21972,"module":21973,"summary":21974},"\u002Fcalculus\u002Fparametric-and-polar\u002Fparametric-curves-and-their-calculus","Parametric Curves and Their Calculus","Parametric Equations and Polar Coordinates","A parametric curve gives x and y separately as functions of a third variable, recording not only a path but the direction and timing with which it is traced. Eliminating the parameter recovers a Cartesian equation; the slope, area, arc-length, and surface-area formulas run directly on the parameter, with the cycloid and astroid as worked examples.\n",{"path":21976,"title":21977,"module":21973,"summary":21978},"\u002Fcalculus\u002Fparametric-and-polar\u002Fpolar-coordinates","Polar Coordinates","Polar coordinates locate a point by a distance from the pole and an angle from the polar axis, giving circles, spirals, and flower-shaped curves short equations. Conversion between the two systems is right-triangle trigonometry, and treating a polar curve as a parametric curve in the angle yields the tangent, area, and arc-length formulas.\n",{"path":21980,"title":21981,"module":21973,"summary":21982},"\u002Fcalculus\u002Fparametric-and-polar\u002Fconic-sections","Conic Sections","Parabolas, ellipses, and hyperbolas are the plane curves cut from a double cone. Each has a focus-based geometric definition and a standard Cartesian equation. A single number, the eccentricity, ties the three together, and placing a focus at the pole gives all of them one polar equation that describes planetary orbits.\n",{"path":21984,"title":21985,"module":21986,"summary":21987},"\u002Fcalculus\u002Fsequences-and-series\u002Fsequences","Sequences","Infinite Sequences and Series","A sequence is a function on the positive integers, and its limit is defined almost exactly like a limit at infinity. The Limit Laws and Squeeze Theorem carry over from functions, monotonic and bounded sequences give a convergence criterion, and the Monotonic Sequence Theorem guarantees a limit exists without naming it.\n",{"path":21989,"title":21990,"module":21986,"summary":21991},"\u002Fcalculus\u002Fsequences-and-series\u002Fseries-and-the-integral-test","Series and the Integral Test","Adding infinitely many terms is made precise as the limit of partial sums. The two series with closed-form partial sums are geometric and telescoping; the harmonic series diverges even as its terms shrink to zero. The Integral Test compares a positive series to an improper integral, settling the p-series and supplying a remainder bound for estimating sums.\n",{"path":21993,"title":21994,"module":21986,"summary":21995},"\u002Fcalculus\u002Fsequences-and-series\u002Fthe-convergence-tests","The Convergence Tests","The comparison, alternating-series, ratio, and root tests decide convergence without a closed-form partial sum. Absolute convergence is stronger than conditional convergence and is preserved under rearrangement; an alternating series errs by less than its first omitted term. A test is chosen from the shape of the general term.\n",{"path":21997,"title":21998,"module":21986,"summary":21999},"\u002Fcalculus\u002Fsequences-and-series\u002Fpower-series","Power Series","A power series is a polynomial of infinite degree whose convergence set is an interval centered at $a$, with a radius the Ratio Test finds and endpoints that must be tested by hand. Inside that interval the series represents a function that can be differentiated and integrated term by term, generating new representations from the geometric series.\n",{"path":22001,"title":22002,"module":21986,"summary":22003},"\u002Fcalculus\u002Fsequences-and-series\u002Ftaylor-and-maclaurin-series","Taylor and Maclaurin Series","If a function equals a power series, its coefficients are forced: the nth is the nth derivative at the center over n factorial. We derive that formula, use Taylor's Inequality to prove the standard series for the exponential, sine, and cosine, record the binomial series and a reference table, and bound the error when a Taylor polynomial replaces a function.\n",{"path":22005,"title":22006,"module":22007,"summary":22008},"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvectors-and-the-dot-product","Three-Dimensional Coordinates, Vectors, and the Dot Product","Vectors and the Geometry of Space","Space needs three coordinates, so we set up the rectangular system, the distance formula, and the equation of a sphere. Vectors then package magnitude and direction into a single algebraic object with its own arithmetic. The dot product turns two vectors into a number that measures the angle between them, gives a clean test for orthogonality, and produces the projection of one vector onto another.\n",{"path":22010,"title":22011,"module":22007,"summary":22012},"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fthe-cross-product-lines-and-planes","The Cross Product, Lines, and Planes","The cross product multiplies two vectors into a third perpendicular to both, with length equal to the area of the parallelogram they span. That one construction supplies the direction of a line, the normal of a plane, and, through the scalar triple product, the volume of a parallelepiped. Lines carry a point and a direction vector; planes carry a point and a normal, which fixes the angle between planes and the distance from a point to a plane.\n",{"path":22014,"title":22015,"module":22007,"summary":22016},"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fcylinders-and-quadric-surfaces","Cylinders and Quadric Surfaces","A surface whose equation omits one variable is a cylinder: the graph of a plane curve swept along the missing axis. A second-degree equation in three variables is a quadric, and translation and rotation reduce every one to a short standard list. Traces — the curves cut by planes parallel to the coordinate planes — sort the six quadrics into ellipsoid, the two paraboloids, the cone, and the two hyperboloids.\n",{"path":22018,"title":22019,"module":22007,"summary":22020},"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvector-functions-and-space-curves","Vector Functions and Space Curves","A vector function assigns a vector to each value of a parameter, and as the parameter runs its tip traces a space curve. Taking limits, derivatives, and integrals component by component carries all of single-variable calculus into three dimensions. The derivative of a vector function is the tangent vector to its curve, and normalizing it gives the unit tangent that points the way along the path.\n",{"path":22022,"title":22023,"module":22007,"summary":22024},"\u002Fcalculus\u002Fvectors-and-space-curves\u002Farc-length-curvature-and-motion","Arc Length, Curvature, and Motion in Space","Integrating the speed of a vector function gives the length of its curve and a natural parameter, arc length, that depends only on the curve's shape. Curvature measures how fast the unit tangent turns, and together with the normal and binormal it builds the moving TNB frame. Reading the same vector function as a trajectory, its first two derivatives are velocity and acceleration, and acceleration splits cleanly into tangential and normal parts.\n",{"path":22026,"title":22027,"module":22028,"summary":22029},"\u002Fcalculus\u002Fpartial-derivatives\u002Ffunctions-of-several-variables","Functions of Several Variables, Limits, and Continuity","Partial Derivatives","A function of several variables assigns one number to each point of a region in the plane or in space. Domain, graph, level curve, and level surface describe it; limits and continuity extend to two variables, where a limit must agree along every path of approach, not just from the left and the right.\n",{"path":22031,"title":22028,"module":22028,"summary":22032},"\u002Fcalculus\u002Fpartial-derivatives\u002Fpartial-derivatives","A partial derivative holds every variable but one fixed and differentiates in the ordinary sense. Geometrically it is the slope of a trace curve cut from the surface by a coordinate plane. The freeze-and-differentiate rule computes the two first partials; the four second partials follow, and the two mixed ones agree under Clairaut's Theorem when they are continuous.\n",{"path":22034,"title":22035,"module":22028,"summary":22036},"\u002Fcalculus\u002Fpartial-derivatives\u002Ftangent-planes-and-the-chain-rule","Tangent Planes, Linear Approximation, and the Chain Rule","Near a point, a smooth surface looks like its tangent plane, and the plane's equation is built from the two partial derivatives. That linearization defines the total differential and the meaning of differentiability in two variables. The chain rule then propagates derivatives through composed functions, tracked by a tree diagram, and yields clean formulas for implicit differentiation.\n",{"path":22038,"title":22039,"module":22028,"summary":22040},"\u002Fcalculus\u002Fpartial-derivatives\u002Fdirectional-derivatives-and-the-gradient","Directional Derivatives and the Gradient","The partial derivatives measure slope along the two axes; the directional derivative measures slope along any chosen direction, and equals the gradient dotted with a unit vector. The gradient points in the direction of steepest increase, its length is the greatest rate, and it stands perpendicular to level curves and surfaces, which fixes the tangent plane to a level surface.\n",{"path":22042,"title":22043,"module":22028,"summary":22044},"\u002Fcalculus\u002Fpartial-derivatives\u002Foptimization-and-lagrange-multipliers","Optimization and Lagrange Multipliers","Extrema of a two-variable function sit at critical points where the gradient vanishes; the Second Derivatives Test sorts them into peaks, valleys, and saddles by the sign of a discriminant. Absolute extrema on a closed region also need the boundary. When the domain is itself a constraint curve, Lagrange multipliers set the two gradients parallel and solve the constrained problem.\n",{"path":22046,"title":22047,"module":22048,"summary":22049},"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fdouble-integrals","Double Integrals","Multiple Integrals and Vector Calculus","The double integral extends the definite integral to functions of two variables: a limit of Riemann sums that measures signed volume under a surface. Fubini's Theorem turns it into two ordinary integrations done one after the other, general regions of type I and type II fix the inner limits, polar coordinates absorb circular symmetry through the factor r, and the same machine computes mass, center of mass, and moments of a lamina.\n",{"path":22051,"title":22052,"module":22048,"summary":22053},"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Ftriple-integrals-and-coordinate-systems","Triple Integrals and Coordinate Systems","The triple integral integrates a function of three variables over a solid, as a limit of Riemann sums evaluated by three nested single integrations. Cylindrical coordinates add the factor r to handle axial symmetry, spherical coordinates add rho-squared sine-phi for radial symmetry, and the general change of variables shows both volume elements are Jacobian determinants of the coordinate map. Surface area for a graph completes the measurement toolkit.\n",{"path":22055,"title":22056,"module":22048,"summary":22057},"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fvector-fields-and-line-integrals","Vector Fields and Line Integrals","A vector field assigns a vector to every point of space; the line integral of a field along a curve accumulates its tangential component, measuring work. Conservative fields are gradients of a potential, and for them the Fundamental Theorem for Line Integrals makes the integral depend only on the endpoints. Path independence, closed-loop integrals of zero, and the component test for a potential are three faces of the same property.\n",{"path":22059,"title":22060,"module":22048,"summary":22061},"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fgreens-theorem-curl-and-divergence","Green's Theorem, Curl, and Divergence","Green's Theorem equates the line integral of a field around a positively oriented closed curve with a double integral over the enclosed region, turning a boundary computation into an area computation and vice versa. Curl measures local circulation and divergence measures local outflow; the two vector forms of Green's Theorem express the boundary integral as the integrated curl or divergence, the planar case of Stokes' and the Divergence Theorem.\n",{"path":22063,"title":22064,"module":22048,"summary":22065},"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fsurface-integrals","Parametric Surfaces and Surface Integrals","A parametric surface is the image of a two-variable vector function; its area element is the magnitude of the cross product of the two tangent vectors. The surface integral of a scalar function sums it over that area, and the flux integral of a vector field sums the field's normal component, measuring flow through the surface. Orientation by a choice of unit normal makes flux well-defined, the integral Stokes' and the Divergence Theorem operate on.\n",{"path":22067,"title":22068,"module":22048,"summary":22069},"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fstokes-and-the-divergence-theorem","Stokes' Theorem and the Divergence Theorem","Stokes' Theorem lifts Green's Theorem into space: the line integral of a field around the boundary of a surface equals the flux of its curl through the surface. The Divergence Theorem relates the outward flux across a closed surface to the triple integral of divergence over the solid it encloses. Together with the Fundamental Theorem of Calculus and its line-integral and Green counterparts, they are one theorem: the integral of a derivative over a region equals the integral of the field over its oriented boundary.\n",{"path":22071,"title":22072,"module":6,"summary":6},"\u002Fcalculus","Calculus",{"path":22074,"title":22075,"module":21519,"summary":22076},"\u002Fmechanics\u002Ffoundations\u002Fmeasurement-and-dimensions","Measurement and Dimensions","Every physical quantity is a number attached to a unit, and that pairing is what lets you check an equation before computing anything, since terms that add together must carry the same dimensions. We build the SI base units and the notion of dimension, then use dimensional analysis to test a proposed relation and form scaling groups — a method that fixes a formula's shape but never its numerical constants. The lesson also sets how precisely a result may be stated, through significant figures, propagated uncertainty, and order-of-magnitude checks that catch errors a raw calculator answer hides.\n",{"path":22078,"title":22079,"module":21519,"summary":22080},"\u002Fmechanics\u002Ffoundations\u002Fvector-algebra","Vector Algebra","Force, velocity, and displacement all carry a direction, so mechanics needs an arithmetic that respects it; adding magnitudes alone gives the wrong answer the moment two arrows point different ways. We set up vectors and their components in a chosen basis, then build the two products that carry most of the physics — the dot product, which extracts the part of one vector along another and yields work and power, and the cross product, which measures oriented area and yields torque and angular momentum. Rotating the axes changes the components while leaving the vector itself untouched, and the same component method resolves a force along whatever directions a constraint picks out.\n",{"path":22082,"title":22083,"module":22084,"summary":22085},"\u002Fmechanics\u002Fkinematics\u002Fone-dimensional-motion","One-Dimensional Motion","Kinematics","Motion along a line already forces the two questions the whole of kinematics repeats: how fast is the object moving now, and where will it be next? Velocity and acceleration answer the first as derivatives of position; integrating them back — the signed area under a graph — answers the second. We derive the constant-acceleration equations, mark exactly where the \"constant\" assumption is load-bearing, and see why sign, not magnitude, is what carries direction.\n",{"path":22087,"title":22088,"module":22084,"summary":22089},"\u002Fmechanics\u002Fkinematics\u002Fmotion-graphs","Motion Graphs","Draw a motion as a graph and its two most useful facts turn geometric: the slope of the position curve is the velocity, and the area under the velocity curve is the displacement. We read motion in both directions — differentiating a graph for the next rate, integrating it back to recover position — and handle the curved, piecewise, and noisy graphs that real measurements produce. Along the way we see why a velocity estimated from two positions belongs to the midpoint of their interval, not its end.\n",{"path":22091,"title":22092,"module":22084,"summary":22093},"\u002Fmechanics\u002Fkinematics\u002Fprojectile-motion","Projectile Motion","Throw an object and it seems to trace one curved path, but the motion is really two independent one-dimensional motions running at once: constant velocity across the ground and free fall in the vertical. Splitting it that way turns every projectile question — how long it stays up, how far it lands, how high it climbs, whether it clears an obstacle — into a pair of equations you already know. We derive the parabolic trajectory, work both the forward and the inverse problems, and show why the familiar $45^\\circ$ range-maximizing angle holds only when launch and landing heights match.\n",{"path":22095,"title":22096,"module":22084,"summary":22097},"\u002Fmechanics\u002Fkinematics\u002Frelative-motion","Relative Motion","A velocity is only ever measured relative to some observer, so a boat's speed through the water, over the ground, and as seen from another boat are three different vectors. Choosing the right frame — and subtracting one motion from another — collapses river crossings, crosswind headings, pursuit, and closest-approach problems into a single vector equation. We build the relative-velocity and relative-position relations for uniformly moving frames, show why acceleration is the one quantity all such observers agree on, and note where rotating frames break the simple subtraction.\n",{"path":22099,"title":22100,"module":22084,"summary":22101},"\u002Fmechanics\u002Fkinematics\u002Fcircular-motion","Circular Motion","An object going around a circle at a steady speed is still accelerating, because its velocity is forever changing direction — the fact that governs everything from a car on a curve to a satellite in orbit. We tie the angular description (angle, angular velocity, angular acceleration) to the linear one through $v=r\\omega$, split the acceleration into an inward part that turns the velocity and a tangential part that changes its speed, and extend the inward $v^2\u002Fr$ result to any curved path through its local radius of curvature. Constant angular acceleration then mirrors straight-line motion equation for equation.\n",{"path":22103,"title":22104,"module":22105,"summary":22106},"\u002Fmechanics\u002Fdynamics\u002Fnewtons-laws","Newton's Laws","Dynamics","What makes a body change its motion, and in which frames does the answer take its simplest form? Newton's three laws settle both: inertial frames are the ones where a force-free body coasts, force is whatever changes momentum, and every interaction pushes back on its source. We write the second law as $\\sum\\vec F=\\d\\vec p\u002F\\d t$, reduce it to $m\\vec a$ at constant mass, and separate what a scale actually reads — the support force — from the weight $m\\vec g$ it is so often mistaken for.\n",{"path":22108,"title":22109,"module":22105,"summary":22110},"\u002Fmechanics\u002Fdynamics\u002Ffree-body-diagrams","Free-Body Diagrams","Once several forces act on a body at once, the reliable way to predict its motion is to isolate that one body and draw every external push and pull on it — nothing more, nothing less. The free-body diagram is that discipline. We fix a system boundary, resolve $\\sum\\vec F=m\\vec a$ into components along axes chosen to fit the geometry, and solve for the unknowns a problem hands us — normal forces, tensions, friction, and the acceleration a constraint permits — seeing why internal forces drop out only when the boundary encloses both bodies that share them.\n",{"path":22112,"title":22113,"module":22105,"summary":22114},"\u002Fmechanics\u002Fdynamics\u002Ffriction-and-curved-motion","Friction and Curved Motion","Real surfaces grip before they slip, fluids push back harder the faster you move through them, and anything rounding a bend must be pulled toward the inside of the curve by something. This lesson supplies the force laws for those three cases. We bound static friction by $|f_s|\\leq\\mu_sN$ and switch to kinetic friction $\\mu_kN$ once sliding starts, model drag as a speed-dependent resistance that levels off at a terminal speed, and show that circular motion demands an inward net force $mv^2\u002Fr$ furnished by real interactions — friction, a banked normal force, tension — never by an invented outward one.\n",{"path":22116,"title":22117,"module":22105,"summary":22118},"\u002Fmechanics\u002Fdynamics\u002Fnumerical-dynamics","Numerical Dynamics","Most force laws — quadratic drag, coupled oscillators, anything nonlinear — admit no closed-form trajectory, so we advance the motion one small time step at a time and let arithmetic do what algebra cannot. This lesson turns $\\d\\vec y\u002F\\d t=f(t,\\vec y)$ into a marching rule. We derive the Euler, Euler--Cromer, midpoint, and Verlet updates, weigh their accuracy and stability, watch a drifting energy expose a bad scheme, and use step-halving and conserved quantities to separate the error of the method from the error of the model.\n",{"path":22120,"title":22121,"module":22105,"summary":22122},"\u002Fmechanics\u002Fdynamics\u002Fcenter-of-mass-systems","Center-of-Mass Systems","A firework bursts into a dozen fragments, yet one point keeps gliding along the original parabola as though nothing had happened. That point is the centre of mass, and following it collapses a many-body tangle into a single equation of motion. We define $\\vec R=\\frac1M\\sum_i m_i\\vec r_i$ and its continuous form, show that internal forces cancel so that only external ones move it, $M\\vec A_{\\rm cm}=\\sum\\vec F_{\\rm ext}$, and put the result to work on recoil, collisions viewed from the centre-of-mass frame, and rocket propulsion, where mass leaving the boundary carries momentum with it.\n",{"path":22124,"title":22125,"module":22126,"summary":22127},"\u002Fmechanics\u002Fenergy\u002Fwork-and-kinetic-energy","Work and Kinetic Energy","Energy","A constant push along a straight path is trivial to score, but real forces vary and bend along curved trajectories, and only the component along the motion transfers any energy. Work captures exactly that transfer as the line integral $W=\\int\\vec F\\cdot\\d\\vec r$, and the work-kinetic-energy theorem turns it into a statement about speed: the net work on a particle equals the change in its $\\tfrac12 mv^2$. We build work up from the dot product to the signed area under a force curve, derive the theorem from Newton's second law, and read power as its instantaneous rate $P=\\vec F\\cdot\\vec v$.\n",{"path":22129,"title":22130,"module":22126,"summary":22131},"\u002Fmechanics\u002Fenergy\u002Fpotential-energy","Potential Energy","When a force does the same work no matter which path a particle takes, that work can be stored as a function of position alone, and solving for the motion becomes bookkeeping instead of integration. We single out the forces that qualify — the conservative ones, for which $\\oint\\vec F\\cdot\\d\\vec r=0$ — define their potential energy through $\\vec F=-\\nabla U$, and use conservation of mechanical energy to read speeds, turning points, and equilibria straight off a potential curve. Friction breaks the shortcut, so we also track where mechanical energy leaks away as heat.\n",{"path":22133,"title":22134,"module":22126,"summary":22135},"\u002Fmechanics\u002Fenergy\u002Fmultiparticle-work","Multiparticle Work","A single particle has one velocity and one kinetic energy; a system of many can spin, deform, explode, and warm up while its centre of mass glides along as if nothing happened. Splitting the motion into a centre-of-mass part and an internal part separates the energy that momentum already fixes from the energy left free for relative motion, $K=\\tfrac12MV_{\\rm cm}^2+K'$. We derive the centre-of-mass work theorem, see why an explosion or a released spring can raise total kinetic energy with no external work at all, and use the reduced-mass and centre-of-mass frames to make collisions and internal transfers clean.\n",{"path":22137,"title":22138,"module":22126,"summary":22139},"\u002Fmechanics\u002Fenergy\u002Fmass-energy-and-binding","Mass-Energy and Binding","Relativity puts rest itself on the energy ledger: a mass $m$ carries energy $mc^2$ even when it sits still, so weighing a system's separated pieces and weighing the assembled whole give different answers, and the gap is binding energy. We convert freely between mass units and MeV, compute the energy that holds a nucleus together, and read the binding-energy-per-nucleon curve that explains why fusing light nuclei and splitting heavy ones both release energy. Reaction $Q$ values, thresholds, and recoil then follow from the same mass-difference accounting, once the frame and mass convention are fixed.\n",{"path":22141,"title":22142,"module":22126,"summary":22143},"\u002Fmechanics\u002Fenergy\u002Fphotons-and-quantization","Photons and Quantization","Light delivers its energy in indivisible lumps: a photon of frequency $f$ carries exactly $hf$, and this one fact explains why a dim blue lamp ejects electrons that an intense red one cannot. We fix a photon's energy and momentum from its wavelength, follow the quanta through emission, absorption, and the photoelectric threshold $K_{\\rm max}=hf-\\phi$, and watch energy and momentum conservation together produce the Compton wavelength shift when a photon scatters from an electron. The recurring discipline is unit and frame care, where a stray factor of $10^9$ or a forgotten rest energy quietly ruins an answer.\n",{"path":22145,"title":22146,"module":22147,"summary":22148},"\u002Fmechanics\u002Fmomentum\u002Fmomentum-and-collisions","Momentum and Collisions","Momentum","When two objects collide, the forces between them are too brief and too tangled to integrate directly, yet the result is fixed by one conserved quantity. Linear momentum $\\vec p=m\\vec v$ turns Newton's second law into the impulse-momentum theorem $\\vec J=\\Delta\\vec p$, and for an isolated system into a conservation law that holds through any internal collision, however dissipative. We use it to separate elastic from inelastic collisions, follow the centre of mass, and read impulse as the signed area under a force-time curve — always tracking which external impulses the chosen system and interval let us drop.\n",{"path":22150,"title":22151,"module":22147,"summary":22152},"\u002Fmechanics\u002Fmomentum\u002Fcenter-of-mass-collisions","Center-of-Mass Collisions","A two-body collision that looks asymmetric in the laboratory becomes almost trivial in the frame that rides along with the centre of mass, where the total momentum is zero and the two momenta stay equal and opposite. We build that frame, reduce the pair to a single relative coordinate carrying the reduced mass $\\mu$, and show that an elastic collision there only rotates one momentum vector while its length holds fixed. Transforming back to the laboratory then handles elastic and inelastic collisions, scattering angles, and reaction thresholds with the same construction — and shows why relative speed, not laboratory kinetic energy, measures what a collision can convert.\n",{"path":22154,"title":22155,"module":22147,"summary":22156},"\u002Fmechanics\u002Fmomentum\u002Frocket-propulsion","Rocket Propulsion","A rocket speeds up by throwing mass backward, so its own mass drops as it flies and $\\vec F=m\\vec a$ no longer applies to a fixed body. Tracking the momentum the exhaust carries across the vehicle boundary gives thrust $T=Ru_e$ and, for a force-free burn, the rocket equation $\\Delta v=u_e\\ln(m_i\u002Fm_f)$ — a logarithm that makes large velocity changes expensive in propellant and forces staging. We then add the forces a real ascent cannot ignore, gravity, drag, and steering, and show how thrust and mass-flow records are cross-checked to infer the exhaust speed.\n",{"path":22158,"title":22159,"module":22160,"summary":22161},"\u002Fmechanics\u002Frotation\u002Frotational-inertia","Rotational Inertia","Rotation","Push a wheel and a merry-go-round with the same force and they speed up at wildly different rates: the same mass resists rotation differently depending on where it sits relative to the axis. That single fact is the moment of inertia, $I=\\int r_\\perp^2\\,\\d m$, and this lesson builds it from the ground up. We tie angular motion to linear through $s=r\\theta$, $v=r\\omega$, and $a_t=r\\alpha$, derive $I$ for rods, disks, and spheres, and use the parallel- and perpendicular-axis theorems to move between axes — always naming the axis, because the same body has as many moments of inertia as it has lines to spin about.\n",{"path":22163,"title":22164,"module":22160,"summary":22165},"\u002Fmechanics\u002Frotation\u002Frotational-dynamics","Rotational Dynamics","A force applied to a wheel does nothing unless it acts off the axis: what turns a rigid body is torque, force times lever arm. This lesson makes that precise and turns it into the rotational Newton's second law, $\\sum\\tau=I\\alpha$ about a fixed axis, the exact analogue of $\\sum F=ma$. From there we get rotational work $W=\\int\\tau\\,\\d\\theta$ and power $P=\\tau\\omega$, size a motor to a load, and solve pulleys and Atwood machines where the pulley's own inertia can no longer be ignored — always insisting that every torque be measured about the same axis.\n",{"path":22167,"title":22168,"module":22160,"summary":22169},"\u002Fmechanics\u002Frotation\u002Frolling-motion","Rolling Motion","A rolling wheel is doing two things at once — translating and spinning — but the no-slip condition $v_{cm}=R\\omega$ locks them together, and that single constraint is what makes rolling tractable. We use it to split the kinetic energy into $\\tfrac12Mv_{cm}^2+\\tfrac12I\\omega^2$, find how fast a cylinder reaches the bottom of an incline, and show why the contact point is instantaneously at rest. The static friction that enforces rolling does no work; we track its direction from the tendency to slip, and mark exactly where the model breaks once the required friction exceeds $\\mu_sN$.\n",{"path":22171,"title":22172,"module":22160,"summary":22173},"\u002Fmechanics\u002Frotation\u002Fangular-momentum","Angular Momentum","A skater pulls in her arms and spins faster, with no torque acting: that is angular momentum conservation, and it lets us answer questions that would be hopeless force by force. We build $\\vec L=\\vec r\\times\\vec p$, show it obeys $\\vec\\tau_{ext}=\\d\\vec L\u002F\\d t$, and use its conservation under zero external torque to link before and after in collisions, reconfigurations, and coupled rotors without ever resolving the internal forces. The catch is bookkeeping: the origin, the system boundary, and the frame must be fixed first, and a change in total $\\vec L$ always points to an external impulse someone forgot.\n",{"path":22175,"title":22176,"module":22160,"summary":22177},"\u002Fmechanics\u002Frotation\u002Frolling-resistance","Rolling Resistance","Ideal rolling should coast forever, yet every real wheel slows down. The reason is that a deformable tire and road do not press through a single point: the contact patch spreads, the normal-force resultant shifts ahead of the axle, and that offset is a resisting moment even with no gross sliding. We package it as an equivalent force $F_{rr}=C_{rr}N$, tie the coefficient to load, surface, speed, and temperature, and use coast-down, towing, and traction tests to separate this contact loss from aerodynamic drag, bearing friction, and the adhesion limit where rolling gives way to skidding.\n",{"path":22179,"title":22180,"module":22160,"summary":22181},"\u002Fmechanics\u002Frotation\u002Fgyroscopic-precession","Gyroscopic Precession","A spinning top leans over but does not fall — it swings its axis in a slow horizontal circle instead. The paradox dissolves once torque is read as the rate of change of a vector: gravity's torque is perpendicular to the spin angular momentum, so it turns $\\vec L$ rather than toppling it. We derive the steady precession rate $\\Omega\\simeq Mgr\u002F(I_s\\omega_s)$ in the fast-top limit, state the assumptions it leans on — dominant spin, slow tilt, negligible bearing torque — and read nutation, support motion, and a decaying spin as the ways real gyroscopes depart from it.\n",{"path":22183,"title":22184,"module":22185,"summary":22186},"\u002Fmechanics\u002Fgravity-and-matter\u002Fkeplerian-orbits","Keplerian Orbits","Gravitation and Matter","Why do the planets trace ellipses rather than any other curve? Newton's inverse-square law collapses the two-body problem onto a single conic section, and the answer falls out of two conserved quantities: a central force can exert no torque, so angular momentum is fixed, and gravity is conservative, so energy is fixed. We read an orbit's size and shape straight off those invariants, recover all three of Kepler's laws, and derive escape speed, the vis-viva relation, and the timing of a pass. We also mark where the ideal ellipse breaks down — drag, oblateness, and a third body slowly move a real orbit.\n",{"path":22188,"title":22189,"module":22185,"summary":22190},"\u002Fmechanics\u002Fgravity-and-matter\u002Fgravitational-fields","Gravitational Fields","Instead of tracking the force between every pair of masses, we attach a field to the source and ask a test mass to read it off locally. That move pays off because gravity is conservative: the field is the gradient of a single scalar potential, and potentials from many sources simply add. We build the field-potential picture, use spherical symmetry and the shell theorem to get the point-mass exterior field and the zero interior field of a shell, and read tides straight out of the field's gradient. Along the way we mark exactly when the constant-$g$ and point-mass shortcuts hold and when a shape correction is needed.\n",{"path":22192,"title":22193,"module":22185,"summary":22194},"\u002Fmechanics\u002Fgravity-and-matter\u002Fstatic-equilibrium","Static Equilibrium","What does it take for a loaded structure to stay put? A body at rest needs its forces to cancel and its turning effects to cancel — $\\sum\\vec F=0$ and $\\sum\\vec\\tau=0$ about any point — and almost all of statics is the craft of turning a physical setup into those equations. We build free-body diagrams, replace supports, cables, friction, couples, and distributed loads with their idealized reactions, and locate the centre of gravity that decides whether a body tips. We also count equations against unknowns to separate a determinate problem from one that needs the material's deformation to resolve, and read every negative or inconsistent reaction as a sign that a contact or a boundary was chosen wrong.\n",{"path":22196,"title":22197,"module":22185,"summary":22198},"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-statics","Fluid Statics","A fluid at rest cannot support a shear, so the only stress it carries is a pressure that must grow with depth to hold up the fluid above it. That single balance, $\\d p\u002F\\d z=-\\rho g$, runs the whole subject: it sets manometer readings, the force on a dam, and — integrated over a submerged boundary — Archimedes' buoyant force $F_B=\\rho g V_{\\rm disp}$. We derive these, use them to decide when a body floats and whether it floats upright, and mark where acceleration, rotation, compressibility, or capillarity forces a richer pressure model.\n",{"path":22200,"title":22201,"module":22185,"summary":22202},"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-flow","Fluid Flow","Two accounting rules carry most of steady flow: mass cannot pile up, so the same volume crosses every section each second, and mechanical energy is conserved along a streamline when the fluid is ideal. From those we get continuity, Bernoulli's relation between pressure, speed, and height, and the results that follow — Torricelli's efflux speed, the Venturi meter, the Pitot tube. We then let go of the ideal assumptions one at a time: viscosity adds wall shear and head loss, Reynolds number decides laminar versus turbulent, and Mach number marks where a gas stops behaving as incompressible.\n",{"path":22204,"title":22205,"module":22185,"summary":22206},"\u002Fmechanics\u002Fgravity-and-matter\u002Forbital-motion","Orbital Motion","A circular orbit is nothing but free fall with enough sideways speed to keep missing the ground, and setting gravity equal to the centripetal requirement fixes that speed and the period once and for all. From the same energy bookkeeping we read off escape speed, sort orbits into bound, parabolic, and hyperbolic by the sign of their specific energy, and see why a tangential burn is the efficient way to change an orbit. We build the Hohmann transfer and its launch window, work the numbers for a geostationary orbit and an escape burn, and mark where finite thrust, perturbations, and an uncertain initial state pull a real trajectory off the ideal.\n",{"path":22208,"title":22209,"module":22185,"summary":22210},"\u002Fmechanics\u002Fgravity-and-matter\u002Fstress-and-elasticity","Stress and Elasticity","Rigid bodies are a fiction; every real material stretches, shears, or squeezes under load, and the useful question is how much. We define stress as force per area and strain as fractional deformation, then find that for small deformations the two are simply proportional — Hooke's law — with Young's, shear, and bulk moduli as the constants for stretch, twist, and volume change. From these we compute extensions, torsional twist, and stored elastic energy, and read a tensile curve for the yield, ultimate, and fracture points where linear elasticity ends. We also mark the practical limits: stress concentrations, fatigue, and the multiaxial states a single uniaxial modulus cannot capture.\n",{"path":22212,"title":22213,"module":22214,"summary":22215},"\u002Fmechanics\u002Foscillations-waves\u002Fdamped-oscillators","Damped Oscillators","Oscillations and Waves","Every real oscillator eventually stops: friction, drag, and internal loss drain its energy, so free motion is a decay rather than a permanent swing. Adding a velocity-proportional resistance to the spring-mass equation produces one dimensionless number, $b\u002F(2\\sqrt{mk})$, that decides whether the mass rings down through many cycles, returns once without overshoot, or crawls back slowly. We solve the three regimes, tie the observed decay to the power balance $b\\dot x^2$, and turn a measured ring-down into the decay rate and quality factor of the apparatus — reading damping off the data instead of assuming it.\n",{"path":22217,"title":22218,"module":22214,"summary":22219},"\u002Fmechanics\u002Foscillations-waves\u002Ftravelling-waves","Travelling Waves","A wave carries a shape, not the material: each element of a rope or air column oscillates in place while the disturbance travels through it. Writing that shape as $f(x\\mp vt)$ turns \"the pattern moves\" into a statement about the cosine's argument, and a local force balance on one string segment fixes the speed at $v=\\sqrt{T\u002F\\mu}$ — restoring stiffness over inertia, with amplitude nowhere in it. We build the sinusoidal wave and its phase, derive the wave equation from Newton's second law, and follow the energy a travelling wave transports, then check speed and power against those predictions.\n",{"path":22221,"title":22222,"module":22214,"summary":22223},"\u002Fmechanics\u002Foscillations-waves\u002Fwave-superposition","Wave Superposition","When two waves cross the same point, what does a probe read? In a linear medium the answer is arithmetic: the displacements add, $y=y_1+y_2$, and the pulses pass through each other unchanged. That one rule produces interference — reinforcement where the signs agree, cancellation where they oppose — and it guards against a common mistake, since displacement can vanish at an instant while the energy sits in transverse motion instead. We work out the signed sum, the phase bookkeeping for equal-frequency components, and why a null in the record is not a null in the wave.\n",{"path":22225,"title":22226,"module":22214,"summary":22227},"\u002Fmechanics\u002Foscillations-waves\u002Fstanding-waves","Standing Waves","Clamp a string at both ends and only certain frequencies survive: the ends must be nodes, and that single geometric demand quantizes the wave into a discrete set of modes $f_n=nv\u002F(2L)$. The travelling wave becomes a fixed pattern of nodes and antinodes — standing, not moving — because equal waves running in opposite directions superpose. We build the standing wave from its counter-propagating pieces, read the harmonic sequence off the boundary conditions (half-wavelengths for a fixed-fixed string, odd quarter-wavelengths for a closed pipe), and test the ideal model against node scans and resonance peaks.\n",{"path":22229,"title":22230,"module":22214,"summary":22231},"\u002Fmechanics\u002Foscillations-waves\u002Fsound-waves","Sound Waves","Sound is a pressure wave so small that a loud tone displaces air molecules by less than the width of an atom, yet a microphone reads it easily — because pressure, not displacement, is what the ear and the instrument sense. The acoustic impedance $Z=\\rho c$ ties pressure, density, and particle velocity together, fixes the intensity a wave carries, and sets the reference for the decibel, a logarithm that tames a $10^{12}$ range in power. We derive the sound speed from the gas's stiffness, convert between pressure and intensity levels, and treat the measurement itself — calibration, geometry, background, averaging — as part of the physics.\n",{"path":22233,"title":22234,"module":22214,"summary":22235},"\u002Fmechanics\u002Foscillations-waves\u002Fdoppler-effect","Doppler Effect","A passing siren drops in pitch not because the source changes but because motion repacks the wavefronts: an approaching source crowds its crests, a receding one stretches them, and a moving listener samples them at a different rate. For mechanical waves every velocity is measured against the medium, and one signed ratio $f_r=f_s(v-u_r)\u002F(v-u_s)$ captures both effects at once. We separate source motion, which sets crest spacing, from receiver motion, which sets arrival rate, invert the shift to recover radial velocity, and mark where the model breaks — supersonic sources, moving air, and reflected paths that carry two shifts, not one.\n",{"path":22237,"title":22238,"module":22214,"summary":22239},"\u002Fmechanics\u002Foscillations-waves\u002Fwave-packets","Wave Packets","No real signal is a single frequency: a disturbance that starts and stops is built from a band of wave numbers, and the width of that band is what makes it local. We ask how such a packet moves — carrier crests at the phase velocity $v_\\mathrm p=\\omega\u002Fk$, the envelope at the group velocity $v_\\mathrm g=\\d\\omega\u002F\\d k$ — and why the two differ once a medium is dispersive. Curvature $\\d^2\\omega\u002F\\d k^2$ spreads and chirps the packet as it travels, and the Fourier reciprocity that ties bandwidth to duration explains why a finite record, aliasing, or a coarse probe can imitate that spreading unless the sampling limits are respected.\n",{"path":22241,"title":22242,"module":22214,"summary":22243},"\u002Fmechanics\u002Foscillations-waves\u002Fbeats-and-coupling","Beats and Coupling","Add two tones a few hertz apart and the sum swells and fades at their difference frequency — a beat — though neither source is changing. We work out that envelope, then ask the mechanical version of the same question: join two oscillators and a single resonance splits into normal modes, with energy sloshing between the coordinates at their frequency difference. The lesson identifies when a slow amplitude envelope signals genuine coupling rather than two independent sources, drift, or deliberate modulation, reading it from envelope timing, spectral sidebands, and the mode shapes.\n",{"path":22245,"title":22246,"module":22214,"summary":22247},"\u002Fmechanics\u002Foscillations-waves\u002Fsimple-harmonic-motion","Simple Harmonic Motion","Any system pushed back toward equilibrium by a force proportional to its displacement obeys one equation, $\\ddot x+\\omega_0^2x=0$, and so moves sinusoidally at $\\omega_0=\\sqrt{k\u002Fm}$ whatever the amplitude. We derive that motion, follow its energy $E=mv^2\u002F2+kx^2\u002F2$ trading between kinetic and potential form at constant total, and read the elliptical phase-space orbit Hooke's law implies. Period, amplitude, velocity, and acceleration then supply redundant checks: an amplitude-dependent period or a curved force residual is the signature that the linear model has failed, and mass-loading and offset tests separate a calibration error from a real frequency shift.\n",{"path":22249,"title":22250,"module":22214,"summary":22251},"\u002Fmechanics\u002Foscillations-waves\u002Fpendulum-motion","Pendulum Motion","A pendulum keeps time only because, for small swings, gravity supplies a restoring torque proportional to the angle — and $T=2\\pi\\sqrt{L\u002Fg}$ then follows without the mass appearing at all. We derive that result, mark exactly which assumptions carry it (small angle, negligible pivot loss, a rigid support), then relax them: finite amplitude lengthens the period through an elliptic integral, and an extended body replaces $L$ with the ratio of its moment of inertia to its center-of-mass distance. How the period drifts with amplitude or pivot position is what diagnoses the geometric, damping, and distributed-mass corrections.\n",{"path":22253,"title":22254,"module":22214,"summary":22255},"\u002Fmechanics\u002Foscillations-waves\u002Fdriven-oscillators","Driven Oscillators","Drive a damped oscillator at a frequency you control and it eventually forgets its own: $m\\ddot x+b\\dot x+kx=F_0\\cos\\omega t$ settles into a steady response whose amplitude and phase depend sharply on how close the drive sits to resonance. We solve for that response, show how damping alone fixes the resonance width, the peak power, and the settling time, and treat base excitation as the same problem with a different input. The steady-state formulas hold only for constant $m$, $b$, and $k$; level-dependent peaks or hysteresis between up- and down-sweeps are how nonlinearity or an extra mode announces itself.\n",{"path":22257,"title":22258,"module":22214,"summary":22259},"\u002Fmechanics\u002Foscillations-waves\u002Fwave-boundaries","Wave Boundaries","A pulse traveling along a string does something abrupt where the string's properties change: part reflects, part transmits, and which is which is set by the impedance mismatch alone. We impose continuity of displacement and transverse force at the join to get the reflection and transmission coefficients in terms of $Z=\\sqrt{T\\mu}$, fix their signs and the polarity flip, and balance the energy. The clean result assumes linear, nondispersive segments meeting at a localized join; pulse polarity, return timing, and energy ratios are the measurements that expose a real connector's mass, loss, or distributed transition.\n",{"path":22261,"title":22262,"module":22263,"summary":22264},"\u002Fmechanics\u002Fthermodynamics\u002Fkinetic-theory-of-ideal-gases","Kinetic Theory of Ideal Gases","Thermodynamics","A gas has no springs and no gears, yet it pushes on its container with a definite pressure and stores energy in a lawful way. Kinetic theory explains both from the motion of the molecules alone: pressure is the accumulated recoil of countless elastic impacts, and temperature is the average translational kinetic energy each molecule carries. We derive $pV=\\tfrac13Nm\\overline{v^2}$ from momentum transfer, read off $\\overline{K}_{\\rm tr}=\\tfrac32kT$, and use the Maxwell–Boltzmann distribution to separate the most probable, mean, and rms speeds — each the right average for a different question — while marking where the dilute, classical assumptions stop holding.\n",{"path":22266,"title":22267,"module":22263,"summary":22268},"\u002Fmechanics\u002Fthermodynamics\u002Ffirst-law-of-thermodynamics","First Law of Thermodynamics","Heat a gas and it may warm, expand, or both; compress it and the same energy can reappear as a temperature rise. The first law settles the bookkeeping: internal energy is a state property whose change equals the heat added plus the work done on the system, $\\Delta E_{\\rm int}=Q_{\\rm in}+W_{\\rm on}$. We fix a system boundary and one sign convention, compute boundary work as $\\int p\\,\\d V$ along a path, and use calorimetry to measure heat and heat capacities. The recurring point is that heat and work are path-dependent transfers while their sum is not, so an energy ledger closes only once every boundary crossing is named.\n",{"path":22270,"title":22271,"module":22263,"summary":22272},"\u002Fmechanics\u002Fthermodynamics\u002Fentropy-and-the-second-law","Entropy and the Second Law","The first law lets energy flow either way; it never says which way heat actually goes. The second law supplies the missing arrow. Entropy, defined through the reversible transfer $\\d S=\\delta Q_{\\rm rev}\u002FT$, can only increase in an isolated system, and that single inequality fixes the direction of heat flow and caps every engine, refrigerator, and heat pump at its Carnot value. We build entropy ledgers for reservoirs and working substances, separate the entropy carried by heat from the entropy generated by irreversibility, and read the sign of the total as a hard check on any proposed thermal machine.\n",{"path":22274,"title":22275,"module":22263,"summary":22276},"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-processes","Thermal Processes","Heat rarely sits still: it stretches solids, pushes real gases off their ideal isotherms, and leaks across walls by conduction, convection, and radiation. Each behavior becomes a number a designer can use. Thermal expansion sets the gaps in a bridge and the stress in a clamped rod; the van der Waals equation and a phase diagram fix when $pV=nRT$ or a latent-heat term applies; Fourier's law, Newton cooling, and Stefan–Boltzmann radiation give the rate of heat flow. We assemble these into thermal-resistance networks and transient time constants, then mark where contact resistance, phase change, or a hidden thermal bridge breaks the simple model.\n",{"path":22278,"title":22279,"module":22263,"summary":22280},"\u002Fmechanics\u002Fthermodynamics\u002Fphase-changes","Phase Changes","Add heat to ice and its temperature climbs — until it reaches $0\\ ^\\circ\\mathrm C$, where the thermometer stalls while the ice melts. That plateau is the whole subject: at a phase boundary the energy rearranges molecules, $Q=mL$, instead of raising temperature, which resumes only once one phase is gone. We stage a heating path into sensible-heat legs ($Q=mc\\Delta T$) and latent plateaus, use the Clausius–Clapeyron relation to track how a boiling point moves with pressure, and solve calorimetry by testing each coexistence endpoint — so a melt fraction that lands outside $[0,1]$ flags a wrong final-state guess rather than a real state.\n",{"path":22282,"title":22283,"module":22263,"summary":22284},"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-machines","Thermal Machines","An engine, a refrigerator, and a heat pump are one machine read three ways: each shuttles heat between a hot and a cold reservoir while trading work at the boundary, and only the flow you call useful separates them. A heat engine turns part of $Q_h$ into work, $W=Q_h-Q_c$; a refrigerator spends work to pull $Q_c$ from the cold side; a heat pump counts the warm-side delivery instead. We measure each with its own ratio — efficiency or coefficient of performance — bound them all by the Carnot limit that reservoir temperatures alone set, and track how finite temperature differences, throttling, and friction generate entropy and pull real machines below that bound.\n",{"path":22286,"title":22287,"module":6,"summary":6},"\u002Fmechanics","Mechanics & Dynamics",{"path":22289,"title":22290,"module":22291,"summary":22292},"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcharge-and-conductors","Charge and Conductors","Electric Fields","Rub two objects together and one pulls electrons from the other; nothing is created, only moved. We define what electric charge is — conserved, additive, and quantized in units of $e$ — and why a conductor's mobile carriers rearrange until its interior field vanishes and its surface sits at one potential. We follow charge through contact, induction, and grounding, treat the field-free cavity that turns a conductor into a shield, and mark where finite conductivity and leakage set the limits of the electrostatic picture.\n",{"path":22294,"title":22295,"module":22291,"summary":22296},"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcoulombs-law","Coulomb's Law","Two charges at rest push or pull along the line joining them, and the whole of electrostatics is assembled by adding up such pairs. We measure that force — its inverse-square falloff, its linear dependence on each charge, the sign that says attract or repel — and write it as a vector so direction survives superposition. We work the magnitude and component forms on real numbers, check them against limiting cases and dimensions, and fix the point-charge approximation to source sizes small against every separation.\n",{"path":22298,"title":22299,"module":22291,"summary":22300},"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-and-force","Electric Field and Force","Rather than ask how one charge reaches across empty space to another, we credit the source with a field that fills the space and let a second charge respond to whatever field sits at its own location. Electric field is force per unit positive test charge, $\\vec E=kq\\hat r\u002Fr^2$ for a point source, and source fields add before any receiving charge is placed. We compute those fields and the force $\\vec F=q\\vec E$ they exert, then follow a charge along its parabolic path through a uniform field and into nonuniform fields where the dynamics turn position-dependent.\n",{"path":22302,"title":22303,"module":22291,"summary":22304},"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-maps","Electric Field Maps","A field is a vector at every point of space, and the quickest way to grasp one is to draw it. We build the two standard pictures — continuous field lines tangent to $\\vec E$, and scaled vector arrows — and read direction, magnitude, and the location of nulls straight off them. We fix what a line drawing can and cannot say: density encodes magnitude only under a stated seeding rule, and integral curves never cross at a regular point. From there we work the topology near sources, sinks, and conductor surfaces, and state the step-size and interpolation checks a numerical map must pass.\n",{"path":22306,"title":22307,"module":22291,"summary":22308},"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-dipoles","Electric Dipoles","Most neutral matter carries no net charge yet still responds to an electric field, because its positive and negative charge sit slightly apart. That separation is a dipole, moment $\\vec p=q\\vec d$ pointing from the negative to the positive charge, and it is the leading term in how any neutral distribution looks from far away. We derive the torque $\\vec p\\times\\vec E$ and energy $-\\vec p\\cdot\\vec E$ a uniform field imposes, the net force a field gradient adds, and the axial and equatorial $1\u002Fr^3$ fields the pair produces, then measure how far out the point-dipole approximation still holds.\n",{"path":22310,"title":22311,"module":22312,"summary":22313},"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fcontinuous-charge-fields","Continuous Charge Fields","Continuous Charge Distributions","A charged rod, ring, or disk is not a point, yet its field is still nothing but Coulomb's law added up over the charge it carries. We replace the discrete sum by an integral, with $\\d q=\\lambda\\d\\ell$, $\\sigma\\d A$, or $\\rho\\d V$, so the real work becomes geometry: writing the vector from each source element to the field point, and letting symmetry cancel the components that must cancel before any integral is attempted. We carry the line, ring, and disk fields through in full, then check each result against its near field, its far field, and its dimensions.\n",{"path":22315,"title":22316,"module":22312,"summary":22317},"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fgauss-law-and-conductors","Gauss's Law and Conductors","Adding up Coulomb's law over a whole distribution is laborious; Gauss's law trades that sum for a single statement, that the flux of $\\vec E$ out of any closed surface counts the charge inside, $\\oint\\vec E\\cdot\\d\\vec A=Q_{\\rm enc}\u002F\\varepsilon_0$. The law is always true, but it hands over the field only when the source is symmetric enough to pull $E$ outside the integral. We apply it to spheres, lines, and sheets, then turn it on conductors, where the zero interior field drives every excess charge to the surface and fixes the normal-field jump $\\sigma\u002F\\varepsilon_0$, the charge induced on a cavity wall, and electrostatic shielding.\n",{"path":22319,"title":22320,"module":22321,"summary":22322},"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpoint-charge-potential","Point-Charge Potential","Electric Potential","The electrostatic force is conservative, so the work it does between two points\ndepends only on the endpoints. That lets us trade the vector field for a single\nscalar attached to each point, the electric potential, the potential energy a unit\ncharge would have there. We build potential from the work integral, fix the usual\nreference at infinity, and add point sources as scalars, $V=k\\sum_i q_i\u002Fr_i$,\navoiding the vector bookkeeping the field demands. Signed charges, the reference\nchoice, equipotential motion, and far-field expansions each give an independent\ncheck on a result.\n",{"path":22324,"title":22325,"module":22321,"summary":22326},"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpotential-gradients-and-equipotentials","Potential Gradients and Equipotentials","Given the potential everywhere, how do we recover the field? The field is the\nnegative gradient, $\\vec E=-\\nabla V$: it points down the steepest local drop in\npotential, and its magnitude is set by how fast $V$ changes, not by the shape of a\ncontour. We read off components with directional derivatives, reconstruct fields\nfrom measured potential grids using centered differences, and use closed-loop\nintegrals and grid refinement to test whether a reconstructed field is physically\nconsistent.\n",{"path":22328,"title":22329,"module":22321,"summary":22330},"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Felectrostatic-energy-and-pressure","Electrostatic Energy and Pressure","Assembling a charge configuration takes work, and that work is stored, but where\nis it kept and how much is there? We total it two ways: as a sum over the charges,\n$U=\\tfrac12\\sum_i q_iV_i$, and as an integral over the field itself,\n$u_E=\\tfrac12\\varepsilon_0E^2$, energy the field carries in every region it fills.\nDifferentiating the stored energy at fixed charge or at fixed voltage recovers the\nmechanical force on a conductor, and at a charged surface the same field scale\nappears as an outward electrostatic pressure. We work the parallel-plate case in\nfull and mark where curvature and fringing make the pressure nonuniform.\n",{"path":22332,"title":22333,"module":22321,"summary":22334},"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Flaplace-boundary-problems","Laplace Boundary Problems","Often the charges are not given, only the conductors and the voltages held on\nthem, and the potential in the empty space between has to be found. There $V$ obeys\nLaplace's equation $\\nabla^2V=0$, and the boundary data alone determine a unique solution.\nWe solve it two ways: separation of variables into boundary-matched modes, whose\nhigher spatial frequencies die away with depth into the domain, and finite-difference\nrelaxation for boundaries no analytic mode fits. Residual norms, boundary error, and\nflux balance tell us when the computed potential and its field can be trusted.\n",{"path":22336,"title":22337,"module":22321,"summary":22338},"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fcontinuous-charge-potentials","Continuous Charge Potentials","When charge is spread over a line, a surface, or a volume, the sum over point\nsources becomes an integral, $V(\\vec r)=k\\int \\d q\u002F|\\vec r-\\vec r'|$. Because\npotential is a scalar, this integral sidesteps the component algebra the field\nwould force, until the field is actually wanted through $\\vec E=-\\nabla V$. We set\nup the right density element for each geometry, choose a workable reference, handle\nthe integrable singularities that arise when the observation point sits on the\ncharge, and check every result against symmetry, dimensions, and the far-field\nmultipole limit.\n",{"path":22340,"title":22341,"module":22342,"summary":22343},"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitance-fundamentals","Capacitance Fundamentals","Capacitance","How much charge must you separate onto two conductors to hold a given voltage between\nthem? That ratio, $C=Q\u002F\\Delta V$, is fixed by the conductor geometry and the medium,\nnot by how much charge is presently stored. We compute it from the field for the\nparallel-plate, isolated-sphere, concentric-sphere, and coaxial geometries, trace how\nsurface charge and boundary conditions set each result, and see where fringing,\nguarding, and stray coupling separate the ideal formula from what a bridge measures.\n",{"path":22345,"title":22346,"module":22342,"summary":22347},"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-networks","Capacitor Networks","Wire several capacitors together and the source sees one equivalent capacitance — but\nwhich? The answer comes not from how the symbols are drawn but from which conductors\nshare a node: parallel branches hold a common voltage and add, $C_{\\rm eq}=\\sum_iC_i$,\nwhile series branches share a common charge and add reciprocally. We derive both rules\nfrom charge conservation on the floating internal node, then extend the node-charge\nmethod to unequal, precharged, and stray-coupled branches and carry a worked reduction\nthrough to the charge and voltage on every element.\n",{"path":22349,"title":22350,"module":22342,"summary":22351},"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-energy-and-force","Capacitor Energy and Force","Charging a capacitor takes work, because every increment of charge is pushed through\nthe voltage the earlier charge already established. We total that work three\nequivalent ways, $U=Q^2\u002F(2C)=Q\\Delta V\u002F2=C(\\Delta V)^2\u002F2$, locate it in the field as\na density $u=\\tfrac12\\epsilon_0E^2$, then let the plates move. Differentiating the\nstored energy at fixed charge, or the coenergy at fixed voltage, gives the mechanical\nforce; the two boundaries differ only by the work the source supplies. We work the\nparallel-plate attraction and its electrostatic pressure in full, and follow the same\ngradient into pull-in, tilt, comb drives, and traceable force calibration.\n",{"path":22353,"title":22354,"module":22342,"summary":22355},"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fdielectric-polarization-and-breakdown","Dielectric Polarization and Breakdown","Slide a dielectric between the plates and the capacitance rises — but why, and how\nhard can you drive it before the insulator fails? Bound charge answers the first:\npolarization $\\vec P$ sets up surface and volume charge that partly cancels the\napplied field, so $\\vec D=\\varepsilon_0\\vec E+\\vec P$ separates what the circuit\ncontrols from what the material contributes. We follow the field across layered\ndielectrics and interfaces, tie permittivity and loss to their frequency dependence,\nand treat dielectric strength as a measured, geometry-dependent limit rather than one\nmaterial number.\n",{"path":22357,"title":22358,"module":22359,"summary":22360},"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fcurrent-and-resistance","Current and Resistance","Direct-Current Circuits","What does it mean, physically, for charge to flow, and what sets how hard a wire resists that flow? Current counts charge crossing a surface, $I=\\int\\vec J\\cdot\\d\\vec A$, and traces back to a slow drift of many carriers, $\\vec J=nq\\vec v_d$. We establish when the linear law $V=IR$ actually holds, how resistivity and geometry combine into bulk resistance, why real sources sag under load through their internal resistance, and how the three power forms $P=IV=I^2R=V^2\u002FR$ tie electrical work to heating and component ratings.\n",{"path":22362,"title":22363,"module":22359,"summary":22364},"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fkirchhoff-network-analysis","Kirchhoff Network Analysis","Once a circuit has more than one loop, no amount of series-parallel folding will reduce it — you need the two conservation laws written as equations. Kirchhoff's junction law is charge conservation at a node; his loop law is energy conservation around a closed path. We turn a labelled network into a linear system in node voltages or mesh currents, fix the sign conventions so a negative answer just means a reversed arrow, and use power balance as an independent check that the algebra describes the circuit that was actually built.\n",{"path":22366,"title":22367,"module":22359,"summary":22368},"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Frc-transients","RC Transients","How does a circuit get from one steady state to the next when a capacitor refuses to change its voltage all at once? Because a jump would demand infinite current, an RC circuit slides between states exponentially, with a single time constant $\\tau=RC$ that sets the whole schedule: charging fills as $1-e^{-t\u002F\\tau}$, discharge empties as $e^{-t\u002F\\tau}$. We solve the first-order loop equation, read the response off three numbers — the switch-instant voltage, the final dc voltage, and the Thevenin resistance the capacitor sees — and mark where source and probe resistance shift $\\tau$ or where a second storage element hides a mode a one-$\\tau$ fit misses.\n",{"path":22370,"title":22371,"module":22372,"summary":22373},"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-trajectories","Magnetic Trajectories","Magnetic Field","A charged particle in a magnetic field never speeds up or slows down, yet its path curves relentlessly. We work out why: the magnetic force is always perpendicular to velocity, so it does no work and bends the transverse motion into a circle of radius $r=mv_\\perp\u002F(|q|B)$ while leaving the parallel motion untouched, producing a helix. We derive the cyclotron frequency, show why it is independent of speed until relativity intervenes, and turn the geometry around: a measured curvature reads back a particle's momentum, which is how tracking detectors weigh what they cannot see.\n",{"path":22375,"title":22376,"module":22372,"summary":22377},"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fhall-effect","Hall Effect","Current tells you charge is moving, but not whether the movers are positive or negative, nor how many there are. A magnetic field settles both questions. Push current through a strip in a transverse field and the carriers pile up on one edge until a transverse electric field just balances the magnetic deflection; the sign of the resulting Hall voltage names the carrier's charge and its size counts the carriers per volume. We derive the balance $q\\vec E+q\\vec v_d\\times\\vec B=0$, read off $V_H=IB\u002F(nqt)$, and see why field-and-current reversal is what separates the real Hall signal from the offsets that mimic it.\n",{"path":22379,"title":22380,"module":22372,"summary":22381},"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-force-on-conductors","Magnetic Force on Conductors","A magnet pushes on a current-carrying wire even though the wire is electrically neutral. The reason is that each moving carrier feels the Lorentz force, and those microscopic pushes add up to a force the wire's supports must hold. We sum them into $\\d\\vec F=I\\,\\d\\vec\\ell\\times\\vec B$, collapse it to $\\vec F=I\\vec L\\times\\vec B$ for a straight segment in a uniform field, and see exactly when that shortcut fails and the full path integral is needed. The same law runs backward as a measurement: a force-versus-current slope weighs a magnetic field against a known length.\n",{"path":22383,"title":22384,"module":22372,"summary":22385},"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-dipoles","Magnetic Dipoles","A compass needle turns to point north; a current loop in a field does the same thing, and for the same reason. Both are magnetic dipoles, and a uniform field cannot push a dipole anywhere, only twist it. We package a loop's response into one vector, the magnetic moment $\\vec\\mu=IA\\hat n$, from which torque $\\vec\\tau=\\vec\\mu\\times\\vec B$ and orientation energy $U=-\\vec\\mu\\cdot\\vec B$ both follow. Stable alignment sits at the energy minimum, a field gradient is what it takes to produce a net force $\\vec F=\\nabla(\\vec\\mu\\cdot\\vec B)$, and the same moment reappears whenever anything from an electron to a planet acts magnetic.\n",{"path":22387,"title":22388,"module":22372,"summary":22389},"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmass-spectrometry","Mass Spectrometry","To weigh a single atom you cannot use a scale, so you use a magnetic field instead. A charged ion of unknown mass bends in a field by an amount that depends on its momentum and charge, so if every ion enters with the same velocity, its landing position reads off its mass-to-charge ratio directly. We build the instrument in two stages: crossed electric and magnetic fields that pass only ions with $v=E\u002FB$, and a magnetic sector that bends the survivors along $r=mv\u002F(|q|B)$. Then we ask what blurs a spectral line and how reference ions turn a position into a calibrated mass.\n",{"path":22391,"title":22392,"module":22393,"summary":22394},"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmoving-charge-fields","Moving-Charge Fields","Magnetic Sources","Every magnetic field comes from charge in motion, and the simplest source is a single point charge drifting past. We work out the field it produces — normal to both the velocity and the line of sight, falling off as the inverse square — and read off why it vanishes straight ahead of the charge and peaks broadside. Summing many such charges is the bridge to steady currents, valid while speeds stay far below $c$ and the motion changes little during the time its field takes to propagate outward.\n",{"path":22396,"title":22397,"module":22393,"summary":22398},"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fbiot-savart-law","Biot–Savart Law","A steady current is a continuous stream of current elements, and the Biot–Savart law hands each one a magnetic contribution — a right-hand cross product that falls off as the inverse square of distance. Summing the contributions along a conductor is a vector line integral, which we carry out for the straight wire to get the endpoint-angle formula. The infinite-wire field $B=\\mu_0 I\u002F2\\pi s$ falls out as the limit where both ends recede, and we mark how fast a finite wire departs from it and when a thin-filament model is safe.\n",{"path":22400,"title":22401,"module":22393,"summary":22402},"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fcircular-current-loops","Circular Current Loops","A ring of current is the simplest source with a well-defined magnetic axis, and it is the building block of every coil and electromagnet. Symmetry kills the transverse Biot–Savart contributions along that axis and leaves a single clean integral; we evaluate it to get $B_z=\\mu_0 I R^2\u002F[2(R^2+z^2)^{3\u002F2}]$, read off the centre field $\\mu_0 I\u002F2R$, and watch it fall into the $1\u002Fz^3$ tail of a magnetic dipole far away. Stacking turns just adds their axial contributions, which is what makes a solenoid out of a pile of loops.\n",{"path":22404,"title":22405,"module":22393,"summary":22406},"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Famperes-law","Ampère’s Law","When a current arrangement is symmetric enough, the Biot–Savart integral is overkill: Ampère's law, $\\oint_C\\vec B\\cdot\\d\\vec\\ell=\\mu_0 I_{\\rm enc}$, gets the field from a single line of reasoning about how much current a loop encloses. We see why the law holds for any steady current, then use cylindrical, planar, and toroidal symmetry to turn the circulation into simple algebra — the field inside and outside a wire, an infinite sheet, a solenoid, and a toroid. We also mark the catch: without symmetry the law still holds but no longer hands you the field pointwise.\n",{"path":22408,"title":22409,"module":22393,"summary":22410},"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fgauss-law-for-magnetism","Gauss’s Law for Magnetism","Electric field lines start and end on charges; magnetic field lines do neither, because no one has ever found an isolated magnetic pole. That single experimental fact is Gauss's law for magnetism: the flux of $\\vec B$ through any closed surface is zero, $\\oint\\vec B\\cdot\\d\\vec A=0$, or in differential form $\\nabla\\cdot\\vec B=0$. We work through what it says — every field line that enters a closed surface must leave it, so field lines close on themselves — and, just as important, what it does not say, since flux through an open surface is generally nonzero.\n",{"path":22412,"title":22413,"module":22393,"summary":22414},"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmagnetic-materials","Magnetic Materials","Put matter in a magnetic field and its atoms respond, each acting as a tiny current loop; the aligned moments per unit volume are the magnetization $\\vec M$, whose bound currents add to the field. Separating what we control (the free current) from what the material supplies leads to $\\vec H$ and the relation $\\vec B=\\mu_0(\\vec H+\\vec M)$. We sort materials into diamagnets, paramagnets, and ferromagnets by how $\\vec M$ answers, follow a ferromagnet around its hysteresis loop, and see why the loop's area is the energy dissipated per cycle and why a sample's shape changes the field it actually feels.\n",{"path":22416,"title":22417,"module":22418,"summary":22419},"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-flux","Magnetic Flux","Electromagnetic Induction","A magnetic field threading a loop collapses to one signed number, the flux, and every induced voltage in this module turns out to be a rate of change of that number — so defining the flux and its sign comes first. We define it as the surface integral of $\\vec B$ over an oriented surface, reduce it to $BA\\cos\\theta$ for a uniform field on a flat loop, and carry the flux linkage $N\\Phi_B$ of a coil. The chosen normal fixes the sign; reversing it flips the sign without touching the field. Nonuniform fields and curved surfaces force the integral, so we also build the numerical estimate and the checks that separate a reliable value from a nominal field-times-area product.\n",{"path":22421,"title":22422,"module":22418,"summary":22423},"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Ffaradays-law","Faraday's Law","Move a magnet toward a coil, or ramp the current in a nearby circuit, and a voltage appears with no battery in sight. Faraday's law names the cause: the emf around a loop equals minus the rate of change of the magnetic flux through it, so any change of field, area, orientation, or position that alters the flux drives an emf. We separate the emf, which lives around the boundary whether or not current can flow, from the current that follows only when the path is closed; fix the single sign convention that ties flux to loop orientation; and read the emf off rotating coils and off flux sampled at discrete times.\n",{"path":22425,"title":22426,"module":22418,"summary":22427},"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Flenzs-law","Lenz's Law","The minus sign in Faraday's law is not decoration: it decides which way the induced current flows, and it always chooses the direction that fights the change that produced it. Lenz's law reads that sign off energy conservation — a current that aided the change would be free energy — and turns it into a repeatable procedure. We fix a surface normal and a positive loop direction so the sign is calculable, then work through approaching magnets, expanding loops, coupled coils, and rotating generators, using mechanical work and Joule heating as an independent check on every direction we draw.\n",{"path":22429,"title":22430,"module":22418,"summary":22431},"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmotional-emf","Motional EMF","Push a wire through a magnetic field and its free charges feel a sideways magnetic force that piles them up at the ends — a battery made of motion. Motional emf is that effect: the work per unit charge a moving conductor supplies is the line integral of $\\vec v\\times\\vec B$ along it, which for a rod moving perpendicular to both its length and the field collapses to $B\\ell v$. We chase where the energy comes from — the hand or motor fighting the magnetic drag, never the magnetic force itself — solve the sliding-rail circuit from both flux and carrier forces, and carry the idea into rotating rods, homopolar disks, generators, and the back emf of a motor.\n",{"path":22433,"title":22434,"module":22418,"summary":22435},"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Feddy-currents","Eddy Currents","A wire carries current along one path; a solid block of metal offers a continuum of them, and any changing flux threading that block sets charge circulating in closed loops it chooses for itself. We ask what those eddy currents do — where they heat, where they drag, and how Lenz's law fixes their direction — and why the same circulation is a feature in an induction furnace and a loss to be suppressed in a transformer core. From a representative-loop estimate we get the scaling (heating grows with the square of frequency and flux rate) and the two design levers, lamination and resistivity, that break the paths a solid conductor would otherwise hand the current.\n",{"path":22437,"title":22438,"module":22418,"summary":22439},"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fself-inductance","Self-Inductance","A coil resists changes to its own current. Drive current through it and the flux it produces threads its own turns; change that current and Faraday's law turns the coil against the source with a back emf $\\mathcal E_L=-L\\,\\d I\u002F\\d t$. We define self-inductance as the flux linkage per ampere fixed by winding and core geometry, derive the long-solenoid value $L=\\mu_0 N^2A\u002F\\ell$, and follow the consequence that dominates circuits: because a finite voltage can only sustain a finite $\\d I\u002F\\d t$, an inductor's current cannot jump — which is why opening a switch on a live coil throws a spark.\n",{"path":22441,"title":22442,"module":22418,"summary":22443},"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-energy","Magnetic Energy","Building current in a coil means working against its back emf, and that work does not vanish — it sits in the magnetic field as recoverable energy $U_B=\\tfrac12LI^2$, spread through space at density $u_B=B^2\u002F(2\\mu_0)$. We derive both forms, show they agree for a solenoid, and read a force out of the same energy: an armature is pulled toward higher inductance, and $B^2\u002F(2\\mu_0)$ doubles as a magnetic pressure. The lesson closes on the accounting a real switching event demands, where recoverable energy, copper heating, core loss, and clamp dissipation must balance a single ledger.\n",{"path":22445,"title":22446,"module":22418,"summary":22447},"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Frl-circuits","RL Circuits","Put a resistor and an inductor in series and the current cannot switch on or off at will: it climbs to $V_0\u002FR$ and falls away exponentially on a single time scale $\\tau=L\u002FR$ set by how much flux the coil hoards against how fast the resistor bleeds it. We solve the turn-on and turn-off, then confront the practical sting — because the coil's current refuses to stop instantly, breaking its path throws up a large voltage, which is why real inductive circuits carry freewheel diodes and clamps that trade voltage stress against how quickly the current dies.\n",{"path":22449,"title":22450,"module":22451,"summary":22452},"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-fundamentals","AC Fundamentals","Alternating Current","A wall socket delivers a voltage that averages to zero over each cycle, yet it still heats a filament and runs a motor. The resolution is that dissipation follows the mean of the square, not the mean, so we define the root-mean-square value that makes an alternating source the equal of a DC one for resistive heating. We show a sinusoid's RMS is its peak divided by $\\sqrt2$, work out the average power an ideal resistor draws when its current stays in phase with the applied voltage, and separate the peak, average, and RMS descriptions that a single number cannot combine.\n",{"path":22454,"title":22455,"module":22451,"summary":22456},"\u002Felectricity-and-magnetism\u002Falternating-current\u002Freactance","Reactance","A resistor obeys Ohm's law instant by instant, but a capacitor responds to how fast its voltage changes and an inductor to how fast its current changes. Under a steady sinusoid that rate-dependence collapses to a fixed quarter-cycle phase shift and a frequency-dependent amplitude ratio, the reactance. We derive $X_C=1\u002F(\\omega C)$ and $X_L=\\omega L$, adopt phasors to turn the defining derivatives into multiplication by $j\\omega$ so a single complex impedance carries amplitude and phase together, and track the energy an ideal reactance stores and returns without dissipating it. Real windings and dielectrics add loss, leakage, and self-resonance that bound where the ideal formulas hold.\n",{"path":22458,"title":22459,"module":22451,"summary":22460},"\u002Felectricity-and-magnetism\u002Falternating-current\u002Frlc-resonance","RLC Resonance","Put a resistor, inductor, and capacitor in one loop and their reactances work against each other: inductive reactance grows with frequency while capacitive reactance shrinks, and at one frequency they cancel exactly. There the branch looks purely resistive, the current peaks, and the inductor and capacitor voltages can swing far above the source. We locate that resonance at $\\omega_0=1\u002F\\sqrt{LC}$, measure how sharp the peak is with the quality factor $Q=\\omega_0L\u002FR$, tie its half-power bandwidth $R\u002FL$ to the ringdown of the unforced circuit, and read the same poles off as bandpass and peaked filters at the R, L, or C terminals.\n",{"path":22462,"title":22463,"module":22451,"summary":22464},"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-power","AC Power","Multiply an AC load's RMS voltage by its RMS current and you get an answer in volt-amperes that the wiring must carry, but not in general the watts the load consumes. The phase between voltage and current splits that product into a part that does net work and a part that merely sloshes energy back and forth. We derive the average power $P=V_{\\rm rms}I_{\\rm rms}\\cos\\phi$, package amplitude and phase into complex power $S=P+jQ$ so that real, reactive, and apparent power form one right triangle, and see why a harmonic-rich current forces the time-domain definition $P=\\langle vi\\rangle$ in place of a single phase angle.\n",{"path":22466,"title":22467,"module":22451,"summary":22468},"\u002Felectricity-and-magnetism\u002Falternating-current\u002Ftransformers","Transformers","Two coils sharing an iron core exchange no charge, yet a changing current in one drives a voltage in the other, and the ratio of their turns sets how voltage and current trade off between the windings. That lets a transformer step a voltage up or down, isolate two circuits, and make a load look larger or smaller to the source by the square of the turns ratio. We build the ideal ratio element from Faraday's law and the dot convention, derive the reflected-impedance rule, then add the winding resistance, leakage, magnetizing current, and core loss that turn the ideal ratios into real regulation, efficiency, and a bounded voltage-frequency range.\n",{"path":22470,"title":22471,"module":22472,"summary":22473},"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdisplacement-current","Displacement Current","Maxwell’s Equations and Electromagnetic Waves","Ampère's law asks for the current through a surface bounded by a loop, but a charging capacitor breaks it: slide the surface off the wire and into the gap and the enclosed conduction current drops to zero, while the magnetic field around the loop plainly does not. Maxwell's repair is to count a changing electric flux as itself a source of magnetic circulation. We derive the displacement-current term $\\varepsilon_0\\,\\d\\Phi_E\u002F\\d t$, show that charge continuity demands it, compute the magnetic field it produces inside a charging capacitor, and see how it closes the Ampère–Maxwell law so that electric and magnetic fields can sustain one another as a wave.\n",{"path":22475,"title":22476,"module":22472,"summary":22477},"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-waves","Electromagnetic Waves","Once a changing electric flux can drive a magnetic field, the two curl laws feed each other: a disturbance in one regenerates the other, and the pair walks off through empty space with no medium holding it up. We take the curl of Faraday's law, land on a wave equation whose speed is fixed entirely by $\\mu_0$ and $\\varepsilon_0$, and find that $c=1\u002F\\sqrt{\\mu_0\\varepsilon_0}$ falls out of purely electric and magnetic constants. The plane-wave solution then fixes the geometry — $\\vec E$, $\\vec B$, and the propagation direction mutually perpendicular, oscillating in phase, with amplitudes locked at $E=cB$ — a set of independent predictions any real measurement must meet at once.\n",{"path":22479,"title":22480,"module":22472,"summary":22481},"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-momentum","Electromagnetic Momentum","A light beam carries no mass, yet it pushes: shine it on a surface and the surface feels a force. We trace that force back to the fields, which store energy with density $\\varepsilon_0E^2$ and carry it along the Poynting vector $\\vec S=\\vec E\\times\\vec B\u002F\\mu_0$. Because that energy also carries momentum $U\u002Fc$, an absorbed beam presses with $I\u002Fc$ and a mirror with $2I\u002Fc$. We derive the Poynting theorem as local energy conservation, tie intensity to field amplitude, and work the momentum balance carefully enough that oblique incidence, partial reflection, and finite beams all drop out of one accounting.\n",{"path":22483,"title":22484,"module":22472,"summary":22485},"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdipole-radiation","Dipole Radiation","Only accelerating charge radiates, and the simplest accelerator is a charge sloshing back and forth: an oscillating electric dipole. We work out the field it throws off, keeping the part that survives to large distance — the $1\u002Fr$ radiation field whose intensity goes as $\\sin^2\\theta\u002Fr^2$, zero along the dipole axis and strongest broadside. From it follow the $\\omega^4$ scaling of total radiated power, radiation resistance as the feed's view of that escaping power, and, through reciprocity, the fact that a good transmitter receives well in the same directions. The near-zone terms that fall off faster carry no net power, and we mark carefully where each description is allowed to be used.\n",{"path":22487,"title":22488,"module":22472,"summary":22489},"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fpolarization","Polarization","A plane wave still leaves one thing free: which way its electric field points as it oscillates. That freedom is polarization, set entirely by the relative amplitude and phase of the two transverse field components — in phase gives a line, equal amplitudes a quarter cycle apart give a circle, everything else an ellipse. We work out how a linear analyzer reads a state through Malus's law $I=I_0\\cos^2\\theta$, why that scan alone cannot tell circular light from unpolarized, and how a quarter-wave plate plus a few analyzer settings recover the full Stokes vector and the degree of polarization.\n",{"path":22491,"title":22492,"module":22493,"summary":22494},"\u002Felectricity-and-magnetism\u002Foptics\u002Freflection-and-refraction","Reflection and Refraction","Geometrical Optics","Light meeting a boundary between two transparent media splits into a reflected ray and a bent transmitted one, and predicting where those rays go is the whole starting point of geometrical optics. Fixing one convention — every angle measured from the surface normal — we get reflection's equal angles and derive Snell's law $n_1\\sin\\theta_1=n_2\\sin\\theta_2$ from wavefront timing. That single relation, applied once or twice, yields the critical angle and total internal reflection, prism deviation, the lateral shift through a window, apparent depth, and a fiber's acceptance cone; a wavelength-dependent index then adds dispersion. We mark throughout where the ray picture is trustworthy: feature sizes large against the wavelength and clean interface geometry.\n",{"path":22496,"title":22497,"module":22493,"summary":22498},"\u002Felectricity-and-magnetism\u002Foptics\u002Fthin-lenses","Thin Lenses","A lens gathers the light spreading from one point back onto another, and a single paraxial relation $1\u002Fs+1\u002Fs'=1\u002Ff$ predicts where that image lands and how large it is. We collapse two refractions into one bending plane, read image position and orientation off the three principal rays, and trace focal length back to glass and curvature through the lensmaker equation. Sign conventions carry the physics here — they separate real from virtual images and upright from inverted — so we drill them before chaining lenses in sequence and in contact. The lesson ends on how focal length is actually measured on a bench, and where finite thickness, aperture, and dispersion break the thin-lens picture.\n",{"path":22500,"title":22501,"module":22493,"summary":22502},"\u002Felectricity-and-magnetism\u002Foptics\u002Fspherical-mirrors","Spherical Mirrors","Curve a mirror and it stops merely reflecting an image and starts forming one: the same $1\u002Fs+1\u002Fs'=1\u002Ff$ that governs lenses reappears, now with $f=R\u002F2$ and reflected rays and object sharing one side of the glass. We derive the mirror equation from the reflection geometry of a single paraxial ray, then let signed distances do the sorting — real inverted images on the near branch, virtual upright ones behind the surface — and check the concave, convex, and plane-mirror limits against each other. The second half turns to how focal length is actually measured on a bench, by finite conjugates, distant targets, return imaging, and sagitta, and to the aperture and off-axis aberrations the single paraxial focus cannot capture.\n",{"path":22504,"title":22505,"module":6,"summary":6},"\u002Felectricity-and-magnetism","Electricity & Magnetism",{"path":22507,"title":22508,"module":22509,"summary":22510},"\u002Flinear-algebra\u002Flinear-systems\u002Fsystems-and-echelon-forms","Systems of Linear Equations and Row Reduction","Linear Equations in Linear Algebra","A linear system is a finite set of linear equations in shared variables. Elementary row operations rewrite it without changing its solution set, and reducing the augmented matrix to echelon form decides both existence and uniqueness. Pivot positions say whether the solution set is empty, a single point, or infinite.\n",{"path":22512,"title":22513,"module":22509,"summary":22514},"\u002Flinear-algebra\u002Flinear-systems\u002Fvector-and-matrix-equations","Vector Equations and the Matrix Equation Ax = b","The same linear system reads three equivalent ways: a system of equations, a vector equation asking whether b is a linear combination of fixed vectors, and a matrix equation Ax = b. Ax is the linear combination of A's columns weighted by x, so consistency for a given b means b lies in the span of the columns, and consistency for every b means the columns span all of R^m.\n",{"path":22516,"title":22517,"module":22509,"summary":22518},"\u002Flinear-algebra\u002Flinear-systems\u002Fsolution-sets-and-applications","Solution Sets and Applied Linear Systems","A homogeneous system Ax = 0 has a solution set that is a span through the origin; a consistent Ax = b has that same span translated by any one particular solution. Parametric vector form writes both explicitly. The structure shows up in applied systems with many solutions: equilibrium prices, balanced chemical reactions, network flows, weight-loss diets, and migration models.\n",{"path":22520,"title":22521,"module":22509,"summary":22522},"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-independence","Linear Independence","A set of vectors is linearly independent when the only linear combination equal to zero is the trivial one; otherwise a dependence relation writes one vector in terms of the others. For the columns of A the question becomes whether Ax = 0 has only the trivial solution — a pivot in every column. Counting pivots settles independence, and any set with more vectors than entries is automatically dependent.\n",{"path":22524,"title":22525,"module":22509,"summary":22526},"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-transformations","Linear Transformations and Their Matrices","Reading A as an action rather than an array, x maps to Ax is a transformation from R^n to R^m. The ones that preserve addition and scalar multiplication are the linear transformations, and every one is x maps to Ax for a unique standard matrix whose columns are the images of the standard basis vectors. Onto and one-to-one translate into the span and independence of those columns.\n",{"path":22528,"title":22529,"module":22530,"summary":22531},"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-operations","Matrix Operations","Matrix Algebra","Matrices add and scale entrywise, but their product is defined so that multiplication corresponds to composition of linear maps: the columns of AB are A applied to the columns of B. From that requirement follow the row-column rule, the algebra of products (associative and distributive but not commutative), powers, and the transpose.\n",{"path":22533,"title":22534,"module":22530,"summary":22535},"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-inverse-and-invertibility","The Inverse and the Invertible Matrix Theorem","The inverse of a square matrix is the matrix analogue of a reciprocal, defined by AA⁻¹ = I. A closed form settles the 2×2 case; the Gauss–Jordan algorithm row reduces [A | I] to [I | A⁻¹] in general; and elementary matrices record single row operations. The Invertible Matrix Theorem collects a dozen equivalent conditions for invertibility into one statement.\n",{"path":22537,"title":22538,"module":22530,"summary":22539},"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fpartitioned-matrices-and-lu","Block Matrices and the LU Factorization","Partitioning a matrix into blocks lets sums, products, and inverses be computed block by block, as if the submatrices were scalars. Block structure also underlies the LU factorization A = LU, which splits solving Ax = b into two fast triangular solves and repays the cost whenever many systems share one coefficient matrix.\n",{"path":22541,"title":22542,"module":22530,"summary":22543},"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fsubspaces-dimension-rank","Subspaces of Rⁿ, Dimension, and Rank","A subspace is a set closed under addition and scalar multiplication. Every matrix carries two: the column space of all attainable outputs Ax, and the null space of all solutions of Ax = 0. A basis measures each with a minimal spanning set, dimension counts it, and the Rank Theorem ties pivots and free variables together as rank + nullity = n.\n",{"path":22545,"title":22546,"module":22530,"summary":22547},"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fapplications-leontief-and-graphics","Applications: Leontief Economics and Computer Graphics","The Leontief input–output model balances an economy through (I − C)x = d and expands the inverse as a geometric series in the consumption matrix. Computer graphics moves figures with matrix products, using homogeneous coordinates so that translation and perspective projection become matrix multiplications too.\n",{"path":22549,"title":22550,"module":22551,"summary":22552},"\u002Flinear-algebra\u002Fdeterminants\u002Fdeterminants-and-cofactors","Introduction to Determinants","Determinants","The determinant of a square matrix is defined recursively by cofactor expansion: an n-by-n determinant is a signed sum of (n-1)-by-(n-1) determinants built from the first row. The expansion can equally run along any row or down any column, and a triangular matrix has determinant equal to the product of its diagonal.\n",{"path":22554,"title":22555,"module":22551,"summary":22556},"\u002Flinear-algebra\u002Fdeterminants\u002Fproperties-of-determinants","Properties of Determinants","Row operations act on the determinant in three predictable ways, and this turns row reduction into a fast algorithm: the determinant is the product of the pivots times a sign for the interchanges. The same properties yield the invertibility test det A is nonzero, the transpose identity, and the multiplicative law det(AB) equals det A times det B.\n",{"path":22558,"title":22559,"module":22551,"summary":22560},"\u002Flinear-algebra\u002Fdeterminants\u002Fcramer-volume-and-area","Cramer's Rule, Volume, and Linear Transformations","Cramer's rule writes each unknown of an invertible system as a ratio of determinants, and the same idea gives a closed formula for the inverse through the adjugate. Geometrically the absolute determinant is the area of the parallelogram or the volume of the parallelepiped spanned by the columns, so a linear map scales every region's measure by that factor.\n",{"path":22562,"title":22563,"module":22564,"summary":22565},"\u002Flinear-algebra\u002Fvector-spaces\u002Fvector-spaces-and-subspaces","Vector Spaces and Subspaces","Vector Spaces","A vector space is any set closed under addition and scalar multiplication that obeys ten algebraic axioms. The same axioms that govern arrows in the plane govern polynomials, functions, matrices, and infinite signals, so one theory covers them all. A subspace is a subset that is a vector space in its own right, tested by three conditions, and the span of any set of vectors is the smallest subspace containing them.\n",{"path":22567,"title":22568,"module":22564,"summary":22569},"\u002Flinear-algebra\u002Fvector-spaces\u002Fnull-and-column-spaces","Null Spaces, Column Spaces, and Linear Transformations","Two subspaces sit inside every matrix. The null space collects all solutions of $Ax = 0$ and lives in the domain; the column space collects every attainable $Ax$ and lives in the codomain. One is defined implicitly by a condition, the other explicitly by a spanning set, and the same pair appears for an abstract linear transformation as its kernel and range.\n",{"path":22571,"title":22572,"module":22564,"summary":22573},"\u002Flinear-algebra\u002Fvector-spaces\u002Fbases-and-independent-sets","Linearly Independent Sets and Bases","A basis is a spanning set with no redundancy: linearly independent and still large enough to reach every vector. The spanning-set theorem shows any spanning set can be trimmed to a basis by discarding dependent vectors, and the pivot columns of a matrix give a basis for its column space. Independence and spanning are defined for abstract spaces exactly as in $\\mathbb{R}^n$.\n",{"path":22575,"title":22576,"module":22564,"summary":22577},"\u002Flinear-algebra\u002Fvector-spaces\u002Fcoordinate-systems","Coordinate Systems","Fixing a basis assigns every vector a unique list of coordinates, turning an abstract space into $\\mathbb{R}^n$. The coordinate mapping is a one-to-one linear transformation onto $\\mathbb{R}^n$ — an isomorphism — so any $n$-dimensional space is indistinguishable from $\\mathbb{R}^n$ as far as vector-space computations go. In $\\mathbb{R}^n$ the change-of-coordinates matrix $P_B$ and its inverse convert between basis coordinates and standard coordinates.\n",{"path":22579,"title":22580,"module":22564,"summary":22581},"\u002Flinear-algebra\u002Fvector-spaces\u002Fdimension-and-rank","The Dimension of a Vector Space and Rank","Every basis of a space has the same number of vectors, and that number is the dimension. Rank is the dimension of the column space, equal to the dimension of the row space and to the number of pivots. The Rank Theorem, rank plus nullity equals the number of columns, ties the four fundamental subspaces of a matrix together and adds six lines to the Invertible Matrix Theorem.\n",{"path":22583,"title":22584,"module":22564,"summary":22585},"\u002Flinear-algebra\u002Fvector-spaces\u002Fchange-of-basis","Change of Basis","Two bases give the same vector two different coordinate vectors, and a single invertible matrix converts between them. Its columns are the coordinate vectors of the old basis expressed in the new one, and its inverse reverses the conversion. In $\\mathbb{R}^n$ the change-of-coordinates matrix between two bases is found by one row reduction.\n",{"path":22587,"title":22588,"module":22564,"summary":22589},"\u002Flinear-algebra\u002Fvector-spaces\u002Fdifference-equations-and-markov","Applications: Difference Equations and Markov Chains","The solutions of an nth-order linear difference equation form an $n$-dimensional vector space, so finding $n$ independent solutions gives them all. A Markov chain evolves a probability distribution by repeated multiplication by a stochastic matrix, and a regular chain converges to a unique steady-state vector fixed by that matrix. Both applications turn a dynamic process into a subspace or a fixed-point question.\n",{"path":22591,"title":22592,"module":22593,"summary":22594},"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-eigenvalues","Eigenvectors and Eigenvalues","Eigenvalues and Eigenvectors","An eigenvector of a square matrix is a nonzero vector the matrix only stretches; its eigenvalue is the stretch factor. The eigenspace of an eigenvalue is the null space of A minus lambda times the identity, the eigenvalues of a triangular matrix are its diagonal entries, and eigenvectors for distinct eigenvalues are linearly independent.\n",{"path":22596,"title":22597,"module":22593,"summary":22598},"\u002Flinear-algebra\u002Feigenvalues\u002Fthe-characteristic-equation","The Characteristic Equation","The eigenvalues of a matrix are the roots of its characteristic polynomial det(A minus lambda I). This degree-n polynomial carries an algebraic multiplicity at each repeated root, a nonzero determinant is equivalent to zero not being an eigenvalue, and similar matrices share a characteristic polynomial and hence the same eigenvalues.\n",{"path":22600,"title":22601,"module":22593,"summary":22602},"\u002Flinear-algebra\u002Feigenvalues\u002Fdiagonalization","Diagonalization","A matrix is diagonalizable when it factors as A equals P D P inverse with D diagonal, which happens exactly when it has n linearly independent eigenvectors. The factorization computes matrix powers cheaply, distinct eigenvalues guarantee it, and a repeated eigenvalue permits it only when its eigenspace dimension equals its multiplicity.\n",{"path":22604,"title":22605,"module":22593,"summary":22606},"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-linear-transformations","Eigenvectors and Linear Transformations","Every linear transformation between finite-dimensional spaces has a matrix relative to chosen bases, built from the coordinate vectors of the images of the basis vectors. For a map from a space to itself, an eigenvector basis makes that matrix diagonal, and that change of basis is diagonalization; the matrices similar to A are the representations of the map in every basis.\n",{"path":22608,"title":22609,"module":22593,"summary":22610},"\u002Flinear-algebra\u002Feigenvalues\u002Fcomplex-eigenvalues","Complex Eigenvalues","A real matrix with no real eigenvalues still has complex ones, occurring in conjugate pairs. A real 2-by-2 matrix with eigenvalue a plus b i is similar to a rotation-scaling matrix, whose rotation angle is the argument of the eigenvalue and whose scale factor is its modulus; the modulus decides whether the trajectories close up, spiral in, or spiral out.\n",{"path":22612,"title":22613,"module":22593,"summary":22614},"\u002Flinear-algebra\u002Feigenvalues\u002Fdynamical-systems","Discrete and Continuous Dynamical Systems","Eigenvalues govern the long-term behavior of a system that evolves by x becomes A x or by x prime equals A x. An eigenvector basis decouples both kinds of system into independent scalar equations; the eigenvalues then classify the origin as attractor, repeller, saddle, or spiral, and the dominant eigenpair fixes the growth rate and limiting direction.\n",{"path":22616,"title":22617,"module":22593,"summary":22618},"\u002Flinear-algebra\u002Feigenvalues\u002Fpower-method","Iterative Estimates for Eigenvalues","When only a numerical eigenvalue is needed, iteration is preferred over the characteristic polynomial. The power method repeatedly multiplies by A to converge on the dominant eigenvalue and its eigenvector; the Rayleigh quotient sharpens the estimate for symmetric matrices; and the inverse power method targets any eigenvalue near a known guess.\n",{"path":22620,"title":22621,"module":22622,"summary":22623},"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-length-orthogonality","Inner Product, Length, and Orthogonality","Orthogonality and Least Squares","The dot product turns the algebra of vectors in R^n into geometry: length, distance, and perpendicularity. The inner product yields the norm, the Pythagorean theorem, and the orthogonal complement, and the null space of a matrix is the orthogonal complement of its row space.\n",{"path":22625,"title":22626,"module":22622,"summary":22627},"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Forthogonal-sets-and-projections","Orthogonal Sets and Orthogonal Projections","An orthogonal basis makes coordinates trivial: each weight is a single dot product, no linear system required. Orthogonal and orthonormal bases give a direct projection formula onto a line and onto a subspace, the orthogonal decomposition and best-approximation theorems, and the matrix form U U-transpose of a projection.\n",{"path":22629,"title":22630,"module":22622,"summary":22631},"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fgram-schmidt-and-qr","The Gram-Schmidt Process and QR Factorization","Gram-Schmidt turns any basis into an orthogonal one by repeatedly subtracting off projections onto the span already built. Normalizing the result and recording the coefficients factors the matrix as A = QR, with Q orthonormal and R upper triangular, the factorization behind stable least-squares and eigenvalue algorithms.\n",{"path":22633,"title":22634,"module":22622,"summary":22635},"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-problems","Least-Squares Problems","When Ax = b has no solution, the least-squares solution makes Ax as close to b as possible. The closest Ax is the projection of b onto the column space, and the vector that produces it solves the normal equations A-transpose A x = A-transpose b. Uniqueness, the residual error, and the stabler QR route follow.\n",{"path":22637,"title":22638,"module":22622,"summary":22639},"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-applications","Applications to Linear Models","Curve fitting is a least-squares problem in statistical notation. The least-squares line, polynomial fits, and multiple regression all reduce to X beta = y with a design matrix X built from the data, solved by the same normal equations.\n",{"path":22641,"title":22642,"module":22622,"summary":22643},"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-spaces","Inner Product Spaces","Promoting the four properties of the dot product to axioms defines an inner product on any vector space, including spaces of functions. Length, distance, orthogonality, Gram-Schmidt, and best approximation all carry over, along with the Cauchy-Schwarz and triangle inequalities and the integral inner product behind Fourier approximation.\n",{"path":22645,"title":22646,"module":22647,"summary":22648},"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fdiagonalizing-symmetric-matrices","Diagonalization of Symmetric Matrices","Symmetric Matrices, Quadratic Forms, and the SVD","A symmetric matrix is one that equals its own transpose. Every such matrix can be diagonalized by an orthogonal change of basis, A = PDPᵀ, with real eigenvalues and perpendicular eigenvectors. This is the Spectral Theorem, and it rewrites A as a weighted sum of rank-one projections onto its eigenvectors.\n",{"path":22650,"title":22651,"module":22647,"summary":22652},"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fquadratic-forms","Quadratic Forms","A quadratic form xᵀAx is the second-degree analogue of a linear map, attached to a symmetric matrix A. Orthogonal diagonalization changes variables to the eigenbasis, removing all cross-terms and rotating the form into standard position. The signs of the eigenvalues then classify it as definite or indefinite.\n",{"path":22654,"title":22655,"module":22647,"summary":22656},"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fconstrained-optimization","Constrained Optimization","Maximizing a quadratic form xᵀAx over the unit sphere has an exact answer: the maximum is the largest eigenvalue of A, attained at its eigenvector, and the minimum is the smallest eigenvalue. Adding orthogonality constraints peels off the eigenvalues in order, characterizing the whole spectrum by optimization.\n",{"path":22658,"title":22659,"module":22647,"summary":22660},"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsingular-value-decomposition","The Singular Value Decomposition","The singular value decomposition factors any m×n matrix as A = UΣVᵀ, with orthogonal U and V and a nonnegative diagonal Σ of singular values. The singular values are the square roots of the eigenvalues of AᵀA, and they describe the matrix geometrically as a rotation, an axiswise stretch, and another rotation, exposing rank, the four fundamental subspaces, and a best low-rank approximation.\n",{"path":22662,"title":22663,"module":22647,"summary":22664},"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsvd-applications-pca-imaging","Applications: Image Processing and Statistics","Principal component analysis diagonalizes the covariance matrix of a data set, producing uncorrelated variables ordered by variance. The leading components capture most of the variation, which reduces dimension, compresses images through low-rank SVD approximation, and connects directly to the singular values of the data matrix.\n",{"path":22666,"title":22667,"module":22668,"summary":22669},"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-thinking-and-matrix-computation","Numerical Thinking and Matrix Computation","Numerical Linear Algebra","Numerical analysis builds efficient discrete algorithms for continuous problems, and its cost is dominated as much by memory traffic as by arithmetic. Block matrix calculus, flop counts, and the BLAS efficiency ratio fix the cost model; triangular and unitary matrices are the two computational building blocks every factorization rests on.\n",{"path":22671,"title":22672,"module":22668,"summary":22673},"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Flu-and-cholesky","LU and Cholesky Factorization in Practice","Gaussian elimination, read as a factorization A = LU, turns a linear system into two triangular solves. A single near-zero pivot wrecks it, so partial pivoting reorders rows to pick the largest available pivot and makes the method work for every invertible matrix. For symmetric positive-definite systems, Cholesky halves the cost and needs no pivoting.\n",{"path":22675,"title":22676,"module":22668,"summary":22677},"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fconditioning-and-floating-point","Conditioning and Floating-Point Arithmetic","A problem's condition number measures how much its answer moves when its data is perturbed, independent of any algorithm. Subtraction is ill-conditioned under cancellation, and for a linear system the amplifier is the matrix condition number κ(A). Floating-point arithmetic supplies the perturbation: every real number is rounded to within a relative machine precision, so even perfect computation inherits an error of order κ times the unit roundoff.\n",{"path":22679,"title":22680,"module":22668,"summary":22681},"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fstability-and-error-analysis","Numerical Stability and Backward Error Analysis","An algorithm is backward stable when its computed answer is the exact answer to a slightly perturbed problem. Combined with the condition number this gives the governing rule of thumb: forward error is at most condition times stability. Three cancellation case studies make the point, then the residual-based backward error applies it to Ax = b and shows why partial pivoting keeps Gaussian elimination stable.\n",{"path":22683,"title":22684,"module":22668,"summary":22685},"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fqr-and-numerical-least-squares","QR, Householder, and Numerical Least Squares","The least-squares problem reduces to the normal equations, but forming AᵀA squares the condition number and can wreck accuracy. The stable route computes a QR factorization directly on A and solves Rx = Qᵀb. Householder reflectors build that QR one column at a time using length-preserving reflections, the unconditionally backward-stable building block behind every serious least-squares solver.\n",{"path":22687,"title":22688,"module":22668,"summary":22689},"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-eigenvalues-and-svd","Numerical Eigenvalue Problems and the SVD","Eigenvalues cannot be found by a formula for large matrices, so they are found by iteration. Power and inverse iteration converge to one eigenvector at a rate set by the eigenvalue gap; the QR algorithm sweeps a matrix to Schur form and, with a good shift and a Hessenberg reduction, computes the whole spectrum in cubic time. Singular values follow from the same machinery applied without ever forming AᵀA.\n",{"path":22691,"title":22692,"module":22693,"summary":22694},"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-combinations","Affine Combinations","Geometry of Vector Spaces","An affine combination is a linear combination whose weights sum to one. The affine hull of a set is the smallest flat containing it: a point, a line, a plane, or a translated subspace. Homogeneous coordinates turn every affine combination into an ordinary linear combination one dimension up.\n",{"path":22696,"title":22697,"module":22693,"summary":22698},"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-independence-and-barycentric-coordinates","Affine Independence and Barycentric Coordinates","Affine independence is linear independence for the translated or lifted points, and it guarantees each point of an affine hull a unique weight vector. Those weights are barycentric coordinates: centers of mass, ratios of triangle areas, and the interpolation rule behind smooth shading in computer graphics.\n",{"path":22700,"title":22701,"module":22693,"summary":22702},"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fconvex-combinations-and-convex-sets","Convex Combinations and Convex Sets","A convex combination is an affine combination with nonnegative weights, and the convex hull of a set is the smallest convex set containing it. Convex sets are closed under intersection, and Carathéodory's theorem bounds how many points a convex combination in $\\mathbb{R}^n$ ever needs: at most $n+1$.\n",{"path":22704,"title":22705,"module":22693,"summary":22706},"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fhyperplanes-and-polytopes","Hyperplanes and Polytopes","A hyperplane is a level set of a linear functional, the set where an inner product equals a constant. Hyperplanes separate disjoint convex sets and support them at their boundaries. Polytopes are convex hulls of finite point sets; their vertices are the extreme points, and a linear functional attains its extremes there.\n",{"path":22708,"title":22709,"module":22693,"summary":22710},"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fcurves-and-surfaces","Curves and Surfaces","Bézier curves are affine combinations of control points with polynomial weights, so they lie in the convex hull of those points and bend toward them. The de Casteljau algorithm evaluates them by repeated interpolation, a matrix form factors them for computation, and matching endpoints and tangents joins segments into smooth curves and surfaces.\n",{"path":22712,"title":22713,"module":6,"summary":6},"\u002Flinear-algebra","Linear Algebra",{"path":22715,"title":22716,"module":6,"summary":6},"\u002Ftheory-of-computation","Theory of Computation",{"path":22718,"title":22719,"module":21519,"summary":22720},"\u002Fcomputer-architecture\u002Ffoundations\u002Fbits-bytes-and-words","Bits, Bytes, and Words","Everything a machine stores is a string of bits grouped into bytes. We set out binary and hexadecimal, the byte as the unit of addressing, the word as the machine's natural integer size, and byte ordering — why the same four bytes read as 0x01234567 on one machine and 0x67452301 on another.\n",{"path":22722,"title":22723,"module":21519,"summary":22724},"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-representation","Integer Representation","A fixed-width byte string is just a pattern; what makes it a number is the rule we read it by. We define unsigned encoding and two's complement — where the top bit carries a negative weight — derive the ranges UMax, TMin, and TMax, and show how the same bits reinterpret between signed and unsigned, how widening sign-extends, and what truncation throws away.\n",{"path":22726,"title":22727,"module":21519,"summary":22728},"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-arithmetic","Integer Arithmetic","Fixed-width integer arithmetic is arithmetic modulo a power of two: add past the top and the result wraps. We work out unsigned and two's-complement addition and the rules that detect their overflow, why negation is a complement-plus-one, how multiplication truncates to the low-order bits and how compilers turn constant multiplies into shifts and adds, why C declares signed overflow undefined, and the bias fix that keeps shift-based signed division rounding toward zero.\n",{"path":22730,"title":22731,"module":21519,"summary":22732},"\u002Fcomputer-architecture\u002Ffoundations\u002Ffloating-point","Floating Point","IEEE-754 trades the exactness of integers for enormous range by storing numbers as sign, exponent, and fraction — scientific notation in binary. We lay out the single and double formats, the bias that encodes the exponent, the three regimes (normalized, denormalized, special), a worked encode\u002Fdecode, the four rounding modes and round-to-even at the bit level, why addition is not associative, the pitfalls of float-int conversion, and why 0.1 has no exact binary representation.\n",{"path":22734,"title":22735,"module":21519,"summary":22736},"\u002Fcomputer-architecture\u002Ffoundations\u002Fboolean-algebra-and-bit-manipulation","Boolean Algebra and Bit Manipulation","Treat a word as a vector of independent bits and the bitwise operators become an algebra. We define AND, OR, NOT, and XOR as bit vectors, build the masking idioms that set, clear, toggle, and test individual bits, extract fields with zero- and sign-extension, count set bits three ways, derive the classic x & (x - 1) family of tricks, and distinguish bitwise operators from C's short-circuiting logical operators.\n",{"path":22738,"title":22739,"module":22740,"summary":22741},"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fthe-machines-view","The Machine's View","Machine-Level Programming","The instruction set architecture is the contract a compiler writes against: the program counter, sixteen integer registers with their sub-register widths, and the condition codes. We follow one C function down through gcc to assembly, learn to read an instruction as operation plus operands, and fix the vocabulary the rest of the module uses.\n",{"path":22743,"title":22744,"module":22740,"summary":22745},"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fdata-movement","Data Movement","Most instructions a program runs simply move data. We cover the mov family and its size suffixes, the three operand forms, the full memory addressing mode D(Rb,Ri,S) and its special cases, lea for address arithmetic, and how push and pop manipulate the stack pointer %rsp on a stack that grows toward lower addresses.\n",{"path":22747,"title":22748,"module":22740,"summary":22749},"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farithmetic-and-logic","Arithmetic and Logic","The ALU instructions that compute on register and memory values: add, sub, and imul; the unary inc\u002Fdec\u002Fneg\u002Fnot; the shifts sal\u002Fshr\u002Fsar; the bitwise and\u002For\u002Fxor; and lea reused as a fast arithmetic trick. Each binary operation also sets the condition-code flags CF, ZF, SF, and OF, which cmp and test compute without keeping a result.\n",{"path":22751,"title":22752,"module":22740,"summary":22753},"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fcontrol-flow","Control Flow","How a flat instruction stream realizes branches and loops. The conditional jumps read the condition-code flags; set instructions turn flags into a 0\u002F1 byte. We translate if\u002Felse into the standard compare-and-branch pattern, while\u002Ffor loops into the guarded-do form, and dense switches into jump tables that index a target directly.\n",{"path":22755,"title":22756,"module":22740,"summary":22757},"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fprocedures","Procedures","How a function call works at the machine level: the run-time stack, call and ret passing control through a saved return address, the System V convention that routes the first six arguments through %rdi..%r9 and the result through %rax, the caller-saved versus callee-saved split, the stack frame, and a recursive factorial traced through its frames.\n",{"path":22759,"title":22760,"module":22740,"summary":22761},"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farrays-structs-and-alignment","Arrays, Structs, and Alignment","How aggregate data lays out in memory. Arrays as base-plus-scaled-index, the row-major ordering of multidimensional arrays, pointer arithmetic in units of the pointed-to type, struct fields at fixed byte offsets, the overlapping storage of unions, and the alignment rules that force padding into a struct.\n",{"path":22763,"title":22764,"module":22740,"summary":22765},"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fmemory-layout-and-buffer-overflows","Memory Layout and Buffer Overflows","The process address space — text, data, heap, and stack — and the classic vulnerability it enables. A stack buffer that is written past its end can overwrite the saved return address and redirect ret, so we sketch the mechanism defensively and then the three standard protections: stack canaries, a non-executable stack, and address-space layout randomization.\n",{"path":22767,"title":22768,"module":22769,"summary":22770},"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fwhat-an-isa-is","What an ISA Is","Instruction Set Architecture","The instruction set architecture is the contract that lets a compiler and a chip be written by people who never meet: the stable interface software targets and hardware implements. We separate architecture from microarchitecture, read RISC and CISC as opposite answers to where complexity should live, price out what each choice costs in decode hardware, code density, and pipeline friendliness, and see how x86-64 endures by translating its instructions into RISC-like operations on the fly.\n",{"path":22772,"title":22773,"module":22769,"summary":22774},"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Finstruction-formats-and-operands","Instruction Formats and Operands","An instruction is an opcode plus a way to name its operands. We count operands — 3-address, 2-address, 1-address accumulator, and 0-address stack machines — by writing the same C = A + B four ways, weigh register operands against memory operands, then lay out the same add byte by byte in x86-64 (REX prefix, opcode, ModRM) and in Y86-64, and what fixed versus variable length costs at fetch time.\n",{"path":22776,"title":22777,"module":22769,"summary":22778},"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Faddressing-modes","Addressing Modes","Once an operand field exists, it needs a rule for turning its bits into the data it names. That rule is the addressing mode. We walk the standard set — immediate, register, direct, register-indirect, displacement, scaled-indexed, and PC-relative — fixing the effective-address computation for each, run every mode against one concrete machine state, and price out what Y86-64 loses by keeping only base plus displacement.\n",{"path":22780,"title":22781,"module":22769,"summary":22782},"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fthe-y86-64-instruction-set","The Y86-64 Instruction Set","Y86-64 is a teaching ISA — a stripped-down x86-64 simple enough to implement by hand yet real enough to compile to. We fix its programmer-visible state (fifteen registers, three condition codes, the PC, memory, and a status code), give the instruction set with exact byte encodings, spell out how the condition codes decide every jXX and cmovXX, and run the encoding both directions: assembly to bytes and raw bytes back to meaning.\n",{"path":22784,"title":22785,"module":22769,"summary":22786},"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fy86-64-programming","Y86-64 Programming","With the encodings fixed, we write real Y86-64 assembly: the .pos, .align, and .quad directives, the calling convention borrowed from x86-64, a stack set up by hand, and complete programs — an array sum and a branch-free max. We watch the assembler turn the listing into the exact byte image the processor will execute, and trace the stack across the call.\n",{"path":22788,"title":22789,"module":22790,"summary":22791},"\u002Fcomputer-architecture\u002Fdigital-logic\u002Ftransistors-gates-and-boolean-functions","Transistors, Gates, and Boolean Functions","Digital Logic","A processor is built from millions of transistor switches. We start at the MOS transistor as a voltage-controlled switch, build the CMOS inverter and NAND transistor by transistor, meet the seven standard gates with their truth tables, show that NAND alone is functionally complete, price each gate in transistors and in time, and turn any truth table into a sum-of-products circuit.\n",{"path":22793,"title":22794,"module":22790,"summary":22795},"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fcombinational-logic-and-hcl","Combinational Logic and HCL","A combinational circuit is a pure Boolean function of its current inputs — no memory, no clock. We draw the line between combinational and sequential logic, do the gate-delay accounting that finds a circuit's critical path and bounds the clock, meet don't-cares, then introduce CS:APP's Hardware Control Language: bit-level operators, word-level signals, equality nets, and the case expression that compiles to a multiplexer tree.\n",{"path":22797,"title":22798,"module":22790,"summary":22799},"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmultiplexers-decoders-and-the-alu","Multiplexers, Decoders, and the ALU","The combinational building blocks that make a datapath. We build the 2:1 and 4:1 multiplexer and tie it back to HCL's case expression, the n-to-2^n decoder, a one-bit full adder (sum is XOR, carry is majority), the ripple-carry adder that chains them, and finally the ALU — a function unit that selects among add, sub, and, and xor under a control input and exposes condition flags.\n",{"path":22801,"title":22802,"module":22790,"summary":22803},"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmemory-elements-latches-flip-flops-and-clocking","Memory Elements: Latches, Flip-Flops, and Clocking","A combinational circuit holds no state; feeding a circuit's output back to its input creates memory. We build the SR latch from cross-coupled gates, the level-sensitive D latch, and the master\u002Fslave edge-triggered D flip-flop, then introduce the clock and the synchronous design discipline, the setup\u002Fhold timing window, clock skew, metastability, and the register as n flip-flops sharing one clock.\n",{"path":22805,"title":22806,"module":22790,"summary":22807},"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fregister-files-and-random-access-memory","Register Files and Random-Access Memory","Storage organized for access by address. We build the register file (a small bank of registers with addressed read ports and clocked write ports, the exact structure Y86-64's decode and write-back stages use), then descend to the SRAM and DRAM cells of main memory, why one is fast and dear and the other dense and slow, and how a row decoder picks a word out of a memory array.\n",{"path":22809,"title":22810,"module":22811,"summary":22812},"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-fetch-decode-execute-cycle","The Fetch-Decode-Execute Cycle","Processor Design","A processor is a machine that repeats one loop forever: read the next instruction from memory, figure out what it asks for, do it, and advance. We fix the stored-program idea, lay out the datapath at a high level — PC, instruction memory, register file, ALU, data memory — and the control unit that sequences them, break the work into the six stages the rest of the module builds in hardware, and work out exactly how fetch parses variable-length instructions and computes the next PC.\n",{"path":22814,"title":22815,"module":22811,"summary":22816},"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-seq-stages","The SEQ Stages","The six SEQ stages, made exact. For every Y86-64 instruction — halt, nop, the moves, OPq, the jumps, call and ret, pushq and popq — we write down what Fetch, Decode, Execute, Memory, Write-back, and PC update each compute, as per-instruction stage tables with every row justified. Once the tables are filled in, the processor is fully specified; the remaining lessons turn them into wires.\n",{"path":22818,"title":22819,"module":22811,"summary":22820},"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fcontrol-logic-and-sequencing","Control Logic and Sequencing","The stage tables say what each instruction needs; the control logic computes it from icode. We write the HCL for the register-port selections (srcA, srcB, dstE, dstM), the ALU function and input selection, the memory read\u002Fwrite and address, the branch condition, and the next-PC mux — each a case expression on icode that compiles to a mux — and see how one blob of combinational logic serves every instruction at once. We close by contrasting hardwired control with the microprogrammed alternative.\n",{"path":22822,"title":22823,"module":22811,"summary":22824},"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fassembling-seq","Assembling SEQ","We wire the whole thing together. The functional units from digital logic and the control signals from the last lesson assemble into the complete SEQ datapath, laid out the way CS:APP draws it — six stages stacked bottom to top, Fetch at the floor and PC update at the ceiling, signals flowing up the margins. Then the timing analysis: why everything must settle in one cycle, the no-reading-back principle that makes single-cycle execution consistent, and the critical path that sets the clock. We close by walking an OPq and a ret through the assembled machine.\n",{"path":22826,"title":22827,"module":22811,"summary":22828},"\u002Fcomputer-architecture\u002Fprocessor-design\u002Ftracing-a-program","Tracing a Program","To close the module, we take a complete Y86-64 program — a loop that sums 1 through 3 — and run it through SEQ one cycle at a time, recording the PC, the fetched instruction, every stage computation, and the registers, condition codes, and memory after each cycle. Then we examine single cycles in detail: every named signal of an OPq in concrete hex, and a second program whose call and ret we trace through the stack. The traces confirm that the assembled datapath and control logic behave as a processor.\n",{"path":22830,"title":22831,"module":22832,"summary":22833},"\u002Fcomputer-architecture\u002Fpipelining\u002Fpipelining-principles","Pipelining Principles","Pipelining","A processor that runs one instruction to completion before starting the next wastes most of its hardware most of the time. Pipelining splits the work into stages separated by registers so several instructions are in flight at once. We separate throughput from latency, work the 300 ps example through one, two, and three stages, and derive the three ceilings on the gain: uneven stages, register overhead, and the dependencies between instructions.\n",{"path":22835,"title":22836,"module":22832,"summary":22837},"\u002Fcomputer-architecture\u002Fpipelining\u002Ffrom-seq-to-pipe","From SEQ to PIPE","We turn the sequential Y86-64 processor into a pipelined one by inserting pipeline registers between its stages so each cycle holds one instruction per stage. Doing it correctly forces a rearrangement: the next-PC computation must move into Fetch as a prediction, because the later stages that used to compute it are now busy with other instructions. We walk SEQ to SEQ+ to PIPE, spell out exactly what each pipeline register carries, and fix the naming discipline (D_stat versus d_stat) that keeps five in-flight instructions straight.\n",{"path":22839,"title":22840,"module":22832,"summary":22841},"\u002Fcomputer-architecture\u002Fpipelining\u002Fdata-hazards-stalling-and-forwarding","Data Hazards: Stalling and Forwarding","Overlapping instructions collide when a later one needs a value an earlier one has not finished computing: a read-after-write data hazard. We map exactly which instruction distances are dangerous, fix hazards the slow way by stalling (three bubbles), then the fast way by forwarding from five distinct sources into Decode, in a priority order that sequential semantics forces. Forwarding handles almost everything; the load-use hazard still needs exactly one stall.\n",{"path":22843,"title":22844,"module":22832,"summary":22845},"\u002Fcomputer-architecture\u002Fpipelining\u002Fcontrol-hazards-and-branch-prediction","Control Hazards and Branch Prediction","A pipeline must fetch an instruction every cycle, but after a conditional jump or a ret the next address is not yet known: a control hazard. We measure the branch penalty, weigh predict-taken against its alternatives with real loop arithmetic, watch PIPE detect a misprediction in Execute and squash the two wrong-path instructions, and meet the ret hazard, which has nothing to predict and stalls three cycles. A 2-bit counter gives a taste of dynamic prediction.\n",{"path":22847,"title":22848,"module":22832,"summary":22849},"\u002Fcomputer-architecture\u002Fpipelining\u002Fthe-complete-pipe-processor","The Complete PIPE Processor","We assemble the full pipelined Y86-64: five stages, five pipeline registers, forwarding paths, and a small control unit that decides, each cycle, whether to stall or bubble each register. The subtle part is when hazards combine: one pairing hides a genuine bug. A fourth control case reads stat and keeps exceptions precise. Performance reduces to CPI = 1 + lp + mp + rp, worked out to 1.27 with realistic frequencies, and PIPE beats SEQ by several times despite every penalty.\n",{"path":22851,"title":22852,"module":22853,"summary":22854},"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fstorage-technologies-and-the-latency-gap","Storage Technologies and the Latency Gap","The Memory Hierarchy","No single memory is both fast and large and cheap. We survey the technologies a machine can store bits in — SRAM, DRAM, flash, and rotating disk — open up a DRAM chip to find the row buffer, work a disk access down to the millisecond, and rank everything by speed, density, and cost per bit. Then we watch the processor outrun memory decade after decade. That widening gap is the whole reason a machine stacks fast small storage on top of slow large storage into a hierarchy.\n",{"path":22856,"title":22857,"module":22853,"summary":22858},"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Flocality","Locality","A hierarchy only pays off because programs do not touch memory at random. They reuse recently-used data (temporal locality) and touch nearby data soon after (spatial locality). We make both precise and then quantitative: miss rates for stride-1 and stride-k traversals against a concrete block size, and the loop-order pair on a 2-D array where the same sum misses 16 times one way and 64 times the other — why row-major versus column-major order can change a program's speed by an order of magnitude.\n",{"path":22860,"title":22861,"module":22853,"summary":22862},"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-memories-direct-mapped","Cache Memories and Direct Mapping","A cache is fast SRAM that holds copies of recently-used blocks of main memory. We fix its organization — S sets, E lines per set, B bytes per block — and the way it dissects an address into tag, set index, and block offset, worked bit by bit on a concrete 16-byte cache. Then we run the direct-mapped (E=1) access algorithm end to end on a seven-access trace: index to a set, compare the tag, hit or miss, evict. Cold and conflict misses fall out of the structure, and a two-array ping-pong shows conflict thrashing and its padding fix.\n",{"path":22864,"title":22865,"module":22853,"summary":22866},"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fset-associative-and-write-policies","Set-Associative Caches and Write Policies","Give each set several lines and a block has a choice of homes — fewer conflict misses, at the cost of comparing E tags in parallel and choosing a victim to evict. We re-run the direct-mapped ping-pong trace on a 2-way cache and watch the conflicts vanish, weigh LRU against random replacement, then turn to writes: write-through versus write-back with a dirty bit on a hit, write-allocate versus no-write-allocate on a miss, and a worked traffic count showing when each pairing wins.\n",{"path":22868,"title":22869,"module":22853,"summary":22870},"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-performance-and-cache-friendly-code","Cache Performance and Cache-Friendly Code","Turn the cache mechanism into a number. Hit time, miss rate, and miss penalty combine into the average memory access time; we compute AMAT for a two-level hierarchy with real numbers, weigh the design knobs against each other, and read the memory mountain. Then we write cache-friendly code — the matrix-multiply loop-order case study (ijk versus kij, misses counted per iteration) and loop blocking, where cache-sized tiles turn evicted reuse back into hits.\n",{"path":22872,"title":22873,"module":22874,"summary":22875},"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Faddress-spaces-and-translation","Address Spaces and Translation","Virtual Memory","Every process runs as if it owns a private, contiguous span of memory — its virtual address space — while the hardware maps those addresses onto a single shared physical memory. We fix virtual memory's three jobs (a cache for disk, a memory manager, a protection boundary), the page as the unit of mapping, and the MMU replacing the virtual page number while the offset passes through untouched — then run one translation end to end at the bit level and trace the control flow of a page hit against a page fault.\n",{"path":22877,"title":22878,"module":22874,"summary":22879},"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fpage-tables-and-page-faults","Page Tables and Page Faults","The page table is an array of page-table entries indexed by virtual page number; each entry's valid bit says whether the page is in DRAM, on disk, or unallocated, and its permission, reference, and dirty bits drive protection and replacement. We walk translation as a table lookup, the page fault and demand paging, the clock algorithm the OS uses to approximate LRU, memory mapping and copy-on-write (why fork is cheap), the taxonomy of bad references, and thrashing.\n",{"path":22881,"title":22882,"module":22874,"summary":22883},"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fthe-tlb-and-multi-level-page-tables","The TLB and Multi-Level Page Tables","A page-table read on every access would double memory traffic; a flat table for a 48-bit space would occupy 512 GB per process. The TLB fixes the first: a small set-associative cache of PTEs inside the MMU whose tag and index come from the VPN. Multi-level page tables fix the second, allocating only the sub-tables a process uses; x86-64 walks four levels with a 9+9+9+9+12 split. We trace one reference end to end through TLB, walk, and cache, and close with the overlap trick that lets the L1 cache start before translation ends.\n",{"path":22885,"title":22886,"module":22887,"summary":22888},"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Fexceptional-control-flow","Exceptional Control Flow","Exceptions & I\u002FO","Beyond the sequential, branch, and call flow a program controls itself, the hardware can divert the processor in response to events. We sort these into four classes — interrupts (asynchronous, from devices), traps (intentional syscalls), faults (recoverable, like a page fault), and aborts (unrecoverable) — then take the mechanism apart: exception numbers and the table dispatch, what the hardware pushes and why it differs from a procedure call, the divide-error \u002F page-fault \u002F general-protection trio on x86-64, the full syscall round trip with a worked write in assembly, and processes and signals as the abstractions ECF makes possible.\n",{"path":22890,"title":22891,"module":22887,"summary":22892},"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Finterrupts-and-the-kernel","Interrupts and the Kernel","An I\u002FO device signals completion by raising an interrupt, crossing the privilege boundary from user mode into the kernel. We fix that boundary, follow an interrupt from device through the interrupt controller to its vectored handler, and use the timer interrupt to drive preemptive scheduling and the context switch. Then the I\u002FO mechanics: polling versus interrupt-driven I\u002FO with a cycle count, device registers and memory-mapped I\u002FO versus port I\u002FO, DMA's full transfer walkthrough and its cache hazard, and a disk read traced end to end, from the read syscall to the completion interrupt.\n",{"path":22894,"title":22895,"module":22896,"summary":22897},"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fprocesses-threads-and-parallelism","Processes, Threads, and Parallelism","Multithreading & Multicore","Around 2004 the single core stopped getting faster, and the industry's answer was to hand programmers more cores instead. This lesson builds the vocabulary that shift demands: process versus thread and exactly which hardware state each one owns, concurrency versus parallelism, the three kinds of parallelism a machine can exploit, why Dennard scaling ended and forced the multicore turn, and Amdahl's law — the arithmetic that bounds the speedup those cores can deliver.\n",{"path":22899,"title":22900,"module":22896,"summary":22901},"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fhardware-multithreading","Hardware Multithreading","A pipeline spends much of its life waiting — on cache misses, on dependences, on branches. Hardware multithreading fills the dead cycles with instructions from another thread. We compare coarse-grained switching (change threads on a long stall), fine-grained interleaving (change every cycle), and simultaneous multithreading (mix threads inside a single cycle), work out exactly which hardware a second thread context duplicates and which it shares, and weigh when SMT pays off and when two threads just fight over one cache.\n",{"path":22903,"title":22904,"module":22896,"summary":22905},"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fcache-coherence","Cache Coherence","Give each core its own cache and the same address can live in two places at once, with copies that disagree. We reproduce the stale-copy bug with a two-core trace, then fix it the way hardware does: snooping caches that watch a shared bus and keep every line in a protocol state. We build MSI in full, upgrade it to MESI, contrast invalidation with updating, add coherence misses as the fourth C, and end with false sharing: the performance bug where cores fight over a line while never touching the same byte.\n",{"path":22907,"title":22908,"module":22896,"summary":22909},"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmemory-consistency-and-synchronization","Memory Consistency and Synchronization","Coherence keeps cores agreeing about one location; consistency is the contract about many. We define sequential consistency, then watch real hardware break it: the store buffer lets a load slip ahead of an older store, and the classic two-thread litmus test ends with both sides reading zero. We state x86-TSO precisely, restore order with mfence, build atomic read-modify-write from the lock prefix, xchg, and cmpxchg, and write a spinlock twice — once naively, once bus-friendly — closing with what lock-free progress actually guarantees.\n",{"path":22911,"title":22912,"module":22896,"summary":22913},"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmulticore-organization","Multicore Organization","Where everything sits on the die. A modern die gives each core private L1 and L2 caches, spreads a shared last-level cache across slices, and wires it all together with a ring or mesh; multi-socket servers add NUMA, where memory is local to one socket and every remote access pays a latency penalty. We walk the floorplan, put numbers on local versus remote latency, meet thread affinity, and account for the two shared resources — coherence traffic and LLC capacity — that decide how far a parallel program scales.\n",{"path":22915,"title":22916,"module":22917,"summary":22918},"\u002Fcomputer-architecture\u002Fcapstone\u002Fthe-whole-machine","The Whole Machine","Capstone","We take one line of C down the whole tower the course built — compiler to assembly, assembly to machine-code bytes, the bytes into the fetch–decode–execute datapath — then trace one load and one add through the pipelined, cached, translated, interruptible machine, each step cross-linked to the lesson that built it. We close with the map of the course as a stack of layers and an accounting of what we simplified: out-of-order execution, superscalar issue, and speculation past the branch predictor.\n",{"path":22920,"title":22921,"module":22917,"summary":22922},"\u002Fcomputer-architecture\u002Fcapstone\u002Fassembling-a-complete-cpu","Assembling a Complete CPU","We bolt the parts the course built — PC, instruction memory and its fetch logic, register file, ALU, condition codes, data memory, and the control unit — into one complete CPU, name the lesson that built each, wire them in a deliberate order, and power the machine on from reset. Then we assemble a real test program (sum a four-element array through a call\u002Fret procedure), give its exact bytes and memory layout, and trace it cycle by cycle to the answer 0xabcdabcdabcd. We close with how to validate such a machine, and what it takes to put two of them on one die.\n",{"path":22924,"title":22925,"module":6,"summary":6},"\u002Fcomputer-architecture","Computer Architecture",{"path":22927,"title":22928,"module":21519,"summary":22929},"\u002Fdifferential-equations\u002Ffoundations\u002Fmodels-and-direction-fields","Models, Direction Fields, and Solution Curves","A differential equation relates an unknown function to its own rates of change. Three first-order models — a falling body, a cooling object, a population under predation — share the form dy\u002Fdt = ay - b; the slope field fixes their equilibria and long-run behavior before any formula is found. Solving the linear case gives the general solution, its integral curves, and the particular solution selected by an initial condition.\n",{"path":22931,"title":22932,"module":21519,"summary":22933},"\u002Fdifferential-equations\u002Ffoundations\u002Fclassification-and-terminology","Classifying Equations: Order, Linearity, ODE vs. PDE","Every solution method targets a specific class of equation, so the first question about any differential equation is which classes it belongs to. Four independent axes sort them: ordinary versus partial, order, linear versus nonlinear, and homogeneous versus nonhomogeneous. Systems, verification of a solution by substitution, and the split between initial and boundary value problems complete the vocabulary.\n",{"path":22935,"title":22936,"module":22937,"summary":22938},"\u002Fdifferential-equations\u002Ffirst-order\u002Flinear-first-order-integrating-factors","Linear Equations and Integrating Factors","First-Order Equations","A first-order linear equation has the unknown and its derivative to the first power only. Multiplying by an integrating factor collapses the left side into a single derivative, and one integration gives the general solution in closed form. The solution exists wherever the coefficients are continuous, and for a constant coefficient it splits into a decaying transient and a steady state set by the forcing.\n",{"path":22940,"title":22941,"module":22937,"summary":22942},"\u002Fdifferential-equations\u002Ffirst-order\u002Fseparable-and-exact","Separable and Exact Equations","Two nonlinear first-order classes solve by direct integration. A separable equation splits so that each variable can be integrated on its own side, giving an implicit relation. An exact equation is the total differential of a hidden potential function, recognized by a symmetry test on its coefficients; when the test fails, an integrating factor can sometimes restore exactness. A change of variable brings homogeneous equations into the separable class.\n",{"path":22944,"title":22945,"module":22937,"summary":22946},"\u002Fdifferential-equations\u002Ffirst-order\u002Fmodeling-first-order","Modeling with First-Order Equations","A rate law is a differential equation. Each first-order model starts from one governing principle: conservation of mass for a mixing tank, proportional change for interest and radioactive decay, Newton's law of cooling, a force balance for a body falling against drag, and Kirchhoff's law for a series circuit. Setting the derivative to zero recovers the steady state, and the transient records how the initial condition relaxes toward it.\n",{"path":22948,"title":22949,"module":22937,"summary":22950},"\u002Fdifferential-equations\u002Ffirst-order\u002Fautonomous-and-population-dynamics","Autonomous Equations, Phase Lines, and Population Dynamics","An autonomous equation y' = f(y) can be analyzed qualitatively without being solved. Its constant solutions are the zeros of f, and the sign of f between them fixes whether nearby solutions rise or fall, which the phase line records as a column of arrows. The logistic and threshold models, constant- and effort-proportional harvesting, and the properties nonlinear equations lose all follow from this reading.\n",{"path":22952,"title":22953,"module":22937,"summary":22954},"\u002Fdifferential-equations\u002Ffirst-order\u002Fexistence-uniqueness-euler","Existence, Uniqueness, and Euler's Method","Existence and uniqueness can be settled before any attempt to solve. The existence-uniqueness theorem gives sufficient conditions on f, and a standard example shows what fails when they do not hold. Picard's successive approximations build the solution as the limit of an iteration, and Euler's method turns the same tangent-line idea into a numerical procedure for the equations no formula reaches.\n",{"path":22956,"title":22957,"module":22937,"summary":22958},"\u002Fdifferential-equations\u002Ffirst-order\u002Ffirst-order-difference-equations","First-Order Difference Equations","A difference equation advances a sequence one index at a time by a rule y_{n+1} = f(y_n). The linear case y_{n+1} = rho*y_n + b solves in closed form and converges to its equilibrium exactly when the ratio has magnitude below one, which underlies compound-interest and loan calculations. The logistic difference equation shows the nonlinear counterpart: an exchange of stability, a cascade of period doublings, and the onset of chaos.\n",{"path":22960,"title":22961,"module":22962,"summary":22963},"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhomogeneous-constant-coefficients","Homogeneous Equations, the Wronskian, and Real Roots","Second-Order Linear Equations","A second-order linear homogeneous equation with constant coefficients is solved by guessing an exponential and reducing to the quadratic characteristic equation. Two solutions span every solution exactly when their Wronskian is nonzero; that condition, superposition, and Abel's formula give the full structure of the general solution for the case of two distinct real roots.\n",{"path":22965,"title":22966,"module":22962,"summary":22967},"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fcomplex-and-repeated-roots","Complex Roots, Repeated Roots, and Reduction of Order","When the characteristic equation has complex conjugate roots, Euler's formula converts the complex exponentials into a real fundamental set of decaying or growing oscillations. When it has a repeated root, one exponential is lost and reduction of order recovers the missing second solution as $t\\,e^{rt}$. The same substitution $y = v(t)y_1(t)$ finds a second solution from any known one.\n",{"path":22969,"title":22970,"module":22962,"summary":22971},"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fnonhomogeneous-undetermined-coefficients","Nonhomogeneous Equations: Undetermined Coefficients","The general solution of a nonhomogeneous linear equation is a complementary solution plus any one particular solution. When the forcing term is a polynomial, exponential, sine, or cosine, a particular solution can be found by assuming a trial form of the same shape with unknown coefficients and solving for them. The one complication is resonance, handled by multiplying the trial by a power of $t$.\n",{"path":22973,"title":22974,"module":22962,"summary":22975},"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fvariation-of-parameters","Variation of Parameters","Variation of parameters finds a particular solution of any nonhomogeneous linear equation from a fundamental set of the homogeneous one. Replacing the constants in the complementary solution by functions and imposing one convenient constraint reduces the problem to a two-by-two linear system whose solution is expressed through the Wronskian, giving an integral formula that works for forcing terms undetermined coefficients cannot touch.\n",{"path":22977,"title":22978,"module":22962,"summary":22979},"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fmechanical-electrical-vibrations","Mechanical and Electrical Vibrations","A spring-mass-damper obeys a second-order linear equation, and so does a series RLC circuit, with the same mathematics governing both. Free undamped motion is a pure sinusoid; damping adds a decaying envelope with three regimes; periodic forcing produces a transient that dies out and a steady-state oscillation whose amplitude peaks sharply near the natural frequency, the phenomenon of resonance.\n",{"path":22981,"title":22982,"module":22962,"summary":22983},"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhigher-order-linear","Higher-Order Linear Equations","The second-order theory extends directly to order $n$: the solution space is $n$-dimensional, spanned by any $n$ solutions with nonzero Wronskian. For constant coefficients the characteristic polynomial has degree $n$, and its roots (counted with multiplicity, real and complex) build the basis by the same rules as before. Coupled oscillators are the natural application that raises the order.\n",{"path":22985,"title":22986,"module":22987,"summary":22988},"\u002Fdifferential-equations\u002Fseries-solutions\u002Fpower-series-ordinary-points","Power Series Solutions Near Ordinary Points","Series Solutions and Special Functions","A linear equation with variable coefficients has no characteristic equation. A power series substituted into the equation matches coefficients to a recurrence relation, which near an ordinary point yields two independent analytic solutions. The radius of convergence is at least the distance from the expansion point to the nearest singular point in the complex plane.\n",{"path":22990,"title":22991,"module":22987,"summary":22992},"\u002Fdifferential-equations\u002Fseries-solutions\u002Fregular-singular-frobenius","Euler Equations, Regular Singular Points, and Frobenius","The Euler equation x^2 y'' + a x y' + b y = 0 is solved outright by y = x^r, and its three root cases fix the behavior at any regular singular point. The Frobenius method multiplies x^r by a power series; the indicial equation chooses the exponents, and equal or integer-separated roots force a logarithm in the second solution. Gauss's hypergeometric equation is the archetype containing most classical functions as special cases.\n",{"path":22994,"title":22995,"module":22987,"summary":22996},"\u002Fdifferential-equations\u002Fseries-solutions\u002Fbessel-and-special-functions","Bessel's Equation, Legendre Polynomials, and Special Functions","Bessel's equation puts the Frobenius machinery through all three of its cases and produces the functions J and Y that govern anything vibrating or diffusing with circular symmetry. The gamma function extends the factorial so that Bessel functions of every order make sense; Legendre's equation, run through the hypergeometric form, yields the polynomials that play the same role in spherical geometry. Orthogonality ties both families to the eigenfunction expansions of Sturm–Liouville theory.\n",{"path":22998,"title":22999,"module":23000,"summary":23001},"\u002Fdifferential-equations\u002Flaplace\u002Flaplace-definition-ivps","The Laplace Transform: Definition, Properties, and Solving IVPs","The Laplace Transform","The Laplace transform sends a function of time to a function of a complex frequency by integrating it against the kernel e^{-st}. Differentiation in t becomes multiplication by s, so a linear constant-coefficient initial value problem turns into an algebraic equation. Existence rests on piecewise continuity and exponential order; the derivative rule folds in the initial data; and inversion runs through a transform table and partial fractions.\n",{"path":23003,"title":23004,"module":23000,"summary":23005},"\u002Fdifferential-equations\u002Flaplace\u002Fstep-impulse-convolution","Step Functions, Discontinuous Forcing, Impulses, and Convolution","The Heaviside step function and the second shifting theorem transform switches and discontinuous forcing into exponential factors on the transform. The Dirac delta idealizes an instantaneous impulse and transforms to a pure exponential. The convolution theorem inverts a product of transforms, writes the forced response as the impulse response convolved with the input, and solves Abel's tautochrone by transform.\n",{"path":23007,"title":23008,"module":23009,"summary":23010},"\u002Fdifferential-equations\u002Fsystems\u002Fmatrices-eigenvalues-review","Matrices, Linear Systems, and the Eigenvalue Toolkit","Systems of First-Order Linear Equations","Any nth-order linear equation, and any coupled collection of them, rewrites as a single first-order system x' = P(t)x + g(t). The matrix and vector algebra behind that form, the eigenvalue problem det(A - λI) = 0 that drives every solution method, and the fundamental theory — superposition, the Wronskian, Abel's theorem — together establish that n independent solutions span all solutions.\n",{"path":23012,"title":23013,"module":23009,"summary":23014},"\u002Fdifferential-equations\u002Fsystems\u002Fconstant-coefficient-systems-phase-portraits","Homogeneous Constant-Coefficient Systems and Phase Portraits","For x' = Ax with A constant, the trial x = ξe^{rt} turns the differential equation into the eigenvalue problem Aξ = rξ. The eigenvalues fix the geometry of the phase plane: real opposite signs give a saddle, real same sign a node, complex a spiral, purely imaginary a center. Worked in the plane, these cases form the eigenvalue-type classification of equilibria.\n",{"path":23016,"title":23017,"module":23009,"summary":23018},"\u002Fdifferential-equations\u002Fsystems\u002Frepeated-eigenvalues-fundamental-matrices","Repeated Eigenvalues, Fundamental Matrices, and Nonhomogeneous Systems","When a repeated eigenvalue supplies too few eigenvectors, a generalized eigenvector supplies the missing solution as ξte^{ρt} + ηe^{ρt}, giving an improper node. A fundamental set packaged as a matrix Φ(t) yields the matrix exponential e^{At}, the propagator mapping initial states to later ones. Variation of parameters solves the nonhomogeneous system x' = Ax + g(t).\n",{"path":23020,"title":23021,"module":23022,"summary":23023},"\u002Fdifferential-equations\u002Fnumerical\u002Feuler-and-runge-kutta","Euler, Improved Euler, and Runge–Kutta","Numerical Methods","Most initial value problems have no closed-form solution, so the solution is approximated on a grid. Euler's method steps along the tangent line, the improved Euler method averages two slopes, and the classical Runge–Kutta method averages four. Each added stage raises the order of accuracy at the cost of more evaluations per step, measured by how the local and global truncation errors scale with the step size.\n",{"path":23025,"title":23026,"module":23022,"summary":23027},"\u002Fdifferential-equations\u002Fnumerical\u002Fmultistep-systems-stability","Multistep Methods, Systems, and Stability","One-step methods discard everything but the last point. Multistep methods fit a polynomial to several past values and integrate it forward: the explicit Adams–Bashforth formulas, the implicit and more accurate Adams–Moulton formulas, and predictor–corrector pairs that combine them. The same rules extend verbatim to systems in vector form. A separate concern is stability: round-off can dominate truncation, and stiff equations force a tiny step for stability even when accuracy would allow a large one.\n",{"path":23029,"title":23030,"module":23031,"summary":23032},"\u002Fdifferential-equations\u002Fnonlinear\u002Fphase-plane-autonomous-stability","The Phase Plane, Critical Points, and Stability","Nonlinear Systems and Stability","Most nonlinear systems cannot be solved in closed form, so they are studied geometrically. The phase plane turns an autonomous planar system into a family of trajectories; the five archetypes of critical point follow from the eigenvalues of the coefficient matrix; the trace-determinant plane reads off type and stability directly; and epsilon-delta definitions make stability, asymptotic stability, and instability precise.\n",{"path":23034,"title":23035,"module":23031,"summary":23036},"\u002Fdifferential-equations\u002Fnonlinear\u002Flocally-linear-and-liapunov","Locally Linear Systems and Liapunov's Method","Near a critical point a nonlinear system looks linear, and the linear part is the Jacobian. The linearization fixes the type and stability of the nonlinear critical point in every case except a center or a repeated eigenvalue. Liapunov's direct method settles those cases and bounds the basin of attraction by constructing an energy-like function, without solving the system.\n",{"path":23038,"title":23039,"module":23031,"summary":23040},"\u002Fdifferential-equations\u002Fnonlinear\u002Fcompeting-species-predator-prey-limit-cycles","Population Models, Limit Cycles, and Chaos","The phase-plane methods apply directly to interacting-population models. Competing species either coexist or drive one another to extinction, decided by a single inequality among the interaction constants; the Lotka-Volterra predator-prey system produces closed population cycles. Limit cycles and the Poincaré-Bendixson theorem, the van der Pol oscillator, and the Lorenz equations with their strange attractor carry the theory into chaos.\n",{"path":23042,"title":23043,"module":23044,"summary":23045},"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Ffourier-series","Fourier Series and Convergence","PDEs, Fourier Series, and Boundary Value Problems","A two-point boundary value problem has nontrivial solutions only at a discrete set of eigenvalues, the same trichotomy that governs a singular linear system. For y'' + lambda y = 0 with zero endpoints the eigenfunctions are sines and cosines, and their orthogonality gives the Euler-Fourier coefficient formulas. The convergence theorem fixes when the series returns the function, the Gibbs phenomenon measures the overshoot at a jump, and even\u002Fodd symmetry produces half-range sine and cosine series.\n",{"path":23047,"title":23048,"module":23044,"summary":23049},"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fheat-wave-laplace-equations","Separation of Variables: Heat, Wave, and Laplace Equations","Separation of variables replaces a partial differential equation by a pair of ordinary ones joined through a shared separation constant. Applied to the heat equation it produces the eigenvalue problem X'' + lambda X = 0, and the solution assembles as a Fourier series in the eigenfunctions. The same steps solve the wave equation, whose modes are standing waves, and Laplace's equation, the steady-state limit posed on a region rather than an interval.\n",{"path":23051,"title":23052,"module":23044,"summary":23053},"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fsturm-liouville","Sturm-Liouville Theory","The eigenvalue problem behind separation of variables generalizes to the self-adjoint Sturm-Liouville form. Lagrange's identity makes the operator symmetric, and from that one fact follow real eigenvalues, orthogonal eigenfunctions, and eigenfunction expansions that behave like Fourier series. Singular problems admit Bessel and Legendre functions, and Sturm's separation and comparison theorems describe how the eigenfunctions oscillate.\n",{"path":23055,"title":23056,"module":23057,"summary":23058},"\u002Fdifferential-equations\u002Fhistory-variations\u002Fcalculus-of-variations","The Calculus of Variations","Historical Notes and the Calculus of Variations","Ordinary calculus finds the point where a function is stationary; the calculus of variations finds the whole curve where an integral is stationary. Euler's differential equation is the necessary condition for an extremal, and it becomes integrable in three cases, solving the shortest-path, minimal-surface, and brachistochrone problems. Lagrange multipliers extend the method to isoperimetric constraints, and Hamilton's principle recovers Newton's law from a single stationary integral.\n",{"path":23060,"title":23061,"module":23057,"summary":23062},"\u002Fdifferential-equations\u002Fhistory-variations\u002Fhistorical-notes","Great Problems and the People Who Solved Them","Differential equations grew out of specific problems, not a plan: the invention of calculus by Newton and Leibniz, the Bernoulli brachistochrone challenge, Euler's flood of methods, Lagrange's analytical mechanics, Gauss and Riemann's rigor, Laplace's celestial mechanics, and Poincaré's qualitative theory. Each method descends from a named problem, and reading the subject forward from those problems explains why its parts fit together.\n",{"path":23064,"title":23065,"module":6,"summary":6},"\u002Fdifferential-equations","Differential Equations",{"path":23067,"title":23068,"module":23069,"summary":23070},"\u002Frelativity\u002Ffoundations\u002Fspecial-relativity-postulates","The Postulates of Special Relativity","Foundations of Relativity","Newton's laws are the same in every inertial frame, but Maxwell's are not: the equations of electromagnetism single out one speed, c, and the nineteenth century read that as the speed of light relative to a medium, the ether. The Michelson-Morley experiment looked for Earth's motion through that medium and found nothing. Einstein's two postulates replace the ether, and their first consequence is that simultaneity is frame-dependent.\n",{"path":23072,"title":23073,"module":23069,"summary":23074},"\u002Frelativity\u002Ffoundations\u002Florentz-transformation-spacetime","The Lorentz Transformation and Spacetime","Requiring that a light sphere stay a light sphere in every inertial frame fixes the coordinate change between frames uniquely: the Lorentz transformation, with its factor gamma. Differentiating it gives relativistic velocity addition, which caps composed speeds at c. Plotting the same events on skewed spacetime axes turns the algebra into geometry, with calibration hyperbolae, an invariant interval, and a light cone that sorts events into past, future, and elsewhere.\n",{"path":23076,"title":23077,"module":23069,"summary":23078},"\u002Frelativity\u002Ffoundations\u002Ftime-dilation-length-contraction","Time Dilation, Length Contraction, and Paradoxes","A light clock and the constancy of c give the two headline effects directly: a moving clock runs slow by gamma, and a moving rod is short by the same factor. Cosmic-ray muons reaching sea level are the standing experimental proof. The relativistic Doppler effect adds the time-dilation factor to the classical shift, and the twin and pole-barn paradoxes dissolve once the relativity of simultaneity is taken seriously.\n",{"path":23080,"title":23081,"module":23069,"summary":23082},"\u002Frelativity\u002Ffoundations\u002Frelativistic-momentum-energy","Relativistic Momentum and Energy","Conserving momentum in every inertial frame forces the redefinition p = gamma m u, which diverges as the speed approaches c. Integrating the corresponding force gives the total energy E = gamma m c-squared, whose rest term m c-squared is Einstein's mass-energy equivalence. Energy and momentum join into a four-vector whose invariant length is the rest energy, giving E-squared = (pc)-squared + (m c-squared)-squared, massless particles, and nuclear binding energy.\n",{"path":23084,"title":23085,"module":23069,"summary":23086},"\u002Frelativity\u002Ffoundations\u002Fgeneral-relativity","A Taste of General Relativity","Einstein's happiest thought was that a freely falling observer feels no gravity: a uniform gravitational field is locally indistinguishable from an accelerating frame. That equivalence principle predicts that light bends near a mass, that clocks run slow deep in a gravitational well, that Mercury's orbit precesses, and that radar echoes are delayed. Every prediction has been confirmed, and pushing the redshift to its limit gives the black hole.\n",{"path":23088,"title":23089,"module":23090,"summary":23091},"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fminkowski-spacetime-and-the-interval","Minkowski Spacetime and the Interval","Spacetime and the Lorentz Group","The Lorentz transformation of the foundations module is repackaged as the geometry of a four-dimensional space whose invariant is not a distance but the spacetime interval. Events, worldlines, and the metric signature define a causal structure that every observer shares. Proper time is the length of a timelike worldline, and the twin paradox becomes the statement that a straight worldline accumulates the most proper time.\n",{"path":23093,"title":23094,"module":23090,"summary":23095},"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Ffour-vectors-and-index-notation","Four-Vectors and Index Notation","The index calculus that the rest of the course runs on. Contravariant and covariant components, the Minkowski metric as the machine that raises and lowers indices, and the Einstein summation convention are assembled into scalar products that are the same in every frame. The four-velocity and four-acceleration follow, together with the identity that the four-velocity has constant invariant length.\n",{"path":23097,"title":23098,"module":23090,"summary":23099},"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fthe-lorentz-group-and-rapidity","The Lorentz Group and Rapidity","The Lorentz transformations are the linear maps that preserve the Minkowski metric, and they form the group O(1,3). Boosts are hyperbolic rotations parametrized by rapidity, which adds along a line where velocity does not. The boost and rotation generators fix the group's local structure; its four disconnected components are set by two signs; and two non-collinear boosts compose into a boost plus a rotation, the Wigner rotation behind Thomas precession.\n",{"path":23101,"title":23102,"module":23090,"summary":23103},"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fdoppler-aberration-and-appearance","Doppler, Aberration, and Appearance","Light carries a null four-momentum, and boosting it produces every optical effect of relativity at once. The covariant Doppler formula follows from the transformation of frequency, aberration from the transformation of direction, and the headlight effect from the resulting concentration of light forward. The Terrell-Penrose result shows that a fast object photographs as rotated, not contracted.\n",{"path":23105,"title":23106,"module":23107,"summary":23108},"\u002Frelativity\u002Frelativistic-dynamics\u002Ffour-momentum-force-and-accelerated-motion","Four-Momentum, Four-Force, and Accelerated Motion","Relativistic Dynamics","The four-momentum packages energy and momentum into a single vector whose invariant length is the rest mass. Its proper-time derivative is the four-force, always orthogonal to the four-velocity, and a constant orthogonal four-force produces hyperbolic motion. Constant proper acceleration gives rapidity linear in proper time, the relativistic rocket equation, and the Rindler horizon behind an eternally accelerating observer.\n",{"path":23110,"title":23111,"module":23107,"summary":23112},"\u002Frelativity\u002Frelativistic-dynamics\u002Fparticle-decays-and-two-body-kinematics","Particle Decays and Two-Body Kinematics","Conservation of four-momentum fixes the kinematics of a decay from the masses alone. In the center-of-momentum frame a parent breaks into two daughters with equal and opposite momenta and energies set by the Kallen triangle function. Boosting to the lab opens the decay into a cone, and the invariant mass built from the daughters reconstructs the parent as a peak. Worked cases: the two-photon decay of the neutral pion and a heavy two-body hadronic decay.\n",{"path":23114,"title":23115,"module":23107,"summary":23116},"\u002Frelativity\u002Frelativistic-dynamics\u002Fcollisions-thresholds-and-the-cm-frame","Relativistic Collisions and Threshold Energies","Two-body collisions run on the same conserved four-momentum as decays. The invariant s sets the total energy available in the center-of-momentum frame and therefore the threshold for producing new particles. Fixed-target energy grows only as the square root of beam energy while a collider grows linearly, which is why colliders reach high energy. Compton scattering follows as a worked photon-electron collision giving the wavelength shift.\n",{"path":23118,"title":23119,"module":23107,"summary":23120},"\u002Frelativity\u002Frelativistic-dynamics\u002Fmandelstam-variables-and-invariants","Mandelstam Variables and Lorentz Invariants","For a two-to-two process the three Mandelstam invariants s, t, and u encode all the kinematics in frame-independent form. They obey a single linear constraint, the sum of the four squared masses, so only two are independent. s is the center-of-momentum energy squared, t and u are momentum transfers tied to the scattering angle, and crossing symmetry relates one amplitude across three channels through these variables.\n",{"path":23122,"title":23123,"module":23124,"summary":23125},"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ffour-current-and-the-four-potential","The Four-Current and Four-Potential","Covariant Electromagnetism","Charge density and current combine into a single four-vector whose divergence is charge conservation. The scalar and vector potentials combine likewise into the four-potential, whose gauge freedom fixes to the Lorenz condition, reducing Maxwell's equations for the potentials to a single wave equation sourced by the four-current.\n",{"path":23127,"title":23128,"module":23124,"summary":23129},"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fthe-electromagnetic-field-tensor","The Electromagnetic Field Tensor","The antisymmetric derivative of the four-potential is the field-strength tensor F, gauge invariant by construction, with the electric and magnetic fields as its components. Its dual exchanges E and B, and its two contractions form the Lorentz invariants that classify a field as electric, magnetic, or radiative in every frame.\n",{"path":23131,"title":23132,"module":23124,"summary":23133},"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ftransformation-of-electric-and-magnetic-fields","How E and B Transform","Transforming the field tensor under a boost gives explicit rules for the electric and magnetic fields: components along the motion are unchanged, transverse components mix and pick up a gamma. The field of a uniformly moving charge compresses transversely, and the force between a current and a moving charge shows that magnetism is the relativistic shadow of electrostatics.\n",{"path":23135,"title":23136,"module":23124,"summary":23137},"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fcovariant-maxwell-and-the-stress-energy-tensor","Covariant Maxwell and the Stress–Energy Tensor","Maxwell's four equations collapse into two tensor equations, one sourced by the four-current and one an identity on the field strength, with charge conservation automatic. The Lorentz force becomes a four-vector law, and the field's energy, momentum, and stress assemble into a symmetric, conserved stress–energy tensor — the object that will source gravity.\n",{"path":23139,"title":23140,"module":23141,"summary":23142},"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-equivalence-principle-formalized","The Equivalence Principle","Curved Spacetime","The equality of gravitational and inertial mass promotes to a physical principle in three graded strengths — weak, Einstein, and strong. A freely falling laboratory is locally indistinguishable from an inertial frame, but the qualifier \"locally\" is essential: the size of the patch over which gravity vanishes is set by the tidal field, which no change of frame can remove. Tidal forces are the true, coordinate-independent signature of gravity, and they are what curvature will measure.\n",{"path":23144,"title":23145,"module":23141,"summary":23146},"\u002Frelativity\u002Fcurved-spacetime\u002Fmanifolds-vectors-and-the-metric","Manifolds, Vectors, and the Metric","A manifold is a space that looks locally like flat space, described by overlapping coordinate charts. Tangent vectors are directional derivatives with the coordinate basis vectors as partial-derivative operators; one-forms live in the dual space; and the metric tensor turns a coordinate line element into an invariant length. The 2-sphere and Rindler metrics serve as worked examples, including the coordinate singularities that are artefacts of the chart, not of the geometry.\n",{"path":23148,"title":23149,"module":23141,"summary":23150},"\u002Frelativity\u002Fcurved-spacetime\u002Fcovariant-derivative-and-christoffel-symbols","Parallel Transport and the Covariant Derivative","The ordinary derivative of a vector field is not a tensor, because it subtracts vectors living in different tangent spaces. A connection supplies the missing comparison: the covariant derivative adds Christoffel-symbol correction terms that cancel the coordinate artefacts. Requiring the connection to be torsion-free and to preserve the metric fixes the Christoffel symbols uniquely in terms of derivatives of the metric, giving the Levi-Civita connection that general relativity uses.\n",{"path":23152,"title":23153,"module":23141,"summary":23154},"\u002Frelativity\u002Fcurved-spacetime\u002Fgeodesics-and-the-geodesic-equation","Geodesics and the Newtonian Limit","Free fall is geodesic motion: a freely falling particle follows the straightest possible worldline, obtained either by parallel-transporting its own tangent vector or by extremizing proper time. Both routes give the geodesic equation. Affine parameters, and conserved quantities from symmetries via Killing vectors, make it solvable. In the weak-field slow-motion limit the geodesic equation reproduces Newton's law of gravity, fixing the time-time metric component as the Newtonian potential.\n",{"path":23156,"title":23157,"module":23141,"summary":23158},"\u002Frelativity\u002Fcurved-spacetime\u002Fcurvature-riemann-and-geodesic-deviation","Curvature and the Riemann Tensor","Curvature is the failure of parallel transport to commute: carrying a vector around an infinitesimal loop returns it rotated, and the rotation per unit area is the Riemann tensor. Its symmetries cut the components to twenty in four dimensions. Geodesic deviation makes it the equation of tidal forces, and its contractions — the Ricci tensor, the Ricci scalar, and the divergence-free Einstein tensor — assemble the objects the field equation is built from.\n",{"path":23160,"title":23161,"module":23141,"summary":23162},"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-einstein-field-equations","The Einstein Field Equations","The field equation is assembled from a short list of requirements: a symmetric, divergence-free, second-order geometric tensor set proportional to the stress–energy tensor, with the coefficient fixed by the Newtonian limit. The cosmological constant is the one extra term the requirements allow. The Einstein–Hilbert action gives the same equation from a variational principle, and the coupled system closes the logic of the module: matter curves spacetime, and spacetime tells matter how to move.\n",{"path":23164,"title":23165,"module":23166,"summary":23167},"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fthe-schwarzschild-metric","The Schwarzschild Metric","The Schwarzschild Solution","The first exact solution of Einstein's equation follows from two assumptions, staticity and spherical symmetry, imposed on the vacuum outside a mass. Solving the vacuum field equations fixes two metric functions and produces the Schwarzschild geometry, whose one length scale is the Schwarzschild radius $r_s = 2GM\u002Fc^2$. Birkhoff's theorem shows this is the only spherical vacuum, and the far field reduces to Newtonian gravity.\n",{"path":23169,"title":23170,"module":23166,"summary":23171},"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fgeodesics-and-orbits-in-schwarzschild","Orbits in the Schwarzschild Geometry","The two Killing symmetries of the Schwarzschild metric give a conserved energy and angular momentum per unit mass, reducing geodesic motion to a one-dimensional problem in an effective potential. The potential carries an extra attractive $1\u002Fr^3$ term absent from Newton's, which caps the centrifugal barrier, produces an innermost stable circular orbit at $6GM\u002Fc^2$, and makes bound orbits precess instead of closing.\n",{"path":23173,"title":23174,"module":23166,"summary":23175},"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Flight-bending-and-null-geodesics","Null Geodesics and the Photon Sphere","Light follows null geodesics, governed by a photon effective potential with a single unstable maximum at $3GM\u002Fc^2$, the photon sphere. The impact parameter sorts rays into those that escape with a deflection and those captured, with the critical value $b_c = 3\\sqrt{3}\\,GM\u002Fc^2$ dividing them. A grazing ray bends by $4GM\u002F(c^2 b)$, twice the naive Newtonian value, and the critical impact parameter sets the edge of a black hole's shadow.\n",{"path":23177,"title":23178,"module":23179,"summary":23180},"\u002Frelativity\u002Ftests-of-general-relativity\u002Fperihelion-precession-of-mercury","The Perihelion Precession of Mercury","Tests of General Relativity","A single extra term in the Schwarzschild orbit equation, cubic in the inverse radius, keeps a bound orbit from closing. The perturbation advances the perihelion by 6πGM\u002F(c²a(1−e²)) per revolution, which for Mercury is 43 arcseconds per century — exactly the anomaly left after Newtonian planetary perturbations are subtracted. A note on frame dragging closes the lesson.\n",{"path":23182,"title":23183,"module":23179,"summary":23184},"\u002Frelativity\u002Ftests-of-general-relativity\u002Fdeflection-of-light-and-gravitational-lensing","Light Deflection and Gravitational Lensing","A light ray grazing the Sun bends by 4GM\u002F(c²b), exactly twice the value a Newtonian corpuscle would give; the extra factor is the curvature of space. The 1919 eclipse confirmed it. The same bending focuses light from distant sources into Einstein rings, multiple images, and microlensing brightenings, making lensing a direct probe of mass, including mass that emits no light.\n",{"path":23186,"title":23187,"module":23179,"summary":23188},"\u002Frelativity\u002Ftests-of-general-relativity\u002Fgravitational-redshift-and-shapiro-delay","Gravitational Redshift and the Shapiro Delay","A clock deeper in a gravitational well ticks slower, and a photon climbing out loses frequency by the ratio of the metric's time-time components. Pound and Rebka measured the 2.5×10⁻¹⁵ shift over a 22.5-metre tower. Radar signals grazing the Sun return late by about 250 microseconds, the Shapiro delay. Both probe the time part of the metric directly.\n",{"path":23190,"title":23191,"module":23179,"summary":23192},"\u002Frelativity\u002Ftests-of-general-relativity\u002Frelativity-in-technology-gps","Relativity and the Global Positioning System","A GPS satellite clock runs slow by 7 microseconds a day from its orbital speed and fast by 46 from its higher gravitational potential, a net gain of about 38 microseconds a day. Left uncorrected, the timing error would grow into kilometres of position error within a day and exceed navigation tolerance within minutes. The satellites carry a pre-launch frequency offset to cancel it.\n",{"path":23194,"title":23195,"module":23196,"summary":23197},"\u002Frelativity\u002Fblack-holes\u002Fhorizons-and-coordinate-singularities","Horizons and Coordinate Singularities","Black Holes","The Schwarzschild radius is a coordinate singularity, not a curvature singularity: the metric blows up there only because the static coordinates fail, while the geometry stays finite. Eddington–Finkelstein and Kruskal– Szekeres coordinates cross the horizon smoothly and show the light cones tipping toward the center. A freely falling observer reaches the true singularity at r=0 in finite proper time, while a distant observer sees the infall freeze and redden at the horizon.\n",{"path":23199,"title":23200,"module":23196,"summary":23201},"\u002Frelativity\u002Fblack-holes\u002Frotating-and-charged-black-holes","Rotating and Charged Black Holes","A stationary black hole is fixed by three numbers: mass, angular momentum, and charge. The Reissner–Nordström metric adds charge and splits the horizon in two; the Kerr metric adds rotation, drags inertial frames, and wraps the horizon in an ergosphere where nothing can stay still. Inside the ergosphere the Penrose process extracts rotational energy, and the no-hair theorem states that no other detail of the collapsed matter survives.\n",{"path":23203,"title":23204,"module":23196,"summary":23205},"\u002Frelativity\u002Fblack-holes\u002Fblack-hole-thermodynamics","Black-Hole Thermodynamics","The four laws of black-hole mechanics mirror the four laws of thermodynamics term for term, with horizon area playing the role of entropy and surface gravity the role of temperature. Hawking's calculation makes the analogy literal: a black hole radiates at a temperature set by its surface gravity, carries a real entropy proportional to its horizon area, and slowly evaporates. The thermal spectrum raises the information paradox.\n",{"path":23207,"title":23208,"module":23209,"summary":23210},"\u002Frelativity\u002Fgravitational-waves\u002Flinearized-gravity-and-wave-solutions","Linearized Gravity and Wave Solutions","Gravitational Waves","Weak gravity is a small perturbation of flat spacetime, and the linearized Einstein equation in the Lorenz gauge is an ordinary wave equation propagating at the speed of light. The trace-reversed perturbation carries the dynamics, residual gauge freedom fixes the transverse-traceless form, and the two physical polarizations deform a ring of freely falling masses into oscillating ellipses whose fractional size change is the strain.\n",{"path":23212,"title":23213,"module":23209,"summary":23214},"\u002Frelativity\u002Fgravitational-waves\u002Fgeneration-and-the-quadrupole-formula","The Quadrupole Formula","The retarded solution of the linearized field equation gives the field of a moving source, and conservation of mass and momentum forbids monopole and dipole radiation, leaving the mass quadrupole as the leading emitter. The quadrupole formula fixes the strain and the radiated luminosity, and applied to a compact binary it predicts the inspiral chirp of rising frequency and amplitude. The Hulse-Taylor pulsar's orbital decay confirmed it to a fraction of a percent.\n",{"path":23216,"title":23217,"module":23209,"summary":23218},"\u002Frelativity\u002Fgravitational-waves\u002Fdetection-ligo-and-the-first-events","LIGO and the First Detections","A gravitational wave is measured as a differential length change of the two arms of a kilometre-scale Michelson interferometer, a strain of order ten to the minus twenty-one that moves the mirrors by a fraction of a proton radius. GW150914 recorded the inspiral, merger, and ringdown of two black holes, fixing their masses and the energy radiated, and GW170817 with its coincident gamma-ray burst and kilonova opened multimessenger astronomy.\n",{"path":23220,"title":23221,"module":23222,"summary":23223},"\u002Frelativity\u002Fcosmological-bridge\u002Fthe-cosmological-principle-and-flrw-metric","The Cosmological Principle and the FLRW Metric","A Bridge to Cosmology","Homogeneity and isotropy restrict the spacetime of the universe to a single family of metrics: a flat cosmic-time slicing of spatial sections of constant curvature, scaled by a time-dependent factor a(t). This lesson builds the Friedmann–Lemaître–Robertson–Walker metric from those symmetries, separates comoving from proper distance, and derives cosmological redshift as the stretching of wavelengths with the scale factor.\n",{"path":23225,"title":23226,"module":23222,"summary":23227},"\u002Frelativity\u002Fcosmological-bridge\u002Ffriedmann-equations-and-cosmic-dynamics","The Friedmann Equations and Cosmic Dynamics","The Einstein equation applied to the FLRW metric with a perfect-fluid source yields the two Friedmann equations and the conservation law that ties them together. This lesson derives them, defines the critical density and the density parameters that fix the spatial geometry, works out how matter, radiation, and a cosmological constant dilute and drive the expansion, and hands off to a dedicated cosmology subject.\n",{"path":23229,"title":23230,"module":6,"summary":6},"\u002Frelativity","Relativity",{"path":23232,"title":23233,"module":6,"summary":6},"\u002Fphysical-computing","Physical Computing",{"path":23235,"title":23236,"module":23237,"summary":23238},"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fblackbody-radiation-and-the-planck-quantum","Blackbody Radiation and the Planck Quantum","Origins of the Quantum","Millikan's oil-drop experiment fixed the electron charge as an indivisible unit, and the spectrum of thermal radiation forced a second, deeper quantum. Classical physics predicts an infinite energy density at short wavelengths; Planck removed the divergence by allowing a cavity oscillator to hold only energies that are integer multiples of hf, the first appearance of the quantum of action.\n",{"path":23240,"title":23241,"module":23237,"summary":23242},"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-photoelectric-effect-and-the-photon","The Photoelectric Effect and the Photon","Light shone on a clean metal ejects electrons, but the details defied the wave theory: the electrons' maximum energy depends on the light's frequency, not its brightness, and there is a sharp threshold frequency below which nothing happens. Einstein resolved every anomaly by treating light as a stream of energy quanta hf, each absorbed whole by one electron, and Millikan's measurement of the stopping-potential slope confirmed h to a decade before anyone expected.\n",{"path":23244,"title":23245,"module":23237,"summary":23246},"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fx-rays-and-the-compton-effect","X-Rays and the Compton Effect","X-rays are short-wavelength electromagnetic waves produced when fast electrons are braked in a target, and their diffraction by crystals lets Bragg's law measure atomic spacings. Compton then scattered X-rays off electrons and found the wavelength shifted by an amount that only a photon carrying momentum hf\u002Fc could explain, closing the case for the particle nature of light.\n",{"path":23248,"title":23249,"module":23237,"summary":23250},"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-old-quantum-theory-bohr-and-sommerfeld","The Old Quantum Theory: Bohr, Sommerfeld, and Correspondence","Between Bohr's 1913 atom and Schrödinger's 1926 equation, physics ran on a provisional recipe: keep classical orbits, but admit only those whose action integral is a whole multiple of Planck's constant. This lesson develops the Wilson-Sommerfeld phase-integral rule, applies it to the oscillator and to the elliptical Kepler orbits of hydrogen, derives Sommerfeld's relativistic fine structure and the quantization of orbit orientation, and shows how the correspondence principle fixed intensities and selection rules. The systematic failures — helium, line intensities, the anomalous Zeeman effect — mark exactly where a theory of orbits had to give way to a theory of waves.\n",{"path":23252,"title":23253,"module":23254,"summary":23255},"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fde-broglie-waves-and-electron-diffraction","De Broglie Waves and Electron Diffraction","The Wave Nature of Matter","In 1924 de Broglie proposed that every particle carries a wave of wavelength h\u002Fp. The hypothesis explains Bohr's quantized orbits as standing waves, and Davisson and Germer, then G. P. Thomson, confirmed it by diffracting electrons from crystals exactly as X-rays diffract. We derive the electron wavelength, work the Bragg analysis of the data, and give the relativistic form.\n",{"path":23257,"title":23258,"module":23254,"summary":23259},"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fwave-packets-and-the-probability-interpretation","Wave Packets and the Probabilistic Wave Function","A single de Broglie wave fills all space, but a particle is localized. Adding many waves of nearby wavelength builds a wave packet that is confined and moves at the group velocity, which equals the particle velocity. Born's rule reads the squared amplitude of the wave function as a probability density, the meaning confirmed by electron interference building up one detection at a time.\n",{"path":23261,"title":23262,"module":23254,"summary":23263},"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fthe-uncertainty-principle","The Uncertainty Principle and Wave-Particle Duality","The packet relations delta-k delta-x about 1 become Heisenberg's principle once momentum is hbar times wave number: position and momentum cannot both be sharp, nor energy and time. The gamma-ray microscope shows the limit is physical, not technical. It fixes the zero-point energy of a confined particle, the size of the hydrogen atom, and the natural width of spectral lines, and it frames the wave-particle duality of all matter and radiation.\n",{"path":23265,"title":23266,"module":23267,"summary":23268},"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-schrodinger-equation-in-one-dimension","The Schrödinger Equation in One Dimension","Wave Mechanics in One Dimension","The wave equation for matter cannot be derived; it is postulated and judged by experiment. We build the time-dependent Schrödinger equation from the de Broglie relations, read Born's probability rule off the complex wave function, and separate the time and space dependence to get the time-independent equation whose bound-state solutions are the stationary states. The five acceptability conditions on the wave function are what force energy to be quantized.\n",{"path":23270,"title":23271,"module":23267,"summary":23272},"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-free-particle-and-wave-packet-dynamics","The Free Particle and Wave-Packet Dynamics","The free particle has no bound states: its stationary solutions are non-normalizable plane waves forming a continuum. Physical states are wave packets built by superposing them, and the superposition is a Fourier transform. We delta-normalize the plane waves, assemble a Gaussian packet, solve for its exact time evolution, and read off the two facts that reconcile the wave picture with mechanics: the packet moves at the group velocity ħk\u002Fm, the classical velocity, and it spreads because its component momenta travel at different speeds.\n",{"path":23274,"title":23275,"module":23267,"summary":23276},"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fparticle-in-infinite-and-finite-square-wells","Particle in Infinite and Finite Square Wells","The infinite square well is the simplest bound-state problem: two boundary conditions quantize the energy into a ladder E_n = n² E_1, and the eigenfunctions are the standing waves of a string fixed at both ends. Relaxing the walls to a finite depth lets the wave function leak into the classically forbidden region, keeps the number of bound states finite, and turns the eigenvalue condition into a transcendental equation solved graphically.\n",{"path":23278,"title":23279,"module":23267,"summary":23280},"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Foperators-expectation-values-and-the-harmonic-oscillator","Operators, Expectation Values, and the Harmonic Oscillator","Measurable quantities are extracted from the wave function as expectation values, and each observable is represented by an operator that acts between Ψ* and Ψ — position by multiplication, momentum by a derivative, energy by the Hamiltonian. Applied to the harmonic oscillator, the machinery yields evenly spaced levels E_n = (n+½)ℏω, Gaussian- times-Hermite eigenfunctions of definite parity, and the selection rule Δn = ±1.\n",{"path":23282,"title":23283,"module":23267,"summary":23284},"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-dirac-delta-potential","The Dirac-Delta Potential: A Single Bound State and Scattering","A potential concentrated at a single point is solvable in closed form and isolates the physics of matching a wave function across a discontinuity. Integrating the Schrödinger equation across the spike gives a jump condition on the derivative; the attractive delta well then supports exactly one bound state, of energy set by the strength alone, while the same spike scatters an incoming beam with a transmission that rises from zero to one. The attractive well and the repulsive barrier scatter identically yet only the well binds.\n",{"path":23286,"title":23287,"module":23267,"summary":23288},"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fbarrier-penetration-and-quantum-tunneling","Barrier Penetration and Quantum Tunneling","Unbound states scatter rather than bind. A particle meeting a step is partly reflected even when it has more than enough energy to pass, and a particle meeting a barrier taller than its energy has a nonzero chance of appearing on the far side. Matching the wave function across the boundaries gives the reflection and transmission coefficients and the exponential tunneling probability that explains alpha decay, the scanning tunneling microscope, and the ammonia clock.\n",{"path":23290,"title":23291,"module":23292,"summary":23293},"\u002Fquantum-mechanics\u002Fformalism\u002Fhilbert-space-and-dirac-notation","Hilbert Space and Dirac Bra–Ket Notation","The Formalism of Quantum Mechanics","Wave mechanics is one representation of a deeper structure: quantum states are vectors in a complex inner-product space, and observables act on them as linear operators. We build that space from the axioms, introduce Dirac's kets and bras as vectors and the linear functionals that measure them, and identify the wavefunction as the components of an abstract state in the position basis. The resolution of the identity is the single algebraic tool that ties every basis, expansion, and matrix element together.\n",{"path":23295,"title":23296,"module":23292,"summary":23297},"\u002Fquantum-mechanics\u002Fformalism\u002Fobservables-hermitian-operators-and-eigenvalues","Observables, Hermitian Operators, and the Spectral Theorem","Every measurable quantity is represented by a Hermitian operator, and the reason is forced: a measurement needs real eigenvalues, orthogonal eigenvectors, and a complete eigenbasis, and Hermiticity delivers precisely those. We derive those properties from self-adjointness, state the spectral theorem, handle degeneracy, and show that two observables share an eigenbasis precisely when they commute — the algebraic condition behind compatible and incompatible measurements.\n",{"path":23299,"title":23300,"module":23292,"summary":23301},"\u002Fquantum-mechanics\u002Fformalism\u002Fthe-postulates-and-quantum-measurement","The Postulates and Quantum Measurement","With states as vectors and observables as Hermitian operators, the physical content of quantum mechanics reduces to a short list of postulates. We state them precisely, derive the Born probability rule for discrete and continuous spectra, work out projective collapse and its idempotence, compute expectation values and their variance, and state the measurement problem cleanly — the one place the postulates split unitary evolution from measurement without explaining the seam.\n",{"path":23303,"title":23304,"module":23292,"summary":23305},"\u002Fquantum-mechanics\u002Fformalism\u002Fposition-momentum-and-continuous-spectra","Position, Momentum, and Continuous Spectra","Position and momentum are the observables with no normalizable eigenstates: their spectra are continuous, their eigenkets are delta-normalized, and the two are Fourier conjugates. We derive the canonical commutator from the momentum operator, build the continuous-basis machinery (Dirac deltas replacing Kronecker deltas), show the position and momentum wavefunctions are a Fourier-transform pair, and compute expectation values in either representation.\n",{"path":23307,"title":23308,"module":23292,"summary":23309},"\u002Fquantum-mechanics\u002Fformalism\u002Fcommutators-and-the-generalized-uncertainty-principle","Commutators and the Generalized Uncertainty Principle","The commutator of two observables measures the obstruction to sharing an eigenbasis, and it bounds how sharply both can be known at once. We derive the generalized uncertainty relation from the Schwarz inequality, recover the position–momentum bound as a special case, characterize the minimum-uncertainty states that saturate it as Gaussians, and give the energy–time relation its correct reading as a lifetime bound rather than a commutator relation.\n",{"path":23311,"title":23312,"module":23292,"summary":23313},"\u002Fquantum-mechanics\u002Fformalism\u002Ftime-evolution-schrodinger-and-heisenberg-pictures","Time Evolution, Propagators, and the Heisenberg Picture","Time evolution is generated by the Hamiltonian and implemented by a unitary operator that preserves probability. We build that operator, expand a state in stationary states to see why probability densities freeze while phases wind, introduce the propagator, transfer the time dependence onto operators in the Heisenberg picture, and derive Ehrenfest's theorem — which recovers classical equations of motion for expectation values and identifies conserved quantities as observables commuting with the Hamiltonian.\n",{"path":23315,"title":23316,"module":23317,"summary":23318},"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fladder-operators-and-the-number-states","Ladder Operators and the Number States","The Oscillator Algebraically, and Symmetry","The harmonic oscillator can be solved without touching a differential equation. Factoring the Hamiltonian into a lowering operator and its adjoint turns the spectrum into pure algebra: the commutator relation fixes the ladder, the vacuum condition fixes the ground state, and the energies fall out as equally spaced rungs. The same operators give the matrix elements of position and momentum for free.\n",{"path":23320,"title":23321,"module":23317,"summary":23322},"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fcoherent-and-squeezed-states","Coherent and Squeezed States","A single number state never moves — its position expectation is pinned at the origin. The superposition that oscillates like a classical particle is the eigenstate of the annihilation operator: the coherent state. It is a displaced vacuum, carries Poissonian photon statistics, saturates the uncertainty bound, and traces a rigid Gaussian orbit in phase space. Squeezing deforms that circle, trading precision in one quadrature for noise in the other.\n",{"path":23324,"title":23325,"module":23317,"summary":23326},"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fsymmetries-generators-and-conservation-laws","Symmetries, Generators, and Conservation Laws","Every continuous symmetry of a quantum system is a unitary operator built by exponentiating a Hermitian generator: momentum generates translations, angular momentum generates rotations, the Hamiltonian generates time evolution. When a generator commutes with the Hamiltonian, the transformation leaves the dynamics unchanged and the generator is conserved — the quantum form of Noether's theorem — and any symmetry that mixes states within a level forces degeneracy.\n",{"path":23328,"title":23329,"module":23317,"summary":23330},"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fparity-time-reversal-and-discrete-symmetries","Parity, Time Reversal, and Discrete Symmetries","Parity and time reversal are symmetries no continuous generator can reach. Parity is a unitary involution whose eigenvalues label states even or odd, fixing the dipole selection rules. Time reversal is antiunitary: it conjugates i, flips momenta and spins, and for half-integer spin squares to minus one, which by Kramers' theorem makes every level of a time-reversal-invariant Hamiltonian at least doubly degenerate.\n",{"path":23332,"title":23333,"module":22172,"summary":23334},"\u002Fquantum-mechanics\u002Fangular-momentum\u002Forbital-angular-momentum-and-spherical-harmonics","Orbital Angular Momentum and Spherical Harmonics","Orbital angular momentum is the operator triple built from position and momentum. Its components fail to commute, so no state carries sharp values of more than one of them, but each commutes with the total square. Solving the common eigenvalue problem in spherical coordinates quantizes both the magnitude and the projection and produces the spherical harmonics, the angular part of every central-force wavefunction.\n",{"path":23336,"title":23337,"module":22172,"summary":23338},"\u002Fquantum-mechanics\u002Fangular-momentum\u002Fthe-angular-momentum-algebra","The Angular-Momentum Algebra and Ladder Operators","The eigenvalues of angular momentum follow from the commutation relations alone, with no reference to coordinates or wavefunctions. Raising and lowering operators built from the components generate finite multiplets, force the total quantum number to be a non-negative integer or half-integer, and fix the matrix elements of every component. The half-integer values excluded by orbital motion appear here, and they are what spin realizes.\n",{"path":23340,"title":23341,"module":22172,"summary":23342},"\u002Fquantum-mechanics\u002Fangular-momentum\u002Faddition-of-angular-momenta-and-clebsch-gordan","Addition of Angular Momenta and Clebsch–Gordan Coefficients","Two angular momenta combine into a total whose allowed magnitudes run from the difference to the sum of the parts in integer steps. The change from the uncoupled product basis to the coupled total-angular-momentum basis is carried out with the lowering operator and orthogonality, and its matrix of overlaps is the table of Clebsch–Gordan coefficients. Two spin-halves split into a triplet and a singlet, the prototype for every composite spin.\n",{"path":23344,"title":23345,"module":23346,"summary":23347},"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-schrodinger-equation-in-three-dimensions","The Schrödinger Equation in Three Dimensions","Central Potentials","A central potential depends only on the distance from a force center, so the three-dimensional Schrödinger equation separates in spherical coordinates. The angular factor is a spherical harmonic; the radial factor obeys a one-dimensional equation with an effective potential whose centrifugal barrier depends on the angular-momentum quantum number. The free particle and the spherical box fix the two limiting cases through the spherical Bessel functions.\n",{"path":23349,"title":23350,"module":23346,"summary":23351},"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-hydrogen-atom","The Hydrogen Atom","The Coulomb potential turns the radial equation into one whose bound states exist only for a discrete set of energies. A power-series solution truncated to keep the wavefunction normalizable forces the principal quantum number, and the energy comes out proportional to minus one over its square, recovering the Rydberg spectrum. The bound states are the associated Laguerre functions times spherical harmonics, and their energy depends on the principal number alone, giving an n-squared degeneracy larger than rotational symmetry can explain.\n",{"path":23353,"title":23354,"module":23346,"summary":23355},"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-isotropic-oscillator-and-hidden-symmetry","The Isotropic Oscillator and Hidden Symmetry","The three-dimensional isotropic harmonic oscillator solves in both Cartesian and spherical bases, and the two solutions must agree on the degeneracy of every level. That agreement, and the accidental degeneracy of hydrogen, both come from a symmetry larger than rotation: the oscillator carries an SU(3) invariance built from a conserved quadrupole tensor, and the Coulomb problem carries an SO(4) invariance built from the conserved Runge–Lenz vector. These hidden symmetries pin the degeneracies that rotational invariance alone leaves unexplained.\n",{"path":23357,"title":23358,"module":23359,"summary":23360},"\u002Fquantum-mechanics\u002Fspin\u002Fspin-half-pauli-matrices-and-stern-gerlach","Spin-½, the Pauli Matrices, and Stern–Gerlach","Spin","A silver atom passing through an inhomogeneous magnetic field splits into two beams, not a smear. That single fact fixes the internal angular momentum of the electron to a two-valued quantity with no spatial wavefunction. We build the two-dimensional spin space, the Pauli matrices and their algebra, the spinor for measurement along an arbitrary axis, and the sequential Stern–Gerlach filters that expose measurement disturbance.\n",{"path":23362,"title":23363,"module":23359,"summary":23364},"\u002Fquantum-mechanics\u002Fspin\u002Fspin-in-a-magnetic-field-precession-and-resonance","Spin in a Magnetic Field: Precession and Resonance","A spin coupled to a magnetic field is the simplest nontrivial quantum dynamics. A static field makes the spin expectation precess on a cone at the Larmor frequency while the energy levels split linearly. Adding a weak oscillating field and passing to the rotating frame produces Rabi oscillations and a resonance lineshape — the physics of NMR and ESR, and the driven qubit.\n",{"path":23366,"title":23367,"module":23359,"summary":23368},"\u002Fquantum-mechanics\u002Fspin\u002Ftwo-level-systems-and-the-bloch-sphere","Two-Level Systems and the Bloch Sphere","Every two-state quantum system is a spin-½ in disguise. Its Hamiltonian is an effective magnetic field, its pure states are points on the Bloch sphere, and its unitary evolution is a rigid rotation of that sphere. The same structure produces avoided level crossings, the ammonia inversion doublet and its maser, and the qubit.\n",{"path":23370,"title":23371,"module":23372,"summary":23373},"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fidentical-particles-and-exchange-symmetry","Identical Particles and Exchange Symmetry","Identical Particles","Two electrons carry no label that distinguishes one from the other, and that bare fact reshapes the state space. The exchange operator that swaps particle labels commutes with any Hamiltonian built from identical particles, so its eigenvalue is conserved, and nature admits only its two extremes: totally symmetric states for bosons and totally antisymmetric states for fermions. The antisymmetry forces a statistical correlation, the exchange \"force,\" that keeps fermions apart and draws bosons together even with no interaction between them.\n",{"path":23375,"title":23376,"module":23372,"summary":23377},"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fthe-pauli-principle-atoms-and-the-periodic-table","The Pauli Principle, Atoms, and the Periodic Table","Antisymmetry packaged as a Slater determinant turns the exclusion principle into an operating rule for building atoms. Helium shows the machinery in full: the electron-electron repulsion splits into a direct Coulomb integral and an exchange integral, and the exchange term alone pushes the spin-triplet (orthohelium) below the spin-singlet (parahelium) with no magnetic interaction in sight. Screening, the aufbau order, and Hund's rules then assemble the whole periodic table from the same antisymmetry.\n",{"path":23379,"title":23380,"module":23381,"summary":23382},"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ftime-independent-perturbation-theory","Time-Independent Perturbation Theory","Approximation Methods for Bound States","Almost no realistic Hamiltonian can be solved exactly. Perturbation theory treats a hard Hamiltonian as a solvable one plus a small correction and expands the eigenvalues and eigenstates in powers of that correction. We derive the first- and second-order energy shifts and the first-order state correction for a nondegenerate level, expose the small-denominator failure that degeneracy forces, and fix it by diagonalizing the perturbation inside the degenerate subspace to find the \"good\" zeroth-order states.\n",{"path":23384,"title":23385,"module":23381,"summary":23386},"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ffine-structure-and-the-real-hydrogen-atom","Fine Structure and the Real Hydrogen Atom","The Bohr spectrum is only the leading term. Two relativistic corrections of order alpha-squared — the relativistic kinetic-energy correction and spin–orbit coupling, joined by the Darwin term for s states — split the hydrogen levels into fine structure that depends on the total angular momentum j. We derive each shift as a first-order perturbation, combine them into a formula depending only on n and j, and continue down the energy ladder to the Lamb shift and the hyperfine 21 cm line.\n",{"path":23388,"title":23389,"module":23381,"summary":23390},"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-zeeman-and-stark-effects","The Zeeman and Stark Effects","An atom in an external field is a perturbation problem whose good basis depends on which interaction wins. A magnetic field competes with the internal spin–orbit coupling: the weak-field limit gives the anomalous Zeeman splitting set by the Landé g-factor, the strong-field limit gives the Paschen–Back pattern in the uncoupled basis, and the intermediate regime is a matrix diagonalization. An electric field gives a quadratic shift for the nondegenerate ground state and a linear splitting for the degenerate n = 2 level.\n",{"path":23392,"title":23393,"module":23381,"summary":23394},"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-variational-method","The Variational Method","The expectation of the Hamiltonian in any trial state is an upper bound on the true ground-state energy. Minimizing that expectation over a parametrized family of trial functions turns the ground-state problem into ordinary calculus and needs no small parameter. We prove the bound, apply it to the helium atom with a screened effective charge, use a two-center trial to predict binding in the hydrogen molecular ion, and extend the method to excited states through orthogonality.\n",{"path":23396,"title":23397,"module":23381,"summary":23398},"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-wkb-approximation","The WKB Approximation","When the potential varies slowly on the scale of the de Broglie wavelength, the wavefunction is locally a plane wave with a position-dependent wavelength. This semiclassical picture builds the wavefunction from the classical momentum, breaks down at the turning points where the momentum vanishes, and is repaired there by connection formulas. The result recovers the Bohr–Sommerfeld quantization rule with its half-integer correction and gives the exponential tunneling rate through a smooth barrier, the Gamow factor.\n",{"path":23400,"title":23401,"module":6,"summary":6},"\u002Fquantum-mechanics","Quantum Mechanics",{"path":23403,"title":23404,"module":23405,"summary":23406},"\u002Freal-analysis\u002Ffoundations\u002Fsets-logic-functions","Sets, Logic, and Functions","Foundations and the Real Number System","The working language of analysis: quantifiers and the proof patterns (contrapositive, contradiction, induction), sets and their operations, relations and equivalence classes, and functions with their images, injections, surjections, and bijections. Cardinality is measured by bijection, and Cantor's theorem that no set surjects onto its power set forces uncountable sets to exist.\n",{"path":23408,"title":23409,"module":23405,"summary":23410},"\u002Freal-analysis\u002Ffoundations\u002Fordered-fields-completeness","Ordered Fields and the Completeness Axiom","The real numbers are the unique ordered field with the least-upper-bound property. The field and order axioms, the exact failure of the rationals (no supremum for the set of rationals below √2), and completeness as the defining axiom of ℝ lead to the first consequences: the existence of √2, the Archimedean property, and the density of ℚ in ℝ.\n",{"path":23412,"title":23413,"module":23405,"summary":23414},"\u002Freal-analysis\u002Ffoundations\u002Fabsolute-value-bounds","Absolute Value, Bounded Sets, and Inequalities","The absolute value turns the order on ℝ into a notion of distance, with the triangle inequality as the estimate underlying most later proofs. Covered: its algebra, the triangle and reverse-triangle inequalities, and the extension of the sup\u002Finf vocabulary from sets to bounded functions.\n",{"path":23416,"title":23417,"module":23405,"summary":23418},"\u002Freal-analysis\u002Ffoundations\u002Fintervals-uncountability","Intervals, Uncountability, and Decimals","Intervals are classified, and ℝ is proved uncountable two ways: a nested-interval construction and the decimal diagonal argument. Decimal expansions are built as suprema of truncations, which pins the source of their non-uniqueness (the 0.4999… equals 0.5000… identity) and the identification of the rationals with the eventually-repeating expansions. The middle-thirds Cantor set is an uncountable set of measure zero.\n",{"path":23420,"title":23421,"module":23422,"summary":23423},"\u002Freal-analysis\u002Fsequences-series\u002Fsequences-limits","Sequences and Their Limits","Sequences and Series","A sequence is a function on the natural numbers; it converges to a limit when its terms eventually stay within any prescribed tolerance of that number. The epsilon-M definition fixes the order of the quantifiers, and from it the limit is unique, every convergent sequence is bounded, and only the tail matters. Divergence to plus or minus infinity records terms that outgrow every bound.\n",{"path":23425,"title":23426,"module":23422,"summary":23427},"\u002Freal-analysis\u002Fsequences-series\u002Flimit-laws-monotone","Limit Laws and Monotone Convergence","Limits commute with sums, products, quotients, roots, and absolute values and preserve non-strict inequalities, so a limit can be assembled from the limits of its parts without returning to epsilon and M. The squeeze lemma transfers a limit through two envelopes; the monotone convergence theorem produces a limit from boundedness alone; and the ratio test settles the geometric and factorial standard limits.\n",{"path":23429,"title":23430,"module":23422,"summary":23431},"\u002Freal-analysis\u002Fsequences-series\u002Flimsup-bolzano-weierstrass","Subsequences, Limit Superior, and Bolzano–Weierstrass","A bounded sequence need not converge, but it always has convergent subsequences, and its terms cluster between two extreme values. The limit superior and inferior are the limits of the tail suprema and infima; they always exist for a bounded sequence, coincide exactly when it converges, and are its largest and smallest subsequential limits. Bolzano–Weierstrass extracts a convergent subsequence from boundedness alone.\n",{"path":23433,"title":23434,"module":23422,"summary":23435},"\u002Freal-analysis\u002Fsequences-series\u002Fcauchy-completeness","Cauchy Sequences and the Completeness of the Reals","The Cauchy criterion tests convergence without knowing the limit: a sequence converges exactly when its terms eventually all lie within any tolerance of one another. Cauchy sequences are bounded, in the reals Cauchy and convergent are equivalent, and this completeness property is interchangeable with the least-upper-bound axiom — the single feature that separates the real line from the rationals.\n",{"path":23437,"title":23438,"module":23422,"summary":23439},"\u002Freal-analysis\u002Fsequences-series\u002Fseries-convergence","Series and Convergence Tests","A series converges when its sequence of partial sums does, so every fact about sequences transfers. Geometric and telescoping series sum in closed form; the n-th term test rejects series whose terms miss zero, though the harmonic series shows the converse fails; and the comparison test against the geometric and p-series benchmarks settles most nonnegative-term series.\n",{"path":23441,"title":23442,"module":23422,"summary":23443},"\u002Freal-analysis\u002Fsequences-series\u002Fabsolute-conditional-rearrangement","Absolute Convergence, the Ratio and Root Tests, and Rearrangements","Absolute convergence is the strong form of convergence that permits free manipulation; conditional convergence is fragile. Absolute convergence implies convergence, and the ratio and root tests detect it by comparison with the geometric series. The alternating series test supplies conditionally convergent series, Riemann's theorem rearranges any of them to any sum, and Mertens' theorem multiplies series when at least one converges absolutely.\n",{"path":23445,"title":23446,"module":23447,"summary":23448},"\u002Freal-analysis\u002Fmetric-spaces\u002Fmetric-spaces-norms","Metric Spaces, Norms, and Examples","Metric Spaces and Topology","A metric is a function $d(x,y)$ obeying four axioms: nonnegativity, identity of indiscernibles, symmetry, and the triangle inequality. The Euclidean, taxicab, sup, discrete, and great-circle metrics all qualify, as does the sup metric on $C[a,b]$. Every norm induces a metric, and strongly equivalent metrics share the same open sets.\n",{"path":23450,"title":23451,"module":23447,"summary":23452},"\u002Freal-analysis\u002Fmetric-spaces\u002Fopen-closed-sets","Open and Closed Sets, Interior, Closure","Open sets are those in which every point has room to move; closed sets are their complements. From the single ball construction come the topology axioms (arbitrary unions, finite intersections), the interior, closure, and boundary of a set, and the fact that openness is always relative to the ambient space.\n",{"path":23454,"title":23455,"module":23447,"summary":23456},"\u002Freal-analysis\u002Fmetric-spaces\u002Fconvergence-completeness","Convergence, Cauchy Sequences, and Completeness","The $\\varepsilon$-$N$ definition of a limit transfers verbatim to any metric space once $|x-y|$ is replaced by $d(x,y)$. Convergent sequences characterize closed sets and closures; Cauchy sequences and completeness capture spaces with no missing limits, with $\\mathbb{R}^n$ and $C[a,b]$ complete and $\\mathbb{Q}$ and $(0,1]$ not.\n",{"path":23458,"title":23459,"module":23447,"summary":23460},"\u002Freal-analysis\u002Fmetric-spaces\u002Fcompactness","Compactness","A set is compact if every open cover has a finite subcover. In a metric space this is equivalent to sequential compactness and to being complete and totally bounded. Compact sets are closed and bounded; the Heine–Borel theorem gives the converse in $\\mathbb{R}^n$ but nowhere else in general.\n",{"path":23462,"title":23463,"module":23447,"summary":23464},"\u002Freal-analysis\u002Fmetric-spaces\u002Fconnectedness","Connectedness","A space is connected when it cannot be split into two nonempty open pieces. The connected subsets of $\\mathbb{R}$ are precisely the intervals, path- connectedness gives a constructive sufficient condition, and connectedness is a topological invariant preserved by continuous maps, the fact behind the intermediate value theorem.\n",{"path":23466,"title":23467,"module":21866,"summary":23468},"\u002Freal-analysis\u002Fcontinuity\u002Flimits-of-functions","Limits of Functions","The limit of a function at a point is an epsilon–delta condition pinning one value L as the target of f(x) as x approaches c, mirroring the sequence definition with distance replacing index. It is stated only at cluster points of the domain, is unique when it exists, and reduces to sequential limits through the Heine criterion. The algebra of limits and one-sided limits follow from that reduction.\n",{"path":23470,"title":23471,"module":21866,"summary":23472},"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuous-functions","Continuous Functions","A function is continuous at c when its limit there equals its own value, lim f(x) = f(c). The epsilon–delta and sequential forms agree; sums, products, quotients, and compositions of continuous functions are continuous; and the failures split into jump, Dirichlet, popcorn, and removable types. The topological reading is that preimages of open sets are open.\n",{"path":23474,"title":23475,"module":21866,"summary":23476},"\u002Freal-analysis\u002Fcontinuity\u002Fevt-ivt","Extreme and Intermediate Value Theorems","On a closed bounded interval a continuous function attains an absolute maximum and minimum (the extreme value theorem, compactness preserved by continuity) and takes every value between its endpoint values (the intermediate value theorem, connectedness preserved). Both proofs run through Bolzano–Weierstrass and bisection, and yield root-finding, existence of k-th roots, and fixed-point theorems.\n",{"path":23478,"title":23479,"module":21866,"summary":23480},"\u002Freal-analysis\u002Fcontinuity\u002Funiform-continuity","Uniform Continuity","Uniform continuity strengthens continuity by demanding one delta that works at every point of the domain, not a delta re-chosen at each point. It separates x^2 on a compact interval from x^2 on the whole line and from 1\u002Fx near zero; continuity on a closed bounded interval is automatically uniform; uniformly continuous functions preserve Cauchy sequences and extend to endpoints; and Lipschitz continuity is the strongest of the three, through its secant-slope bound.\n",{"path":23482,"title":23483,"module":21866,"summary":23484},"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuity-metric-spaces","Continuity on Metric Spaces","The epsilon–delta definition used only distances, so continuity transfers to maps between metric spaces by replacing absolute values with the two metrics. In this generality continuity still admits a sequential form, preserves compactness and connectedness, is uniform on a compact domain, and reads topologically as preimages of open sets being open, the formulation that defines homeomorphisms.\n",{"path":23486,"title":23487,"module":21866,"summary":23488},"\u002Freal-analysis\u002Fcontinuity\u002Flimits-infinity-monotone","Limits at Infinity and Monotone Functions","Treating infinity as a cluster point extends the epsilon–delta limit to x approaching plus or minus infinity, giving horizontal asymptotes and infinite limits. For monotone functions the one-sided limits always exist as suprema and infima, the discontinuities are jumps and at most countably many, the continuity is equivalent to the image being an interval, and a strictly monotone function always has a continuous inverse.\n",{"path":23490,"title":23491,"module":23492,"summary":23493},"\u002Freal-analysis\u002Fdifferentiation\u002Fthe-derivative","The Derivative","Differentiation","The derivative is the limit of the difference quotient, the slope the secant lines approach as the second point slides into the first. Differentiability forces continuity; linearity and the product, quotient, and chain rules follow from the definition; and a continuous function can fail to be differentiable, as the absolute value does at the origin.\n",{"path":23495,"title":23496,"module":23492,"summary":23497},"\u002Freal-analysis\u002Fdifferentiation\u002Fmean-value-theorem","The Mean Value Theorem","A relative extremum in the interior forces the derivative to vanish; Rolle's theorem and the mean value theorem turn that local fact into global control. The sign of the derivative fixes monotonicity, a bounded derivative yields a Lipschitz bound, and Darboux's theorem shows derivatives have the intermediate value property even where they are discontinuous.\n",{"path":23499,"title":23500,"module":23492,"summary":23501},"\u002Freal-analysis\u002Fdifferentiation\u002Ftaylors-theorem","Taylor's Theorem","Taylor's theorem generalizes the mean value theorem: an n-times differentiable function is matched near a point by a degree-n polynomial, with a Lagrange remainder that names the error exactly through one higher derivative. Iterating the mean value theorem proves it; the second-derivative test is the order-one case; and a smooth non-analytic bump separates a Taylor series from the function it fails to represent.\n",{"path":23503,"title":23504,"module":23492,"summary":23505},"\u002Freal-analysis\u002Fdifferentiation\u002Finverse-function-1d","The Inverse Function Theorem in One Variable","A nonzero derivative certifies a local inverse and fixes its slope. A strictly monotone differentiable function has a differentiable inverse whose derivative is the reciprocal of the original; the inverse function theorem removes the monotonicity hypothesis, and the reciprocal formula constructs nth roots and the logarithm's derivative, failing exactly where the derivative vanishes.\n",{"path":23507,"title":23508,"module":23509,"summary":23510},"\u002Freal-analysis\u002Friemann-integration\u002Fdarboux-integral","Partitions, Darboux Sums, and Integrability","The Riemann Integral","The Riemann integral is defined by trapping the area under a bounded function between under- and over-estimates. Partitions cut the domain into strips; lower and upper Darboux sums bracket the area; refining a partition tightens the bracket. A function is integrable exactly when the bracket can be made arbitrarily thin, and the tagged Riemann-sum limit gives the same number.\n",{"path":23512,"title":23513,"module":23509,"summary":23514},"\u002Freal-analysis\u002Friemann-integration\u002Fintegrability-classes","Which Functions Are Integrable","The Cauchy criterion certifies whole classes of functions as integrable. Continuous functions are integrable because uniform continuity makes every oscillation cap small; monotone functions are integrable because their caps telescope to a single total jump; bounded functions with finitely many discontinuities are integrable by isolating the bad points. The Dirichlet function fails, and the Lebesgue criterion names the exact boundary.\n",{"path":23516,"title":23517,"module":23509,"summary":23518},"\u002Freal-analysis\u002Friemann-integration\u002Fproperties-of-the-integral","Properties of the Integral","The integral is a linear, order-preserving, additive operator on the integrable functions. It splits across subintervals, respects inequalities, bounds the size of a function by the integral of its absolute value, and preserves products. The mean value theorem for integrals identifies the integral with an attained average height on a fixed rectangle.\n",{"path":23520,"title":21921,"module":23509,"summary":23521},"\u002Freal-analysis\u002Friemann-integration\u002Ffundamental-theorem","The fundamental theorem ties the integral to the derivative in two forms. The evaluation form computes a definite integral from any antiderivative; the differentiation form shows the area function has derivative equal to the integrand at points of continuity. Together they make differentiation and integration inverse operations, and yield integration by parts and change of variables.\n",{"path":23523,"title":23524,"module":23509,"summary":23525},"\u002Freal-analysis\u002Friemann-integration\u002Flog-exp-improper","The Logarithm, Exponential, and Improper Integrals","The integral defines transcendental functions. The logarithm is the area under 1\u002Ft, the exponential is its inverse, and their calculus properties follow from the fundamental theorem. Improper integrals extend integration to unbounded intervals and unbounded integrands as limits of proper integrals, with a p-test, a comparison test, absolute versus conditional convergence, and the integral test linking integrals to series.\n",{"path":23527,"title":23528,"module":23529,"summary":23530},"\u002Freal-analysis\u002Ffunction-sequences\u002Fpointwise-uniform-convergence","Pointwise and Uniform Convergence","Sequences and Series of Functions","A sequence of functions has two natural notions of limit. Pointwise convergence fixes each input and takes the limit of numbers; uniform convergence demands one rate that works for every input at once. The uniform norm turns the second into a statement about a single sequence of numbers, and the uniform Cauchy criterion and the Weierstrass M-test let us certify it.\n",{"path":23532,"title":23533,"module":23529,"summary":23534},"\u002Freal-analysis\u002Ffunction-sequences\u002Finterchange-of-limits","Interchange of Limits: Continuity, Integration, Differentiation","Passing to a limit inside a continuity statement, an integral, or a derivative is an interchange of two limits, and the two limits do not always commute. Uniform convergence licenses the first two swaps: the uniform limit of continuous functions is continuous, and the limit of the integrals is the integral of the limit. Differentiation needs uniform convergence of the derivatives, and counterexamples show why each hypothesis is required.\n",{"path":23536,"title":23537,"module":23529,"summary":23538},"\u002Freal-analysis\u002Ffunction-sequences\u002Fpower-series-weierstrass","Power Series and the Weierstrass Approximation Theorem","A power series converges uniformly on every closed subinterval inside its radius of convergence, together with all of its derivatives. That makes it continuous, differentiable, and integrable term by term, so a power series defines an infinitely differentiable function. The Weierstrass approximation theorem then shows that polynomials come uniformly close to any continuous function on a closed bounded interval.\n",{"path":23540,"title":23541,"module":23529,"summary":23542},"\u002Freal-analysis\u002Ffunction-sequences\u002Fpicard-ode","Picard's Existence and Uniqueness Theorem","The Banach fixed-point theorem says a contraction of a complete metric space has exactly one fixed point, found by iterating from any start. Applied to the space of continuous functions with the uniform norm, it proves Picard's theorem: a first-order differential equation with a Lipschitz right-hand side has a unique local solution. Picard iteration constructs that solution explicitly, and worked examples show the Lipschitz condition is not optional.\n",{"path":23544,"title":23545,"module":23546,"summary":23547},"\u002Freal-analysis\u002Fseveral-variables\u002Fdifferentiability-rn","The Derivative of a Map ℝⁿ → ℝᵐ","Functions of Several Variables (Introduction)","The derivative of a map between Euclidean spaces is the linear transformation of vanishing relative error, unique when it exists and represented in coordinates by the Jacobian matrix of partial derivatives. Differentiability forces continuity through a local Lipschitz bound. Existence of the partial derivatives alone does not suffice; continuity of the partials does.\n",{"path":23549,"title":23550,"module":23546,"summary":23551},"\u002Freal-analysis\u002Fseveral-variables\u002Fgradient-chain-rule","Directional Derivatives, the Gradient, and the Chain Rule","The directional derivative measures the rate of change of a scalar field along a chosen heading and equals the derivative applied to that direction. The gradient collects these into a vector that points along steepest ascent and sits orthogonal to level sets. The chain rule composes derivatives by multiplying Jacobians, and a mean value theorem holds for scalar fields but fails for vector-valued maps.\n",{"path":23553,"title":23554,"module":23546,"summary":23555},"\u002Freal-analysis\u002Fseveral-variables\u002Fhigher-derivatives-taylor-extrema","Higher Derivatives, Taylor's Theorem, and Extrema","Iterating the derivative gives a symmetric second derivative, the Hessian, whose mixed partials agree when they are continuous. Taylor's theorem expands a smooth map to any order with a Lagrange-type remainder, and at a critical point the definiteness of the Hessian decides between a local minimum, a local maximum, and a saddle.\n",{"path":23557,"title":23558,"module":23546,"summary":23559},"\u002Freal-analysis\u002Fseveral-variables\u002Finverse-implicit-theorems","The Inverse and Implicit Function Theorems","A nonlinear map with a nonsingular Jacobian is locally invertible, with the inverse's derivative given by the inverse matrix. The contraction mapping principle supplies the local inverse; the implicit function theorem then solves a system for some variables in terms of the rest whenever the relevant Jacobian block is invertible. Worked coordinate changes show both theorems in use.\n",{"path":23561,"title":23562,"module":23546,"summary":23563},"\u002Freal-analysis\u002Fseveral-variables\u002Fmultiple-integrals","Multiple Integrals","The Riemann integral of a bounded function over a closed rectangle in Euclidean space is built from Darboux upper and lower sums on a grid of subrectangles, with the same squeeze criterion that governs the one-variable integral. Continuous integrands are integrable, and a set of content zero can be ignored. Fubini's theorem evaluates a multiple integral as an iterated one in either order, and the indicator trick extends the theory to regions bounded by curves.\n",{"path":23565,"title":23566,"module":6,"summary":6},"\u002Freal-analysis","Real Analysis",{"path":23568,"title":23569,"module":21519,"summary":23570},"\u002Fabstract-algebra\u002Ffoundations\u002Fsets-functions-relations","Sets, Functions, and Equivalence Relations","Algebra is built on three prior notions: the set, the map between sets, and the equivalence relation that reorganizes a set into disjoint classes. Sets, maps (injective, surjective, bijective), fibers and preimages, and the correspondence between equivalence relations and partitions — the one structural fact reused in every later quotient construction.\n",{"path":23572,"title":23573,"module":21519,"summary":23574},"\u002Fabstract-algebra\u002Ffoundations\u002Fintegers-and-modular-arithmetic","The Integers and Modular Arithmetic","The integers carry the template every ring later imitates: well-ordering drives induction, induction drives the division algorithm, and division drives the Euclidean algorithm, gcd, Bézout's identity, and unique factorization into primes. Quotienting by congruence mod n builds the first finite arithmetic, Z\u002FnZ, whose invertible elements form the group of units.\n",{"path":23576,"title":23577,"module":23578,"summary":23579},"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fgroup-axioms-and-first-examples","Group Axioms and First Examples","Groups and Symmetry","A group is a set with one associative operation that has an identity and inverses. We state the axioms, prove that the identity, inverses, and cancellation behave as expected, define the order of a group and of an element, and catalogue the running examples: the integers, the additive group of residues mod n, and the multiplicative group of units mod n.\n",{"path":23581,"title":23582,"module":23578,"summary":23583},"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fdihedral-and-symmetric-groups","Dihedral and Symmetric Groups","The dihedral group D_{2n} is the symmetries of a regular n-gon, generated by a rotation r and a reflection s subject to three relations. The symmetric group S_n is all permutations of n objects, written in cycle notation. Orders, generators and relations, cycle decomposition, the order of a permutation from its cycle type, and the parity that splits S_n in half.\n",{"path":23585,"title":23586,"module":23578,"summary":23587},"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fmatrix-and-quaternion-groups","Matrix and Quaternion Groups","Invertible matrices over a field form the general linear group GL_n(F), with the determinant-one matrices as the subgroup SL_n(F). Over a finite field the order of GL_n(F) has a clean product formula. The quaternion group Q_8 is a second small nonabelian group, distinct from the dihedral group of the same order; its multiplication and subgroup structure sharpen the contrast between the two.\n",{"path":23589,"title":23590,"module":23578,"summary":23591},"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fhomomorphisms-and-group-actions","Homomorphisms, Isomorphisms, and Actions","A homomorphism is a map between groups that respects the operation; an isomorphism is a bijective one, making two groups the same up to relabeling. The kernel and image measure how far a homomorphism is from injective and surjective. A group action realizes a group as permutations of a set, and actions correspond exactly to homomorphisms into a symmetric group, with orbits and stabilizers as the first tools for counting.\n",{"path":23593,"title":23594,"module":23595,"summary":23596},"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fsubgroups-and-substructures","Subgroups and Their Substructures","Subgroups and Quotients","A subgroup is a subset that is a group under the inherited operation. One test decides it: nonempty and closed under the map $(x,y) \\mapsto xy^{-1}$. From an arbitrary subset $A$ we build the centralizer, normalizer, and center, and from an action the stabilizer and kernel, all of them subgroups nested in a fixed chain inside $G$.\n",{"path":23598,"title":23599,"module":23595,"summary":23600},"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcyclic-groups","Cyclic Groups","A cyclic group is generated by one element. Two facts organize the whole theory: the order of an element equals the order of the subgroup it generates, and cyclic groups of equal order are isomorphic, so $\\mathbb{Z}$ and $\\mathbb{Z}\u002Fn\\mathbb{Z}$ are the only ones. From there the generators ($\\varphi(n)$ of them), the subgroups (one per divisor of $n$), and a fast exponentiation algorithm all follow.\n",{"path":23602,"title":23603,"module":23595,"summary":23604},"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fgeneration-and-subgroup-lattices","Generation and the Lattice of Subgroups","The subgroup generated by a subset $A$ is the smallest subgroup containing it, described top-down as an intersection and bottom-up as the set of words in $A$ and its inverses. Collecting all subgroups and ordering them by containment produces the subgroup lattice, whose Hasse diagram shows the joins, meets, and containment relations among all subgroups.\n",{"path":23606,"title":23607,"module":23595,"summary":23608},"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcosets-lagrange-and-normal-subgroups","Cosets, Lagrange, and Normal Subgroups","The left cosets of a subgroup partition a group into equal-sized blocks, so the order of a subgroup divides the order of the group: Lagrange's theorem. When the blocks can be multiplied consistently — exactly when the subgroup is normal — they form the quotient group $G\u002FN$. Fermat's and Euler's theorems fall out as index computations.\n",{"path":23610,"title":23611,"module":23595,"summary":23612},"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fisomorphism-theorems","The Isomorphism Theorems","Four theorems relate homomorphisms, quotients, and subgroup lattices. The first identifies the image of a homomorphism with the quotient by its kernel; the second and third compute quotients built from two subgroups and quotients of quotients; the fourth matches the subgroups of $G\u002FN$ with the subgroups of $G$ lying above $N$. Together they make quotient groups computable.\n",{"path":23614,"title":23615,"module":23595,"summary":23616},"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcomposition-series-and-the-alternating-group","Composition Series and the Alternating Group","A composition series breaks a finite group into simple quotient factors, and Jordan-Hölder says those factors are unique up to order. This turns classification into two problems: list the simple groups, and describe how to reassemble them. The sign homomorphism splits $S_n$ into even and odd permutations, defining the alternating group $A_n$, simple for $n \\ge 5$.\n",{"path":23618,"title":23619,"module":23620,"summary":23621},"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Factions-and-cayleys-theorem","Actions, Orbits, and Cayley's Theorem","Group Actions and Sylow Theory","A group action turns abstract elements into permutations of a set. The action splits the set into orbits, and the orbit-stabilizer theorem ties each orbit's size to the index of a stabilizer. Applied to a group acting on itself by left multiplication, this gives Cayley's theorem: every group is a group of permutations.\n",{"path":23623,"title":23624,"module":23620,"summary":23625},"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fconjugation-and-the-class-equation","Conjugation and the Class Equation","A group acts on itself by conjugation, and the orbits are the conjugacy classes. Orbit-stabilizer turns the resulting partition into the class equation, which forces every group of prime-power order to have a nontrivial center. Conjugacy in the symmetric group is cycle type, and Burnside's lemma counts orbits by averaging fixed points.\n",{"path":23627,"title":23628,"module":23620,"summary":23629},"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fsylow-theorems","The Sylow Theorems","Lagrange's theorem forbids subgroups whose order fails to divide the group order; Sylow's theorems supply a partial converse for prime powers. A Sylow p-subgroup always exists, all of them are conjugate, and their count satisfies two congruence-and-divisibility constraints tight enough to prove many groups non-simple from their order alone.\n",{"path":23631,"title":23632,"module":23620,"summary":23633},"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fautomorphisms-and-simple-groups","Automorphisms and Simplicity of Aₙ","Conjugation makes a group act on itself and on its normal subgroups by automorphisms, giving the inner automorphism group G\u002FZ(G) and the embedding of N(H)\u002FC(H) into Aut(H). Characteristic subgroups are those every automorphism fixes, and the automorphism group of a cyclic group is its unit group. The lesson closes by proving the alternating group Aₙ is simple for n ≥ 5.\n",{"path":23635,"title":23636,"module":23637,"summary":23638},"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fdirect-products-and-finite-abelian-groups","Direct Products and Finite Abelian Groups","Products and Group Structure","The direct product assembles a larger group from componentwise copies of smaller ones, and a recognition theorem reverses the process when two normal subgroups meet trivially and span the group. The Fundamental Theorem of Finitely Generated Abelian Groups then classifies every such group by two equivalent invariants, invariant factors and elementary divisors.\n",{"path":23640,"title":23641,"module":23637,"summary":23642},"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fsemidirect-products","Semidirect Products","The semidirect product relaxes the direct product by requiring only one factor to be normal, with the other acting on it through a homomorphism into its automorphism group. This single twisting map lets abelian pieces assemble into non-abelian groups, realizes the dihedral groups as $\\mathbb{Z}_n \\rtimes \\mathbb{Z}_2$, and, with a recognition theorem, classifies groups of several small orders.\n",{"path":23644,"title":23645,"module":23637,"summary":23646},"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fnilpotent-and-solvable-groups","p-Groups, Nilpotent, and Solvable Groups","Finite p-groups have nontrivial center, and iterating the center upward builds the nilpotent groups, which decompose as the direct product of their Sylow subgroups. Iterating the commutator downward builds the solvable groups, whose factors are abelian. The chain cyclic, abelian, nilpotent, solvable orders these classes, and A_5 breaks the last link.\n",{"path":23648,"title":23649,"module":23637,"summary":23650},"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fclassifying-small-groups","Classifying Groups of Small Order","With Sylow's theorem to force normal subgroups, direct and semidirect products to assemble them, and presentations to name the result, every group up to order fifteen can be listed explicitly. Free groups make presentations precise: generators with no relations, from which any group is a quotient by the normal closure of its relations.\n",{"path":23652,"title":23653,"module":23654,"summary":23655},"\u002Fabstract-algebra\u002Fring-theory\u002Frings-definitions-and-examples","Rings: Definitions and Examples","Ring Theory","A ring carries two operations: an abelian group under addition and an associative multiplication linked by the distributive laws. The named special cases — commutative rings, integral domains, division rings, and fields — differ only in how their multiplication behaves. Standard examples include quadratic integer rings, polynomial rings, matrix rings, and group rings.\n",{"path":23657,"title":23658,"module":23654,"summary":23659},"\u002Fabstract-algebra\u002Fring-theory\u002Fideals-quotients-and-homomorphisms","Ideals, Quotient Rings, and Homomorphisms","Ring homomorphisms have kernels that absorb multiplication; such subsets are ideals, and every ideal is the kernel of the projection onto a quotient ring. The quotient construction yields the ring isomorphism theorems and classifies ideals by their quotients: R\u002FI is a field exactly when I is maximal, an integral domain exactly when I is prime.\n",{"path":23661,"title":23662,"module":23654,"summary":23663},"\u002Fabstract-algebra\u002Fring-theory\u002Ffractions-and-the-chinese-remainder-theorem","Fields of Fractions and the CRT","Rings of fractions invert a multiplicatively closed set, enlarging an integral domain into its field of fractions the way Z becomes Q. The Chinese Remainder Theorem splits a quotient by comaximal ideals into a direct product, generalizing Z\u002FmnZ ≅ Z\u002FmZ × Z\u002FnZ and explaining why the Euler function is multiplicative.\n",{"path":23665,"title":23666,"module":23667,"summary":23668},"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Feuclidean-domains-pids-ufds","Euclidean Domains, PIDs, and UFDs","Factorization and Polynomial Rings","Three classes of integral domain, ordered by how much of elementary arithmetic survives: Euclidean domains carry a division algorithm, principal ideal domains make every ideal a single multiple, and unique factorization domains factor every element into irreducibles in one way. We prove the chain ED implies PID implies UFD, the classes are separated by explicit counterexamples, and irreducible and prime coincide exactly in a UFD.\n",{"path":23670,"title":23671,"module":23667,"summary":23672},"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fpolynomial-rings-over-fields","Polynomial Rings over Fields","When the coefficients form a field, polynomial long division works exactly as it does over the rationals, and it works with a unique quotient and remainder. That single fact makes F[x] a Euclidean domain, hence a PID and a UFD: every ideal is the multiples of one polynomial, roots correspond to linear factors, and F[x]\u002F(f) is a field precisely when f is irreducible.\n",{"path":23674,"title":23675,"module":23667,"summary":23676},"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fgauss-lemma-and-unique-factorization","Gauss's Lemma and Unique Factorization","A UFD is not a field, so its polynomial ring is not a PID — yet unique factorization survives the passage from R to R[x]. Gauss's lemma supplies the passage: a polynomial that factors over the fraction field already factors over R, once content is factored out. This gives the theorem that R[x] is a UFD whenever R is, so Z[x] and Q[x,y] factor uniquely even though neither is a PID.\n",{"path":23678,"title":23679,"module":23667,"summary":23680},"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Firreducibility-criteria-and-groebner","Irreducibility Criteria and Gröbner Bases","Deciding whether a given polynomial is irreducible, and computing in multivariate polynomial rings. In one variable: the rational root test, reduction modulo a prime, and Eisenstein's criterion. In several variables, where division fails, a monomial order gives leading terms, a Gröbner basis restores a well-defined remainder, and Buchberger's algorithm computes it.\n",{"path":23682,"title":23683,"module":23684,"summary":23685},"\u002Fabstract-algebra\u002Fmodule-theory\u002Fintroduction-to-modules","Introduction to Modules","Module Theory","A module is an abelian group on which a ring acts, generalizing both vector spaces (when the ring is a field) and abelian groups (when the ring is the integers). Submodules, homomorphisms, quotients, and the isomorphism theorems carry over from groups, and an F[x]-module is the same datum as a vector space with a chosen linear operator — the correspondence behind the canonical forms.\n",{"path":23687,"title":23688,"module":23684,"summary":23689},"\u002Fabstract-algebra\u002Fmodule-theory\u002Ffree-modules-and-direct-sums","Generation, Direct Sums, and Free Modules","A generating set spans a module by R-linear combinations; a direct sum decomposes it into independent pieces; a free module has a basis and the universal property that a homomorphism is determined by arbitrary values on that basis. Rank is well defined over a commutative ring, torsion blocks a basis, and every module is a quotient of a free one — a presentation by generators and relations.\n",{"path":23691,"title":23692,"module":23684,"summary":23693},"\u002Fabstract-algebra\u002Fmodule-theory\u002Ftensor-products-and-exact-sequences","Tensor Products and Exact Sequences","The tensor product builds a module in which elements of two modules can be multiplied, characterized by a universal property turning bilinear maps into linear ones; extension of scalars is its guiding case. Exact sequences track how a module is assembled from a submodule and a quotient, when that assembly splits, and which modules — projective, injective, flat — make the Hom and tensor functors preserve exactness.\n",{"path":23695,"title":23696,"module":23684,"summary":23697},"\u002Fabstract-algebra\u002Fmodule-theory\u002Fvector-spaces-and-linear-maps","Vector Spaces and Linear Maps","A vector space is a module over a field, and the field hypothesis removes every pathology a general module can have: every vector space is free, so it has a basis, a well-defined dimension, and a coordinate isomorphism with F^n. Linear maps become matrices, change of basis becomes similarity, every space pairs with a dual of the same dimension, and the determinant is the unique alternating multilinear normalized form.\n",{"path":23699,"title":23700,"module":23701,"summary":23702},"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fstructure-theorem-over-pids","The Structure Theorem for Modules over a PID","Modules over PIDs and Canonical Forms","Every finitely generated module over a principal ideal domain splits as a free part plus a direct sum of cyclic torsion pieces, in two canonical ways: invariant factors, tied together by a divisibility chain, and elementary divisors, one prime power at a time. Existence follows from the stacked-basis theorem, both lists are unique, and the case $R = \\mathbb{Z}$ is the classification of finitely generated abelian groups.\n",{"path":23704,"title":23705,"module":23701,"summary":23706},"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Frational-canonical-form","Rational Canonical Form","A linear operator turns its vector space into a module over the polynomial ring $F[x]$, with $x$ acting as the operator. The structure theorem's invariant factors then become polynomials, each cyclic summand becomes a companion matrix, and the block-diagonal assembly is the rational canonical form. It is unique, it is computed inside the base field, and two matrices are similar exactly when their rational canonical forms agree.\n",{"path":23708,"title":23709,"module":23701,"summary":23710},"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fjordan-canonical-form","Jordan Canonical Form","When the base field contains all the eigenvalues, the elementary divisors of an operator are powers of linear polynomials, and each cyclic summand becomes a Jordan block: an eigenvalue on the diagonal with ones just above it. Stacking the blocks gives the Jordan canonical form, unique up to reordering, as close to diagonal as the operator allows. Diagonalizability reads off the minimal polynomial, and the block sizes are counted by ranks of powers of the operator minus the eigenvalue.\n",{"path":23712,"title":23713,"module":23714,"summary":23715},"\u002Fabstract-algebra\u002Ffield-theory\u002Ffield-extensions-and-algebraic-elements","Field Extensions and Algebraic Elements","Field Theory","A field extension makes a larger field K into a vector space over a smaller field F, and its degree [K:F] is that dimension. Adjoining a root of an irreducible polynomial builds a simple extension F(α) isomorphic to F[x]\u002F(m), whose degree is the degree of the minimal polynomial. The tower law makes these degrees multiply, which turns algebra over fields into bookkeeping with integers.\n",{"path":23717,"title":23718,"module":23714,"summary":23719},"\u002Fabstract-algebra\u002Ffield-theory\u002Fstraightedge-and-compass-constructions","Straightedge-and-Compass Constructions","The lengths a straightedge and compass can build from a unit form a field closed under square roots, and every constructible number lies in a tower of quadratic extensions. So its degree over the rationals is a power of two. That single obstruction settles three problems the Greeks left open: doubling the cube, trisecting a general angle, and squaring the circle are all impossible.\n",{"path":23721,"title":23722,"module":23714,"summary":23723},"\u002Fabstract-algebra\u002Ffield-theory\u002Fsplitting-fields-and-algebraic-closure","Splitting Fields and Algebraic Closure","The splitting field of a polynomial is the smallest extension in which it factors into linear pieces, obtained by adjoining all its roots. Every polynomial has one, its degree is at most n factorial, and any two splitting fields are isomorphic. Pushing this to all polynomials at once gives the algebraic closure, a field in which every polynomial splits and which is unique up to isomorphism.\n",{"path":23725,"title":23726,"module":23714,"summary":23727},"\u002Fabstract-algebra\u002Ffield-theory\u002Fseparable-and-cyclotomic-extensions","Separable Extensions and Cyclotomic Fields","A polynomial is separable when its roots are distinct, detected by whether it shares a factor with its formal derivative. Over perfect fields — characteristic zero and finite fields — every irreducible is separable, and the existence and uniqueness of the finite fields follow. Cyclotomic polynomials package the roots of unity by order, are irreducible over the rationals, and give the cyclotomic field its degree phi(n).\n",{"path":23729,"title":23730,"module":23731,"summary":23732},"\u002Fabstract-algebra\u002Fgalois-theory\u002Fthe-galois-correspondence","The Galois Correspondence","Galois Theory","Galois theory attaches to a field extension its group of symmetries and shows that, for the right extensions, the subgroups of that group are in exact order-reversing correspondence with the intermediate fields. The automorphism group, Artin's theorem, the characterization of Galois extensions, and the Fundamental Theorem together turn questions about fields into questions about finite groups.\n",{"path":23734,"title":23735,"module":23731,"summary":23736},"\u002Fabstract-algebra\u002Fgalois-theory\u002Ffinite-fields","Finite Fields","Every finite field has prime-power order, is the splitting field of $x^{p^n} - x$, and is unique up to isomorphism. Its extension over the prime field is Galois with cyclic group generated by the Frobenius map $x \\mapsto x^p$, so the Galois correspondence reduces the subfield lattice to the divisor lattice of $n$. Möbius inversion counts the irreducible polynomials of each degree, and cyclic error-correcting codes are one application.\n",{"path":23738,"title":23739,"module":23731,"summary":23740},"\u002Fabstract-algebra\u002Fgalois-theory\u002Fcyclotomic-and-abelian-extensions","Cyclotomic and Abelian Extensions","The Galois group of the $n$th cyclotomic field over $\\mathbb{Q}$ is the unit group $(\\mathbb{Z}\u002Fn\\mathbb{Z})^\\times$, which makes cyclotomic fields the worked catalogue of abelian extensions of $\\mathbb{Q}$. The isomorphism identifies subfields with subgroups, realizes every finite abelian group as a Galois group over $\\mathbb{Q}$, and leads to Kronecker–Weber. Composites of Galois extensions and the primitive element theorem supply the machinery.\n",{"path":23742,"title":23743,"module":23731,"summary":23744},"\u002Fabstract-algebra\u002Fgalois-theory\u002Fgalois-groups-of-polynomials","Galois Groups of Polynomials","Ordering the roots of a separable polynomial embeds its Galois group in the symmetric group $S_n$, and the group is transitive exactly when the polynomial is irreducible. The discriminant decides membership in $A_n$; for cubics and quartics the resolvent cubic pins the group down; and reduction modulo a prime produces elements of prescribed cycle type, the standard tool for computing Galois groups over $\\mathbb{Q}$.\n",{"path":23746,"title":23747,"module":23731,"summary":23748},"\u002Fabstract-algebra\u002Fgalois-theory\u002Fsolvability-by-radicals-and-the-quintic","Solvability by Radicals and the Quintic","A polynomial is solvable by radicals exactly when its Galois group is solvable. Cyclic extensions are radical extensions once roots of unity are present, which turns a radical tower into a solvable subnormal series. Since $S_n$ is solvable only for $n \\le 4$, the general quintic has no radical formula, and an explicit quintic with Galois group $S_5$ has roots provably not expressible in radicals.\n",{"path":23750,"title":23751,"module":23752,"summary":23753},"\u002Fabstract-algebra\u002Fcapstone\u002Fcommutative-algebra-and-algebraic-geometry","A Glimpse of Commutative Algebra and Algebraic Geometry","Capstone: Where Algebra Goes Next","Commutative algebra reads geometry off the ring of polynomial functions. The dictionary runs through Noetherian rings and the ascending chain condition, Hilbert's Basis Theorem, affine algebraic sets and the two maps connecting ideals to zero sets, radicals, the Zariski topology, and Hilbert's Nullstellensatz, which over an algebraically closed field makes radical ideals and algebraic sets the same object.\n",{"path":23755,"title":23756,"module":23752,"summary":23757},"\u002Fabstract-algebra\u002Fcapstone\u002Frepresentation-and-character-theory","A Glimpse of Representation and Character Theory","Representation theory studies a group by the ways it can act linearly on a vector space. Representations are equivalent to modules over the group ring; Maschke's theorem gives complete reducibility, the Wedderburn consequences bound the irreducible degrees, and character theory reduces a representation to a trace invariant governed by the orthogonality relations and displayed in the character table of a small group.\n",{"path":23759,"title":23760,"module":6,"summary":6},"\u002Fabstract-algebra","Abstract Algebra",{"path":23762,"title":23763,"module":23764,"summary":23765},"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fatomic-spectra-rutherford","Atomic Spectra and Rutherford's Nucleus","Early Atomic Models and the Old Quantum Theory","Atoms emit light only at sharp, reproducible wavelengths, and by 1890 those wavelengths were captured by the Rydberg-Ritz formula. Neither empirical regularity had a mechanical explanation. Rutherford's alpha-scattering experiment supplied the missing structure: the atom's positive charge and nearly all its mass sit in a tiny central nucleus, with the electrons far outside.\n",{"path":23767,"title":23768,"module":23764,"summary":23769},"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-model-hydrogen","The Bohr Model of Hydrogen","Bohr grafted three quantum postulates onto Rutherford's nuclear atom: certain orbits do not radiate, radiation accompanies a jump between them, and quantization must match classical physics for large orbits. Quantizing the angular momentum fixes the orbit radii and energies, reproduces the Rydberg-Ritz formula, and predicts the Rydberg constant from fundamental constants alone.\n",{"path":23771,"title":23772,"module":23764,"summary":23773},"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fx-ray-spectra-franck-hertz","X-Ray Spectra and the Franck-Hertz Experiment","Two 1913-14 experiments confirmed the Bohr-Rutherford atom independently of optical spectra. Moseley found that the square root of a characteristic X-ray frequency is linear in atomic number, fixing Z as nuclear charge and ordering the periodic table. Franck and Hertz measured discrete atomic energy levels directly by scattering electrons through a mercury vapor.\n",{"path":23775,"title":23776,"module":23764,"summary":23777},"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-sommerfeld-old-quantum-theory","The Bohr-Sommerfeld Old Quantum Theory","Bohr fixed the hydrogen levels with a single quantum number by quantizing angular momentum. Sommerfeld replaced that ad hoc rule with a general prescription: quantize the action of each separable coordinate. The rule produces elliptical orbits, a second (azimuthal) quantum number, space quantization, and — once the relativistic mass variation is included — a fine-structure splitting that matches experiment to order alpha squared.\n",{"path":23779,"title":23780,"module":23764,"summary":23781},"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fold-quantum-theory-limits-wkb","Limits of the Old Quantum Theory and the WKB Bridge","The old quantum theory works only where the classical motion is separable into independent periodic coordinates. It fails for helium, forbids the correct zero angular momentum of the hydrogen ground state, and misses the half-integer in the oscillator and in molecular spectra. The WKB quantization condition, derived from the Schrodinger equation, is the modern descendant of the Sommerfeld rule and repairs the half-integer through the Maslov correction.\n",{"path":23783,"title":23784,"module":23785,"summary":23786},"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fschrodinger-3d-hydrogen","The Schrödinger Equation in Three Dimensions and Hydrogen","The Quantum Hydrogen Atom","Extending the Schrödinger equation to three dimensions and separating it in spherical coordinates produces three ordinary differential equations, one per coordinate. Their boundary conditions generate the quantum numbers n, ℓ, and mℓ, quantize the angular momentum to √(ℓ(ℓ+1))ℏ with projections mℏ, and fix the bound-state energies of hydrogen at −Z²(13.6 eV)\u002Fn².\n",{"path":23788,"title":23789,"module":23785,"summary":23790},"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fhydrogen-wave-functions","Hydrogen Wave Functions and Orbitals","The hydrogen wave functions factor into a radial part Rₙℓ(r) and an angular spherical harmonic Yℓm(θ,φ). Squaring gives the probability cloud; the radial distribution P(r) = r²|ψ|² peaks at the Bohr radius for the ground state and at the Bohr orbits for excited states. The angular part fixes the s, p, and d orbital shapes that govern chemical bonding.\n",{"path":23792,"title":23793,"module":23785,"summary":23794},"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fradial-equation-in-full","Solving the Radial Equation in Full","The hydrogen radial equation is solved from the differential equation up. The substitution u = rR turns it into a one-dimensional problem with a centrifugal barrier; matching the asymptotic behaviour at the origin and at infinity peels off the factors r^(ℓ+1) and e^(−r\u002Fna₀); a Frobenius series for the remainder must terminate, and that termination condition yields the quantization n ≥ ℓ+1 with E = −Z²Ry\u002Fn². The surviving polynomials are the associated Laguerre functions, whose degree n−ℓ−1 counts the radial nodes.\n",{"path":23796,"title":23797,"module":23785,"summary":23798},"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fsymmetry-degeneracy-runge-lenz","Accidental Degeneracy and the Runge-Lenz Symmetry","Hydrogen energies depend only on n, so states of different ℓ at the same n are degenerate. This is not a coincidence but the mark of a hidden symmetry: the quantum Runge-Lenz vector is conserved for the 1\u002Fr potential alone, and together with angular momentum it generates the group SO(4). The Casimir invariant of that group reproduces E = −Z²Ry\u002Fn² and its representations count the n² states. Any departure from 1\u002Fr breaks the symmetry and lifts the ℓ-degeneracy.\n",{"path":23800,"title":23801,"module":23785,"summary":23802},"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fexpectation-values-virial","Expectation Values, the Virial Theorem, and Scaling","The radial matrix elements ⟨r^k⟩ of hydrogenic states are the raw material of every later correction. This lesson derives ⟨1\u002Fr⟩ from the virial theorem, builds the full family ⟨r⟩, ⟨r²⟩, ⟨1\u002Fr²⟩, ⟨1\u002Fr³⟩ from Kramers' recursion and the Feynman-Hellmann theorem, and reads off their scaling with n, ℓ, and Z. The virial balance ⟨T⟩ = −½⟨V⟩ = −E fixes the energy budget of every bound state.\n",{"path":23804,"title":23805,"module":23785,"summary":23806},"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fquantum-defects-alkali-spectra","Quantum Defects and Alkali Spectra","An alkali atom is one valence electron outside a closed-shell core, and to a good approximation it is hydrogen with a modified quantum number. Core penetration makes low-ℓ states more bound than the Coulomb formula predicts, and the shortfall is captured by a single number per ℓ, the quantum defect δℓ. The spectrum then follows the Rydberg formula with n replaced by the effective n − δℓ, and the sodium D-line doublet is the worked case.\n",{"path":23808,"title":23809,"module":23785,"summary":23810},"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Frydberg-atoms","Rydberg Atoms","A Rydberg atom is an atom excited to a very high principal quantum number, and every hydrogenic property becomes exaggerated by a power of n. Size grows as n², binding falls as n⁻², radiative lifetime lengthens as n³, and the static polarizability explodes as n⁷. The levels crowd toward the ionization limit, and the enormous dipole interaction between two Rydberg atoms produces the blockade that underlies neutral-atom quantum computing.\n",{"path":23812,"title":23813,"module":23814,"summary":23815},"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Frelativistic-kinetic-correction","The Relativistic Kinetic-Energy Correction","Fine Structure and the Dirac Atom","The Bohr energies treat the electron as slowly moving, but its speed is of order αc, so the kinetic energy needs a relativistic correction. Expanding √(p²c²+m²c⁴) to order (v\u002Fc)² produces the perturbation −p⁴\u002F8m³c², whose first-order shift on a hydrogenic state is evaluated with the trick p²=2m(E−V). The result depends on n and ℓ, is smaller than the gross structure by α²≈5×10⁻⁵, and is one of the three pieces that combine into the fine-structure formula.\n",{"path":23817,"title":23818,"module":23814,"summary":23819},"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fspin-orbit-thomas-precession","Spin-Orbit Coupling and Thomas Precession","In the electron's rest frame the nucleus orbits it, and the resulting current produces a magnetic field that couples to the electron's spin moment. The interaction is ξ(r) L·S, with ξ built from the Coulomb potential and the radial expectation ⟨1\u002Fr³⟩. A relativistic subtlety, Thomas precession, halves the naive coefficient because the electron's rest frame is accelerating. The result splits each ℓ≥1 level into a j=ℓ±½ doublet and makes (n, ℓ, j, mⱼ) the good quantum numbers.\n",{"path":23821,"title":23822,"module":23814,"summary":23823},"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdarwin-term-fine-structure-formula","The Darwin Term and the Fine-Structure Formula","The third fine-structure correction, the Darwin term, is a contact interaction proportional to ∇²V that acts only on s-states, physically a smearing of the electron over a Compton wavelength. Adding the relativistic, spin-orbit, and Darwin shifts, the ℓ-dependence cancels and the total collapses to a formula in n and j alone. The n=2 shell splits into 2S₁\u002F₂, 2P₁\u002F₂, 2P₃\u002F₂, with the two j=½ levels exactly degenerate, a coincidence the Dirac theory explains.\n",{"path":23825,"title":23826,"module":23814,"summary":23827},"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdirac-equation-hydrogen","The Dirac Equation for Hydrogen","The fine-structure formula was assembled from three perturbations; the Dirac equation produces it in one stroke and exactly. A first-order relativistic wave equation forces a four-component spinor, from which spin s=½, the g-factor of 2, the spin-orbit term, and antiparticles all emerge automatically. Its exact Coulomb spectrum depends only on n and j, and expanding in Zα reproduces the perturbative result, including the 2S₁\u002F₂–2P₁\u002F₂ degeneracy that sets up the Lamb shift.\n",{"path":23829,"title":23830,"module":23831,"summary":23832},"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Flamb-shift-qed","The Lamb Shift and QED Radiative Corrections","QED Corrections and Hyperfine Structure","The Dirac equation makes the 2S₁\u002F₂ and 2P₁\u002F₂ levels of hydrogen exactly degenerate. Lamb and Retherford measured a splitting of about 1058 MHz that the Dirac theory cannot produce. The gap comes from the electron's coupling to the quantized electromagnetic field: self-energy, vacuum polarization, and the anomalous magnetic moment. Welton's vacuum-fluctuation estimate reproduces the size and shows why the effect lands almost entirely on s-states, and the same radiative corrections make hydrogen the most stringent test of QED.\n",{"path":23834,"title":23835,"module":23831,"summary":23836},"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fhyperfine-structure-21cm","Hyperfine Structure and the 21 cm Line","The proton carries a magnetic moment, and it interacts with the magnetic field the electron produces at the nucleus. For s-states that interaction is the Fermi contact term, proportional to the electron density at the origin and to the dot product of the nuclear and electronic spins. Coupling I and J into F = I + J splits each level by a Landé interval rule; in hydrogen's ground state it produces the F = 0\u002FF = 1 doublet whose 1420 MHz, 21 cm transition maps neutral hydrogen across the galaxy.\n",{"path":23838,"title":23839,"module":23831,"summary":23840},"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fnuclear-effects-isotope-shift","Nuclear Size, Moments, and Isotope Shifts","A real nucleus has a finite size, a mass that changes between isotopes, and, when its spin is at least one, an electric quadrupole moment. Each leaves a fingerprint in the atomic spectrum: the volume shift from s-electrons sampling the charge distribution, the mass and field isotope shifts that separate on a King plot, the quadrupole interaction that breaks the Landé interval rule, and the hyperfine anomaly from the magnetization distribution. Atomic spectroscopy reads nuclear properties out of these shifts.\n",{"path":23842,"title":23843,"module":23844,"summary":23845},"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fperiodic-table-atomic-spectra","The Periodic Table and Atomic Spectra","Many-Electron Atoms","Identical electrons demand antisymmetric wave functions, which is the Pauli exclusion principle: no two electrons share all four quantum numbers. Filling shells in order of increasing energy — shifted by penetration and shielding — builds the periodic table and its recurring ionization pattern. Selection rules govern optical spectra, and an external field splits lines by the Zeeman effect.\n",{"path":23847,"title":23848,"module":23844,"summary":23849},"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fcentral-field-self-consistent","The Central-Field Approximation and the Self-Consistent Field","The N-electron Hamiltonian does not separate because every pair of electrons repels. The central-field approximation replaces that pairwise repulsion with an averaged spherical potential each electron feels, restoring hydrogen-like orbitals labelled by n and ℓ. The Thomas-Fermi statistical model fixes the shape of the screened charge from Fermi-gas thermodynamics; the Hartree self-consistent field determines it exactly by iterating orbitals against the potential they generate until the two agree.\n",{"path":23851,"title":23852,"module":23844,"summary":23853},"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fidentical-particles-hartree-fock","Exchange, Slater Determinants, and Hartree-Fock","A product wave function ignores that electrons are identical fermions. Enforcing antisymmetry writes the state as a Slater determinant, which vanishes whenever two electrons share a spin-orbital — the exclusion principle made algebraic. The energy of a determinant carries a new term with no classical analogue, the exchange integral, nonzero only for parallel spins; it lowers the energy of aligned electrons and carves a Fermi hole around each one. Adding the exchange operator to the mean field gives the Hartree-Fock equations, and what they still miss defines the correlation energy.\n",{"path":23855,"title":23856,"module":23844,"summary":23857},"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhelium-two-electron-atom","Helium: the Prototype Two-Electron Atom","Helium is the smallest atom the Schrödinger equation cannot solve exactly, and the smallest that shows every many-electron effect. Ignoring the electron repulsion overbinds the ground state by 30 eV; first-order perturbation theory and a one-parameter variational calculation with an effective charge close most of the gap. The excited configurations split into para (singlet) and ortho (triplet) states separated by the exchange integral, with the triplet lower — and the absence of a 1s² triplet is the Pauli principle in its plainest form.\n",{"path":23859,"title":23860,"module":23844,"summary":23861},"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fls-jj-coupling-term-symbols","LS and jj Coupling; Term Symbols","A configuration is not a single energy level. The residual electrostatic repulsion and the spin-orbit interaction split it, and which one dominates fixes the coupling scheme. In light atoms the electrostatic term wins: orbital and spin angular momenta couple separately into L and S, then into J, giving Russell- Saunders term symbols. In heavy atoms spin-orbit wins and each electron's j forms first. The Pauli principle prunes the allowed terms of equivalent electrons, the Landé interval rule spaces the fine-structure multiplet, and the scheme crosses over from LS to jj down a column.\n",{"path":23863,"title":23864,"module":23844,"summary":23865},"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhund-rules-ground-terms","Hund's Rules and Ground-State Terms","A configuration allows several terms; Hund's three rules pick the ground one. Maximize the spin S first, then the orbital L, then set J to |L−S| for a less-than-half shell and L+S for a more-than-half shell. The first two rules come from exchange lowering the energy of apart-kept electrons; the third comes from the sign of the spin-orbit coupling, which flips as a shell passes half-filling and turns the multiplet from normal to inverted. Worked ground terms for carbon, nitrogen, oxygen, and iron show the rules in action.\n",{"path":23867,"title":23868,"module":23869,"summary":23870},"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fzeeman-effect","The Zeeman Effect","Atoms in External Fields","A magnetic field couples to the atom through its magnetic moment, splitting each level into equally spaced sublevels labelled by the projection of the total angular momentum. When spin is present the spacing is not the classical one: it carries the Landé g-factor, a projection of the spin and orbital moments onto the total angular momentum. We derive the weak-field Hamiltonian from minimal coupling, evaluate the shift with the projection theorem, and read off the polarization of the emitted components.\n",{"path":23872,"title":23873,"module":23869,"summary":23874},"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fpaschen-back-intermediate","The Paschen-Back and Intermediate-Field Regimes","When the magnetic interaction grows past the fine-structure coupling, spin and orbital angular momentum decouple and precess independently about the field. The anomalous Zeeman pattern reverts to a simple triplet, the Paschen-Back effect. Between the two limits neither coupling dominates and the level positions follow from diagonalizing the combined spin-orbit and Zeeman Hamiltonian. We build the two-by-two problem for a single valence electron, solve it in closed form, and show both limits emerge from one expression.\n",{"path":23876,"title":23877,"module":23869,"summary":23878},"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fstark-effect-polarizability","The Stark Effect and Field Ionization","An electric field shifts atomic levels by coupling to the electron's position. Parity forbids a first-order shift for a non-degenerate state, so most atoms respond only at second order through their polarizability, a quadratic Stark shift. Hydrogen is the exception: its accidental degeneracy admits a permanent dipole and a linear shift, cleanest in parabolic coordinates. At large fields the Coulomb well develops a saddle, and Rydberg states field-ionize at a threshold that falls as the fourth power of the principal quantum number.\n",{"path":23880,"title":23881,"module":23882,"summary":23883},"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Ftime-dependent-perturbation-golden-rule","Time-Dependent Perturbation Theory and the Golden Rule","Radiative Transitions and Spectral Lines","An atom in a weak oscillating field makes transitions between its stationary states. First-order time-dependent perturbation theory gives the transition amplitude as a Fourier component of the perturbation at the Bohr frequency, and the resulting probability is a sinc-squared resonance that sharpens as the field acts longer. For a two-level system the same coupling produces Rabi oscillations; for a transition into a continuum the long-time limit collapses the sinc-squared into a delta function and yields Fermi's golden rule, a constant transition rate set by the coupling strength and the density of final states.\n",{"path":23885,"title":23886,"module":23882,"summary":23887},"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fdipole-approximation-einstein-coefficients","The Dipole Approximation and Einstein Coefficients","The coupling between an atom and light is the interaction of the electron with the electromagnetic field. Because an optical wavelength dwarfs the atom, the spatial variation of the field across the atom can be dropped, leaving the electric-dipole interaction and its matrix element. That matrix element defines the oscillator strength, which obeys the Thomas-Reiche-Kuhn sum rule. Einstein's three rate coefficients (absorption, stimulated emission, spontaneous emission) follow from detailed balance with thermal radiation, fixing the ratio of spontaneous to stimulated rates and its steep growth with frequency.\n",{"path":23889,"title":23890,"module":23882,"summary":23891},"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fselection-rules-forbidden-transitions","Selection Rules and Forbidden Transitions","The dipole matrix element vanishes for most pairs of states, and the pattern of which survive is the set of selection rules. Parity forces the orbital angular momentum to change by one; the angular integral of three spherical harmonics restricts the magnetic quantum number to change by zero or one; the photon's spin restricts the total angular momentum. When the dipole element vanishes, higher multipoles (magnetic dipole and electric quadrupole) can still drive the transition at rates smaller by powers of the fine-structure constant, and states with no allowed decay become metastable.\n",{"path":23893,"title":23894,"module":23882,"summary":23895},"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Flifetimes-and-line-shapes","Lifetimes, Line Widths, and Line Shapes","A spectral line is never infinitely sharp. The finite lifetime of the excited state gives every line a natural Lorentzian width set by the total decay rate, the Fourier transform of an exponentially damped emission. Thermal motion adds a Gaussian Doppler width that usually dominates in a gas; collisions add a further Lorentzian pressure width; the observed profile is the Voigt convolution of the Gaussian and Lorentzian parts. Strong driving fields broaden the line further through saturation. Each mechanism has a distinct dependence on temperature, density, and intensity that lets it be identified and, where possible, removed.\n",{"path":23897,"title":23898,"module":23899,"summary":23900},"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Flaser-principles","Population Inversion, Gain, and the Laser","Lasers and Spectroscopy","A laser is an optical amplifier placed inside a resonant cavity. Amplification requires that stimulated emission outrun absorption, which requires more atoms in the upper level than the lower one — a population inversion that the Einstein relations forbid in thermal equilibrium and that no two-level pump can produce. Three- and four-level schemes reach it by routing atoms through auxiliary states. The gain coefficient sets how strongly a weak beam grows, the cavity fixes the threshold and selects a comb of longitudinal modes, and gain saturation clamps the steady-state inversion at its threshold value.\n",{"path":23902,"title":23903,"module":23899,"summary":23904},"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fspectroscopy-techniques","Spectroscopic Techniques and Frequency Combs","A tunable laser turns spectroscopy from photographing a spectrum into interrogating a single transition, but at room temperature the Doppler width buries the natural linewidth under a thousandfold-broader Gaussian. Saturated absorption and two-photon spectroscopy defeat the first-order Doppler shift by selecting the zero-velocity class or cancelling the shift between counter-propagating photons, recovering natural-width features. Laser-induced fluorescence pushes sensitivity to single atoms, and the optical frequency comb converts an optical frequency into a countable radio-frequency beat, giving absolute frequency measurement across the visible spectrum.\n",{"path":23906,"title":23907,"module":23899,"summary":23908},"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fline-catalog-nist-asd","Reading Real Spectra with the NIST Database","Every quantity computed in this course — energy levels, transition frequencies, oscillator strengths, lifetimes — is tabulated for real atoms in the NIST Atomic Spectra Database. This lesson reads that data as physics: how levels are labelled by term symbols and energies in wavenumbers, how a transition list encodes wavelength, Einstein coefficient, and line strength, how a Grotrian diagram is reconstructed from the tables, and how a measured spectrum is matched to catalog lines. The residual between computed and tabulated positions is the running score of atomic theory.\n",{"path":23910,"title":23911,"module":23912,"summary":23913},"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Flaser-cooling-doppler","Laser Cooling and Optical Molasses","Modern Atomic Physics","A near-resonant laser beam pushes an atom because every absorbed photon delivers one unit of momentum and the subsequent spontaneous emission averages to zero. Two counter-propagating red-detuned beams turn that push into friction: the Doppler shift brings a moving atom closer to resonance with the beam it moves against, so the net force opposes the velocity. Six beams give optical molasses in three dimensions. The random recoil of spontaneous emission heats against the friction, and the balance sets the Doppler cooling limit. Adding a magnetic-field gradient makes the force position-dependent as well, giving the magneto-optical trap.\n",{"path":23915,"title":23916,"module":23912,"summary":23917},"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fsub-doppler-trapping","Sub-Doppler Cooling and Atom Traps","Optical molasses cools multilevel atoms below the Doppler limit. A polarization gradient plus optical pumping makes an atom repeatedly climb a light-shift hill and be pumped to the valley, losing kinetic energy each cycle — Sisyphus cooling. The floor is the recoil limit, one photon momentum of residual motion. Below it, cooling must avoid scattering photons: conservative magnetic and optical-dipole traps hold the atoms while forced evaporation removes the hot tail, driving the phase-space density up toward quantum degeneracy.\n",{"path":23919,"title":23920,"module":23912,"summary":23921},"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fbose-einstein-condensation","Bose-Einstein Condensation of Atomic Gases","Below a critical temperature a gas of identical bosons places a macroscopic fraction of its atoms in the single lowest-energy state. The transition occurs when the thermal de Broglie wavelength grows to the interparticle spacing, so the phase- space density reaches order unity. The critical temperature follows from the Bose-Einstein distribution and the density of states, the condensate fraction grows as one minus (T\u002FTc) to the three-halves, and the condensate reveals itself in time-of-flight as a sharp bimodal peak in the momentum distribution. The 1995 rubidium and sodium experiments realized it in dilute trapped gases.\n",{"path":23923,"title":23924,"module":23912,"summary":23925},"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Foptical-clocks-precision","Optical Atomic Clocks and Precision Measurement","An atomic clock counts the oscillations of a field locked to an atomic transition. The cesium microwave standard defines the second through the 9.19 GHz ground-state hyperfine transition, interrogated by Ramsey's separated-oscillatory-field method whose fringe width is set by the free-precession time. Optical clocks replace the microwave transition with an optical one five orders of magnitude higher in frequency, raising the quality factor and the fractional stability in proportion. Lattice and single-ion clocks reach fractional uncertainties near ten-to-the-minus- eighteen by trapping the atoms at a magic wavelength that cancels the light shift, and at that level they measure the gravitational redshift over centimetres of height.\n",{"path":23927,"title":23928,"module":6,"summary":6},"\u002Fatomic-physics","Atomic Physics",{"path":23930,"title":23931,"module":6,"summary":6},"\u002Fdatabases","Databases",{"path":23933,"title":23934,"module":21519,"summary":23935},"\u002Fcategory-theory\u002Ffoundations\u002Fwhat-is-a-category","Categories, Objects, and Arrows","A category is objects, arrows between them, a rule for composing arrows, and an identity arrow on every object, subject to associativity and the unit laws. The axioms mention no elements: arrows need not be functions, and an object is known only through the arrows into and out of it. Isomorphism, commutative diagrams, duality, and the terminal object are the first consequences.\n",{"path":23937,"title":23938,"module":21519,"summary":23939},"\u002Fcategory-theory\u002Ffoundations\u002Fexamples-of-categories","A Zoo of Categories","The axioms admit two very different kinds of model: large categories of structured sets and their structure-preserving maps (Set, Mon, Grp, Top, Vect), and small categories that are themselves single algebraic objects — a monoid as a one-object category, a poset as a thin category. The awkward cases Rel and Pfn have sets as objects but relations and partial functions as arrows, and a typed programming language presents its types and programs as a category.\n",{"path":23941,"title":23942,"module":21519,"summary":23943},"\u002Fcategory-theory\u002Ffoundations\u002Fspecial-morphisms","Isomorphisms, Monos, and Epis","Injectivity and surjectivity mention elements, so a general category re-expresses them by cancellation: monomorphisms cancel on the left, epimorphisms on the right. Sections and retractions are the split versions with an explicit one-sided inverse. Mono plus epi does not force an isomorphism, and subobjects are equivalence classes of monos into a fixed object.\n",{"path":23945,"title":23946,"module":21519,"summary":23947},"\u002Fcategory-theory\u002Ffoundations\u002Ffunctors","Functors: Maps Between Categories","A functor sends objects to objects and arrows to arrows while preserving composition and identities. Covariant and contravariant functors, the standard stock (forgetful, free, hom, and powerset), and the classification by faithfulness, fullness, and essential surjectivity all follow. Functors compose, so categories and functors form a category themselves.\n",{"path":23949,"title":23950,"module":21519,"summary":23951},"\u002Fcategory-theory\u002Ffoundations\u002Fnatural-transformations","Natural Transformations and Functor Categories","A natural transformation is a map between two parallel functors: one component arrow per object, subject to a commuting square for every arrow of the source. Naturality is verified for the determinant, the double dual, and list operations; functors and natural transformations form the functor category [C, D]; and vertical and horizontal composition satisfy the Godement interchange law.\n",{"path":23953,"title":23954,"module":21519,"summary":23955},"\u002Fcategory-theory\u002Ffoundations\u002Fsize-and-set-theory","Size: Small, Large, Locally Small","The objects of Set do not form a set, and pretending otherwise reproduces the classical paradoxes. Classes make the small\u002Flarge distinction precise, with locally small and essentially small as the intermediate notions. Cantor's theorem shows Set and its algebraic relatives are large, and the function-based axiomatization of sets is the one category theory prefers to ZFC.\n",{"path":23957,"title":23958,"module":23959,"summary":23960},"\u002Fcategory-theory\u002Funiversal-properties\u002Funiversal-properties","Universal Properties, Initial and Terminal Objects","Universal Properties and Basic Constructions","A universal property characterizes an object by a for-all\u002Fexists-unique condition on the arrows into or out of it, and any two objects satisfying the same property are isomorphic by a unique isomorphism. Initial and terminal objects are the simplest cases; the free vector space, the discrete topology, and the ring of integers show the pattern at work.\n",{"path":23962,"title":23963,"module":23959,"summary":23964},"\u002Fcategory-theory\u002Funiversal-properties\u002Fproducts-and-coproducts","Products and Coproducts","The product of two objects is a wedge of projections through which every other wedge factors uniquely; the coproduct is the dual, built from injections. In Set these are the cartesian product and the disjoint union, in a poset the meet and join, and in abelian groups the two coincide. The mediating-arrow discipline established here is the template for all limits.\n",{"path":23966,"title":23967,"module":23959,"summary":23968},"\u002Fcategory-theory\u002Funiversal-properties\u002Fconstructions-on-categories","Opposite, Product, Slice, and Comma Categories","Categories are themselves mathematical structures, and the standard algebraic constructions apply: opposites, products, subcategories, slices, and the comma category that subsumes them. The opposite category yields the duality principle, halving the subject's proofs; slice and comma categories repackage every universal property as an initial or terminal object.\n",{"path":23970,"title":23971,"module":23972,"summary":23973},"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Frepresentable-functors","Hom-Functors and Representables","Representables and the Yoneda Lemma","Fixing an object A of a locally small category produces a set-valued functor, the hom-functor A(A,-), that records every map out of A. A functor is representable when it is naturally isomorphic to such a hom-functor. We define the covariant and contravariant hom-functors, collect the standard representables (identity, forgetful, powerset), and read maps as generalized elements of varying shape.\n",{"path":23975,"title":23976,"module":23972,"summary":23977},"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-lemma","The Yoneda Lemma","The Yoneda lemma computes the natural transformations out of a representable presheaf: they form a set in natural bijection with X(A). The proof fixes a single degree of freedom, the image of the identity arrow, and shows naturality forces everything else. We prove the bijection, verify naturality in both variables, and read off that a natural transformation out of a representable is just one element.\n",{"path":23979,"title":23980,"module":23972,"summary":23981},"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-consequences","The Yoneda Embedding and Its Uses","Three corollaries turn the Yoneda lemma into working machinery. A representation of a presheaf is the same thing as a universal element; the Yoneda embedding of a category into its presheaf category is full and faithful; and two objects are isomorphic exactly when their representables are. Together they justify constructing arrows by constructing natural transformations between hom-functors, and they contain Cayley's theorem as the one-object case.\n",{"path":23983,"title":23984,"module":23985,"summary":23986},"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits","Cones and Limits","Limits and Colimits","A diagram is a functor from a small shape category; a cone over it is an object with compatible legs to every node; and a limit is the terminal cone, the one every other cone factors through uniquely. Products and terminal objects reappear as limits over particular shapes, and the whole construction is unique up to a single isomorphism.\n",{"path":23988,"title":23989,"module":23985,"summary":23990},"\u002Fcategory-theory\u002Flimits-colimits\u002Fproducts-equalizers-pullbacks","Equalizers and Pullbacks","The equalizer of a parallel pair is the universal arrow that makes the two composites agree; the pullback of a cospan is the universal commutative square. In Set they are solution sets and fibered products, every equalizer is monic, monics are stable under pullback, and products plus equalizers together generate all limits.\n",{"path":23992,"title":23993,"module":23985,"summary":23994},"\u002Fcategory-theory\u002Flimits-colimits\u002Fcolimits","Colimits: Coproducts, Coequalizers, Pushouts","Colimits are limits in the opposite category: cocones replace cones, and the universal cocone is initial rather than terminal. Coproducts glue objects side by side, coequalizers impose relations and produce quotients, pushouts glue along a shared part, and in Set every colimit is a quotient of a disjoint union. Directed colimits admit a clean elementwise description.\n",{"path":23996,"title":23997,"module":23985,"summary":23998},"\u002Fcategory-theory\u002Flimits-colimits\u002Fcomputing-limits","Computing Limits in Concrete Categories","In Set the limit of any diagram is the set of threads: choice functions through the nodes that commute with every edge. In Pos, Mon, and Top the recipe is the same limit downstairs plus the unique structure that makes the projections structure-preserving — pointwise order, componentwise operations, the topology generated by the projections. The pattern is what \"the forgetful functor creates limits\" means concretely.\n",{"path":24000,"title":24001,"module":23985,"summary":24002},"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits-and-functors","Preservation, Reflection, and Creation of Limits","A functor preserves limits if it sends limit cones to limit cones, reflects them if it recognizes them, and creates them if limits downstairs lift uniquely upstairs. Representable functors preserve all limits, forgetful functors from algebra create them, and limits in functor categories are computed pointwise, one evaluation at a time.\n",{"path":24004,"title":24005,"module":24006,"summary":24007},"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions","Adjoint Functors via Hom-Set Bijections","Adjunctions","An adjunction is a natural bijection between two hom-sets: maps out of $F(A)$ in one category correspond to maps into $G(B)$ in the other. We give the definition, spell out the naturality axioms that make the correspondence compatible with composition, and work the flagship examples — free vector spaces, free groups, discrete and indiscrete topologies, and currying.\n",{"path":24009,"title":24010,"module":24006,"summary":24011},"\u002Fcategory-theory\u002Fadjunctions\u002Funits-and-counits","Units, Counits, and the Triangle Identities","The whole hom-set bijection of an adjunction is generated by two natural transformations: the unit, obtained by transposing identity maps on one side, and the counit, by transposing them on the other. Two triangle identities are all they must satisfy, and any pair satisfying them determines a unique adjunction. The same correspondence specializes to order-preserving maps between posets and to free constructions.\n",{"path":24013,"title":24014,"module":24006,"summary":24015},"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions-via-universal-arrows","Adjunctions from Universal Arrows","The unit component at a single object is an initial object of a comma category, and this universal property alone rebuilds the whole adjunction. A functor has a left adjoint exactly when every object admits such a universal arrow, and the left adjoint is assembled from them one object at a time. We prove the equivalence of all three formulations of adjointness.\n",{"path":24017,"title":24018,"module":24006,"summary":24019},"\u002Fcategory-theory\u002Fadjunctions\u002Ffree-forgetful-adjunctions","Free Constructions and Free–Forgetful Adjunctions","Free monoids, free groups, and free vector spaces are left adjoints to forgetful functors, and the universal mapping property is all one needs to prove it. Some forgetful functors also have right adjoints (co-free constructions like the indiscrete topology), producing three-functor chains. Contravariant adjunctions, symmetric in their two functors, close the lesson with the pattern behind duality and representation theorems.\n",{"path":24021,"title":24022,"module":24023,"summary":24024},"\u002Fcategory-theory\u002Fadjoints-limits\u002Flimits-via-adjoints","Limits as Adjoints and as Representables","Adjoints, Representables, and Limits Together","A cone on a diagram is a natural transformation from a constant diagram, so a limit is a representation of the cone functor and, equivalently, a value of the right adjoint to the diagonal functor. We prove both rephrasings, derive uniqueness and functoriality of limits from them, and record the dual statement that a colimit is the left adjoint to the diagonal.\n",{"path":24026,"title":24027,"module":24023,"summary":24028},"\u002Fcategory-theory\u002Fadjoints-limits\u002Fpresheaf-limits-colimits","Limits and Colimits of Presheaves","Representables preserve limits, and limits in a functor category are computed one object at a time, so a presheaf category is complete and cocomplete with all its structure inherited pointwise from Set. The Yoneda embedding then preserves limits but not colimits, and the density theorem repairs the colimit side: every presheaf is a canonical colimit of representables.\n",{"path":24030,"title":24031,"module":24023,"summary":24032},"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoints-preserve-limits","Right Adjoints Preserve Limits (RAPL)","A functor with a left adjoint preserves every limit that exists, and dually a functor with a right adjoint preserves colimits. The proof is a four-line chain of natural isomorphisms through the adjunction and the continuity of representables. The theorem yields product-and-exponential arithmetic in Set, another proof that limits commute with limits, and a standard test for proving that a functor has no adjoint.\n",{"path":24034,"title":24035,"module":24023,"summary":24036},"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoint-functor-theorem","The Adjoint Functor Theorem","RAPL makes limit preservation necessary for having a left adjoint; the adjoint functor theorems identify when it is sufficient. For ordered sets no extra hypothesis is needed. In general the candidate adjoint is a limit over a comma category that may be large, and the general adjoint functor theorem tames it with a weakly initial set. We prove GAFT in full and apply it to free groups and, through the special adjoint functor theorem, the Stone–Čech compactification.\n",{"path":24038,"title":24039,"module":24040,"summary":24041},"\u002Fcategory-theory\u002Fmonads-algebras\u002Fmonads","Monads from Adjunctions","Monads and Algebras","A monad on a category is an endofunctor equipped with a unit and a multiplication satisfying associativity and unit laws — the data of a monoid, written internally to the category of endofunctors. Every adjunction induces one, and the list, exception, and state constructions that model computational effects are all monads on Set.\n",{"path":24043,"title":24044,"module":24040,"summary":24045},"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-eilenberg-moore","Algebras for a Monad","An algebra for a monad is an object with a structure map that interacts correctly with the unit and multiplication. The algebras form the Eilenberg–Moore category, whose free–forgetful adjunction induces the monad back; a comparison functor relates any other inducing adjunction to it, and for the list monad the algebras are exactly monoids.\n",{"path":24047,"title":24048,"module":24040,"summary":24049},"\u002Fcategory-theory\u002Fmonads-algebras\u002Fkleisli-and-programming","The Kleisli Category and Monads in Programming","The Kleisli category of a monad has the same objects as the base but takes arrows A to TB, composed by mapping and flattening. These arrows are effectful programs, Kleisli composition is the bind of functional programming, and the Kleisli adjunction is the initial resolution of the monad, with Eilenberg–Moore at the terminal end.\n",{"path":24051,"title":24052,"module":24040,"summary":24053},"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-for-endofunctors","Algebras for an Endofunctor and Recursion","Dropping the monad laws leaves algebras for a bare endofunctor, whose initial objects are the least fixed points of the functor by Lambek's lemma. The natural numbers, lists, and trees are initial algebras; the unique map out of an initial algebra is the fold of functional programming; and the Smyth–Plotkin fixed-point technique builds Scott domains the same way.\n",{"path":24055,"title":24056,"module":24057,"summary":24058},"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Fcartesian-closed-categories","Cartesian Closed Categories","Cartesian Closed Categories and Typed Lambda Calculus","A cartesian closed category has a terminal object, binary products, and for every pair of objects an exponential object that internalizes the hom-set as an object of the category. The defining data is an evaluation arrow and a currying operation, packaged by the adjunction between product-with-A and exponential-by-A. Set, Boolean and Heyting algebras, functor categories, and Cat are all cartesian closed.\n",{"path":24060,"title":24061,"module":24057,"summary":24062},"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Flambda-calculus-correspondence","Typed Lambda Calculus and CCCs","The typed lambda calculus and the cartesian closed category are two presentations of the same theory. Types become objects, terms with one free variable become arrows, product types become products, and function types become exponentials, with abstraction matching currying and application matching evaluation. Building the category of a lambda theory and the internal language of a category are mutually inverse up to equivalence.\n",{"path":24064,"title":24065,"module":24057,"summary":24066},"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Ffixed-points-and-recursion","Fixed Points in Cartesian Closed Categories","The untyped lambda calculus has a fixed-point combinator; the typed calculus cannot, and Lawvere's fixed-point theorem explains why: any point-surjection onto an exponential forces every endomap to have a fixed point, which is the abstract form of Cantor's diagonal argument. Recursion is recovered instead by restricting to omega-complete partially ordered objects, where every continuous endomap has a least fixed point built by iterating from bottom. This gives While loops a semantics.\n",{"path":24068,"title":24069,"module":6,"summary":6},"\u002Fcategory-theory","Category Theory",{"path":24071,"title":22713,"module":24072,"summary":24073},"\u002Fdeep-learning\u002Fmathematical-background\u002Flinear-algebra-for-deep-learning","Mathematical Background","Every quantity a network touches is a tensor, and every layer is a matrix acting on one. This lesson compiles the linear algebra deep learning actually uses: products and norms, the system $Ax=b$ and when it is solvable, the two decompositions (eigen and SVD) that diagonalize a transformation, and the pseudoinverse that solves what cannot be solved exactly. It then derives PCA as the worked example that ties it all together.\n",{"path":24075,"title":24076,"module":24072,"summary":24077},"\u002Fdeep-learning\u002Fmathematical-background\u002Fprobability-and-information-theory","Probability & Information Theory","This lesson assembles the probabilistic vocabulary a network is trained in (random variables, densities, the chain rule, expectation and covariance, the handful of distributions that recur everywhere) and then the information theory that turns a probabilistic model into a loss: self-information, entropy, and the KL divergence whose asymmetry is the cross-entropy objective itself.\n",{"path":24079,"title":24080,"module":24072,"summary":24081},"\u002Fdeep-learning\u002Fmathematical-background\u002Fnumerical-computation","Numerical Computation","Machine learning runs on finite-precision arithmetic, where every number is approximated and every operation rounds. This lesson sets the numerical ground rules: overflow and underflow and the standard stabilizations, the condition number that measures how much a problem amplifies error, and the gradient-based optimization (first and second order, constrained and unconstrained) that every training loop runs.\n",{"path":24083,"title":22072,"module":24072,"summary":24084},"\u002Fdeep-learning\u002Fmathematical-background\u002Fcalculus","This lesson assembles the differential calculus used in training networks: the gradient and directional derivative, the Jacobian and Hessian, and the chain rule in scalar, vector, and matrix form. From the chain rule it derives back-propagation as a single sweep over the computational graph, tabulates the matrix-calculus identities that recur in layer gradients, reads optimization off a second-order Taylor expansion, and ends with why reverse-mode automatic differentiation is the algorithm every framework runs.\n",{"path":24086,"title":24087,"module":21519,"summary":24088},"\u002Fdeep-learning\u002Ffoundations\u002Fwhat-is-deep-learning","What Is Deep Learning?","Deep learning is representation learning by composition: stack simple differentiable layers, define a loss, and let gradient descent discover the features a human would otherwise have to engineer by hand. We set up the whole vocabulary (model, loss, optimizer, data), the training loop that ties them together, and the three reasons the approach became practical.\n",{"path":24090,"title":24091,"module":21519,"summary":24092},"\u002Fdeep-learning\u002Ffoundations\u002Fmachine-learning-refresher","A Machine-Learning Refresher","The statistical framework the networks live in: data drawn from an unknown distribution, a loss to minimize, and the central question of generalization: will it work on data we have not seen? We set up empirical risk, capacity, the bias–variance tradeoff, and maximum likelihood.\n",{"path":24094,"title":24095,"module":21519,"summary":24096},"\u002Fdeep-learning\u002Ffoundations\u002Flinear-models-and-the-perceptron","Linear Models & the Perceptron","The simplest learners (linear regression, logistic regression, the perceptron) already contain the whole template: a weighted sum, a loss, a gradient step. They also fail on the XOR problem, which no linear model can solve — the limitation that motivates deep learning.\n",{"path":24098,"title":24099,"module":24100,"summary":24101},"\u002Fdeep-learning\u002Fneural-networks\u002Fthe-multilayer-perceptron","The Multilayer Perceptron","Neural Networks","Stacking linear layers with a nonlinearity between them removes the limitation that stopped the perceptron. We build the multilayer perceptron in explicit matrix form (the forward pass, its dimensions, a worked XOR network with concrete weights) and prove why the nonlinearity is essential: without it the deepest stack collapses to a single hyperplane.\n",{"path":24103,"title":24104,"module":24100,"summary":24105},"\u002Fdeep-learning\u002Fneural-networks\u002Factivation-functions","Activation Functions","The activation is the only nonlinear part of a layer, and the reason depth adds expressive power. We catalog the standard hidden units (sigmoid, tanh, ReLU and its descendants, plus GELU, softplus, swish and maxout), derive each unit's derivative in full, make the vanishing-gradient problem quantitative with the chain-rule product, work numeric examples, and explain why the saturating units gave way to ReLU and why ReLU's own dead-unit failure gave way to Leaky\u002FPReLU\u002FELU\u002FGELU.\n",{"path":24107,"title":24108,"module":24100,"summary":24109},"\u002Fdeep-learning\u002Fneural-networks\u002Funiversal-approximation","Universal Approximation","One hidden layer with a non-polynomial activation can approximate any continuous function on a compact set to arbitrary accuracy: the universal approximation theorem. We prove it constructively (two sigmoids make a bump; sums of bumps make any curve), then show the limitation: existence is not efficiency. Depth-separation results exhibit functions a deep net represents with $O(n)$ units that a shallow net needs $\\exp(n)$ units to match.\n",{"path":16668,"title":24111,"module":24100,"summary":24112},"Backpropagation","Backpropagation is the chain rule run backward over a computational graph. We formalize the graph, derive the four backprop equations for an MLP, present the forward and backward passes as algorithms, and work a tiny two-layer net by hand with explicit numbers. The result: one scalar loss, reverse-mode autodiff, and a gradient for every parameter at twice the cost of a forward pass.\n",{"path":24114,"title":24115,"module":24100,"summary":24116},"\u002Fdeep-learning\u002Fneural-networks\u002Floss-functions-and-output-units","Loss Functions & Output Units","The last layer is where a network's hidden representation meets the task. Choosing an output unit and a loss is not two independent choices; maximum likelihood fixes the pair. We derive the standard couplings (linear\u002FMSE, sigmoid\u002FBCE, softmax\u002Fcross-entropy), show why softmax and cross-entropy were built to cancel into the residual $\\hat y - y$, and prove why squared error is the wrong loss for a saturating classifier.\n",{"path":17,"title":24118,"module":20579,"summary":24119},"Gradient Descent & SGD","Training is descent on the empirical risk: step the parameters against the gradient. We derive the minibatch gradient as an unbiased estimator whose variance falls as $1\u002FB$, derive the learning-rate ceiling from the smoothness-stability bound $\\eta \u003C 2\u002FL$, and lay out the schedules (step, exponential, cosine, warmup) that anneal it over training.\n",{"path":22,"title":24121,"module":20579,"summary":24122},"Momentum & Adaptive Methods","Plain gradient descent zig-zags across ravines and moves slowly along flat valleys, because one global learning rate cannot suit a surface with wildly different curvature in different directions. Two fixes address the two problems: momentum accumulates a velocity that damps the oscillation and accelerates the drift, and adaptive methods give every parameter its own learning rate scaled by the history of its gradients. Adam fuses both, and is the default optimizer of modern deep learning.\n",{"path":24124,"title":24125,"module":20579,"summary":24126},"\u002Fdeep-learning\u002Foptimization\u002Finitialization","Weight Initialization","The initial weights determine whether training can succeed before the first gradient step. Initialize every weight equal and all hidden units compute the same function forever; initialize too small or too large and the signal vanishes or explodes as it crosses depth. A single variance condition, $n_{\\text{in}}\\mathrm{Var}(W)=1$, fixes both, and reading it off the forward and backward passes yields Xavier and He initialization directly.\n",{"path":257,"title":24128,"module":20579,"summary":24129},"The Optimization Landscape","The loss of a deep network is a non-convex surface in millions of dimensions, so local search carries no global guarantee, yet it works. We classify critical points by the eigenvalues of the Hessian, show that in high dimension nearly all of them are saddle points rather than bad local minima, and read off the practical terrain — plateaus, cliffs, ill-conditioning, and the sharp-versus-flat distinction that ties the geometry of a minimum to how well it generalizes.\n",{"path":20581,"title":5,"module":20579,"summary":20600},{"path":24132,"title":24133,"module":24134,"summary":24135},"\u002Fdeep-learning\u002Fregularization\u002Fregularization-overview","Regularization Overview","Regularization","Regularization is any modification to a learning algorithm meant to lower test error at the possible expense of training error. We derive the bias–variance decomposition that explains why it helps, set up the two parameter-norm penalties, $L^2$ weight decay and $L^1$, derive their update rules and eigenbasis shrinkage, show geometrically why $L^1$ alone produces sparse weights (soft-thresholding), distinguish weight decay from loss-added $L^2$ under AdamW, and read both penalties through the two lenses that recur across the chapter: a norm-ball constraint via KKT, and a prior via MAP estimation.\n",{"path":24137,"title":24138,"module":24134,"summary":24139},"\u002Fdeep-learning\u002Fregularization\u002Fdropout-and-data-augmentation","Dropout & Data Augmentation","Two of the most effective regularizers add no penalty term at all; they perturb the computation instead. Dropout multiplies hidden units by a random Bernoulli mask, training an exponential ensemble of thinned subnetworks that share weights; inverted scaling collapses that ensemble into one cheap forward pass at test time. Data augmentation enlarges the training set with label-preserving transforms, injecting the invariances the task demands, and noise injection (input, weight, label smoothing, Mixup) generalizes the same idea into a continuous family.\n",{"path":24141,"title":24142,"module":24134,"summary":24143},"\u002Fdeep-learning\u002Fregularization\u002Fearly-stopping-and-parameter-sharing","Early Stopping & Parameter Sharing","Two cheap regularizers that cost no extra term in the loss. Early stopping treats training time itself as a hyperparameter (watch the validation curve, halt at its minimum, keep the best checkpoint), and for a quadratic objective it is provably equivalent to $L^2$ weight decay. Parameter sharing goes the other way: it constrains many weights to be _equal_, the prior behind every convolution and every recurrent step, and the reason a CNN has orders of magnitude fewer parameters than the dense net it replaces.\n",{"path":24145,"title":24146,"module":24134,"summary":24147},"\u002Fdeep-learning\u002Fregularization\u002Fnormalization","Normalization","Normalization layers standardize activations to zero mean and unit variance inside the network, then hand the model a learnable scale and shift to undo the constraint when it pays to. Batch normalization does this across the batch and must keep separate train-time and test-time statistics; layer, instance, and group norm change only the axes they average over. The result is faster, better-conditioned optimization and a free dose of regularizing batch noise.\n",{"path":24149,"title":24150,"module":24151,"summary":24152},"\u002Fdeep-learning\u002Farchitectures\u002Fconvolutional-networks","Convolutional Networks","Architectures","A convolutional network replaces the dense layer's all-to-all weight matrix with a small kernel slid across the input. Three structural commitments (sparse connectivity, parameter sharing, and translation equivariance) collapse the parameter count by orders of magnitude and bake the right prior for images directly into the architecture. We derive the convolution arithmetic, the output geometry, pooling, and the receptive field, then assemble the canonical stack.\n",{"path":24154,"title":24155,"module":24151,"summary":24156},"\u002Fdeep-learning\u002Farchitectures\u002Fcnn-architectures","CNN Architectures","Six landmark networks, each contributing exactly one idea: LeNet's conv-pool stack, AlexNet's ReLU-and-dropout scale, VGG's $3\\times3$ uniformity, Inception's multi-scale module, ResNet's residual skip, and DenseNet's dense connectivity. The common thread is the degradation problem (why plain deeper nets train worse, not just overfit) and the residual block that solved it by keeping a $+1$ path open for the gradient.\n",{"path":24158,"title":24159,"module":24151,"summary":24160},"\u002Fdeep-learning\u002Farchitectures\u002Frecurrent-networks","Recurrent Networks","A recurrent network folds a sequence into a fixed-size hidden state, reusing one set of weights at every time step, the architectural prior that the same rule applies wherever it lands in time. Unrolling the recurrence exposes a deep feed-forward graph; backpropagation through it sums gradient contributions across all steps and chains a product of Jacobians, and that product is why long-range gradients vanish or explode. That failure motivates gated architectures.\n",{"path":24162,"title":24163,"module":24151,"summary":24164},"\u002Fdeep-learning\u002Farchitectures\u002Flstm-and-gru","LSTM & GRU","A plain recurrent network propagates its hidden state through a repeated weight-matrix multiply, and the Jacobian product that results vanishes or explodes long before a useful gradient can reach the early steps. Gated RNNs fix this with an additive memory path: a cell state that is carried forward almost unchanged, past which the gradient flows along a near-identity highway. We derive that highway, give the full LSTM and GRU equations, and compare the two.\n",{"path":24166,"title":24167,"module":24151,"summary":24168},"\u002Fdeep-learning\u002Farchitectures\u002Fattention-and-transformers","Attention & Transformers","Attention replaces fixed wiring with content-based routing: every position reads from every other through a soft, learned dot-product lookup. We derive scaled dot-product attention and its $\\sqrt{d_k}$ correction, build it into multi-head self-attention, inject order with positional encodings, and stack the whole thing into the Transformer block that displaced recurrence and convolution alike.\n",{"path":24170,"title":24171,"module":24151,"summary":24172},"\u002Fdeep-learning\u002Farchitectures\u002Fthe-transformer-architecture","The Transformer Architecture","The Transformer is the architecture built around the attention mechanism. This first part assembles the full encoder–decoder of \"Attention Is All You Need\" — embeddings and positional encoding, stacked self-attention and feed-forward sublayers wrapped in residual connections and LayerNorm, masked decoding and cross-attention — works through causal masking and the three modern families (encoder-only, decoder-only, encoder–decoder), and accounts for where the parameters and the $O(n^2)$ compute actually go.\n",{"path":24174,"title":24175,"module":24151,"summary":24176},"\u002Fdeep-learning\u002Farchitectures\u002Ftransformers-in-practice","Transformers in Practice","The Transformer makes no assumption about what a token represents. This part follows the architecture out of language: image patches feed a plain encoder (the Vision Transformer), the decoder-only half scales into the GPT line of large language models, and one substrate covers translation, retrieval, and multimodal grounding. We work the ViT patch arithmetic and a GPT parameter count by hand, then close on the empirical scaling laws — power-law loss, the Chinchilla compute-optimal balance, and emergent behavior — that made scale the dominant lever.\n",{"path":24178,"title":24179,"module":24151,"summary":24180},"\u002Fdeep-learning\u002Farchitectures\u002Fgraph-neural-networks","Graph Neural Networks","A graph neural network learns on data with no grid and no canonical ordering: atoms in a molecule, users in a social network, road segments in a map. The unifying idea is message passing — each node repeatedly aggregates its neighbors' states and updates its own — built to respect the one symmetry graphs demand, permutation equivariance. We derive the message-passing framework, specialize it into GCN, GraphSAGE, GAT, and GIN, read off graph-level outputs, and bound what message passing can and cannot tell apart.\n",{"path":24182,"title":24183,"module":24151,"summary":24184},"\u002Fdeep-learning\u002Farchitectures\u002Fstate-space-models","State-Space Models and Mamba","A state-space model carries a continuous linear hidden state through a sequence, and that linearity buys two equivalent algorithms from one set of weights: a recurrence that runs in linear time with constant memory, and a global convolution that trains in parallel. Long-range memory comes from how the transition matrix is initialized (HiPPO) and parameterized (S4's diagonal-plus-low-rank form). Mamba breaks the convolution on purpose, making the parameters input-dependent so the model can select what to remember, recovered at speed by a hardware-aware parallel scan.\n",{"path":24186,"title":24187,"module":24188,"summary":24189},"\u002Fdeep-learning\u002Ftheory\u002Fgeneralization-theory","Generalization Theory","Theory & Frontiers","Classical learning theory bounds the gap between training and test error by a model's capacity (VC dimension, Rademacher complexity), and predicts that a model with more parameters than data should overfit catastrophically. Modern networks do the opposite: they interpolate, even fit pure noise, and still generalize. We derive the classical bounds, work the bias-variance decomposition, show why the bounds go vacuous, and survey what replaced them: double descent, the interpolation threshold, margin and norm-based bounds, and the implicit bias of the optimizer itself.\n",{"path":24191,"title":24192,"module":24188,"summary":24193},"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-robustness","Adversarial Robustness","A trained network can be fooled by a perturbation too small for a human to see: add a carefully aimed vector of magnitude $\\epsilon$ to a correctly classified image and the prediction flips. We derive the fast gradient sign method as the first-order-optimal step inside an $L_\\infty$ ball, explain the linearity hypothesis that makes high-dimensional models so easy to push around, build up to projected gradient descent, and frame adversarial training as a min-max robust-optimization problem with its own accuracy cost. Defenses beyond training continue in the next lesson.\n",{"path":24195,"title":24196,"module":24188,"summary":24197},"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-defenses","Adversarial Defenses","Defending a network against an adversary is far harder than attacking one. This lesson covers the defense side: certified guarantees via randomized smoothing, the transferability that makes black-box attacks possible, and the recurring failure of gradient masking, where a defense hides the attacker's gradient instead of moving the decision boundary. It ends with the adaptive-attack discipline (BPDA, EOT, transfer) that every robustness claim must be tested against.\n",{"path":24199,"title":24200,"module":24188,"summary":24201},"\u002Fdeep-learning\u002Ftheory\u002Fbayesian-and-ensemble-methods","Bayesian & Ensemble Methods","A trained network returns a single point prediction and, with the softmax, a confidence, but that confidence is usually miscalibrated, collapsing to near- certainty even on inputs the model has never seen. This lesson covers uncertainty estimation for networks: the two kinds of uncertainty, the Bayesian posterior over weights and its tractable stand-ins (MC dropout, deep ensembles), and how to check whether a model's reported confidences match observed frequencies.\n",{"path":24203,"title":24204,"module":24188,"summary":24205},"\u002Fdeep-learning\u002Ftheory\u002Fdeep-equilibrium-models","Deep Equilibrium Models","A deep network need not be a fixed stack of layers; it can be a single weight-tied layer iterated to convergence, its output defined implicitly as the fixed point $z^\\star = f_\\theta(z^\\star, x)$. The forward pass becomes root-finding and the backward pass becomes implicit differentiation, so training costs O(1) memory regardless of effective depth. We derive both passes from the implicit function theorem and close the course on defining a layer by a fixed-point condition rather than an explicit stack.\n",{"path":24207,"title":24208,"module":24209,"summary":24210},"\u002Fdeep-learning\u002Fgenerative-models\u002Flinear-factor-models","Linear Factor Models","Generative Models","The simplest generative models share one template: a latent variable drawn from a fixed prior, run through a linear decoder, plus noise. Probabilistic PCA, factor analysis, independent component analysis, and sparse coding are all this template with a different prior on the latents and a different noise model. We derive each marginal, see why ICA needs non-Gaussianity to identify its sources, and show how sparse coding learns Gabor-like dictionary atoms.\n",{"path":24212,"title":24213,"module":24209,"summary":24214},"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoencoders","Autoencoders","An autoencoder is a network trained to copy its input to its output through a narrow channel; the useful product is the bottleneck representation $h$, not the reconstruction. We derive the undercomplete autoencoder and prove its linear case recovers PCA, then trade the bottleneck for explicit regularization (sparse, denoising, contractive) and show how a denoising autoencoder learns the low-dimensional manifold the data lives on.\n",{"path":24216,"title":24217,"module":24209,"summary":24218},"\u002Fdeep-learning\u002Fgenerative-models\u002Fvariational-autoencoders","Variational Autoencoders","An autoencoder compresses, but its latent space has gaps: sample a point between two encodings and the decoder produces noise. The variational autoencoder fixes this by training a probabilistic encoder against a prior, so the latent space becomes a smooth, samplable density. We derive the evidence lower bound it maximizes, the reparameterization trick that lets gradients flow through a random sample, and the closed-form Gaussian regularizer that pulls the posterior toward the prior.\n",{"path":24220,"title":24221,"module":24209,"summary":24222},"\u002Fdeep-learning\u002Fgenerative-models\u002Fgenerative-adversarial-networks","Generative Adversarial Networks","A generative adversarial network trains two networks against each other: a generator that turns noise into samples, and a discriminator that tries to tell real data from forgeries. The game has a clean theory: the optimal discriminator is a likelihood ratio, and at equilibrium the generator minimizes the Jensen–Shannon divergence to the data, with a global optimum exactly when its distribution matches the data. We derive that result, fix the saturating loss that breaks training, and catalogue the failure modes (mode collapse, instability, vanishing gradients) and the architectural fixes.\n",{"path":24224,"title":24225,"module":24209,"summary":24226},"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoregressive-and-normalizing-flows","Autoregressive Models & Normalizing Flows","Two families that provide exact likelihoods, each at a cost. Autoregressive models factor the joint by the probability chain rule and learn each conditional with a masked network: exact $\\log p(x)$, but sampling proceeds one coordinate at a time. Normalizing flows push a simple base density through an invertible map and read $\\log p(x)$ off the change-of-variables formula, trading architectural freedom for a cheap Jacobian determinant via triangular coupling layers.\n",{"path":24228,"title":24229,"module":24209,"summary":24230},"\u002Fdeep-learning\u002Fgenerative-models\u002Fenergy-based-and-boltzmann-machines","Energy-Based & Boltzmann Machines","Energy-based models replace an explicit density with a scalar energy and a Boltzmann normalization, $p(x) = e^{-E(x)}\u002FZ$: simple to specify, but with an intractable partition function $Z$. The Boltzmann machine and its restricted variant make the energy bilinear so the hidden units factorize, and contrastive divergence sidesteps $Z$ by replacing the model expectation with a few Gibbs steps started at the data. We close on the undirected deep models (DBNs and DBMs) and how they differ from the directed VAE.\n",{"path":24232,"title":24233,"module":24209,"summary":24234},"\u002Fdeep-learning\u002Fgenerative-models\u002Fdiffusion-and-score-based-models","Diffusion and Score-Based Models","Corrupt a data point with Gaussian noise in small steps until only noise remains, then train a network to undo one step at a time. We derive the forward process and its closed-form marginal, reduce the variational bound to the single noise-prediction objective that makes diffusion trainable, and show the score-matching view that unifies it with Langevin sampling and the continuous SDE. The lesson closes with DDIM fast sampling, classifier-free guidance, and the latent diffusion that powers modern text-to-image systems.\n",{"path":24236,"title":24237,"module":24238,"summary":24239},"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fstructured-probabilistic-models","Structured Probabilistic Models","Probabilistic Methods","A joint distribution over $n$ variables is a table with exponentially many entries; nobody can store it, fit it, or sample from it directly. Structure fixes this: a graph whose missing edges encode conditional independencies that factor the joint into small local pieces. We build the two dialects, directed (Bayesian networks) and undirected (Markov random fields), read independence off the graph, and connect the machinery to the latent-variable and energy-based models that power deep generative learning.\n",{"path":24241,"title":24242,"module":24238,"summary":24243},"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fmonte-carlo-and-mcmc","Monte Carlo & MCMC","Most quantities of interest in a probabilistic model are integrals nobody can compute in closed form: expectations, marginals, partition functions. Monte Carlo replaces the integral with an average over samples; importance sampling reweights samples from a tractable proposal; and when even sampling the target is hard, Markov-chain Monte Carlo builds a chain whose stationary distribution _is_ the target. We derive Metropolis–Hastings and Gibbs, analyze mixing, and close on the partition-function gradient that powers energy-based learning.\n",{"path":24245,"title":24246,"module":24238,"summary":24247},"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fapproximate-inference","Approximate Inference","In a latent-variable model the quantity we need, the posterior $p(h\\mid v)$ over hidden causes, is almost never computable, because its normalizer is an intractable sum over configurations. Approximate inference reframes the problem as optimization: maximize the evidence lower bound, a tractable functional whose gap to the true log-evidence equals a KL divergence. From that single bound fall expectation–maximization, mean-field variational inference, MAP, and the learned encoders behind variational autoencoders.\n",{"path":24249,"title":24250,"module":24251,"summary":24252},"\u002Fdeep-learning\u002Fpractical\u002Fpractical-methodology","Practical Methodology","Practical Deep Learning","Knowing the algorithms is half the job; the other half is a disciplined loop. Fix a goal and a metric, stand up an end-to-end baseline, then read the train\u002Fvalidation gap to decide whether the next move is more data or a bigger model. We detail that loop: choosing metrics under class imbalance, default baselines by data type, extrapolating the data a target needs, and guarding the data pipeline against the leaks and label bugs that corrupt every gradient. Hyperparameter tuning, debugging, and deployment continue in the sequel.\n",{"path":24254,"title":24255,"module":24251,"summary":24256},"\u002Fdeep-learning\u002Fpractical\u002Fhyperparameters-and-debugging","Hyperparameters & Debugging","The tuning half of the methodology loop. The learning rate is the one hyperparameter that dominates, so we tune it first, on a log scale, coarse to fine, and prefer random search to grid when only a few dials matter. Then an ordered debugging playbook — overfit one batch, check the loss at initialization against ln C, watch the gradient norm, gradient-check against centered finite differences — and, after launch, monitoring for train-test skew and distribution drift with confidence-based abstention.\n",{"path":24258,"title":24259,"module":24251,"summary":24260},"\u002Fdeep-learning\u002Fpractical\u002Frepresentation-learning","Representation Learning","A good representation makes a hard task easy by changing coordinates: it disentangles the factors of variation, spends its bits as a distributed code, and respects the low-dimensional manifold the data lives on. We make those three properties precise, recover the manifold hypothesis, and close on the first method that turned them into training practice — greedy layer-wise unsupervised pretraining — before the sequel picks up how the field learned to reuse those features.\n",{"path":24262,"title":24263,"module":24251,"summary":24264},"\u002Fdeep-learning\u002Fpractical\u002Ftransfer-learning","Transfer Learning","A representation learned once can be reused everywhere. We cover the main mechanisms of reuse: feature extraction versus fine-tuning, the generic-to-specific gradient of features that sets the freeze boundary, the learning-rate discipline that keeps borrowed weights from being erased, domain adaptation when only the input distribution shifts, and the modern arc from supervised transfer to self-supervised foundation models.\n",{"path":24266,"title":24267,"module":24251,"summary":24268},"\u002Fdeep-learning\u002Fpractical\u002Fapplications","Applications","We survey large-scale training (the hardware, the two axes of parallelism, mixed precision, and the compression tricks that shrink a model after it is trained), then specialize the same gradient loop to vision, language, speech, and recommendation. Each domain is a different prior bolted onto one optimizer: convolutional invariance for pixels, distributed word vectors for tokens, sequence transduction for audio, low-rank factorization for the user–item matrix.\n",{"path":24270,"title":24271,"module":24251,"summary":24272},"\u002Fdeep-learning\u002Fpractical\u002Fmodel-compression-and-distillation","Model Compression and Distillation","A trained network and a deployable one are rarely the same object. This lesson is the toolkit for closing that gap: knowledge distillation transfers a large teacher's soft, information-rich logits into a small student; pruning deletes the weights that contribute least; quantization swaps 32-bit floats for 8- or 4-bit integers; and low-rank factorization replaces a fat matrix with two thin ones. We derive each method, show what it costs in accuracy, and lay out which combinations win on which hardware.\n",{"path":24274,"title":24275,"module":24251,"summary":24276},"\u002Fdeep-learning\u002Fpractical\u002Fmeta-learning-and-few-shot","Meta-Learning and Few-Shot Learning","A deep network trained on one example per class overfits. Meta-learning targets this few-shot regime by training across a distribution of tasks so that a new task is learnable from a handful of examples. We formalize the $N$-way $K$-shot episode, then derive the two dominant families: metric methods that learn an embedding where distance classifies (Prototypical Networks), and optimization methods that learn an initialization a few gradient steps can adapt (MAML). We close on the link to transfer learning and to the in-context few-shot behavior of large language models.\n",{"path":24278,"title":24279,"module":24280,"summary":24281},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models","Large Language Models","Large Models & Agents","A large language model is a decoder-only Transformer trained on one objective, next-token prediction, then scaled until new behavior appears. This first part builds the object itself: the equivalence between next-token prediction and lossless compression, subword tokenization (BPE, WordPiece, Unigram, SentencePiece) worked on a real sentence, the four pretraining objectives and the attention masks that distinguish them, and the three model families (encoder-only, decoder-only, encoder--decoder) with their parameter budgets. Scaling, decoding, the KV cache, and alignment continue in part two.\n",{"path":24283,"title":24284,"module":24280,"summary":24285},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment","Scaling, Inference, and Alignment of Language Models","Once a language model is built, three questions remain: how does it improve as it grows, how is it decoded and served affordably, and how is a raw next-token predictor turned into an assistant. We derive the Kaplan power laws and the Chinchilla compute-optimal balance, trace emergent abilities and in-context learning, catalog the decoding strategies from greedy to nucleus sampling, work the KV cache that makes generation quadratic instead of cubic, cover parameter-efficient adaptation by low-rank updates (LoRA), and close on the alignment stack: instruction tuning, RLHF, and DPO.\n",{"path":24287,"title":24288,"module":24280,"summary":24289},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fseq2seq-pretraining-and-bart","Denoising Sequence-to-Sequence Pretraining: BART","BERT corrupts and reconstructs; GPT predicts the next token. Sequence-to-sequence pretraining unifies both by training a full encoder–decoder as a denoising autoencoder: corrupt the text with a noise function, then reconstruct the original through a bidirectional encoder and an autoregressive decoder. This first part derives the denoising objective, catalogs BART's five noise functions (with a worked Poisson-infilling budget), proves BART specializes to both BERT and GPT, and traces a dimension-annotated forward pass through its encoder--decoder. T5, PEGASUS, fine-tuning, and decoding continue in part two.\n",{"path":24291,"title":24292,"module":24280,"summary":24293},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ftext-to-text-transfer-and-conditional-generation","Text-to-Text Transfer and Conditional Generation","BART reconstructs a corrupted document; T5 pushes the same denoising idea into a single interface where every task is a string-to-string map. This second part covers T5's span corruption with sentinel tokens (with a worked token budget), PEGASUS's summarization-matched gap sentences and the MASS midpoint, supervised fine-tuning and beam-search decoding with a length penalty, the exposure-bias failure modes of autoregressive decoding, and a theorem showing why a bidirectional encoder--decoder strictly dominates a decoder-only model when the output is conditioned on a full input.\n",{"path":24295,"title":24296,"module":24280,"summary":24297},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fspeech-and-audio-models","Speech Recognition: Front-Ends and Alignment","Speech is a long, high-rate sequence whose label is short and unaligned, so the whole subject turns on bridging that mismatch. This first part builds the spectral front-ends that compress a waveform into frames (STFT, mel spectrogram, MFCC, with a worked frame-count), derives CTC's marginalization over alignments and its forward-backward recursion with a two-frame numeric example, and contrasts it with attention-based seq2seq (LAS) and the RNN transducer. Self-supervised and weakly-supervised models, and text-to-speech, continue in part two.\n",{"path":24299,"title":24300,"module":24280,"summary":24301},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fself-supervised-speech-and-synthesis","Self-Supervised Speech Models and Synthesis","The recognition front-ends and alignment losses of part one all need transcribed audio, which is scarce. This second part removes that dependence: wav2vec 2.0 learns speech representations from unlabeled audio by a masked contrastive objective, HuBERT swaps the contrast for masked prediction of clustered units, and Whisper trades curation for scale with weakly-supervised web audio and a multitask token interface. We close with text-to-speech (the same length mismatch run backwards) and a tour of speech foundation models, discrete audio codecs, and neural TTS.\n",{"path":24303,"title":24304,"module":24280,"summary":24305},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents","AI Agents: Tools and Reasoning","A language model that only emits text is a function from prompt to prompt; an agent closes the loop, letting that model act on an environment, read back the result, and decide again. This first part formalizes the agent as a policy over interaction histories, builds out tool calling and the executor trust boundary, the ReAct interleaving of reasoning and action (with concrete traces), and search over thoughts: chain-of-thought, self-consistency, least-to-most, and Tree of Thoughts. Memory, retrieval, reflection, and multi-agent orchestration continue in part two.\n",{"path":24307,"title":24308,"module":24280,"summary":24309},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fagent-memory-retrieval-and-orchestration","Agent Memory, Retrieval, and Orchestration","An agent's reasoning and tool use only matter if it can remember what it learned and coordinate work larger than one context window. This second part builds the systems around the loop: short-term scratchpad versus long-term vector store, retrieval-augmented generation with a worked softmax over passage scores, reflection (Reflexion, Self-Refine), and multi-agent orchestration. It closes on the failure modes that bound agents — invalid tool calls, horizon-error compounding, context overflow, non-terminating loops — and the benchmarks that score the full loop.\n",{"path":24311,"title":24312,"module":24280,"summary":24313},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmixture-of-experts","Mixture-of-Experts","A mixture-of-experts layer replaces one feed-forward network with many and a router that sends each token to only a few of them, so the parameter count and the per-token compute become separate dials. We derive the gated output, sparse top-$k$ routing softmax, the load-balancing loss that stops the router from collapsing onto a single expert, and expert\u002Ftoken capacity with dropping, then work the dimension-annotated tensor shapes and FLOP arithmetic. We trace the architectures from the sparsely-gated LSTM through GShard, Switch Transformer, and Mixtral, cover distributed expert parallelism, and close on the training dynamics, failure modes, and serving costs of a sparse model.\n",{"path":24315,"title":24316,"module":24280,"summary":24317},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmultimodal-models","Multimodal Contrastive Learning","A multimodal model places images, text, and audio in one representation space, so a picture and its caption land close together. This first part builds the contrastive route: the shared embedding space and its residual modality gap, the Vision Transformer image encoder (patch embedding, CLS token, position embeddings, with shapes), the symmetric InfoNCE loss that trains the CLIP dual encoder from a batch similarity matrix (with a worked numeric step), and zero-shot classification as a softmax over class-prompt embeddings. Fusion and vision-language models continue in part two.\n",{"path":24319,"title":24320,"module":24280,"summary":24321},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ffusion-and-vision-language-models","Fusion and Vision-Language Models","A contrastive model compares modalities but never lets one read another. This second part builds the fusion route: early, late, and cross-attention fusion, then the three designs that connect a frozen vision encoder to a frozen language model — Flamingo's zero-initialized gated cross-attention, BLIP-2's Q-Former, and LLaVA's linear projector. We work the token-budget arithmetic that separates them, name the object-hallucination and fine-detail failure modes, cover the contrastive-then- instruction-tune recipe and its retrieval\u002Fcaptioning\u002FVQA benchmarks, and close on natively multimodal models.\n",{"path":24323,"title":24324,"module":24325,"summary":24326},"\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning","Foundations of Reinforcement Learning","Reinforcement Learning","Reinforcement learning is the third paradigm: an agent learns to act by interacting with an environment that returns rewards, not labels. We formalize the interaction as a Markov decision process, define the value functions that rank states and actions, and derive the Bellman expectation and optimality equations that every method downstream solves. Dynamic programming gives the exact answer when the model is known, and its convergence rests on a single fact: the Bellman operator is a contraction.\n",{"path":24328,"title":24329,"module":24325,"summary":24330},"\u002Fdeep-learning\u002Freinforcement-learning\u002Fmodel-free-prediction-and-control","Model-Free Prediction and Control","When the dynamics are unknown, an agent cannot plan against a model; it must learn directly from sampled experience. We build prediction and control from two estimators of the same return: Monte Carlo averages whole episodes, while temporal-difference learning bootstraps from its own next estimate. We trace the bias-variance contrast between them, derive SARSA and Q-learning as the on-policy and off-policy forms of control, unify everything through n-step returns and eligibility traces, and close on the deadly triad that makes off-policy bootstrapping with function approximation diverge.\n",{"path":24332,"title":24333,"module":24325,"summary":24334},"\u002Fdeep-learning\u002Freinforcement-learning\u002Fdeep-q-networks","Deep Q-Networks","A Deep Q-Network replaces the tabular action-value function with a neural approximator $Q(s,a;\\theta)$ and trains it by regression toward a bootstrapped target. Naive online Q-learning with a network diverges, so DQN adds two stabilizers: an experience-replay buffer that decorrelates samples, and a periodically-frozen target network that holds the regression target still. We derive the loss, give the full algorithm and the Atari pipeline, and then layer on Double DQN, the dueling split, prioritized replay, and the Rainbow combination.\n",{"path":24336,"title":24337,"module":24325,"summary":24338},"\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic","Policy Gradients and Actor-Critic Methods","Value-based reinforcement learning learns what each state is worth and acts greedily; policy-gradient methods skip the detour and optimize a parameterized policy directly by ascending the gradient of expected return. The policy gradient theorem makes this tractable through the log-derivative trick, turning an intractable gradient of an expectation into an expectation of a gradient. REINFORCE realizes the idea but suffers high variance; baselines, the advantage function, and actor-critic learning reduce it, and trust-region methods (TRPO, PPO) keep each update from destroying the policy it just learned.\n",{"path":24340,"title":24341,"module":24325,"summary":24342},"\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback","Reinforcement Learning from Human Feedback","Many objectives we want from a model, that it be helpful and harmless, are hard to write down but easy to judge by comparison. RLHF turns that asymmetry into a training signal: fit a reward model to pairwise human preferences under the Bradley-Terry likelihood, then fine-tune the policy to maximize that reward under a KL penalty toward a reference. We derive the reward loss, the KL-regularized RL objective and its closed-form optimum, then show how DPO inverts that optimum to collapse the whole pipeline into one supervised log-sigmoid loss, and survey IPO, KTO, RLAIF, and GRPO.\n",{"path":24344,"title":20246,"module":6,"summary":6},"\u002Fdeep-learning",{"path":24346,"title":24347,"module":22263,"summary":24348},"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fequilibrium-state-variables-zeroth-law","Equilibrium, State Variables, and the Zeroth Law","Thermodynamics describes a many-body system by a handful of macroscopic variables and the equilibrium relations among them. This lesson fixes the vocabulary: systems and the walls that separate them, state variables versus path-dependent process quantities, quasi-static and reversible idealizations, and the zeroth law, whose transitivity of thermal equilibrium is what lets temperature exist as a number. The ideal-gas thermometer turns that number into a scale, and an equation of state ties the variables into a surface.\n",{"path":24350,"title":24351,"module":22263,"summary":24352},"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Ffirst-law-heat-and-work","The First Law: Internal Energy, Heat, and Work","The first law is energy conservation for a system that exchanges energy as both heat and work. Internal energy is a state function with an exact differential; heat and work are path-dependent process quantities. This lesson states $\\d U=\\delta Q+\\delta W$, computes compression work as an area on the $P$–$V$ plane, defines the heat capacities $C_V$ and $C_P$ and the enthalpy that makes $C_P$ natural, and works the isothermal and adiabatic processes of an ideal gas, including the adiabat $PV^\\gamma=\\text{const}$.\n",{"path":24354,"title":24355,"module":22263,"summary":24356},"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fsecond-law-entropy-and-the-carnot-bound","The Second Law, Carnot Cycles, and Entropy","The second law forbids the free conversion of heat into work. This lesson states the Kelvin and Clausius forms, proves them equivalent, and analyzes the Carnot cycle to get the efficiency bound $1-T_c\u002FT_h$. Carnot's theorem makes that bound universal and defines the thermodynamic temperature scale. The Clausius inequality $\\oint \\delta Q\u002FT\\le 0$ then constructs entropy as a state function, $\\d S=\\delta Q_{\\rm rev}\u002FT$, whose non-decrease in isolated systems is the arrow of time.\n",{"path":24358,"title":24359,"module":22263,"summary":24360},"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fthermodynamic-potentials-and-maxwell-relations","Thermodynamic Potentials and Maxwell Relations","The fundamental relation $\\d U=T\\,\\d S-P\\,\\d V+\\mu\\,\\d N$ packages the first and second laws into one exact differential. Legendre transforms swap each conjugate pair to produce the Helmholtz, enthalpy, Gibbs, and grand potentials, each minimized under its own natural variables. Equality of mixed second partials of these potentials gives the Maxwell relations, which convert unmeasurable entropy derivatives into measurable ones from the equation of state.\n",{"path":24362,"title":24363,"module":22263,"summary":24364},"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fstability-response-functions-and-the-third-law","Response Functions, Stability, and the Third Law","Response functions — heat capacities, compressibilities, thermal expansion — are the second derivatives of the potentials and the quantities an experiment actually measures. This lesson derives the general relation $C_P-C_V=TV\\alpha^2\u002F\\kappa_T$, shows that convexity of the potentials forces the stability conditions $C_V>0$ and $\\kappa_T>0$, and states the third law: entropy approaches a constant as $T\\to0$, so heat capacities and expansion coefficients vanish there and absolute zero is unattainable.\n",{"path":24366,"title":24367,"module":24368,"summary":24369},"\u002Fstatistical-mechanics\u002Ffoundations\u002Fclassical-statistics-and-equipartition","Classical Statistics and Equipartition","Microstates, Phase Space, and Statistical Entropy","A liter of gas holds on the order of a trillion trillion molecules, far too many to track by their equations of motion. Classical statistical mechanics replaces the trajectories with a single probability law, the Boltzmann distribution, and reads the measurable properties of matter off it: the Maxwell speed distribution, the average energy per degree of freedom, and the heat capacities of gases and solids — together with the low-temperature failures that forced the quantum revision.\n",{"path":24371,"title":24372,"module":24368,"summary":24373},"\u002Fstatistical-mechanics\u002Ffoundations\u002Fphase-space-and-liouvilles-theorem","Phase Space, Trajectories, and Liouville's Theorem","A classical system of N particles is one point in a 6N-dimensional phase space, and its evolution is a single trajectory driven by Hamilton's equations. This lesson builds that geometric picture, introduces the phase-space density of an ensemble, and proves Liouville's theorem: the density is carried by the flow as an incompressible fluid, so phase-space volume is conserved. The stationary densities of equilibrium follow as functions of the conserved quantities alone.\n",{"path":24375,"title":24376,"module":24368,"summary":24377},"\u002Fstatistical-mechanics\u002Ffoundations\u002Fensembles-and-the-equal-probability-postulate","Ensembles and the Postulate of Equal a Priori Probabilities","An ensemble is a probability distribution over the microstates of a system. This lesson states the single postulate on which equilibrium statistical mechanics rests — that an isolated system in equilibrium is equally likely to be in any of its accessible microstates — and works out its consequences: the accessible phase-space volume, the overwhelming dominance of the most probable macrostate as the particle number grows, and the ergodic hypothesis that lets a time average be replaced by an ensemble average.\n",{"path":24379,"title":24380,"module":24368,"summary":24381},"\u002Fstatistical-mechanics\u002Ffoundations\u002Fstatistical-entropy-boltzmann-and-gibbs","Statistical Entropy: Boltzmann and Gibbs","Entropy is the logarithm of the number of accessible microstates. This lesson builds the two statistical entropies — Boltzmann's S = k ln Omega for an isolated system and Gibbs's S = -k sum p ln p for any ensemble — proves they agree for a uniform distribution, and connects both to Shannon's measure of missing information. The second law emerges as the drift toward maximum multiplicity, and maximizing the Gibbs entropy under constraints previews the canonical distribution.\n",{"path":24383,"title":24384,"module":24385,"summary":24386},"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fmicrocanonical-ensemble-and-entropy","The Microcanonical Ensemble and Statistical Entropy","The Microcanonical Ensemble","An isolated system holds its energy, volume, and particle number fixed, and the fundamental postulate assigns equal probability to every microstate on its energy shell. This lesson builds the microcanonical distribution, defines the enclosed phase-space volume $\\Gamma(E)$, the surface density of states $\\omega(E)=\\d\\Gamma\u002F\\d E$, and the shell count $\\Omega(E)$, shows their logarithms agree to $O(\\ln N)$ for large $N$, and reads the Boltzmann entropy $S=k\\ln\\Omega$ off the count. The measure factors $h^{3N}$ and $N!$ enter here and make $S$ extensive.\n",{"path":24388,"title":24389,"module":24385,"summary":24390},"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fequilibrium-conditions-temperature-pressure-chemical-potential","Thermal, Mechanical, and Diffusive Equilibrium","Two isolated subsystems that can exchange energy, volume, or particles reach equilibrium at the partition that maximizes their combined entropy. Setting the derivative of the total entropy to zero identifies the statistical definitions $1\u002FT=(\\partial S\u002F\\partial E)$, $P\u002FT=(\\partial S\u002F\\partial V)$, and $-\\mu\u002FT=(\\partial S\u002F\\partial N)$, shows heat flows from hot to cold as an entropy increase, and recovers the fundamental relation $\\d S=(\\d E+P\\,\\d V-\\mu\\,\\d N)\u002FT$ from pure counting.\n",{"path":24392,"title":24393,"module":24385,"summary":24394},"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fideal-gas-phase-space-and-the-sackur-tetrode-entropy","The Ideal Gas, Phase-Space Volume, and the Sackur–Tetrode Entropy","The monatomic ideal gas is the first system whose microcanonical count can be done in closed form. The momentum integral is the volume of a $3N$-dimensional ball of radius $\\sqrt{2mE}$, the configuration integral is $V^N$, and together they give the Sackur–Tetrode entropy $S=Nk[\\ln(V\u002FN\\lambda^3)+5\u002F2]$ with the thermal wavelength $\\lambda=h\u002F\\sqrt{2\\pi mkT}$. The formula matches the measured entropy of helium, fixes the classical regime $n\\ll n_Q$, and shows why the $N!$ is needed for extensivity.\n",{"path":24396,"title":24397,"module":24385,"summary":24398},"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Ftwo-state-systems-paramagnets-and-negative-temperature","Two-State Systems, Paramagnets, and Negative Temperature","The ideal two-state paramagnet has a multiplicity counted by the binomial coefficient, an entropy that is an inverted dome in the energy, and a temperature read from the slope $1\u002FT=\\partial S\u002F\\partial E$. Because the energy is bounded above, the slope changes sign past the entropy maximum: a population-inverted spin system has a negative absolute temperature, which is hotter than any positive temperature. Nuclear-spin experiments and lasers realize the inverted state.\n",{"path":24400,"title":24401,"module":24402,"summary":24403},"\u002Fstatistical-mechanics\u002Fcanonical\u002Fcanonical-ensemble-and-the-boltzmann-distribution","The Canonical Ensemble and the Boltzmann Distribution","The Canonical Ensemble","A system held at fixed temperature by contact with a heat reservoir is described by the canonical ensemble. Expanding the reservoir entropy to first order in the system energy gives the Boltzmann distribution $p_i\\propto e^{-\\beta E_i}$, and the same law follows from maximizing the Gibbs entropy at fixed mean energy. Both routes identify $\\beta=1\u002Fk_BT$ and fix the probability of every microstate from the temperature alone.\n",{"path":24405,"title":24406,"module":24402,"summary":24407},"\u002Fstatistical-mechanics\u002Fcanonical\u002Fpartition-function-and-the-helmholtz-free-energy","The Partition Function and the Helmholtz Free Energy","The normalizing sum of the Boltzmann distribution, the partition function $Z=\\sum_i e^{-\\beta E_i}$, is a generating function for the thermodynamics. The mean energy is $-\\partial\\ln Z\u002F\\partial\\beta$, and the Gibbs entropy of the canonical distribution collapses to the bridge relation $F=-k_BT\\ln Z$. From $F$ every thermodynamic quantity follows by differentiation, and $Z$ factorizes over independent degrees of freedom.\n",{"path":24409,"title":24410,"module":24402,"summary":24411},"\u002Fstatistical-mechanics\u002Fcanonical\u002Fenergy-fluctuations-and-ensemble-equivalence","Energy Fluctuations and the Equivalence of Ensembles","In the canonical ensemble the energy fluctuates, and the second derivative of $\\ln Z$ gives its variance. The fluctuation–response identity $\\langle\\Delta E^2\\rangle = k_BT^2C_V$ ties the spread of the energy to the heat capacity, and the relative fluctuation falls as $1\u002F\\sqrt{N}$. In the thermodynamic limit the canonical energy distribution is a sharp spike, and the canonical and microcanonical ensembles predict the same thermodynamics.\n",{"path":24413,"title":24414,"module":24402,"summary":24415},"\u002Fstatistical-mechanics\u002Fcanonical\u002Fthe-einstein-solid-and-harmonic-systems","Harmonic Systems: The Einstein Solid and Vibrational Heat Capacity","A quantum harmonic oscillator has a geometric partition function summed in closed form, giving a mean energy $\\hbar\\omega(\\tfrac12+\\langle n\\rangle)$ with the Bose occupation factor. Modeling a solid as $3N$ independent oscillators yields a heat capacity that rises from zero and saturates at the Dulong–Petit value $3Nk_B$. The Einstein temperature sets the crossover, and the model's exponential low-temperature falloff, too steep against the observed $T^3$, motivates the Debye theory.\n",{"path":24417,"title":24418,"module":24402,"summary":24419},"\u002Fstatistical-mechanics\u002Fcanonical\u002Fparamagnetism-and-the-schottky-anomaly","Paramagnetism, Two-Level Systems, and the Schottky Anomaly","A magnetic moment in a field is a two-level system whose partition function is a hyperbolic cosine. The magnetization of a spin-$\\tfrac12$ paramagnet is $N\\mu\\tanh(\\mu B\u002Fk_BT)$, generalizing to the Brillouin function for spin $J$; it gives Curie's law $\\chi\\propto 1\u002FT$ at high temperature and saturates at low temperature. A finite level gap produces the Schottky heat-capacity peak, and the temperature dependence of the entropy on the field is the basis of adiabatic demagnetization cooling.\n",{"path":24421,"title":24422,"module":24423,"summary":24424},"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fideal-gas-partition-function-and-the-gibbs-paradox","The Ideal Gas Partition Function and the Gibbs Paradox","The Classical Ideal Gas","The classical monatomic ideal gas built from the partition function. The single-particle sum is $z_1=V\u002F\\lambda^3$ with the thermal de Broglie wavelength $\\lambda$; the $N$-particle partition function is $z_1^N\u002FN!$, and the $N!$ is forced by indistinguishability. From $Z$ the ideal-gas law, $U=\\tfrac32 Nk_BT$, and the Sackur–Tetrode entropy follow. The $N!$ makes the entropy extensive and resolves the Gibbs paradox: mixing identical gases produces no entropy change.\n",{"path":24426,"title":24427,"module":24423,"summary":24428},"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fequipartition-and-the-virial-theorem","Equipartition and the Virial Theorem","The equipartition theorem derived from the canonical ensemble: every phase-space coordinate that enters the Hamiltonian quadratically carries a mean energy $\\tfrac12 k_BT$. The generalized form $\\langle x_i\\,\\partial H\u002F\\partial x_j\\rangle = k_BT\\,\\delta_{ij}$ contains equipartition and the classical virial theorem as special cases. Equipartition fixes the classical heat capacities, fails by quantum freeze-out when a level gap exceeds $k_BT$, and shifts for a relativistic gas whose energy is linear rather than quadratic in momentum.\n",{"path":24430,"title":24431,"module":24423,"summary":24432},"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fmolecular-gases-rotation-and-vibration","Molecular Gases: Rotational and Vibrational Degrees of Freedom","The internal partition function of a diatomic gas factorizes into translational, rotational, vibrational, and electronic parts. The rigid rotor gives a rotational temperature $\\theta_{\\rm rot}$; the harmonic bond gives a vibrational temperature $\\theta_{\\rm vib}$. Each mode contributes to the heat capacity only above its characteristic temperature, producing the diatomic $C_V$ staircase from $\\tfrac32 R$ to $\\tfrac52 R$ to $\\tfrac72 R$. Homonuclear molecules carry a symmetry number, and hydrogen splits into ortho and para species.\n",{"path":24434,"title":24435,"module":24436,"summary":24437},"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fgrand-canonical-ensemble-and-the-grand-partition-function","The Grand Canonical Ensemble","Grand Canonical Ensemble","When a system exchanges both energy and particles with a reservoir, the reservoir fixes its temperature and its chemical potential. Expanding the reservoir entropy to first order in the exchanged energy and particle number gives the Gibbs factor $e^{-\\beta(E-\\mu N)}$, and summing it over every microstate of every particle number gives the grand partition function $\\Xi$. The grand potential $\\Phi = -k_BT\\ln\\Xi = -PV$ generates the mean particle number, energy, entropy, and pressure by differentiation.\n",{"path":24439,"title":24440,"module":24436,"summary":24441},"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fchemical-potential-fugacity-and-number-fluctuations","Chemical Potential, Fugacity, and Number Fluctuations","The chemical potential is the energy to add one particle at fixed entropy and volume, equal to the slope of the free energy in the particle number. For the classical ideal gas $\\mu=k_BT\\ln(n\\lambda^3)$ is large and negative, and the fugacity $z=n\\lambda^3$ is small. The grand ensemble makes the particle number fluctuate; its variance $\\langle\\Delta N^2\\rangle=k_BT(\\partial N\u002F\\partial\\mu)$ equals $k_BT\\,N^2\\kappa_T\u002FV$, tying density fluctuations to the isothermal compressibility. Equality of $\\mu$ is the condition for diffusive equilibrium and phase coexistence.\n",{"path":24443,"title":24444,"module":24436,"summary":24445},"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fensemble-summary-and-the-thermodynamic-web","The Three Ensembles and the Thermodynamic Web","The microcanonical, canonical, and grand canonical ensembles hold different variables fixed and generate different potentials — the entropy $S$, the Helmholtz free energy $F$, and the grand potential $\\Phi$ — linked by Legendre transforms that trade each fixed variable for its conjugate. Each successive ensemble lets one more quantity fluctuate. In the thermodynamic limit the three agree, the relative fluctuations vanishing as $1\u002F\\sqrt{N}$; the ideal gas gives the same equation of state in all three. The choice of ensemble is a matter of convenience, set by which sum is easiest.\n",{"path":24447,"title":24448,"module":24449,"summary":24450},"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fquantum-statistics-bose-einstein-and-fermi-dirac","Quantum Statistics — Bose-Einstein and Fermi-Dirac","Quantum Statistics","Quantum particles of the same kind are genuinely indistinguishable: no label survives an overlap of their wave functions. Counting states with that constraint replaces the Boltzmann distribution with two quantum laws — the Bose-Einstein distribution for integer-spin particles, which clump into shared states, and the Fermi-Dirac distribution for half-integer-spin particles, which exclude one another. Both reduce to Boltzmann in the dilute, hot limit, and a de Broglie criterion says exactly when.\n",{"path":24452,"title":24453,"module":24449,"summary":24454},"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fderiving-the-quantum-distributions","Deriving the Quantum Distributions from the Grand Ensemble","The Bose-Einstein and Fermi-Dirac distributions follow from one observation: in the occupation-number representation the single-particle modes are independent, so the grand partition function factorizes into one factor per mode. A boson mode sums a geometric series over all occupancies; a fermion mode sums two terms. Differentiating each factor gives the mean occupation $1\u002F(e^{\\beta(\\varepsilon-\\mu)}\\mp 1)$, the Maxwell-Boltzmann limit when occupancies are small, and the occupation fluctuations that distinguish bunching from anti-bunching.\n",{"path":24456,"title":24457,"module":24449,"summary":24458},"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fthe-classical-limit-and-quantum-concentration","The Classical Limit and Quantum Concentration","When every single-particle level is nearly empty, both quantum distributions collapse to the Maxwell-Boltzmann form, and the fugacity equals the ratio of the number density to the quantum concentration $n_Q = 1\u002F\\lambda^3$. The gas is classical when $n \\ll n_Q$, degenerate when $n \\gtrsim n_Q$. The chemical potential is large and negative in the classical regime and rises through zero as the gas degenerates. The leading quantum correction to the ideal-gas law is a second virial term that lowers the pressure for bosons and raises it for fermions — a statistical attraction and repulsion with no interaction behind it.\n",{"path":24460,"title":24461,"module":24449,"summary":24462},"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fideal-quantum-gases-general-framework","Ideal Quantum Gases: The General Framework","Every ideal quantum gas is handled by one calculation. The sum over single-particle modes becomes an energy integral weighted by a density of states $g(\\varepsilon)\\propto\\varepsilon^{1\u002F2}$, and the number and pressure reduce to the Bose and Fermi functions $g_\\nu(z)$ and $f_\\nu(z)$ of the fugacity. An integration by parts fixes $PV=\\tfrac23 U$ for a nonrelativistic gas and $PV=\\tfrac13 U$ for an ultrarelativistic one, independent of statistics. Specializing the density of states and the chemical potential then produces the photon gas, phonons, the Bose gas, and the Fermi gas as four branches of the same framework.\n",{"path":24464,"title":24465,"module":24466,"summary":24467},"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-and-the-fermion-gas","Bose-Einstein Condensation and the Fermion Gas","Bosonic Systems","Below a critical temperature a boson gas drops a macroscopic fraction of its particles into the single ground state — Bose-Einstein condensation, the mechanism behind superfluid helium and the dilute-atom condensates cooled to nanokelvin. The same statistics applied to a photon gas reproduces Planck's blackbody spectrum. Fermions do the opposite: forbidden from sharing states, they fill every level up to the Fermi energy, and that filled sea governs the electrons in metals and the pressure that holds up a white dwarf.\n",{"path":24469,"title":24470,"module":24466,"summary":24471},"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthe-photon-gas-and-plancks-radiation-law","The Photon Gas and Planck's Radiation Law","Electromagnetic radiation in equilibrium with cavity walls is a gas of non-conserved bosons, and non-conservation forces the chemical potential to zero. Counting standing-wave modes with two polarizations and weighting each by the Bose occupation gives the Planck spectral energy density. Its low-frequency tail reproduces the classical Rayleigh-Jeans law and the ultraviolet catastrophe; the Bose factor cuts the divergence off at high frequency and the peak obeys Wien's displacement law.\n",{"path":24473,"title":24474,"module":24466,"summary":24475},"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fblackbody-thermodynamics-and-radiation-pressure","Blackbody Thermodynamics and Radiation Pressure","Integrating the Planck spectrum over all frequencies gives the total energy density proportional to the fourth power of temperature — the Stefan-Boltzmann law — and the isotropy of a relativistic gas fixes the radiation pressure at one third of the energy density. From the free energy follow the entropy and heat capacity, both proportional to T cubed, and the adiabatic law for radiation. The results govern the pressure inside stars and the cooling of the cosmic microwave background as the universe expands.\n",{"path":24477,"title":24478,"module":24466,"summary":24479},"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fphonons-and-the-debye-model","Phonons and the Debye Model","The vibrations of a crystal lattice are quantized into phonons — bosons of zero chemical potential, counted exactly like cavity photons but with three polarizations, a finite sound speed, and a total of 3N modes. The Debye model replaces the true dispersion by a linear one cut off at a frequency that enforces that count. It gives the correct low-temperature T-cubed heat capacity the Einstein model missed and recovers the Dulong-Petit value at high temperature.\n",{"path":24481,"title":24482,"module":24466,"summary":24483},"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-derived","Bose-Einstein Condensation Derived","For a gas of conserved bosons the excited states can hold only a finite number of particles at fixed temperature, set by the Bose function at unit fugacity. When the total exceeds that ceiling the surplus collapses into the single ground state, which the continuum density-of-states integral misses and which must be restored by hand. This fixes the critical temperature, the condensate fraction, and the fact that a uniform gas condenses only in three or more dimensions.\n",{"path":24485,"title":24486,"module":24466,"summary":24487},"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthermodynamics-of-the-bose-gas-and-superfluidity","Thermodynamics of the Bose Gas and Superfluidity","The energy and pressure of the ideal Bose gas follow from the Bose function at the order above the density, and below the critical temperature the pressure depends on temperature alone because the condensate carries none. The heat capacity rises to a cusp at the transition. Real superfluid helium departs from the ideal gas because interactions matter: the Landau criterion ties frictionless flow to the phonon-roton excitation spectrum, and the two-fluid model carries a second sound.\n",{"path":24489,"title":24490,"module":24491,"summary":24492},"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fthe-ideal-fermi-gas-at-zero-temperature","The Ideal Fermi Gas at Zero Temperature","Degenerate Fermi Gas","At absolute zero a gas of non-interacting fermions fills every single-particle state up to the Fermi energy and leaves the rest empty, a filled Fermi sphere in momentum space. This lesson computes the Fermi momentum, energy, and temperature from the density, the density of states, the total ground-state energy, and the degeneracy pressure that grows as $n^{5\u002F3}$. Numerical Fermi energies for metals set the scale: they are electron-volts, so room temperature is deep in the degenerate regime.\n",{"path":24494,"title":24495,"module":24491,"summary":24496},"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fsommerfeld-expansion-and-electrons-in-metals","The Sommerfeld Expansion and Electrons in Metals","Turning on a small temperature blurs the Fermi step over a shell of width $k_BT$ around $\\epsilon_F$. The Sommerfeld expansion turns integrals over the Fermi function into a power series in $(k_BT\u002F\\epsilon_F)^2$, giving the shift of the chemical potential and a heat capacity linear in $T$. This resolves the old puzzle of the missing electronic heat capacity, predicts the combined $C=\\gamma T+AT^3$ of a metal, and gives the temperature-independent Pauli paramagnetism of the electron gas.\n",{"path":24498,"title":24499,"module":24491,"summary":24500},"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fwhite-dwarfs-and-the-chandrasekhar-limit","White Dwarfs and the Chandrasekhar Limit","A white dwarf is held up against its own gravity by the degeneracy pressure of its electrons. Balancing that pressure against gravity gives a mass-radius relation $R\\propto M^{-1\u002F3}$: heavier white dwarfs are smaller and denser. As the density rises the electrons turn relativistic, the pressure softens from $n^{5\u002F3}$ to $n^{4\u002F3}$, and the star can no longer support itself above a critical mass. This lesson derives that Chandrasekhar mass, about $1.4\\,M_\\odot$, and what lies beyond it.\n",{"path":24502,"title":24503,"module":24491,"summary":24504},"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fneutron-stars-and-nuclear-matter","Neutron Stars and Dense Matter","When a collapsing core passes nuclear density, electron capture converts the matter to neutrons and their degeneracy pressure takes over. The same balance that fixes a white dwarf, rescaled by the neutron mass, gives a neutron star of a few solar masses in a ten-kilometre radius. General relativity is no longer a correction: the Tolman-Oppenheimer-Volkoff equation replaces the Newtonian balance and sets a maximum mass around two solar masses. This lesson rescales the Fermi-gas argument, states where it breaks, and places the compact objects in one stability sequence.\n",{"path":24506,"title":24507,"module":24508,"summary":24509},"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-cluster-expansion-and-virial-coefficients","The Cluster Expansion and Virial Coefficients","Interacting Gases","A real gas departs from $PV=Nk_BT$ because its molecules interact. The configuration integral factors through the Mayer function $f_{ij}=e^{-\\beta u_{ij}}-1$, and expanding it in powers of density produces the virial expansion $PV\u002FNk_BT = 1 + B_2(T)n + B_3(T)n^2 + \\cdots$. The second virial coefficient $B_2(T)=-\\tfrac12\\int f\\,\\d^3r$ is a single integral over the pair potential; it is positive for a hard core, negative for an attractive well, and vanishes at the Boyle temperature where the two balance.\n",{"path":24511,"title":24512,"module":24508,"summary":24513},"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-van-der-waals-gas-and-liquid-gas-coexistence","The van der Waals Gas and Liquid-Gas Coexistence","Resumming the second virial coefficient $B_2=b-a\u002Fk_BT$ into an equation of state gives the van der Waals model $(P+a\u002Fv^2)(v-b)=k_BT$, the simplest theory of a fluid that condenses. Below the critical temperature its isotherms develop a mechanically unstable loop; the Maxwell equal-area construction replaces the loop with a coexistence tie line. The critical point sits at $v_c=3b$, $k_BT_c=8a\u002F27b$, $P_c=a\u002F27b^2$, and the model predicts universal but incorrect critical exponents because it ignores fluctuations.\n",{"path":24515,"title":24516,"module":24508,"summary":24517},"\u002Fstatistical-mechanics\u002Finteractions\u002Fquantum-gases-with-interactions-and-exchange","Quantum Gases with Interactions and Statistical Exchange","A quantum gas has a nonzero second virial coefficient even with no forces between the particles: symmetrization alone produces an effective statistical interaction, attractive for bosons and repulsive for fermions, with range the thermal wavelength $\\lambda$. This lesson derives that exchange contribution $B_2=\\mp\\lambda^3\u002F2^{5\u002F2}g$, writes it as a statistical potential $v_s(r)=-k_BT\\ln(1\\pm e^{-2\\pi r^2\u002F\\lambda^2})$, and shows how real interactions add on top through the Beth-Uhlenbeck phase-shift formula, reducing at low temperature to a single scattering length.\n",{"path":24519,"title":24520,"module":24521,"summary":24522},"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fphases-coexistence-and-classification","Phases, Coexistence, and the Classification of Transitions","Phase Transitions","A phase transition is a point where the free energy of a substance loses analyticity, so a small change in temperature or pressure produces a qualitative change of state. This lesson maps the coexistence curves of a pure substance, derives the Clausius-Clapeyron relation between the slope of a coexistence line and its latent heat, and separates first-order transitions (discontinuous entropy and density) from continuous ones (a vanishing order parameter and divergent response). The Ehrenfest scheme, the order parameter, and the triple and critical points fix the vocabulary the rest of the module builds on.\n",{"path":24524,"title":24525,"module":24521,"summary":24526},"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-ising-model-and-exact-solutions","The Ising Model and Exact Results","The Ising model reduces cooperative ordering to spins on a lattice coupled to their neighbors, and the same Hamiltonian describes uniaxial magnets, the liquid-gas critical point through the lattice gas, and binary alloys. This lesson solves the one-dimensional chain exactly with the transfer matrix, shows by a domain-wall argument why one dimension has no ordered phase at any positive temperature, contrasts the survival of order in two dimensions, and quotes Onsager's exact two-dimensional results: the critical temperature, the logarithmically divergent heat capacity, and the magnetization exponent one eighth.\n",{"path":24528,"title":24529,"module":24521,"summary":24530},"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fmean-field-theory-and-the-weiss-model","Mean-Field Theory and Spontaneous Symmetry Breaking","Mean-field theory replaces the neighbors of each spin by their average, turning the interacting Ising model into a single spin in a self-consistent field. The resulting equation m = tanh(beta J z m + beta h) has only the zero solution above a critical temperature and gains a nonzero root below it, giving spontaneous magnetization and a mean-field critical temperature k T_c = J z. The Bragg-Williams free energy turns single-welled above T_c and double-welled below, the picture of spontaneous symmetry breaking. The approximation is exact in high dimension and fails below the upper critical dimension four, quantified by the Ginzburg criterion.\n",{"path":24532,"title":24533,"module":24521,"summary":24534},"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fcritical-exponents-and-landau-theory","Critical Exponents, Scaling, and Landau Theory","Near a continuous transition every singular quantity follows a power law in the reduced temperature, and the exponents alpha, beta, gamma, delta, nu, and eta encode the transition more sharply than T_c itself. Landau theory expands the free energy in the order parameter and delivers the mean-field exponents in a few lines. They disagree with experiment and with the exact two-dimensional Ising values, but the exponents are not independent: the scaling relations of Rushbrooke, Widom, Fisher, and Josephson tie them together, and the correlation length sets the length scale that organizes universality classes.\n",{"path":24536,"title":24537,"module":24521,"summary":24538},"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-renormalization-group-idea","Scaling and the Renormalization-Group Idea","At a critical point fluctuations exist on every length scale, so the system looks the same after coarse-graining. The renormalization group makes this self-similarity a computation: group spins into blocks, integrate out the short scales, and track how the couplings change. The transformation has fixed points, and the flow near a critical fixed point separates relevant couplings that grow from irrelevant ones that shrink, which is why only dimension and symmetry survive to set the exponents. The one-dimensional Ising decimation carries the whole scheme through in closed form and reproduces the absence of a finite-temperature transition.\n",{"path":24540,"title":24541,"module":24542,"summary":24543},"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fthermodynamic-fluctuations-and-response","Thermodynamic Fluctuations and Response Functions","Fluctuations and Response","Thermodynamic variables are sharp only on average; a macroscopic system in equilibrium fluctuates about its mean values. Einstein inverted Boltzmann's $S=k_B\\ln\\Omega$ into a Gaussian probability for a fluctuation, $w\\propto e^{\\Delta S\u002Fk_B}$, and the second moments it predicts reproduce the response functions: $\\langle\\Delta E^2\\rangle=k_BT^2C_V$, $\\langle\\Delta V^2\\rangle=k_BTV\\kappa_T$, $\\langle\\Delta M^2\\rangle=k_BT\\chi_T$. The variances diverge where the responses diverge, at a critical point, producing critical opalescence and the breakdown of the thermodynamic description.\n",{"path":24545,"title":24546,"module":24542,"summary":24547},"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fbrownian-motion-and-the-langevin-equation","Brownian Motion and the Langevin Equation","A pollen grain in water executes a random walk driven by molecular collisions. Einstein tied its diffusion constant to its mobility, $D=\\mu_{\\mathrm{mob}}k_BT$, turning a visible motion into a measurement of Avogadro's number. The Langevin equation splits the collisions into a systematic drag and a random force whose strength is fixed by the drag through $\\langle\\xi(t)\\xi(t')\\rangle=2\\gamma k_BT\\,\\delta(t-t')$ — the first fluctuation–dissipation relation. The mean-square displacement grows ballistically at short times and linearly, $\\langle r^2\\rangle=2dDt$, at long times, and the Stokes–Einstein relation $D=k_BT\u002F6\\pi\\eta a$ closes the loop to Perrin's experiments.\n",{"path":24549,"title":24550,"module":24542,"summary":24551},"\u002Fstatistical-mechanics\u002Ffluctuations\u002Flinear-response-and-the-fluctuation-dissipation-theorem","Linear Response and the Fluctuation-Dissipation Theorem","A system driven by a weak external field responds through a generalized susceptibility $\\chi(\\omega)$ whose imaginary part measures dissipation. The Wiener–Khinchin theorem makes the power spectrum of equilibrium fluctuations the Fourier transform of their correlation function, and the fluctuation–dissipation theorem ties the two together: $S_x(\\omega)=(2k_BT\u002F\\omega)\\,\\chi''(\\omega)$, so the spectrum of spontaneous fluctuations is fixed by the dissipative response. The Johnson–Nyquist noise of a resistor, $\\langle V^2\\rangle=4k_BTR\\,\\Delta f$, is the canonical example, and Onsager reciprocity closes the subject.\n",{"path":24553,"title":24554,"module":6,"summary":6},"\u002Fstatistical-mechanics","Statistical Mechanics",{"path":24556,"title":24557,"module":24558,"summary":24559},"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fbonding-mechanisms","Bonding Mechanisms","Molecules and Chemical Bonding","A molecule forms when the total energy of two atoms drops below the energy of the separated pair. This lesson works through the four mechanisms that produce that minimum: the ionic bond from charge transfer, the covalent bond from shared electron wave functions, the metallic bond, and the weak dipole-dipole and hydrogen bonds, computing bond lengths and dissociation energies for NaCl, H₂, and H₂⁺.\n",{"path":24561,"title":24562,"module":24558,"summary":24563},"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fmolecular-orbitals-and-h2-plus","The Molecular-Orbital Method and H₂⁺","The hydrogen molecule ion is the two-center problem that fixes the language of chemical bonding. This lesson builds the molecular orbital as a linear combination of atomic orbitals, minimizes the energy through the variational secular equation, and reduces the result to three two-center integrals: the overlap, the Coulomb term, and the exchange (resonance) integral. The bonding and antibonding levels, their potential-energy curves, and the charge piled between the nuclei follow from those integrals.\n",{"path":24565,"title":24566,"module":24558,"summary":24567},"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fhydrogen-molecule-and-exchange","The Hydrogen Molecule, Exchange, and Hybridization","Adding the second electron turns the one-electron ion into the two-electron hydrogen molecule, where electron-electron repulsion and the Pauli principle govern the bond. This lesson contrasts the Heitler-London valence-bond and molecular-orbital wave functions, derives the singlet-triplet splitting as an exchange energy, shows why naive molecular orbitals fail at dissociation, and builds the sp, sp², and sp³ hybrids that fix the directed geometry of covalent bonds.\n",{"path":24569,"title":24570,"module":24558,"summary":24571},"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fvan-der-waals-forces","Van der Waals Forces","The bond of last resort acts between all atoms, even closed-shell noble gases with no permanent moment. This lesson separates the three van der Waals contributions — Keesom orientation, Debye induction, and London dispersion — derives the London 1\u002Fr⁶ attraction from the coupled-oscillator and second-order perturbation pictures, and assembles the Lennard-Jones potential to compute the equilibrium spacing and cohesive energy of the noble-gas crystals, argon in particular.\n",{"path":24573,"title":24574,"module":24575,"summary":24576},"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Frotational-vibrational-spectra","Rotational and Vibrational Spectra of Molecules","Molecular Spectra","A diatomic molecule stores energy in three well-separated ledgers: electronic, vibrational, and rotational. Quantizing the rigid rotor gives levels spaced as ℓ(ℓ+1); quantizing the bond as a harmonic oscillator gives equally spaced vibrational levels. Their combination produces the P and R branches of an infrared absorption band, from which the bond length and force constant are read directly.\n",{"path":24578,"title":24579,"module":24575,"summary":24580},"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Fanharmonicity-and-rovibrational-structure","Anharmonicity and Rovibrational Structure","The rigid rotor and harmonic oscillator are first approximations. A real bond follows the Morse potential, whose levels converge toward dissociation; a real rotor stretches centrifugally; and vibration couples to rotation, so the rotational constant depends on the vibrational level. This lesson works out the anharmonic and centrifugal corrections, the Birge-Sponer route to the dissociation energy, the isotope shift, and the thermal band envelope.\n",{"path":24582,"title":24583,"module":24575,"summary":24584},"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Framan-and-electronic-bands","Raman Scattering and Electronic Bands","Not every vibration absorbs in the infrared. Raman scattering reaches modes that modulate the polarizability, giving Stokes and anti-Stokes lines whose intensity ratio measures temperature, and the mutual-exclusion rule pairs it with infrared absorption. Electronic transitions add the vibronic structure of band spectra, governed by the Franck-Condon principle, and the radiative fates of an excited state are sorted by the Jablonski diagram into fluorescence and phosphorescence.\n",{"path":24586,"title":24587,"module":24575,"summary":24588},"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Flasers-and-masers","Lasers, Masers, and Stimulated Emission","Einstein's three radiative processes — absorption, spontaneous emission, and stimulated emission — and the coefficients that relate them. Stimulated emission produces coherent photons, and inverting the level populations turns it into net amplification. We build the ruby three-level laser and the helium-neon four-level laser, and show why the fourth level makes inversion easy.\n",{"path":24590,"title":24591,"module":24592,"summary":24593},"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fstructure-of-solids","The Structure of Solids","Crystal Structure","A crystal is a unit cell repeated in three dimensions. We classify the common cubic lattices, compute the Coulomb energy of an ionic crystal through the Madelung constant, and show how the divergent naive lattice sum is tamed by cubic shells. The cohesive energy that results predicts melting points and connects the diatomic bond of an earlier lesson to the bulk solid.\n",{"path":24595,"title":24596,"module":24592,"summary":24597},"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fbravais-lattices-and-crystal-systems","Bravais Lattices, Bases, and Crystal Structures","A crystal is a Bravais lattice decorated by a basis. This lesson separates the two, builds primitive and Wigner-Seitz cells, enumerates the seven crystal systems and fourteen Bravais lattices, and fixes the language of point and space groups. Miller indices label planes and directions, and the packing fractions of the close-packed, cubic, and diamond structures follow from the geometry.\n",{"path":24599,"title":24600,"module":24592,"summary":24601},"\u002Fcondensed-matter\u002Fcrystal-structure\u002Freciprocal-lattice-and-brillouin-zones","The Reciprocal Lattice and Brillouin Zones","Every periodic crystal has a dual lattice in wavevector space. This lesson defines the reciprocal lattice through the condition b_i dot a_j equals two pi delta, derives its properties, shows the reciprocal of fcc is bcc, links reciprocal vectors to families of lattice planes, and builds the first Brillouin zone as the Wigner-Seitz cell of the reciprocal lattice, including the higher zones.\n",{"path":24603,"title":24604,"module":24592,"summary":24605},"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fdiffraction-and-structure-factors","X-ray and Neutron Diffraction","A crystal diffracts radiation whose wavelength matches its atomic spacing. This lesson derives the Bragg condition, the equivalent Laue condition 2k dot G equals G squared, and the Ewald-sphere construction, then computes the geometric structure factor that produces systematic absences for bcc and fcc, the atomic form factor, and the powder method. It closes on why neutrons and electrons complement X-rays.\n",{"path":24607,"title":24608,"module":24609,"summary":24610},"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonon-dispersion","The Harmonic Crystal and Phonon Dispersion","Lattice Dynamics","Atoms in a crystal vibrate about their equilibrium sites, and expanding the potential to second order turns the whole lattice into a set of coupled harmonic oscillators. This lesson sets up the harmonic approximation and the dynamical matrix, solves the monatomic linear chain for its dispersion omega(k) = 2 sqrt(K\u002FM) times the absolute sine of ka over two, explains why wavevectors outside the first Brillouin zone are redundant, and extends the chain to two atoms per cell to produce acoustic and optical branches with a frequency gap.\n",{"path":24612,"title":24613,"module":24609,"summary":24614},"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonons-quantization-and-dos","Phonons, Density of States, and Crystal Momentum","Quantizing the normal modes of a harmonic crystal turns each vibrational mode into a quantum oscillator whose excitations are phonons. This lesson counts phonons with Bose-Einstein statistics, defines crystal momentum and the normal versus Umklapp distinction in momentum conservation, builds the density of states with its van Hove singularities, and shows how inelastic neutron scattering measures a dispersion curve point by point.\n",{"path":24616,"title":24617,"module":24609,"summary":24618},"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fdebye-einstein-heat-capacity","Thermal Properties — Einstein and Debye Models","The lattice heat capacity follows the classical Dulong-Petit value at high temperature but collapses toward zero as T approaches zero, a purely quantum effect. This lesson derives that behavior from the Einstein model of a single frequency, then the Debye model of a linear phonon spectrum with a cutoff, obtaining the Debye T-cubed law at low temperature and the Debye interpolation across all temperatures, and closes with thermal expansion and the Gruneisen parameter.\n",{"path":24620,"title":24621,"module":24609,"summary":24622},"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fanharmonicity-and-thermal-transport","Anharmonicity, Thermal Expansion, and Heat Conduction","A perfectly harmonic crystal neither expands when heated nor resists heat flow. Both effects come from the cubic and higher terms the harmonic approximation discards. This lesson derives thermal expansion from an asymmetric interatomic potential, treats phonon-phonon scattering as the decay channel these terms open, shows why Umklapp processes are what make lattice thermal conductivity finite, and traces the temperature dependence of the conductivity and the phonon mean free path.\n",{"path":24624,"title":24625,"module":24626,"summary":24627},"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ffree-electron-gas-and-conduction","Conduction and the Free-Electron Gas","Free-Electron Fermi Gas","Drude's classical free-electron model gets Ohm's law right but the resistivity, its temperature dependence, and the heat capacity wrong. Replacing the Maxwell-Boltzmann distribution with the Fermi-Dirac distribution and treating electron-lattice collisions as wave scattering repairs all three: the Fermi energy, Fermi speed, and a mean free path set by thermal lattice vibrations.\n",{"path":24629,"title":24630,"module":24626,"summary":24631},"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fsommerfeld-model-and-heat-capacity","The Sommerfeld Model: Ground State and Heat Capacity","Quantizing the free-electron gas in a box fills a Fermi sphere in k-space. The density of states grows as the square root of energy in three dimensions, and the Fermi energy, temperature, and wavevector follow for real metals. The Sommerfeld expansion shows only a thermal shell of width k_BT near E_F is excited, giving an electronic heat capacity linear in T that sits beneath the phonon T-cubed term.\n",{"path":24633,"title":24634,"module":24626,"summary":24635},"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ftransport-and-the-hall-effect","Transport, Wiedemann–Franz, and the Hall Effect","The relaxation-time picture displaces the Fermi sphere under an applied field and gives the electrical conductivity ne-squared-tau over m. The same electrons carry heat, and their ratio yields the Wiedemann–Franz law with the universal Lorenz number. A magnetic field bends the carriers into cyclotron orbits and produces the Hall voltage, whose sign reveals the charge of the carriers.\n",{"path":24637,"title":24638,"module":24626,"summary":24639},"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fscreening-and-plasmons","Screening, Plasmons, and the Limits of Free Electrons","A mobile electron gas rearranges to screen any foreign charge, turning the bare Coulomb potential into a short-ranged Yukawa form over the Thomas–Fermi length. Displaced collectively, the gas rings at the plasma frequency, whose quantum is the plasmon and whose value sets the reflectivity edge of metals. A ledger of free-electron successes and failures then motivates band theory.\n",{"path":24641,"title":24642,"module":24643,"summary":24644},"\u002Fcondensed-matter\u002Fband-theory\u002Fblochs-theorem-and-energy-bands","Bloch's Theorem and Energy Bands","Band Theory","An electron in a periodic potential has stationary states that are plane waves modulated by a lattice-periodic envelope. This lesson proves Bloch's theorem two ways, defines crystal momentum and the band index, counts the allowed wavevectors from Born–von Kármán boundary conditions, and sets up the extended, reduced, and repeated-zone descriptions of a band.\n",{"path":24646,"title":24647,"module":24643,"summary":24648},"\u002Fcondensed-matter\u002Fband-theory\u002Fnearly-free-electron-model","The Nearly-Free-Electron Model","A weak periodic potential leaves the free-electron parabola almost intact except near Brillouin-zone boundaries, where two nearly degenerate plane waves mix. This lesson solves the resulting two-by-two secular problem, shows the gap of size twice the potential component opening at each boundary, identifies the two standing waves that pile charge on and between the ions, and works the exactly solvable Kronig–Penney model.\n",{"path":24650,"title":24651,"module":24643,"summary":24652},"\u002Fcondensed-matter\u002Fband-theory\u002Ftight-binding-method","The Tight-Binding Method","The opposite limit to nearly-free electrons builds bands from atomic orbitals. A Bloch sum of one orbital per site gives a dispersion set by the hopping integral between neighbours; the band widens from a sharp atomic level as the atoms approach. This lesson derives the s-band cosine dispersion, extends it to p-bands, and introduces Wannier functions as the localized dual of Bloch states.\n",{"path":24654,"title":24655,"module":24643,"summary":24656},"\u002Fcondensed-matter\u002Fband-theory\u002Ffermi-surfaces-and-semiclassical-dynamics","Fermi Surfaces, Effective Mass, and Metals vs Insulators","Filling the bands settles which crystals conduct. A filled band carries no current, so a crystal with filled bands and a gap is an insulator, while a partly filled band makes a metal. This lesson derives the no-current theorem for a filled band, defines the Fermi surface and Harrison's construction, introduces holes and the effective mass from band curvature, and states the semiclassical equations of motion that lead to Bloch oscillations.\n",{"path":24658,"title":24659,"module":24660,"summary":24661},"\u002Fcondensed-matter\u002Fsemiconductors\u002Fsemiconductor-bands-and-junctions","Band Theory and Semiconductors","Semiconductors","The periodic lattice splits atomic levels into allowed energy bands separated by forbidden gaps. Whether the highest occupied band is full or partly full, and how wide the gap above it is, sorts every solid into conductor, insulator, or semiconductor. Doping adds donor or acceptor levels inside the gap, and a p-n junction built from doped regions gives the diode, the solar cell, the LED, and the transistor.\n",{"path":24663,"title":24664,"module":24660,"summary":24665},"\u002Fcondensed-matter\u002Fsemiconductors\u002Fintrinsic-and-extrinsic-semiconductors","Carrier Statistics: Intrinsic and Extrinsic Semiconductors","The number of mobile electrons and holes in a semiconductor follows from the density of states near each band edge and the Fermi-Dirac tail that reaches into it. This lesson derives the effective densities of states, the intrinsic concentration and its exponential gap dependence, the law of mass action, the temperature march of the Fermi level, and the freeze-out, saturation, and intrinsic regimes of a doped crystal.\n",{"path":24667,"title":24668,"module":24660,"summary":24669},"\u002Fcondensed-matter\u002Fsemiconductors\u002Fcarrier-transport-and-recombination","Carrier Transport and Recombination","Carriers move by drift in a field and by diffusion down a concentration gradient, the two tied together by the Einstein relation. This lesson derives mobility and its scattering-limited temperature dependence, the drift and diffusion currents, the continuity equations, band-to-band and trap-assisted recombination, and the minority-carrier lifetime and diffusion length that set the length scale of every junction device.\n",{"path":24671,"title":24672,"module":24660,"summary":24673},"\u002Fcondensed-matter\u002Fsemiconductors\u002Fthe-pn-junction","The p-n Junction in Depth","Joining p-type and n-type silicon aligns their Fermi levels and leaves a depletion region of fixed charge with a built-in potential. This lesson derives the space-charge field and potential from Poisson's equation in the depletion approximation, the built-in voltage from Fermi-level alignment, the Shockley diode equation from minority-carrier diffusion, junction and diffusion capacitance, and the avalanche and Zener breakdown mechanisms.\n",{"path":24675,"title":24676,"module":24660,"summary":24677},"\u002Fcondensed-matter\u002Fsemiconductors\u002Ftransistors-and-optoelectronics","Transistors and Optoelectronic Devices","Two junctions in series make a bipolar transistor whose thin base gives current gain; a gate over an oxide makes a MOSFET whose inversion channel switches digital logic. Run in reverse, a junction converts photons to current. This lesson derives the transistor current gain and the MOSFET channel current, then treats the LED, the diode laser, and the illuminated solar-cell characteristic.\n",{"path":24679,"title":24680,"module":24681,"summary":24682},"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fdielectrics-and-polarization","Dielectrics, Polarization, and the Local Field","Dielectrics and Ferroelectrics","An insulator responds to an electric field by polarizing. This lesson builds the macroscopic polarization and the dielectric constant, sorts the microscopic response into electronic, ionic, and orientational polarizability, and corrects the field an atom actually feels to the Lorentz local field E + P\u002F3 epsilon-0. The Clausius-Mossotti relation links the measured permittivity to the atomic polarizability, and the frequency dependence of each mechanism explains why the static and optical dielectric constants differ.\n",{"path":24684,"title":24685,"module":24681,"summary":24686},"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fferroelectrics-and-piezoelectrics","Ferroelectrics, Piezoelectrics, and Structural Transitions","Some crystals carry a polarization with no applied field and switch it under a reversing field, tracing a hysteresis loop. This lesson develops the ferroelectric transition through the perovskite BaTiO3 displacive instability and its soft transverse-optical mode, builds the Landau free-energy theory of first- and second-order polar transitions, derives the Curie-Weiss divergence of the dielectric constant, and closes with piezoelectricity and pyroelectricity and their devices.\n",{"path":24688,"title":24689,"module":24690,"summary":24691},"\u002Fcondensed-matter\u002Fmagnetism\u002Fdiamagnetism-and-paramagnetism","Diamagnetism and Paramagnetism","Magnetism in Solids","Every solid responds to a magnetic field. Filled shells give a small negative diamagnetic susceptibility from induced Larmor currents; localized moments give a positive Curie paramagnetism described by the Brillouin function, with the ground-state moment fixed by Hund's rules. The conduction electrons add a temperature-independent Pauli paramagnetism from the thermal shell near the Fermi surface, partly cancelled by Landau diamagnetism of their orbital motion.\n",{"path":24693,"title":24694,"module":24690,"summary":24695},"\u002Fcondensed-matter\u002Fmagnetism\u002Fexchange-and-ferromagnetism","Exchange and Ferromagnetism","Magnetic ordering at hundreds of kelvin cannot be dipolar; it is an exchange effect, the Coulomb repulsion sorted by the Pauli principle into a spin-dependent energy captured by the Heisenberg Hamiltonian. Weiss molecular-field theory replaces the exchange field by an average proportional to the magnetization, giving a self-consistent equation whose solution is spontaneous magnetization below a Curie temperature and a Curie–Weiss susceptibility above it. Itinerant ferromagnetism follows from the Stoner criterion on the band density of states.\n",{"path":24697,"title":24698,"module":24690,"summary":24699},"\u002Fcondensed-matter\u002Fmagnetism\u002Fantiferromagnetism-and-domains","Antiferromagnetism, Ferrimagnetism, and Domains","A negative exchange coupling orders neighboring spins antiparallel. Two-sublattice molecular-field theory gives a Néel temperature marked by a cusp in the susceptibility, and unequal sublattices leave a net moment — ferrimagnetism, the magnetism of the ferrites. A ferromagnet breaks into domains to reduce its magnetostatic energy, separated by Bloch walls whose width is set by the competition between exchange and magnetocrystalline anisotropy, and the irreversible motion of those walls produces the hysteresis loop.\n",{"path":24701,"title":24702,"module":24690,"summary":24703},"\u002Fcondensed-matter\u002Fmagnetism\u002Fspin-waves-and-magnons","Spin Waves and Magnons","The lowest excitations of a ferromagnet are not single flipped spins but collective precessions in which every moment tips slightly and its phase advances along the crystal. These spin waves have a quadratic dispersion at long wavelength, quantize into magnons obeying Bose statistics, and their thermal population removes magnetization as the Bloch T-to-the-three-halves law. Antiferromagnetic magnons disperse linearly, and inelastic neutron scattering measures both.\n",{"path":24705,"title":24706,"module":24707,"summary":24708},"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fsuperconductivity-phenomenology","Superconductivity: Phenomenology and BCS","Superconductivity","Below a critical temperature some materials lose all resistance and expel magnetic flux — the Meissner effect that defines the state. The isotope effect points to lattice vibrations, and BCS theory binds electrons into Cooper pairs through phonon exchange. The paired condensate opens an energy gap, quantizes magnetic flux, and drives the Josephson effects.\n",{"path":24710,"title":24711,"module":24707,"summary":24712},"\u002Fcondensed-matter\u002Fsuperconductivity\u002Flondon-theory-and-the-meissner-effect","London Theory and the Meissner Effect","A perfect conductor freezes the field it was cooled in; a superconductor expels it. The distinction needs a constitutive law beyond zero resistance — the two London equations — whose solution is exponential flux decay over the penetration depth. The same rigidity follows from a macroscopic condensate wave function, and the thermodynamics of the critical field fixes the condensation energy, the latent heat, and the specific-heat jump.\n",{"path":24714,"title":24715,"module":24707,"summary":24716},"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fginzburg-landau-theory","Ginzburg–Landau Theory, Vortices, and Type-II","A complex order parameter and a free-energy expansion turn the superconducting transition into a Landau theory. Two lengths emerge — the coherence length and the penetration depth — whose ratio kappa sorts superconductors into type I and type II. Type-II materials admit flux as an Abrikosov lattice of vortices, each threading exactly one quantum h\u002F2e, between a lower and an upper critical field.\n",{"path":24718,"title":24719,"module":24707,"summary":24720},"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fbcs-theory","Microscopic BCS Theory","A phonon-mediated attraction, however weak, binds two electrons above the Fermi sea — the Cooper problem shows the sea is unstable. The BCS variational ground state pairs all electrons near the Fermi surface and, through a self-consistent gap equation, opens an energy gap. Weak-coupling solution gives the exponential T_c and the universal ratios 2 Delta(0) = 3.53 k_B T_c and Delta C \u002F C_n = 1.43.\n",{"path":24722,"title":24723,"module":24707,"summary":24724},"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fjosephson-and-high-tc","Josephson Effects and Unconventional Superconductors","Two superconductors joined by a thin barrier carry a supercurrent set by their phase difference — the dc Josephson effect — and oscillate at 2eV\u002Fh under a voltage. A two-junction loop turns flux quantization into a magnetometer of single-quantum sensitivity. The cuprates superconduct in CuO2 planes with a doping-dependent dome, d-wave pairing, and a pseudogap that lie outside the phonon picture.\n",{"path":24726,"title":24727,"module":24728,"summary":24729},"\u002Fcondensed-matter\u002Fnanostructures\u002Fquantum-wells-wires-and-dots","Quantum Wells, Wires, and Dots","Nanostructures","When a crystal is shrunk until one or more of its dimensions approaches the electron wavelength, the continuous bands of the bulk break into discrete subbands. Confining in one direction gives a quantum well with a step-like density of states, in two directions a quantum wire with inverse-square-root singularities, and in all three a quantum dot whose levels are sharp like an atom's. This lesson derives the density of states in each case and applies it to size-tunable dot emission and the Coulomb blockade of a single-electron transistor.\n",{"path":24731,"title":24732,"module":24728,"summary":24733},"\u002Fcondensed-matter\u002Fnanostructures\u002Finteger-quantum-hall-effect","The 2D Electron Gas and the Integer Quantum Hall Effect","A two-dimensional electron gas in a strong perpendicular magnetic field has its continuous density of states collapse into macroscopically degenerate Landau levels. As the field is swept, the Hall resistance locks onto exact plateaus at h over an integer times e squared, while the longitudinal resistance drops to zero. This lesson derives the Landau levels and their degeneracy, explains the plateaus through disorder-localized states and current-carrying edge channels, and states why the von Klitzing constant is now a resistance standard.\n",{"path":24735,"title":24736,"module":24728,"summary":24737},"\u002Fcondensed-matter\u002Fnanostructures\u002Ffractional-quantum-hall-and-topology","The Fractional Quantum Hall Effect and Topological Order","When the lowest Landau level is only partly filled, the non-interacting theory predicts no gap, yet a plateau appears at filling one-third. It is a many-body effect: Coulomb repulsion selects a correlated ground state, the Laughlin wavefunction, whose excitations carry a fraction of the electron charge. This lesson builds the Laughlin state, introduces composite fermions that map the fractional effect onto an integer one, and explains how the quantum Hall effect brought the Chern number and topology into condensed-matter physics.\n",{"path":24739,"title":24740,"module":24728,"summary":24741},"\u002Fcondensed-matter\u002Fnanostructures\u002Fgraphene-and-dirac-materials","Graphene and Dirac Materials","Graphene is one atomic layer of carbon on a honeycomb lattice. A tight-binding calculation on its two-atom basis gives valence and conduction bands that touch at the corners of the Brillouin zone, where the dispersion is linear and the electrons behave as massless two-dimensional Dirac particles with a fixed speed. This lesson derives the Dirac cones, the Berry phase of pi and the sublattice chirality, the anomalous half-integer quantum Hall effect that follows, and how opening a gap in a Dirac cone points toward topological insulators.\n",{"path":24743,"title":24744,"module":6,"summary":6},"\u002Fcondensed-matter","Condensed Matter Physics",{"path":24746,"title":24747,"module":21519,"summary":24748},"\u002Flogic\u002Ffoundations\u002Flogic-as-a-mathematical-model","Logic as a Mathematical Model of Deduction","Symbolic logic models deductive reasoning the way probability theory models chance: it keeps the form of a correct deduction and discards its content. A deduction is valid when its conclusion follows from the form of the premises alone, independent of what the non-logical words mean. Two models carry the subject — coarse sentential logic and fine first-order logic — and four questions organize it: logical consequence, methods of proof, the gap between provable and true, and the link between logic and computability. Tuples, relations, functions, equivalence classes, and cardinality supply the set-theoretic vocabulary every later chapter uses.\n",{"path":24750,"title":24751,"module":24752,"summary":24753},"\u002Flogic\u002Fsentential-logic\u002Fformal-languages-and-well-formed-formulas","Formal Languages and Well-Formed Formulas","Sentential Logic","The language of sentential logic has an alphabet of sentence symbols, five connectives, and two parentheses, with formation rules that pick out the well-formed formulas. The wffs are the least set of expressions closed under the five formula-building operations, and every such generated set carries an induction principle.\n",{"path":24755,"title":24756,"module":24752,"summary":24757},"\u002Flogic\u002Fsentential-logic\u002Ftruth-assignments-and-tautologies","Truth Assignments, Tautologies, and Consequence","A truth assignment fixes the sentence symbols true or false, and a recursion extends it uniquely to every formula. Satisfaction, tautologies, and tautological implication — one formula following semantically from others — rest on that extension, and the truth-table procedure decides implication for finite premise sets.\n",{"path":24759,"title":24760,"module":24752,"summary":24761},"\u002Flogic\u002Fsentential-logic\u002Funique-readability-and-parsing","Unique Readability and a Parsing Algorithm","Parentheses keep a formula from being read two ways. The parenthesis lemmas and a top-down parsing algorithm recover a formula's structure and yield unique readability: every wff has exactly one formation tree, which is what makes the truth recursion well defined.\n",{"path":24763,"title":24764,"module":24752,"summary":24765},"\u002Flogic\u002Fsentential-logic\u002Finduction-and-recursion","Induction and Recursion on Formulas","Two principles govern any set generated from initial elements by operations: prove a property of all its members by covering the initial elements and the closure steps, and define a function on it by recursion on structure. The recursion theorem needs the set to be freely generated, and unique readability supplies that condition for the well-formed formulas.\n",{"path":24767,"title":24768,"module":24752,"summary":24769},"\u002Flogic\u002Fsentential-logic\u002Fexpressive-completeness-and-normal-forms","Sentential Connectives and Normal Forms","Every formula computes a Boolean function of its atoms, and Post's theorem gives the converse: every Boolean function is realized by a wff in disjunctive normal form, so the five connectives are more than enough. Minimal complete sets follow, down to the single connectives NAND and NOR, together with a method for proving a set of connectives incomplete.\n",{"path":24771,"title":24772,"module":24752,"summary":24773},"\u002Flogic\u002Fsentential-logic\u002Fboolean-circuits","Switching Circuits","A memoryless two-valued circuit computes a Boolean function, so every formula names a gate network and every network a formula. Cost and delay are read off the formula by recursion, and tautological equivalence and normal forms design and simplify circuits realizing a given specification.\n",{"path":24775,"title":24776,"module":24752,"summary":24777},"\u002Flogic\u002Fsentential-logic\u002Fcompactness-and-effectiveness","Compactness and Effectiveness","The compactness theorem reduces satisfiability of an infinite set of formulas to its finite subsets, proved by extension to a maximal finitely satisfiable set and applied to color infinite graphs. Effectiveness fixes what \"decidable\" and \"effectively enumerable\" mean and settles the decidability of tautologyhood.\n",{"path":24779,"title":24780,"module":24781,"summary":24782},"\u002Flogic\u002Ffirst-order-languages\u002Ffirst-order-languages","First-Order Languages","First-Order Languages and Structures","Sentential logic cannot see inside a simple statement, so it misses valid arguments that turn on quantifiers and predicates. A first-order language adds a quantifier, variables, and a chosen vocabulary of predicate, function, and constant symbols. Terms and well-formed formulas are built by recursion over this alphabet, and a variable occurs free or bound according to the quantifiers that reach it.\n",{"path":24784,"title":24785,"module":24781,"summary":24786},"\u002Flogic\u002Ffirst-order-languages\u002Fstructures-truth-and-satisfaction","Structures, Truth, and Satisfaction","A structure interprets a language: a nonempty universe plus a meaning for every predicate, function, and constant symbol. Tarski's recursion defines when a structure satisfies a formula under a variable assignment, and hence when a sentence is true. From satisfaction we recover logical implication, validity, and logical equivalence for first-order logic.\n",{"path":24788,"title":24789,"module":24781,"summary":24790},"\u002Flogic\u002Ffirst-order-languages\u002Fdefinability-and-elementary-equivalence","Definability and Elementary Equivalence","Fix a structure and ask which relations a formula can pick out: the definable ones. A set of sentences picks out a class of structures, the elementary classes. Homomorphisms and isomorphisms compare structures, and the homomorphism theorem shows isomorphic structures satisfy the same sentences. Automorphisms bound what first-order logic can distinguish, giving a tool for proving relations undefinable.\n",{"path":24792,"title":24793,"module":24781,"summary":24794},"\u002Flogic\u002Ffirst-order-languages\u002Fterms-substitution-and-parsing","Parsing, Substitution, and Substitutability","Every recursion on first-order syntax rests on unique readability. A parenthesis-counting function proves that terms and formulas decompose in exactly one way, and a parsing algorithm recovers the decomposition. Substituting a term for a free variable can capture it under a quantifier; the substitutability condition rules that out, and the substitution lemma trades syntactic substitution for a change of assignment.\n",{"path":24796,"title":24797,"module":24798,"summary":24799},"\u002Flogic\u002Fdeductive-calculus\u002Fa-deductive-calculus","A Deductive Calculus for First-Order Logic","The Deductive Calculus and Its Metatheorems","A proof must be finite and mechanically checkable. A Hilbert-style calculus meets both demands: six schemas of logical axioms, a single rule of inference (modus ponens), and the syntactic consequence relation they generate. Substitution and substitutability are defined by recursion, and the bridge theorem reduces deducibility to tautological implication from the axioms.\n",{"path":24801,"title":24802,"module":24798,"summary":24803},"\u002Flogic\u002Fdeductive-calculus\u002Fdeduction-theorem-and-derived-rules","The Deduction Theorem and Derived Rules","Raw deductions from axioms are unusable by hand. The generalization theorem, the deduction theorem, contraposition, reductio ad absurdum, and rule T reduce the calculus to the moves of ordinary mathematics, each proved once to license a block of axiom-level steps. Generalization on constants and alphabetic variants handle the quantifier and substitution bookkeeping.\n",{"path":24805,"title":24806,"module":24798,"summary":24807},"\u002Flogic\u002Fdeductive-calculus\u002Fsoundness","The Soundness Theorem","Soundness is the easy half of the match between proof and truth. Whatever the calculus deduces is logically implied, by an induction on deduction length that rests on one lemma: every logical axiom is valid. The only hard case, quantifier instantiation, needs the substitution lemma. The contrapositive corollary states that every satisfiable set is consistent.\n",{"path":24809,"title":24810,"module":24798,"summary":24811},"\u002Flogic\u002Fdeductive-calculus\u002Fcompleteness-and-consistency","The Completeness Theorem","Gödel's completeness theorem is the deep converse of soundness: whatever is logically implied can be deduced. Equivalently, every consistent set has a model. The Henkin proof manufactures that model out of syntax alone: add witnessing constants, extend to a maximal consistent set, and read a term model off the formulas it contains. Compactness and the enumerability theorem drop out.\n",{"path":24813,"title":24814,"module":24815,"summary":24816},"\u002Flogic\u002Fmodels-and-theories\u002Fcompactness-and-lowenheim-skolem","Compactness and the Löwenheim–Skolem Theorems","Models, Compactness, and Theories","A set of first-order sentences has a model whenever each of its finite subsets does. This compactness theorem follows from completeness and yields the finiteness limitation, the downward and upward Löwenheim–Skolem theorems, models of every infinite cardinality, and nonstandard models of arithmetic.\n",{"path":24818,"title":24819,"module":24815,"summary":24820},"\u002Flogic\u002Fmodels-and-theories\u002Ftheories-elementary-classes-and-categoricity","Theories, Elementary Classes, and Categoricity","A theory is a set of sentences closed under logical consequence. Theories correspond to classes of models; a theory may be complete, axiomatizable, or finitely axiomatizable, and completeness together with axiomatizability yields decidability. The Łoś–Vaught test derives completeness from categoricity in a cardinal, applied to dense linear orders and to algebraically closed fields.\n",{"path":24822,"title":24823,"module":24815,"summary":24824},"\u002Flogic\u002Fmodels-and-theories\u002Finterpretations-between-theories","Interpretations Between Theories","An interpretation translates the vocabulary of one theory into formulas of another, relativizing quantifiers to a definable domain and mapping symbols to defining formulas. Defined function symbols meet a noncreativity criterion; the syntactic translation of formulas carries theoremhood forward, and a faithful interpretation transfers decidability and undecidability between theories.\n",{"path":24826,"title":24827,"module":24815,"summary":24828},"\u002Flogic\u002Fmodels-and-theories\u002Fnonstandard-analysis","Nonstandard Analysis","Compactness builds a model of the real ordered field containing infinite elements and nonzero infinitesimals. The transfer principle carries every first-order truth from the reals to this extension, the standard-part map collapses finite hyperreals back onto the reals, and continuity and the derivative are rederived by working with infinitely small quantities directly.\n",{"path":24830,"title":24831,"module":24832,"summary":24833},"\u002Flogic\u002Farithmetic-and-definability\u002Fdefinability-in-arithmetic","The Structure of Arithmetic and Definability","Number Theory and Definability","Number theory is the theory of one fixed structure, the natural numbers under successor, order, addition, multiplication, and exponentiation. Every number is named by a numeral, and a relation is definable when a single formula picks out exactly its tuples. The central gap separates the sentences true in that structure from those any reasonable set of axioms can prove.\n",{"path":24835,"title":24836,"module":24832,"summary":24837},"\u002Flogic\u002Farithmetic-and-definability\u002Fnatural-numbers-with-successor","Natural Numbers with Successor","The weakest reduct keeps only zero and successor. Its models are a standard chain together with disjoint copies of the integers, which makes the theory categorical in every uncountable power, hence complete and decidable. A quantifier-elimination procedure gives a practical decision method and shows a subset is definable if and only if it is finite or cofinite.\n",{"path":24839,"title":24840,"module":24832,"summary":24841},"\u002Flogic\u002Farithmetic-and-definability\u002Fpresburger-and-reducts","Reducts: Order, Addition, and Multiplication","Adding order to the successor reduct keeps decidability and makes the theory finitely axiomatizable; adding addition gives Presburger arithmetic, still decidable by quantifier elimination once congruence predicates are included, with definable sets exactly the eventually periodic ones. Multiplication is the break point: neither addition nor order can define it, and once it joins addition the theory stops being decidable.\n",{"path":24843,"title":24844,"module":24832,"summary":24845},"\u002Flogic\u002Farithmetic-and-definability\u002Fa-subtheory-and-representability","A Subtheory of Number Theory and Representability","A finite set of eleven axioms, the recursion equations for successor, order, addition, multiplication, and exponentiation, already proves every true quantifier-free and existential sentence. Representability asks a theory to prove the right instances of a formula rather than merely make them true, and a relation is defined to be recursive exactly when some consistent finite theory represents it. Church's thesis identifies that with decidability, and closure under composition, minimization, and primitive recursion builds the catalog the incompleteness proofs need.\n",{"path":24847,"title":24848,"module":24849,"summary":24850},"\u002Flogic\u002Fincompleteness\u002Farithmetization-of-syntax","Arithmetization of Syntax","Arithmetization and the Incompleteness Theorems","Gödel numbering assigns a natural number to every symbol, expression, formula, and deduction, turning statements about syntax into statements about numbers. The syntactic operations — substitution, \"is a wff\", \"is an axiom\", \"d codes a deduction of a\" — come out primitive recursive and hence representable in the subtheory, which lets a formula of arithmetic talk about formulas, including itself.\n",{"path":24852,"title":24853,"module":24849,"summary":24854},"\u002Flogic\u002Fincompleteness\u002Fincompleteness-and-undecidability","Incompleteness, Undecidability, and Church's Theorem","The fixed-point lemma manufactures a sentence that talks about its own Gödel number. Pointed at truth it gives Tarski's theorem — arithmetic truth is not arithmetically definable; pointed at provability it gives Gödel's first incompleteness theorem and the undecidability of the theory of the natural numbers, and, applied to validity, Church's theorem that first-order logic is undecidable. The set of theorems of a recursive theory is only recursively enumerable — the gap between provable and true.\n",{"path":24856,"title":24857,"module":24849,"summary":24858},"\u002Flogic\u002Fincompleteness\u002Fsecond-incompleteness-theorem","The Second Incompleteness Theorem","Consistency of a recursively axiomatized theory is itself an arithmetic sentence, built from a provability predicate. When the theory is strong enough to formalize its own reflection and modus ponens — the Hilbert–Bernays–Löb derivability conditions — it cannot prove that sentence unless it is inconsistent. Löb's theorem is the companion result, and set theory is the case that closes Hilbert's program.\n",{"path":24860,"title":24861,"module":24862,"summary":24863},"\u002Flogic\u002Fcomputability-and-representability\u002Frecursive-functions","Recursive Functions and Church's Thesis","Recursive Functions and Representability","The recursive functions are the formal counterpart of the effectively computable ones: built from three initial functions by composition, primitive recursion, and minimization, and equivalently the functions representable in a finitely axiomatized arithmetic. Church's thesis identifies the class with effective calculability; Kleene's normal form theorem and the unsolvable halting problem place the recursive sets strictly inside the recursively enumerable ones.\n",{"path":24865,"title":24866,"module":24862,"summary":24867},"\u002Flogic\u002Fcomputability-and-representability\u002Frepresenting-exponentiation","Representing Exponentiation and the β-Function","Coding finite sequences by prime-power exponents already assumes exponentiation, so representing exponentiation from addition and multiplication alone needs a different encoder. Gödel's β-function, built from a pairing function and the Chinese remainder theorem, reads back arbitrary finite sequences using only plus and times. This represents exponentiation in the addition-multiplication arithmetic and closes the last gap in the representability of every recursive syntactic operation.\n",{"path":24869,"title":24870,"module":24871,"summary":24872},"\u002Flogic\u002Fsecond-order-logic\u002Fsecond-order-languages","Second-Order Languages","Second-Order Logic and Beyond","Second-order logic quantifies over relations and functions, not just individuals. Second-order Peano arithmetic and the second-order theory of the reals become categorical, and finiteness is definable by a single sentence. Compactness, completeness, and the Löwenheim–Skolem theorems all fail for the standard semantics.\n",{"path":24874,"title":24875,"module":24871,"summary":24876},"\u002Flogic\u002Fsecond-order-logic\u002Fskolem-functions-and-many-sorted-logic","Skolem Functions and Many-Sorted Logic","Skolem functions replace existential quantifiers with named witnesses, putting any first-order formula into a prenex form with all existentials — now over functions — pulled to the front. The Skolemized formula is equisatisfiable with the original, which reduces satisfiability to universal sentences and, through Herbrand expansions, to sentential logic. Many-sorted logic then adds several universes at once and reduces cleanly to ordinary one-sorted logic.\n",{"path":24878,"title":24879,"module":24871,"summary":24880},"\u002Flogic\u002Fsecond-order-logic\u002Fgeneral-structures","General (Henkin) Structures","General semantics reinterprets second-order logic by letting the predicate and function quantifiers range over a designated collection of relations and functions rather than all of them. Recast as many-sorted first-order logic with comprehension axioms, general second-order logic recovers a sound and complete calculus together with compactness and Löwenheim–Skolem, giving up the categoricity of the standard semantics. The ω-models of analysis show the trade.\n",{"path":24882,"title":24883,"module":6,"summary":6},"\u002Flogic","Logic",{"path":24885,"title":24886,"module":21519,"summary":24887},"\u002Freinforcement-learning\u002Ffoundations\u002Fwhat-is-reinforcement-learning","What Is Reinforcement Learning?","Reinforcement learning is learning what to do — how to map situations to actions — so as to maximize a numerical reward signal, discovered by trial and error rather than told. We set up the agent–environment loop, separate it from supervised and unsupervised learning, name the four elements (policy, reward, value, and an optional model), and train a tic-tac-toe player with a temporal-difference value update.\n",{"path":24889,"title":24890,"module":21519,"summary":24891},"\u002Freinforcement-learning\u002Ffoundations\u002Fa-brief-history-of-rl","A Brief History of Reinforcement Learning","The origins of reinforcement learning. Three threads — trial-and-error learning from animal psychology, optimal control and dynamic programming, and temporal-difference learning — ran independently for decades and merged around 1989 into the modern field. Replacing the lookup table with a neural network then produced deep reinforcement learning: DQN, AlphaGo, AlphaZero, MuZero, and RLHF.\n",{"path":24893,"title":24894,"module":21519,"summary":24895},"\u002Freinforcement-learning\u002Ffoundations\u002Fmulti-armed-bandits","Multi-Armed Bandits","A bandit is reinforcement learning stripped to a single decision, repeated: no state, no consequences, only the tension between exploiting the arm that looks best and exploring the ones that might be better. We build up the whole toolkit — sample-average value estimates, the incremental update rule, ε-greedy, optimistic initialization, UCB, and gradient bandits — and use it to study exploration in isolation, the one problem that carries over to the full setting.\n",{"path":24897,"title":24898,"module":21519,"summary":24899},"\u002Freinforcement-learning\u002Ffoundations\u002Fbandit-exploration-algorithms","Bandit Exploration Algorithms","Better ways to explore than picking at random. Upper-confidence-bound selection explores by optimism about what it hasn't measured; gradient bandits learn action preferences by stochastic gradient ascent on reward. We then add context to get the contextual bandit, the bridge to full RL, and measure everything by regret — where UCB1 and Thompson sampling reach the logarithmic optimum that fixed-ε greedy cannot.\n",{"path":24901,"title":24902,"module":21519,"summary":24903},"\u002Freinforcement-learning\u002Ffoundations\u002Fmarkov-decision-processes","Markov Decision Processes","A Markov decision process is the formal interface between an agent and its environment: at each step the agent reads a state, chooses an action, and receives a reward and a next state. We fix that loop, the dynamics function that governs it, and the Markov property that makes the state sufficient; then turn goals into a scalar reward and rewards into a discounted return, with one notation that covers both episodic and continuing tasks.\n",{"path":24905,"title":24906,"module":21519,"summary":24907},"\u002Freinforcement-learning\u002Ffoundations\u002Fvalue-functions-and-optimality","Value Functions and Optimality","A value function scores how good a state (or state–action pair) is under a policy: the expected return from there onward. Its defining property is the Bellman equation, a self-consistency condition linking a state's value to its successors' values, which we derive from the return and the dynamics. Pushing the same idea to the best-achievable value gives the Bellman optimality equations, whose solution yields an optimal policy — and whose intractability is what the rest of the course is about.\n",{"path":24909,"title":21738,"module":24910,"summary":24911},"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdynamic-programming","Tabular Solution Methods","Dynamic programming computes optimal policies when a perfect model of the MDP is given, by turning the Bellman equations into assignment statements. We build up iterative policy evaluation (the expected update), the policy improvement theorem, and the two classic algorithms that alternate them — policy iteration and value iteration — worked on the gridworld, a two-state MDP, Jack's car rental, and the gambler's problem.\n",{"path":24913,"title":24914,"module":24910,"summary":24915},"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdp-async-and-gpi","Dynamic Programming: Asynchronous DP and Generalized Policy Iteration","Policy and value iteration both sweep the entire state set on every pass, which is impossible once the state space is huge. This lesson loosens the schedule: asynchronous DP updates states in any order, generalized policy iteration names the alternation of evaluation and improvement that underlies nearly every RL method, and a look at efficiency and the curse of dimensionality places DP among the alternatives. We close past Sutton & Barto with prioritized sweeping, neuro-dynamic programming, value-iteration networks, and MuZero.\n",{"path":24917,"title":24918,"module":24910,"summary":24919},"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods","Monte Carlo Methods","Monte Carlo methods learn value functions and optimal policies from complete sampled episodes, with no model of the environment: they simply average the returns that actually followed each state. We build prediction (first-visit and every-visit averaging), see why estimating action values forces the exploration question, and answer it two ways on-policy — exploring starts and epsilon-soft control. Throughout, Monte Carlo samples one whole trajectory to termination and never bootstraps.\n",{"path":24921,"title":24922,"module":24910,"summary":24923},"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-off-policy","Monte Carlo Methods: Off-Policy Learning","On-policy Monte Carlo can only reach the best exploring policy, not the true optimum. Off-policy methods remove that ceiling by learning about a greedy target policy from data generated by a soft behavior policy, corrected with importance sampling. We derive the importance-sampling ratio, weigh ordinary against weighted estimators on real numbers, give the incremental off-policy algorithm, sharpen it with discounting-aware sampling, and close by placing Monte Carlo on the model\u002Fbootstrap map beside DP and temporal-difference learning.\n",{"path":24925,"title":24926,"module":24910,"summary":24927},"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftemporal-difference-learning","Temporal-Difference Learning","Temporal-difference learning is the one idea most central to reinforcement learning: learn a value directly from experience, like Monte Carlo, but update each guess toward the next guess before the episode ends, like dynamic programming. We derive the TD(0) prediction rule and its reward-prediction error, contrast its one-step backup with MC and DP, work the driving-home and random-walk examples, and show the batch-updating optimality that makes TD approximate the certainty-equivalence estimate.\n",{"path":24929,"title":24930,"module":24910,"summary":24931},"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftd-control-sarsa-and-q-learning","TD Control: Sarsa, Q-learning, and Double Learning","With TD prediction in hand, control follows the generalized-policy-iteration pattern with TD as the evaluation step. We build Sarsa (on-policy), Q-learning (off-policy, targeting the optimal policy), and Expected Sarsa that spans the two, then confront the maximization bias every max-based method inherits and fix it with Double Q-learning. We close past Sutton & Barto, following each one-step tabular update into its deep-RL descendant — DQN, Double DQN, and Rainbow.\n",{"path":24933,"title":24934,"module":24910,"summary":24935},"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping","n-Step Bootstrapping","Monte Carlo waits for the full return; one-step TD bootstraps after a single reward. Between them lies a whole spectrum, indexed by one integer n: look ahead n real rewards, then bootstrap from the value n steps out. The n-step return unifies the previous two lessons, and — on the random walk — an intermediate n beats both extremes. We build the n-step return, the n-step TD update, the backup-diagram spectrum, and n-step Sarsa for control.\n",{"path":24937,"title":24938,"module":24910,"summary":24939},"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-off-policy-methods","n-Step Bootstrapping: Off-Policy Methods","Taking the n-step family off-policy raises the same importance-sampling questions Monte Carlo did, now over a window of exactly n actions. We reweight n-step returns by the policy ratio, watch the ratio product inflate variance on real numbers, then build the tree-backup algorithm that learns off-policy with no ratios at all — and finally n-step Q(sigma), one algorithm whose per-step switch recovers Sarsa, tree backup, and Expected Sarsa as special cases.\n",{"path":24941,"title":24942,"module":24910,"summary":24943},"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-and-learning","Planning and Learning","Planning and learning are the same operation run on two kinds of experience. A model turns states and actions into simulated transitions; planning backs up values over that simulated experience exactly as learning backs them up over real experience. We build the Dyna architecture that interleaves acting, model-learning, direct RL, and planning in one loop, trace a single Dyna-Q step by hand, and patch the architecture for when the model goes stale.\n",{"path":24945,"title":24946,"module":24910,"summary":24947},"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-focusing-and-decision-time","Planning: Focusing Updates and Decision-Time Search","Dyna plans by replaying remembered transitions, but sampling them uniformly wastes most of the effort. This lesson sharpens planning: prioritized sweeping works backward from states whose value just changed, expected versus sample updates weigh thoroughness against cost, and trajectory sampling and real-time DP focus updates on the states the policy actually visits. We trace Dyna forward to model-based deep RL, then turn to decision-time planning — heuristic search, rollouts, and Monte Carlo Tree Search.\n",{"path":24949,"title":24950,"module":24910,"summary":24951},"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdecision-time-planning","Decision-Time Planning","Planning need not build a global policy. Decision-time planning runs a fresh lookahead every time a state arrives and returns just one action, then throws the work away. We start from real-time dynamic programming — asynchronous value iteration on the states the agent actually visits — then move through heuristic search and rollout algorithms, each a one-step policy improvement applied on the fly to the current state.\n",{"path":24953,"title":24954,"module":24910,"summary":24955},"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-tree-search","Monte Carlo Tree Search","Monte Carlo Tree Search is a rollout algorithm with memory: it accumulates value estimates across simulations and steers later ones toward promising branches. We work through the four steps — selection, expansion, simulation, backup — the UCT selection rule computed on real numbers, the asymmetric growing tree, and the full pseudocode. We close past Sutton & Barto with the lineage from UCT to AlphaGo, AlphaZero, and MuZero, where a learned network stands in for the leaf value and the rollout.\n",{"path":24957,"title":24958,"module":24959,"summary":24960},"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-prediction","On-Policy Prediction with Approximation","Approximate Solution Methods","Every tabular method so far stored one number per state, which fails once the state space is large or continuous. We replace the table with a parameterized value function $\\hat v(s,\\mathbf{w})$, define the mean squared value error it should minimize under the on-policy distribution, and derive stochastic- and semi-gradient learning rules — the semi-gradient TD(0) update that bootstraps and so is not a true gradient. Linear methods make the analysis clean and give the TD fixed point; feature construction (polynomials, Fourier basis, coarse and tile coding, RBFs) supplies the vectors $\\mathbf{x}(s)$, and neural networks are the nonlinear bridge to deep RL.\n",{"path":24962,"title":24963,"module":24959,"summary":24964},"\u002Freinforcement-learning\u002Fapproximation\u002Ffeature-construction-and-nonlinear","Feature Construction and Nonlinear Approximation","Linear methods are only as good as the feature vectors $\\mathbf{x}(s)$ fed to them, and this lesson builds those vectors. Polynomials and the Fourier basis turn a state's coordinates into smooth global features; coarse coding, tile coding, and radial basis functions cover a continuous space with overlapping local receptive fields whose size sets the reach of generalization. Then we stop designing features by hand: a neural network learns the representation itself by gradient descent, trading the convergence guarantees of the linear case for expressiveness — the bridge to deep reinforcement learning.\n",{"path":24966,"title":24967,"module":24959,"summary":24968},"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-control","On-Policy Control with Approximation","Prediction learned a value function from features; control learns to act. We carry semi-gradient methods over to action values $\\hat q(s,a,\\mathbf{w})$, giving episodic semi-gradient Sarsa and its n-step form, and solve Mountain Car by descending a cost-to-go surface. In the continuing case, function approximation makes discounting unable to affect which policy is best, so we replace it with the average-reward setting — the differential return, differential value functions, and differential semi-gradient Sarsa.\n",{"path":24970,"title":24971,"module":24959,"summary":24972},"\u002Freinforcement-learning\u002Fapproximation\u002Faverage-reward-control","Average-Reward Control for Continuing Tasks","With function approximation, discounting has no effect on a continuing task: averaged over the on-policy distribution, the discounted objective equals the average reward times a policy-independent constant, so $\\gamma$ cannot change which policy is best. This lesson replaces discounting with the average-reward setting — the long-run reward rate $r(\\pi)$, the differential return that measures each state's transient advantage over that rate, differential value functions and TD error, and differential semi-gradient Sarsa, the control method for continuing tasks that never invokes a discount factor.\n",{"path":24974,"title":24975,"module":24959,"summary":24976},"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad","Off-Policy Methods and the Deadly Triad","Off-policy learning with function approximation is where the convergence guarantees of reinforcement learning fail. We extend the tabular off-policy updates to semi-gradient form with per-step importance sampling, show Baird's counterexample driving the weights to infinity, and identify the cause: the deadly triad of function approximation, bootstrapping, and off-policy training — any two are safe, all three can diverge. The divergence is not caused by sampling noise: a fully synchronous dynamic-programming update blows up just the same, which is what makes the triad a structural hazard rather than a fluke.\n",{"path":24978,"title":24979,"module":24959,"summary":24980},"\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td","Value-Function Geometry and Gradient-TD Methods","Why does the deadly triad diverge, and how do you stop it? This lesson develops the geometry that explains the failure: value functions as vectors, the projection operator onto the representable subspace, and the split between the Bellman error, the value error, and the projected Bellman error: the three objectives have different minimizers. The projected Bellman error is the learnable one, and Gradient-TD methods (GTD2, TDC) do true stochastic gradient descent on it, staying stable even off-policy at $O(d)$ cost. Emphatic TD reweights states instead, and a survey of variance-reduction techniques closes the gap between stability and usable learning.\n",{"path":24982,"title":24983,"module":24959,"summary":24984},"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces","Eligibility Traces","n-step methods unify TD and Monte Carlo by storing the last n feature vectors; eligibility traces do the same job with a single short-term memory vector. The λ-return averages every n-step return under a geometric weighting; the forward view looks ahead to that average, and the backward view produces nearly the same updates online through a decaying trace vector. We build the λ-return, TD(λ) with its trace, the two ways λ recovers TD(0) and Monte Carlo, a note on the exact equivalence of true online TD(λ), and Sarsa(λ) for control.\n",{"path":24986,"title":24987,"module":24959,"summary":24988},"\u002Freinforcement-learning\u002Fapproximation\u002Ftrue-online-and-sarsa-lambda","True Online TD(λ) and Sarsa(λ)","Plain TD(λ) makes the forward and backward views nearly agree; this lesson closes the gap. True online TD(λ) uses a dutch trace and a small correction term to produce exactly the same weight sequence as the online λ-return algorithm, at the same memory and only a constant factor more compute — the sharpest statement of the forward\u002Fbackward duality. The whole apparatus then lifts to control unchanged: Sarsa(λ) threads a single delayed reward back along an entire trajectory in one sweep, and the λ-weighting reappears in modern deep RL as generalized advantage estimation.\n",{"path":24990,"title":24991,"module":24959,"summary":24992},"\u002Freinforcement-learning\u002Fapproximation\u002Fpolicy-gradient-methods","Policy Gradient Methods","Every method so far learned values and read a policy off them. Policy gradient methods drop the intermediary: parameterize the policy directly and climb the performance gradient. We build the softmax-in-preferences parameterization, prove the policy gradient theorem that makes the gradient computable without the unknown state distribution, and derive REINFORCE and its variance-cutting state-value baseline — the launch point for the bootstrapping actor-critic that follows.\n",{"path":24994,"title":24995,"module":24959,"summary":24996},"\u002Freinforcement-learning\u002Fapproximation\u002Factor-critic-and-continuous-actions","Actor-Critic Methods and Continuous Actions","REINFORCE with a baseline learns a value function but never bootstraps; this lesson adds the bootstrapping critic that completes the actor-critic architecture. The critic scores each transition into a single TD error that steers both the actor's policy step and its own value step, trading a little bias for much lower variance and fully online, continuing-task learning. The policy gradient theorem carries over unchanged to the average-reward setting, a Gaussian policy handles real-valued actions with self-tuning exploration, and the natural policy gradient leads straight to TRPO, PPO, and the deep actor-critic methods that train today's agents.\n",{"path":24998,"title":24999,"module":24959,"summary":25000},"\u002Freinforcement-learning\u002Fapproximation\u002Fleast-squares-and-memory-based-methods","Least-Squares TD","Semi-gradient TD spends one cheap step per example and needs many examples; this lesson makes the opposite tradeoff. Least-Squares TD (LSTD) accumulates the matrices $\\mathbf{A}$ and $\\mathbf{b}$ and solves the TD fixed point $\\mathbf{w} = \\mathbf{A}^{-1}\\mathbf{b}$ directly, using the Sherman-Morrison identity to maintain the inverse in $O(d^2)$ — the most data-efficient linear TD method, at a quadratic cost. We work a solve by hand, weigh the quadratic cost against semi-gradient TD's cheap steps, and note that LSTD never forgets — a problem in control, where least-squares policy iteration is the natural extension.\n",{"path":25002,"title":25003,"module":24959,"summary":25004},"\u002Freinforcement-learning\u002Fapproximation\u002Fmemory-and-kernel-methods","Memory-Based and Kernel Methods","Least-squares TD spent more compute to extract more from each example; this lesson drops the parametric form entirely. Memory-based methods store training examples untouched and answer a query locally at retrieval time — nearest neighbor, weighted average, locally weighted regression — so accuracy grows with the data and effort concentrates where the agent actually goes. Kernel-based methods weight stored examples by a similarity kernel $k(s,s')$, and every linear method turns out to be a kernel method. Interest and emphasis, finally, make the on-policy weighting itself a design choice, aiming scarce approximation capacity at the states that matter.\n",{"path":25006,"title":25007,"module":24959,"summary":25008},"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces","Off-Policy Eligibility Traces","Eligibility traces meet off-policy learning and function approximation — the corner where stability gets hard. We first let the bootstrapping and discounting parameters vary with state, so a single generalized return covers episodic and continuing tasks and folds termination into the discount. Then we fold the per-decision importance ratio into the trace with a control-variate correction, and build Watkins's Q(λ) and its importance-sampling-free successor Tree-Backup(λ) — all correct in expectation, but still semi-gradient, so the deadly triad and its fixes wait for the next lesson.\n",{"path":25010,"title":25011,"module":24959,"summary":25012},"\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces","Stable Off-Policy Methods with Traces","Off-policy traces get the expected target right, but with $\\lambda \u003C 1$ they bootstrap, so off-policy plus bootstrapping plus function approximation is the deadly triad and the weights can diverge. This lesson carries the two one-step fixes to traces: GTD(λ) and GQ(λ) add a second weight vector and a gradient correction for true gradient descent on the projected Bellman error, while Emphatic TD(λ) reweights updates through a followon trace and interest to recover the on-policy stability. It closes with the implementation reality that traces are cheap because they are sparse, and with Retrace and V-trace — the clipped-ratio descendants that make off-policy traces work at deep-RL scale.\n",{"path":25014,"title":24333,"module":25015,"summary":25016},"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks","Deep Reinforcement Learning","Deep Q-networks replace the linear value function with a neural network $Q(s,a;\\mathbf{w})$ and confront the fact that a nonlinear approximator, off-policy bootstrapping, and correlated online data — the deadly triad — make naive Q-learning diverge. DQN counters this empirically with two stabilizers: an experience replay buffer that decorrelates and reuses samples, and a periodically-frozen target network that fixes the bootstrap target. We derive the DQN loss and gradient, walk through the Atari convolutional architecture and its results, and then add the three refinements that define modern value-based deep RL — Double DQN, dueling networks, and prioritized experience replay.\n",{"path":25018,"title":25019,"module":25015,"summary":25020},"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdqn-improvements","DQN Improvements: Double, Dueling, and Prioritized Replay","Three refinements that turn plain DQN into the standard modern value-based agent, each touching a different part of the system. Double DQN fixes the maximization bias in the target by splitting action selection from evaluation; dueling networks restructure the network around a state value and per-action advantages; prioritized replay changes which transitions are learned from. We close with Rainbow, which combines them, and the distributional view that predicts the whole return distribution rather than its mean.\n",{"path":25022,"title":25023,"module":25015,"summary":25024},"\u002Freinforcement-learning\u002Fdeep-rl\u002Factor-critic-and-ppo","Actor–Critic and GAE","Make the actor and the critic deep networks and the policy-gradient architecture becomes modern deep RL. We build the neural actor-critic, the advantage estimate that replaces the raw return, and Generalized Advantage Estimation as a λ-blend of n-step advantages, then the parallel-worker methods A3C and A2C that decorrelate on-policy data. The step-size constraints — trust regions, PPO, and the continuous-control family — follow in the next lesson.\n",{"path":25026,"title":25027,"module":25015,"summary":25028},"\u002Freinforcement-learning\u002Fdeep-rl\u002Fppo-and-continuous-control","PPO and Continuous Control","Keeping the policy-gradient step from destroying the policy, and the algorithms that result. Trust-region optimization bounds each update by a KL constraint; PPO keeps that goal but replaces the second-order machinery with a first-order clip on the probability ratio, which is why it is the modern default and the optimizer inside RLHF. We then tour the off-policy continuous-control family — DDPG, TD3, and SAC — and where actor-critic went at scale, from OpenAI Five to language-model alignment.\n",{"path":25030,"title":25031,"module":25015,"summary":25032},"\u002Freinforcement-learning\u002Fdeep-rl\u002Fcase-studies","Case Studies: Learning to Play","The game-playing systems that turned reinforcement learning from a theory into a track record: Samuel's checkers player, TD-Gammon, Watson's Daily-Double wagering, a reinforcement-learning memory controller, DQN, and AlphaGo through AlphaGo Zero. Read as a set they draw one line — a value function, learned by self-play or interaction, refined by search, carried by a deep network — that runs from a 1959 checkers program to superhuman Go.\n",{"path":25034,"title":25035,"module":25015,"summary":25036},"\u002Freinforcement-learning\u002Fdeep-rl\u002Frl-beyond-games","Reinforcement Learning Beyond Games","The same value-and-reward machinery, pointed at problems with no opponent. Web personalization as a contextual bandit and then a full MDP for life-time value; thermal soaring, where a glider learns to climb on turbulent air and reward design does most of the work; and the industrial-scale systems that carried the same design past Sutton & Barto — AlphaStar, OpenAI Five, GT Sophy, and RLHF, where the reward itself is learned from human preference.\n",{"path":25038,"title":25039,"module":25015,"summary":25040},"\u002Freinforcement-learning\u002Fdeep-rl\u002Ffrontiers","Frontiers: Beyond the Standard MDP","The standard MDP fixes three things — state, reward, and single-step actions — and this lesson loosens two of them. We generalize the value function into a general value function that predicts any signal, and use those predictions as auxiliary tasks that shape representations; we extend actions in time with the options framework; and we treat state as a construction the agent builds from a stream of observations. Reward design and the open problems follow in the next lesson.\n",{"path":25042,"title":25043,"module":25015,"summary":25044},"\u002Freinforcement-learning\u002Fdeep-rl\u002Freward-design-and-open-problems","Reward Design and Open Problems","How to design a reward signal that encodes the intended goal — sparse reward, shaping, and reward hacking — and the problems the whole tabular, approximate, and deep arc leaves unsolved. We close with how the frontiers were pushed after Sutton & Barto: auxiliary tasks, learned options, intrinsic-motivation bonuses, learned world models, and offline RL, then the two concerns of reward hacking and safety that any real-world agent must address.\n",{"path":25046,"title":25047,"module":25048,"summary":25049},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow","Sharpening DQN: Improvements and the Distributional Idea","Modern Deep Reinforcement Learning","In the years after the 2015 DQN paper, a stream of focused improvements each fixed one weakness of the baseline without disturbing its frame. This lesson recaps five that keep the scalar $Q$-value — Double DQN, multi-step returns, dueling networks, prioritized replay, and NoisyNets, each changing a different slot of the same Q-learning loop — then develops the sixth, distributional RL, which changes the objective itself: learn the whole return distribution $Z(s,a)$. We build the distributional Bellman equation and the C51 categorical algorithm, projection step and all, worked end to end on real numbers. A companion lesson takes up QR-DQN, Rainbow, and the modern distributional line.\n",{"path":25051,"title":25052,"module":25048,"summary":25053},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow-part-2","Distributional RL and Rainbow","A companion to the DQN improvements lesson. C51 fixed the return atoms and learned their probabilities; QR-DQN does the reverse — fix the probabilities, learn the values — which removes the projection and trains with a quantile loss. We cover why the distribution helps even when you act on the mean, then assemble Rainbow: all six improvements in one Q-learning loop, with the component ablation that shows each one's real weight. The distributional line then runs on through IQN, FQF, and Agent57, the first agent to beat the human baseline on all 57 Atari games.\n",{"path":25055,"title":25056,"module":25048,"summary":25057},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control","Continuous Control: DDPG and TD3","When actions are real-valued, the $\\arg\\max_a Q(s,a)$ in Q-learning becomes an optimization problem on every step. This lesson builds the off-policy actor-critic family that sidesteps it: the deterministic policy gradient and DDPG, which replaces the max with a learned actor, and the three fixes of TD3 that counter the value overestimation DDPG inherits. A companion lesson takes up SAC's maximum-entropy objective and the methods built on this template.\n",{"path":25059,"title":25060,"module":25048,"summary":25061},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control-part-2","Continuous Control: SAC and Beyond","A companion to the DDPG and TD3 lesson. Where those actors are deterministic and explore with bolted-on noise, soft actor-critic (SAC) changes the objective itself: maximize return plus the entropy of the policy, so exploration becomes intrinsic and the agent stays robust. We develop the maximum-entropy objective, the reparameterized squashed-Gaussian actor, and automatic temperature tuning, then survey the methods built on this off-policy template — distributional critics (D4PG), critic ensembles (REDQ), and control from pixels (DrQ, RAD).\n",{"path":25063,"title":25064,"module":25048,"summary":25065},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl","Model-Based Deep RL: Sample Efficiency and PETS","A model turns experience into imagined planning. This lesson makes the sample-efficiency case for learning a dynamics model, works through why a learned model's errors compound over the planning horizon, and builds the most direct model-based method: PETS plans online with a probabilistic ensemble under model-predictive control, distrusting the model exactly where its members disagree. A companion lesson takes up latent world models (Dreamer) and MuZero.\n",{"path":25067,"title":25068,"module":25048,"summary":25069},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl-part-2","Model-Based Deep RL: World Models, Dreamer, and MuZero","A companion to the PETS lesson. PETS plans in the environment's native state space; these methods change what the model represents. World Models and Dreamer learn a compact latent state and do almost all their learning by imagining inside it, with value gradients flowing through the differentiable dynamics. MuZero predicts neither states nor pixels — only the reward, value, and policy that MCTS reads — and plans with search against that learned model, AlphaZero without the rules. We close with MBPO, TD-MPC, and EfficientZero.\n",{"path":25071,"title":25072,"module":25048,"summary":25073},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration","Exploration in Deep RL: Novelty as Reward","When the state space is enormous and reward is rare, ε-greedy amounts to a random walk that almost never reaches the first reward. This lesson scales the bandit's exploration ideas up to deep RL through the dominant approach — manufacture a reward for novelty and let the agent chase it: optimism and pseudo-counts from density models, and intrinsic motivation and curiosity (the Intrinsic Curiosity Module and Random Network Distillation). A companion lesson takes up posterior sampling, Go-Explore, and the modern methods.\n",{"path":25075,"title":25076,"module":25048,"summary":25077},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration-part-2","Exploration in Deep RL: Posterior Sampling and Go-Explore","A companion to the novelty-as-reward lesson. Pseudo-counts and curiosity reward the unfamiliar after the agent stumbles into it; this lesson covers two ideas that go further. Bootstrapped DQN keeps an ensemble that approximates a posterior over value functions and explores by committing to one sampled hypothesis per episode — the deep, directed exploration ε-greedy cannot manage. Go-Explore remembers and returns to the frontier, defeating detachment and derailment to solve Montezuma's Revenge. We close with episodic memory (Never Give Up), Agent57, and model-based exploration.\n",{"path":25079,"title":25080,"module":25048,"summary":25081},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl","Offline RL: The Problem and Value-Based Fixes","Offline reinforcement learning learns a policy from a fixed logged dataset with no further environment interaction — off-policy learning pushed to the extreme, and it breaks for the extreme version of the same reason. Bootstrapping queries the value function at out-of-distribution actions the data never covers, those errors are optimistic, and with no online feedback to correct them they compound through the Bellman backup. This lesson sets up the failure and off-policy evaluation, then builds the first two families of pessimistic fixes: policy constraint (BCQ) and conservative value estimation (CQL). A companion lesson takes up implicit methods, model-based offline RL, and Decision Transformer.\n",{"path":25083,"title":25084,"module":25048,"summary":25085},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2","Offline RL: Implicit Methods, Sequence Models, and Beyond","A companion to the offline-RL problem lesson. Policy constraint and conservative value estimation both still query a learned value function; implicit methods (IQL) avoid querying it off the data at all, using an in-sample expectile backup. We then build pessimism into a learned model (MOPO, COMBO) and drop bootstrapping entirely with Decision Transformer's return-conditioned sequence modeling, closing with offline-to-online fine-tuning, diffusion planners, and the offline view of RLHF. The one rule throughout: without online correction, be pessimistic about what you cannot verify.\n",{"path":25087,"title":25088,"module":25048,"summary":25089},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl","Imitation Learning: Cloning, DAgger, and Inverse RL","When a reward is hard to specify but an expert is easy to watch, learn from demonstrations instead. Behavioral cloning treats control as supervised learning of the expert's state-to-action map, and fails through compounding error: small mistakes carry the agent off the expert's distribution, where it was never trained. DAgger fixes the mismatch by querying the expert on the learner's own states. Inverse RL instead recovers the reward the expert seems to optimize — an ill-posed problem that maximum-entropy IRL disambiguates. A companion lesson casts imitation as adversarial occupancy matching (GAIL, AIRL).\n",{"path":25091,"title":25092,"module":25048,"summary":25093},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl-part-2","Imitation as Adversarial Matching: GAIL and AIRL","A companion to the imitation-learning lesson. If the point of recovering a reward is only to re-run RL and match the expert, you can skip the reward and match the behavior directly. GAIL casts imitation as a GAN — a discriminator separating expert from learner state-action pairs supplies the reward a policy-gradient method optimizes — matching occupancy measures without ever naming a reward. AIRL reads a transferable reward back out of the discriminator. We compare all four methods and close with reward models in RLHF, scaled cloning, and diffusion policies.\n",{"path":25095,"title":25096,"module":25048,"summary":25097},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl","Multi-Agent RL: Markov Games and Centralized Training","With more than one learning agent in an environment, each agent's world becomes non-stationary because the others are changing too. This lesson builds the Markov-game generalization of the MDP, diagnoses non-stationarity as the central obstacle, shows why the naive baselines fail, and develops the dominant fix — centralized training with decentralized execution (MADDPG, VDN, QMIX). A companion lesson takes up self-play, the landmark game-playing systems, and the equilibrium concepts that define what \"solved\" means.\n",{"path":25099,"title":25100,"module":25048,"summary":25101},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl-part-2","Multi-Agent RL: Self-Play and Solution Concepts","A companion to the Markov-games lesson. In the purely competitive setting, an agent can generate its own training curriculum by playing against copies of itself — self-play, the method behind AlphaGo, OpenAI Five, and AlphaStar. We develop why self-play produces an ever-improving opponent, the systems it built, and then the equilibrium solution concepts (Nash, correlated, coarse-correlated) that define what \"solved\" means once there is an opponent, closing with PSRO, MAPPO, and the language-model-agent frontier.\n",{"path":25103,"title":25104,"module":25048,"summary":25105},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl","Hierarchical RL: Options and the Option-Critic","Flat RL cannot explore a long horizon: reaching reward through hundreds of primitive actions is exponentially unlikely, and every credit-assignment update crawls one step at a time. Hierarchy breaks one hard long-horizon problem into many short ones. This lesson develops temporal abstraction — the options framework and its semi-Markov view, and learning options end to end with the option-critic. A companion lesson takes up goal-conditioned manager\u002Fworker hierarchies (FeUdal Networks and HIRO), hindsight relabeling, and unsupervised skill discovery.\n",{"path":25107,"title":25108,"module":25048,"summary":25109},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl-part-2","Hierarchical RL: Goal-Conditioned Hierarchies and Skills","A companion to the options lesson. Options package a behavior; goal-conditioned hierarchies instead give the top level an explicit language of goals — a manager proposes a target state or a latent direction, and a worker is rewarded for reaching it (FeUdal Networks, HIRO). We develop that architecture, the hindsight relabeling that lets it learn from sparse reward, and unsupervised skill discovery (DIAYN) that learns a repertoire of behaviors with no reward at all. The shared idea throughout: shorten the horizon by inserting a level that decides less often.\n",{"path":25111,"title":25112,"module":25048,"summary":25113},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Frlhf-and-language-models","RLHF and Language Models","A language model trained to predict the next token is fluent but not helpful, honest, or harmless — the objective it was optimized for is not the objective we want. RLHF closes that gap by turning the one thing humans do reliably, comparing two outputs, into a reward. We build the three-stage pipeline: supervised fine-tuning, a Bradley-Terry reward model fit to preference pairs, then PPO against that reward with a KL penalty keeping it near the reference policy. We then cover reward hacking and why the KL penalty matters, Direct Preference Optimization, which folds the reward model into a single classification loss, and the RLAIF and verifiable-reward variants. This pipeline is what makes the largest models usable as assistants.\n",{"path":25115,"title":25116,"module":25048,"summary":25117},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps","Partial Observability: POMDPs and the Belief State","Drop the assumption that the agent sees the state. It sees an observation, a partial and noisy function of a hidden state, and one observation is no longer a Markov signal. This lesson builds the POMDP tuple, shows that the belief state — the posterior over hidden states — is a sufficient statistic that turns a POMDP back into an MDP over beliefs, and works the Bayes-filter belief update step by step. A companion lesson explains why exact planning is intractable and develops the deep-RL answer of recurrent, history-based policies.\n",{"path":25119,"title":25120,"module":25048,"summary":25121},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps-part-2","Partial Observability: Planning and Recurrent Policies","A companion to the belief-state lesson. In principle a POMDP reduces to an MDP over beliefs; in practice two obstacles block that. Exact planning over the belief simplex is intractable — the value function is piecewise-linear-and-convex with a number of pieces that can explode — and computing the belief needs a model the agent rarely has. This lesson develops the intractability, the point-based approximations that address it, and the deep-RL answer: make the policy a function of history with a recurrent network (DRQN, R2D2), with frame-stacking, attention, and world-model latents as learned beliefs.\n",{"path":25123,"title":25124,"module":25048,"summary":25125},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl","Safe and Constrained RL: The CMDP and Policy Methods","Maximizing a scalar reward is not the same as behaving well: a capable optimizer will find and exploit any gap between the reward and what its designer actually meant, a failure called specification gaming or reward hacking. The remedy is to add explicit cost constraints — the constrained MDP — maximizing return subject to an expected-cost budget. This lesson builds the core toolkit: the CMDP itself, Lagrangian primal-dual methods that learn a multiplier on the constraint (RCPO), and constrained policy optimization (CPO) with its trust-region cost bound. A companion lesson covers risk-sensitivity, safe exploration, and the alignment framing.\n",{"path":25127,"title":25128,"module":25048,"summary":25129},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl-part-2","Safe RL: Risk, Safe Exploration, and Alignment","A companion to the constrained-MDP lesson. Constraining the mean cost is not enough: a policy safe on average can be catastrophic in the tail, and a policy safe at convergence can violate its limits wildly while learning. This lesson optimizes the tail with risk-sensitive objectives (CVaR), then makes exploration itself safe with shields, Lyapunov methods, and safety layers that project unsafe actions onto the feasible set — closing with benchmarks, safe RLHF, robustness, and the alignment framing that ties safety back to the problem of incompletely specified reward.\n",{"path":25131,"title":25132,"module":25048,"summary":25133},"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmeta-rl-and-generalization","Meta-RL and Generalization","An agent that masters one task often fails on the next; it has overfit to a single environment. This lesson treats fast adaptation as a meta-problem over a distribution of tasks: meta-train so that a few episodes at meta-test time suffice. We cover the two families — optimization-based (MAML learns an initialization) and context-based (RL-squared and PEARL infer a latent task) — the exploration cost of adaptation, and the parallel problem of generalization: why deep RL memorizes environments and what fixes it (domain randomization, procedural generation, augmentation, regularization). It closes on foundation models and sequence-model agents as the generalist endpoint.\n",{"path":25135,"title":25136,"module":25137,"summary":25138},"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fpsychology-of-reinforcement","The Psychology of Reinforcement","Reinforcement Learning in Minds and Brains","Reinforcement learning is both an engineering method and a theory of how animals learn. The prediction\u002Fcontrol split of the algorithms mirrors the psychologist's split between classical and instrumental conditioning. We trace the correspondence: the Rescorla–Wagner model as a prediction-error rule that explains blocking, its real-time TD extension, Thorndike's Law of Effect behind trial-and-error control, and the habitual\u002Fgoal-directed distinction that maps onto model-free versus model-based learning.\n",{"path":25140,"title":25141,"module":25137,"summary":25142},"\u002Freinforcement-learning\u002Fminds-and-brains\u002Finstrumental-conditioning-and-control","The Psychology of Reinforcement: Instrumental Control","Classical conditioning was prediction; instrumental conditioning is control. Thorndike's Law of Effect is trial-and-error control — selection plus association, search plus memory — and Skinner's shaping and schedules are reward engineering. The habitual\u002Fgoal-directed distinction maps onto model-free versus model-based control, dissociated by outcome devaluation and arbitrated by uncertainty. Delayed reinforcement is the credit-assignment problem, and the stimulus traces and secondary reinforcers of animal-learning theory are eligibility traces and value functions.\n",{"path":25144,"title":25145,"module":25137,"summary":25146},"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-and-td-error","Dopamine and the TD Error","The TD error was invented as an algorithm; a decade later it turned out to closely describe the firing of the brain's dopamine neurons. We follow Schultz's experiments — dopamine fires at an unpredicted reward, shifts to the earliest predictive cue, and dips below baseline when a predicted reward is withheld — and match each result to the TD error term by term. We then read the basal ganglia as a neural actor–critic with dopamine as its shared training signal, and close on addiction as a hijacking of that signal.\n",{"path":25148,"title":25149,"module":25137,"summary":25150},"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-in-the-brain","Dopamine in the Brain: The Neural Actor–Critic","If phasic dopamine is a TD error, where does it go and what does it change? We follow the axons into the basal ganglia, read the corticostriatal synapse as the place where state, action, and error meet, and map the ventral and dorsal striatum onto the critic and the actor of an actor–critic. Addiction becomes a broken cancellation in the same learning signal, and distributional dopamine extends the scalar RPE into a population code.\n",{"path":25152,"title":25153,"module":25137,"summary":25154},"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fanimal-learning-and-cognition","Animal Learning and Cognition","Three classic associative phenomena turn out to be reinforcement-learning mechanisms seen in behavior. Blocking says learning is driven by prediction error, not co-occurrence, and reduces to least-squares regression fitting a collinear feature. Higher-order conditioning and conditioned reinforcement make a value estimate a secondary reinforcer — bootstrapping in an animal. Delayed reinforcement is the credit-assignment problem, and the stimulus traces and goal gradients of Pavlov and Hull are eligibility traces and TD-learned value functions.\n",{"path":25156,"title":25157,"module":25137,"summary":25158},"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fcognitive-maps-and-planning","Cognitive Maps and Model-Based Learning","Tolman's rats learned the layout of a maze with no reward, then used it the moment food appeared — latent learning, a cognitive map, and the behavioral face of model-based reinforcement learning. The map is learned by system identification (stimulus–stimulus associations), which fills in whether or not reward is present, and queried by planning, which re-solves a route from a single changed reward. The successor representation sits between cache and model, and hippocampal predictive maps and scaled-up world models carry the same idea into brain and machine.\n",{"path":25160,"title":25161,"module":25137,"summary":25162},"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fneuroscience-of-reinforcement","The Neuroscience of Reinforcement","The dopamine story is one contact point between reinforcement learning and the brain; this lesson fills in the surrounding neuroscience so the mapping stands on its own. We build a working primer of neurons, synapses, and neuromodulation; separate four signals that casual usage conflates — reward, reinforcement, value, and prediction error; and read the actor and critic as corticostriatal synapses updated by two- and three-factor rules, grounded in spike-timing-dependent and reward-modulated plasticity.\n",{"path":25164,"title":25165,"module":25137,"summary":25166},"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fseveral-learning-systems","The Brain's Several Learning Systems","The actor's three-factor rule has an ancestor in Klopf's hedonistic neuron — a single cell as a reinforcement-seeking agent — and a bacterium's run-and-twiddle shows the Law of Effect with no synapses at all. Teams of such neurons implement policy gradient collectively, the broadcast reward replacing backpropagation. And the brain is not only model-free: outcome devaluation, prefrontal value coding, and hippocampal forward sweeps localize a model-based system. The recurring conclusion is that the brain is several interacting learning systems, not one algorithm.\n",{"path":25168,"title":24325,"module":6,"summary":6},"\u002Freinforcement-learning",{"path":25170,"title":25171,"module":21519,"summary":25172},"\u002Fartificial-intelligence\u002Ffoundations\u002Fwhat-is-ai","What Is Artificial Intelligence?","Eight definitions of AI fall into a two-by-two grid: think versus act, and measure success against human performance versus an ideal standard of rationality. We work through all four schools — the Turing test, cognitive modelling, the laws of thought, and the rational agent — and adopt the last as the frame for the whole course: AI is the study and design of rational agents.\n",{"path":25174,"title":25175,"module":21519,"summary":25176},"\u002Fartificial-intelligence\u002Ffoundations\u002Ffoundations-of-ai","The Foundations of AI","Where the rational-agent idea came from and what surrounds it. AI inherited its core tools from eight older disciplines — philosophy, mathematics, economics, neuroscience, psychology, computer engineering, control theory, and linguistics. Its history runs in cycles of boom and winter, from the 1956 Dartmouth workshop through expert systems to the statistical turn. And the deep-learning era — AlexNet, the Transformer, GPT-3, AlphaGo — is a new way of computing the agent function at scale, not a new definition of AI.\n",{"path":25178,"title":25179,"module":21519,"summary":25180},"\u002Fartificial-intelligence\u002Ffoundations\u002Fintelligent-agents","Intelligent Agents","An agent perceives an environment through sensors and acts on it through actuators; its behavior is an agent function mapping percept sequences to actions. A rational agent chooses, for each percept sequence, the action that maximizes its expected performance measure given its knowledge. We build the first half of the vocabulary the whole course rests on — the agent function, rationality, PEAS task specifications, and the six axes along which task environments vary.\n",{"path":25182,"title":25183,"module":21519,"summary":25184},"\u002Fartificial-intelligence\u002Ffoundations\u002Fagent-architectures","Agent Architectures","How to build a program that computes a good agent function without storing an astronomically large lookup table. Four skeleton architectures in order of increasing power — simple reflex, model-based, goal-based, and utility-based — plus the learning agent that improves any of them, the scale of world representations (atomic, factored, structured) they rest on, and how a modern language-model agent fits the same frame.\n",{"path":25186,"title":25187,"module":25188,"summary":25189},"\u002Fartificial-intelligence\u002Fsearch\u002Funinformed-search","Uninformed Search","Search","A goal-based agent that cannot see which action is best turns the problem into a state space — an initial state, a set of actions, a transition model, a goal test, and a path cost — and searches for a sequence of actions reaching the goal. We build the state-space formulation on the 8-puzzle and route-finding, give the one TREE-SEARCH \u002F GRAPH-SEARCH skeleton every algorithm specializes, and measure strategies by completeness, optimality, and complexity. This lesson develops the first two frontier disciplines — breadth-first and uniform-cost search; the rest follow in the next lesson.\n",{"path":25191,"title":25192,"module":25188,"summary":25193},"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-strategies-compared","Search Strategies Compared","Breadth-first and uniform-cost search pay for optimality in memory. This lesson develops the strategies that trade memory for depth: depth-first search, which keeps only the current path; depth-limited and iterative-deepening search, which fix DFS's failure on infinite paths; and bidirectional search, which meets in the middle for a square-root saving. It closes by lining up all six uninformed strategies against completeness, optimality, and complexity, and tracing where the algorithms came from and where they went.\n",{"path":25195,"title":25196,"module":25188,"summary":25197},"\u002Fartificial-intelligence\u002Fsearch\u002Finformed-search","Informed Search and A*","An informed search uses a heuristic $h(n)$, an estimate of the cost from a node to the goal, to decide what to expand next. Greedy best-first search follows the heuristic blindly and gives up optimality; A* corrects it by ranking nodes on $f(n) = g(n) + h(n)$, and is optimal when the heuristic is admissible (tree search) or consistent (graph search). This lesson defines the heuristic, builds best-first search, and proves why A* is optimal, with the contour picture that explains its pruning. Where good heuristics come from is the next lesson.\n",{"path":25199,"title":25200,"module":25188,"summary":25201},"\u002Fartificial-intelligence\u002Fsearch\u002Fheuristic-functions","Heuristic Functions and Memory-Bounded Search","A* is only as good as its heuristic, so this lesson answers where good heuristics come from: relaxed problems, whose exact solution cost is an admissible heuristic, and pattern databases, which precompute subproblem costs. It measures heuristic quality with dominance and the effective branching factor, then tackles A*'s memory problem with IDA*, RBFS, and SMA*. It closes with modern heuristic search — weighted A*, learned and disjoint pattern-database heuristics, and bidirectional A*.\n",{"path":25203,"title":25204,"module":25188,"summary":25205},"\u002Fartificial-intelligence\u002Fsearch\u002Flocal-search","Local Search and Optimization","When the path to a goal is irrelevant and only the final state matters, we can discard the search tree entirely and keep just the current state, moving to a better neighbor at each step. This lesson builds the state-space landscape metaphor, works through hill climbing and the three obstacles that defeat it (local maxima, ridges, plateaus), then develops the first escapes: random restarts and simulated annealing with its temperature schedule. The population-based methods and continuous-space calculus follow in the next lesson.\n",{"path":25207,"title":25208,"module":25188,"summary":25209},"\u002Fartificial-intelligence\u002Fsearch\u002Fpopulation-and-continuous-search","Population and Continuous Search","Single-state local search escapes a trap by restarting or tolerating downhill moves. This lesson develops the alternatives that keep several states at once — local beam search, which shares successors across parallel threads, and genetic algorithms, which recombine two parents through crossover and mutation — then crosses into continuous spaces, where calculus replaces the finite neighbor set: gradient ascent, line search, and Newton's method. It closes with the industrial descendants of these methods and the loop they all share.\n",{"path":25211,"title":25212,"module":25188,"summary":25213},"\u002Fartificial-intelligence\u002Fsearch\u002Fadversarial-search","Adversarial Search and Games","When another agent plans against you, search becomes a game. We formalize two-player, zero-sum, perfect-information games as search problems, define the minimax value that optimal play backs up through the game tree, and give the MINIMAX algorithm that computes it. Alpha–beta pruning then cuts the cost of that search roughly in half in the exponent without changing the answer, and a heuristic evaluation function plus a cutoff test turns the exact algorithm into a real-time player that copes with the horizon effect.\n",{"path":25215,"title":25216,"module":25188,"summary":25217},"\u002Fartificial-intelligence\u002Fsearch\u002Fgames-of-chance-and-imperfect-information","Games of Chance and Imperfect Information","Minimax and alpha–beta assume a deterministic game both players can see in full. Drop either assumption and search must change. This lesson adds chance nodes and the expectiminimax value for games with dice, then belief-state reasoning for partially observable games — Kriegspiel and card games — where averaging over clairvoyance both helps and misleads. It closes with the line from Deep Blue's alpha–beta to AlphaGo's learned evaluation and Monte Carlo tree search, and the provable-pruning and self-play research around each end of that story.\n",{"path":25219,"title":25220,"module":25188,"summary":25221},"\u002Fartificial-intelligence\u002Fsearch\u002Fconstraint-satisfaction","Constraint Satisfaction Problems","A constraint satisfaction problem replaces the black-box state with a factored one: variables, domains, and constraints. That structure supports inference before any search runs. This lesson defines the CSP on map coloring, Sudoku, and scheduling, then develops constraint propagation: node and arc consistency, the AC-3 algorithm that makes a whole network arc-consistent, and the way one deleted value cascades across the graph to prune impossible options ahead of search.\n",{"path":25223,"title":25224,"module":25188,"summary":25225},"\u002Fartificial-intelligence\u002Fsearch\u002Fcsp-search-and-structure","CSP Search and Structure","Propagation prunes a CSP but rarely finishes it, so we search. This lesson builds backtracking search over partial assignments and the general-purpose heuristics that make it fast — MRV, degree, least-constraining-value, forward checking, MAC, and intelligent backtracking. It then shows how the shape of the constraint graph controls difficulty: tree-structured problems fall in linear time, cutset conditioning handles the rest, and min-conflicts local search solves a million queens in a constant number of steps.\n",{"path":25227,"title":25228,"module":25188,"summary":25229},"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-under-uncertainty","Search Under Uncertainty","Classical search assumes the agent knows the state it is in and exactly what each action does. Drop the second assumption and a plan can no longer be a fixed sequence of actions. This lesson develops the first response: AND-OR search over nondeterministic actions, which returns a branching contingency plan rather than a straight line. We build it on the erratic vacuum world, show how OR nodes (the agent's choices) alternate with AND nodes (nature's outcomes), trace the recursion that finds a plan, and handle the case where the only solution is a cyclic \"try, try again.\"\n",{"path":25231,"title":25232,"module":25188,"summary":25233},"\u002Fartificial-intelligence\u002Fsearch\u002Fbelief-state-and-online-search","Belief-State and Online Search","When the agent cannot see the full state, a plan can no longer test where it actually is — it must reason over the set of states it might be in. This lesson develops belief-state search, from sensorless (conformant) planning that coerces an unknown world into a goal, through the predict-observe-update cycle of contingent planning with percepts, to online search in unknown environments, where the agent must act in order to learn. It closes with LRTA*, which refines its own heuristic as it explores, one step from reinforcement learning.\n",{"path":25235,"title":25236,"module":25237,"summary":25238},"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-logic","Logical Agents and Propositional Logic","Logic and Planning","A knowledge-based agent keeps a store of sentences and acts by asking it what to do. To make \"asking\" mean something we need entailment — the relation $KB \\models \\alpha$ that holds when every model of the knowledge base is a model of the query. Propositional logic gives a syntax and a truth-table semantics for which entailment is decidable. This first part builds the foundations: the agent loop, the Wumpus World, models and entailment, the connectives and truth tables, theorem proving by refutation, and the resolution rule with its CNF conversion — a single complete inference procedure for all of propositional logic.\n",{"path":25240,"title":25241,"module":25237,"summary":25242},"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-inference","Propositional Inference and Logical Agents","Model checking and resolution decide entailment, but both can blow up. This part turns propositional logic into a practical engine and a working agent. Horn clauses give linear-time forward and backward chaining — the basis of logic programming. DPLL and WalkSAT make satisfiability testing fast in the common case. Then we make the agent situated: time-indexed fluents, the frame problem and its solution by successor-state axioms, a hybrid agent that deduces a safe map and plans a route through it, and SATPlan, which finds a plan by asking a SAT solver for a satisfying model.\n",{"path":25244,"title":25245,"module":25237,"summary":25246},"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic","First-Order Logic","Propositional logic can only say that facts hold; it cannot talk about the objects a fact is about, or state a rule once and have it cover every object. First-order logic fixes this by committing to a world of objects, relations, and functions. This first part builds the language from the ground up: the ontology it commits to, the model that gives a sentence a truth value, the syntax of terms and sentences, the two quantifiers with their standard mistakes, and equality.\n",{"path":25248,"title":25249,"module":25237,"summary":25250},"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic-in-use","First-Order Logic in Use","With the language of first-order logic in hand, this part is about using it well. Database semantics trades expressive power for the convenience of a single intended model; higher-order logic shows what first-order logic gives up for decidability. Then we put the language to work: the Tell\u002FAsk interface, the kinship domain axiomatized from scratch, and the seven-step knowledge-engineering process applied to a digital circuit.\n",{"path":25252,"title":25253,"module":25237,"summary":25254},"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Finference-and-resolution","Inference in First-Order Logic","Propositional inference lifts to first-order logic once we can make terms match. Unification is that machinery: the algorithm that finds the substitution making two expressions identical, and the basis of generalized modus ponens. This first part builds the lifted inference rules and the two chaining algorithms they drive — forward chaining, the data-driven procedure behind production systems and Datalog, and backward chaining, the goal-driven procedure behind Prolog.\n",{"path":25256,"title":25257,"module":25237,"summary":25258},"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-resolution","First-Order Resolution","Chaining is complete only for Horn knowledge bases. General first-order sentences — with disjunctive conclusions and negations — need a single sound and complete rule: resolution. This part converts arbitrary sentences to CNF by skolemizing away the existentials, lifts the resolution rule with unification, and proves entailment by refuting the negated goal. The result is the proof procedure Gödel's completeness theorem guarantees will find any entailment, together with the search strategies that make it usable.\n",{"path":25260,"title":25261,"module":25237,"summary":25262},"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fclassical-planning","Classical Planning","Classical planning represents a problem in a factored language, PDDL: states are sets of ground fluents, and actions are lifted schemas with a precondition and an effect. That structure turns planning into search — forward through states or backward through goals — and lets a program read heuristics straight off the schemas by relaxing the problem. This first part develops the representation, the two search directions, and the domain-independent heuristics that come from ignoring preconditions or delete lists.\n",{"path":25264,"title":25265,"module":25237,"summary":25266},"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-graphs-and-graphplan","Planning Heuristics and GraphPlan","Every relaxation heuristic can be inaccurate, and none can tell how far apart subgoals sit. The planning graph is a polynomial-size structure that does better: leveled off the problem, it yields admissible distance estimates and a record of which actions and fluents cannot coexist. This part builds the graph, reads heuristics from it, extracts plans with GraphPlan, and closes with the other classical approaches — SATPlan and partial-order planning — and the representational trade that makes all of it work.\n",{"path":25268,"title":25269,"module":25237,"summary":25270},"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-in-the-real-world","Planning and Acting in the Real World","Classical planning's clean theory rests on four assumptions: time is ignored, actions are atomic, the world is deterministic and fully observable, and the agent is alone. This first part drops the first two. We add durations and resource constraints — turning a plan into a schedule, solved by the critical-path method and, once resources contend, by NP-hard job-shop scheduling — and let a planner reason at multiple levels of abstraction through high-level actions and their angelic reachable sets.\n",{"path":25272,"title":25273,"module":25237,"summary":25274},"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-under-uncertainty","Planning Under Uncertainty","Classical planning assumed the world was deterministic, fully observable, and the agent alone. This part drops the last two assumptions. When the agent cannot see or predict the world, planning moves into belief-state space: sensorless plans that coerce the world into the goal without sensing, contingent plans that branch on what is sensed, and online agents that monitor and replan when execution diverges. Then we add other agents — joint plans, the coordination problem, and the conventions that let a team act without constant negotiation.\n",{"path":25276,"title":25277,"module":25237,"summary":25278},"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fknowledge-representation","Knowledge Representation","First-order logic gives you the language; this lesson is about what to say in it. This first part builds the content: a general upper ontology from the top down, categories as first-class objects with taxonomies and inheritance, physical composition and the count-noun\u002Fmass-noun split, events and time reified through the event calculus, and belief modeled with modal logic — the machinery for representing the world an agent reasons about.\n",{"path":25280,"title":25281,"module":25237,"summary":25282},"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Freasoning-systems-and-defaults","Reasoning Systems and Default Logic","Having represented the world, this part is about reasoning with it at scale. Semantic networks give a graphical notation with fast inheritance; description logics keep subsumption and classification tractable by design. Then we confront the fact that most useful rules hold only by default: circumscription and default logic give a logical account of nonmonotonic reasoning, and truth maintenance systems retract conclusions cleanly when the beliefs beneath them change.\n",{"path":25284,"title":25285,"module":25286,"summary":25287},"\u002Fartificial-intelligence\u002Funcertainty\u002Fprobability-and-bayes","Quantifying Uncertainty","Uncertainty","Logic breaks down in any domain where the rules have exceptions you cannot enumerate — the qualification problem. Probability replaces truth values with degrees of belief that obey Kolmogorov's axioms, and the full joint distribution becomes a knowledge base from which any query is answered by summing entries: marginalization, conditioning, and normalization. Independence factors that joint into smaller pieces — the first step toward a calculus of rational belief that an agent can actually compute with.\n",{"path":25289,"title":25290,"module":25286,"summary":25291},"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayes-rule-and-naive-bayes","Bayes' Rule and Naive Bayes","Bayes' rule inverts a causal model into a diagnostic one, turning \"how a cause produces its symptoms\" into \"which cause explains what I observed.\" Ignoring the prior is the base-rate fallacy behind overconfident test results. Conditional independence then lets several pieces of evidence combine by multiplying likelihood ratios instead of building an exponential joint, giving the naive Bayes model and pointing directly at Bayesian networks.\n",{"path":25293,"title":25294,"module":25286,"summary":25295},"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayesian-networks","Bayesian Networks","A Bayesian network is a directed acyclic graph of random variables in which each node carries a conditional probability table for itself given its parents. That structure factors the full joint distribution into a product of local terms, turning an exponential table into a linear one, and it makes the conditional independences of the domain explicit. We build the canonical burglary–alarm network, read compactness and d-separation off the graph, run exact inference by variable elimination, and, where that is intractable, estimate answers by sampling.\n",{"path":25297,"title":25298,"module":25286,"summary":25299},"\u002Fartificial-intelligence\u002Funcertainty\u002Finference-in-bayesian-networks","Bayesian Networks: Inference and Relational Models","When exact inference is intractable, sampling estimates the posterior instead: prior and rejection sampling, likelihood weighting, and Gibbs\u002FMCMC, whose error shrinks as one over the square root of the sample count. The same graphical idea then lifts from a fixed set of variables to whole populations — relational and open-universe probability models write dependencies once and unroll them over objects — and we close by placing probability against the rule-based, Dempster–Shafer, and fuzzy alternatives it displaced.\n",{"path":25301,"title":25302,"module":25286,"summary":25303},"\u002Fartificial-intelligence\u002Funcertainty\u002Freasoning-over-time","Probabilistic Reasoning over Time","A world that changes needs a state variable at every point in time. The Markov assumption cuts the dependence on history down to the previous slice, leaving a transition model and a sensor model that define a temporal Bayesian network. Four recursive tasks fall out — filtering, prediction, smoothing, and the most likely explanation — each a message passed along the sequence. We ground them in hidden Markov models and their matrix form, sketch the Kalman filter for continuous state, and reach dynamic Bayesian networks with particle filtering as the general approximate method.\n",{"path":25305,"title":25306,"module":25286,"summary":25307},"\u002Fartificial-intelligence\u002Funcertainty\u002Ftracking-and-data-association","Reasoning over Time: Tracking and Data Association","Dynamic Bayesian networks generalize HMMs and Kalman filters to arbitrarily many state variables per slice, and when exact inference blows up, particle filtering approximates the belief state with a population of weighted samples that propagate, reweight, and resample. Tracking several objects at once adds the data-association problem — which observation came from which object — whose combinatorics defeat any exact filter, so particle filters and MCMC keep many hypotheses alive. We close with SLAM and learned state-space models.\n",{"path":25309,"title":25310,"module":25286,"summary":25311},"\u002Fartificial-intelligence\u002Funcertainty\u002Fmaking-decisions","Making Decisions: Utility Theory","A rational agent chooses the action that maximizes expected utility — the probability of each outcome weighted by how much the agent wants it. We derive the utility function from six axioms on preferences, so maximizing expected utility is forced by consistency rather than assumed; look at risk aversion in the utility-of-money curve; package one-shot choices into decision networks; and quantify what an observation is worth with the value of information.\n",{"path":25313,"title":24902,"module":25286,"summary":25314},"\u002Fartificial-intelligence\u002Funcertainty\u002Fmarkov-decision-processes","When an agent must act repeatedly in a stochastic world, a fixed plan is useless — it needs a policy, an action for every state. The Markov decision process makes this precise with a transition model, a reward, and a discount factor; the Bellman equation characterizes the optimal state utilities, and value iteration and policy iteration solve it. Partial observability lifts the problem to belief states, and bandits, Monte-Carlo tree search, and scalable POMDP solvers extend it — this is the model-known half of reinforcement learning.\n",{"path":25316,"title":25317,"module":25286,"summary":25318},"\u002Fartificial-intelligence\u002Funcertainty\u002Fdecision-networks-and-game-theory","Decision Analysis: Multi-Attribute Utility and Decision Networks","Decision analysis takes the single-agent utility framework and makes it practical: utility over several attributes, dominance and additive value functions, influence diagrams that fold Bayesian networks together with decision and utility nodes, and the value of information that tells an agent which questions are worth asking. Structure in an agent's preferences — dominance, preferential and utility independence — collapses an exponential utility table into a few one-dimensional functions, the same move that made Bayesian networks compact.\n",{"path":25320,"title":25321,"module":25286,"summary":25322},"\u002Fartificial-intelligence\u002Funcertainty\u002Fgame-theory-and-mechanism-design","Game Theory and Mechanism Design","When outcomes depend on other rational agents, single-agent utility maximization no longer suffices. Game theory studies decisions among agents — normal-form games, dominant strategies, Nash and maximin equilibria, and repeated games — and mechanism design runs the logic backwards, engineering rules (auctions, VCG) so that self-interested play produces a good collective outcome. Algorithmic game theory then asks whether equilibria can be computed, what selfishness costs society, and how the mechanisms deployed at internet scale actually behave.\n",{"path":25324,"title":25325,"module":25326,"summary":25327},"\u002Fartificial-intelligence\u002Flearning\u002Flearning-from-examples","Learning from Examples","Learning","An agent that improves with experience does not need its designer to anticipate every situation. Inductive learning takes that ambition and narrows it to one tractable problem: from labelled input-output pairs, recover a function that predicts the output for inputs never seen. This first part builds the foundation around a single organizing question — generalization — through decision trees and information gain, and the training\u002Fvalidation\u002Ftest discipline for evaluating and choosing hypotheses. A second part takes up the theory of learning and the main model families.\n",{"path":25329,"title":25330,"module":25326,"summary":25331},"\u002Fartificial-intelligence\u002Flearning\u002Ftheory-and-model-families","The Theory of Learning and Model Families","Cross-validation measures generalization but does not explain it. This part supplies the theory — PAC learning, sample complexity, and the VC dimension — that says when a hypothesis consistent with enough data is probably approximately correct, and why an unrestricted hypothesis space can never generalize. It then surveys the model families a practitioner reaches for: linear regression and gradient descent, the perceptron and logistic regression, support vector machines and the kernel trick, and ensembles by bagging and boosting — closing with what deep learning changed about the classical picture.\n",{"path":25333,"title":25334,"module":25326,"summary":25335},"\u002Fartificial-intelligence\u002Flearning\u002Fprobabilistic-learning","Learning Probabilistic Models","A [Bayesian network](\u002Fartificial-intelligence\u002Funcertainty\u002Fbayesian-networks) is useless until its numbers are filled in, and those numbers come from data. This first part casts learning itself as probabilistic inference: hypotheses carry a prior, data update it to a posterior, and predictions average over what remains. From that frame fall the standard estimators — maximum likelihood by counting, MAP with a conjugate prior, full Bayesian updating — for the case where every variable is observed. A second part takes up the harder case of hidden variables and the EM algorithm.\n",{"path":25337,"title":25338,"module":25326,"summary":25339},"\u002Fartificial-intelligence\u002Flearning\u002Fexpectation-maximization","Learning with Hidden Variables: The EM Algorithm","Complete data can be learned by counting; real data usually hide some variables — the disease behind the symptoms, the cluster behind the points. This part develops the expectation-maximization algorithm, which learns those models by alternating an expected completion of the missing data with a re-estimation of the parameters. It works the idea through mixtures of Gaussians, Bayesian networks, and hidden Markov models, proves the monotone-likelihood guarantee from the evidence lower bound, and traces the line from EM to variational inference and the variational autoencoder.\n",{"path":25341,"title":24325,"module":25326,"summary":25342},"\u002Fartificial-intelligence\u002Flearning\u002Freinforcement-learning","Reinforcement learning is an MDP with the model unknown: the agent knows neither how its actions move the world nor which states are rewarded, and must recover good behaviour from experienced transitions and rewards alone. This first part builds the classical tabular theory — passive learning (fix a policy, learn its value, by direct estimation, adaptive dynamic programming, and temporal differences) and active learning (choose actions, trade exploration against exploitation, and learn control with Q-learning and SARSA). A second part lifts it off the lookup table with function approximation and policy search.\n",{"path":25344,"title":25345,"module":25326,"summary":25346},"\u002Fartificial-intelligence\u002Flearning\u002Fgeneralization-and-policy-search","Reinforcement Learning: Generalization and Policy Search","Tabular reinforcement learning stores one number per state, which is hopeless for backgammon or chess. This part lifts RL off the lookup table with function approximation, so that updating one state generalizes to related ones, then turns to policy search — representing and optimizing the policy directly, up to the REINFORCE policy gradient and correlated sampling. It closes with the bridge to deep reinforcement learning (deep Q-networks, actor-critic, PPO), the classic applications, and the hand-off to the dedicated RL subject.\n",{"path":25348,"title":25349,"module":25326,"summary":25350},"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-in-learning","Knowledge in Learning","Pure induction learns a function from labelled examples while knowing almost nothing to begin with. This first part brings prior knowledge into the loop by recasting learning as logical inference — hypotheses, examples, and classifications as sentences. It develops current-best-hypothesis search, the version space and its general\u002Fspecific boundary maintained by candidate elimination, and states the three entailment constraints that fix how background knowledge enters. A second part builds the three knowledge-based methods those constraints define.\n",{"path":25352,"title":25353,"module":25326,"summary":25354},"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-based-learning-methods","Knowledge-Based Learning: EBL, Relevance, and ILP","Once learning is cast as logical inference, three methods follow from the three ways prior knowledge can enter. Explanation-based learning generalizes a single example by explaining it with the domain theory, gaining speed but nothing new. Relevance-based learning uses determinations to shrink the hypothesis space and converge from fewer examples. Inductive logic programming learns genuinely new first-order rules — top-down with FOIL, bottom-up by inverting resolution, even inventing new predicates — and connects to modern statistical relational and neuro-symbolic learning.\n",{"path":25356,"title":25357,"module":25358,"summary":25359},"\u002Fartificial-intelligence\u002Ffrontiers\u002Fvision-and-perception","Vision and Perception","Frontiers","Perception connects an agent to the physical world. We follow one modality — vision — from the physics of image formation (the pinhole camera, perspective projection, lenses, shading, color) through the early operations that turn a pixel array into edges, texture, and motion, and into recognition by appearance. The recurring problem is inversion: a camera collapses a 3-D world onto a 2-D grid, and an agent that wants to act must build the scene back up. Rebuilding the scene is the subject of the companion lesson.\n",{"path":25361,"title":25362,"module":25358,"summary":25363},"\u002Fartificial-intelligence\u002Ffrontiers\u002Freconstructing-the-3d-world","Vision: Reconstructing the 3D World","A camera collapses a three-dimensional world onto a flat grid; this lesson inverts that collapse. We build the camera projection matrix (intrinsics and extrinsics), triangulate a point from two views, then work through the toolbox of depth cues — motion parallax, binocular stereopsis, multiple views, texture, shading, and contour — that turn an ambiguous image back into a scene. We add structural recognition (pictorial-structure \"cardboard people\"), the task-driven use of vision in cars and robots, and the shift from hand-built pipelines to learned deep-vision networks.\n",{"path":25365,"title":25366,"module":25358,"summary":25367},"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobotics","Robotics","A robot is an agent with a body: sensors that read the physical world and effectors that push back on it. This lesson grounds the abstract AI machinery in that body. We build up the hardware (range finders, proprioception, degrees of freedom), then cast perception as probabilistic filtering — the kinematic motion and sensor models, Monte Carlo localization, the extended Kalman filter, and simultaneous localization and mapping (SLAM). The companion lesson takes the estimated pose forward into planning and control.\n",{"path":25369,"title":25370,"module":25358,"summary":25371},"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobot-planning-and-control","Robotics: Planning and Control","A robot that knows where it is still has to decide how to move, and then make a slipping, sensing-imperfect body actually go there. This lesson takes the pose estimate forward: planning motion in configuration space with cell decomposition and sampling-based roadmaps (PRMs and RRTs), planning under uncertainty with most-likely-state and online replanning, closing the loop with P\u002FPD\u002FPID control and potential fields, and finally the software architectures — subsumption, three-layer, and pipeline — that assemble it all, plus the learning-based turn in modern robotics.\n",{"path":25373,"title":25374,"module":25358,"summary":25375},"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnatural-language-in-ai","Natural Language for AI Agents","Language is how agents acquire the knowledge already written down and how they communicate with the humans they serve. This lesson gives the classical AI account of language as a source of information: n-gram language models and the information-seeking tasks built on them — text classification, information retrieval (BM25, the inverted index, PageRank), and information extraction with finite-state templates and hidden Markov models. Throughout, we point to the dedicated NLP subject for the modern deep-learning treatment; the companion lesson takes up grammar, translation, and speech.\n",{"path":25377,"title":25378,"module":25358,"summary":25379},"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnlp-grammar-translation-and-speech","Language for AI Agents: Grammar, Translation, and Speech","N-gram models see only a local window; they cannot say why \"black dog\" is well-formed English and \"dog black\" is not, because that is a fact about structure. This lesson takes up structure: phrase-structure and probabilistic context-free grammars, syntactic analysis by chart parsing and CYK, augmented grammars and compositional semantics, then the two major statistical successes — machine translation and speech recognition — cast as noisy-channel problems. It closes with the bridge from n-grams to transformers and where the classical account sits relative to modern NLP.\n",{"path":25381,"title":25382,"module":25358,"summary":25383},"\u002Fartificial-intelligence\u002Ffrontiers\u002Fphilosophy-and-future","Philosophy, Ethics, and the Future of AI","Two questions have shadowed the field since its founding: can machines act intelligently (weak AI), and can they really think (strong AI)? We work through Turing's objections and their rebuttals — the arguments from disability, mathematics, and informality — then the strong-AI debate: the mind-body problem, functionalism and the brain prosthesis, Searle's Chinese Room and the systems reply, and consciousness and qualia. The companion lesson turns from what AI can do to what it should, and closes the course.\n",{"path":25385,"title":25386,"module":25358,"summary":25387},"\u002Fartificial-intelligence\u002Ffrontiers\u002Fai-ethics-and-future","The Ethics and Future of AI","Having asked whether machines can act intelligently and really think, we turn to whether we should build them at all. This lesson works through the six ethical risks — lost jobs, autonomous weapons, surveillance and privacy, biased decisions, the safety of superintelligence, and the erosion of accountability — then the value-alignment problem in the LLM era, and where the classical agent components could go next. It closes the course by tying search, logic, probability, and learning into a single picture of intelligence as rational agency.\n",{"path":25389,"title":25390,"module":6,"summary":6},"\u002Fartificial-intelligence","Artificial Intelligence",{"path":25392,"title":25393,"module":25394,"summary":25395},"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-constituents-nuclide-chart","Nuclear Composition and Ground-State Properties","Nuclear Properties","The nucleus is a bound assembly of Z protons and N neutrons packed to a radius R = R0 A^(1\u002F3) at a nearly constant density of about 10^17 kg\u002Fm^3. We fix the vocabulary of nuclides, derive nuclear size from mirror-nuclide and electron-scattering data, read the binding-energy-per-nucleon curve, and model it with the liquid-drop semiempirical mass formula.\n",{"path":25397,"title":25398,"module":25394,"summary":25399},"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-size-charge-distributions","Nuclear Size, Shape, and Charge Distributions","Elastic electron scattering resolves the nucleus by its de Broglie wavelength. The measured cross section is the Mott point-charge cross section modulated by a form factor, and that form factor is the Fourier transform of the charge density. Diffraction minima fix the radius, the small-angle slope fixes the mean-square radius, and the fitted Woods-Saxon profile gives a central density and a skin thickness. Mirror-nucleus Coulomb energies, muonic-atom X-rays, and optical isotope shifts give independent radii that all track R = R0 A^(1\u002F3).\n",{"path":25401,"title":25402,"module":25394,"summary":25403},"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-masses-binding-energy","Nuclear Masses, Mass Excess, and Separation Energies","The atomic mass unit fixes the scale, and the mass excess collects the small binding-driven deviation from the integer mass number. Penning-trap cyclotron frequencies now measure masses to parts in a billion, and every decay and reaction Q-value is a difference of these masses. One- and two-nucleon separation energies read the binding difference between neighbouring nuclides directly, showing the even-odd pairing stagger and the sharp drops at magic numbers, and their vanishing marks the neutron and proton drip lines that bound the chart of the nuclides.\n",{"path":25405,"title":25406,"module":25394,"summary":25407},"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fsemi-empirical-mass-formula","The Semi-Empirical Mass Formula and the Valley of Stability","Five physical terms reproduce nuclear binding across the chart: a volume term from saturation, a surface term from the deficit of edge neighbours, a Coulomb term from the electrostatic self-energy of a charged sphere, an asymmetry term from the Pauli cost of unequal proton and neutron filling, and a pairing term. The formula is quadratic in Z at fixed A, so isobars lie on a mass parabola whose minimum sets the most stable charge and whose slope dictates the direction of beta decay. The same competition between surface and Coulomb energy defines the fissility parameter and the onset of fission.\n",{"path":25409,"title":25410,"module":25394,"summary":25411},"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-moments-multipoles","Nuclear Spin, Magnetic Dipole, and Electric Quadrupole Moments","The ground state of a nucleus carries a definite spin and parity, a magnetic dipole moment of order the nuclear magneton, and, when its spin exceeds one-half, an electric quadrupole moment that measures its shape. The single-particle Schmidt lines predict the magnetic moment of an odd-A nucleus from the last unpaired nucleon, and the measured moments fall between them. The quadrupole moment distinguishes prolate from oblate deformation, and hyperfine structure is the experimental handle that fixes the spin and the moments from an atomic spectrum.\n",{"path":25413,"title":25414,"module":25415,"summary":25416},"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnuclear-force-shell-overview","The Nuclear Force and the Shell Model","The Nuclear Force","The strong force between nucleons is short-range, charge-independent, saturated, and repulsive at its core, about a hundred times stronger than Coulomb. Yukawa explained it as an exchange of massive mesons, tying the force's range to the meson mass through the uncertainty principle. Layered on top, an independent-particle shell model with strong spin-orbit coupling reproduces the magic numbers 2, 8, 20, 28, 50, 82, 126.\n",{"path":25418,"title":25419,"module":25415,"summary":25420},"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fthe-deuteron","The Deuteron and the Tensor Force","The deuteron is the only bound two-nucleon state: one shallow level at 2.22 MeV, no excited states. A square-well fit fixes a depth near 35 MeV over a 2 fm range, yet the wavefunction leaks so far past the edge that most of the probability lies outside the force. Its spin-1 ground state, magnetic moment close to the sum of the free-nucleon moments, and small but nonzero electric quadrupole moment together force a D-state admixture and a non-central tensor force.\n",{"path":25422,"title":25423,"module":25415,"summary":25424},"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnucleon-nucleon-scattering","Nucleon-Nucleon Scattering and the Interaction's Structure","Scattering probes the nuclear force above threshold. Partial-wave analysis reduces low-energy data to a single s-wave phase shift, and the effective-range expansion packages that into a scattering length and an effective range. The triplet channel binds (the deuteron) while the singlet is only virtual, which together explain the anomalously large free neutron-proton cross section. Comparing pp, nn, and np results establishes charge symmetry and charge independence, and polarization experiments expose the spin-orbit and tensor pieces.\n",{"path":25426,"title":25427,"module":25415,"summary":25428},"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fmeson-theory-isospin","Meson Exchange, the Yukawa Potential, and Isospin","Yukawa's massive-field propagator turns the range of the nuclear force into a meson mass: the exchanged quantum's Compton wavelength is the range. One-pion exchange fixes the long-range tail, complete with the tensor structure the deuteron demanded, while heavier mesons build the intermediate attraction and the repulsive core. Charge independence becomes an isospin symmetry, the force is diagonalized by the total isospin through a tau-dot-tau interaction, and the whole picture sits inside QCD as a residual color force between color-neutral nucleons.\n",{"path":25430,"title":25431,"module":25432,"summary":25433},"\u002Fnuclear-physics\u002Fnuclear-models\u002Ffermi-gas-model","The Fermi Gas Model","Nuclear Models","Treating the nucleus as two degenerate Fermi gases of protons and neutrons confined in a common well fixes the Fermi momentum near 250 MeV\u002Fc and the Fermi energy near 33 MeV from the nuclear density alone. The average kinetic energy per nucleon is about 20 MeV, the well depth is the Fermi energy plus the separation energy, and unequal proton and neutron Fermi levels reproduce the asymmetry term of the mass formula.\n",{"path":25435,"title":25436,"module":25432,"summary":25437},"\u002Fnuclear-physics\u002Fnuclear-models\u002Fliquid-drop-collective-coordinates","The Liquid-Drop Model and Collective Deformation","Deforming a charged liquid drop into a spheroid raises its surface energy and lowers its Coulomb energy; the two effects compete through the deformation parameter to set a stability minimum and a fission barrier. The ratio of Coulomb to twice the surface energy is the fissility Z-squared over A, which crosses one near 49 and marks the point where the sphere is unstable. The same surface tension that restores small deformations quantizes into collective vibrations, carrying the static mass formula into dynamic collective motion.\n",{"path":25439,"title":25440,"module":25432,"summary":25441},"\u002Fnuclear-physics\u002Fnuclear-models\u002Fshell-model-single-particle","The Shell Model: Single-Particle States and Spin-Orbit Coupling","A harmonic-oscillator well reproduces the first three magic numbers but fails above twenty; adding a strong inverted spin-orbit term that drives the stretched j equals l plus one-half level down closes the gaps at 28, 50, 82, and 126. The filled shells couple to zero, so the last unpaired nucleon fixes the ground-state spin and parity, and its single-particle magnetic moment falls on the Schmidt lines. Configuration mixing sets the limits of the extreme single-particle model.\n",{"path":25443,"title":25444,"module":25432,"summary":25445},"\u002Fnuclear-physics\u002Fnuclear-models\u002Fcollective-model-rotations-vibrations","The Collective Model: Rotations, Vibrations, and Deformed Nuclei","Deformed nuclei rotate with energies proportional to I times I plus one, giving the ground-state band its characteristic level ratios, while near-spherical nuclei vibrate in quantized surface phonons that build one- and two-phonon multiplets. The Nilsson model tracks single-particle levels as the well deforms, moments of inertia fall between the rigid and irrotational limits, backbending marks the sudden alignment of a broken pair, and giant resonances are the bulk dipole and quadrupole modes of the whole nucleus.\n",{"path":25447,"title":25448,"module":25449,"summary":25450},"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-law-modes","Radioactivity and Decay Modes","Radioactive Decay","Unstable nuclei decay at a rate proportional to how many remain, giving the exponential law N(t) = N0 e^(-lambda t) with half-life t = 0.693\u002Flambda. We work through the three common modes: alpha decay as Coulomb-barrier tunneling with the Geiger-Nuttall rule, beta decay whose continuous spectrum demands the neutrino, and gamma de-excitation, and follow a decay chain across the chart of nuclides.\n",{"path":25452,"title":25453,"module":25449,"summary":25454},"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-kinetics-equilibrium","Serial Decay, the Bateman Equations, and Radioactive Equilibrium","A radioactive parent that decays into a radioactive daughter obeys a coupled pair of rate equations whose solution is the Bateman formula. Depending on the half-life ordering the chain settles into secular equilibrium (equal activities), transient equilibrium (a fixed activity ratio), or no equilibrium. Constant production under irradiation drives the activity toward a saturation value equal to the production rate, competing decay modes split the total decay constant into partial constants, and the natural decay series in secular equilibrium underpin radiometric dating.\n",{"path":25456,"title":25457,"module":25458,"summary":25459},"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-decay-gamow-theory","Alpha Decay and the Gamow Theory of Tunneling","Alpha Decay","The alpha Q-value turns positive above mass number 150 because the emitted helium-4 is exceptionally tightly bound. Emission proceeds by quantum tunneling through the Coulomb barrier: a WKB integral from the nuclear surface to the outer turning point gives the Gamow factor, and multiplying its penetrability by the assault frequency yields half-lives spanning more than twenty orders of magnitude. The leading term reproduces the Geiger-Nuttall relation, log t½ proportional to the daughter charge over the square root of Q.\n",{"path":25461,"title":25462,"module":25458,"summary":25463},"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-fine-structure-hindrance","Fine Structure, Angular Momentum, and Hindrance Factors","A single parent emits several alpha groups of slightly different energy, each feeding a distinct level of the daughter, so the alpha spectrum maps the daughter's low-lying states. Emission with orbital angular momentum L raises the barrier by a centrifugal term and is allowed only when angular-momentum and parity selection rules permit. Comparing the measured partial half-life to the Gamow estimate defines a hindrance factor near unity for even-even ground-state transitions and large for odd-A decays that must rearrange the unpaired nucleon.\n",{"path":25465,"title":25466,"module":25467,"summary":25468},"\u002Fnuclear-physics\u002Fbeta-decay\u002Fbeta-decay-energetics-neutrino","Beta Decay Energetics and the Neutrino","Beta Decay and the Weak Interaction","Beta decay converts a neutron into a proton or the reverse, adjusting Z at fixed A along an isobaric mass parabola. We write the three processes (beta-minus, beta-plus, electron capture), reduce every Q-value to a difference of neutral atomic masses, and read the continuous electron spectrum as the fingerprint of a third, nearly massless particle. Pauli's neutrino, its detection by Reines and Cowan, and the endpoint bound on its mass close the lesson.\n",{"path":25470,"title":25471,"module":25467,"summary":25472},"\u002Fnuclear-physics\u002Fbeta-decay\u002Ffermi-theory-beta-decay","Fermi's Theory: Kurie Plots and ft Values","Fermi treated beta decay as a point-contact weak transition and read its rate from the golden rule. The electron spectrum then follows from phase space and the Coulomb Fermi function; the Kurie plot straightens it to a line whose intercept is the endpoint. Integrating the spectrum gives the comparative half-life ft, whose logarithm sorts transitions into superallowed, allowed, and forbidden classes governed by the Fermi and Gamow-Teller selection rules.\n",{"path":25474,"title":25475,"module":25467,"summary":25476},"\u002Fnuclear-physics\u002Fbeta-decay\u002Fweak-interaction-parity-violation","The Weak Interaction and Parity Violation","Beta decay violates mirror symmetry. The Wu experiment on polarized cobalt-60 showed electrons emitted preferentially against the nuclear spin, a pseudoscalar correlation forbidden if parity were conserved. The result fixes the weak charged current as left-handed V minus A, forces neutrinos to be left-handed and antineutrinos right-handed (measured by Goldhaber), and places beta decay within the electroweak theory as W-boson exchange turning a down quark into an up quark.\n",{"path":25478,"title":25479,"module":25467,"summary":25480},"\u002Fnuclear-physics\u002Fbeta-decay\u002Fdouble-beta-decay-neutrino-mass","Double Beta Decay and Neutrino Mass","For even-A isobars the pairing term splits the mass parabola into two curves, and a handful of even-even nuclides sit below their odd-odd neighbor yet above the next even-even one: single beta decay is forbidden but second-order double beta decay is allowed. The two-neutrino mode is a standard-model process with the longest measured lifetimes in nature; the neutrinoless mode would require the neutrino to be its own antiparticle and its rate measures the effective Majorana mass, the sharpest probe of the absolute neutrino mass scale.\n",{"path":25482,"title":25483,"module":25484,"summary":25485},"\u002Fnuclear-physics\u002Fgamma-decay\u002Fgamma-multipole-radiation","Multipole Radiation and Selection Rules","Gamma Decay","Gamma decay carries a nucleus from an excited state to a lower one by emitting a photon of definite angular momentum and parity. We correct the photon energy for nuclear recoil, expand the radiation field into electric and magnetic multipoles, and read off how the transition rate collapses with each increase in multipole order. The Weisskopf single-particle estimates set the scale, and angular-momentum and parity conservation fix which multipole dominates.\n",{"path":25487,"title":25488,"module":25484,"summary":25489},"\u002Fnuclear-physics\u002Fgamma-decay\u002Finternal-conversion-isomers","Internal Conversion and Isomers","A nucleus can shed excitation energy without emitting a photon by handing it directly to an atomic electron. We define the internal-conversion coefficient, trace its growth with atomic number, multipole order, and decreasing energy, and treat the electron-only E0 transitions and internal pair formation. When the lowest allowed multipole is high and the energy low, the gamma rate falls so far that the excited state survives as a metastable isomer.\n",{"path":25491,"title":25492,"module":25484,"summary":25493},"\u002Fnuclear-physics\u002Fgamma-decay\u002Fangular-correlations-mossbauer","Angular Correlations and the Mössbauer Effect","Two gammas emitted in cascade are not independent in direction: detecting the first selects magnetic substates of the intermediate level and makes the second anisotropic, so the correlation function fixes the intermediate spin. The same nuclear resonance that recoil normally destroys is recovered when the emitter is locked in a lattice, giving the Mössbauer effect and its part-in-a-trillion resolution of isomer shifts and hyperfine fields.\n",{"path":25495,"title":25496,"module":25497,"summary":25498},"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Freaction-kinematics-cross-sections","Nuclear Reactions, Fission, and Fusion","Nuclear Reactions","A nuclear reaction X(x, y)Y is governed by its Q value and its cross section, the effective target area for a given process. Splitting the curve of binding energy near iron in either direction releases energy: fission of heavy nuclei by neutron capture and a chain reaction, and fusion of light nuclei that powers the Sun and needs Lawson's density-confinement criterion to be practical.\n",{"path":25500,"title":25501,"module":25497,"summary":25502},"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fcompound-nucleus-resonances","The Compound Nucleus and Resonance Reactions","Low-energy reactions proceed through a long-lived intermediate state whose decay forgets how it formed. Bohr's independence hypothesis factorizes the cross section into a formation step and a branching ratio, an isolated level gives the single-level Breit-Wigner line shape with total width Γ tied to the lifetime by Γτ = ħ, and at high excitation overlapping levels merge into a statistical continuum described by evaporation spectra and the Hauser-Feshbach average.\n",{"path":25504,"title":25505,"module":25497,"summary":25506},"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fdirect-reactions-optical-model","Direct Reactions and the Optical Model","A complex optical potential replaces the many-body target by a single particle moving in an average field whose imaginary part removes flux into non-elastic channels, reproducing the diffraction pattern of elastic scattering. Direct reactions bypass the compound nucleus, transferring a nucleon in one step: stripping and pickup deposit or remove a single nucleon, the angle of the first peak in the distorted-wave angular distribution fixes the transferred orbital angular momentum, and its magnitude gives the spectroscopic factor.\n",{"path":25508,"title":25509,"module":25510,"summary":25511},"\u002Fnuclear-physics\u002Ffission\u002Ffission-barrier-dynamics","The Fission Barrier and Fragment Energetics","Nuclear Fission","Fission is the large-amplitude collective deformation of a heavy nucleus into two fragments. The liquid-drop model sets a barrier from the competition between rising surface energy and falling Coulomb energy under quadrupole deformation, with the fissility parameter Z²\u002FA measuring how close a nucleus is to instability. Bohr-Wheeler theory separates spontaneous from neutron-induced fission, the fragment mass yield is double-humped and asymmetric, about 200 MeV is released per event, and shell corrections add a second minimum that produces fission isomers.\n",{"path":25513,"title":25514,"module":25510,"summary":25515},"\u002Fnuclear-physics\u002Ffission\u002Fchain-reactions-reactor-physics","Chain Reactions and Reactor Physics","A self-sustaining chain reaction is a fixed point of neutron bookkeeping: the multiplication factor k counts the neutrons in one generation per neutron in the last, and criticality is k = 1. The four-factor formula tracks a neutron through fast fission, resonance escape, thermal utilization, and reproduction; moderation slows fission neutrons to the thermal energies where the fission cross section is largest; and the small delayed-neutron fraction sets the timescale that makes a reactor controllable. Breeding converts fertile U-238 and Th-232 into new fissile fuel.\n",{"path":25517,"title":25518,"module":25519,"summary":25520},"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Ffusion-reactions-confinement","Fusion Reactions and Confinement","Fusion and Nucleosynthesis","Light nuclei release energy when they fuse because binding per nucleon rises steeply toward the iron peak, but the Coulomb barrier suppresses the rate at reactor temperatures. The thermonuclear rate is a convolution of the Maxwell distribution with the tunneling probability, sharply peaked at the Gamow energy. The deuterium-tritium reaction has the lowest barrier and largest cross section; sustained energy gain requires the Lawson triple product of density, temperature, and confinement time, reached by magnetic or inertial confinement.\n",{"path":25522,"title":25523,"module":25519,"summary":25524},"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fstellar-nucleosynthesis","Stellar Nucleosynthesis","Main-sequence stars burn hydrogen to helium through the proton-proton chain and the CNO cycle, both releasing 26.7 MeV per helium nucleus. Helium burning bridges the mass-5 and mass-8 gaps by the triple-alpha process through the Beryllium-8 and Hoyle resonances, and successive carbon-to-silicon burning stages climb to the iron peak, where fusion stops. The elements beyond iron are built by slow and rapid neutron capture, and the solar neutrino flux confirms the reactions directly.\n",{"path":25526,"title":25527,"module":25519,"summary":25528},"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fbig-bang-nucleosynthesis","Big-Bang Nucleosynthesis","In the first three minutes the expanding universe forged the light elements. The weak interaction froze the neutron-to-proton ratio near one in six when the reaction rate fell below the expansion rate, and free-neutron decay lowered it to about one in seven before the deuterium bottleneck broke. Almost every surviving neutron ended in helium-4, fixing the primordial helium mass fraction near 0.25, with trace deuterium, helium-3, and lithium-7. The deuterium abundance measures the cosmic baryon density.\n",{"path":25530,"title":25531,"module":25532,"summary":25533},"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fcharged-particle-stopping-power","Stopping Power and the Range of Charged Particles","Radiation and Applications","A heavy charged particle loses energy in a dense sequence of small Coulomb collisions with atomic electrons, at a rate the Bethe-Bloch formula fixes from the particle's charge and speed and the medium's electron density and mean excitation energy. The rate scales as the inverse square of the speed, so most energy is deposited at the end of the track in the Bragg peak, and integrating the reciprocal rate gives a sharp range. Electrons differ: they also radiate, and above a critical energy bremsstrahlung dominates. Fast particles above the phase velocity of light in the medium emit Cherenkov radiation.\n",{"path":25535,"title":25536,"module":25532,"summary":25537},"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fphoton-neutron-interactions","Interactions of Photons and Neutrons","Photons are removed from a beam in single events, so their intensity falls exponentially with a linear attenuation coefficient built from three processes: the photoelectric effect at low energy, Compton scattering at intermediate energy, and pair production above twice the electron rest energy, each with its own atomic-number and energy dependence. Neutrons carry no charge and interact only with nuclei, moderating by elastic scattering and being captured with a cross section that rises as one over speed away from resonances.\n",{"path":25539,"title":25540,"module":25532,"summary":25541},"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fradiation-detectors","Radiation Detectors and Nuclear Spectroscopy","Every detector converts the energy a radiation deposits into a measurable electrical signal. Gas counters read the ionization directly, in three operating regions set by the applied voltage; scintillators convert the energy to light read out by a photomultiplier; semiconductor detectors collect electron-hole pairs and give the best energy resolution because so many carriers are made per event. The resolution is governed by the number of independent charge carriers, and the pulse-height spectrum of a gamma line shows a full-energy photopeak, a Compton continuum with its edge, and escape peaks.\n",{"path":25543,"title":25544,"module":25532,"summary":25545},"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fdosimetry-radiation-biology","Dosimetry, Radiation Biology, and Protection","Absorbed dose is the energy deposited per unit mass, measured in gray. Equal absorbed doses do unequal biological damage because densely ionizing radiation deposits its energy along short tracks: weighting the dose by a radiation factor gives the equivalent dose, and weighting by tissue sensitivity gives the effective dose, both in sieverts. Deterministic effects have a threshold and a severity that grows with dose; stochastic effects are assumed to follow a linear-no-threshold probability. Natural background dominates the dose to the population, and protection rests on time, distance, and shielding.\n",{"path":25547,"title":25548,"module":25532,"summary":25549},"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fnuclear-applications-dating-medicine","Applications — Dating, Analysis, and Nuclear Medicine","Charged particles lose energy continuously and stop at a well-defined range with a Bragg peak, while gamma rays are attenuated exponentially. These interactions define radiation detectors and dosimetry (gray and sievert) and drive the applications: neutron activation analysis, magnetic resonance imaging, PET, and radiometric dating with carbon-14 and long-lived rock clocks.\n",{"path":25551,"title":25552,"module":6,"summary":6},"\u002Fnuclear-physics","Nuclear Physics",{"path":25554,"title":25555,"module":21519,"summary":25556},"\u002Fnatural-language-processing\u002Ffoundations\u002Fwhat-is-nlp","What Is Natural Language Processing?","Natural language processing is the computational treatment of human language: reading it, representing it, and generating it. We set up why the problem is hard — ambiguity at every level, from sound to intent — trace the field from ELIZA's pattern-matching through statistical methods to today's neural models, lay out the linguistic levels and task families the course covers, and fix the vocabulary of tokens, types, and corpora the rest of the notes rely on.\n",{"path":25558,"title":25559,"module":21519,"summary":25560},"\u002Fnatural-language-processing\u002Ffoundations\u002Fregex-and-text-normalization","Regular Expressions and Text Normalization","Before any model touches text, the text has to be found and cleaned. Regular expressions give an algebra for describing string patterns; tokenization, case folding, and stemming turn raw characters into the units a model counts; and byte-pair encoding builds a subword vocabulary that spells out any word. Measuring how far apart two strings are — minimum edit distance — is the next lesson.\n",{"path":25562,"title":25563,"module":21519,"summary":25564},"\u002Fnatural-language-processing\u002Ffoundations\u002Fminimum-edit-distance","Minimum Edit Distance","Much of language processing needs to measure how similar two strings are — a speller ranking corrections, a diff tool, a coreference resolver. Minimum edit distance counts the insertions, deletions, and substitutions that turn one string into another, computed by a dynamic-programming table. We fill the table for intention to execution, backtrace to recover the alignment, and see how the same machinery generalizes to weighted edits, Viterbi, and biological sequence alignment.\n",{"path":25566,"title":25567,"module":21519,"summary":25568},"\u002Fnatural-language-processing\u002Ffoundations\u002Fn-gram-language-models","N-Gram Language Models","A language model assigns a probability to a sequence of words and, equivalently, predicts the next word from its history. The n-gram model makes this tractable by truncating the history to the last few words, estimates the resulting conditional probabilities by counting, and is scored by perplexity. We build the model from the chain rule, work a bigram example on a small corpus, and read perplexity as a branching factor. The next lesson covers the zero counts that break this model and the smoothing that repairs them.\n",{"path":25570,"title":25571,"module":21519,"summary":25572},"\u002Fnatural-language-processing\u002Ffoundations\u002Fsmoothing-and-backoff","Smoothing and Backoff","Every finite corpus is missing good word sequences it simply never saw, so a raw n-gram model assigns them probability zero and breaks. Smoothing repairs the zeros: add-one and add-k shave mass off seen events, backoff and interpolation fall back on shorter contexts, and Kneser-Ney — worked here by hand — replaces raw frequency with how many contexts a word completes. We close on web-scale stupid backoff and the neural models that dissolve the zero problem rather than patch it.\n",{"path":25574,"title":25575,"module":25576,"summary":25577},"\u002Fnatural-language-processing\u002Fclassification\u002Fnaive-bayes-and-sentiment","Naive Bayes and Sentiment Classification","Text Classification","Text classification assigns a category to a document — positive or negative, spam or not, one topic among many. Naive Bayes is a generative solution: apply Bayes' rule, assume the words are conditionally independent given the class, and the winning class is the one maximizing the product of a prior and per-word likelihoods. We train it by counting with add-one smoothing, work a full sentiment example by hand, sharpen it for sentiment (binary counts, negation, lexicons), and place it among the transformer classifiers that came after.\n",{"path":25579,"title":25580,"module":25576,"summary":25581},"\u002Fnatural-language-processing\u002Fclassification\u002Fevaluating-classifiers","Evaluating Classifiers","A trained classifier is only useful once we can measure how good it is. We build the confusion matrix, see why accuracy misleads on unbalanced data, and define precision, recall, and the F-measure that balances them. Multi-class tasks need macro- versus micro-averaging; reliable estimates need cross-validation. We close on statistical significance — the paired bootstrap test for whether one system's lead over another is significant.\n",{"path":25583,"title":25584,"module":25576,"summary":25585},"\u002Fnatural-language-processing\u002Fclassification\u002Flogistic-regression","Logistic Regression","Logistic regression is the discriminative counterpart to naive Bayes: instead of modelling how a document is generated, it learns weights that directly separate the classes. We build it from the sigmoid, derive the cross-entropy loss from maximum likelihood, learn the weights by stochastic gradient descent, regularize to curb overfitting, and generalize to many classes with the softmax. The two-class model is already a one-neuron network, so this is the bridge to neural language models.\n",{"path":25587,"title":25588,"module":25576,"summary":25589},"\u002Fnatural-language-processing\u002Fclassification\u002Fsentiment-and-affect-lexicons","Sentiment and Affect Lexicons","A sentiment lexicon is a list of words annotated with the affective meaning they carry — positive or negative, or scores along valence, arousal, and dominance. We fix what \"emotion\" means (basic-emotion versus dimensional models), survey the standard lexicons, and then build lexicons three ways: by human labeling with best-worst scaling, by semi-supervised induction from seed words over an embedding space, and by supervised learning from starred reviews. We close on connotation frames, which record the sentiment a verb implies about each of its arguments.\n",{"path":25591,"title":25592,"module":25593,"summary":25594},"\u002Fnatural-language-processing\u002Fsemantics\u002Fvector-semantics-and-embeddings","Vector Semantics and Embeddings","Semantics","Vector semantics represents a word's meaning as a point in space, derived from the company the word keeps. This first part builds the count-based side: the distributional hypothesis, co-occurrence matrices in their term-document and word-word forms, cosine as the similarity measure, and the two weightings — tf-idf and PPMI — that fix what raw counts get wrong. The result is a sparse, interpretable vector for every word, and the setup for the dense embeddings of the next lesson.\n",{"path":25596,"title":25597,"module":25593,"summary":25598},"\u002Fnatural-language-processing\u002Fsemantics\u002Fstatic-word-embeddings","Static Word Embeddings: word2vec and After","Count-based vectors are long and sparse; embeddings are the short, dense alternative. This lesson builds them with word2vec's skip-gram and negative sampling — a classifier whose learned weights are the vectors — derives its gradient, and works one update by hand. It then reads relations off the analogy parallelogram, surveys the papers that framed the static-embedding era (word2vec, GloVe, the SGNS-as-PPMI equivalence, fastText, ELMo), and closes on the biases embeddings inherit and the single-vector-per-word ceiling that contextual models break.\n",{"path":25600,"title":25601,"module":25593,"summary":25602},"\u002Fnatural-language-processing\u002Fsemantics\u002Fneural-language-models","Neural Networks and Neural Language Models","A neural network is a stack of units, each a weighted sum passed through a non-linearity — a single unit on its own is logistic regression. We build the network up from that unit: the activation functions that give it power, the XOR problem that forces a hidden layer, the feedforward forward pass in matrix form, and the Bengio-style feedforward neural language model that concatenates word embeddings and predicts the next word with a softmax. Training is cross-entropy minimized by gradient descent, with backpropagation supplying the gradient. Embeddings let the model share statistical strength across similar words, avoiding the sparsity that limits n-gram models.\n",{"path":25604,"title":25605,"module":21985,"summary":25606},"\u002Fnatural-language-processing\u002Fsequences\u002Fsequence-labeling","Sequence Labeling: POS and NER","Sequence labeling assigns one tag to every token in a sentence. This first part sets up the task through its two canonical cases — part-of-speech tagging over the Penn Treebank tagset, and named-entity recognition reframed as token labeling with the BIO scheme — then builds the hidden Markov model, the classic probabilistic tagger. The HMM tags by Bayesian inference: transition and emission probabilities under two Markov assumptions, reducing tagging to an argmax over tag sequences. That argmax is exponential to enumerate, which sets up the Viterbi decoder, the CRF, and neural taggers of the next lesson.\n",{"path":25608,"title":25609,"module":21985,"summary":25610},"\u002Fnatural-language-processing\u002Fsequences\u002Fcrfs-and-neural-taggers","Viterbi Decoding, CRFs, and Neural Taggers","The HMM reduced tagging to an argmax over exponentially many tag sequences. This lesson builds the decoder that makes it tractable — the Viterbi dynamic program, worked through a full numeric trace on real WSJ probabilities — then keeps that same decoder while replacing the HMM's rigid tables. The linear-chain conditional random field is a discriminative log-linear model whose global feature functions can inspect any part of the input, which is why CRFs win for NER. Finally it traces the shift to neural taggers (biLSTM-CRF, character-aware NER, ELMo), where hand-built features become learned representations while the Viterbi decoder carries over unchanged.\n",{"path":25612,"title":25613,"module":21985,"summary":25614},"\u002Fnatural-language-processing\u002Fsequences\u002Frnns-and-lstms","RNNs and LSTMs","A feedforward neural language model sees a fixed window of words and can look no further back. The recurrent neural network removes that limit: it carries a hidden state across time, so each word is read in the context of everything before it. We build the RNN from its one recurrent equation, use it as a language model, train it by backpropagation through time, and diagnose the vanishing-gradient problem that makes plain RNNs forget. The LSTM fixes the forgetting with a cell state and three gates, and the encoder-decoder stacks two RNNs into a sequence-to-sequence model — and its single-vector bottleneck is the problem attention was invented to remove.\n",{"path":25616,"title":25617,"module":22467,"summary":25618},"\u002Fnatural-language-processing\u002Ftransformers\u002Ftransformers-and-attention","Transformers and Self-Attention","Recurrence forced language models to read one word at a time and to squeeze every dependency through a chain of hidden states. Self-attention removes the recurrence: at every layer each position compares itself to every other and reads a weighted mixture of them, in a single parallel step. This first part builds the attention operation from the ground up — the soft lookup, queries and keys and values, the scaled dot-product, the numeric trace, the matrix form, and the causal mask — and sets up the full transformer architecture that follows.\n",{"path":25620,"title":24171,"module":22467,"summary":25621},"\u002Fnatural-language-processing\u002Ftransformers\u002Fthe-transformer-architecture","This part takes the scaled dot-product attention of the previous lesson and assembles the full transformer architecture around it: multi-head attention so several relations can be read at once, the transformer block of residual connections and layer norm that makes deep stacks trainable, positional embeddings that restore word order, the decoder-only language model, and the encoder, decoder, and encoder-decoder shapes — closing with the 2017 paper and the pre-norm, FlashAttention, and RoPE refinements that scaled it up.\n",{"path":25623,"title":24279,"module":22467,"summary":25624},"\u002Fnatural-language-processing\u002Ftransformers\u002Flarge-language-models","A large language model is a decoder-only transformer trained on one objective — predict the next token. This first part assembles the inference side: the language-modeling head that turns a hidden state into a distribution over the vocabulary, autoregressive generation, and the decoding strategies — greedy, beam, and sampling with temperature, top-k, and nucleus — that read text back out of that distribution. Training the distribution at web scale comes next.\n",{"path":25626,"title":25627,"module":22467,"summary":25628},"\u002Fnatural-language-processing\u002Ftransformers\u002Fllm-pretraining-and-scaling","Large Language Models: Pretraining and Scaling","A language model's next-token distribution is only as good as the parameters behind it. This part is where those parameters come from: self-supervised pretraining on web-scale text with teacher forcing and cross-entropy, the scaling laws that make test loss a predictable power law in parameters, data, and compute, the KV cache that keeps long-context inference affordable, and how a finished model is evaluated by perplexity and benchmarks — closing with the Kaplan, Chinchilla, GPT-3, and emergence papers behind the scaling story.\n",{"path":25630,"title":25631,"module":22467,"summary":25632},"\u002Fnatural-language-processing\u002Ftransformers\u002Ffine-tuning-and-prompting","Fine-Tuning and Prompting","A pretrained transformer is a general-purpose knowledge source; a task is what you do with it. There are two ways to adapt one, and this first part covers the one that updates the weights: fine-tuning. A bidirectional encoder like BERT is pretrained by masked language modeling, then a small task head is bolted on and the whole thing is trained on labelled data for classification, sequence labeling, or span-based question answering — with parameter-efficient variants (adapters, LoRA) that touch only a sliver of the weights. Prompting, the family that leaves the weights frozen, comes next.\n",{"path":25634,"title":25635,"module":22467,"summary":25636},"\u002Fnatural-language-processing\u002Ftransformers\u002Fprompting-and-alignment","Prompting and Alignment","Fine-tuning adapts a model by changing its weights. The second family of adaptation changes nothing: a large frozen model performs a task from an instruction and a few examples placed in its context. This part covers prompting and in-context learning, chain-of-thought that elicits reasoning, and the two training stages — instruction tuning and RLHF — that turn a fluent base predictor into an aligned assistant, closing with the BERT, LoRA, chain-of-thought, InstructGPT, and retrieval-augmentation papers behind the modern adaptation pipeline.\n",{"path":25638,"title":25639,"module":25640,"summary":25641},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-parsing","Constituency Parsing","Linguistic Structure","A constituency parse groups a sentence into nested phrases described by a context-free grammar. We build the CFG formalism, read the phrase structure of English off a treebank, confront the structural ambiguity that makes parsing hard, convert to Chomsky normal form, and then solve it with CKY — the dynamic-programming chart that fills a triangular table bottom-up. Probabilistic and neural span parsers, evaluation, and shallow parsing follow in the companion lesson.\n",{"path":25643,"title":25644,"module":25640,"summary":25645},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcky-scoring-and-evaluation","CKY Scoring, Evaluation, and Shallow Parsing","The CKY chart returns every parse but does not say which is correct. Disambiguation needs a score on trees. This lesson attaches probabilities to a grammar (the PCFG and lexicalization), replaces the grammar with a neural span scorer over a pretrained encoder, states the self-attentive results that made it the state of the art, evaluates parsers against a treebank with PARSEVAL, and closes with chunking and shallow parsing for tasks that need only the flat phrases.\n",{"path":25647,"title":25648,"module":25640,"summary":25649},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdependency-parsing","Dependency Parsing","A dependency parse throws away phrases and keeps only directed, labeled arcs from heads to their dependents, so the subject and object of a verb hang off the verb directly. We fix the formalism (rooted trees, typed Universal-Dependency relations, projectivity), then build the first parser family: transition-based arc-standard and arc-eager parsing, a greedy stack-and-buffer machine trained from an oracle. Graph-based and neural dependency parsing follow in the companion lesson.\n",{"path":25651,"title":25652,"module":25640,"summary":25653},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fgraph-based-and-neural-dependency-parsing","Graph-Based and Neural Dependency Parsing","Greedy transition parsing commits locally; the graph-based family scores whole trees instead. This lesson scores every candidate head-dependent edge and extracts the maximum spanning tree with Chu-Liu\u002FEdmonds, develops the biaffine neural scorer that made graph-based parsing the accuracy leader, evaluates parsers with the unlabeled and labeled attachment scores (UAS and LAS), and closes on where the two parser families sit and what they feed downstream.\n",{"path":25655,"title":25656,"module":25640,"summary":25657},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fword-senses-and-wsd","Word Senses and Disambiguation","A word is not an atom of meaning: \"bass\" names a fish, a voice, and an instrument, and one static embedding blurs them into a single point. This lesson pulls those senses apart. We define polysemy and the relations that organize senses — synonymy, antonymy, hyponymy, meronymy — build them into WordNet's synset graph, measure similarity along that graph, and then solve the core of word sense disambiguation: the most-frequent-sense baseline, the Lesk gloss-overlap algorithm, feature-based classifiers, and the nearest-neighbor method over BERT embeddings. WSD variants, embeddings, and evaluation follow in the companion lesson.\n",{"path":25659,"title":25660,"module":25640,"summary":25661},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fwsd-in-practice-and-induction","WSD in Practice and Word Sense Induction","Beyond core word sense disambiguation lie the variants and loose ends: the sense-inventory-free Word-in-Context task, retrofitting static embeddings to a thesaurus, discovering senses without a fixed inventory (word sense induction), the gloss-aware and bi-encoder neural systems that hold the state of the art, and how WSD and its cousins are evaluated. Together they connect one-vector-per-word embeddings to sense-aware contextual representations.\n",{"path":25663,"title":25664,"module":25640,"summary":25665},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-roles-and-information-extraction","Semantic Roles and Information Extraction","Semantic roles answer \"who did what to whom\" for a single event, abstracting away the syntax that expresses it. We show why syntax alone is not enough, generalize over diathesis alternations with thematic roles, number a predicate's arguments with PropBank and group predicates into frames with FrameNet, tag each argument automatically with semantic role labeling, and factor predicates into primitives. Information extraction scales the idea to a corpus in the companion lesson.\n",{"path":25667,"title":25668,"module":25640,"summary":25669},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Frelations-events-and-templates","Relations, Events, and Templates","Semantic roles answer \"who did what\" for one predicate; information extraction scales the idea to a whole corpus. This lesson turns unstructured text into structured data: relation extraction pulls entity-relation-entity triples out of sentences by patterns, supervision, and distant supervision; event and temporal extraction place those facts on a timeline; and template filling and knowledge-base population assemble them into a database a downstream system can query.\n",{"path":25671,"title":25672,"module":25640,"summary":25673},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoreference-and-discourse","Coreference and Discourse","A text is more than a bag of sentences: entities recur under different names. Coreference resolution links every mention to the discourse entity it evokes — the linguistic background of pronouns, definite NPs, and names; mention detection; the mention-pair, mention-ranking, and entity-based architectures; a neural end-to-end span model that scores candidate antecedents; features, evaluation by the CoNLL F1, gender bias, and the neural coreference lineage. Discourse coherence follows in the companion lesson.\n",{"path":25675,"title":25676,"module":25640,"summary":25677},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoherence-and-discourse-structure","Coherence and Discourse Structure","Coherence is what makes a run of sentences a discourse rather than an arbitrary collection. This lesson develops coherence relations and Rhetorical Structure Theory trees, discourse-structure parsing, Centering and the entity grid for entity-based coherence, and representation-learning models of local coherence, measured in part over the coreference chains recovered in the companion lesson.\n",{"path":25679,"title":25680,"module":25640,"summary":25681},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Flogical-semantics","Logical Representations of Meaning","A meaning representation turns a sentence into a formal structure a machine can check against a world and reason over. We set the desiderata a good representation must meet, ground truth in a model, build up first-order logic for sentences with its connectives, quantifiers, and inference, and reify events with the neo-Davidsonian event variable to escape fixed predicate arity. The compositional lambda calculus, quantifier scope, and description logics follow in the companion lesson.\n",{"path":25683,"title":25684,"module":25640,"summary":25685},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcompositional-semantics-and-description-logics","Compositional Semantics and Description Logics","How do you compute a logical form from a sentence automatically? This lesson builds the compositional machinery: the lambda calculus that assembles a formula from a parse tree one beta-reduction at a time, the quantifier-scope ambiguity a single syntax tree leaves open, and the decidable description logics — TBox, ABox, subsumption, role restrictions — behind the Web Ontology Language, closing with how the map from string to logical form can be learned.\n",{"path":25687,"title":25688,"module":25640,"summary":25689},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-parsing","Semantic Parsing","Turning a sentence into a structured, executable meaning, the grammar-based way. We take the logical forms defined earlier and build them compositionally: a rule-based parser that walks a syntax tree applying lambda terms, then Combinatory Categorial Grammar (CCG), which fuses syntax and semantics so one lexicalized derivation produces both — including supertagging and A* parsing. Learned and neural semantic parsers follow in the companion lesson.\n",{"path":25691,"title":25692,"module":25640,"summary":25693},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fneural-semantic-parsing","Learned and Neural Semantic Parsing","Hand-writing a lexicon of lambda terms does not scale, so this lesson learns the parser instead. We cover the two supervision regimes (from logical forms and from denotations), Abstract Meaning Representation as a rooted concept graph, neural sequence-to-sequence parsing with constrained decoding and copy mechanisms, executable text-to-SQL and knowledge-based question answering, the practical systems that made learned parsers accurate, and how the task is evaluated.\n",{"path":25695,"title":25696,"module":25640,"summary":25697},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Finformation-extraction","Information Extraction","Information extraction turns free text into a database, and the first step is relation extraction: pulling entity-relation-entity triples out of sentences. We cover all five families — hand-built patterns, supervised classifiers, semi-supervised bootstrapping, distant supervision, and unsupervised Open IE — with worked bootstrapping and distant-supervision traces, then the neural and LLM systems that extended them. Times, events, and templates follow in the companion lesson.\n",{"path":25699,"title":25700,"module":25640,"summary":25701},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftimes-events-and-templates","Extracting Times, Events, and Templates","Once relation extraction has produced typed triples, the information-extraction pipeline still has to place facts in time and assemble them into records. This lesson detects and normalizes temporal expressions to ISO 8601 values, detects events and orders them on a timeline with the 13 Allen relations, and fills slot-and-filler templates — flat and hierarchical — for stereotyped situations, closing the loop from text to a queryable database.\n",{"path":25703,"title":25704,"module":25640,"summary":25705},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdiscourse-coherence","Discourse Coherence","A text is more than a set of sentences. What binds a run of sentences into a discourse is coherence, and one of its sources is structured relations between clauses. This lesson develops relational coherence — RST and the PDTB models of coherence relations — and discourse-structure parsing: EDU segmentation and shift-reduce RST parsing, then PDTB relation classification. Entity-based and global coherence follow in the companion lesson.\n",{"path":25707,"title":25708,"module":25640,"summary":25709},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fentity-based-and-global-coherence","Entity-Based and Global Coherence","A text coheres not only through relations between clauses but by staying about the same entities and the same topic, and by obeying the macro-structure of its genre. This lesson develops Centering Theory and the entity grid for entity-based coherence, representation-learning models of local coherence, and global coherence — topic segmentation, narrative and argumentation structure, and scientific discourse — then the neural models that learn each.\n",{"path":25711,"title":25712,"module":25640,"summary":25713},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-grammars","Constituency Grammars","A constituency grammar is the declarative theory of sentence structure that a parser operates on. We build the context-free grammar formalism from its four parts, show how derivations become parse trees, and work through the phrase structure of English — noun phrases, verb phrases and their subcategorization frames, agreement, coordination, and long-distance dependencies. The treebank, normal-form, and lexicalized views follow in the companion lesson.\n",{"path":25715,"title":25716,"module":25640,"summary":25717},"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftreebanks-and-lexicalized-grammars","Treebanks and Lexicalized Grammars","Where does a grammar come from, and how is it prepared for a parser? We read a context-free grammar off the Penn Treebank, normalize it to Chomsky Normal Form for the CKY chart, then invert the phrase-structure emphasis with lexicalized grammars — Combinatory Categorial Grammar and its slash categories — and close with the grammar's fate in the neural era: span scoring, self-attention, and grammar induction.\n",{"path":25719,"title":25720,"module":24267,"summary":25721},"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation","Machine Translation","Machine translation is the task that built the modern toolkit: the encoder-decoder was invented for it, attention was invented to fix its fixed-context bottleneck, and both were later folded into the general transformer. We work through why translation is hard (word order, morphology, lexical and structural divergences), the sequence-to-sequence model and its attention mechanism, transformer-based NMT with cross-attention, subword tokenization with a shared vocabulary, beam-search decoding, and evaluation by BLEU and its successors chrF, BERTScore, and COMET — closing on multilingual and low-resource translation and backtranslation.\n",{"path":25723,"title":25724,"module":24267,"summary":25725},"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation-decoding-and-evaluation","Machine Translation: Decoding, Evaluation, and Scale","Having built the transformer translation model, we now decode from it and measure the output. Beam search turns the decoder's per-step distributions into a single output string; length normalization keeps it from favoring short translations. We then score translations automatically — BLEU with its n-gram precision, clipping, and brevity penalty, worked through by hand, then its successors chrF, BERTScore, and COMET — and close on the parts of MT that scale beyond one language pair: multilingual and low-resource translation, backtranslation, gender bias, and the lineage from the Transformer to massively multilingual models like NLLB-200.\n",{"path":25727,"title":25728,"module":24267,"summary":25729},"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering","Question Answering","A question-answering system takes a natural-language question and returns an answer, not a ranked list of documents. Almost every modern system is built on one pattern: retrieve then read. We start with the information-retrieval machinery that finds candidate text — tf-idf and BM25 term weighting, a worked ranking example, the inverted index, and dense embedding retrieval — then build the retriever-reader pipeline that extracts an answer span with BERT and trace a full retrieve-and-read example end to end.\n",{"path":25731,"title":25732,"module":24267,"summary":25733},"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering-knowledge-and-llms","Question Answering: Knowledge Bases and Language Models","The retrieve-and-read pipeline extracts an answer span from prose, but not all knowledge lives in prose. This part covers the rest of the QA stack: entity linking (Wikification) that grounds a question's entities to a knowledge base, knowledge-based QA by semantic parsing a question into an executable query, and the modern default — closed-book QA and retrieval-augmented generation with a large language model — closing on the DPR\u002FRAG\u002Ffusion-in-decoder lineage and how factoid answers are scored by exact match and F1.\n",{"path":25735,"title":25736,"module":24267,"summary":25737},"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-and-chatbots","Dialogue and Chatbots","Conversation is the most natural interface to a machine and one of the hardest to build. We set up what makes human dialogue work — turns, speech acts, grounding, and the local structure of adjacency pairs — then trace the two traditions that answer it: chatbots built to chat (ELIZA's pattern-matching, corpus retrieval, and seq2seq generation with its blandness problem) and task-oriented systems built to get something done (the GUS frame-and-slot architecture and the modern NLU \u002F state-tracker \u002F policy \u002F NLG pipeline that accumulates a frame across turns).\n",{"path":25739,"title":25740,"module":24267,"summary":25741},"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-systems-and-assistants","Dialogue Systems: LLM Assistants, Evaluation, and Design","Two dialogue traditions — chatbots built to chat and task-oriented frame systems built to get something done — met in the aligned LLM assistant. Instruction tuning plus RLHF fold chit-chat and task dialogue into one model; the LaMDA \u002F InstructGPT \u002F ChatGPT lineage fills in how. The lesson then turns to evaluation (human ratings and acute-eval for chatbots, task success and slot error rate for task systems), user-centered design with Wizard-of-Oz prototyping, and the ethical stakes of building agents people talk to.\n",{"path":25743,"title":25744,"module":24267,"summary":25745},"\u002Fnatural-language-processing\u002Fapplications\u002Ftext-summarization","Text Summarization","Summarization compresses a document to its essential meaning, by either selecting sentences to keep (extractive) or writing new ones (abstractive). This part fixes the task and its flavors — single vs. multi-document, generic vs. query-focused, extractive vs. abstractive — then works through extractive summarization in full: scoring by position and centrality, the TextRank\u002FLexRank graph algorithm run as PageRank over a sentence-similarity graph with a worked iteration, and supervised sentence selection.\n",{"path":25747,"title":25748,"module":24267,"summary":25749},"\u002Fnatural-language-processing\u002Fapplications\u002Fabstractive-summarization-and-evaluation","Abstractive Summarization and Evaluation","Extractive methods can only reuse the source's own sentences; to compress within a sentence or paraphrase, a summarizer has to generate. This part covers abstractive summarization: the sequence-to-sequence approach, the pointer-generator's copy switch and coverage mechanism, pretrained summarizers (BART, PEGASUS) and zero-shot LLM prompting, the long-document and factuality problems, and ROUGE evaluation with a worked example and its limits — closing on the abstractive lineage from See 2017 through faithfulness metrics.\n",{"path":25751,"title":25752,"module":25753,"summary":25754},"\u002Fnatural-language-processing\u002Fspeech\u002Fphonetics","Phonetics","Speech","Before a recognizer can read speech it has to know what speech is. This first part covers the linguistic substrate: phones and their transcription in the IPA and ARPAbet; articulatory phonetics — how the vocal tract shapes airflow into consonants and vowels; and prosody — stress, tune, and the F0 contour. The acoustic side — the waveform, its spectrum, formants, and the spectrogram — is the second part.\n",{"path":25756,"title":25757,"module":25753,"summary":25758},"\u002Fnatural-language-processing\u002Fspeech\u002Facoustic-phonetics","Acoustic Phonetics","Articulation is the cause; the acoustic signal is the effect, and the effect is all a microphone ever gets. This part follows the sound out of the mouth: waves, sampling and the Nyquist limit, F0 and the pitch track, the mel scale, the spectrum and Fourier analysis, the source-filter model that explains why each vowel carries its own formants, and the spectrogram the log-mel front end of every ASR system sits directly on top of — closing with neural TTS, wav2vec, HuBERT, and Whisper, where phonetics went in neural speech.\n",{"path":25760,"title":25761,"module":25753,"summary":25762},"\u002Fnatural-language-processing\u002Fspeech\u002Fautomatic-speech-recognition","Automatic Speech Recognition","Speech recognition maps an acoustic waveform to a string of words, and once the waveform is turned into a sequence of log-mel spectrogram frames the problem is the same sequence-to-sequence transduction the rest of the course already solved. This first part builds the feature front end (framing, the DFT, the mel filterbank, the log), then the modern architectures: the attention-based encoder-decoder, the CTC alignment trick that collapses repeated and blank frames, and RNN-T for streaming. Training-data advances, evaluation, TTS, and the other speech tasks come next.\n",{"path":25764,"title":25765,"module":25753,"summary":25766},"\u002Fnatural-language-processing\u002Fspeech\u002Fasr-evaluation-and-applications","ASR Evaluation and Speech Applications","A recognizer turns a waveform into text; this part scores that text and puts the same machinery to other uses. It opens with the self-supervised and weakly- supervised systems (wav2vec 2.0, HuBERT, Whisper) that made ASR error rates fall. Word error rate reuses the edit distance from the first module, run over words. Text-to-speech runs the whole pipeline in reverse — text to mel spectrogram to waveform. And a family of smaller tasks — wake-word detection, speaker recognition and diarization, language identification — reuse the same log-mel front end without the decoder.\n",{"path":25768,"title":25769,"module":6,"summary":6},"\u002Fnatural-language-processing","Natural Language Processing",{"path":25771,"title":25772,"module":21519,"summary":25773},"\u002Fparticle-physics\u002Ffoundations\u002Fhistorical-overview-particle-zoo","From the Electron to the Particle Zoo","A timeline of the subject, from J. J. Thomson's electron in 1897 to the Higgs boson in 2012. The electron, photon, nucleus, proton, and neutron gave a tidy picture that Yukawa's meson prediction and the muon–pion confusion complicated; strange particles in cosmic rays and the accelerator-era flood of hadrons then produced a \"particle zoo\" that only the quark model organized.\n",{"path":25775,"title":25776,"module":21519,"summary":25777},"\u002Fparticle-physics\u002Ffoundations\u002Fparticle-physics-basic-concepts","Basic Concepts and Particle Classification","Every particle has an antiparticle of equal mass and opposite charge, a consequence of the Dirac equation confirmed by the positron. Feynman diagrams track interactions in spacetime; the material particles sort into leptons and the composite hadrons built from quarks, with baryons carrying three quarks and mesons a quark-antiquark pair.\n",{"path":25779,"title":25780,"module":21519,"summary":25781},"\u002Fparticle-physics\u002Ffoundations\u002Ffundamental-interactions-force-carriers","Fundamental Interactions and Force Carriers","Four interactions account for every force in nature: strong, electromagnetic, weak, and gravitational, in decreasing strength. Each is carried by a boson — the gluon, photon, W and Z, and the graviton — with a range fixed by the carrier's mass through the Yukawa relation, and a coupling constant that itself varies with distance.\n",{"path":25783,"title":25784,"module":25785,"summary":25786},"\u002Fparticle-physics\u002Funits-kinematics\u002Fnatural-units-and-scales","Natural Units and Scales","Units and Kinematics","Setting $\\hbar = c = 1$ collapses mass, momentum, and energy into a single unit, the GeV, and turns lengths and times into inverse energies through the conversion $\\hbar c = 197.3$ MeV·fm. This lesson fixes the natural-unit conventions used for the rest of the course, converts cross sections between barns and GeV$^{-2}$, and shows how to restore factors of $\\hbar$ and $c$ by dimensional analysis.\n",{"path":25788,"title":25789,"module":25785,"summary":25790},"\u002Fparticle-physics\u002Funits-kinematics\u002Ffour-vectors-invariant-mass","Four-Vectors and Invariant Mass","The energy and momentum of a particle form a four-vector whose square is the frame-independent quantity $p^2 = m^2$. This lesson develops the metric and four-vector products, the invariant mass of a multiparticle system, the center-of-momentum and laboratory frames, and the description of collinear boosts by rapidity, whose additivity replaces the awkward velocity-addition law.\n",{"path":25792,"title":25793,"module":25785,"summary":25794},"\u002Fparticle-physics\u002Funits-kinematics\u002Fdecay-scattering-kinematics-mandelstam","Decay, Scattering, and Mandelstam Variables","Two-body decay in the rest frame fixes the daughter momenta from the three masses alone; production thresholds follow from the minimum invariant mass. This lesson works both, then introduces the Mandelstam invariants $s$, $t$, $u$ for $2\\to2$ scattering, proves the identity $s+t+u=\\sum m_i^2$, and maps the physical regions and the crossing that relates channels.\n",{"path":25796,"title":25797,"module":25785,"summary":25798},"\u002Fparticle-physics\u002Funits-kinematics\u002Fcross-sections-golden-rule","Cross Sections and the Golden Rule","The cross section measures how often a scattering happens and the decay width how fast a particle disintegrates. This lesson defines both, relates event rate to luminosity through $R=\\mathcal L\\,\\sigma$ and lifetime to width through $\\tau=\\hbar\u002F\\Gamma$, and states Fermi's golden rule with Lorentz-invariant phase space, giving the master formulas that turn an amplitude $\\mathcal M$ into a measurable rate for $1\\to2$ decay and $2\\to2$ scattering.\n",{"path":25800,"title":25801,"module":25802,"summary":25803},"\u002Fparticle-physics\u002Fsymmetries\u002Fconservation-laws-symmetries","Conservation Laws and Symmetries","Symmetries and Conservation Laws","Which decays occur is decided by conservation laws, each tied by Noether's theorem to a symmetry of physical law. Energy, charge, baryon number, and lepton number are conserved universally; strangeness, isospin, and parity hold in the strong and electromagnetic interactions but break in the weak one, whose parity and CP violation distinguish matter from antimatter.\n",{"path":25805,"title":25806,"module":25802,"summary":25807},"\u002Fparticle-physics\u002Fsymmetries\u002Fdiscrete-symmetries-cpt","Discrete Symmetries — C, P, T, and CPT","Parity reflects space, charge conjugation swaps particle for antiparticle, and time reversal runs the clock backward. Each assigns multiplicative quantum numbers that act as selection rules — intrinsic parities, the photon's C = −1, the C-parity argument fixing the pion's two-photon decay. Their product CPT is a theorem of any local relativistic field theory, forcing particle and antiparticle to share mass and lifetime.\n",{"path":25809,"title":25810,"module":25802,"summary":25811},"\u002Fparticle-physics\u002Fsymmetries\u002Fparity-violation-weak","Parity Violation and the Weak Force","The tau–theta puzzle forced a choice: two particles with identical mass but opposite parity, or one particle whose decay violates parity. Lee and Yang proposed the latter, Wu's polarized cobalt-60 confirmed it, and the violation proved maximal. The charged weak force couples only to left-handed chirality — the Goldhaber experiment showed the neutrino is left-handed — which is why the mirror image of a weak decay is something nature never produces.\n",{"path":25813,"title":25814,"module":25802,"summary":25815},"\u002Fparticle-physics\u002Fsymmetries\u002Fsu2-su3-flavor-symmetry","Isospin, SU(2), and Flavor SU(3)","The near-equal masses of the proton and neutron, and of the three pions, signal a continuous internal symmetry of the strong force: isospin, an SU(2) whose ladder operators move between the members of a multiplet. Adding strangeness enlarges it to an approximate SU(3) flavor symmetry, and the Gell-Mann–Nishijima relation Q = I3 + Y\u002F2 places every hadron on a weight diagram in the isospin–hypercharge plane — the language in which the quark model is written.\n",{"path":25817,"title":25818,"module":25819,"summary":25820},"\u002Fparticle-physics\u002Fquark-model\u002Feightfold-way-su3","The Eightfold Way and SU(3) Flavor","The Quark Model","Gell-Mann and Ne'eman's classification of the hadrons into geometric multiplets, read as representations of an approximate flavor SU(3). The fundamental triplet (u, d, s) and its antitriplet combine into the meson nonet from 3⊗3̄ = 8⊕1 and the baryon octet and decuplet from 3⊗3⊗3, and the empty corner of the decuplet forecast the Ω⁻.\n",{"path":25822,"title":25823,"module":25819,"summary":25824},"\u002Fparticle-physics\u002Fquark-model\u002Fmeson-spectroscopy","Meson Multiplets and Quantum Numbers","Mesons as quark–antiquark bound states. The spin singlet and triplet, orbital excitations, and the assignment of J^PC from the quark spins and orbital angular momentum, giving the pseudoscalar and vector nonets. The η–η' and ω–φ mixing problems, and the charmonium and bottomonium spectra read as heavy-quark positronium.\n",{"path":25826,"title":25827,"module":25819,"summary":25828},"\u002Fparticle-physics\u002Fquark-model\u002Fbaryon-spectroscopy","Baryon Multiplets, Spin, and the Color Puzzle","Baryons as three-quark states, with a wavefunction factored into space, spin, flavor, and color. The spin-3\u002F2 Δ⁺⁺ = uuu forces a totally symmetric state that the Pauli principle forbids, and the resolution is an antisymmetric color factor — the first evidence for color. The octet and decuplet spin content, and baryon magnetic moments as a quantitative test of the model.\n",{"path":25830,"title":25831,"module":25819,"summary":25832},"\u002Fparticle-physics\u002Fquark-model\u002Fcolor-confinement-exotics","Color, Confinement, and Exotic Hadrons","Color as the gauged SU(3) charge, and the requirement that every physical hadron be a color singlet — which selects q-qbar mesons and qqq baryons as the simplest states. The R-ratio of e⁺e⁻ annihilation measures three colors directly. Beyond the simplest singlets lie glueballs, tetraquarks, and pentaquarks, and the recent XYZ states, read as either compact multiquarks or loose hadronic molecules.\n",{"path":25834,"title":25835,"module":25836,"summary":25837},"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fklein-gordon-equation","The Klein-Gordon Equation","Relativistic Wave Equations","Quantizing the relativistic energy relation $E^2 = p^2 + m^2$ produces the Klein-Gordon equation for a scalar field. Its plane-wave solutions come in positive- and negative-energy branches, and the conserved density it supplies is not positive-definite — the two difficulties that first drove physicists to seek a first-order equation. The static Klein-Gordon equation with a point source gives the Yukawa potential, and the free equation gives the scalar propagator that later modules attach to exchanged lines.\n",{"path":25839,"title":25840,"module":25836,"summary":25841},"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fdirac-equation-spinors","The Dirac Equation and Spinors","Dirac demanded a wave equation first order in time to fix the Klein-Gordon density problem. Factorizing $E^2 = p^2 + m^2$ into a linear form forces the coefficients to be anticommuting matrices — the gamma matrices of the Clifford algebra — so the wavefunction becomes a four-component spinor. The plane-wave solutions split into two particle and two antiparticle states, spin appears automatically with the correct $g = 2$ magnetic moment, and the chirality projectors that the weak interaction later needs fall straight out of the fifth gamma matrix.\n",{"path":25843,"title":25844,"module":25836,"summary":25845},"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fantiparticles-hole-theory","Antiparticles and Hole Theory","The negative-energy solutions of the Dirac equation refuse to go away, so they must mean something. Dirac read them as a filled sea of occupied negative-energy states whose holes are positive-energy antiparticles, predicting the positron before its discovery. The picture works for fermions but not bosons, and the Feynman-Stückelberg interpretation replaces it: an antiparticle is a negative-energy solution propagating backward in time, equivalent to a positive-energy antiparticle going forward. Crossing symmetry ties incoming particles to outgoing antiparticles in a single amplitude.\n",{"path":25847,"title":25848,"module":25849,"summary":25850},"\u002Fparticle-physics\u002Fqed\u002Ffeynman-rules-qed","Feynman Rules for QED","Quantum Electrodynamics","Quantum electrodynamics computes a process by summing diagrams, each a term in a power series in the coupling. Every diagram translates into an amplitude by a fixed dictionary: spinors and polarization vectors for external lines, propagators for internal lines, and the vertex factor $ie\\gamma^\\mu$ for each photon-fermion junction. Squaring the amplitude and feeding it to Fermi's golden rule produces a cross section or decay rate, with each extra vertex costing one power of $\\alpha$.\n",{"path":25852,"title":25853,"module":25849,"summary":25854},"\u002Fparticle-physics\u002Fqed\u002Fqed-tree-processes","Tree-Level QED Processes","The Feynman rules become numbers on the reference reactions of QED. Muon pair production $e^+e^-\\to\\mu^+\\mu^-$ sets the scale with its $1+\\cos^2\\theta$ distribution and $4\\pi\\alpha^2\u002F3s$ total cross section, and its ratio to hadron production counts colors. Compton scattering gives the Klein-Nishina formula and the Thomson limit; Bhabha scattering shows $s$- and $t$-channel interference. Casimir's trick turns every spin-averaged square into a trace of gamma matrices.\n",{"path":25856,"title":25857,"module":25849,"summary":25858},"\u002Fparticle-physics\u002Fqed\u002Frenormalization-running-coupling","Renormalization and the Running Coupling","Beyond tree level, QED loops diverge. The three primitive one-loop diagrams — vacuum polarization, electron self-energy, and vertex correction — carry ultraviolet divergences that regularization exposes as logarithms of a cutoff. Renormalization absorbs them into the measured mass, charge, and field normalization, leaving finite predictions. The surviving physical content is that the coupling runs: vacuum polarization screens charge, so $\\alpha$ grows from $1\u002F137$ at low energy to about $1\u002F128$ at the $Z$ mass.\n",{"path":25860,"title":25861,"module":25849,"summary":25862},"\u002Fparticle-physics\u002Fqed\u002Felectron-g-2","The Anomalous Magnetic Moment","The Dirac equation predicts $g=2$; loops shift it. Schwinger's one-loop vertex correction gives the anomaly $a=(g-2)\u002F2=\\alpha\u002F2\\pi$, and the QED series continues to five loops. The electron $a_e$ agrees with theory to better than a part in a billion, the most precise confrontation of theory and experiment in physics. The muon $a_\\mu$, heavier and so more sensitive to virtual heavy states, is dominated by hadronic uncertainty and sits at the center of a long-running comparison with the Standard Model prediction.\n",{"path":25864,"title":25865,"module":25866,"summary":25867},"\u002Fparticle-physics\u002Fweak-interaction\u002Fva-structure-weak","The V–A Charged Weak Current","The Weak Interaction","Fermi modelled beta decay as a four-fermion contact interaction, but a coupling with dimensions of inverse mass squared makes cross sections grow without bound and the theory fails near 300 GeV. The cure is a heavy mediator: the $W$ boson, whose propagator collapses to Fermi's contact term at low energy and fixes $G_F\u002F\\sqrt2 = g^2\u002F8M_W^2$. Parity violation dictates the current's form — vector minus axial-vector, coupling only to left-chiral fields — and universality of the coupling ties muon decay, beta decay, and pion decay to one constant. Pion decay's helicity suppression of the electron channel is the sharpest test.\n",{"path":25869,"title":25870,"module":25866,"summary":25871},"\u002Fparticle-physics\u002Fweak-interaction\u002Fw-z-bosons-decays","The W and Z Bosons","The contact theory hides a massive mediator. The charged $W^\\pm$ carries the current that changes flavour; the neutral $Z^0$ carries a current that does not. Both were found at CERN's proton–antiproton collider in 1983 at the masses the electroweak theory demanded. Their decay widths partition into leptonic and hadronic channels, and the $Z$ carries a decisive extra: an invisible width from decays to neutrinos that counts the number of light generations at exactly three. Beta decay and muon decay are re-read at the parton level as $W$ exchange.\n",{"path":25873,"title":25874,"module":25866,"summary":25875},"\u002Fparticle-physics\u002Fweak-interaction\u002Fckm-matrix","Quark Mixing and the CKM Matrix","The quark eigenstates the weak force acts on are not the mass eigenstates. Cabibbo captured this with one rotation angle; the GIM mechanism added a fourth quark to cancel dangerous flavour-changing neutral currents and predicted charm before its discovery. Three generations promote the rotation to the unitary Cabibbo–Kobayashi–Maskawa matrix — three angles and one irreducible complex phase, the sole source of Standard-Model CP violation. The Wolfenstein parametrization exposes its steep hierarchy, and unitarity closes into a triangle whose area measures the phase.\n",{"path":25877,"title":25878,"module":25866,"summary":25879},"\u002Fparticle-physics\u002Fweak-interaction\u002Fcp-violation-kaons-b-mesons","CP Violation in Kaons and B Mesons","The neutral kaon is its own laboratory for CP. Weak box diagrams mix $K^0$ and its antiparticle into short- and long-lived states that should be pure CP eigenstates decaying to two and three pions. In 1964 Cronin and Fitch caught the long-lived kaon decaying to two pions — CP is violated, at the two-per-mille level of $\\epsilon$. Direct violation ($\\epsilon'$) followed, and the $B$ factories turned the CKM phase into a large, clean time-dependent asymmetry measuring $\\sin 2\\beta$. The effect is real but far too small to explain why the universe is made of matter.\n",{"path":25881,"title":25882,"module":25883,"summary":25884},"\u002Fparticle-physics\u002Fqcd\u002Fcolor-su3-gluons","Color SU(3), Gluons, and the QCD Lagrangian","Quantum Chromodynamics","Color is the exact gauged SU(3) charge of the strong force. Gauging it forces eight massless gluons in the adjoint representation and, because the gauge group is non-abelian, three- and four-gluon self-couplings absent from QED. This lesson builds the QCD Lagrangian from the covariant derivative and the non-abelian field strength, states the Feynman rules with their color factors, and computes the Casimir invariants that set the strength of quark-gluon and gluon-gluon coupling.\n",{"path":25886,"title":25887,"module":25883,"summary":25888},"\u002Fparticle-physics\u002Fqcd\u002Fasymptotic-freedom-confinement","Asymptotic Freedom and Confinement","The QCD beta function is negative: gluon self-interaction antiscreens color, so the coupling weakens at short distance (asymptotic freedom) and strengthens at long distance (confinement). This lesson computes the one-loop beta coefficient, solves for the running of alpha_s and the emergent scale Lambda_QCD, and reads the strong-coupling regime as the linear quark-antiquark potential of a color flux tube that breaks by pair creation.\n",{"path":25890,"title":25891,"module":25883,"summary":25892},"\u002Fparticle-physics\u002Fqcd\u002Fdeep-inelastic-scattering-partons","Deep Inelastic Scattering and the Parton Model","Scattering electrons hard off a proton resolves pointlike constituents. This lesson sets up the deep-inelastic kinematics, defines the structure functions F1 and F2, and reads Bjorken scaling as the signature of free spin-half partons. The Callan-Gross relation fixes the parton spin, the structure function becomes a charge-weighted sum of parton distributions, and the slow logarithmic scaling violations expose the gluon through DGLAP evolution.\n",{"path":25894,"title":25895,"module":25883,"summary":25896},"\u002Fparticle-physics\u002Fqcd\u002Fjets-hadronization","Jets, Hadronization, and Testing QCD","Quarks and gluons produced in a collision fragment into collimated sprays of hadrons — jets — whose directions track the underlying partons. This lesson reads two-jet events as the quark and antiquark of electron-positron annihilation, three-jet events as direct evidence of the radiated gluon, and the hadronization step as the flux tube breaking into color singlets. Jet algorithms and event-shape variables turn the pattern into precision measurements of alpha_s.\n",{"path":25898,"title":25899,"module":25900,"summary":25901},"\u002Fparticle-physics\u002Felectroweak-higgs\u002Felectroweak-su2-u1","The Electroweak Theory","Electroweak Unification and the Higgs","The electromagnetic and weak interactions are two faces of a single gauge theory built on $SU(2)_L \\times U(1)_Y$. Left-handed fermions sit in weak-isospin doublets and right-handed fermions in singlets, each carrying a hypercharge fixed by the Gell-Mann–Nishijima relation $Q = T_3 + Y\u002F2$. The four gauge fields $W^{1,2,3}$ and $B$ mix: the charged combinations $W^\\pm$ mediate the charged current, while $W^3$ and $B$ rotate through the Weinberg angle into the massless photon and the massive $Z$. The single angle $\\theta_W$ ties the couplings, the boson masses, and the neutral-current strengths together.\n",{"path":25903,"title":25904,"module":25900,"summary":25905},"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fspontaneous-symmetry-breaking","Spontaneous Symmetry Breaking","A symmetry of the Lagrangian need not be a symmetry of the ground state. When the lowest-energy configuration sits away from the symmetric point, the symmetry is spontaneously broken and the vacuum is one of a degenerate family. Breaking a continuous global symmetry produces one massless scalar — a Goldstone boson — for every broken generator, the flat direction along the vacuum manifold. The Mexican-hat potential and the ferromagnet below its Curie point are the working pictures.\n",{"path":25907,"title":25908,"module":25900,"summary":25909},"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-mechanism","The Higgs Mechanism","Gauging a spontaneously broken symmetry converts the would-be Goldstone bosons into the longitudinal polarizations of the gauge fields, which thereby acquire mass. Applied to $SU(2)_L \\times U(1)_Y$ with a single Higgs doublet, three of the four scalar degrees of freedom are eaten by the $W^\\pm$ and $Z$; the fourth survives as the physical Higgs boson, and the photon stays massless. Fermion masses come from Yukawa couplings to the same field, each mass proportional to its coupling times the vacuum expectation value $v \\approx 246$ GeV.\n",{"path":25911,"title":25912,"module":25900,"summary":25913},"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-boson-discovery","The Higgs Boson","The Higgs boson is produced at the LHC chiefly through gluon fusion, with vector-boson fusion and associated production as cleaner but rarer channels. It decays most often to $b\\bar b$ and $WW^\\ast$, but the discovery rested on two rare clean modes, $H \\to \\gamma\\gamma$ and $H \\to ZZ^\\ast \\to 4\\ell$, whose narrow invariant-mass peaks emerged over smooth backgrounds. ATLAS and CMS announced a boson near 125 GeV in 2012; its measured spin-parity $0^+$ and its couplings, which scale with particle mass, identify it as the Standard Model Higgs.\n",{"path":25915,"title":25916,"module":25900,"summary":25917},"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fstandard-model","The Standard Model","The Standard Model combines the quark model, quantum chromodynamics, and the electroweak theory. SU(3) symmetry sorts the hadrons and predicted the omega; color explains why only colorless quark combinations exist; QCD gives asymptotic freedom and confinement; and spontaneous symmetry breaking through the Higgs field gives the weak bosons their mass.\n",{"path":25919,"title":25920,"module":25921,"summary":25922},"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-oscillations","Neutrino Oscillations","Neutrino Physics","Neutrinos are produced and detected in flavour states, but they propagate as mass states, and the two bases are misaligned. A flavour therefore evolves coherently into a superposition of other flavours with a probability set by the mass-squared splitting and the ratio L\u002FE. This lesson derives the two-flavour oscillation formula, applies it to the solar and atmospheric neutrino deficits, shows how the SNO neutral-current measurement resolved the solar problem, and works out the MSW resonance that amplifies mixing inside the Sun.\n",{"path":25924,"title":25925,"module":25921,"summary":25926},"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-mass-pmns","Neutrino Mass and the PMNS Matrix","Three-flavour mixing promotes the single oscillation angle to the unitary Pontecorvo–Maki–Nakagawa–Sakata matrix, parametrised by three angles and a Dirac CP phase. This lesson decomposes the PMNS matrix into three rotations, records the measured angles and mass-squared splittings, lays out the normal and inverted mass orderings, contrasts the large leptonic mixing with the near-diagonal CKM matrix, and collects the absolute-mass bounds from beta decay and cosmology.\n",{"path":25928,"title":25929,"module":25921,"summary":25930},"\u002Fparticle-physics\u002Fneutrinos\u002Fdirac-majorana-experiments","Dirac, Majorana, and Neutrino Experiments","A neutral fermion can carry a mass term forbidden to every charged particle, so the neutrino may be its own antiparticle. This lesson contrasts the Dirac and Majorana mass terms and their state content, derives the seesaw mechanism that ties a tiny light mass to a heavy right-handed partner, presents neutrinoless double-beta decay as the decisive lepton-number test, surveys the reactor, accelerator, solar, and atmospheric sources on a baseline–energy map, and explains why neutrino mass is physics beyond the original Standard Model.\n",{"path":25932,"title":25933,"module":25934,"summary":25935},"\u002Fparticle-physics\u002Fexperiment\u002Faccelerators-luminosity","Accelerators, Colliders, and Luminosity","Accelerators and Detectors","Fixed-target machines waste energy in the center-of-mass motion of the whole system, so the reachable $\\sqrt s$ grows only as the square root of the beam energy, while colliders put the full beam energy into the collision. Circular electron machines are limited by synchrotron radiation scaling as $E^4\u002Fm^4R$; proton machines are limited by bending fields. Luminosity, set by beam current and focusing, converts a cross section into an event rate through $R=\\mathcal L\\,\\sigma$, and integrated luminosity sets the total event count.\n",{"path":25937,"title":25938,"module":25934,"summary":25939},"\u002Fparticle-physics\u002Fexperiment\u002Fdetectors-subsystems","Particle Detectors and Subsystems","A detector reads a collision by the energy particles deposit as they cross matter. Charged particles ionize at the Bethe-Bloch rate, radiate in the field of nuclei above a critical energy, and emit Cherenkov light above a velocity threshold; electrons and photons build electromagnetic showers over a radiation length, and hadrons build wider showers over a nuclear interaction length. The onion of tracker, electromagnetic and hadronic calorimeters, and outer muon chambers turns these processes into momentum, energy, and identity, with neutrinos inferred from missing transverse momentum.\n",{"path":25941,"title":25942,"module":25934,"summary":25943},"\u002Fparticle-physics\u002Fexperiment\u002Fhow-discoveries-are-made","From Collisions to Discoveries","A discovery is a peak that survives statistics. Events are reconstructed into invariant masses, a signal accumulates as a bump over a smooth background, and its significance is judged by a p-value; the field's threshold is five sigma. The expected yield is a product — luminosity times cross section times branching ratio times acceptance and efficiency — that must be balanced by a trigger and data-reduction chain against an overwhelming rate. Worked reconstructions of $Z\\to\\ell\\ell$, the $J\u002F\\psi$, and the Higgs show the same peak-over-background logic at three scales.\n",{"path":25945,"title":25946,"module":25946,"summary":25947},"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fbeyond-standard-model","Beyond the Standard Model","The Standard Model leaves the four interactions ununified and the neutrinos massless, both now known to be wrong. Grand unification predicts the couplings merge near ten-to-the-sixteen GeV and the proton decays; supersymmetry pairs each particle with a superpartner; and the confirmed oscillation of neutrinos proves they carry mass, the first crack in the model.\n",{"path":25949,"title":25950,"module":25946,"summary":25951},"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fgrand-unified-theories","Grand Unified Theories and Proton Decay","The Standard Model gauge group is a product of three factors with three independent couplings. A grand unified theory embeds them in a single simple group — SU(5) is the minimal choice — so that one coupling runs into all three and the fractional quark charges follow from a tracelessness condition. The same embedding places quarks and leptons in shared multiplets, mediates baryon-number violation through superheavy gauge bosons, and predicts the proton decays with a lifetime that Super-Kamiokande has pushed past ten-to-the-thirty-four years.\n",{"path":25953,"title":25954,"module":25946,"summary":25955},"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fsupersymmetry","Supersymmetry","Supersymmetry relates fermions and bosons, pairing every Standard Model particle with a superpartner whose spin differs by one half. The pairing makes the scalar and fermion loop corrections to the Higgs mass cancel, removing the quadratic sensitivity to high scales; it sharpens the meeting of the three gauge couplings; and, when R-parity is conserved, it leaves the lightest superpartner stable and neutral, a natural dark-matter candidate. The LHC has excluded gluinos and light squarks below roughly two TeV.\n",{"path":25957,"title":25958,"module":25946,"summary":25959},"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fhierarchy-problem-naturalness","The Hierarchy Problem and Naturalness","The electroweak scale sits sixteen orders of magnitude below the Planck scale, and nothing in the Standard Model protects that gap. The Higgs mass squared picks up quadratic corrections proportional to the highest scale in the theory, so keeping it at the observed value requires the bare mass and its counterterm to cancel to some thirty significant figures. Naturalness treats that cancellation as a symptom of missing physics. Supersymmetry, compositeness, and extra dimensions each remove the quadratic sensitivity, but the LHC has found none of them at the predicted scale.\n",{"path":25961,"title":25962,"module":25946,"summary":25963},"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fdark-matter-candidates","Dark Matter and Particle Candidates","Flat galactic rotation curves, gravitational lensing, the cosmic microwave background, and structure formation all require about five times more matter than the visible baryons, none of it interacting electromagnetically. A stable weakly interacting particle of roughly weak-scale mass freezes out of the early universe with close to the observed abundance — the WIMP miracle — and is the leading candidate, with axions and sterile neutrinos as alternatives. Direct, indirect, and collider searches have so far only tightened the limits.\n",{"path":25965,"title":25966,"module":25946,"summary":25967},"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fmatter-antimatter-open-questions","Matter-Antimatter Asymmetry and Open Questions","The universe is made of matter, with about one extra baryon for every billion photons and no antimatter regions. Sakharov identified the three conditions any dynamical explanation must meet: baryon-number violation, C and CP violation, and a departure from thermal equilibrium. The Standard Model contains all three in principle, but its CP violation falls short by some ten orders of magnitude, so baryogenesis requires new physics — leptogenesis being the leading route. A closing survey collects the open questions and the experiments aimed at them.\n",{"path":25969,"title":25970,"module":6,"summary":6},"\u002Fparticle-physics","Particle Physics",{"path":25972,"title":25973,"module":25974,"summary":25975},"\u002Fastrophysics-cosmology\u002Forientation\u002Fthe-sun-and-stars","The Sun and the Life of Stars","Orientation","The Sun is the one star close enough to study in detail: its luminosity fixes a surface temperature of 5780 K, and the proton-proton fusion cycle in its 1.5-million-kelvin core supplies its power. Measuring other stars needs the magnitude scale, parallax, and the distance ladder; plotting luminosity against temperature builds the Hertzsprung-Russell diagram, on which a star's mass sets its lifetime and its evolutionary track off the main sequence.\n",{"path":25977,"title":25978,"module":25974,"summary":25979},"\u002Fastrophysics-cosmology\u002Forientation\u002Fstellar-death-final-states","Cataclysmic Events and the Final States of Stars","A star's death is set by its mass. In close binaries, matter poured across the Roche lobe onto a white dwarf produces novae and, at the Chandrasekhar limit of 1.4 solar masses, a Type Ia supernova; a massive star fusing to an iron core collapses into a Type II supernova. The remnant is a white dwarf held by electron degeneracy, a neutron star held by neutron degeneracy, or, above the neutron-star limit, a black hole inside its Schwarzschild radius.\n",{"path":25981,"title":25982,"module":25974,"summary":25983},"\u002Fastrophysics-cosmology\u002Forientation\u002Fgalaxies-and-cosmology","Galaxies, Cosmology, and the Evolving Universe","Galaxies come in elliptical, spiral, and irregular forms, and their redshifts obey Hubble's law, evidence that space itself is expanding. The critical density and the density parameter decide whether the universe is open, flat, or closed; baryons, dark matter, and dark energy each contribute. The cosmic microwave background and primordial helium anchor the Big Bang, whose thermal history runs from inflation through nucleosynthesis to the atoms of today.\n",{"path":25985,"title":25986,"module":25987,"summary":25988},"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fmagnitudes-fluxes-and-the-distance-modulus","Magnitudes, Fluxes, and the Distance Modulus","Observational Foundations","The brightness of a star reaches us as a radiant flux that falls off as the inverse square of distance. The magnitude scale encodes flux logarithmically through the Pogson ratio; the apparent and absolute magnitudes differ by the distance modulus, which converts a measured brightness into a distance. The bolometric correction folds a filtered magnitude into a total luminosity, and the difference of two magnitudes in different bands, the color index, measures surface temperature.\n",{"path":25990,"title":25991,"module":25987,"summary":25992},"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fstellar-spectra-and-spectral-classification","Stellar Spectra and Spectral Classification","A stellar spectrum is a continuum crossed by absorption lines whose strengths are set by the temperature of the atmosphere. The Boltzmann factor governs how atoms populate excited states, and the Saha equation governs how they ionize; their product explains why each line, such as the hydrogen Balmer series, peaks in strength at a characteristic temperature. This behavior orders stars into the OBAFGKM sequence, and the luminosity classes of the MK system add a second dimension for surface gravity.\n",{"path":25994,"title":25995,"module":25987,"summary":25996},"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Ftelescopes-and-detectors-across-the-spectrum","Telescopes and Detectors Across the Spectrum","A telescope collects light in proportion to its collecting area and resolves detail down to the diffraction limit set by its aperture and the observing wavelength. The atmosphere blurs and blocks large parts of the spectrum, which drives the choice between ground and space and between refractors, reflectors, and radio dishes. CCDs record the light with high quantum efficiency, and interferometry synthesizes an aperture as large as the separation of two telescopes.\n",{"path":25998,"title":25999,"module":25987,"summary":26000},"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fthe-cosmic-distance-ladder","The Cosmic Distance Ladder","No single method measures distances from the nearest stars to the far reaches of the universe. Instead a ladder of overlapping techniques, each calibrated by the one below it, extends the scale rung by rung: trigonometric parallax, main-sequence fitting, pulsating variables, the tip of the red-giant branch, the Tully-Fisher relation, and Type Ia supernovae. Each rung inherits the uncertainty of every rung beneath it, so the whole chain sets the accuracy of the Hubble constant.\n",{"path":26002,"title":26003,"module":26004,"summary":26005},"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fblackbody-radiation-and-specific-intensity","Blackbody Radiation and Specific Intensity","Radiation and Matter","Specific intensity is the fundamental measure of a radiation field: energy per unit area, time, frequency, and solid angle. It is conserved along a ray in empty space, and its angular moments give the mean intensity, flux, and radiation pressure. In thermal equilibrium the intensity equals the Planck function, whose limits and integrals reproduce the Rayleigh-Jeans law, the Wien law, Stefan-Boltzmann, and Wien's displacement law.\n",{"path":26007,"title":26008,"module":26004,"summary":26009},"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fradiative-transfer-and-the-transfer-equation","Radiative Transfer and the Transfer Equation","Along a ray, matter adds intensity through emission and removes it through absorption. Measuring path length in optical depth turns this into the transfer equation, whose formal solution superposes an attenuated background on the source function integrated along the line of sight. In local thermodynamic equilibrium the source function is the Planck function, and the Eddington-Barbier relation shows that the emergent intensity samples the source function at optical depth of order unity, explaining absorption lines and solar limb darkening.\n",{"path":26011,"title":26012,"module":26004,"summary":26013},"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fspectral-line-formation-and-broadening","Spectral-Line Formation and Broadening","A spectral line is a bound-bound transition whose strength is set by an oscillator strength and whose shape is set by three broadening mechanisms: the Lorentzian natural and collisional wings, the Gaussian thermal Doppler core, and their Voigt convolution. Equivalent width measures the total absorption, and the curve of growth relates it to the number of absorbers through a linear, saturated, and damping regime, turning line strengths into abundances.\n",{"path":26015,"title":26016,"module":26004,"summary":26017},"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fopacity-and-the-rosseland-mean","Opacity Sources and the Rosseland Mean","Stellar opacity comes from four processes: bound-bound line absorption, bound-free photoionization, free-free absorption, and electron scattering. The bound-free and free-free terms follow a Kramers law, electron scattering sets a frequency-flat floor, and the negative hydrogen ion dominates cool photospheres. The Rosseland mean averages these harmonically, weighting transparent frequencies because they carry the flux, and its value fixes the radiative temperature gradient and decides where a star becomes convective.\n",{"path":26019,"title":26020,"module":26021,"summary":26022},"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fhydrostatic-equilibrium-and-the-virial-theorem","Hydrostatic Equilibrium and the Virial Theorem","Stellar Structure","A star holds itself up by balancing the inward pull of gravity against an outward pressure gradient. This balance, hydrostatic equilibrium, fixes a lower bound on the central pressure and, combined with the gravitational potential energy, yields the virial theorem. The virial relation gives a star a negative heat capacity, so that losing energy makes it hotter, and sets the Kelvin-Helmholtz timescale over which contraction alone can power the Sun.\n",{"path":26024,"title":26025,"module":26021,"summary":26026},"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equations-of-stellar-structure","The Equations of Stellar Structure","A static star is described by four coupled first-order differential equations in the interior mass or radius: mass conservation, hydrostatic equilibrium, energy generation, and energy transport. Closed with an equation of state, opacity, and reaction rates, and subject to central and surface boundary conditions, they determine the structure uniquely from mass and composition, the Vogt-Russell theorem. Energy moves by radiation until the temperature gradient exceeds the Schwarzschild limit, where convection takes over.\n",{"path":26028,"title":26029,"module":26021,"summary":26030},"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equation-of-state-and-polytropes","The Equation of State and Polytropes","Stellar pressure comes from gas, radiation, and, at high density, degenerate electrons. When pressure depends on density as a power law, hydrostatic equilibrium reduces to the Lane-Emden equation, whose solutions describe polytropes of index n. The relativistic degenerate case, n equal to three, gives a mass independent of radius, the Chandrasekhar mass. Eddington's standard model treats a radiation-supported star as an n equal to three polytrope and yields the quartic relating radiation fraction to mass.\n",{"path":26032,"title":26033,"module":26021,"summary":26034},"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-standard-solar-model","The Standard Solar Model","The standard solar model integrates the structure equations for one solar mass and calibrates the composition and convection parameter to reproduce the Sun's observed luminosity, radius, and age. Helioseismology tests the model's sound speed through the Sun's acoustic p-mode oscillations, and the model predicts a neutrino flux by production channel. The measured deficit, the solar-neutrino problem, is resolved by matter-enhanced flavor oscillation, confirmed when SNO measured the total flux across all flavors.\n",{"path":26036,"title":26037,"module":26038,"summary":26039},"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fthermonuclear-reaction-rates-and-the-gamow-peak","Thermonuclear Reaction Rates and the Gamow Peak","Nuclear Astrophysics","Stellar fusion proceeds only by quantum tunneling through the Coulomb barrier, because thermal energies are a thousand times smaller than the barrier height. The reaction rate is an integral over the Maxwell–Boltzmann distribution and the tunneling probability, whose product is sharply peaked at the Gamow energy. The astrophysical S-factor isolates the nuclear physics from the barrier penetration, and the steep temperature dependence follows from the width and position of the Gamow peak.\n",{"path":26041,"title":26042,"module":26038,"summary":26043},"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhydrogen-burning-pp-chains-and-cno","Hydrogen Burning: pp Chains and the CNO Cycle","Four protons fuse into one helium-4 nucleus, releasing 26.7 MeV, through two competing networks. The pp chain begins with a weak-interaction bottleneck and branches three ways; the CNO cycle uses carbon, nitrogen, and oxygen as catalysts and is limited by nitrogen-14 proton capture. Their steep and gentle temperature dependences cross near 1.8e7 K, which divides pp-powered lower-main-sequence stars from CNO-powered upper-main-sequence stars.\n",{"path":26045,"title":26046,"module":26038,"summary":26047},"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhelium-burning-and-the-triple-alpha-process","Helium Burning and the Triple-Alpha Process","Helium fuses to carbon in two steps through the unbound beryllium-8 nucleus and a resonant excited state of carbon-12, the Hoyle state, whose existence was predicted from the observed carbon abundance. The rate scales as roughly the fortieth power of temperature, and in a degenerate low-mass core this drives the runaway helium flash. A competing alpha capture on carbon-12 sets the carbon-to-oxygen ratio and the composition of the resulting white dwarf.\n",{"path":26049,"title":26050,"module":26038,"summary":26051},"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fadvanced-burning-and-neutron-capture-nucleosynthesis","Advanced Burning, the Iron Peak, and the s\u002Fr Processes","Massive stars burn carbon, neon, oxygen, and silicon in ever-shorter stages, building an onion-shell interior and reaching nuclear statistical equilibrium at the iron peak, where the binding-energy-per-nucleon curve turns over and fusion can release no more energy. Elements beyond iron form by neutron capture: the slow s-process in AGB stars tracks the valley of stability, while the rapid r-process in supernovae and neutron-star mergers builds the heaviest nuclei far from it.\n",{"path":26053,"title":26054,"module":26055,"summary":26056},"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fphases-of-the-interstellar-medium","The Phases of the Interstellar Medium","The Interstellar Medium","The gas between the stars separates into distinct thermal phases, from cold molecular clouds at 10 K to a diffuse million-degree corona, held near a common pressure by a balance of photoelectric heating and radiative cooling. Neutral hydrogen is traced by the 21-cm hyperfine line, dust reddens and extinguishes starlight along a characteristic wavelength law, and the ultraviolet output of hot stars carves ionized Strömgren spheres out of the surrounding gas.\n",{"path":26058,"title":26059,"module":26055,"summary":26060},"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fmolecular-clouds-and-gravitational-collapse","Molecular Clouds and Gravitational Collapse","Stars form in cold, dense molecular clouds when self-gravity overcomes thermal and magnetic support. The virial theorem fixes the Jeans mass and length at which a clump becomes unstable, the free-fall time sets how fast it collapses, and a fragmentation cascade — cut off at a minimum mass by the onset of opacity — turns one cloud into a whole cluster, imprinting the stellar initial mass function.\n",{"path":26062,"title":26063,"module":26055,"summary":26064},"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fprotostars-and-the-pre-main-sequence","Protostars and Pre-Main-Sequence Evolution","A collapsing core becomes optically thick and forms a protostar that grows by accretion through a disk while driving bipolar outflows. The newborn star appears on the birthline and contracts down the fully convective Hayashi track, then crosses the radiative Henyey track to the zero-age main sequence, powered by gravitational contraction until hydrogen ignites. Below about 0.08 solar masses degeneracy halts contraction before ignition, dividing stars from brown dwarfs.\n",{"path":26066,"title":26067,"module":26068,"summary":26069},"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-main-sequence-and-its-structure","The Main Sequence and Its Structure","Stellar Evolution","A star settles onto the zero-age main sequence when core hydrogen ignition halts contraction. Homology scaling of the structure equations reproduces the mass–luminosity relation, and the burning mode splits the sequence into an upper branch with a convective core and a lower branch with a convective envelope. The main-sequence lifetime falls steeply with mass, and the turnoff of a coeval cluster serves as a clock.\n",{"path":26071,"title":26072,"module":26068,"summary":26073},"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fpost-main-sequence-low-mass-evolution","Post-Main-Sequence Evolution of Low-Mass Stars","When a low-mass star exhausts core hydrogen, burning moves to a shell, the core contracts, and the envelope swells into a red giant. A degenerate helium core ignites in a flash, settles onto the horizontal branch, and after a second contraction the star climbs the asymptotic giant branch with two burning shells. Thermal pulses and dredge-up enrich the surface, and mass loss ejects a planetary nebula, leaving a carbon–oxygen white dwarf.\n",{"path":26075,"title":26076,"module":26068,"summary":26077},"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-evolution-of-massive-stars","The Evolution of Massive Stars","Stars above about eight solar masses burn through hydrogen, helium, carbon, neon, oxygen, and silicon in stages that grow shorter as neutrino losses accelerate contraction. The interior becomes an onion of concentric burning shells around an inert iron core. Radiation pressure near the Eddington limit drives fierce winds that can strip the hydrogen envelope entirely, and silicon burning builds an iron core toward the threshold of collapse.\n",{"path":26079,"title":26080,"module":26068,"summary":26081},"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fstellar-pulsation-and-the-instability-strip","Stellar Pulsation and the Instability Strip","Radial pulsation is a standing sound wave whose period scales inversely with the square root of the mean density. The kappa mechanism, an opacity valve seated in the helium partial-ionization zone, turns a star into a heat engine that pumps the oscillation. Stars in the instability strip pulsate as Cepheids, RR Lyrae, and Mira variables, and the Cepheid period–luminosity relation calibrates the distance ladder.\n",{"path":26083,"title":24499,"module":26084,"summary":26085},"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fwhite-dwarfs-and-the-chandrasekhar-limit","Stellar Death and Compact Remnants","A white dwarf is held up by the degeneracy pressure of its electrons, a quantum-mechanical stiffness that survives to zero temperature. Filling the Fermi sea sets a pressure that scales as density to the five-thirds power when the electrons are slow and only four-thirds when they are relativistic. The softer relativistic law produces the inverted mass-radius relation and a maximum mass, the Chandrasekhar limit near 1.4 solar masses, above which no cold equilibrium exists. Cooling and crystallization then turn the white-dwarf population into a clock for the Galactic disk.\n",{"path":26087,"title":26088,"module":26084,"summary":26089},"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fcore-collapse-supernovae","Core-Collapse Supernovae","When a massive star builds an iron core past the Chandrasekhar mass, degeneracy fails and the core collapses in less than a second. Photodisintegration and electron capture remove pressure support and neutronize the matter; the collapse halts abruptly at nuclear density, launching a shock that stalls and is revived by neutrino heating. The event is a Type II or stripped-envelope Ib\u002FIc supernova, and the neutrinos from SN 1987A confirmed the picture directly.\n",{"path":26091,"title":26092,"module":26084,"summary":26093},"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fthermonuclear-supernovae-type-ia","Thermonuclear Supernovae","A carbon-oxygen white dwarf driven toward the Chandrasekhar mass ignites its degenerate fuel and unbinds itself in a thermonuclear runaway, the Type Ia supernova. The light curve is powered by the radioactive decay of nickel-56 to cobalt-56 to iron-56, and the Phillips relation between peak brightness and decline rate makes these events standardizable candles. Their near-uniform luminosity turns them into the distance indicators that revealed cosmic acceleration.\n",{"path":26095,"title":26096,"module":26084,"summary":26097},"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fneutron-stars-and-pulsars","Neutron Stars and Pulsars","A neutron star is held up by neutron degeneracy and the repulsive nuclear force, with a maximum mass, the Tolman-Oppenheimer-Volkoff limit, set by an uncertain dense-matter equation of state. Its rotating magnetic dipole sweeps a beam past Earth as a pulsar, and magnetic braking traces a track across the period-period- derivative diagram. Millisecond pulsars, magnetars, glitches, and the orbital decay of the Hulse-Taylor binary follow from the same structure.\n",{"path":26099,"title":26100,"module":26084,"summary":26101},"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fblack-holes-schwarzschild-and-kerr","Black Holes, Schwarzschild and Kerr","Above the neutron-star mass limit gravity wins completely and the remnant is a black hole. The Schwarzschild solution gives the event horizon, gravitational redshift, and time dilation; the innermost stable circular orbit sets the efficiency of accretion. Rotating Kerr black holes drag spacetime and carry an ergosphere. Stellar-mass black holes are found in X-ray binaries, and the Event Horizon Telescope has imaged the shadow of a supermassive one.\n",{"path":26103,"title":26104,"module":26105,"summary":26106},"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fbinary-systems-and-mass-transfer","Binary Systems and Mass Transfer","Binaries and Gravitational Waves","Most stars are born in pairs, and a binary is the only setting where a stellar mass can be measured directly. Visual, spectroscopic, and eclipsing binaries each expose a different combination of the orbital elements, and together they calibrate the mass-luminosity relation. When one star swells to fill its Roche lobe, gas streams through the inner Lagrange point onto its companion. Conservative transfer widens or shrinks the orbit depending on the mass ratio, and the sign of that response explains the Algol paradox.\n",{"path":26108,"title":26109,"module":26105,"summary":26110},"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Faccreting-compact-objects","Accreting Compact Objects","Gas falling onto a compact object converts gravitational binding energy into radiation with an efficiency set by the depth of the potential well, up to tens of percent of the rest mass for a neutron star or black hole. Angular momentum forces the flow into a disk, and viscous dissipation gives a temperature profile that falls as radius to the minus three-quarters, producing a multicolor blackbody spectrum. Radiation pressure caps the steady luminosity at the Eddington limit. Unstable nuclear burning of the accreted fuel powers classical novae on white dwarfs and Type I X-ray bursts on neutron stars.\n",{"path":26112,"title":26113,"module":26105,"summary":26114},"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fgravitational-waves-from-inspiraling-binaries","Gravitational Waves from Inspiraling Binaries","A time-varying mass quadrupole radiates gravitational waves, ripples in spacetime that stretch and squeeze a ring of free masses along two polarizations. The radiated power drains a binary's orbital energy, shrinking the orbit and sweeping the wave frequency upward in a chirp whose rate fixes the chirp mass. Laser interferometers with kilometre arms measure the resulting strain of order ten to the minus twenty-one. The first detection, GW150914, matched a template for two merging black holes near thirty solar masses each.\n",{"path":26116,"title":26117,"module":26105,"summary":26118},"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fmultimessenger-astronomy-and-gamma-ray-bursts","Multimessenger Astronomy and Gamma-Ray Bursts","Gamma-ray bursts split into two populations: long bursts from the collapse of massive stars and short bursts from merging compact objects. The compactness problem forces the emitting plasma to move at ultra-relativistic speed, beaming the radiation into a narrow jet. The neutron-star merger GW170817 tied a gravitational chirp to a short gamma-ray burst, a radioactive kilonova, and a broadband afterglow, confirming that mergers forge r-process elements. A merger with a measured redshift is a standard siren that reads the Hubble constant from gravitational data alone.\n",{"path":26120,"title":26121,"module":26122,"summary":26123},"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fthe-milky-way","The Milky Way Galaxy","Galaxies and Dark Matter","The Galaxy resolves into a thin disk of gas and young stars, a central bar and bulge, and a diffuse old halo studded with globular clusters. Star counts and the reddening of distant light map these components, while the differential rotation of the disk — encoded in the Oort constants and the flat rotation curve — measures the enclosed mass and reveals more than the stars can account for. Spiral arms are density waves, not material structures, and the innermost stellar orbits around Sgr A* weigh a four-million-solar-mass black hole.\n",{"path":26125,"title":26126,"module":26122,"summary":26127},"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-morphology-and-classification","Galaxy Morphology and Classification","Galaxies sort along the Hubble tuning fork from smooth ellipticals through lenticulars to grand-design and barred spirals, with irregulars off the end. The light of a spheroid follows the de Vaucouleurs quarter-power law while a disk fades exponentially, and the general Sérsic profile interpolates between them. Virial scaling relations — Tully–Fisher for disks, Faber–Jackson and the fundamental plane for spheroids — tie luminosity to internal motions, and the Schechter function fixes the abundance of galaxies as a function of luminosity.\n",{"path":26129,"title":26130,"module":26122,"summary":26131},"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-rotation-curves-and-dark-matter","Galaxy Rotation Curves and Dark Matter","The rotation curves of disk galaxies stay flat far beyond the light, demanding an extended halo whose density falls as the inverse square of radius. Decomposing the curve into disk, bulge, and halo, and fitting isothermal or NFW profiles, quantifies the missing mass. Gravitational lensing weighs the same mass without dynamics, the mass-to-light ratio climbs from stars to clusters, and the Bullet Cluster separates the collisionless dark matter from the colliding gas — evidence that MOND strains to match.\n",{"path":26133,"title":26134,"module":26122,"summary":26135},"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Factive-galactic-nuclei-and-supermassive-black-holes","Active Galactic Nuclei","A small fraction of galaxies pour out enormous luminosity from a region smaller than the solar system. Accretion onto a supermassive black hole, limited by the Eddington balance of radiation pressure and gravity, powers the Seyferts, quasars, radio galaxies, and blazars — one engine seen from different angles through an obscuring torus. Relativistic jets produce apparent superluminal motion, reverberation mapping and stellar dynamics weigh the central mass, and the M–sigma relation ties that mass to the host bulge.\n",{"path":26137,"title":26138,"module":26122,"summary":26139},"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-clusters-and-large-scale-structure","Galaxy Clusters and Large-Scale Structure","Galaxies gather into groups and rich clusters bound by a common dark halo and filled with hot X-ray gas. Three independent probes — the virial theorem, the hydrostatic X-ray temperature, and gravitational lensing — agree on a mass that dwarfs the stars. On the largest scales galaxies trace a cosmic web of filaments, walls, and voids, quantified by the two-point correlation function, whose baryon acoustic oscillation bump provides a standard ruler for cosmology.\n",{"path":26141,"title":26142,"module":26143,"summary":26144},"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-expanding-universe-and-hubbles-law","The Expanding Universe and Hubble's Law","Cosmic Expansion and Dynamics","The universe is homogeneous and isotropic on large scales, so its expansion is captured by a single function of time, the scale factor. Comoving coordinates stay fixed while proper distances grow in proportion to the scale factor, producing Hubble's law and a cosmological redshift that measures stretched space rather than a Doppler shift. A Newtonian energy argument reproduces the dynamics, and the same finite, expanding cosmos resolves Olbers' paradox.\n",{"path":26146,"title":26147,"module":26143,"summary":26148},"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-frw-metric-and-cosmological-redshift","The FRW Metric and Cosmological Redshift","The geometry of a homogeneous, isotropic universe is fixed by symmetry to the Robertson-Walker metric, with the entire freedom reduced to a scale factor and a single curvature constant selecting an open, flat, or closed space. From the metric the null geodesic of light gives comoving distance, the exact cosmological redshift, and the distinction between the proper distance we cannot measure and the redshift we can.\n",{"path":26150,"title":23226,"module":26143,"summary":26151},"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-friedmann-equations-and-cosmic-dynamics","The scale factor obeys the Friedmann equation, the acceleration equation, and the fluid equation, only two of which are independent. An equation of state fixes how each component behaves under expansion, so radiation dilutes as the inverse fourth power of the scale factor, matter as the inverse cube, and vacuum energy not at all. The critical density defines the density parameters, and the deceleration parameter encodes whether gravity or dark energy is winning.\n",{"path":26153,"title":26154,"module":26143,"summary":26155},"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fcosmological-models-and-distances","Cosmological Models and Distances","Integrating the Friedmann equation for particular mixtures gives the benchmark models, from the matter-only Einstein-de Sitter universe to the concordance Lambda-CDM, each with its own scale-factor history and age. Because the redshift is the only direct observable, several distance measures diverge at high redshift, and the angular-diameter distance even turns over so that the most distant objects look larger. The horizon and lookback time set what is causally and observationally reachable.\n",{"path":26157,"title":26158,"module":26143,"summary":26159},"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fdark-energy-and-the-accelerating-universe","Dark Energy and the Accelerating Universe","In 1998 two teams found that distant Type Ia supernovae are fainter than a decelerating universe predicts, revealing that the expansion is accelerating and that a component with negative pressure dominates the energy budget. The simplest candidate is the cosmological constant, or vacuum energy, with an equation of state near minus one. It works observationally but leaves two deep puzzles: why the vacuum energy is a hundred and twenty orders of magnitude smaller than expected, and why it is comparable to the matter density just now.\n",{"path":26161,"title":26162,"module":26163,"summary":26164},"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fthe-thermal-history-of-the-universe","The Thermal History of the Universe","The Hot Big Bang","Running the expansion backward compresses and heats the universe, so its past is a sequence of thermal epochs set by temperature. Temperature scales as the inverse scale factor; species stay in equilibrium while their interaction rate exceeds the expansion rate and freeze out when it drops below. The effective degrees of freedom count the relativistic species and step down through mass thresholds, and neutrino decoupling just before electron-positron annihilation leaves a relic neutrino background slightly cooler than the photons.\n",{"path":26166,"title":26167,"module":26163,"summary":26168},"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fbig-bang-nucleosynthesis","Big Bang Nucleosynthesis","In the first three minutes the weak interactions freeze out the neutron-to-proton ratio, and once deuterium survives photodissociation a fast reaction network converts nearly all free neutrons into helium-4. The primordial abundances of deuterium, helium-3, helium-4, and lithium-7 depend on a single free parameter, the baryon-to-photon ratio, so measuring them fixes the baryon density. The predictions match observation across nine decades of abundance, with a persistent discrepancy in lithium-7.\n",{"path":26170,"title":26171,"module":26163,"summary":26172},"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Frecombination-and-the-cosmic-microwave-background","Recombination and the Cosmic Microwave Background","As the universe cooled through a few thousand kelvin the free electrons bound to protons, and the Saha equation tracks the falling ionization fraction. Once the plasma neutralized, photons stopped scattering and streamed freely from a spherical surface of last scattering at redshift about 1100. Those photons are the cosmic microwave background, an almost perfect blackbody at 2.725 kelvin with a dipole from our motion through it.\n",{"path":26174,"title":26175,"module":26163,"summary":26176},"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcmb-anisotropies-and-cosmological-parameters","CMB Anisotropies and Cosmological Parameters","The cosmic microwave background carries temperature fluctuations at the ten-parts-per-million level, imprinted by sound waves in the photon-baryon plasma before recombination. Decomposed into spherical harmonics, the fluctuations form an angular power spectrum whose acoustic peaks encode the geometry and contents of the universe: the first peak fixes spatial flatness, the odd-even peak ratio the baryon density, and the third peak the dark-matter density. Polarization adds an independent channel, and the Planck measurements pin the concordance parameters.\n",{"path":26178,"title":26179,"module":26163,"summary":26180},"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcosmic-inflation","Cosmic Inflation","The hot Big Bang leaves three initial-condition puzzles unexplained: why causally disconnected patches share a temperature, why the geometry is so nearly flat, and why no magnetic monopoles are seen. A brief epoch of accelerated expansion driven by a slowly rolling scalar field solves all three by stretching a small causal patch across the observable universe. The same accelerated expansion freezes quantum fluctuations into a near-scale-invariant spectrum of density perturbations, seeding all later structure.\n",{"path":26182,"title":26183,"module":26163,"summary":26184},"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fstructure-formation-and-the-growth-of-perturbations","Structure Formation and the Growth of Perturbations","The near-uniform early universe grew its galaxies and clusters by gravitational instability acting on the tiny inflationary perturbations. In an expanding background the growth is slowed to a power law rather than the exponential of a static medium; perturbations stall during radiation domination and grow with the scale factor once matter dominates. The transfer function turns the primordial spectrum into the processed matter power spectrum, and cold dark matter builds structure from the bottom up.\n",{"path":26186,"title":26187,"module":26163,"summary":26188},"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fdark-matter-dark-energy-and-open-questions","Dark Matter, Dark Energy, and Open Questions","Five independent lines of evidence converge on a universe whose energy budget is dominated by dark energy and dark matter, with ordinary baryons a small remainder. The candidate particles for dark matter range from WIMPs to axions to sterile neutrinos, each with its own detection strategy. The concordance model fits the data with six parameters but leaves the nature of dark energy, the Hubble tension, small-scale structure, and the matter-antimatter asymmetry unexplained.\n",{"path":26190,"title":26191,"module":6,"summary":6},"\u002Fastrophysics-cosmology","Astrophysics & Cosmology",{"path":26193,"title":26194,"module":6,"summary":6},"\u002Fcolophon","Colophon",{"path":1971,"title":26196,"module":6,"summary":6},"Study Notes",[26198,26209,26218,26231,26244,26255,26276,26289,26306,26315,26332,26357],{"module":24072,"moduleNumber":9648,"slug":26199,"lessons":26200},"mathematical-background",[26201,26203,26205,26207],{"title":22713,"path":24071,"lessonNumber":9648,"topics":26202,"summary":24073},[24072],{"title":24076,"path":24075,"lessonNumber":9654,"topics":26204,"summary":24077},[24072],{"title":24080,"path":24079,"lessonNumber":9660,"topics":26206,"summary":24081},[24072],{"title":22072,"path":24083,"lessonNumber":9666,"topics":26208,"summary":24084},[24072],{"module":21519,"moduleNumber":9654,"slug":26210,"lessons":26211},"foundations",[26212,26214,26216],{"title":24087,"path":24086,"lessonNumber":9648,"topics":26213,"summary":24088},[21519],{"title":24091,"path":24090,"lessonNumber":9654,"topics":26215,"summary":24092},[21519],{"title":24095,"path":24094,"lessonNumber":9660,"topics":26217,"summary":24096},[21519],{"module":24100,"moduleNumber":9660,"slug":26219,"lessons":26220},"neural-networks",[26221,26223,26225,26227,26229],{"title":24099,"path":24098,"lessonNumber":9648,"topics":26222,"summary":24101},[24100],{"title":24104,"path":24103,"lessonNumber":9654,"topics":26224,"summary":24105},[24100],{"title":24108,"path":24107,"lessonNumber":9660,"topics":26226,"summary":24109},[24100],{"title":24111,"path":16668,"lessonNumber":9666,"topics":26228,"summary":24112},[24100],{"title":24115,"path":24114,"lessonNumber":9672,"topics":26230,"summary":24116},[24100],{"module":20579,"moduleNumber":9666,"slug":26232,"lessons":26233},"optimization",[26234,26236,26238,26240,26242],{"title":24118,"path":17,"lessonNumber":9648,"topics":26235,"summary":24119},[20579],{"title":24121,"path":22,"lessonNumber":9654,"topics":26237,"summary":24122},[20579],{"title":24125,"path":24124,"lessonNumber":9660,"topics":26239,"summary":24126},[20579],{"title":24128,"path":257,"lessonNumber":9666,"topics":26241,"summary":24129},[20579],{"title":5,"path":20581,"lessonNumber":9672,"topics":26243,"summary":20600},[20579],{"module":24134,"moduleNumber":9672,"slug":26245,"lessons":26246},"regularization",[26247,26249,26251,26253],{"title":24133,"path":24132,"lessonNumber":9648,"topics":26248,"summary":24135},[24134],{"title":24138,"path":24137,"lessonNumber":9654,"topics":26250,"summary":24139},[24134],{"title":24142,"path":24141,"lessonNumber":9660,"topics":26252,"summary":24143},[24134],{"title":24146,"path":24145,"lessonNumber":9666,"topics":26254,"summary":24147},[24134],{"module":24151,"moduleNumber":9678,"slug":26256,"lessons":26257},"architectures",[26258,26260,26262,26264,26266,26268,26270,26272,26274],{"title":24150,"path":24149,"lessonNumber":9648,"topics":26259,"summary":24152},[24151],{"title":24155,"path":24154,"lessonNumber":9654,"topics":26261,"summary":24156},[24151],{"title":24159,"path":24158,"lessonNumber":9660,"topics":26263,"summary":24160},[24151],{"title":24163,"path":24162,"lessonNumber":9666,"topics":26265,"summary":24164},[24151],{"title":24167,"path":24166,"lessonNumber":9672,"topics":26267,"summary":24168},[24151],{"title":24171,"path":24170,"lessonNumber":9678,"topics":26269,"summary":24172},[24151],{"title":24175,"path":24174,"lessonNumber":9684,"topics":26271,"summary":24176},[24151],{"title":24179,"path":24178,"lessonNumber":9690,"topics":26273,"summary":24180},[24151],{"title":24183,"path":24182,"lessonNumber":9696,"topics":26275,"summary":24184},[24151],{"module":24188,"moduleNumber":9684,"slug":26277,"lessons":26278},"theory",[26279,26281,26283,26285,26287],{"title":24187,"path":24186,"lessonNumber":9648,"topics":26280,"summary":24189},[24188],{"title":24192,"path":24191,"lessonNumber":9654,"topics":26282,"summary":24193},[24188],{"title":24196,"path":24195,"lessonNumber":9660,"topics":26284,"summary":24197},[24188],{"title":24200,"path":24199,"lessonNumber":9666,"topics":26286,"summary":24201},[24188],{"title":24204,"path":24203,"lessonNumber":9672,"topics":26288,"summary":24205},[24188],{"module":24209,"moduleNumber":9690,"slug":26290,"lessons":26291},"generative-models",[26292,26294,26296,26298,26300,26302,26304],{"title":24208,"path":24207,"lessonNumber":9648,"topics":26293,"summary":24210},[24209],{"title":24213,"path":24212,"lessonNumber":9654,"topics":26295,"summary":24214},[24209],{"title":24217,"path":24216,"lessonNumber":9660,"topics":26297,"summary":24218},[24209],{"title":24221,"path":24220,"lessonNumber":9666,"topics":26299,"summary":24222},[24209],{"title":24225,"path":24224,"lessonNumber":9672,"topics":26301,"summary":24226},[24209],{"title":24229,"path":24228,"lessonNumber":9678,"topics":26303,"summary":24230},[24209],{"title":24233,"path":24232,"lessonNumber":9684,"topics":26305,"summary":24234},[24209],{"module":24238,"moduleNumber":9696,"slug":26307,"lessons":26308},"probabilistic-methods",[26309,26311,26313],{"title":24237,"path":24236,"lessonNumber":9648,"topics":26310,"summary":24239},[24238],{"title":24242,"path":24241,"lessonNumber":9654,"topics":26312,"summary":24243},[24238],{"title":24246,"path":24245,"lessonNumber":9660,"topics":26314,"summary":24247},[24238],{"module":24251,"moduleNumber":9702,"slug":26316,"lessons":26317},"practical",[26318,26320,26322,26324,26326,26328,26330],{"title":24250,"path":24249,"lessonNumber":9648,"topics":26319,"summary":24252},[24251],{"title":24255,"path":24254,"lessonNumber":9654,"topics":26321,"summary":24256},[24251],{"title":24259,"path":24258,"lessonNumber":9660,"topics":26323,"summary":24260},[24251],{"title":24263,"path":24262,"lessonNumber":9666,"topics":26325,"summary":24264},[24251],{"title":24267,"path":24266,"lessonNumber":9672,"topics":26327,"summary":24268},[24251],{"title":24271,"path":24270,"lessonNumber":9678,"topics":26329,"summary":24272},[24251],{"title":24275,"path":24274,"lessonNumber":9684,"topics":26331,"summary":24276},[24251],{"module":24280,"moduleNumber":9708,"slug":26333,"lessons":26334},"large-models-and-agents",[26335,26337,26339,26341,26343,26345,26347,26349,26351,26353,26355],{"title":24279,"path":24278,"lessonNumber":9648,"topics":26336,"summary":24281},[24280],{"title":24284,"path":24283,"lessonNumber":9654,"topics":26338,"summary":24285},[24280],{"title":24288,"path":24287,"lessonNumber":9660,"topics":26340,"summary":24289},[24280],{"title":24292,"path":24291,"lessonNumber":9666,"topics":26342,"summary":24293},[24280],{"title":24296,"path":24295,"lessonNumber":9672,"topics":26344,"summary":24297},[24280],{"title":24300,"path":24299,"lessonNumber":9678,"topics":26346,"summary":24301},[24280],{"title":24304,"path":24303,"lessonNumber":9684,"topics":26348,"summary":24305},[24280],{"title":24308,"path":24307,"lessonNumber":9690,"topics":26350,"summary":24309},[24280],{"title":24312,"path":24311,"lessonNumber":9696,"topics":26352,"summary":24313},[24280],{"title":24316,"path":24315,"lessonNumber":9702,"topics":26354,"summary":24317},[24280],{"title":24320,"path":24319,"lessonNumber":9708,"topics":26356,"summary":24321},[24280],{"module":24325,"moduleNumber":26358,"slug":26359,"lessons":26360},12,"reinforcement-learning",[26361,26363,26365,26367,26369],{"title":24324,"path":24323,"lessonNumber":9648,"topics":26362,"summary":24326},[24325],{"title":24329,"path":24328,"lessonNumber":9654,"topics":26364,"summary":24330},[24325],{"title":24333,"path":24332,"lessonNumber":9660,"topics":26366,"summary":24334},[24325],{"title":24337,"path":24336,"lessonNumber":9666,"topics":26368,"summary":24338},[24325],{"title":24341,"path":24340,"lessonNumber":9672,"topics":26370,"summary":24342},[24325],"\u003Csvg style=\"width:100%;max-width:384.948px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 288.711 179.202\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" stroke=\"var(--tk-accent)\" d=\"m14.525-61.337 1.435 2.988 1.434 2.936 1.435 2.886 1.434 2.835 1.434 2.785 1.435 2.734 1.434 2.683 1.435 2.633 1.434 2.581 1.435 2.533 1.434 2.48 1.435 2.43 1.434 2.38 1.434 2.328 1.435 2.28 1.434 2.227 1.435 2.177 1.434 2.126 1.435 2.076 1.434 2.026L44.65-9.24l1.434 1.924 1.434 1.873 1.435 1.823 1.434 1.772 1.435 1.722 1.434 1.67 1.435 1.62 1.434 1.57 1.435 1.52 1.434 1.468 1.434 1.418 1.435 1.366 1.434 1.317 1.435 1.266 1.434 1.215L67.6 15.47l1.434 1.114 1.435 1.064 1.434 1.012 1.434.963 1.435.91 1.434.862 1.435.81 1.434.76 1.435.709 1.434.658 1.435.608 1.434.557 1.434.507 1.435.455 1.434.406 1.435.354 1.434.304 1.435.254 1.434.203 1.435.152 1.434.101 1.434.051h1.435l1.434-.05 1.435-.1 1.434-.152 1.435-.203 1.434-.252 1.435-.304 1.434-.354 1.435-.405 1.434-.455 1.434-.506 1.435-.556 1.434-.607 1.435-.658 1.434-.709 1.435-.759 1.434-.81 1.435-.86 1.434-.91 1.434-.962 1.435-1.012 1.434-1.063 1.435-1.113 1.434-1.164 1.435-1.215 1.434-1.265 1.435-1.316 1.434-1.366 1.434-1.417 1.435-1.469 1.434-1.517 1.435-1.57 1.434-1.62 1.435-1.67 1.434-1.72 1.435-1.772 1.434-1.822 1.434-1.873 1.435-1.924 1.434-1.973 1.435-2.025 1.434-2.075 1.435-2.126 1.434-2.177 1.435-2.226 1.434-2.279 1.434-2.328 1.435-2.378 1.434-2.43 1.435-2.48 1.434-2.531 1.435-2.581 1.434-2.633 1.435-2.682 1.434-2.733 1.434-2.785 1.435-2.834 1.434-2.886 1.435-2.935 1.434-2.986\" style=\"stroke-width:1.2\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(80.355 -87.182)\">\u003Cpath d=\"M89.701 34.299L89.701 32.517L88.951 32.517Q88.752 32.494 88.701 32.275L88.701 32.189Q88.752 31.978 88.951 31.955L89.701 31.955L89.701 31.205Q89.752 30.998 89.951 30.970L90.096 30.970Q90.291 30.998 90.342 31.205L90.342 31.955L91.701 31.955Q91.893 31.974 91.951 32.189L91.951 32.275Q91.897 32.494 91.701 32.517L90.342 32.517L90.342 34.267Q90.342 34.884 90.916 34.884Q91.166 34.884 91.330 34.699Q91.494 34.513 91.494 34.267Q91.494 34.174 91.566 34.103Q91.639 34.033 91.740 34.021L91.885 34.021Q92.084 34.045 92.135 34.252L92.135 34.299Q92.135 34.623 91.951 34.886Q91.768 35.150 91.475 35.297Q91.182 35.443 90.861 35.443Q90.350 35.443 90.025 35.133Q89.701 34.822 89.701 34.299M92.971 35.166L92.971 35.076Q93.029 34.869 93.221 34.845L93.932 34.845L93.932 32.517L93.221 32.517Q93.025 32.494 92.971 32.275L92.971 32.189Q93.029 31.978 93.221 31.955L94.322 31.955Q94.522 31.974 94.572 32.189L94.572 32.517Q94.834 32.232 95.189 32.074Q95.545 31.916 95.932 31.916Q96.225 31.916 96.459 32.050Q96.693 32.185 96.693 32.451Q96.693 32.619 96.584 32.736Q96.475 32.853 96.307 32.853Q96.154 32.853 96.039 32.742Q95.924 32.631 95.924 32.474Q95.549 32.474 95.234 32.675Q94.920 32.877 94.746 33.211Q94.572 33.545 94.572 33.924L94.572 34.845L95.518 34.845Q95.725 34.869 95.764 35.076L95.764 35.166Q95.725 35.381 95.518 35.404L93.221 35.404Q93.029 35.381 92.971 35.166M97.736 34.549L97.736 32.517L97.314 32.517Q97.107 32.494 97.064 32.275L97.064 32.189Q97.111 31.978 97.314 31.955L98.131 31.955Q98.326 31.978 98.377 32.189L98.377 34.517Q98.377 34.752 98.547 34.818Q98.717 34.884 99.002 34.884Q99.209 34.884 99.404 34.808Q99.600 34.732 99.725 34.582Q99.850 34.431 99.850 34.220L99.850 32.517L99.428 32.517Q99.217 32.494 99.178 32.275L99.178 32.189Q99.217 31.978 99.428 31.955L100.240 31.955Q100.439 31.978 100.490 32.189L100.490 34.845L100.916 34.845Q101.123 34.869 101.162 35.076L101.162 35.166Q101.123 35.381 100.916 35.404L100.100 35.404Q99.900 35.381 99.850 35.181Q99.447 35.443 98.939 35.443Q98.705 35.443 98.490 35.402Q98.275 35.361 98.109 35.259Q97.943 35.158 97.840 34.980Q97.736 34.802 97.736 34.549M104.752 33.916L102.311 33.916Q102.365 34.193 102.563 34.416Q102.760 34.638 103.037 34.761Q103.314 34.884 103.600 34.884Q104.072 34.884 104.295 34.595Q104.303 34.584 104.359 34.478Q104.416 34.373 104.465 34.330Q104.514 34.287 104.607 34.275L104.752 34.275Q104.943 34.295 105.002 34.509L105.002 34.564Q104.936 34.865 104.705 35.062Q104.475 35.259 104.162 35.351Q103.850 35.443 103.545 35.443Q103.061 35.443 102.621 35.215Q102.182 34.986 101.914 34.586Q101.647 34.185 101.647 33.693L101.647 33.634Q101.647 33.166 101.893 32.763Q102.139 32.361 102.547 32.127Q102.955 31.892 103.424 31.892Q103.928 31.892 104.281 32.115Q104.635 32.338 104.818 32.726Q105.002 33.115 105.002 33.619L105.002 33.677Q104.943 33.892 104.752 33.916M102.318 33.365L104.346 33.365Q104.299 32.955 104.061 32.703Q103.822 32.451 103.424 32.451Q103.029 32.451 102.723 32.713Q102.416 32.974 102.318 33.365\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(80.355 -87.182)\">\u003Cpath d=\"M110.202 35.166L110.202 35.076Q110.253 34.869 110.448 34.845L111.554 34.845L111.554 31.076L110.448 31.076Q110.253 31.052 110.202 30.838L110.202 30.748Q110.253 30.541 110.448 30.517L111.944 30.517Q112.136 30.541 112.194 30.748L112.194 34.845L113.296 34.845Q113.495 34.869 113.546 35.076L113.546 35.166Q113.495 35.381 113.296 35.404L110.448 35.404Q110.253 35.381 110.202 35.166M116.120 35.443Q115.648 35.443 115.263 35.199Q114.878 34.955 114.655 34.545Q114.433 34.134 114.433 33.677Q114.433 33.334 114.558 33.011Q114.683 32.689 114.913 32.435Q115.144 32.181 115.450 32.037Q115.757 31.892 116.120 31.892Q116.483 31.892 116.796 32.039Q117.108 32.185 117.331 32.431Q117.554 32.677 117.681 32.998Q117.808 33.318 117.808 33.677Q117.808 34.134 117.583 34.547Q117.358 34.959 116.974 35.201Q116.589 35.443 116.120 35.443M116.120 34.884Q116.585 34.884 116.876 34.490Q117.167 34.095 117.167 33.611Q117.167 33.318 117.032 33.050Q116.898 32.783 116.657 32.617Q116.417 32.451 116.120 32.451Q115.816 32.451 115.577 32.617Q115.339 32.783 115.204 33.050Q115.069 33.318 115.069 33.611Q115.069 34.092 115.362 34.488Q115.655 34.884 116.120 34.884M118.831 35.205L118.831 34.291Q118.858 34.084 119.069 34.060L119.237 34.060Q119.401 34.084 119.460 34.244Q119.663 34.884 120.390 34.884Q120.597 34.884 120.825 34.849Q121.054 34.814 121.222 34.699Q121.390 34.584 121.390 34.381Q121.390 34.170 121.167 34.056Q120.944 33.943 120.671 33.900L119.972 33.787Q118.831 33.576 118.831 32.853Q118.831 32.564 118.976 32.375Q119.120 32.185 119.360 32.078Q119.601 31.970 119.857 31.931Q120.112 31.892 120.390 31.892Q120.640 31.892 120.833 31.922Q121.026 31.951 121.191 32.029Q121.269 31.912 121.398 31.892L121.476 31.892Q121.573 31.904 121.636 31.967Q121.698 32.029 121.710 32.123L121.710 32.830Q121.698 32.924 121.636 32.990Q121.573 33.056 121.476 33.068L121.308 33.068Q121.214 33.056 121.148 32.990Q121.081 32.924 121.069 32.830Q121.069 32.451 120.374 32.451Q120.026 32.451 119.708 32.533Q119.390 32.615 119.390 32.861Q119.390 33.127 120.062 33.236L120.765 33.357Q121.249 33.439 121.599 33.687Q121.948 33.935 121.948 34.381Q121.948 34.771 121.712 35.013Q121.476 35.256 121.126 35.349Q120.776 35.443 120.390 35.443Q119.812 35.443 119.413 35.189Q119.343 35.314 119.294 35.371Q119.245 35.427 119.140 35.443L119.069 35.443Q118.855 35.420 118.831 35.205M123.077 35.205L123.077 34.291Q123.105 34.084 123.316 34.060L123.483 34.060Q123.648 34.084 123.706 34.244Q123.909 34.884 124.636 34.884Q124.843 34.884 125.071 34.849Q125.300 34.814 125.468 34.699Q125.636 34.584 125.636 34.381Q125.636 34.170 125.413 34.056Q125.191 33.943 124.917 33.900L124.218 33.787Q123.077 33.576 123.077 32.853Q123.077 32.564 123.222 32.375Q123.366 32.185 123.607 32.078Q123.847 31.970 124.103 31.931Q124.358 31.892 124.636 31.892Q124.886 31.892 125.079 31.922Q125.273 31.951 125.437 32.029Q125.515 31.912 125.644 31.892L125.722 31.892Q125.819 31.904 125.882 31.967Q125.944 32.029 125.956 32.123L125.956 32.830Q125.944 32.924 125.882 32.990Q125.819 33.056 125.722 33.068L125.554 33.068Q125.460 33.056 125.394 32.990Q125.327 32.924 125.316 32.830Q125.316 32.451 124.620 32.451Q124.273 32.451 123.954 32.533Q123.636 32.615 123.636 32.861Q123.636 33.127 124.308 33.236L125.011 33.357Q125.495 33.439 125.845 33.687Q126.194 33.935 126.194 34.381Q126.194 34.771 125.958 35.013Q125.722 35.256 125.372 35.349Q125.023 35.443 124.636 35.443Q124.058 35.443 123.659 35.189Q123.589 35.314 123.540 35.371Q123.491 35.427 123.386 35.443L123.316 35.443Q123.101 35.420 123.077 35.205\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath stroke=\"none\" d=\"M33.997-23.209a2.4 2.4 0 1 0-4.8 0 2.4 2.4 0 0 0 4.8 0m-2.4 0\"\u002F>\u003Cg transform=\"translate(-83.13 -56.168)\">\u003Cpath d=\"M88.572 35.166L88.572 35.076Q88.615 34.869 88.822 34.845L89.244 34.845L89.244 31.076L88.822 31.076Q88.615 31.052 88.572 30.838L88.572 30.748Q88.615 30.541 88.822 30.517L89.639 30.517Q89.834 30.541 89.885 30.748L89.885 32.299Q90.346 31.916 90.943 31.916Q91.291 31.916 91.529 32.056Q91.768 32.197 91.883 32.455Q91.998 32.713 91.998 33.068L91.998 34.845L92.424 34.845Q92.631 34.869 92.670 35.076L92.670 35.166Q92.631 35.381 92.424 35.404L91.029 35.404Q90.834 35.381 90.783 35.166L90.783 35.076Q90.834 34.865 91.029 34.845L91.357 34.845L91.357 33.099Q91.357 32.791 91.268 32.633Q91.178 32.474 90.885 32.474Q90.615 32.474 90.387 32.605Q90.158 32.736 90.022 32.965Q89.885 33.193 89.885 33.459L89.885 34.845L90.311 34.845Q90.518 34.869 90.557 35.076L90.557 35.166Q90.518 35.381 90.311 35.404L88.822 35.404Q88.615 35.381 88.572 35.166M96.260 33.916L93.818 33.916Q93.873 34.193 94.070 34.416Q94.268 34.638 94.545 34.761Q94.822 34.884 95.107 34.884Q95.580 34.884 95.803 34.595Q95.811 34.584 95.867 34.478Q95.924 34.373 95.973 34.330Q96.022 34.287 96.115 34.275L96.260 34.275Q96.451 34.295 96.510 34.509L96.510 34.564Q96.443 34.865 96.213 35.062Q95.982 35.259 95.670 35.351Q95.357 35.443 95.053 35.443Q94.568 35.443 94.129 35.215Q93.689 34.986 93.422 34.586Q93.154 34.185 93.154 33.693L93.154 33.634Q93.154 33.166 93.400 32.763Q93.647 32.361 94.055 32.127Q94.463 31.892 94.932 31.892Q95.436 31.892 95.789 32.115Q96.143 32.338 96.326 32.726Q96.510 33.115 96.510 33.619L96.510 33.677Q96.451 33.892 96.260 33.916M93.826 33.365L95.854 33.365Q95.807 32.955 95.568 32.703Q95.330 32.451 94.932 32.451Q94.537 32.451 94.231 32.713Q93.924 32.974 93.826 33.365M97.217 35.166L97.217 35.076Q97.275 34.869 97.467 34.845L98.178 34.845L98.178 32.517L97.467 32.517Q97.272 32.494 97.217 32.275L97.217 32.189Q97.275 31.978 97.467 31.955L98.568 31.955Q98.768 31.974 98.818 32.189L98.818 32.517Q99.080 32.232 99.436 32.074Q99.791 31.916 100.178 31.916Q100.471 31.916 100.705 32.050Q100.939 32.185 100.939 32.451Q100.939 32.619 100.830 32.736Q100.721 32.853 100.553 32.853Q100.400 32.853 100.285 32.742Q100.170 32.631 100.170 32.474Q99.795 32.474 99.481 32.675Q99.166 32.877 98.992 33.211Q98.818 33.545 98.818 33.924L98.818 34.845L99.764 34.845Q99.971 34.869 100.010 35.076L100.010 35.166Q99.971 35.381 99.764 35.404L97.467 35.404Q97.275 35.381 97.217 35.166M104.752 33.916L102.311 33.916Q102.365 34.193 102.563 34.416Q102.760 34.638 103.037 34.761Q103.314 34.884 103.600 34.884Q104.072 34.884 104.295 34.595Q104.303 34.584 104.359 34.478Q104.416 34.373 104.465 34.330Q104.514 34.287 104.607 34.275L104.752 34.275Q104.943 34.295 105.002 34.509L105.002 34.564Q104.936 34.865 104.705 35.062Q104.475 35.259 104.162 35.351Q103.850 35.443 103.545 35.443Q103.061 35.443 102.621 35.215Q102.182 34.986 101.914 34.586Q101.647 34.185 101.647 33.693L101.647 33.634Q101.647 33.166 101.893 32.763Q102.139 32.361 102.547 32.127Q102.955 31.892 103.424 31.892Q103.928 31.892 104.281 32.115Q104.635 32.338 104.818 32.726Q105.002 33.115 105.002 33.619L105.002 33.677Q104.943 33.892 104.752 33.916M102.318 33.365L104.346 33.365Q104.299 32.955 104.061 32.703Q103.822 32.451 103.424 32.451Q103.029 32.451 102.723 32.713Q102.416 32.974 102.318 33.365\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"M3.144-71.01 104.152 96.577\" style=\"stroke-width:.8\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(-137.029 -99.452)\">\u003Cpath d=\"M89.701 34.299L89.701 32.517L88.951 32.517Q88.752 32.494 88.701 32.275L88.701 32.189Q88.752 31.978 88.951 31.955L89.701 31.955L89.701 31.205Q89.752 30.998 89.951 30.970L90.096 30.970Q90.291 30.998 90.342 31.205L90.342 31.955L91.701 31.955Q91.893 31.974 91.951 32.189L91.951 32.275Q91.897 32.494 91.701 32.517L90.342 32.517L90.342 34.267Q90.342 34.884 90.916 34.884Q91.166 34.884 91.330 34.699Q91.494 34.513 91.494 34.267Q91.494 34.174 91.566 34.103Q91.639 34.033 91.740 34.021L91.885 34.021Q92.084 34.045 92.135 34.252L92.135 34.299Q92.135 34.623 91.951 34.886Q91.768 35.150 91.475 35.297Q91.182 35.443 90.861 35.443Q90.350 35.443 90.025 35.133Q89.701 34.822 89.701 34.299M93.154 34.291Q93.154 33.845 93.568 33.588Q93.982 33.330 94.523 33.230Q95.064 33.131 95.572 33.123Q95.572 32.908 95.438 32.756Q95.303 32.603 95.096 32.527Q94.889 32.451 94.678 32.451Q94.334 32.451 94.174 32.474L94.174 32.533Q94.174 32.701 94.055 32.816Q93.936 32.931 93.772 32.931Q93.596 32.931 93.481 32.808Q93.365 32.685 93.365 32.517Q93.365 32.111 93.746 32.002Q94.127 31.892 94.686 31.892Q94.955 31.892 95.223 31.970Q95.490 32.049 95.715 32.199Q95.939 32.349 96.076 32.570Q96.213 32.791 96.213 33.068L96.213 34.787Q96.213 34.845 96.740 34.845Q96.936 34.865 96.986 35.076L96.986 35.166Q96.936 35.381 96.740 35.404L96.596 35.404Q96.252 35.404 96.023 35.357Q95.795 35.310 95.650 35.123Q95.189 35.443 94.482 35.443Q94.147 35.443 93.842 35.302Q93.537 35.162 93.346 34.900Q93.154 34.638 93.154 34.291M93.795 34.299Q93.795 34.572 94.037 34.728Q94.279 34.884 94.564 34.884Q94.783 34.884 95.016 34.826Q95.248 34.767 95.410 34.629Q95.572 34.490 95.572 34.267L95.572 33.677Q95.291 33.677 94.875 33.734Q94.459 33.791 94.127 33.929Q93.795 34.068 93.795 34.299M97.064 35.166L97.064 35.076Q97.107 34.869 97.314 34.845L97.736 34.845L97.736 32.517L97.314 32.517Q97.107 32.494 97.064 32.275L97.064 32.189Q97.111 31.978 97.314 31.955L98.131 31.955Q98.326 31.978 98.377 32.189L98.377 32.275L98.369 32.299Q98.596 32.119 98.869 32.017Q99.143 31.916 99.436 31.916Q99.783 31.916 100.022 32.056Q100.260 32.197 100.375 32.455Q100.490 32.713 100.490 33.068L100.490 34.845L100.916 34.845Q101.123 34.869 101.162 35.076L101.162 35.166Q101.123 35.381 100.916 35.404L99.522 35.404Q99.326 35.381 99.275 35.166L99.275 35.076Q99.326 34.865 99.522 34.845L99.850 34.845L99.850 33.099Q99.850 32.791 99.760 32.633Q99.670 32.474 99.377 32.474Q99.107 32.474 98.879 32.605Q98.650 32.736 98.514 32.965Q98.377 33.193 98.377 33.459L98.377 34.845L98.803 34.845Q99.010 34.869 99.049 35.076L99.049 35.166Q99.010 35.381 98.803 35.404L97.314 35.404Q97.107 35.381 97.064 35.166M101.479 36.052Q101.479 35.752 101.627 35.490Q101.775 35.228 102.025 35.068Q101.842 34.814 101.842 34.494Q101.842 34.209 101.994 33.947Q101.744 33.623 101.744 33.205Q101.744 32.845 101.936 32.549Q102.127 32.252 102.445 32.084Q102.764 31.916 103.127 31.916Q103.334 31.916 103.537 31.976Q103.740 32.037 103.904 32.138Q104.287 31.877 104.760 31.877Q104.998 31.877 105.180 32.008Q105.361 32.138 105.361 32.365Q105.361 32.513 105.260 32.619Q105.158 32.724 105.002 32.724Q104.869 32.724 104.773 32.646Q104.678 32.568 104.647 32.443Q104.502 32.451 104.303 32.541Q104.506 32.853 104.506 33.205Q104.506 33.486 104.395 33.718Q104.283 33.951 104.088 34.129Q103.893 34.306 103.641 34.404Q103.389 34.502 103.127 34.502Q102.740 34.502 102.408 34.314Q102.397 34.314 102.387 34.386Q102.377 34.459 102.377 34.494Q102.377 34.615 102.443 34.722Q102.510 34.830 102.631 34.869Q102.647 34.865 102.660 34.863Q102.674 34.861 102.697 34.861Q102.713 34.861 102.744 34.869Q102.775 34.877 102.783 34.877L103.377 34.877Q104.158 34.877 104.699 35.119Q105.240 35.361 105.240 36.052Q105.240 36.353 105.061 36.580Q104.881 36.806 104.584 36.951Q104.287 37.095 103.967 37.162Q103.647 37.228 103.361 37.228Q102.971 37.228 102.529 37.105Q102.088 36.982 101.783 36.715Q101.479 36.447 101.479 36.052M102.018 36.045Q102.018 36.263 102.258 36.404Q102.498 36.545 102.822 36.611Q103.147 36.677 103.361 36.677Q103.576 36.677 103.900 36.611Q104.225 36.545 104.465 36.404Q104.705 36.263 104.705 36.045Q104.705 35.756 104.481 35.617Q104.256 35.478 103.975 35.445Q103.693 35.412 103.346 35.412L102.729 35.412Q102.545 35.412 102.383 35.492Q102.221 35.572 102.119 35.718Q102.018 35.865 102.018 36.045M103.127 33.947Q103.428 33.947 103.647 33.728Q103.865 33.509 103.865 33.205Q103.865 33.049 103.811 32.918Q103.756 32.787 103.650 32.681Q103.545 32.576 103.414 32.521Q103.283 32.467 103.127 32.467Q102.822 32.467 102.604 32.685Q102.385 32.904 102.385 33.205Q102.385 33.502 102.607 33.724Q102.830 33.947 103.127 33.947M108.998 33.916L106.557 33.916Q106.611 34.193 106.809 34.416Q107.006 34.638 107.283 34.761Q107.561 34.884 107.846 34.884Q108.318 34.884 108.541 34.595Q108.549 34.584 108.606 34.478Q108.662 34.373 108.711 34.330Q108.760 34.287 108.854 34.275L108.998 34.275Q109.189 34.295 109.248 34.509L109.248 34.564Q109.182 34.865 108.951 35.062Q108.721 35.259 108.408 35.351Q108.096 35.443 107.791 35.443Q107.307 35.443 106.867 35.215Q106.428 34.986 106.160 34.586Q105.893 34.185 105.893 33.693L105.893 33.634Q105.893 33.166 106.139 32.763Q106.385 32.361 106.793 32.127Q107.201 31.892 107.670 31.892Q108.174 31.892 108.527 32.115Q108.881 32.338 109.064 32.726Q109.248 33.115 109.248 33.619L109.248 33.677Q109.189 33.892 108.998 33.916M106.564 33.365L108.592 33.365Q108.545 32.955 108.307 32.703Q108.068 32.451 107.670 32.451Q107.275 32.451 106.969 32.713Q106.662 32.974 106.564 33.365M109.803 35.166L109.803 35.076Q109.846 34.869 110.053 34.845L110.475 34.845L110.475 32.517L110.053 32.517Q109.846 32.494 109.803 32.275L109.803 32.189Q109.850 31.978 110.053 31.955L110.869 31.955Q111.064 31.978 111.115 32.189L111.115 32.275L111.107 32.299Q111.334 32.119 111.607 32.017Q111.881 31.916 112.174 31.916Q112.522 31.916 112.760 32.056Q112.998 32.197 113.113 32.455Q113.229 32.713 113.229 33.068L113.229 34.845L113.654 34.845Q113.861 34.869 113.900 35.076L113.900 35.166Q113.861 35.381 113.654 35.404L112.260 35.404Q112.064 35.381 112.014 35.166L112.014 35.076Q112.064 34.865 112.260 34.845L112.588 34.845L112.588 33.099Q112.588 32.791 112.498 32.633Q112.408 32.474 112.115 32.474Q111.846 32.474 111.617 32.605Q111.389 32.736 111.252 32.965Q111.115 33.193 111.115 33.459L111.115 34.845L111.541 34.845Q111.748 34.869 111.787 35.076L111.787 35.166Q111.748 35.381 111.541 35.404L110.053 35.404Q109.846 35.381 109.803 35.166M115.178 34.299L115.178 32.517L114.428 32.517Q114.229 32.494 114.178 32.275L114.178 32.189Q114.229 31.978 114.428 31.955L115.178 31.955L115.178 31.205Q115.229 30.998 115.428 30.970L115.572 30.970Q115.768 30.998 115.818 31.205L115.818 31.955L117.178 31.955Q117.369 31.974 117.428 32.189L117.428 32.275Q117.373 32.494 117.178 32.517L115.818 32.517L115.818 34.267Q115.818 34.884 116.393 34.884Q116.643 34.884 116.807 34.699Q116.971 34.513 116.971 34.267Q116.971 34.174 117.043 34.103Q117.115 34.033 117.217 34.021L117.361 34.021Q117.561 34.045 117.611 34.252L117.611 34.299Q117.611 34.623 117.428 34.886Q117.244 35.150 116.951 35.297Q116.658 35.443 116.338 35.443Q115.826 35.443 115.502 35.133Q115.178 34.822 115.178 34.299\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-137.029 -99.452)\">\u003Cpath d=\"M122.573 36.947L122.573 36.861Q122.616 36.642 122.823 36.619L123.245 36.619L123.245 32.517L122.823 32.517Q122.616 32.494 122.573 32.275L122.573 32.189Q122.620 31.978 122.823 31.955L123.640 31.955Q123.835 31.974 123.886 32.189L123.886 32.259Q124.097 32.092 124.360 32.004Q124.624 31.916 124.894 31.916Q125.233 31.916 125.530 32.060Q125.827 32.205 126.042 32.457Q126.257 32.709 126.372 33.021Q126.487 33.334 126.487 33.677Q126.487 34.142 126.261 34.552Q126.034 34.963 125.648 35.203Q125.261 35.443 124.792 35.443Q124.273 35.443 123.886 35.076L123.886 36.619L124.312 36.619Q124.519 36.642 124.558 36.861L124.558 36.947Q124.519 37.158 124.312 37.181L122.823 37.181Q122.620 37.158 122.573 36.947M124.749 34.884Q125.058 34.884 125.308 34.715Q125.558 34.545 125.702 34.263Q125.847 33.982 125.847 33.677Q125.847 33.388 125.722 33.109Q125.597 32.830 125.364 32.652Q125.132 32.474 124.831 32.474Q124.511 32.474 124.249 32.660Q123.987 32.845 123.886 33.146L123.886 33.998Q123.976 34.365 124.196 34.625Q124.417 34.884 124.749 34.884M127.198 35.166L127.198 35.076Q127.249 34.869 127.444 34.845L128.550 34.845L128.550 31.076L127.444 31.076Q127.249 31.052 127.198 30.838L127.198 30.748Q127.249 30.541 127.444 30.517L128.940 30.517Q129.132 30.541 129.190 30.748L129.190 34.845L130.292 34.845Q130.491 34.869 130.542 35.076L130.542 35.166Q130.491 35.381 130.292 35.404L127.444 35.404Q127.249 35.381 127.198 35.166M131.401 34.291Q131.401 33.845 131.815 33.588Q132.230 33.330 132.771 33.230Q133.312 33.131 133.819 33.123Q133.819 32.908 133.685 32.756Q133.550 32.603 133.343 32.527Q133.136 32.451 132.925 32.451Q132.581 32.451 132.421 32.474L132.421 32.533Q132.421 32.701 132.302 32.816Q132.183 32.931 132.019 32.931Q131.843 32.931 131.728 32.808Q131.612 32.685 131.612 32.517Q131.612 32.111 131.993 32.002Q132.374 31.892 132.933 31.892Q133.202 31.892 133.470 31.970Q133.737 32.049 133.962 32.199Q134.187 32.349 134.323 32.570Q134.460 32.791 134.460 33.068L134.460 34.787Q134.460 34.845 134.987 34.845Q135.183 34.865 135.233 35.076L135.233 35.166Q135.183 35.381 134.987 35.404L134.843 35.404Q134.499 35.404 134.271 35.357Q134.042 35.310 133.898 35.123Q133.437 35.443 132.730 35.443Q132.394 35.443 132.089 35.302Q131.784 35.162 131.593 34.900Q131.401 34.638 131.401 34.291M132.042 34.299Q132.042 34.572 132.284 34.728Q132.526 34.884 132.812 34.884Q133.030 34.884 133.263 34.826Q133.495 34.767 133.657 34.629Q133.819 34.490 133.819 34.267L133.819 33.677Q133.538 33.677 133.122 33.734Q132.706 33.791 132.374 33.929Q132.042 34.068 132.042 34.299M135.312 35.166L135.312 35.076Q135.355 34.869 135.562 34.845L135.983 34.845L135.983 32.517L135.562 32.517Q135.355 32.494 135.312 32.275L135.312 32.189Q135.358 31.978 135.562 31.955L136.378 31.955Q136.573 31.978 136.624 32.189L136.624 32.275L136.616 32.299Q136.843 32.119 137.116 32.017Q137.390 31.916 137.683 31.916Q138.030 31.916 138.269 32.056Q138.507 32.197 138.622 32.455Q138.737 32.713 138.737 33.068L138.737 34.845L139.163 34.845Q139.370 34.869 139.409 35.076L139.409 35.166Q139.370 35.381 139.163 35.404L137.769 35.404Q137.573 35.381 137.523 35.166L137.523 35.076Q137.573 34.865 137.769 34.845L138.097 34.845L138.097 33.099Q138.097 32.791 138.007 32.633Q137.917 32.474 137.624 32.474Q137.355 32.474 137.126 32.605Q136.898 32.736 136.761 32.965Q136.624 33.193 136.624 33.459L136.624 34.845L137.050 34.845Q137.257 34.869 137.296 35.076L137.296 35.166Q137.257 35.381 137.050 35.404L135.562 35.404Q135.355 35.381 135.312 35.166M142.999 33.916L140.558 33.916Q140.612 34.193 140.810 34.416Q141.007 34.638 141.284 34.761Q141.562 34.884 141.847 34.884Q142.319 34.884 142.542 34.595Q142.550 34.584 142.607 34.478Q142.663 34.373 142.712 34.330Q142.761 34.287 142.855 34.275L142.999 34.275Q143.190 34.295 143.249 34.509L143.249 34.564Q143.183 34.865 142.952 35.062Q142.722 35.259 142.409 35.351Q142.097 35.443 141.792 35.443Q141.308 35.443 140.868 35.215Q140.429 34.986 140.161 34.586Q139.894 34.185 139.894 33.693L139.894 33.634Q139.894 33.166 140.140 32.763Q140.386 32.361 140.794 32.127Q141.202 31.892 141.671 31.892Q142.175 31.892 142.528 32.115Q142.882 32.338 143.065 32.726Q143.249 33.115 143.249 33.619L143.249 33.677Q143.190 33.892 142.999 33.916M140.565 33.365L142.593 33.365Q142.546 32.955 142.308 32.703Q142.069 32.451 141.671 32.451Q141.276 32.451 140.970 32.713Q140.663 32.974 140.565 33.365\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(22.027 62.195)\">\u003Cpath d=\"M88.572 35.166L88.572 35.076Q88.615 34.869 88.822 34.845L89.244 34.845L89.244 32.517L88.822 32.517Q88.615 32.494 88.572 32.275L88.572 32.189Q88.619 31.978 88.822 31.955L89.639 31.955Q89.834 31.978 89.885 32.189L89.885 32.275L89.877 32.299Q90.104 32.119 90.377 32.017Q90.650 31.916 90.943 31.916Q91.291 31.916 91.529 32.056Q91.768 32.197 91.883 32.455Q91.998 32.713 91.998 33.068L91.998 34.845L92.424 34.845Q92.631 34.869 92.670 35.076L92.670 35.166Q92.631 35.381 92.424 35.404L91.029 35.404Q90.834 35.381 90.783 35.166L90.783 35.076Q90.834 34.865 91.029 34.845L91.357 34.845L91.357 33.099Q91.357 32.791 91.268 32.633Q91.178 32.474 90.885 32.474Q90.615 32.474 90.387 32.605Q90.158 32.736 90.022 32.965Q89.885 33.193 89.885 33.459L89.885 34.845L90.311 34.845Q90.518 34.869 90.557 35.076L90.557 35.166Q90.518 35.381 90.311 35.404L88.822 35.404Q88.615 35.381 88.572 35.166M94.869 35.443Q94.397 35.443 94.012 35.199Q93.627 34.955 93.404 34.545Q93.182 34.134 93.182 33.677Q93.182 33.334 93.307 33.011Q93.432 32.689 93.662 32.435Q93.893 32.181 94.199 32.037Q94.506 31.892 94.869 31.892Q95.232 31.892 95.545 32.039Q95.857 32.185 96.080 32.431Q96.303 32.677 96.430 32.998Q96.557 33.318 96.557 33.677Q96.557 34.134 96.332 34.547Q96.107 34.959 95.723 35.201Q95.338 35.443 94.869 35.443M94.869 34.884Q95.334 34.884 95.625 34.490Q95.916 34.095 95.916 33.611Q95.916 33.318 95.781 33.050Q95.647 32.783 95.406 32.617Q95.166 32.451 94.869 32.451Q94.564 32.451 94.326 32.617Q94.088 32.783 93.953 33.050Q93.818 33.318 93.818 33.611Q93.818 34.092 94.111 34.488Q94.404 34.884 94.869 34.884\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(22.027 62.195)\">\u003Cpath d=\"M101.995 35.166L101.995 31.076L101.573 31.076Q101.366 31.052 101.323 30.838L101.323 30.748Q101.366 30.541 101.573 30.517L102.390 30.517Q102.585 30.541 102.636 30.748L102.636 32.259Q102.847 32.092 103.110 32.004Q103.374 31.916 103.644 31.916Q103.983 31.916 104.280 32.060Q104.577 32.205 104.792 32.457Q105.007 32.709 105.122 33.021Q105.237 33.334 105.237 33.677Q105.237 34.142 105.011 34.552Q104.784 34.963 104.398 35.203Q104.011 35.443 103.542 35.443Q103.023 35.443 102.636 35.076L102.636 35.166Q102.585 35.384 102.390 35.404L102.245 35.404Q102.054 35.381 101.995 35.166M103.499 34.884Q103.808 34.884 104.058 34.715Q104.308 34.545 104.452 34.263Q104.597 33.982 104.597 33.677Q104.597 33.388 104.472 33.109Q104.347 32.830 104.114 32.652Q103.882 32.474 103.581 32.474Q103.261 32.474 102.999 32.660Q102.737 32.845 102.636 33.146L102.636 33.998Q102.726 34.365 102.946 34.625Q103.167 34.884 103.499 34.884M107.620 35.443Q107.148 35.443 106.763 35.199Q106.378 34.955 106.155 34.545Q105.933 34.134 105.933 33.677Q105.933 33.334 106.058 33.011Q106.183 32.689 106.413 32.435Q106.644 32.181 106.950 32.037Q107.257 31.892 107.620 31.892Q107.983 31.892 108.296 32.039Q108.608 32.185 108.831 32.431Q109.054 32.677 109.181 32.998Q109.308 33.318 109.308 33.677Q109.308 34.134 109.083 34.547Q108.858 34.959 108.474 35.201Q108.089 35.443 107.620 35.443M107.620 34.884Q108.085 34.884 108.376 34.490Q108.667 34.095 108.667 33.611Q108.667 33.318 108.532 33.050Q108.398 32.783 108.157 32.617Q107.917 32.451 107.620 32.451Q107.316 32.451 107.077 32.617Q106.839 32.783 106.704 33.050Q106.569 33.318 106.569 33.611Q106.569 34.092 106.862 34.488Q107.155 34.884 107.620 34.884M110.944 34.299L110.944 32.517L110.194 32.517Q109.995 32.494 109.944 32.275L109.944 32.189Q109.995 31.978 110.194 31.955L110.944 31.955L110.944 31.205Q110.995 30.998 111.194 30.970L111.339 30.970Q111.534 30.998 111.585 31.205L111.585 31.955L112.944 31.955Q113.136 31.974 113.194 32.189L113.194 32.275Q113.140 32.494 112.944 32.517L111.585 32.517L111.585 34.267Q111.585 34.884 112.159 34.884Q112.409 34.884 112.573 34.699Q112.737 34.513 112.737 34.267Q112.737 34.174 112.810 34.103Q112.882 34.033 112.983 34.021L113.128 34.021Q113.327 34.045 113.378 34.252L113.378 34.299Q113.378 34.623 113.194 34.886Q113.011 35.150 112.718 35.297Q112.425 35.443 112.105 35.443Q111.593 35.443 111.269 35.133Q110.944 34.822 110.944 34.299M115.191 34.299L115.191 32.517L114.441 32.517Q114.241 32.494 114.191 32.275L114.191 32.189Q114.241 31.978 114.441 31.955L115.191 31.955L115.191 31.205Q115.241 30.998 115.441 30.970L115.585 30.970Q115.780 30.998 115.831 31.205L115.831 31.955L117.191 31.955Q117.382 31.974 117.441 32.189L117.441 32.275Q117.386 32.494 117.191 32.517L115.831 32.517L115.831 34.267Q115.831 34.884 116.405 34.884Q116.655 34.884 116.819 34.699Q116.983 34.513 116.983 34.267Q116.983 34.174 117.056 34.103Q117.128 34.033 117.230 34.021L117.374 34.021Q117.573 34.045 117.624 34.252L117.624 34.299Q117.624 34.623 117.441 34.886Q117.257 35.150 116.964 35.297Q116.671 35.443 116.351 35.443Q115.839 35.443 115.515 35.133Q115.191 34.822 115.191 34.299M120.358 35.443Q119.886 35.443 119.501 35.199Q119.116 34.955 118.894 34.545Q118.671 34.134 118.671 33.677Q118.671 33.334 118.796 33.011Q118.921 32.689 119.151 32.435Q119.382 32.181 119.689 32.037Q119.995 31.892 120.358 31.892Q120.722 31.892 121.034 32.039Q121.347 32.185 121.569 32.431Q121.792 32.677 121.919 32.998Q122.046 33.318 122.046 33.677Q122.046 34.134 121.821 34.547Q121.597 34.959 121.212 35.201Q120.827 35.443 120.358 35.443M120.358 34.884Q120.823 34.884 121.114 34.490Q121.405 34.095 121.405 33.611Q121.405 33.318 121.271 33.050Q121.136 32.783 120.896 32.617Q120.655 32.451 120.358 32.451Q120.054 32.451 119.816 32.617Q119.577 32.783 119.442 33.050Q119.308 33.318 119.308 33.611Q119.308 34.092 119.601 34.488Q119.894 34.884 120.358 34.884M122.429 35.166L122.429 35.076Q122.480 34.865 122.675 34.845L122.874 34.845L122.874 32.517L122.675 32.517Q122.468 32.494 122.429 32.275L122.429 32.189Q122.480 31.974 122.675 31.955L123.155 31.955Q123.347 31.978 123.405 32.189Q123.726 31.916 124.140 31.916Q124.331 31.916 124.499 32.025Q124.667 32.134 124.741 32.306Q124.917 32.115 125.140 32.015Q125.362 31.916 125.605 31.916Q126.023 31.916 126.177 32.258Q126.331 32.599 126.331 33.068L126.331 34.845L126.530 34.845Q126.741 34.869 126.780 35.076L126.780 35.166Q126.730 35.381 126.530 35.404L125.714 35.404Q125.519 35.381 125.468 35.166L125.468 35.076Q125.519 34.869 125.714 34.845L125.804 34.845L125.804 33.099Q125.804 32.474 125.554 32.474Q125.222 32.474 125.044 32.785Q124.866 33.095 124.866 33.459L124.866 34.845L125.069 34.845Q125.276 34.869 125.316 35.076L125.316 35.166Q125.265 35.381 125.069 35.404L124.253 35.404Q124.054 35.381 124.003 35.166L124.003 35.076Q124.054 34.869 124.253 34.845L124.339 34.845L124.339 33.099Q124.339 32.474 124.093 32.474Q123.761 32.474 123.583 32.787Q123.405 33.099 123.405 33.459L123.405 34.845L123.605 34.845Q123.812 34.869 123.851 35.076L123.851 35.166Q123.800 35.384 123.605 35.404L122.675 35.404Q122.468 35.381 122.429 35.166\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-good)\" d=\"m-2.546-62.967 1.584 4.048 1.585 3.95 1.585 3.854 1.584 3.756 1.585 3.66 1.585 3.562 1.584 3.466 1.585 3.368 1.585 3.271 1.584 3.174 1.585 3.077 1.585 2.98 1.584 2.883 1.585 2.785 1.585 2.69 1.584 2.59 1.585 2.495 1.585 2.397 1.584 2.301 1.585 2.203 1.585 2.106 1.584 2.01 1.585 1.911 1.585 1.815 1.584 1.718 1.585 1.62 1.585 1.525 1.584 1.426 1.585 1.33 1.585 1.232 1.584 1.135 1.585 1.038 1.585.941 1.584.844 1.585.748 1.585.65 1.584.552 1.585.456 1.585.359 1.584.261 1.585.165 1.585.067 1.584-.03 1.585-.126 1.585-.225 1.584-.32 1.585-.418 1.585-.515 1.584-.613 1.585-.709 1.585-.806 1.584-.904 1.585-1 1.585-1.098 1.584-1.194 1.585-1.292 1.585-1.389 1.584-1.486 1.585-1.583 1.585-1.68 1.584-1.777 1.585-1.875 1.585-1.97 1.584-2.07 1.585-2.165 1.585-2.262 1.584-2.36 1.585-2.457 1.585-2.553 1.584-2.652 1.585-2.747 1.585-2.846 1.584-2.942 1.585-3.04 1.585-3.135 1.584-3.234 1.585-3.33 1.585-3.428 1.584-3.524\" style=\"stroke-width:.8\"\u002F>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\">\u003Cg fill=\"var(--tk-good)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(-144.106 -64.133)\">\u003Cpath d=\"M90.814 36.947L90.814 36.861Q90.865 36.642 91.061 36.619L91.525 36.619L91.525 35.021Q91.072 35.443 90.463 35.443Q90.107 35.443 89.803 35.300Q89.498 35.158 89.266 34.908Q89.033 34.658 88.908 34.336Q88.783 34.013 88.783 33.677Q88.783 33.193 89.022 32.793Q89.260 32.392 89.666 32.154Q90.072 31.916 90.549 31.916Q91.111 31.916 91.525 32.306L91.525 32.146Q91.576 31.935 91.775 31.916L91.916 31.916Q92.115 31.939 92.166 32.146L92.166 36.619L92.631 36.619Q92.826 36.642 92.877 36.861L92.877 36.947Q92.826 37.158 92.631 37.181L91.061 37.181Q90.865 37.158 90.814 36.947M90.510 34.884Q90.881 34.884 91.156 34.631Q91.432 34.377 91.525 34.013L91.525 33.396Q91.482 33.158 91.359 32.945Q91.236 32.732 91.039 32.603Q90.842 32.474 90.600 32.474Q90.283 32.474 90.012 32.640Q89.740 32.806 89.582 33.088Q89.424 33.369 89.424 33.685Q89.424 34.150 89.738 34.517Q90.053 34.884 90.510 34.884M93.490 34.549L93.490 32.517L93.068 32.517Q92.861 32.494 92.818 32.275L92.818 32.189Q92.865 31.978 93.068 31.955L93.885 31.955Q94.080 31.978 94.131 32.189L94.131 34.517Q94.131 34.752 94.301 34.818Q94.471 34.884 94.756 34.884Q94.963 34.884 95.158 34.808Q95.354 34.732 95.479 34.582Q95.604 34.431 95.604 34.220L95.604 32.517L95.182 32.517Q94.971 32.494 94.932 32.275L94.932 32.189Q94.971 31.978 95.182 31.955L95.994 31.955Q96.193 31.978 96.244 32.189L96.244 34.845L96.670 34.845Q96.877 34.869 96.916 35.076L96.916 35.166Q96.877 35.381 96.670 35.404L95.854 35.404Q95.654 35.381 95.604 35.181Q95.201 35.443 94.693 35.443Q94.459 35.443 94.244 35.402Q94.029 35.361 93.863 35.259Q93.697 35.158 93.594 34.980Q93.490 34.802 93.490 34.549M97.400 34.291Q97.400 33.845 97.814 33.588Q98.229 33.330 98.770 33.230Q99.311 33.131 99.818 33.123Q99.818 32.908 99.684 32.756Q99.549 32.603 99.342 32.527Q99.135 32.451 98.924 32.451Q98.580 32.451 98.420 32.474L98.420 32.533Q98.420 32.701 98.301 32.816Q98.182 32.931 98.018 32.931Q97.842 32.931 97.727 32.808Q97.611 32.685 97.611 32.517Q97.611 32.111 97.992 32.002Q98.373 31.892 98.932 31.892Q99.201 31.892 99.469 31.970Q99.736 32.049 99.961 32.199Q100.186 32.349 100.322 32.570Q100.459 32.791 100.459 33.068L100.459 34.787Q100.459 34.845 100.986 34.845Q101.182 34.865 101.232 35.076L101.232 35.166Q101.182 35.381 100.986 35.404L100.842 35.404Q100.498 35.404 100.270 35.357Q100.041 35.310 99.897 35.123Q99.436 35.443 98.729 35.443Q98.393 35.443 98.088 35.302Q97.783 35.162 97.592 34.900Q97.400 34.638 97.400 34.291M98.041 34.299Q98.041 34.572 98.283 34.728Q98.525 34.884 98.811 34.884Q99.029 34.884 99.262 34.826Q99.494 34.767 99.656 34.629Q99.818 34.490 99.818 34.267L99.818 33.677Q99.537 33.677 99.121 33.734Q98.705 33.791 98.373 33.929Q98.041 34.068 98.041 34.299M103.104 35.443Q102.639 35.443 102.273 35.193Q101.908 34.943 101.703 34.539Q101.498 34.134 101.498 33.677Q101.498 33.334 101.623 33.013Q101.748 32.693 101.981 32.443Q102.213 32.193 102.518 32.054Q102.822 31.916 103.178 31.916Q103.689 31.916 104.096 32.236L104.096 31.076L103.674 31.076Q103.463 31.052 103.424 30.838L103.424 30.748Q103.463 30.541 103.674 30.517L104.486 30.517Q104.686 30.541 104.736 30.748L104.736 34.845L105.162 34.845Q105.369 34.869 105.408 35.076L105.408 35.166Q105.369 35.381 105.162 35.404L104.346 35.404Q104.147 35.381 104.096 35.166L104.096 35.037Q103.900 35.228 103.639 35.336Q103.377 35.443 103.104 35.443M103.143 34.884Q103.502 34.884 103.754 34.615Q104.006 34.345 104.096 33.970L104.096 33.138Q104.037 32.951 103.910 32.800Q103.783 32.650 103.606 32.562Q103.428 32.474 103.232 32.474Q102.924 32.474 102.674 32.644Q102.424 32.814 102.279 33.099Q102.135 33.384 102.135 33.685Q102.135 34.134 102.420 34.509Q102.705 34.884 103.143 34.884M105.709 35.166L105.709 35.076Q105.768 34.869 105.959 34.845L106.670 34.845L106.670 32.517L105.959 32.517Q105.764 32.494 105.709 32.275L105.709 32.189Q105.768 31.978 105.959 31.955L107.061 31.955Q107.260 31.974 107.311 32.189L107.311 32.517Q107.572 32.232 107.928 32.074Q108.283 31.916 108.670 31.916Q108.963 31.916 109.197 32.050Q109.432 32.185 109.432 32.451Q109.432 32.619 109.322 32.736Q109.213 32.853 109.045 32.853Q108.893 32.853 108.777 32.742Q108.662 32.631 108.662 32.474Q108.287 32.474 107.973 32.675Q107.658 32.877 107.484 33.211Q107.311 33.545 107.311 33.924L107.311 34.845L108.256 34.845Q108.463 34.869 108.502 35.076L108.502 35.166Q108.463 35.381 108.256 35.404L105.959 35.404Q105.768 35.381 105.709 35.166M110.139 34.291Q110.139 33.845 110.553 33.588Q110.967 33.330 111.508 33.230Q112.049 33.131 112.557 33.123Q112.557 32.908 112.422 32.756Q112.287 32.603 112.080 32.527Q111.873 32.451 111.662 32.451Q111.318 32.451 111.158 32.474L111.158 32.533Q111.158 32.701 111.039 32.816Q110.920 32.931 110.756 32.931Q110.580 32.931 110.465 32.808Q110.350 32.685 110.350 32.517Q110.350 32.111 110.731 32.002Q111.111 31.892 111.670 31.892Q111.939 31.892 112.207 31.970Q112.475 32.049 112.699 32.199Q112.924 32.349 113.061 32.570Q113.197 32.791 113.197 33.068L113.197 34.787Q113.197 34.845 113.725 34.845Q113.920 34.865 113.971 35.076L113.971 35.166Q113.920 35.381 113.725 35.404L113.580 35.404Q113.236 35.404 113.008 35.357Q112.779 35.310 112.635 35.123Q112.174 35.443 111.467 35.443Q111.131 35.443 110.826 35.302Q110.522 35.162 110.330 34.900Q110.139 34.638 110.139 34.291M110.779 34.299Q110.779 34.572 111.022 34.728Q111.264 34.884 111.549 34.884Q111.768 34.884 112 34.826Q112.232 34.767 112.395 34.629Q112.557 34.490 112.557 34.267L112.557 33.677Q112.275 33.677 111.859 33.734Q111.443 33.791 111.111 33.929Q110.779 34.068 110.779 34.299M115.178 34.299L115.178 32.517L114.428 32.517Q114.229 32.494 114.178 32.275L114.178 32.189Q114.229 31.978 114.428 31.955L115.178 31.955L115.178 31.205Q115.229 30.998 115.428 30.970L115.572 30.970Q115.768 30.998 115.818 31.205L115.818 31.955L117.178 31.955Q117.369 31.974 117.428 32.189L117.428 32.275Q117.373 32.494 117.178 32.517L115.818 32.517L115.818 34.267Q115.818 34.884 116.393 34.884Q116.643 34.884 116.807 34.699Q116.971 34.513 116.971 34.267Q116.971 34.174 117.043 34.103Q117.115 34.033 117.217 34.021L117.361 34.021Q117.561 34.045 117.611 34.252L117.611 34.299Q117.611 34.623 117.428 34.886Q117.244 35.150 116.951 35.297Q116.658 35.443 116.338 35.443Q115.826 35.443 115.502 35.133Q115.178 34.822 115.178 34.299M118.842 35.166L118.842 35.076Q118.893 34.869 119.088 34.845L120.127 34.845L120.127 32.517L119.154 32.517Q118.955 32.494 118.904 32.275L118.904 32.189Q118.955 31.978 119.154 31.955L120.522 31.955Q120.717 31.974 120.768 32.189L120.768 34.845L121.682 34.845Q121.877 34.869 121.928 35.076L121.928 35.166Q121.877 35.381 121.682 35.404L119.088 35.404Q118.893 35.381 118.842 35.166M119.873 30.978L119.873 30.924Q119.873 30.752 120.010 30.631Q120.147 30.509 120.322 30.509Q120.494 30.509 120.631 30.631Q120.768 30.752 120.768 30.924L120.768 30.978Q120.768 31.154 120.631 31.275Q120.494 31.396 120.322 31.396Q120.147 31.396 120.010 31.275Q119.873 31.154 119.873 30.978M123.029 33.677Q123.029 33.197 123.273 32.783Q123.518 32.369 123.934 32.131Q124.350 31.892 124.830 31.892Q125.385 31.892 125.764 32.002Q126.143 32.111 126.143 32.517Q126.143 32.685 126.029 32.808Q125.916 32.931 125.744 32.931Q125.572 32.931 125.453 32.816Q125.334 32.701 125.334 32.533L125.334 32.474Q125.174 32.451 124.838 32.451Q124.510 32.451 124.242 32.621Q123.975 32.791 123.822 33.074Q123.670 33.357 123.670 33.677Q123.670 33.998 123.842 34.279Q124.014 34.560 124.299 34.722Q124.584 34.884 124.912 34.884Q125.225 34.884 125.352 34.781Q125.479 34.677 125.596 34.488Q125.713 34.299 125.830 34.283L125.998 34.283Q126.104 34.295 126.168 34.363Q126.232 34.431 126.232 34.533Q126.232 34.580 126.213 34.619Q126.104 34.912 125.900 35.092Q125.697 35.271 125.422 35.357Q125.147 35.443 124.830 35.443Q124.346 35.443 123.930 35.205Q123.514 34.967 123.272 34.564Q123.029 34.162 123.029 33.677\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-144.106 -64.133)\">\u003Cpath d=\"M130.948 35.166L130.948 35.076Q130.999 34.865 131.194 34.845L131.394 34.845L131.394 32.517L131.194 32.517Q130.987 32.494 130.948 32.275L130.948 32.189Q130.999 31.974 131.194 31.955L131.675 31.955Q131.866 31.978 131.925 32.189Q132.245 31.916 132.659 31.916Q132.851 31.916 133.019 32.025Q133.187 32.134 133.261 32.306Q133.437 32.115 133.659 32.015Q133.882 31.916 134.124 31.916Q134.542 31.916 134.696 32.258Q134.851 32.599 134.851 33.068L134.851 34.845L135.050 34.845Q135.261 34.869 135.300 35.076L135.300 35.166Q135.249 35.381 135.050 35.404L134.233 35.404Q134.038 35.381 133.987 35.166L133.987 35.076Q134.038 34.869 134.233 34.845L134.323 34.845L134.323 33.099Q134.323 32.474 134.073 32.474Q133.741 32.474 133.564 32.785Q133.386 33.095 133.386 33.459L133.386 34.845L133.589 34.845Q133.796 34.869 133.835 35.076L133.835 35.166Q133.784 35.381 133.589 35.404L132.773 35.404Q132.573 35.381 132.523 35.166L132.523 35.076Q132.573 34.869 132.773 34.845L132.858 34.845L132.858 33.099Q132.858 32.474 132.612 32.474Q132.280 32.474 132.103 32.787Q131.925 33.099 131.925 33.459L131.925 34.845L132.124 34.845Q132.331 34.869 132.370 35.076L132.370 35.166Q132.319 35.384 132.124 35.404L131.194 35.404Q130.987 35.381 130.948 35.166M137.370 35.443Q136.898 35.443 136.513 35.199Q136.128 34.955 135.905 34.545Q135.683 34.134 135.683 33.677Q135.683 33.334 135.808 33.011Q135.933 32.689 136.163 32.435Q136.394 32.181 136.700 32.037Q137.007 31.892 137.370 31.892Q137.733 31.892 138.046 32.039Q138.358 32.185 138.581 32.431Q138.804 32.677 138.931 32.998Q139.058 33.318 139.058 33.677Q139.058 34.134 138.833 34.547Q138.608 34.959 138.224 35.201Q137.839 35.443 137.370 35.443M137.370 34.884Q137.835 34.884 138.126 34.490Q138.417 34.095 138.417 33.611Q138.417 33.318 138.282 33.050Q138.148 32.783 137.907 32.617Q137.667 32.451 137.370 32.451Q137.065 32.451 136.827 32.617Q136.589 32.783 136.454 33.050Q136.319 33.318 136.319 33.611Q136.319 34.092 136.612 34.488Q136.905 34.884 137.370 34.884M141.358 35.443Q140.894 35.443 140.528 35.193Q140.163 34.943 139.958 34.539Q139.753 34.134 139.753 33.677Q139.753 33.334 139.878 33.013Q140.003 32.693 140.235 32.443Q140.468 32.193 140.773 32.054Q141.077 31.916 141.433 31.916Q141.944 31.916 142.351 32.236L142.351 31.076L141.929 31.076Q141.718 31.052 141.679 30.838L141.679 30.748Q141.718 30.541 141.929 30.517L142.741 30.517Q142.940 30.541 142.991 30.748L142.991 34.845L143.417 34.845Q143.624 34.869 143.663 35.076L143.663 35.166Q143.624 35.381 143.417 35.404L142.601 35.404Q142.401 35.381 142.351 35.166L142.351 35.037Q142.155 35.228 141.894 35.336Q141.632 35.443 141.358 35.443M141.398 34.884Q141.757 34.884 142.009 34.615Q142.261 34.345 142.351 33.970L142.351 33.138Q142.292 32.951 142.165 32.800Q142.038 32.650 141.860 32.562Q141.683 32.474 141.487 32.474Q141.179 32.474 140.929 32.644Q140.679 32.814 140.534 33.099Q140.390 33.384 140.390 33.685Q140.390 34.134 140.675 34.509Q140.960 34.884 141.398 34.884M147.253 33.916L144.812 33.916Q144.866 34.193 145.064 34.416Q145.261 34.638 145.538 34.761Q145.815 34.884 146.101 34.884Q146.573 34.884 146.796 34.595Q146.804 34.584 146.860 34.478Q146.917 34.373 146.966 34.330Q147.015 34.287 147.108 34.275L147.253 34.275Q147.444 34.295 147.503 34.509L147.503 34.564Q147.437 34.865 147.206 35.062Q146.976 35.259 146.663 35.351Q146.351 35.443 146.046 35.443Q145.562 35.443 145.122 35.215Q144.683 34.986 144.415 34.586Q144.148 34.185 144.148 33.693L144.148 33.634Q144.148 33.166 144.394 32.763Q144.640 32.361 145.048 32.127Q145.456 31.892 145.925 31.892Q146.429 31.892 146.782 32.115Q147.136 32.338 147.319 32.726Q147.503 33.115 147.503 33.619L147.503 33.677Q147.444 33.892 147.253 33.916M144.819 33.365L146.847 33.365Q146.800 32.955 146.562 32.703Q146.323 32.451 145.925 32.451Q145.530 32.451 145.224 32.713Q144.917 32.974 144.819 33.365M148.437 35.166L148.437 35.076Q148.487 34.869 148.683 34.845L149.788 34.845L149.788 31.076L148.683 31.076Q148.487 31.052 148.437 30.838L148.437 30.748Q148.487 30.541 148.683 30.517L150.179 30.517Q150.370 30.541 150.429 30.748L150.429 34.845L151.530 34.845Q151.730 34.869 151.780 35.076L151.780 35.166Q151.730 35.381 151.530 35.404L148.683 35.404Q148.487 35.381 148.437 35.166\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-good)\" stroke=\"none\" d=\"M66.517 23.454a2.2 2.2 0 1 0-4.4 0 2.2 2.2 0 0 0 4.4 0m-2.2 0\"\u002F>\u003Cpath fill=\"none\" stroke=\"var(--tk-good)\" d=\"M64.317 23.454 34.442 51.053\" style=\"stroke-dasharray:3.0,2.0\"\u002F>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\">\u003Cg fill=\"var(--tk-good)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(-95.844 18.094)\">\u003Cpath d=\"M88.447 35.166L88.447 35.076Q88.498 34.865 88.693 34.845L88.893 34.845L88.893 32.517L88.693 32.517Q88.486 32.494 88.447 32.275L88.447 32.189Q88.498 31.974 88.693 31.955L89.174 31.955Q89.365 31.978 89.424 32.189Q89.744 31.916 90.158 31.916Q90.350 31.916 90.518 32.025Q90.686 32.134 90.760 32.306Q90.936 32.115 91.158 32.015Q91.381 31.916 91.623 31.916Q92.041 31.916 92.195 32.258Q92.350 32.599 92.350 33.068L92.350 34.845L92.549 34.845Q92.760 34.869 92.799 35.076L92.799 35.166Q92.748 35.381 92.549 35.404L91.732 35.404Q91.537 35.381 91.486 35.166L91.486 35.076Q91.537 34.869 91.732 34.845L91.822 34.845L91.822 33.099Q91.822 32.474 91.572 32.474Q91.240 32.474 91.063 32.785Q90.885 33.095 90.885 33.459L90.885 34.845L91.088 34.845Q91.295 34.869 91.334 35.076L91.334 35.166Q91.283 35.381 91.088 35.404L90.272 35.404Q90.072 35.381 90.022 35.166L90.022 35.076Q90.072 34.869 90.272 34.845L90.357 34.845L90.357 33.099Q90.357 32.474 90.111 32.474Q89.779 32.474 89.602 32.787Q89.424 33.099 89.424 33.459L89.424 34.845L89.623 34.845Q89.830 34.869 89.869 35.076L89.869 35.166Q89.818 35.384 89.623 35.404L88.693 35.404Q88.486 35.381 88.447 35.166M94.869 35.443Q94.397 35.443 94.012 35.199Q93.627 34.955 93.404 34.545Q93.182 34.134 93.182 33.677Q93.182 33.334 93.307 33.011Q93.432 32.689 93.662 32.435Q93.893 32.181 94.199 32.037Q94.506 31.892 94.869 31.892Q95.232 31.892 95.545 32.039Q95.857 32.185 96.080 32.431Q96.303 32.677 96.430 32.998Q96.557 33.318 96.557 33.677Q96.557 34.134 96.332 34.547Q96.107 34.959 95.723 35.201Q95.338 35.443 94.869 35.443M94.869 34.884Q95.334 34.884 95.625 34.490Q95.916 34.095 95.916 33.611Q95.916 33.318 95.781 33.050Q95.647 32.783 95.406 32.617Q95.166 32.451 94.869 32.451Q94.564 32.451 94.326 32.617Q94.088 32.783 93.953 33.050Q93.818 33.318 93.818 33.611Q93.818 34.092 94.111 34.488Q94.404 34.884 94.869 34.884M98.857 35.443Q98.393 35.443 98.027 35.193Q97.662 34.943 97.457 34.539Q97.252 34.134 97.252 33.677Q97.252 33.334 97.377 33.013Q97.502 32.693 97.734 32.443Q97.967 32.193 98.272 32.054Q98.576 31.916 98.932 31.916Q99.443 31.916 99.850 32.236L99.850 31.076L99.428 31.076Q99.217 31.052 99.178 30.838L99.178 30.748Q99.217 30.541 99.428 30.517L100.240 30.517Q100.439 30.541 100.490 30.748L100.490 34.845L100.916 34.845Q101.123 34.869 101.162 35.076L101.162 35.166Q101.123 35.381 100.916 35.404L100.100 35.404Q99.900 35.381 99.850 35.166L99.850 35.037Q99.654 35.228 99.393 35.336Q99.131 35.443 98.857 35.443M98.897 34.884Q99.256 34.884 99.508 34.615Q99.760 34.345 99.850 33.970L99.850 33.138Q99.791 32.951 99.664 32.800Q99.537 32.650 99.359 32.562Q99.182 32.474 98.986 32.474Q98.678 32.474 98.428 32.644Q98.178 32.814 98.033 33.099Q97.889 33.384 97.889 33.685Q97.889 34.134 98.174 34.509Q98.459 34.884 98.897 34.884M104.752 33.916L102.311 33.916Q102.365 34.193 102.563 34.416Q102.760 34.638 103.037 34.761Q103.314 34.884 103.600 34.884Q104.072 34.884 104.295 34.595Q104.303 34.584 104.359 34.478Q104.416 34.373 104.465 34.330Q104.514 34.287 104.607 34.275L104.752 34.275Q104.943 34.295 105.002 34.509L105.002 34.564Q104.936 34.865 104.705 35.062Q104.475 35.259 104.162 35.351Q103.850 35.443 103.545 35.443Q103.061 35.443 102.621 35.215Q102.182 34.986 101.914 34.586Q101.647 34.185 101.647 33.693L101.647 33.634Q101.647 33.166 101.893 32.763Q102.139 32.361 102.547 32.127Q102.955 31.892 103.424 31.892Q103.928 31.892 104.281 32.115Q104.635 32.338 104.818 32.726Q105.002 33.115 105.002 33.619L105.002 33.677Q104.943 33.892 104.752 33.916M102.318 33.365L104.346 33.365Q104.299 32.955 104.061 32.703Q103.822 32.451 103.424 32.451Q103.029 32.451 102.723 32.713Q102.416 32.974 102.318 33.365M105.936 35.166L105.936 35.076Q105.986 34.869 106.182 34.845L107.287 34.845L107.287 31.076L106.182 31.076Q105.986 31.052 105.936 30.838L105.936 30.748Q105.986 30.541 106.182 30.517L107.678 30.517Q107.869 30.541 107.928 30.748L107.928 34.845L109.029 34.845Q109.229 34.869 109.279 35.076L109.279 35.166Q109.229 35.381 109.029 35.404L106.182 35.404Q105.986 35.381 105.936 35.166\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-95.844 18.094)\">\u003Cpath d=\"M113.948 35.166L113.948 35.076Q113.999 34.865 114.194 34.845L114.394 34.845L114.394 32.517L114.194 32.517Q113.987 32.494 113.948 32.275L113.948 32.189Q113.999 31.974 114.194 31.955L114.675 31.955Q114.866 31.978 114.925 32.189Q115.245 31.916 115.659 31.916Q115.851 31.916 116.019 32.025Q116.187 32.134 116.261 32.306Q116.437 32.115 116.659 32.015Q116.882 31.916 117.124 31.916Q117.542 31.916 117.696 32.258Q117.851 32.599 117.851 33.068L117.851 34.845L118.050 34.845Q118.261 34.869 118.300 35.076L118.300 35.166Q118.249 35.381 118.050 35.404L117.233 35.404Q117.038 35.381 116.987 35.166L116.987 35.076Q117.038 34.869 117.233 34.845L117.323 34.845L117.323 33.099Q117.323 32.474 117.073 32.474Q116.741 32.474 116.564 32.785Q116.386 33.095 116.386 33.459L116.386 34.845L116.589 34.845Q116.796 34.869 116.835 35.076L116.835 35.166Q116.784 35.381 116.589 35.404L115.773 35.404Q115.573 35.381 115.523 35.166L115.523 35.076Q115.573 34.869 115.773 34.845L115.858 34.845L115.858 33.099Q115.858 32.474 115.612 32.474Q115.280 32.474 115.103 32.787Q114.925 33.099 114.925 33.459L114.925 34.845L115.124 34.845Q115.331 34.869 115.370 35.076L115.370 35.166Q115.319 35.384 115.124 35.404L114.194 35.404Q113.987 35.381 113.948 35.166M118.866 35.166L118.866 35.076Q118.917 34.869 119.112 34.845L120.151 34.845L120.151 32.517L119.179 32.517Q118.980 32.494 118.929 32.275L118.929 32.189Q118.980 31.978 119.179 31.955L120.546 31.955Q120.741 31.974 120.792 32.189L120.792 34.845L121.706 34.845Q121.901 34.869 121.952 35.076L121.952 35.166Q121.901 35.381 121.706 35.404L119.112 35.404Q118.917 35.381 118.866 35.166M119.898 30.978L119.898 30.924Q119.898 30.752 120.034 30.631Q120.171 30.509 120.347 30.509Q120.519 30.509 120.655 30.631Q120.792 30.752 120.792 30.924L120.792 30.978Q120.792 31.154 120.655 31.275Q120.519 31.396 120.347 31.396Q120.171 31.396 120.034 31.275Q119.898 31.154 119.898 30.978M122.566 35.166L122.566 35.076Q122.608 34.869 122.816 34.845L123.237 34.845L123.237 32.517L122.816 32.517Q122.608 32.494 122.566 32.275L122.566 32.189Q122.612 31.978 122.816 31.955L123.632 31.955Q123.827 31.978 123.878 32.189L123.878 32.275L123.870 32.299Q124.097 32.119 124.370 32.017Q124.644 31.916 124.937 31.916Q125.284 31.916 125.523 32.056Q125.761 32.197 125.876 32.455Q125.991 32.713 125.991 33.068L125.991 34.845L126.417 34.845Q126.624 34.869 126.663 35.076L126.663 35.166Q126.624 35.381 126.417 35.404L125.023 35.404Q124.827 35.381 124.776 35.166L124.776 35.076Q124.827 34.865 125.023 34.845L125.351 34.845L125.351 33.099Q125.351 32.791 125.261 32.633Q125.171 32.474 124.878 32.474Q124.608 32.474 124.380 32.605Q124.151 32.736 124.015 32.965Q123.878 33.193 123.878 33.459L123.878 34.845L124.304 34.845Q124.511 34.869 124.550 35.076L124.550 35.166Q124.511 35.381 124.304 35.404L122.816 35.404Q122.608 35.381 122.566 35.166\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M31.597-8.698h28.698\"\u002F>\u003Cpath stroke=\"none\" d=\"m62.895-8.698-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-49.757 -49.413)\">\u003Cpath d=\"M89.088 35.205L89.088 34.291Q89.115 34.084 89.326 34.060L89.494 34.060Q89.658 34.084 89.717 34.244Q89.920 34.884 90.647 34.884Q90.854 34.884 91.082 34.849Q91.311 34.814 91.479 34.699Q91.647 34.584 91.647 34.381Q91.647 34.170 91.424 34.056Q91.201 33.943 90.928 33.900L90.229 33.787Q89.088 33.576 89.088 32.853Q89.088 32.564 89.232 32.375Q89.377 32.185 89.617 32.078Q89.857 31.970 90.113 31.931Q90.369 31.892 90.647 31.892Q90.897 31.892 91.090 31.922Q91.283 31.951 91.447 32.029Q91.525 31.912 91.654 31.892L91.732 31.892Q91.830 31.904 91.893 31.967Q91.955 32.029 91.967 32.123L91.967 32.830Q91.955 32.924 91.893 32.990Q91.830 33.056 91.732 33.068L91.564 33.068Q91.471 33.056 91.404 32.990Q91.338 32.924 91.326 32.830Q91.326 32.451 90.631 32.451Q90.283 32.451 89.965 32.533Q89.647 32.615 89.647 32.861Q89.647 33.127 90.318 33.236L91.022 33.357Q91.506 33.439 91.856 33.687Q92.205 33.935 92.205 34.381Q92.205 34.771 91.969 35.013Q91.732 35.256 91.383 35.349Q91.033 35.443 90.647 35.443Q90.068 35.443 89.670 35.189Q89.600 35.314 89.551 35.371Q89.502 35.427 89.397 35.443L89.326 35.443Q89.111 35.420 89.088 35.205M93.947 34.299L93.947 32.517L93.197 32.517Q92.998 32.494 92.947 32.275L92.947 32.189Q92.998 31.978 93.197 31.955L93.947 31.955L93.947 31.205Q93.998 30.998 94.197 30.970L94.342 30.970Q94.537 30.998 94.588 31.205L94.588 31.955L95.947 31.955Q96.139 31.974 96.197 32.189L96.197 32.275Q96.143 32.494 95.947 32.517L94.588 32.517L94.588 34.267Q94.588 34.884 95.162 34.884Q95.412 34.884 95.576 34.699Q95.740 34.513 95.740 34.267Q95.740 34.174 95.813 34.103Q95.885 34.033 95.986 34.021L96.131 34.021Q96.330 34.045 96.381 34.252L96.381 34.299Q96.381 34.623 96.197 34.886Q96.014 35.150 95.721 35.297Q95.428 35.443 95.107 35.443Q94.596 35.443 94.272 35.133Q93.947 34.822 93.947 34.299M100.506 33.916L98.064 33.916Q98.119 34.193 98.316 34.416Q98.514 34.638 98.791 34.761Q99.068 34.884 99.354 34.884Q99.826 34.884 100.049 34.595Q100.057 34.584 100.113 34.478Q100.170 34.373 100.219 34.330Q100.268 34.287 100.361 34.275L100.506 34.275Q100.697 34.295 100.756 34.509L100.756 34.564Q100.689 34.865 100.459 35.062Q100.229 35.259 99.916 35.351Q99.604 35.443 99.299 35.443Q98.814 35.443 98.375 35.215Q97.936 34.986 97.668 34.586Q97.400 34.185 97.400 33.693L97.400 33.634Q97.400 33.166 97.647 32.763Q97.893 32.361 98.301 32.127Q98.709 31.892 99.178 31.892Q99.682 31.892 100.035 32.115Q100.389 32.338 100.572 32.726Q100.756 33.115 100.756 33.619L100.756 33.677Q100.697 33.892 100.506 33.916M98.072 33.365L100.100 33.365Q100.053 32.955 99.814 32.703Q99.576 32.451 99.178 32.451Q98.783 32.451 98.477 32.713Q98.170 32.974 98.072 33.365M101.311 36.947L101.311 36.861Q101.354 36.642 101.561 36.619L101.982 36.619L101.982 32.517L101.561 32.517Q101.354 32.494 101.311 32.275L101.311 32.189Q101.357 31.978 101.561 31.955L102.377 31.955Q102.572 31.974 102.623 32.189L102.623 32.259Q102.834 32.092 103.098 32.004Q103.361 31.916 103.631 31.916Q103.971 31.916 104.268 32.060Q104.564 32.205 104.779 32.457Q104.994 32.709 105.109 33.021Q105.225 33.334 105.225 33.677Q105.225 34.142 104.998 34.552Q104.772 34.963 104.385 35.203Q103.998 35.443 103.529 35.443Q103.010 35.443 102.623 35.076L102.623 36.619L103.049 36.619Q103.256 36.642 103.295 36.861L103.295 36.947Q103.256 37.158 103.049 37.181L101.561 37.181Q101.357 37.158 101.311 36.947M103.486 34.884Q103.795 34.884 104.045 34.715Q104.295 34.545 104.439 34.263Q104.584 33.982 104.584 33.677Q104.584 33.388 104.459 33.109Q104.334 32.830 104.102 32.652Q103.869 32.474 103.568 32.474Q103.248 32.474 102.986 32.660Q102.725 32.845 102.623 33.146L102.623 33.998Q102.713 34.365 102.934 34.625Q103.154 34.884 103.486 34.884\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">First order fits a tangent plane (no bottom, needs a learning rate); second order fits a bowl \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7667em;vertical-align:-0.0833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathcal\">L\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0435em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8491em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">Δ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1941em;vertical-align:-0.345em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8451em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.394em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.345em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">Δ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8491em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0813em;\">H\u003C\u002Fspan>\u003Cspan class=\"mord\">Δ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> whose minimum \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6887em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">Δ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mbin mtight\">⋆\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0085em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">−\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0813em;\">H\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">−\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> sets both the direction and the length of the step.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:405.681px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 304.261 146.876\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" d=\"M111.98-1.832c0-5.657-12.739-10.243-28.453-10.243S55.074-7.489 55.074-1.832 67.813 8.411 83.527 8.411 111.98 3.825 111.98-1.832Zm-28.453 0\"\u002F>\u003Cpath fill=\"none\" d=\"M140.433-1.832c0-11.314-25.478-20.486-56.906-20.486S26.622-13.146 26.622-1.832s25.477 20.486 56.905 20.486 56.906-9.172 56.906-20.486Zm-56.906 0\"\u002F>\u003Cpath fill=\"none\" d=\"M168.885-1.832c0-16.971-38.215-30.729-85.358-30.729S-1.831-18.803-1.831-1.83c0 16.97 38.215 30.728 85.358 30.728s85.358-13.758 85.358-30.729Zm-85.358 0\"\u002F>\u003Cpath fill=\"none\" d=\"M197.338-1.832c0-22.628-50.954-40.972-113.811-40.972S-30.283-24.46-30.283-1.832c0 22.629 50.953 40.972 113.81 40.972S197.338 20.797 197.338-1.832Zm-113.811 0\"\u002F>\u003Cpath fill=\"none\" d=\"M225.791-1.832c0-28.286-63.693-51.215-142.264-51.215S-58.737-30.117-58.737-1.832 4.956 49.383 83.527 49.383s142.264-22.93 142.264-51.215Zm-142.264 0\"\u002F>\u003Cpath fill=\"var(--tk-good)\" stroke=\"none\" d=\"M86.127-1.832a2.6 2.6 0 1 0-5.2 0 2.6 2.6 0 0 0 5.2 0m-2.6 0\"\u002F>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\">\u003Cg transform=\"translate(27.804 62.482)\">\u003Cpath d=\"M83.472-2.070L83.472-2.160Q83.523-2.371 83.718-2.391L83.918-2.391L83.918-4.719L83.718-4.719Q83.511-4.742 83.472-4.961L83.472-5.047Q83.523-5.262 83.718-5.281L84.199-5.281Q84.390-5.258 84.449-5.047Q84.769-5.320 85.183-5.320Q85.375-5.320 85.543-5.211Q85.711-5.102 85.785-4.930Q85.961-5.121 86.183-5.221Q86.406-5.320 86.648-5.320Q87.066-5.320 87.220-4.978Q87.375-4.637 87.375-4.168L87.375-2.391L87.574-2.391Q87.785-2.367 87.824-2.160L87.824-2.070Q87.773-1.855 87.574-1.832L86.757-1.832Q86.562-1.855 86.511-2.070L86.511-2.160Q86.562-2.367 86.757-2.391L86.847-2.391L86.847-4.137Q86.847-4.762 86.597-4.762Q86.265-4.762 86.088-4.451Q85.910-4.141 85.910-3.777L85.910-2.391L86.113-2.391Q86.320-2.367 86.359-2.160L86.359-2.070Q86.308-1.855 86.113-1.832L85.297-1.832Q85.097-1.855 85.047-2.070L85.047-2.160Q85.097-2.367 85.297-2.391L85.382-2.391L85.382-4.137Q85.382-4.762 85.136-4.762Q84.804-4.762 84.627-4.449Q84.449-4.137 84.449-3.777L84.449-2.391L84.648-2.391Q84.855-2.367 84.894-2.160L84.894-2.070Q84.843-1.852 84.648-1.832L83.718-1.832Q83.511-1.855 83.472-2.070M88.390-2.070L88.390-2.160Q88.441-2.367 88.636-2.391L89.675-2.391L89.675-4.719L88.703-4.719Q88.504-4.742 88.453-4.961L88.453-5.047Q88.504-5.258 88.703-5.281L90.070-5.281Q90.265-5.262 90.316-5.047L90.316-2.391L91.230-2.391Q91.425-2.367 91.476-2.160L91.476-2.070Q91.425-1.855 91.230-1.832L88.636-1.832Q88.441-1.855 88.390-2.070M89.422-6.258L89.422-6.312Q89.422-6.484 89.558-6.605Q89.695-6.727 89.871-6.727Q90.043-6.727 90.179-6.605Q90.316-6.484 90.316-6.312L90.316-6.258Q90.316-6.082 90.179-5.961Q90.043-5.840 89.871-5.840Q89.695-5.840 89.558-5.961Q89.422-6.082 89.422-6.258M92.090-2.070L92.090-2.160Q92.132-2.367 92.340-2.391L92.761-2.391L92.761-4.719L92.340-4.719Q92.132-4.742 92.090-4.961L92.090-5.047Q92.136-5.258 92.340-5.281L93.156-5.281Q93.351-5.258 93.402-5.047L93.402-4.961L93.394-4.937Q93.621-5.117 93.894-5.219Q94.168-5.320 94.461-5.320Q94.808-5.320 95.047-5.180Q95.285-5.039 95.400-4.781Q95.515-4.523 95.515-4.168L95.515-2.391L95.941-2.391Q96.148-2.367 96.187-2.160L96.187-2.070Q96.148-1.855 95.941-1.832L94.547-1.832Q94.351-1.855 94.300-2.070L94.300-2.160Q94.351-2.371 94.547-2.391L94.875-2.391L94.875-4.137Q94.875-4.445 94.785-4.603Q94.695-4.762 94.402-4.762Q94.132-4.762 93.904-4.631Q93.675-4.500 93.539-4.271Q93.402-4.043 93.402-3.777L93.402-2.391L93.828-2.391Q94.035-2.367 94.074-2.160L94.074-2.070Q94.035-1.855 93.828-1.832L92.340-1.832Q92.132-1.855 92.090-2.070M96.882-2.070L96.882-2.160Q96.933-2.367 97.129-2.391L98.168-2.391L98.168-4.719L97.195-4.719Q96.996-4.742 96.945-4.961L96.945-5.047Q96.996-5.258 97.195-5.281L98.562-5.281Q98.757-5.262 98.808-5.047L98.808-2.391L99.722-2.391Q99.918-2.367 99.968-2.160L99.968-2.070Q99.918-1.855 99.722-1.832L97.129-1.832Q96.933-1.855 96.882-2.070M97.914-6.258L97.914-6.312Q97.914-6.484 98.050-6.605Q98.187-6.727 98.363-6.727Q98.535-6.727 98.672-6.605Q98.808-6.484 98.808-6.312L98.808-6.258Q98.808-6.082 98.672-5.961Q98.535-5.840 98.363-5.840Q98.187-5.840 98.050-5.961Q97.914-6.082 97.914-6.258M100.457-2.070L100.457-2.160Q100.507-2.371 100.703-2.391L100.902-2.391L100.902-4.719L100.703-4.719Q100.496-4.742 100.457-4.961L100.457-5.047Q100.507-5.262 100.703-5.281L101.183-5.281Q101.375-5.258 101.433-5.047Q101.754-5.320 102.168-5.320Q102.359-5.320 102.527-5.211Q102.695-5.102 102.769-4.930Q102.945-5.121 103.168-5.221Q103.390-5.320 103.632-5.320Q104.050-5.320 104.205-4.978Q104.359-4.637 104.359-4.168L104.359-2.391L104.558-2.391Q104.769-2.367 104.808-2.160L104.808-2.070Q104.757-1.855 104.558-1.832L103.742-1.832Q103.547-1.855 103.496-2.070L103.496-2.160Q103.547-2.367 103.742-2.391L103.832-2.391L103.832-4.137Q103.832-4.762 103.582-4.762Q103.250-4.762 103.072-4.451Q102.894-4.141 102.894-3.777L102.894-2.391L103.097-2.391Q103.304-2.367 103.343-2.160L103.343-2.070Q103.293-1.855 103.097-1.832L102.281-1.832Q102.082-1.855 102.031-2.070L102.031-2.160Q102.082-2.367 102.281-2.391L102.367-2.391L102.367-4.137Q102.367-4.762 102.121-4.762Q101.789-4.762 101.611-4.449Q101.433-4.137 101.433-3.777L101.433-2.391L101.632-2.391Q101.840-2.367 101.879-2.160L101.879-2.070Q101.828-1.852 101.632-1.832L100.703-1.832Q100.496-1.855 100.457-2.070M105.500-2.687L105.500-4.719L105.078-4.719Q104.871-4.742 104.828-4.961L104.828-5.047Q104.875-5.258 105.078-5.281L105.894-5.281Q106.090-5.258 106.140-5.047L106.140-2.719Q106.140-2.484 106.310-2.418Q106.480-2.352 106.765-2.352Q106.972-2.352 107.168-2.428Q107.363-2.504 107.488-2.654Q107.613-2.805 107.613-3.016L107.613-4.719L107.191-4.719Q106.980-4.742 106.941-4.961L106.941-5.047Q106.980-5.258 107.191-5.281L108.004-5.281Q108.203-5.258 108.254-5.047L108.254-2.391L108.679-2.391Q108.886-2.367 108.925-2.160L108.925-2.070Q108.886-1.855 108.679-1.832L107.863-1.832Q107.664-1.855 107.613-2.055Q107.211-1.793 106.703-1.793Q106.468-1.793 106.254-1.834Q106.039-1.875 105.873-1.977Q105.707-2.078 105.603-2.256Q105.500-2.434 105.500-2.687M108.949-2.070L108.949-2.160Q109-2.371 109.195-2.391L109.394-2.391L109.394-4.719L109.195-4.719Q108.988-4.742 108.949-4.961L108.949-5.047Q109-5.262 109.195-5.281L109.675-5.281Q109.867-5.258 109.925-5.047Q110.246-5.320 110.660-5.320Q110.851-5.320 111.019-5.211Q111.187-5.102 111.261-4.930Q111.437-5.121 111.660-5.221Q111.882-5.320 112.125-5.320Q112.543-5.320 112.697-4.978Q112.851-4.637 112.851-4.168L112.851-2.391L113.050-2.391Q113.261-2.367 113.300-2.160L113.300-2.070Q113.250-1.855 113.050-1.832L112.234-1.832Q112.039-1.855 111.988-2.070L111.988-2.160Q112.039-2.367 112.234-2.391L112.324-2.391L112.324-4.137Q112.324-4.762 112.074-4.762Q111.742-4.762 111.564-4.451Q111.386-4.141 111.386-3.777L111.386-2.391L111.590-2.391Q111.797-2.367 111.836-2.160L111.836-2.070Q111.785-1.855 111.590-1.832L110.773-1.832Q110.574-1.855 110.523-2.070L110.523-2.160Q110.574-2.367 110.773-2.391L110.859-2.391L110.859-4.137Q110.859-4.762 110.613-4.762Q110.281-4.762 110.103-4.449Q109.925-4.137 109.925-3.777L109.925-2.391L110.125-2.391Q110.332-2.367 110.371-2.160L110.371-2.070Q110.320-1.852 110.125-1.832L109.195-1.832Q108.988-1.855 108.949-2.070\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath stroke=\"none\" d=\"M-19.348-40.243a2.4 2.4 0 1 0-4.8 0 2.4 2.4 0 0 0 4.8 0m-2.4 0\"\u002F>\u003Cg transform=\"translate(-134.327 -36.197)\">\u003Cpath d=\"M84.113-2.031L84.113-2.945Q84.140-3.152 84.351-3.176L84.519-3.176Q84.683-3.152 84.742-2.992Q84.945-2.352 85.672-2.352Q85.879-2.352 86.107-2.387Q86.336-2.422 86.504-2.537Q86.672-2.652 86.672-2.855Q86.672-3.066 86.449-3.180Q86.226-3.293 85.953-3.336L85.254-3.449Q84.113-3.660 84.113-4.383Q84.113-4.672 84.257-4.861Q84.402-5.051 84.642-5.158Q84.882-5.266 85.138-5.305Q85.394-5.344 85.672-5.344Q85.922-5.344 86.115-5.314Q86.308-5.285 86.472-5.207Q86.550-5.324 86.679-5.344L86.757-5.344Q86.855-5.332 86.918-5.269Q86.980-5.207 86.992-5.113L86.992-4.406Q86.980-4.312 86.918-4.246Q86.855-4.180 86.757-4.168L86.590-4.168Q86.496-4.180 86.429-4.246Q86.363-4.312 86.351-4.406Q86.351-4.785 85.656-4.785Q85.308-4.785 84.990-4.703Q84.672-4.621 84.672-4.375Q84.672-4.109 85.343-4L86.047-3.879Q86.531-3.797 86.881-3.549Q87.230-3.301 87.230-2.855Q87.230-2.465 86.994-2.223Q86.757-1.980 86.408-1.887Q86.058-1.793 85.672-1.793Q85.093-1.793 84.695-2.047Q84.625-1.922 84.576-1.865Q84.527-1.809 84.422-1.793L84.351-1.793Q84.136-1.816 84.113-2.031M88.972-2.937L88.972-4.719L88.222-4.719Q88.023-4.742 87.972-4.961L87.972-5.047Q88.023-5.258 88.222-5.281L88.972-5.281L88.972-6.031Q89.023-6.238 89.222-6.266L89.367-6.266Q89.562-6.238 89.613-6.031L89.613-5.281L90.972-5.281Q91.164-5.262 91.222-5.047L91.222-4.961Q91.168-4.742 90.972-4.719L89.613-4.719L89.613-2.969Q89.613-2.352 90.187-2.352Q90.437-2.352 90.601-2.537Q90.765-2.723 90.765-2.969Q90.765-3.062 90.838-3.133Q90.910-3.203 91.011-3.215L91.156-3.215Q91.355-3.191 91.406-2.984L91.406-2.937Q91.406-2.613 91.222-2.350Q91.039-2.086 90.746-1.939Q90.453-1.793 90.132-1.793Q89.621-1.793 89.297-2.103Q88.972-2.414 88.972-2.937M92.425-2.945Q92.425-3.391 92.840-3.648Q93.254-3.906 93.795-4.006Q94.336-4.105 94.843-4.113Q94.843-4.328 94.709-4.480Q94.574-4.633 94.367-4.709Q94.160-4.785 93.949-4.785Q93.605-4.785 93.445-4.762L93.445-4.703Q93.445-4.535 93.326-4.420Q93.207-4.305 93.043-4.305Q92.867-4.305 92.752-4.428Q92.636-4.551 92.636-4.719Q92.636-5.125 93.017-5.234Q93.398-5.344 93.957-5.344Q94.226-5.344 94.494-5.266Q94.761-5.187 94.986-5.037Q95.211-4.887 95.347-4.666Q95.484-4.445 95.484-4.168L95.484-2.449Q95.484-2.391 96.011-2.391Q96.207-2.371 96.257-2.160L96.257-2.070Q96.207-1.855 96.011-1.832L95.867-1.832Q95.523-1.832 95.295-1.879Q95.066-1.926 94.922-2.113Q94.461-1.793 93.754-1.793Q93.418-1.793 93.113-1.934Q92.808-2.074 92.617-2.336Q92.425-2.598 92.425-2.945M93.066-2.937Q93.066-2.664 93.308-2.508Q93.550-2.352 93.836-2.352Q94.054-2.352 94.287-2.410Q94.519-2.469 94.681-2.607Q94.843-2.746 94.843-2.969L94.843-3.559Q94.562-3.559 94.146-3.502Q93.730-3.445 93.398-3.307Q93.066-3.168 93.066-2.937M96.488-2.070L96.488-2.160Q96.547-2.367 96.738-2.391L97.449-2.391L97.449-4.719L96.738-4.719Q96.543-4.742 96.488-4.961L96.488-5.047Q96.547-5.258 96.738-5.281L97.840-5.281Q98.039-5.262 98.090-5.047L98.090-4.719Q98.351-5.004 98.707-5.162Q99.062-5.320 99.449-5.320Q99.742-5.320 99.976-5.186Q100.211-5.051 100.211-4.785Q100.211-4.617 100.101-4.500Q99.992-4.383 99.824-4.383Q99.672-4.383 99.556-4.494Q99.441-4.605 99.441-4.762Q99.066-4.762 98.752-4.561Q98.437-4.359 98.263-4.025Q98.090-3.691 98.090-3.312L98.090-2.391L99.035-2.391Q99.242-2.367 99.281-2.160L99.281-2.070Q99.242-1.855 99.035-1.832L96.738-1.832Q96.547-1.855 96.488-2.070M101.711-2.937L101.711-4.719L100.961-4.719Q100.761-4.742 100.711-4.961L100.711-5.047Q100.761-5.258 100.961-5.281L101.711-5.281L101.711-6.031Q101.761-6.238 101.961-6.266L102.105-6.266Q102.300-6.238 102.351-6.031L102.351-5.281L103.711-5.281Q103.902-5.262 103.961-5.047L103.961-4.961Q103.906-4.742 103.711-4.719L102.351-4.719L102.351-2.969Q102.351-2.352 102.925-2.352Q103.175-2.352 103.340-2.537Q103.504-2.723 103.504-2.969Q103.504-3.062 103.576-3.133Q103.648-3.203 103.750-3.215L103.894-3.215Q104.093-3.191 104.144-2.984L104.144-2.937Q104.144-2.613 103.961-2.350Q103.777-2.086 103.484-1.939Q103.191-1.793 102.871-1.793Q102.359-1.793 102.035-2.103Q101.711-2.414 101.711-2.937\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"m-21.748-40.243 33.807 51.387\"\u002F>\u003Cpath stroke=\"none\" d=\"m13.818 13.817-.675-5.684-1.084 3.011-3.194-.197\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"m13.818 13.817 23.814-35.192\"\u002F>\u003Cpath stroke=\"none\" d=\"m39.425-24.025-4.99 2.806 3.197-.156 1.044 3.025\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"M39.425-24.025 55.066 7.256\"\u002F>\u003Cpath stroke=\"none\" d=\"M56.497 10.118V4.394l-1.43 2.862-3.15-.572\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"M56.497 10.118 67.638-8.203\"\u002F>\u003Cpath stroke=\"none\" d=\"m69.3-10.937-4.847 3.044 3.185-.31 1.19 2.97\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(-95.805 65.327)\">\u003Cpath d=\"M83.765-1.184Q83.765-1.484 83.914-1.746Q84.062-2.008 84.312-2.168Q84.129-2.422 84.129-2.742Q84.129-3.027 84.281-3.289Q84.031-3.613 84.031-4.031Q84.031-4.391 84.222-4.687Q84.414-4.984 84.732-5.152Q85.050-5.320 85.414-5.320Q85.621-5.320 85.824-5.260Q86.027-5.199 86.191-5.098Q86.574-5.359 87.047-5.359Q87.285-5.359 87.466-5.228Q87.648-5.098 87.648-4.871Q87.648-4.723 87.547-4.617Q87.445-4.512 87.289-4.512Q87.156-4.512 87.060-4.590Q86.965-4.668 86.933-4.793Q86.789-4.785 86.590-4.695Q86.793-4.383 86.793-4.031Q86.793-3.750 86.681-3.518Q86.570-3.285 86.375-3.107Q86.179-2.930 85.927-2.832Q85.675-2.734 85.414-2.734Q85.027-2.734 84.695-2.922Q84.683-2.922 84.673-2.850Q84.664-2.777 84.664-2.742Q84.664-2.621 84.730-2.514Q84.797-2.406 84.918-2.367Q84.933-2.371 84.947-2.373Q84.961-2.375 84.984-2.375Q85-2.375 85.031-2.367Q85.062-2.359 85.070-2.359L85.664-2.359Q86.445-2.359 86.986-2.117Q87.527-1.875 87.527-1.184Q87.527-0.883 87.347-0.656Q87.168-0.430 86.871-0.285Q86.574-0.141 86.254-0.074Q85.933-0.008 85.648-0.008Q85.257-0.008 84.816-0.131Q84.375-0.254 84.070-0.521Q83.765-0.789 83.765-1.184M84.304-1.191Q84.304-0.973 84.545-0.832Q84.785-0.691 85.109-0.625Q85.433-0.559 85.648-0.559Q85.863-0.559 86.187-0.625Q86.511-0.691 86.752-0.832Q86.992-0.973 86.992-1.191Q86.992-1.480 86.767-1.619Q86.543-1.758 86.261-1.791Q85.980-1.824 85.632-1.824L85.015-1.824Q84.832-1.824 84.670-1.744Q84.507-1.664 84.406-1.518Q84.304-1.371 84.304-1.191M85.414-3.289Q85.715-3.289 85.933-3.508Q86.152-3.727 86.152-4.031Q86.152-4.187 86.097-4.318Q86.043-4.449 85.937-4.555Q85.832-4.660 85.701-4.715Q85.570-4.769 85.414-4.769Q85.109-4.769 84.890-4.551Q84.672-4.332 84.672-4.031Q84.672-3.734 84.894-3.512Q85.117-3.289 85.414-3.289M87.996-2.070L87.996-2.160Q88.054-2.367 88.246-2.391L88.957-2.391L88.957-4.719L88.246-4.719Q88.050-4.742 87.996-4.961L87.996-5.047Q88.054-5.258 88.246-5.281L89.347-5.281Q89.547-5.262 89.597-5.047L89.597-4.719Q89.859-5.004 90.215-5.162Q90.570-5.320 90.957-5.320Q91.250-5.320 91.484-5.186Q91.718-5.051 91.718-4.785Q91.718-4.617 91.609-4.500Q91.500-4.383 91.332-4.383Q91.179-4.383 91.064-4.494Q90.949-4.605 90.949-4.762Q90.574-4.762 90.259-4.561Q89.945-4.359 89.771-4.025Q89.597-3.691 89.597-3.312L89.597-2.391L90.543-2.391Q90.750-2.367 90.789-2.160L90.789-2.070Q90.750-1.855 90.543-1.832L88.246-1.832Q88.054-1.855 87.996-2.070M92.425-2.945Q92.425-3.391 92.840-3.648Q93.254-3.906 93.795-4.006Q94.336-4.105 94.843-4.113Q94.843-4.328 94.709-4.480Q94.574-4.633 94.367-4.709Q94.160-4.785 93.949-4.785Q93.605-4.785 93.445-4.762L93.445-4.703Q93.445-4.535 93.326-4.420Q93.207-4.305 93.043-4.305Q92.867-4.305 92.752-4.428Q92.636-4.551 92.636-4.719Q92.636-5.125 93.017-5.234Q93.398-5.344 93.957-5.344Q94.226-5.344 94.494-5.266Q94.761-5.187 94.986-5.037Q95.211-4.887 95.347-4.666Q95.484-4.445 95.484-4.168L95.484-2.449Q95.484-2.391 96.011-2.391Q96.207-2.371 96.257-2.160L96.257-2.070Q96.207-1.855 96.011-1.832L95.867-1.832Q95.523-1.832 95.295-1.879Q95.066-1.926 94.922-2.113Q94.461-1.793 93.754-1.793Q93.418-1.793 93.113-1.934Q92.808-2.074 92.617-2.336Q92.425-2.598 92.425-2.945M93.066-2.937Q93.066-2.664 93.308-2.508Q93.550-2.352 93.836-2.352Q94.054-2.352 94.287-2.410Q94.519-2.469 94.681-2.607Q94.843-2.746 94.843-2.969L94.843-3.559Q94.562-3.559 94.146-3.502Q93.730-3.445 93.398-3.307Q93.066-3.168 93.066-2.937M98.129-1.793Q97.664-1.793 97.298-2.043Q96.933-2.293 96.728-2.697Q96.523-3.102 96.523-3.559Q96.523-3.902 96.648-4.223Q96.773-4.543 97.006-4.793Q97.238-5.043 97.543-5.182Q97.847-5.320 98.203-5.320Q98.715-5.320 99.121-5L99.121-6.160L98.699-6.160Q98.488-6.184 98.449-6.398L98.449-6.488Q98.488-6.695 98.699-6.719L99.511-6.719Q99.711-6.695 99.761-6.488L99.761-2.391L100.187-2.391Q100.394-2.367 100.433-2.160L100.433-2.070Q100.394-1.855 100.187-1.832L99.371-1.832Q99.172-1.855 99.121-2.070L99.121-2.199Q98.925-2.008 98.664-1.900Q98.402-1.793 98.129-1.793M98.168-2.352Q98.527-2.352 98.779-2.621Q99.031-2.891 99.121-3.266L99.121-4.098Q99.062-4.285 98.935-4.436Q98.808-4.586 98.631-4.674Q98.453-4.762 98.257-4.762Q97.949-4.762 97.699-4.592Q97.449-4.422 97.304-4.137Q97.160-3.852 97.160-3.551Q97.160-3.102 97.445-2.727Q97.730-2.352 98.168-2.352M101.129-2.070L101.129-2.160Q101.179-2.367 101.375-2.391L102.414-2.391L102.414-4.719L101.441-4.719Q101.242-4.742 101.191-4.961L101.191-5.047Q101.242-5.258 101.441-5.281L102.808-5.281Q103.004-5.262 103.054-5.047L103.054-2.391L103.968-2.391Q104.164-2.367 104.215-2.160L104.215-2.070Q104.164-1.855 103.968-1.832L101.375-1.832Q101.179-1.855 101.129-2.070M102.160-6.258L102.160-6.312Q102.160-6.484 102.297-6.605Q102.433-6.727 102.609-6.727Q102.781-6.727 102.918-6.605Q103.054-6.484 103.054-6.312L103.054-6.258Q103.054-6.082 102.918-5.961Q102.781-5.840 102.609-5.840Q102.433-5.840 102.297-5.961Q102.160-6.082 102.160-6.258M108.269-3.320L105.828-3.320Q105.882-3.043 106.080-2.820Q106.277-2.598 106.554-2.475Q106.832-2.352 107.117-2.352Q107.590-2.352 107.812-2.641Q107.820-2.652 107.877-2.758Q107.933-2.863 107.982-2.906Q108.031-2.949 108.125-2.961L108.269-2.961Q108.461-2.941 108.519-2.727L108.519-2.672Q108.453-2.371 108.222-2.174Q107.992-1.977 107.679-1.885Q107.367-1.793 107.062-1.793Q106.578-1.793 106.138-2.021Q105.699-2.250 105.431-2.650Q105.164-3.051 105.164-3.543L105.164-3.602Q105.164-4.070 105.410-4.473Q105.656-4.875 106.064-5.109Q106.472-5.344 106.941-5.344Q107.445-5.344 107.798-5.121Q108.152-4.898 108.336-4.510Q108.519-4.121 108.519-3.617L108.519-3.559Q108.461-3.344 108.269-3.320M105.836-3.871L107.863-3.871Q107.816-4.281 107.578-4.533Q107.340-4.785 106.941-4.785Q106.547-4.785 106.240-4.523Q105.933-4.262 105.836-3.871M109.074-2.070L109.074-2.160Q109.117-2.367 109.324-2.391L109.746-2.391L109.746-4.719L109.324-4.719Q109.117-4.742 109.074-4.961L109.074-5.047Q109.121-5.258 109.324-5.281L110.140-5.281Q110.336-5.258 110.386-5.047L110.386-4.961L110.379-4.937Q110.605-5.117 110.879-5.219Q111.152-5.320 111.445-5.320Q111.793-5.320 112.031-5.180Q112.269-5.039 112.384-4.781Q112.500-4.523 112.500-4.168L112.500-2.391L112.925-2.391Q113.132-2.367 113.172-2.160L113.172-2.070Q113.132-1.855 112.925-1.832L111.531-1.832Q111.336-1.855 111.285-2.070L111.285-2.160Q111.336-2.371 111.531-2.391L111.859-2.391L111.859-4.137Q111.859-4.445 111.769-4.603Q111.679-4.762 111.386-4.762Q111.117-4.762 110.888-4.631Q110.660-4.500 110.523-4.271Q110.386-4.043 110.386-3.777L110.386-2.391L110.812-2.391Q111.019-2.367 111.058-2.160L111.058-2.070Q111.019-1.855 110.812-1.832L109.324-1.832Q109.117-1.855 109.074-2.070M114.449-2.937L114.449-4.719L113.699-4.719Q113.500-4.742 113.449-4.961L113.449-5.047Q113.500-5.258 113.699-5.281L114.449-5.281L114.449-6.031Q114.500-6.238 114.699-6.266L114.843-6.266Q115.039-6.238 115.090-6.031L115.090-5.281L116.449-5.281Q116.640-5.262 116.699-5.047L116.699-4.961Q116.644-4.742 116.449-4.719L115.090-4.719L115.090-2.969Q115.090-2.352 115.664-2.352Q115.914-2.352 116.078-2.537Q116.242-2.723 116.242-2.969Q116.242-3.062 116.314-3.133Q116.386-3.203 116.488-3.215L116.632-3.215Q116.832-3.191 116.882-2.984L116.882-2.937Q116.882-2.613 116.699-2.350Q116.515-2.086 116.222-1.939Q115.929-1.793 115.609-1.793Q115.097-1.793 114.773-2.103Q114.449-2.414 114.449-2.937\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-95.805 65.327)\">\u003Cpath d=\"M123.641-1.793Q123.176-1.793 122.811-2.043Q122.446-2.293 122.241-2.697Q122.036-3.102 122.036-3.559Q122.036-3.902 122.161-4.223Q122.286-4.543 122.518-4.793Q122.751-5.043 123.055-5.182Q123.360-5.320 123.716-5.320Q124.227-5.320 124.633-5L124.633-6.160L124.212-6.160Q124.001-6.184 123.962-6.398L123.962-6.488Q124.001-6.695 124.212-6.719L125.024-6.719Q125.223-6.695 125.274-6.488L125.274-2.391L125.700-2.391Q125.907-2.367 125.946-2.160L125.946-2.070Q125.907-1.855 125.700-1.832L124.883-1.832Q124.684-1.855 124.633-2.070L124.633-2.199Q124.438-2.008 124.176-1.900Q123.915-1.793 123.641-1.793M123.680-2.352Q124.040-2.352 124.292-2.621Q124.544-2.891 124.633-3.266L124.633-4.098Q124.575-4.285 124.448-4.436Q124.321-4.586 124.143-4.674Q123.966-4.762 123.770-4.762Q123.462-4.762 123.212-4.592Q122.962-4.422 122.817-4.137Q122.673-3.852 122.673-3.551Q122.673-3.102 122.958-2.727Q123.243-2.352 123.680-2.352M129.536-3.320L127.094-3.320Q127.149-3.043 127.346-2.820Q127.544-2.598 127.821-2.475Q128.098-2.352 128.383-2.352Q128.856-2.352 129.079-2.641Q129.087-2.652 129.143-2.758Q129.200-2.863 129.249-2.906Q129.298-2.949 129.391-2.961L129.536-2.961Q129.727-2.941 129.786-2.727L129.786-2.672Q129.719-2.371 129.489-2.174Q129.258-1.977 128.946-1.885Q128.633-1.793 128.329-1.793Q127.844-1.793 127.405-2.021Q126.966-2.250 126.698-2.650Q126.430-3.051 126.430-3.543L126.430-3.602Q126.430-4.070 126.676-4.473Q126.923-4.875 127.331-5.109Q127.739-5.344 128.208-5.344Q128.712-5.344 129.065-5.121Q129.419-4.898 129.602-4.510Q129.786-4.121 129.786-3.617L129.786-3.559Q129.727-3.344 129.536-3.320M127.102-3.871L129.130-3.871Q129.083-4.281 128.844-4.533Q128.606-4.785 128.208-4.785Q127.813-4.785 127.507-4.523Q127.200-4.262 127.102-3.871M130.856-2.031L130.856-2.945Q130.883-3.152 131.094-3.176L131.262-3.176Q131.426-3.152 131.485-2.992Q131.688-2.352 132.415-2.352Q132.622-2.352 132.850-2.387Q133.079-2.422 133.247-2.537Q133.415-2.652 133.415-2.855Q133.415-3.066 133.192-3.180Q132.969-3.293 132.696-3.336L131.997-3.449Q130.856-3.660 130.856-4.383Q130.856-4.672 131.001-4.861Q131.145-5.051 131.385-5.158Q131.626-5.266 131.882-5.305Q132.137-5.344 132.415-5.344Q132.665-5.344 132.858-5.314Q133.051-5.285 133.216-5.207Q133.294-5.324 133.423-5.344L133.501-5.344Q133.598-5.332 133.661-5.269Q133.723-5.207 133.735-5.113L133.735-4.406Q133.723-4.312 133.661-4.246Q133.598-4.180 133.501-4.168L133.333-4.168Q133.239-4.180 133.173-4.246Q133.106-4.312 133.094-4.406Q133.094-4.785 132.399-4.785Q132.051-4.785 131.733-4.703Q131.415-4.621 131.415-4.375Q131.415-4.109 132.087-4L132.790-3.879Q133.274-3.797 133.624-3.549Q133.973-3.301 133.973-2.855Q133.973-2.465 133.737-2.223Q133.501-1.980 133.151-1.887Q132.801-1.793 132.415-1.793Q131.837-1.793 131.438-2.047Q131.368-1.922 131.319-1.865Q131.270-1.809 131.165-1.793L131.094-1.793Q130.880-1.816 130.856-2.031M135.075-3.559Q135.075-4.039 135.319-4.453Q135.563-4.867 135.979-5.105Q136.395-5.344 136.876-5.344Q137.430-5.344 137.809-5.234Q138.188-5.125 138.188-4.719Q138.188-4.551 138.075-4.428Q137.962-4.305 137.790-4.305Q137.618-4.305 137.499-4.420Q137.380-4.535 137.380-4.703L137.380-4.762Q137.219-4.785 136.883-4.785Q136.555-4.785 136.288-4.615Q136.020-4.445 135.868-4.162Q135.716-3.879 135.716-3.559Q135.716-3.238 135.887-2.957Q136.059-2.676 136.344-2.514Q136.630-2.352 136.958-2.352Q137.270-2.352 137.397-2.455Q137.524-2.559 137.641-2.748Q137.758-2.937 137.876-2.953L138.044-2.953Q138.149-2.941 138.214-2.873Q138.278-2.805 138.278-2.703Q138.278-2.656 138.258-2.617Q138.149-2.324 137.946-2.144Q137.743-1.965 137.467-1.879Q137.192-1.793 136.876-1.793Q136.391-1.793 135.975-2.031Q135.559-2.269 135.317-2.672Q135.075-3.074 135.075-3.559M142.274-3.320L139.833-3.320Q139.887-3.043 140.085-2.820Q140.282-2.598 140.559-2.475Q140.837-2.352 141.122-2.352Q141.594-2.352 141.817-2.641Q141.825-2.652 141.882-2.758Q141.938-2.863 141.987-2.906Q142.036-2.949 142.130-2.961L142.274-2.961Q142.466-2.941 142.524-2.727L142.524-2.672Q142.458-2.371 142.227-2.174Q141.997-1.977 141.684-1.885Q141.372-1.793 141.067-1.793Q140.583-1.793 140.143-2.021Q139.704-2.250 139.436-2.650Q139.169-3.051 139.169-3.543L139.169-3.602Q139.169-4.070 139.415-4.473Q139.661-4.875 140.069-5.109Q140.477-5.344 140.946-5.344Q141.450-5.344 141.803-5.121Q142.157-4.898 142.341-4.510Q142.524-4.121 142.524-3.617L142.524-3.559Q142.466-3.344 142.274-3.320M139.841-3.871L141.868-3.871Q141.821-4.281 141.583-4.533Q141.344-4.785 140.946-4.785Q140.551-4.785 140.245-4.523Q139.938-4.262 139.841-3.871M143.079-2.070L143.079-2.160Q143.122-2.367 143.329-2.391L143.751-2.391L143.751-4.719L143.329-4.719Q143.122-4.742 143.079-4.961L143.079-5.047Q143.126-5.258 143.329-5.281L144.145-5.281Q144.341-5.258 144.391-5.047L144.391-4.961L144.383-4.937Q144.610-5.117 144.883-5.219Q145.157-5.320 145.450-5.320Q145.798-5.320 146.036-5.180Q146.274-5.039 146.389-4.781Q146.505-4.523 146.505-4.168L146.505-2.391L146.930-2.391Q147.137-2.367 147.176-2.160L147.176-2.070Q147.137-1.855 146.930-1.832L145.536-1.832Q145.341-1.855 145.290-2.070L145.290-2.160Q145.341-2.371 145.536-2.391L145.864-2.391L145.864-4.137Q145.864-4.445 145.774-4.603Q145.684-4.762 145.391-4.762Q145.122-4.762 144.893-4.631Q144.665-4.500 144.528-4.271Q144.391-4.043 144.391-3.777L144.391-2.391L144.817-2.391Q145.024-2.367 145.063-2.160L145.063-2.070Q145.024-1.855 144.817-1.832L143.329-1.832Q143.122-1.855 143.079-2.070M148.454-2.937L148.454-4.719L147.704-4.719Q147.505-4.742 147.454-4.961L147.454-5.047Q147.505-5.258 147.704-5.281L148.454-5.281L148.454-6.031Q148.505-6.238 148.704-6.266L148.848-6.266Q149.044-6.238 149.094-6.031L149.094-5.281L150.454-5.281Q150.645-5.262 150.704-5.047L150.704-4.961Q150.649-4.742 150.454-4.719L149.094-4.719L149.094-2.969Q149.094-2.352 149.669-2.352Q149.919-2.352 150.083-2.537Q150.247-2.723 150.247-2.969Q150.247-3.062 150.319-3.133Q150.391-3.203 150.493-3.215L150.637-3.215Q150.837-3.191 150.887-2.984L150.887-2.937Q150.887-2.613 150.704-2.350Q150.520-2.086 150.227-1.939Q149.934-1.793 149.614-1.793Q149.102-1.793 148.778-2.103Q148.454-2.414 148.454-2.937\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-95.805 65.327)\">\u003Cpath d=\"M159.372-1.832L156.305-1.832Q156.200-1.832 156.122-1.914Q156.044-1.996 156.044-2.098L156.044-2.207Q156.044-2.324 156.130-2.398L158.626-4.719L156.817-4.719L156.817-4.426Q156.766-4.207 156.571-4.184L156.426-4.184Q156.235-4.203 156.176-4.426L156.176-5.047Q156.235-5.258 156.426-5.281L159.329-5.281Q159.438-5.281 159.516-5.203Q159.594-5.125 159.594-5.016L159.594-4.902Q159.594-4.789 159.505-4.711L157.008-2.391L158.977-2.391L158.977-2.785Q159.028-2.992 159.227-3.016L159.372-3.016Q159.567-2.992 159.618-2.785L159.618-2.070Q159.567-1.855 159.372-1.832M160.641-2.070L160.641-2.160Q160.692-2.367 160.887-2.391L161.926-2.391L161.926-4.719L160.954-4.719Q160.755-4.742 160.704-4.961L160.704-5.047Q160.755-5.258 160.954-5.281L162.321-5.281Q162.516-5.262 162.567-5.047L162.567-2.391L163.481-2.391Q163.676-2.367 163.727-2.160L163.727-2.070Q163.676-1.855 163.481-1.832L160.887-1.832Q160.692-1.855 160.641-2.070M161.673-6.258L161.673-6.312Q161.673-6.484 161.809-6.605Q161.946-6.727 162.122-6.727Q162.294-6.727 162.430-6.605Q162.567-6.484 162.567-6.312L162.567-6.258Q162.567-6.082 162.430-5.961Q162.294-5.840 162.122-5.840Q161.946-5.840 161.809-5.961Q161.673-6.082 161.673-6.258M164.508-1.184Q164.508-1.484 164.657-1.746Q164.805-2.008 165.055-2.168Q164.872-2.422 164.872-2.742Q164.872-3.027 165.024-3.289Q164.774-3.613 164.774-4.031Q164.774-4.391 164.965-4.687Q165.157-4.984 165.475-5.152Q165.794-5.320 166.157-5.320Q166.364-5.320 166.567-5.260Q166.770-5.199 166.934-5.098Q167.317-5.359 167.790-5.359Q168.028-5.359 168.210-5.228Q168.391-5.098 168.391-4.871Q168.391-4.723 168.290-4.617Q168.188-4.512 168.032-4.512Q167.899-4.512 167.803-4.590Q167.708-4.668 167.676-4.793Q167.532-4.785 167.333-4.695Q167.536-4.383 167.536-4.031Q167.536-3.750 167.424-3.518Q167.313-3.285 167.118-3.107Q166.923-2.930 166.671-2.832Q166.419-2.734 166.157-2.734Q165.770-2.734 165.438-2.922Q165.426-2.922 165.417-2.850Q165.407-2.777 165.407-2.742Q165.407-2.621 165.473-2.514Q165.540-2.406 165.661-2.367Q165.676-2.371 165.690-2.373Q165.704-2.375 165.727-2.375Q165.743-2.375 165.774-2.367Q165.805-2.359 165.813-2.359L166.407-2.359Q167.188-2.359 167.729-2.117Q168.270-1.875 168.270-1.184Q168.270-0.883 168.090-0.656Q167.911-0.430 167.614-0.285Q167.317-0.141 166.997-0.074Q166.676-0.008 166.391-0.008Q166.001-0.008 165.559-0.131Q165.118-0.254 164.813-0.521Q164.508-0.789 164.508-1.184M165.048-1.191Q165.048-0.973 165.288-0.832Q165.528-0.691 165.852-0.625Q166.176-0.559 166.391-0.559Q166.606-0.559 166.930-0.625Q167.255-0.691 167.495-0.832Q167.735-0.973 167.735-1.191Q167.735-1.480 167.510-1.619Q167.286-1.758 167.005-1.791Q166.723-1.824 166.376-1.824L165.758-1.824Q165.575-1.824 165.413-1.744Q165.251-1.664 165.149-1.518Q165.048-1.371 165.048-1.191M166.157-3.289Q166.458-3.289 166.676-3.508Q166.895-3.727 166.895-4.031Q166.895-4.187 166.840-4.318Q166.786-4.449 166.680-4.555Q166.575-4.660 166.444-4.715Q166.313-4.769 166.157-4.769Q165.852-4.769 165.633-4.551Q165.415-4.332 165.415-4.031Q165.415-3.734 165.637-3.512Q165.860-3.289 166.157-3.289M172.012-3.969L169.270-3.969Q169.145-3.980 169.059-4.066Q168.973-4.152 168.973-4.281Q168.973-4.410 169.059-4.492Q169.145-4.574 169.270-4.586L172.012-4.586Q172.137-4.574 172.219-4.492Q172.301-4.410 172.301-4.281Q172.301-4.152 172.219-4.066Q172.137-3.980 172.012-3.969M176.356-1.832L173.290-1.832Q173.184-1.832 173.106-1.914Q173.028-1.996 173.028-2.098L173.028-2.207Q173.028-2.324 173.114-2.398L175.610-4.719L173.801-4.719L173.801-4.426Q173.751-4.207 173.555-4.184L173.411-4.184Q173.219-4.203 173.161-4.426L173.161-5.047Q173.219-5.258 173.411-5.281L176.313-5.281Q176.423-5.281 176.501-5.203Q176.579-5.125 176.579-5.016L176.579-4.902Q176.579-4.789 176.489-4.711L173.993-2.391L175.962-2.391L175.962-2.785Q176.012-2.992 176.212-3.016L176.356-3.016Q176.551-2.992 176.602-2.785L176.602-2.070Q176.551-1.855 176.356-1.832M177.415-2.945Q177.415-3.391 177.829-3.648Q178.243-3.906 178.784-4.006Q179.325-4.105 179.833-4.113Q179.833-4.328 179.698-4.480Q179.563-4.633 179.356-4.709Q179.149-4.785 178.938-4.785Q178.594-4.785 178.434-4.762L178.434-4.703Q178.434-4.535 178.315-4.420Q178.196-4.305 178.032-4.305Q177.856-4.305 177.741-4.428Q177.626-4.551 177.626-4.719Q177.626-5.125 178.007-5.234Q178.387-5.344 178.946-5.344Q179.215-5.344 179.483-5.266Q179.751-5.187 179.975-5.037Q180.200-4.887 180.337-4.666Q180.473-4.445 180.473-4.168L180.473-2.449Q180.473-2.391 181.001-2.391Q181.196-2.371 181.247-2.160L181.247-2.070Q181.196-1.855 181.001-1.832L180.856-1.832Q180.512-1.832 180.284-1.879Q180.055-1.926 179.911-2.113Q179.450-1.793 178.743-1.793Q178.407-1.793 178.102-1.934Q177.798-2.074 177.606-2.336Q177.415-2.598 177.415-2.945M178.055-2.937Q178.055-2.664 178.298-2.508Q178.540-2.352 178.825-2.352Q179.044-2.352 179.276-2.410Q179.508-2.469 179.671-2.607Q179.833-2.746 179.833-2.969L179.833-3.559Q179.551-3.559 179.135-3.502Q178.719-3.445 178.387-3.307Q178.055-3.168 178.055-2.937M181.493-1.184Q181.493-1.484 181.641-1.746Q181.790-2.008 182.040-2.168Q181.856-2.422 181.856-2.742Q181.856-3.027 182.008-3.289Q181.758-3.613 181.758-4.031Q181.758-4.391 181.950-4.687Q182.141-4.984 182.460-5.152Q182.778-5.320 183.141-5.320Q183.348-5.320 183.551-5.260Q183.755-5.199 183.919-5.098Q184.301-5.359 184.774-5.359Q185.012-5.359 185.194-5.228Q185.376-5.098 185.376-4.871Q185.376-4.723 185.274-4.617Q185.173-4.512 185.016-4.512Q184.883-4.512 184.788-4.590Q184.692-4.668 184.661-4.793Q184.516-4.785 184.317-4.695Q184.520-4.383 184.520-4.031Q184.520-3.750 184.409-3.518Q184.298-3.285 184.102-3.107Q183.907-2.930 183.655-2.832Q183.403-2.734 183.141-2.734Q182.755-2.734 182.423-2.922Q182.411-2.922 182.401-2.850Q182.391-2.777 182.391-2.742Q182.391-2.621 182.458-2.514Q182.524-2.406 182.645-2.367Q182.661-2.371 182.674-2.373Q182.688-2.375 182.712-2.375Q182.727-2.375 182.758-2.367Q182.790-2.359 182.798-2.359L183.391-2.359Q184.173-2.359 184.714-2.117Q185.255-1.875 185.255-1.184Q185.255-0.883 185.075-0.656Q184.895-0.430 184.598-0.285Q184.301-0.141 183.981-0.074Q183.661-0.008 183.376-0.008Q182.985-0.008 182.544-0.131Q182.102-0.254 181.798-0.521Q181.493-0.789 181.493-1.184M182.032-1.191Q182.032-0.973 182.272-0.832Q182.512-0.691 182.837-0.625Q183.161-0.559 183.376-0.559Q183.590-0.559 183.915-0.625Q184.239-0.691 184.479-0.832Q184.719-0.973 184.719-1.191Q184.719-1.480 184.495-1.619Q184.270-1.758 183.989-1.791Q183.708-1.824 183.360-1.824L182.743-1.824Q182.559-1.824 182.397-1.744Q182.235-1.664 182.133-1.518Q182.032-1.371 182.032-1.191M183.141-3.289Q183.442-3.289 183.661-3.508Q183.880-3.727 183.880-4.031Q183.880-4.187 183.825-4.318Q183.770-4.449 183.665-4.555Q183.559-4.660 183.428-4.715Q183.298-4.769 183.141-4.769Q182.837-4.769 182.618-4.551Q182.399-4.332 182.399-4.031Q182.399-3.734 182.622-3.512Q182.844-3.289 183.141-3.289M186.087-2.031L186.087-2.945Q186.114-3.152 186.325-3.176L186.493-3.176Q186.657-3.152 186.715-2.992Q186.919-2.352 187.645-2.352Q187.852-2.352 188.081-2.387Q188.309-2.422 188.477-2.537Q188.645-2.652 188.645-2.855Q188.645-3.066 188.423-3.180Q188.200-3.293 187.926-3.336L187.227-3.449Q186.087-3.660 186.087-4.383Q186.087-4.672 186.231-4.861Q186.376-5.051 186.616-5.158Q186.856-5.266 187.112-5.305Q187.368-5.344 187.645-5.344Q187.895-5.344 188.089-5.314Q188.282-5.285 188.446-5.207Q188.524-5.324 188.653-5.344L188.731-5.344Q188.829-5.332 188.891-5.269Q188.954-5.207 188.965-5.113L188.965-4.406Q188.954-4.312 188.891-4.246Q188.829-4.180 188.731-4.168L188.563-4.168Q188.469-4.180 188.403-4.246Q188.337-4.312 188.325-4.406Q188.325-4.785 187.630-4.785Q187.282-4.785 186.964-4.703Q186.645-4.621 186.645-4.375Q186.645-4.109 187.317-4L188.020-3.879Q188.505-3.797 188.854-3.549Q189.204-3.301 189.204-2.855Q189.204-2.465 188.967-2.223Q188.731-1.980 188.382-1.887Q188.032-1.793 187.645-1.793Q187.067-1.793 186.669-2.047Q186.598-1.922 186.549-1.865Q186.501-1.809 186.395-1.793L186.325-1.793Q186.110-1.816 186.087-2.031\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"M-21.748-40.243 80.521-2.93\"\u002F>\u003Cpath stroke=\"none\" d=\"m83.527-1.832-3.932-4.16.926 3.063-2.681 1.747\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\">\u003Cg fill=\"var(--tk-good)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(-78.805 -62.216)\">\u003Cpath d=\"M83.726-2.070L83.726-2.160Q83.777-2.367 83.976-2.391L84.246-2.391L84.246-6.160L83.976-6.160Q83.777-6.184 83.726-6.398L83.726-6.488Q83.777-6.695 83.976-6.719L84.757-6.719Q84.886-6.719 84.992-6.644Q85.097-6.570 85.136-6.457Q85.414-5.668 85.574-5.219Q85.734-4.769 85.988-4.051Q86.242-3.332 86.381-2.922Q86.519-2.512 86.519-2.473L86.519-6.160L86.246-6.160Q86.050-6.184 86-6.398L86-6.488Q86.050-6.695 86.246-6.719L87.320-6.719Q87.515-6.695 87.566-6.488L87.566-6.398Q87.515-6.184 87.320-6.160L87.047-6.160L87.047-2.070Q86.996-1.855 86.800-1.832L86.535-1.832Q86.410-1.832 86.306-1.906Q86.203-1.980 86.160-2.098Q85.863-2.945 85.490-3.994Q85.117-5.043 84.947-5.531Q84.777-6.019 84.773-6.078L84.773-2.391L85.047-2.391Q85.238-2.367 85.297-2.160L85.297-2.070Q85.238-1.855 85.047-1.832L83.976-1.832Q83.777-1.855 83.726-2.070M91.285-3.320L88.843-3.320Q88.898-3.043 89.095-2.820Q89.293-2.598 89.570-2.475Q89.847-2.352 90.132-2.352Q90.605-2.352 90.828-2.641Q90.836-2.652 90.892-2.758Q90.949-2.863 90.998-2.906Q91.047-2.949 91.140-2.961L91.285-2.961Q91.476-2.941 91.535-2.727L91.535-2.672Q91.468-2.371 91.238-2.174Q91.007-1.977 90.695-1.885Q90.382-1.793 90.078-1.793Q89.593-1.793 89.154-2.021Q88.715-2.250 88.447-2.650Q88.179-3.051 88.179-3.543L88.179-3.602Q88.179-4.070 88.425-4.473Q88.672-4.875 89.080-5.109Q89.488-5.344 89.957-5.344Q90.461-5.344 90.814-5.121Q91.168-4.898 91.351-4.510Q91.535-4.121 91.535-3.617L91.535-3.559Q91.476-3.344 91.285-3.320M88.851-3.871L90.879-3.871Q90.832-4.281 90.593-4.533Q90.355-4.785 89.957-4.785Q89.562-4.785 89.256-4.523Q88.949-4.262 88.851-3.871M93.027-2.055L92.562-4.719L92.402-4.719Q92.195-4.742 92.156-4.961L92.156-5.047Q92.195-5.258 92.402-5.281L93.570-5.281Q93.781-5.258 93.820-5.047L93.820-4.961Q93.781-4.742 93.570-4.719L93.097-4.719Q93.211-4.074 93.289-3.629Q93.367-3.184 93.418-2.865Q93.468-2.547 93.468-2.473Q93.476-2.644 93.761-3.641Q93.800-3.754 93.898-3.828Q93.996-3.902 94.117-3.902L94.195-3.902Q94.316-3.902 94.414-3.828Q94.511-3.754 94.547-3.641Q94.656-3.258 94.738-2.934Q94.820-2.609 94.820-2.473Q94.832-2.762 95.179-4.719L94.707-4.719Q94.500-4.742 94.461-4.961L94.461-5.047Q94.500-5.258 94.707-5.281L95.882-5.281Q96.074-5.258 96.132-5.047L96.132-4.961Q96.078-4.742 95.882-4.719L95.715-4.719L95.250-2.055Q95.226-1.937 95.138-1.865Q95.050-1.793 94.929-1.793L94.789-1.793Q94.668-1.793 94.572-1.865Q94.476-1.937 94.433-2.055Q94.386-2.234 94.314-2.490Q94.242-2.746 94.199-2.955Q94.156-3.164 94.156-3.266Q94.148-2.984 93.867-2.055Q93.840-1.945 93.742-1.869Q93.644-1.793 93.523-1.793L93.347-1.793Q93.226-1.793 93.138-1.865Q93.050-1.937 93.027-2.055M97.465-2.937L97.465-4.719L96.715-4.719Q96.515-4.742 96.465-4.961L96.465-5.047Q96.515-5.258 96.715-5.281L97.465-5.281L97.465-6.031Q97.515-6.238 97.715-6.266L97.859-6.266Q98.054-6.238 98.105-6.031L98.105-5.281L99.465-5.281Q99.656-5.262 99.715-5.047L99.715-4.961Q99.660-4.742 99.465-4.719L98.105-4.719L98.105-2.969Q98.105-2.352 98.679-2.352Q98.929-2.352 99.093-2.537Q99.257-2.723 99.257-2.969Q99.257-3.062 99.330-3.133Q99.402-3.203 99.504-3.215L99.648-3.215Q99.847-3.191 99.898-2.984L99.898-2.937Q99.898-2.613 99.715-2.350Q99.531-2.086 99.238-1.939Q98.945-1.793 98.625-1.793Q98.113-1.793 97.789-2.103Q97.465-2.414 97.465-2.937M102.632-1.793Q102.160-1.793 101.775-2.037Q101.390-2.281 101.168-2.691Q100.945-3.102 100.945-3.559Q100.945-3.902 101.070-4.225Q101.195-4.547 101.425-4.801Q101.656-5.055 101.963-5.199Q102.269-5.344 102.632-5.344Q102.996-5.344 103.308-5.197Q103.621-5.051 103.843-4.805Q104.066-4.559 104.193-4.238Q104.320-3.918 104.320-3.559Q104.320-3.102 104.095-2.689Q103.871-2.277 103.486-2.035Q103.101-1.793 102.632-1.793M102.632-2.352Q103.097-2.352 103.388-2.746Q103.679-3.141 103.679-3.625Q103.679-3.918 103.545-4.186Q103.410-4.453 103.170-4.619Q102.929-4.785 102.632-4.785Q102.328-4.785 102.090-4.619Q101.851-4.453 101.716-4.186Q101.582-3.918 101.582-3.625Q101.582-3.144 101.875-2.748Q102.168-2.352 102.632-2.352M104.828-2.070L104.828-2.160Q104.871-2.367 105.078-2.391L105.500-2.391L105.500-4.719L105.078-4.719Q104.871-4.742 104.828-4.961L104.828-5.047Q104.875-5.258 105.078-5.281L105.894-5.281Q106.090-5.258 106.140-5.047L106.140-4.961L106.132-4.937Q106.359-5.117 106.632-5.219Q106.906-5.320 107.199-5.320Q107.547-5.320 107.785-5.180Q108.023-5.039 108.138-4.781Q108.254-4.523 108.254-4.168L108.254-2.391L108.679-2.391Q108.886-2.367 108.925-2.160L108.925-2.070Q108.886-1.855 108.679-1.832L107.285-1.832Q107.090-1.855 107.039-2.070L107.039-2.160Q107.090-2.371 107.285-2.391L107.613-2.391L107.613-4.137Q107.613-4.445 107.523-4.603Q107.433-4.762 107.140-4.762Q106.871-4.762 106.642-4.631Q106.414-4.500 106.277-4.271Q106.140-4.043 106.140-3.777L106.140-2.391L106.566-2.391Q106.773-2.367 106.812-2.160L106.812-2.070Q106.773-1.855 106.566-1.832L105.078-1.832Q104.871-1.855 104.828-2.070M110.570-2.391Q110.570-2.613 110.736-2.779Q110.902-2.945 111.132-2.945Q111.281-2.945 111.408-2.867Q111.535-2.789 111.609-2.664Q111.683-2.539 111.683-2.391Q111.683-2.164 111.517-1.998Q111.351-1.832 111.132-1.832Q110.906-1.832 110.738-2Q110.570-2.168 110.570-2.391M110.570-4.727Q110.570-4.949 110.736-5.115Q110.902-5.281 111.132-5.281Q111.281-5.281 111.408-5.203Q111.535-5.125 111.609-5Q111.683-4.875 111.683-4.727Q111.683-4.500 111.517-4.334Q111.351-4.168 111.132-4.168Q110.906-4.168 110.738-4.336Q110.570-4.504 110.570-4.727\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-78.805 -62.216)\">\u003Cpath d=\"M123.899-1.793Q123.426-1.793 123.042-2.037Q122.657-2.281 122.434-2.691Q122.212-3.102 122.212-3.559Q122.212-3.902 122.337-4.225Q122.462-4.547 122.692-4.801Q122.923-5.055 123.229-5.199Q123.536-5.344 123.899-5.344Q124.262-5.344 124.575-5.197Q124.887-5.051 125.110-4.805Q125.333-4.559 125.460-4.238Q125.587-3.918 125.587-3.559Q125.587-3.102 125.362-2.689Q125.137-2.277 124.753-2.035Q124.368-1.793 123.899-1.793M123.899-2.352Q124.364-2.352 124.655-2.746Q124.946-3.141 124.946-3.625Q124.946-3.918 124.811-4.186Q124.676-4.453 124.436-4.619Q124.196-4.785 123.899-4.785Q123.594-4.785 123.356-4.619Q123.118-4.453 122.983-4.186Q122.848-3.918 122.848-3.625Q122.848-3.144 123.141-2.748Q123.434-2.352 123.899-2.352M126.094-2.070L126.094-2.160Q126.137-2.367 126.344-2.391L126.766-2.391L126.766-4.719L126.344-4.719Q126.137-4.742 126.094-4.961L126.094-5.047Q126.141-5.258 126.344-5.281L127.161-5.281Q127.356-5.258 127.407-5.047L127.407-4.961L127.399-4.937Q127.626-5.117 127.899-5.219Q128.173-5.320 128.466-5.320Q128.813-5.320 129.051-5.180Q129.290-5.039 129.405-4.781Q129.520-4.523 129.520-4.168L129.520-2.391L129.946-2.391Q130.153-2.367 130.192-2.160L130.192-2.070Q130.153-1.855 129.946-1.832L128.551-1.832Q128.356-1.855 128.305-2.070L128.305-2.160Q128.356-2.371 128.551-2.391L128.880-2.391L128.880-4.137Q128.880-4.445 128.790-4.603Q128.700-4.762 128.407-4.762Q128.137-4.762 127.909-4.631Q127.680-4.500 127.544-4.271Q127.407-4.043 127.407-3.777L127.407-2.391L127.833-2.391Q128.040-2.367 128.079-2.160L128.079-2.070Q128.040-1.855 127.833-1.832L126.344-1.832Q126.137-1.855 126.094-2.070M133.782-3.320L131.341-3.320Q131.395-3.043 131.592-2.820Q131.790-2.598 132.067-2.475Q132.344-2.352 132.630-2.352Q133.102-2.352 133.325-2.641Q133.333-2.652 133.389-2.758Q133.446-2.863 133.495-2.906Q133.544-2.949 133.637-2.961L133.782-2.961Q133.973-2.941 134.032-2.727L134.032-2.672Q133.966-2.371 133.735-2.174Q133.505-1.977 133.192-1.885Q132.880-1.793 132.575-1.793Q132.091-1.793 131.651-2.021Q131.212-2.250 130.944-2.650Q130.676-3.051 130.676-3.543L130.676-3.602Q130.676-4.070 130.923-4.473Q131.169-4.875 131.577-5.109Q131.985-5.344 132.454-5.344Q132.958-5.344 133.311-5.121Q133.665-4.898 133.848-4.510Q134.032-4.121 134.032-3.617L134.032-3.559Q133.973-3.344 133.782-3.320M131.348-3.871L133.376-3.871Q133.329-4.281 133.091-4.533Q132.852-4.785 132.454-4.785Q132.059-4.785 131.753-4.523Q131.446-4.262 131.348-3.871\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-78.805 -62.216)\">\u003Cpath d=\"M139.364-2.031L139.364-2.945Q139.391-3.152 139.602-3.176L139.770-3.176Q139.934-3.152 139.993-2.992Q140.196-2.352 140.923-2.352Q141.130-2.352 141.358-2.387Q141.587-2.422 141.755-2.537Q141.923-2.652 141.923-2.855Q141.923-3.066 141.700-3.180Q141.477-3.293 141.204-3.336L140.505-3.449Q139.364-3.660 139.364-4.383Q139.364-4.672 139.508-4.861Q139.653-5.051 139.893-5.158Q140.133-5.266 140.389-5.305Q140.645-5.344 140.923-5.344Q141.173-5.344 141.366-5.314Q141.559-5.285 141.723-5.207Q141.801-5.324 141.930-5.344L142.008-5.344Q142.106-5.332 142.169-5.269Q142.231-5.207 142.243-5.113L142.243-4.406Q142.231-4.312 142.169-4.246Q142.106-4.180 142.008-4.168L141.840-4.168Q141.747-4.180 141.680-4.246Q141.614-4.312 141.602-4.406Q141.602-4.785 140.907-4.785Q140.559-4.785 140.241-4.703Q139.923-4.621 139.923-4.375Q139.923-4.109 140.594-4L141.298-3.879Q141.782-3.797 142.132-3.549Q142.481-3.301 142.481-2.855Q142.481-2.465 142.245-2.223Q142.008-1.980 141.659-1.887Q141.309-1.793 140.923-1.793Q140.344-1.793 139.946-2.047Q139.876-1.922 139.827-1.865Q139.778-1.809 139.673-1.793L139.602-1.793Q139.387-1.816 139.364-2.031M144.223-2.937L144.223-4.719L143.473-4.719Q143.274-4.742 143.223-4.961L143.223-5.047Q143.274-5.258 143.473-5.281L144.223-5.281L144.223-6.031Q144.274-6.238 144.473-6.266L144.618-6.266Q144.813-6.238 144.864-6.031L144.864-5.281L146.223-5.281Q146.415-5.262 146.473-5.047L146.473-4.961Q146.419-4.742 146.223-4.719L144.864-4.719L144.864-2.969Q144.864-2.352 145.438-2.352Q145.688-2.352 145.852-2.537Q146.016-2.723 146.016-2.969Q146.016-3.062 146.089-3.133Q146.161-3.203 146.262-3.215L146.407-3.215Q146.606-3.191 146.657-2.984L146.657-2.937Q146.657-2.613 146.473-2.350Q146.290-2.086 145.997-1.939Q145.704-1.793 145.383-1.793Q144.872-1.793 144.548-2.103Q144.223-2.414 144.223-2.937M150.782-3.320L148.340-3.320Q148.395-3.043 148.592-2.820Q148.790-2.598 149.067-2.475Q149.344-2.352 149.630-2.352Q150.102-2.352 150.325-2.641Q150.333-2.652 150.389-2.758Q150.446-2.863 150.495-2.906Q150.544-2.949 150.637-2.961L150.782-2.961Q150.973-2.941 151.032-2.727L151.032-2.672Q150.965-2.371 150.735-2.174Q150.505-1.977 150.192-1.885Q149.880-1.793 149.575-1.793Q149.090-1.793 148.651-2.021Q148.212-2.250 147.944-2.650Q147.676-3.051 147.676-3.543L147.676-3.602Q147.676-4.070 147.923-4.473Q148.169-4.875 148.577-5.109Q148.985-5.344 149.454-5.344Q149.958-5.344 150.311-5.121Q150.665-4.898 150.848-4.510Q151.032-4.121 151.032-3.617L151.032-3.559Q150.973-3.344 150.782-3.320M148.348-3.871L150.376-3.871Q150.329-4.281 150.090-4.533Q149.852-4.785 149.454-4.785Q149.059-4.785 148.753-4.523Q148.446-4.262 148.348-3.871M151.587-0.289L151.587-0.375Q151.630-0.594 151.837-0.617L152.258-0.617L152.258-4.719L151.837-4.719Q151.630-4.742 151.587-4.961L151.587-5.047Q151.633-5.258 151.837-5.281L152.653-5.281Q152.848-5.262 152.899-5.047L152.899-4.977Q153.110-5.144 153.374-5.232Q153.637-5.320 153.907-5.320Q154.247-5.320 154.544-5.176Q154.840-5.031 155.055-4.779Q155.270-4.527 155.385-4.215Q155.501-3.902 155.501-3.559Q155.501-3.094 155.274-2.684Q155.048-2.273 154.661-2.033Q154.274-1.793 153.805-1.793Q153.286-1.793 152.899-2.160L152.899-0.617L153.325-0.617Q153.532-0.594 153.571-0.375L153.571-0.289Q153.532-0.078 153.325-0.055L151.837-0.055Q151.633-0.078 151.587-0.289M153.762-2.352Q154.071-2.352 154.321-2.521Q154.571-2.691 154.715-2.973Q154.860-3.254 154.860-3.559Q154.860-3.848 154.735-4.127Q154.610-4.406 154.378-4.584Q154.145-4.762 153.844-4.762Q153.524-4.762 153.262-4.576Q153.001-4.391 152.899-4.090L152.899-3.238Q152.989-2.871 153.210-2.611Q153.430-2.352 153.762-2.352\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">On an ill-conditioned bowl, gradient descent (red) zig-zags across the contours while the Newton step (green) rescales by curvature and lands in one move.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:450.196px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 337.647 143.653\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg stroke=\"none\" font-size=\"8\">\u003Cg transform=\"translate(-10.624 -76.443)\">\u003Cpath d=\"M-48.075 23.668Q-48.075 23.368-47.926 23.106Q-47.778 22.844-47.528 22.684Q-47.711 22.430-47.711 22.110Q-47.711 21.825-47.559 21.563Q-47.809 21.239-47.809 20.821Q-47.809 20.461-47.618 20.165Q-47.426 19.868-47.108 19.700Q-46.790 19.532-46.426 19.532Q-46.219 19.532-46.016 19.592Q-45.813 19.653-45.649 19.754Q-45.266 19.493-44.793 19.493Q-44.555 19.493-44.374 19.624Q-44.192 19.754-44.192 19.981Q-44.192 20.129-44.293 20.235Q-44.395 20.340-44.551 20.340Q-44.684 20.340-44.780 20.262Q-44.876 20.184-44.907 20.059Q-45.051 20.067-45.251 20.157Q-45.047 20.469-45.047 20.821Q-45.047 21.102-45.159 21.334Q-45.270 21.567-45.465 21.745Q-45.661 21.922-45.913 22.020Q-46.165 22.118-46.426 22.118Q-46.813 22.118-47.145 21.930Q-47.157 21.930-47.167 22.002Q-47.176 22.075-47.176 22.110Q-47.176 22.231-47.110 22.338Q-47.043 22.446-46.922 22.485Q-46.907 22.481-46.893 22.479Q-46.879 22.477-46.856 22.477Q-46.840 22.477-46.809 22.485Q-46.778 22.493-46.770 22.493L-46.176 22.493Q-45.395 22.493-44.854 22.735Q-44.313 22.977-44.313 23.668Q-44.313 23.969-44.493 24.196Q-44.672 24.422-44.969 24.567Q-45.266 24.711-45.586 24.778Q-45.907 24.844-46.192 24.844Q-46.583 24.844-47.024 24.721Q-47.465 24.598-47.770 24.331Q-48.075 24.063-48.075 23.668M-47.536 23.661Q-47.536 23.879-47.295 24.020Q-47.055 24.161-46.731 24.227Q-46.407 24.293-46.192 24.293Q-45.977 24.293-45.653 24.227Q-45.329 24.161-45.088 24.020Q-44.848 23.879-44.848 23.661Q-44.848 23.372-45.073 23.233Q-45.297 23.094-45.579 23.061Q-45.860 23.028-46.208 23.028L-46.825 23.028Q-47.008 23.028-47.170 23.108Q-47.333 23.188-47.434 23.334Q-47.536 23.481-47.536 23.661M-46.426 21.563Q-46.126 21.563-45.907 21.344Q-45.688 21.125-45.688 20.821Q-45.688 20.665-45.743 20.534Q-45.797 20.403-45.903 20.297Q-46.008 20.192-46.139 20.137Q-46.270 20.082-46.426 20.082Q-46.731 20.082-46.950 20.301Q-47.168 20.520-47.168 20.821Q-47.168 21.118-46.946 21.340Q-46.723 21.563-46.426 21.563M-43.844 22.782L-43.844 22.692Q-43.786 22.485-43.594 22.461L-42.883 22.461L-42.883 20.133L-43.594 20.133Q-43.790 20.110-43.844 19.891L-43.844 19.805Q-43.786 19.594-43.594 19.571L-42.493 19.571Q-42.293 19.590-42.243 19.805L-42.243 20.133Q-41.981 19.848-41.626 19.690Q-41.270 19.532-40.883 19.532Q-40.590 19.532-40.356 19.666Q-40.122 19.801-40.122 20.067Q-40.122 20.235-40.231 20.352Q-40.340 20.469-40.508 20.469Q-40.661 20.469-40.776 20.358Q-40.891 20.247-40.891 20.090Q-41.266 20.090-41.581 20.291Q-41.895 20.493-42.069 20.827Q-42.243 21.161-42.243 21.540L-42.243 22.461L-41.297 22.461Q-41.090 22.485-41.051 22.692L-41.051 22.782Q-41.090 22.997-41.297 23.020L-43.594 23.020Q-43.786 22.997-43.844 22.782M-39.415 21.907Q-39.415 21.461-39.001 21.204Q-38.586 20.946-38.045 20.846Q-37.504 20.747-36.997 20.739Q-36.997 20.524-37.131 20.372Q-37.266 20.219-37.473 20.143Q-37.680 20.067-37.891 20.067Q-38.235 20.067-38.395 20.090L-38.395 20.149Q-38.395 20.317-38.514 20.432Q-38.633 20.547-38.797 20.547Q-38.973 20.547-39.088 20.424Q-39.204 20.301-39.204 20.133Q-39.204 19.727-38.823 19.618Q-38.442 19.508-37.883 19.508Q-37.614 19.508-37.346 19.586Q-37.079 19.665-36.854 19.815Q-36.629 19.965-36.493 20.186Q-36.356 20.407-36.356 20.684L-36.356 22.403Q-36.356 22.461-35.829 22.461Q-35.633 22.481-35.583 22.692L-35.583 22.782Q-35.633 22.997-35.829 23.020L-35.973 23.020Q-36.317 23.020-36.545 22.973Q-36.774 22.926-36.918 22.739Q-37.379 23.059-38.086 23.059Q-38.422 23.059-38.727 22.918Q-39.032 22.778-39.223 22.516Q-39.415 22.254-39.415 21.907M-38.774 21.915Q-38.774 22.188-38.532 22.344Q-38.290 22.500-38.004 22.500Q-37.786 22.500-37.553 22.442Q-37.321 22.383-37.159 22.245Q-36.997 22.106-36.997 21.883L-36.997 21.293Q-37.278 21.293-37.694 21.350Q-38.110 21.407-38.442 21.545Q-38.774 21.684-38.774 21.915M-33.711 23.059Q-34.176 23.059-34.542 22.809Q-34.907 22.559-35.112 22.155Q-35.317 21.750-35.317 21.293Q-35.317 20.950-35.192 20.629Q-35.067 20.309-34.834 20.059Q-34.602 19.809-34.297 19.670Q-33.993 19.532-33.637 19.532Q-33.126 19.532-32.719 19.852L-32.719 18.692L-33.141 18.692Q-33.352 18.668-33.391 18.454L-33.391 18.364Q-33.352 18.157-33.141 18.133L-32.329 18.133Q-32.129 18.157-32.079 18.364L-32.079 22.461L-31.653 22.461Q-31.446 22.485-31.407 22.692L-31.407 22.782Q-31.446 22.997-31.653 23.020L-32.469 23.020Q-32.668 22.997-32.719 22.782L-32.719 22.653Q-32.915 22.844-33.176 22.952Q-33.438 23.059-33.711 23.059M-33.672 22.500Q-33.313 22.500-33.061 22.231Q-32.809 21.961-32.719 21.586L-32.719 20.754Q-32.778 20.567-32.905 20.416Q-33.032 20.266-33.209 20.178Q-33.387 20.090-33.583 20.090Q-33.891 20.090-34.141 20.260Q-34.391 20.430-34.536 20.715Q-34.680 21-34.680 21.301Q-34.680 21.750-34.395 22.125Q-34.110 22.500-33.672 22.500M-30.711 22.782L-30.711 22.692Q-30.661 22.485-30.465 22.461L-29.426 22.461L-29.426 20.133L-30.399 20.133Q-30.598 20.110-30.649 19.891L-30.649 19.805Q-30.598 19.594-30.399 19.571L-29.032 19.571Q-28.836 19.590-28.786 19.805L-28.786 22.461L-27.872 22.461Q-27.676 22.485-27.626 22.692L-27.626 22.782Q-27.676 22.997-27.872 23.020L-30.465 23.020Q-30.661 22.997-30.711 22.782M-29.680 18.594L-29.680 18.540Q-29.680 18.368-29.543 18.247Q-29.407 18.125-29.231 18.125Q-29.059 18.125-28.922 18.247Q-28.786 18.368-28.786 18.540L-28.786 18.594Q-28.786 18.770-28.922 18.891Q-29.059 19.012-29.231 19.012Q-29.407 19.012-29.543 18.891Q-29.680 18.770-29.680 18.594M-23.571 21.532L-26.012 21.532Q-25.958 21.809-25.760 22.032Q-25.563 22.254-25.286 22.377Q-25.008 22.500-24.723 22.500Q-24.251 22.500-24.028 22.211Q-24.020 22.200-23.963 22.094Q-23.907 21.989-23.858 21.946Q-23.809 21.903-23.715 21.891L-23.571 21.891Q-23.379 21.911-23.321 22.125L-23.321 22.180Q-23.387 22.481-23.618 22.678Q-23.848 22.875-24.161 22.967Q-24.473 23.059-24.778 23.059Q-25.262 23.059-25.702 22.831Q-26.141 22.602-26.409 22.202Q-26.676 21.801-26.676 21.309L-26.676 21.250Q-26.676 20.782-26.430 20.379Q-26.184 19.977-25.776 19.743Q-25.368 19.508-24.899 19.508Q-24.395 19.508-24.042 19.731Q-23.688 19.954-23.504 20.342Q-23.321 20.731-23.321 21.235L-23.321 21.293Q-23.379 21.508-23.571 21.532M-26.004 20.981L-23.977 20.981Q-24.024 20.571-24.262 20.319Q-24.501 20.067-24.899 20.067Q-25.293 20.067-25.600 20.329Q-25.907 20.590-26.004 20.981M-22.766 22.782L-22.766 22.692Q-22.723 22.485-22.516 22.461L-22.094 22.461L-22.094 20.133L-22.516 20.133Q-22.723 20.110-22.766 19.891L-22.766 19.805Q-22.719 19.594-22.516 19.571L-21.700 19.571Q-21.504 19.594-21.454 19.805L-21.454 19.891L-21.461 19.915Q-21.235 19.735-20.961 19.633Q-20.688 19.532-20.395 19.532Q-20.047 19.532-19.809 19.672Q-19.571 19.813-19.456 20.071Q-19.340 20.329-19.340 20.684L-19.340 22.461L-18.915 22.461Q-18.708 22.485-18.668 22.692L-18.668 22.782Q-18.708 22.997-18.915 23.020L-20.309 23.020Q-20.504 22.997-20.555 22.782L-20.555 22.692Q-20.504 22.481-20.309 22.461L-19.981 22.461L-19.981 20.715Q-19.981 20.407-20.071 20.249Q-20.161 20.090-20.454 20.090Q-20.723 20.090-20.952 20.221Q-21.180 20.352-21.317 20.581Q-21.454 20.809-21.454 21.075L-21.454 22.461L-21.028 22.461Q-20.821 22.485-20.782 22.692L-20.782 22.782Q-20.821 22.997-21.028 23.020L-22.516 23.020Q-22.723 22.997-22.766 22.782M-17.391 21.915L-17.391 20.133L-18.141 20.133Q-18.340 20.110-18.391 19.891L-18.391 19.805Q-18.340 19.594-18.141 19.571L-17.391 19.571L-17.391 18.821Q-17.340 18.614-17.141 18.586L-16.997 18.586Q-16.801 18.614-16.751 18.821L-16.751 19.571L-15.391 19.571Q-15.200 19.590-15.141 19.805L-15.141 19.891Q-15.196 20.110-15.391 20.133L-16.751 20.133L-16.751 21.883Q-16.751 22.500-16.176 22.500Q-15.926 22.500-15.762 22.315Q-15.598 22.129-15.598 21.883Q-15.598 21.790-15.526 21.719Q-15.454 21.649-15.352 21.637L-15.208 21.637Q-15.008 21.661-14.958 21.868L-14.958 21.915Q-14.958 22.239-15.141 22.502Q-15.325 22.766-15.618 22.913Q-15.911 23.059-16.231 23.059Q-16.743 23.059-17.067 22.749Q-17.391 22.438-17.391 21.915\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-10.624 -76.443)\">\u003Cpath d=\"M-11.327 24.149Q-11.327 23.950-11.190 23.801Q-11.053 23.653-10.854 23.653Q-10.717 23.653-10.622 23.739Q-10.526 23.825-10.526 23.957Q-10.526 24.079-10.592 24.188Q-10.659 24.297-10.768 24.356Q-10.674 24.375-10.499 24.385Q-10.323 24.395-10.104 24.395Q-9.729 24.395-9.409 24.135Q-9.088 23.875-8.999 23.508L-8.776 22.614Q-9.225 23.020-9.702 23.020Q-10.061 23.020-10.331 22.844Q-10.600 22.668-10.743 22.375Q-10.885 22.082-10.885 21.723Q-10.885 21.207-10.608 20.684Q-10.331 20.161-9.862 19.827Q-9.393 19.493-8.870 19.493Q-8.713 19.493-8.567 19.545Q-8.420 19.598-8.299 19.702Q-8.178 19.805-8.104 19.930Q-8.077 19.805-7.971 19.725Q-7.866 19.645-7.737 19.645Q-7.624 19.645-7.543 19.715Q-7.463 19.786-7.463 19.899Q-7.463 19.922-7.465 19.938Q-7.467 19.954-7.471 19.973L-8.374 23.563Q-8.463 23.915-8.737 24.159Q-9.010 24.403-9.381 24.528Q-9.752 24.653-10.112 24.653Q-10.600 24.653-10.963 24.565Q-11.327 24.477-11.327 24.149M-9.686 22.762Q-9.389 22.762-9.114 22.563Q-8.838 22.364-8.639 22.075L-8.213 20.387Q-8.276 20.122-8.452 19.934Q-8.627 19.747-8.885 19.747Q-9.225 19.747-9.495 20.020Q-9.764 20.293-9.920 20.668Q-9.999 20.868-10.069 21.114Q-10.139 21.360-10.186 21.624Q-10.233 21.887-10.233 22.051Q-10.233 22.235-10.174 22.397Q-10.116 22.559-9.991 22.661Q-9.866 22.762-9.686 22.762\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" d=\"M-46.89 23.02v-65.442h17.071V23.02Zm17.071-65.442\" style=\"stroke-width:.8\"\u002F>\u003Cg transform=\"translate(7.877 11.24)\">\u003Cpath d=\"M-46.823 23.088Q-47.147 23.088-47.392 22.931Q-47.636 22.774-47.768 22.509Q-47.899 22.244-47.899 21.919Q-47.899 21.451-47.646 20.988Q-47.394 20.525-46.970 20.229Q-46.546 19.934-46.074 19.934Q-45.859 19.934-45.673 20.038Q-45.486 20.142-45.367 20.327L-44.987 18.799Q-44.953 18.696-44.953 18.611Q-44.953 18.519-45.387 18.519Q-45.473 18.491-45.473 18.406L-45.442 18.296Q-45.415 18.249-45.367 18.238L-44.386 18.163Q-44.348 18.163-44.314 18.190Q-44.280 18.218-44.280 18.266L-45.288 22.292Q-45.315 22.449-45.315 22.538Q-45.315 22.665-45.264 22.765Q-45.213 22.866-45.093 22.866Q-44.871 22.866-44.758 22.613Q-44.646 22.360-44.560 21.991Q-44.536 21.930-44.485 21.930L-44.372 21.930Q-44.338 21.930-44.316 21.959Q-44.293 21.988-44.293 22.012Q-44.293 22.025-44.300 22.039Q-44.563 23.088-45.107 23.088Q-45.356 23.088-45.565 22.965Q-45.773 22.842-45.842 22.613Q-46.317 23.088-46.823 23.088M-46.809 22.866Q-46.536 22.866-46.284 22.682Q-46.033 22.497-45.842 22.230L-45.473 20.750Q-45.510 20.590-45.591 20.453Q-45.671 20.316-45.797 20.236Q-45.924 20.156-46.088 20.156Q-46.296 20.156-46.483 20.280Q-46.669 20.405-46.807 20.597Q-46.946 20.788-47.031 20.990Q-47.144 21.284-47.228 21.629Q-47.312 21.974-47.312 22.224Q-47.312 22.480-47.183 22.673Q-47.055 22.866-46.809 22.866\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"none\" font-size=\"8\">\u003Cg transform=\"translate(56.903 -74.665)\">\u003Cpath d=\"M-47.727 22.821L-47.727 21.907Q-47.700 21.700-47.489 21.676L-47.321 21.676Q-47.157 21.700-47.098 21.860Q-46.895 22.500-46.168 22.500Q-45.961 22.500-45.733 22.465Q-45.504 22.430-45.336 22.315Q-45.168 22.200-45.168 21.997Q-45.168 21.786-45.391 21.672Q-45.614 21.559-45.887 21.516L-46.586 21.403Q-47.727 21.192-47.727 20.469Q-47.727 20.180-47.583 19.991Q-47.438 19.801-47.198 19.694Q-46.958 19.586-46.702 19.547Q-46.446 19.508-46.168 19.508Q-45.918 19.508-45.725 19.538Q-45.532 19.567-45.368 19.645Q-45.290 19.528-45.161 19.508L-45.083 19.508Q-44.985 19.520-44.922 19.582Q-44.860 19.645-44.848 19.739L-44.848 20.446Q-44.860 20.540-44.922 20.606Q-44.985 20.672-45.083 20.684L-45.251 20.684Q-45.344 20.672-45.411 20.606Q-45.477 20.540-45.489 20.446Q-45.489 20.067-46.184 20.067Q-46.532 20.067-46.850 20.149Q-47.168 20.231-47.168 20.477Q-47.168 20.743-46.497 20.852L-45.793 20.973Q-45.309 21.055-44.959 21.303Q-44.610 21.551-44.610 21.997Q-44.610 22.387-44.846 22.629Q-45.083 22.872-45.432 22.965Q-45.782 23.059-46.168 23.059Q-46.747 23.059-47.145 22.805Q-47.215 22.930-47.264 22.987Q-47.313 23.043-47.418 23.059L-47.489 23.059Q-47.704 23.036-47.727 22.821M-44.122 22.782L-44.122 22.692Q-44.071 22.481-43.876 22.461L-43.676 22.461L-43.676 20.133L-43.876 20.133Q-44.083 20.110-44.122 19.891L-44.122 19.805Q-44.071 19.590-43.876 19.571L-43.395 19.571Q-43.204 19.594-43.145 19.805Q-42.825 19.532-42.411 19.532Q-42.219 19.532-42.051 19.641Q-41.883 19.750-41.809 19.922Q-41.633 19.731-41.411 19.631Q-41.188 19.532-40.946 19.532Q-40.528 19.532-40.374 19.874Q-40.219 20.215-40.219 20.684L-40.219 22.461L-40.020 22.461Q-39.809 22.485-39.770 22.692L-39.770 22.782Q-39.821 22.997-40.020 23.020L-40.836 23.020Q-41.032 22.997-41.083 22.782L-41.083 22.692Q-41.032 22.485-40.836 22.461L-40.747 22.461L-40.747 20.715Q-40.747 20.090-40.997 20.090Q-41.329 20.090-41.506 20.401Q-41.684 20.711-41.684 21.075L-41.684 22.461L-41.481 22.461Q-41.274 22.485-41.235 22.692L-41.235 22.782Q-41.286 22.997-41.481 23.020L-42.297 23.020Q-42.497 22.997-42.547 22.782L-42.547 22.692Q-42.497 22.485-42.297 22.461L-42.211 22.461L-42.211 20.715Q-42.211 20.090-42.458 20.090Q-42.790 20.090-42.967 20.403Q-43.145 20.715-43.145 21.075L-43.145 22.461L-42.946 22.461Q-42.739 22.485-42.700 22.692L-42.700 22.782Q-42.751 23-42.946 23.020L-43.876 23.020Q-44.083 22.997-44.122 22.782M-39.415 21.907Q-39.415 21.461-39.001 21.204Q-38.586 20.946-38.045 20.846Q-37.504 20.747-36.997 20.739Q-36.997 20.524-37.131 20.372Q-37.266 20.219-37.473 20.143Q-37.680 20.067-37.891 20.067Q-38.235 20.067-38.395 20.090L-38.395 20.149Q-38.395 20.317-38.514 20.432Q-38.633 20.547-38.797 20.547Q-38.973 20.547-39.088 20.424Q-39.204 20.301-39.204 20.133Q-39.204 19.727-38.823 19.618Q-38.442 19.508-37.883 19.508Q-37.614 19.508-37.346 19.586Q-37.079 19.665-36.854 19.815Q-36.629 19.965-36.493 20.186Q-36.356 20.407-36.356 20.684L-36.356 22.403Q-36.356 22.461-35.829 22.461Q-35.633 22.481-35.583 22.692L-35.583 22.782Q-35.633 22.997-35.829 23.020L-35.973 23.020Q-36.317 23.020-36.545 22.973Q-36.774 22.926-36.918 22.739Q-37.379 23.059-38.086 23.059Q-38.422 23.059-38.727 22.918Q-39.032 22.778-39.223 22.516Q-39.415 22.254-39.415 21.907M-38.774 21.915Q-38.774 22.188-38.532 22.344Q-38.290 22.500-38.004 22.500Q-37.786 22.500-37.553 22.442Q-37.321 22.383-37.159 22.245Q-36.997 22.106-36.997 21.883L-36.997 21.293Q-37.278 21.293-37.694 21.350Q-38.110 21.407-38.442 21.545Q-38.774 21.684-38.774 21.915M-35.126 22.782L-35.126 22.692Q-35.075 22.485-34.879 22.461L-33.774 22.461L-33.774 18.692L-34.879 18.692Q-35.075 18.668-35.126 18.454L-35.126 18.364Q-35.075 18.157-34.879 18.133L-33.383 18.133Q-33.192 18.157-33.133 18.364L-33.133 22.461L-32.032 22.461Q-31.833 22.485-31.782 22.692L-31.782 22.782Q-31.833 22.997-32.032 23.020L-34.879 23.020Q-35.075 22.997-35.126 22.782M-30.879 22.782L-30.879 22.692Q-30.829 22.485-30.633 22.461L-29.528 22.461L-29.528 18.692L-30.633 18.692Q-30.829 18.668-30.879 18.454L-30.879 18.364Q-30.829 18.157-30.633 18.133L-29.137 18.133Q-28.946 18.157-28.887 18.364L-28.887 22.461L-27.786 22.461Q-27.586 22.485-27.536 22.692L-27.536 22.782Q-27.586 22.997-27.786 23.020L-30.633 23.020Q-30.829 22.997-30.879 22.782\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(56.903 -74.665)\">\u003Cpath d=\"M-22.725 23.098Q-23.081 23.098-23.350 22.918Q-23.620 22.739-23.760 22.444Q-23.901 22.149-23.901 21.790Q-23.901 21.403-23.739 20.989Q-23.577 20.575-23.293 20.237Q-23.010 19.899-22.641 19.696Q-22.272 19.493-21.870 19.493Q-21.377 19.493-21.100 19.942L-20.663 18.172Q-20.620 18.008-20.620 17.957Q-20.620 17.852-21.116 17.852Q-21.213 17.821-21.213 17.723L-21.190 17.622Q-21.159 17.567-21.100 17.555L-19.999 17.469Q-19.956 17.469-19.916 17.500Q-19.877 17.532-19.877 17.586L-21.030 22.188Q-21.069 22.434-21.069 22.485Q-21.069 22.844-20.823 22.844Q-20.678 22.844-20.577 22.737Q-20.475 22.629-20.411 22.475Q-20.346 22.321-20.297 22.131Q-20.249 21.942-20.229 21.844Q-20.202 21.774-20.139 21.774L-20.038 21.774Q-19.999 21.774-19.973 21.807Q-19.948 21.840-19.948 21.868Q-19.948 21.883-19.956 21.899Q-20.069 22.391-20.268 22.745Q-20.467 23.098-20.838 23.098Q-21.120 23.098-21.346 22.956Q-21.573 22.813-21.655 22.555Q-21.877 22.797-22.151 22.948Q-22.424 23.098-22.725 23.098M-22.709 22.844Q-22.499 22.844-22.290 22.731Q-22.081 22.618-21.911 22.444Q-21.741 22.270-21.612 22.075Q-21.624 22.090-21.633 22.104Q-21.643 22.118-21.655 22.133L-21.221 20.411Q-21.260 20.231-21.346 20.081Q-21.432 19.930-21.569 19.838Q-21.706 19.747-21.885 19.747Q-22.221 19.747-22.493 20.028Q-22.764 20.309-22.924 20.684Q-23.049 21.004-23.147 21.424Q-23.245 21.844-23.245 22.125Q-23.245 22.415-23.112 22.629Q-22.979 22.844-22.709 22.844\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" d=\"M8.593-13.969v-28.453h28.452v28.453Zm28.452-28.453\" style=\"stroke-width:.8\"\u002F>\u003Cg stroke=\"none\" font-size=\"8\">\u003Cg transform=\"translate(130.892 -74.665)\">\u003Cpath d=\"M-48.368 22.782L-48.368 22.692Q-48.317 22.481-48.122 22.461L-47.922 22.461L-47.922 20.133L-48.122 20.133Q-48.329 20.110-48.368 19.891L-48.368 19.805Q-48.317 19.590-48.122 19.571L-47.641 19.571Q-47.450 19.594-47.391 19.805Q-47.071 19.532-46.657 19.532Q-46.465 19.532-46.297 19.641Q-46.129 19.750-46.055 19.922Q-45.879 19.731-45.657 19.631Q-45.434 19.532-45.192 19.532Q-44.774 19.532-44.620 19.874Q-44.465 20.215-44.465 20.684L-44.465 22.461L-44.266 22.461Q-44.055 22.485-44.016 22.692L-44.016 22.782Q-44.067 22.997-44.266 23.020L-45.083 23.020Q-45.278 22.997-45.329 22.782L-45.329 22.692Q-45.278 22.485-45.083 22.461L-44.993 22.461L-44.993 20.715Q-44.993 20.090-45.243 20.090Q-45.575 20.090-45.752 20.401Q-45.930 20.711-45.930 21.075L-45.930 22.461L-45.727 22.461Q-45.520 22.485-45.481 22.692L-45.481 22.782Q-45.532 22.997-45.727 23.020L-46.543 23.020Q-46.743 22.997-46.793 22.782L-46.793 22.692Q-46.743 22.485-46.543 22.461L-46.458 22.461L-46.458 20.715Q-46.458 20.090-46.704 20.090Q-47.036 20.090-47.213 20.403Q-47.391 20.715-47.391 21.075L-47.391 22.461L-47.192 22.461Q-46.985 22.485-46.946 22.692L-46.946 22.782Q-46.997 23-47.192 23.020L-48.122 23.020Q-48.329 22.997-48.368 22.782M-40.555 21.532L-42.997 21.532Q-42.942 21.809-42.745 22.032Q-42.547 22.254-42.270 22.377Q-41.993 22.500-41.708 22.500Q-41.235 22.500-41.012 22.211Q-41.004 22.200-40.948 22.094Q-40.891 21.989-40.842 21.946Q-40.793 21.903-40.700 21.891L-40.555 21.891Q-40.364 21.911-40.305 22.125L-40.305 22.180Q-40.372 22.481-40.602 22.678Q-40.833 22.875-41.145 22.967Q-41.458 23.059-41.762 23.059Q-42.247 23.059-42.686 22.831Q-43.126 22.602-43.393 22.202Q-43.661 21.801-43.661 21.309L-43.661 21.250Q-43.661 20.782-43.415 20.379Q-43.168 19.977-42.760 19.743Q-42.352 19.508-41.883 19.508Q-41.379 19.508-41.026 19.731Q-40.672 19.954-40.489 20.342Q-40.305 20.731-40.305 21.235L-40.305 21.293Q-40.364 21.508-40.555 21.532M-42.989 20.981L-40.961 20.981Q-41.008 20.571-41.247 20.319Q-41.485 20.067-41.883 20.067Q-42.278 20.067-42.584 20.329Q-42.891 20.590-42.989 20.981M-37.958 23.059Q-38.422 23.059-38.788 22.809Q-39.153 22.559-39.358 22.155Q-39.563 21.750-39.563 21.293Q-39.563 20.950-39.438 20.629Q-39.313 20.309-39.081 20.059Q-38.848 19.809-38.543 19.670Q-38.239 19.532-37.883 19.532Q-37.372 19.532-36.965 19.852L-36.965 18.692L-37.387 18.692Q-37.598 18.668-37.637 18.454L-37.637 18.364Q-37.598 18.157-37.387 18.133L-36.575 18.133Q-36.376 18.157-36.325 18.364L-36.325 22.461L-35.899 22.461Q-35.692 22.485-35.653 22.692L-35.653 22.782Q-35.692 22.997-35.899 23.020L-36.715 23.020Q-36.915 22.997-36.965 22.782L-36.965 22.653Q-37.161 22.844-37.422 22.952Q-37.684 23.059-37.958 23.059M-37.918 22.500Q-37.559 22.500-37.307 22.231Q-37.055 21.961-36.965 21.586L-36.965 20.754Q-37.024 20.567-37.151 20.416Q-37.278 20.266-37.456 20.178Q-37.633 20.090-37.829 20.090Q-38.137 20.090-38.387 20.260Q-38.637 20.430-38.782 20.715Q-38.926 21-38.926 21.301Q-38.926 21.750-38.641 22.125Q-38.356 22.500-37.918 22.500M-34.958 22.782L-34.958 22.692Q-34.907 22.485-34.711 22.461L-33.672 22.461L-33.672 20.133L-34.645 20.133Q-34.844 20.110-34.895 19.891L-34.895 19.805Q-34.844 19.594-34.645 19.571L-33.278 19.571Q-33.083 19.590-33.032 19.805L-33.032 22.461L-32.118 22.461Q-31.922 22.485-31.872 22.692L-31.872 22.782Q-31.922 22.997-32.118 23.020L-34.711 23.020Q-34.907 22.997-34.958 22.782M-33.926 18.594L-33.926 18.540Q-33.926 18.368-33.790 18.247Q-33.653 18.125-33.477 18.125Q-33.305 18.125-33.168 18.247Q-33.032 18.368-33.032 18.540L-33.032 18.594Q-33.032 18.770-33.168 18.891Q-33.305 19.012-33.477 19.012Q-33.653 19.012-33.790 18.891Q-33.926 18.770-33.926 18.594M-30.586 22.165L-30.586 20.133L-31.008 20.133Q-31.215 20.110-31.258 19.891L-31.258 19.805Q-31.211 19.594-31.008 19.571L-30.192 19.571Q-29.997 19.594-29.946 19.805L-29.946 22.133Q-29.946 22.368-29.776 22.434Q-29.606 22.500-29.321 22.500Q-29.114 22.500-28.918 22.424Q-28.723 22.348-28.598 22.198Q-28.473 22.047-28.473 21.836L-28.473 20.133L-28.895 20.133Q-29.106 20.110-29.145 19.891L-29.145 19.805Q-29.106 19.594-28.895 19.571L-28.083 19.571Q-27.883 19.594-27.833 19.805L-27.833 22.461L-27.407 22.461Q-27.200 22.485-27.161 22.692L-27.161 22.782Q-27.200 22.997-27.407 23.020L-28.223 23.020Q-28.422 22.997-28.473 22.797Q-28.876 23.059-29.383 23.059Q-29.618 23.059-29.833 23.018Q-30.047 22.977-30.213 22.875Q-30.379 22.774-30.483 22.596Q-30.586 22.418-30.586 22.165M-27.137 22.782L-27.137 22.692Q-27.086 22.481-26.891 22.461L-26.692 22.461L-26.692 20.133L-26.891 20.133Q-27.098 20.110-27.137 19.891L-27.137 19.805Q-27.086 19.590-26.891 19.571L-26.411 19.571Q-26.219 19.594-26.161 19.805Q-25.840 19.532-25.426 19.532Q-25.235 19.532-25.067 19.641Q-24.899 19.750-24.825 19.922Q-24.649 19.731-24.426 19.631Q-24.204 19.532-23.961 19.532Q-23.543 19.532-23.389 19.874Q-23.235 20.215-23.235 20.684L-23.235 22.461L-23.036 22.461Q-22.825 22.485-22.786 22.692L-22.786 22.782Q-22.836 22.997-23.036 23.020L-23.852 23.020Q-24.047 22.997-24.098 22.782L-24.098 22.692Q-24.047 22.485-23.852 22.461L-23.762 22.461L-23.762 20.715Q-23.762 20.090-24.012 20.090Q-24.344 20.090-24.522 20.401Q-24.700 20.711-24.700 21.075L-24.700 22.461L-24.497 22.461Q-24.290 22.485-24.251 22.692L-24.251 22.782Q-24.301 22.997-24.497 23.020L-25.313 23.020Q-25.512 22.997-25.563 22.782L-25.563 22.692Q-25.512 22.485-25.313 22.461L-25.227 22.461L-25.227 20.715Q-25.227 20.090-25.473 20.090Q-25.805 20.090-25.983 20.403Q-26.161 20.715-26.161 21.075L-26.161 22.461L-25.961 22.461Q-25.754 22.485-25.715 22.692L-25.715 22.782Q-25.766 23-25.961 23.020L-26.891 23.020Q-27.098 22.997-27.137 22.782\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(130.892 -74.665)\">\u003Cpath d=\"M-17.058 23.098Q-17.414 23.098-17.683 22.918Q-17.953 22.739-18.093 22.444Q-18.234 22.149-18.234 21.790Q-18.234 21.403-18.072 20.989Q-17.910 20.575-17.626 20.237Q-17.343 19.899-16.974 19.696Q-16.605 19.493-16.203 19.493Q-15.710 19.493-15.433 19.942L-14.996 18.172Q-14.953 18.008-14.953 17.957Q-14.953 17.852-15.449 17.852Q-15.546 17.821-15.546 17.723L-15.523 17.622Q-15.492 17.567-15.433 17.555L-14.332 17.469Q-14.289 17.469-14.250 17.500Q-14.210 17.532-14.210 17.586L-15.363 22.188Q-15.402 22.434-15.402 22.485Q-15.402 22.844-15.156 22.844Q-15.011 22.844-14.910 22.737Q-14.808 22.629-14.744 22.475Q-14.679 22.321-14.630 22.131Q-14.582 21.942-14.562 21.844Q-14.535 21.774-14.472 21.774L-14.371 21.774Q-14.332 21.774-14.306 21.807Q-14.281 21.840-14.281 21.868Q-14.281 21.883-14.289 21.899Q-14.402 22.391-14.601 22.745Q-14.800 23.098-15.171 23.098Q-15.453 23.098-15.679 22.956Q-15.906 22.813-15.988 22.555Q-16.210 22.797-16.484 22.948Q-16.757 23.098-17.058 23.098M-17.042 22.844Q-16.832 22.844-16.623 22.731Q-16.414 22.618-16.244 22.444Q-16.074 22.270-15.945 22.075Q-15.957 22.090-15.966 22.104Q-15.976 22.118-15.988 22.133L-15.554 20.411Q-15.593 20.231-15.679 20.081Q-15.765 19.930-15.902 19.838Q-16.039 19.747-16.218 19.747Q-16.554 19.747-16.826 20.028Q-17.097 20.309-17.257 20.684Q-17.382 21.004-17.480 21.424Q-17.578 21.844-17.578 22.125Q-17.578 22.415-17.445 22.629Q-17.312 22.844-17.042 22.844\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" d=\"M71.189 8.793v-56.905h56.905V8.793Zm56.905-56.905\" style=\"stroke-width:.8\"\u002F>\u003Cg stroke=\"none\" font-size=\"8\">\u003Cg transform=\"translate(244.691 -86.401)\">\u003Cpath d=\"M-47.864 22.782L-47.864 22.692Q-47.813 22.485-47.618 22.461L-46.512 22.461L-46.512 18.692L-47.618 18.692Q-47.813 18.668-47.864 18.454L-47.864 18.364Q-47.813 18.157-47.618 18.133L-46.122 18.133Q-45.930 18.157-45.872 18.364L-45.872 22.461L-44.770 22.461Q-44.571 22.485-44.520 22.692L-44.520 22.782Q-44.571 22.997-44.770 23.020L-47.618 23.020Q-47.813 22.997-47.864 22.782M-43.661 21.907Q-43.661 21.461-43.247 21.204Q-42.833 20.946-42.292 20.846Q-41.751 20.747-41.243 20.739Q-41.243 20.524-41.377 20.372Q-41.512 20.219-41.719 20.143Q-41.926 20.067-42.137 20.067Q-42.481 20.067-42.641 20.090L-42.641 20.149Q-42.641 20.317-42.760 20.432Q-42.879 20.547-43.043 20.547Q-43.219 20.547-43.334 20.424Q-43.450 20.301-43.450 20.133Q-43.450 19.727-43.069 19.618Q-42.688 19.508-42.129 19.508Q-41.860 19.508-41.592 19.586Q-41.325 19.665-41.100 19.815Q-40.876 19.965-40.739 20.186Q-40.602 20.407-40.602 20.684L-40.602 22.403Q-40.602 22.461-40.075 22.461Q-39.879 22.481-39.829 22.692L-39.829 22.782Q-39.879 22.997-40.075 23.020L-40.219 23.020Q-40.563 23.020-40.792 22.973Q-41.020 22.926-41.165 22.739Q-41.626 23.059-42.333 23.059Q-42.668 23.059-42.973 22.918Q-43.278 22.778-43.469 22.516Q-43.661 22.254-43.661 21.907M-43.020 21.915Q-43.020 22.188-42.778 22.344Q-42.536 22.500-42.251 22.500Q-42.032 22.500-41.799 22.442Q-41.567 22.383-41.405 22.245Q-41.243 22.106-41.243 21.883L-41.243 21.293Q-41.524 21.293-41.940 21.350Q-42.356 21.407-42.688 21.545Q-43.020 21.684-43.020 21.915M-39.598 22.782L-39.598 22.692Q-39.540 22.485-39.348 22.461L-38.637 22.461L-38.637 20.133L-39.348 20.133Q-39.543 20.110-39.598 19.891L-39.598 19.805Q-39.540 19.594-39.348 19.571L-38.247 19.571Q-38.047 19.590-37.997 19.805L-37.997 20.133Q-37.735 19.848-37.379 19.690Q-37.024 19.532-36.637 19.532Q-36.344 19.532-36.110 19.666Q-35.876 19.801-35.876 20.067Q-35.876 20.235-35.985 20.352Q-36.094 20.469-36.262 20.469Q-36.415 20.469-36.530 20.358Q-36.645 20.247-36.645 20.090Q-37.020 20.090-37.334 20.291Q-37.649 20.493-37.823 20.827Q-37.997 21.161-37.997 21.540L-37.997 22.461L-37.051 22.461Q-36.844 22.485-36.805 22.692L-36.805 22.782Q-36.844 22.997-37.051 23.020L-39.348 23.020Q-39.540 22.997-39.598 22.782M-35.336 23.668Q-35.336 23.368-35.188 23.106Q-35.040 22.844-34.790 22.684Q-34.973 22.430-34.973 22.110Q-34.973 21.825-34.821 21.563Q-35.071 21.239-35.071 20.821Q-35.071 20.461-34.879 20.165Q-34.688 19.868-34.370 19.700Q-34.051 19.532-33.688 19.532Q-33.481 19.532-33.278 19.592Q-33.075 19.653-32.911 19.754Q-32.528 19.493-32.055 19.493Q-31.817 19.493-31.635 19.624Q-31.454 19.754-31.454 19.981Q-31.454 20.129-31.555 20.235Q-31.657 20.340-31.813 20.340Q-31.946 20.340-32.042 20.262Q-32.137 20.184-32.168 20.059Q-32.313 20.067-32.512 20.157Q-32.309 20.469-32.309 20.821Q-32.309 21.102-32.420 21.334Q-32.532 21.567-32.727 21.745Q-32.922 21.922-33.174 22.020Q-33.426 22.118-33.688 22.118Q-34.075 22.118-34.407 21.930Q-34.418 21.930-34.428 22.002Q-34.438 22.075-34.438 22.110Q-34.438 22.231-34.372 22.338Q-34.305 22.446-34.184 22.485Q-34.168 22.481-34.155 22.479Q-34.141 22.477-34.118 22.477Q-34.102 22.477-34.071 22.485Q-34.040 22.493-34.032 22.493L-33.438 22.493Q-32.657 22.493-32.116 22.735Q-31.575 22.977-31.575 23.668Q-31.575 23.969-31.754 24.196Q-31.934 24.422-32.231 24.567Q-32.528 24.711-32.848 24.778Q-33.168 24.844-33.454 24.844Q-33.844 24.844-34.286 24.721Q-34.727 24.598-35.032 24.331Q-35.336 24.063-35.336 23.668M-34.797 23.661Q-34.797 23.879-34.557 24.020Q-34.317 24.161-33.993 24.227Q-33.668 24.293-33.454 24.293Q-33.239 24.293-32.915 24.227Q-32.590 24.161-32.350 24.020Q-32.110 23.879-32.110 23.661Q-32.110 23.372-32.334 23.233Q-32.559 23.094-32.840 23.061Q-33.122 23.028-33.469 23.028L-34.086 23.028Q-34.270 23.028-34.432 23.108Q-34.594 23.188-34.696 23.334Q-34.797 23.481-34.797 23.661M-33.688 21.563Q-33.387 21.563-33.168 21.344Q-32.950 21.125-32.950 20.821Q-32.950 20.665-33.004 20.534Q-33.059 20.403-33.165 20.297Q-33.270 20.192-33.401 20.137Q-33.532 20.082-33.688 20.082Q-33.993 20.082-34.211 20.301Q-34.430 20.520-34.430 20.821Q-34.430 21.118-34.208 21.340Q-33.985 21.563-33.688 21.563M-27.817 21.532L-30.258 21.532Q-30.204 21.809-30.006 22.032Q-29.809 22.254-29.532 22.377Q-29.254 22.500-28.969 22.500Q-28.497 22.500-28.274 22.211Q-28.266 22.200-28.209 22.094Q-28.153 21.989-28.104 21.946Q-28.055 21.903-27.961 21.891L-27.817 21.891Q-27.626 21.911-27.567 22.125L-27.567 22.180Q-27.633 22.481-27.864 22.678Q-28.094 22.875-28.407 22.967Q-28.719 23.059-29.024 23.059Q-29.508 23.059-29.948 22.831Q-30.387 22.602-30.655 22.202Q-30.922 21.801-30.922 21.309L-30.922 21.250Q-30.922 20.782-30.676 20.379Q-30.430 19.977-30.022 19.743Q-29.614 19.508-29.145 19.508Q-28.641 19.508-28.288 19.731Q-27.934 19.954-27.751 20.342Q-27.567 20.731-27.567 21.235L-27.567 21.293Q-27.626 21.508-27.817 21.532M-30.251 20.981L-28.223 20.981Q-28.270 20.571-28.508 20.319Q-28.747 20.067-29.145 20.067Q-29.540 20.067-29.846 20.329Q-30.153 20.590-30.251 20.981\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(244.691 -86.401)\">\u003Cpath d=\"M-22.725 23.098Q-23.081 23.098-23.350 22.918Q-23.620 22.739-23.760 22.444Q-23.901 22.149-23.901 21.790Q-23.901 21.403-23.739 20.989Q-23.577 20.575-23.293 20.237Q-23.010 19.899-22.641 19.696Q-22.272 19.493-21.870 19.493Q-21.377 19.493-21.100 19.942L-20.663 18.172Q-20.620 18.008-20.620 17.957Q-20.620 17.852-21.116 17.852Q-21.213 17.821-21.213 17.723L-21.190 17.622Q-21.159 17.567-21.100 17.555L-19.999 17.469Q-19.956 17.469-19.916 17.500Q-19.877 17.532-19.877 17.586L-21.030 22.188Q-21.069 22.434-21.069 22.485Q-21.069 22.844-20.823 22.844Q-20.678 22.844-20.577 22.737Q-20.475 22.629-20.411 22.475Q-20.346 22.321-20.297 22.131Q-20.249 21.942-20.229 21.844Q-20.202 21.774-20.139 21.774L-20.038 21.774Q-19.999 21.774-19.973 21.807Q-19.948 21.840-19.948 21.868Q-19.948 21.883-19.956 21.899Q-20.069 22.391-20.268 22.745Q-20.467 23.098-20.838 23.098Q-21.120 23.098-21.346 22.956Q-21.573 22.813-21.655 22.555Q-21.877 22.797-22.151 22.948Q-22.424 23.098-22.725 23.098M-22.709 22.844Q-22.499 22.844-22.290 22.731Q-22.081 22.618-21.911 22.444Q-21.741 22.270-21.612 22.075Q-21.624 22.090-21.633 22.104Q-21.643 22.118-21.655 22.133L-21.221 20.411Q-21.260 20.231-21.346 20.081Q-21.432 19.930-21.569 19.838Q-21.706 19.747-21.885 19.747Q-22.221 19.747-22.493 20.028Q-22.764 20.309-22.924 20.684Q-23.049 21.004-23.147 21.424Q-23.245 21.844-23.245 22.125Q-23.245 22.415-23.112 22.629Q-22.979 22.844-22.709 22.844\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" d=\"M162.237 40.092v-96.74h96.74v96.74Zm96.74-96.74\" style=\"stroke-width:.8\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-size=\"8\">\u003Cg transform=\"translate(247.463 -29.41)\">\u003Cpath d=\"M-46.809 23.098Q-47.165 23.098-47.434 22.918Q-47.704 22.739-47.844 22.444Q-47.985 22.149-47.985 21.790Q-47.985 21.403-47.823 20.989Q-47.661 20.575-47.377 20.237Q-47.094 19.899-46.725 19.696Q-46.356 19.493-45.954 19.493Q-45.461 19.493-45.184 19.942L-44.747 18.172Q-44.704 18.008-44.704 17.957Q-44.704 17.852-45.200 17.852Q-45.297 17.821-45.297 17.723L-45.274 17.622Q-45.243 17.567-45.184 17.555L-44.083 17.469Q-44.040 17.469-44.001 17.500Q-43.961 17.532-43.961 17.586L-45.114 22.188Q-45.153 22.434-45.153 22.485Q-45.153 22.844-44.907 22.844Q-44.762 22.844-44.661 22.737Q-44.559 22.629-44.495 22.475Q-44.430 22.321-44.381 22.131Q-44.333 21.942-44.313 21.844Q-44.286 21.774-44.223 21.774L-44.122 21.774Q-44.083 21.774-44.057 21.807Q-44.032 21.840-44.032 21.868Q-44.032 21.883-44.040 21.899Q-44.153 22.391-44.352 22.745Q-44.551 23.098-44.922 23.098Q-45.204 23.098-45.430 22.956Q-45.657 22.813-45.739 22.555Q-45.961 22.797-46.235 22.948Q-46.508 23.098-46.809 23.098M-46.793 22.844Q-46.583 22.844-46.374 22.731Q-46.165 22.618-45.995 22.444Q-45.825 22.270-45.696 22.075Q-45.708 22.090-45.717 22.104Q-45.727 22.118-45.739 22.133L-45.305 20.411Q-45.344 20.231-45.430 20.081Q-45.516 19.930-45.653 19.838Q-45.790 19.747-45.969 19.747Q-46.305 19.747-46.577 20.028Q-46.848 20.309-47.008 20.684Q-47.133 21.004-47.231 21.424Q-47.329 21.844-47.329 22.125Q-47.329 22.415-47.196 22.629Q-47.063 22.844-46.793 22.844\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(247.463 -29.41)\">\u003Cpath d=\"M-40.364 22.782L-40.364 18.692L-40.786 18.692Q-40.993 18.668-41.036 18.454L-41.036 18.364Q-40.993 18.157-40.786 18.133L-39.969 18.133Q-39.774 18.157-39.723 18.364L-39.723 19.875Q-39.512 19.707-39.249 19.620Q-38.985 19.532-38.715 19.532Q-38.376 19.532-38.079 19.676Q-37.782 19.821-37.567 20.073Q-37.352 20.325-37.237 20.637Q-37.122 20.950-37.122 21.293Q-37.122 21.758-37.348 22.168Q-37.575 22.579-37.961 22.819Q-38.348 23.059-38.817 23.059Q-39.336 23.059-39.723 22.692L-39.723 22.782Q-39.774 23-39.969 23.020L-40.114 23.020Q-40.305 22.997-40.364 22.782M-38.860 22.500Q-38.551 22.500-38.301 22.331Q-38.051 22.161-37.907 21.879Q-37.762 21.598-37.762 21.293Q-37.762 21.004-37.887 20.725Q-38.012 20.446-38.245 20.268Q-38.477 20.090-38.778 20.090Q-39.098 20.090-39.360 20.276Q-39.622 20.461-39.723 20.762L-39.723 21.614Q-39.633 21.981-39.413 22.241Q-39.192 22.500-38.860 22.500M-36.516 24.141Q-36.516 24.032-36.465 23.940Q-36.415 23.848-36.323 23.797Q-36.231 23.747-36.126 23.747Q-36.016 23.747-35.924 23.797Q-35.833 23.848-35.782 23.940Q-35.731 24.032-35.731 24.141L-35.876 24.141Q-35.876 24.278-35.852 24.278Q-35.594 24.278-35.407 24.086Q-35.219 23.895-35.133 23.629L-34.922 23.020L-36.059 20.133L-36.387 20.133Q-36.583 20.110-36.637 19.891L-36.637 19.805Q-36.579 19.594-36.387 19.571L-35.227 19.571Q-35.032 19.594-34.981 19.805L-34.981 19.891Q-35.032 20.110-35.227 20.133L-35.493 20.133Q-35.157 20.981-34.913 21.635Q-34.669 22.290-34.669 22.379L-34.661 22.379Q-34.661 22.321-34.569 22.028Q-34.477 21.735-34.284 21.151Q-34.090 20.567-33.950 20.133L-34.227 20.133Q-34.438 20.110-34.477 19.891L-34.477 19.805Q-34.426 19.590-34.227 19.571L-33.075 19.571Q-32.868 19.594-32.829 19.805L-32.829 19.891Q-32.868 20.110-33.075 20.133L-33.395 20.133L-34.571 23.629Q-34.739 24.129-35.065 24.483Q-35.391 24.836-35.852 24.836Q-36.126 24.836-36.321 24.625Q-36.516 24.415-36.516 24.141\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(247.463 -29.41)\">\u003Cpath d=\"M-28.268 23.098Q-28.624 23.098-28.893 22.918Q-29.163 22.739-29.303 22.444Q-29.444 22.149-29.444 21.790Q-29.444 21.403-29.282 20.989Q-29.120 20.575-28.836 20.237Q-28.553 19.899-28.184 19.696Q-27.815 19.493-27.413 19.493Q-26.920 19.493-26.643 19.942L-26.206 18.172Q-26.163 18.008-26.163 17.957Q-26.163 17.852-26.659 17.852Q-26.756 17.821-26.756 17.723L-26.733 17.622Q-26.702 17.567-26.643 17.555L-25.542 17.469Q-25.499 17.469-25.459 17.500Q-25.420 17.532-25.420 17.586L-26.573 22.188Q-26.612 22.434-26.612 22.485Q-26.612 22.844-26.366 22.844Q-26.221 22.844-26.120 22.737Q-26.018 22.629-25.954 22.475Q-25.889 22.321-25.840 22.131Q-25.792 21.942-25.772 21.844Q-25.745 21.774-25.682 21.774L-25.581 21.774Q-25.542 21.774-25.516 21.807Q-25.491 21.840-25.491 21.868Q-25.491 21.883-25.499 21.899Q-25.612 22.391-25.811 22.745Q-26.010 23.098-26.381 23.098Q-26.663 23.098-26.889 22.956Q-27.116 22.813-27.198 22.555Q-27.420 22.797-27.694 22.948Q-27.967 23.098-28.268 23.098M-28.252 22.844Q-28.042 22.844-27.833 22.731Q-27.624 22.618-27.454 22.444Q-27.284 22.270-27.155 22.075Q-27.167 22.090-27.176 22.104Q-27.186 22.118-27.198 22.133L-26.764 20.411Q-26.803 20.231-26.889 20.081Q-26.975 19.930-27.112 19.838Q-27.249 19.747-27.428 19.747Q-27.764 19.747-28.036 20.028Q-28.307 20.309-28.467 20.684Q-28.592 21.004-28.690 21.424Q-28.788 21.844-28.788 22.125Q-28.788 22.415-28.655 22.629Q-28.522 22.844-28.252 22.844\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\">\u003Cg transform=\"translate(80.173 37.253)\">\u003Cpath d=\"M-47.727 22.821L-47.727 21.907Q-47.700 21.700-47.489 21.676L-47.321 21.676Q-47.157 21.700-47.098 21.860Q-46.895 22.500-46.168 22.500Q-45.961 22.500-45.733 22.465Q-45.504 22.430-45.336 22.315Q-45.168 22.200-45.168 21.997Q-45.168 21.786-45.391 21.672Q-45.614 21.559-45.887 21.516L-46.586 21.403Q-47.727 21.192-47.727 20.469Q-47.727 20.180-47.583 19.991Q-47.438 19.801-47.198 19.694Q-46.958 19.586-46.702 19.547Q-46.446 19.508-46.168 19.508Q-45.918 19.508-45.725 19.538Q-45.532 19.567-45.368 19.645Q-45.290 19.528-45.161 19.508L-45.083 19.508Q-44.985 19.520-44.922 19.582Q-44.860 19.645-44.848 19.739L-44.848 20.446Q-44.860 20.540-44.922 20.606Q-44.985 20.672-45.083 20.684L-45.251 20.684Q-45.344 20.672-45.411 20.606Q-45.477 20.540-45.489 20.446Q-45.489 20.067-46.184 20.067Q-46.532 20.067-46.850 20.149Q-47.168 20.231-47.168 20.477Q-47.168 20.743-46.497 20.852L-45.793 20.973Q-45.309 21.055-44.959 21.303Q-44.610 21.551-44.610 21.997Q-44.610 22.387-44.846 22.629Q-45.083 22.872-45.432 22.965Q-45.782 23.059-46.168 23.059Q-46.747 23.059-47.145 22.805Q-47.215 22.930-47.264 22.987Q-47.313 23.043-47.418 23.059L-47.489 23.059Q-47.704 23.036-47.727 22.821M-42.868 21.915L-42.868 20.133L-43.618 20.133Q-43.817 20.110-43.868 19.891L-43.868 19.805Q-43.817 19.594-43.618 19.571L-42.868 19.571L-42.868 18.821Q-42.817 18.614-42.618 18.586L-42.473 18.586Q-42.278 18.614-42.227 18.821L-42.227 19.571L-40.868 19.571Q-40.676 19.590-40.618 19.805L-40.618 19.891Q-40.672 20.110-40.868 20.133L-42.227 20.133L-42.227 21.883Q-42.227 22.500-41.653 22.500Q-41.403 22.500-41.239 22.315Q-41.075 22.129-41.075 21.883Q-41.075 21.790-41.002 21.719Q-40.930 21.649-40.829 21.637L-40.684 21.637Q-40.485 21.661-40.434 21.868L-40.434 21.915Q-40.434 22.239-40.618 22.502Q-40.801 22.766-41.094 22.913Q-41.387 23.059-41.708 23.059Q-42.219 23.059-42.543 22.749Q-42.868 22.438-42.868 21.915M-37.700 23.059Q-38.172 23.059-38.557 22.815Q-38.942 22.571-39.165 22.161Q-39.387 21.750-39.387 21.293Q-39.387 20.950-39.262 20.627Q-39.137 20.305-38.907 20.051Q-38.676 19.797-38.370 19.653Q-38.063 19.508-37.700 19.508Q-37.336 19.508-37.024 19.655Q-36.711 19.801-36.489 20.047Q-36.266 20.293-36.139 20.614Q-36.012 20.934-36.012 21.293Q-36.012 21.750-36.237 22.163Q-36.461 22.575-36.846 22.817Q-37.231 23.059-37.700 23.059M-37.700 22.500Q-37.235 22.500-36.944 22.106Q-36.653 21.711-36.653 21.227Q-36.653 20.934-36.788 20.666Q-36.922 20.399-37.163 20.233Q-37.403 20.067-37.700 20.067Q-38.004 20.067-38.243 20.233Q-38.481 20.399-38.616 20.666Q-38.751 20.934-38.751 21.227Q-38.751 21.707-38.458 22.104Q-38.165 22.500-37.700 22.500M-35.352 22.782L-35.352 22.692Q-35.293 22.485-35.102 22.461L-34.391 22.461L-34.391 20.133L-35.102 20.133Q-35.297 20.110-35.352 19.891L-35.352 19.805Q-35.293 19.594-35.102 19.571L-34.001 19.571Q-33.801 19.590-33.751 19.805L-33.751 20.133Q-33.489 19.848-33.133 19.690Q-32.778 19.532-32.391 19.532Q-32.098 19.532-31.864 19.666Q-31.629 19.801-31.629 20.067Q-31.629 20.235-31.739 20.352Q-31.848 20.469-32.016 20.469Q-32.168 20.469-32.284 20.358Q-32.399 20.247-32.399 20.090Q-32.774 20.090-33.088 20.291Q-33.403 20.493-33.577 20.827Q-33.751 21.161-33.751 21.540L-33.751 22.461L-32.805 22.461Q-32.598 22.485-32.559 22.692L-32.559 22.782Q-32.598 22.997-32.805 23.020L-35.102 23.020Q-35.293 22.997-35.352 22.782M-30.922 21.907Q-30.922 21.461-30.508 21.204Q-30.094 20.946-29.553 20.846Q-29.012 20.747-28.504 20.739Q-28.504 20.524-28.639 20.372Q-28.774 20.219-28.981 20.143Q-29.188 20.067-29.399 20.067Q-29.743 20.067-29.903 20.090L-29.903 20.149Q-29.903 20.317-30.022 20.432Q-30.141 20.547-30.305 20.547Q-30.481 20.547-30.596 20.424Q-30.711 20.301-30.711 20.133Q-30.711 19.727-30.331 19.618Q-29.950 19.508-29.391 19.508Q-29.122 19.508-28.854 19.586Q-28.586 19.665-28.362 19.815Q-28.137 19.965-28.001 20.186Q-27.864 20.407-27.864 20.684L-27.864 22.403Q-27.864 22.461-27.336 22.461Q-27.141 22.481-27.090 22.692L-27.090 22.782Q-27.141 22.997-27.336 23.020L-27.481 23.020Q-27.825 23.020-28.053 22.973Q-28.282 22.926-28.426 22.739Q-28.887 23.059-29.594 23.059Q-29.930 23.059-30.235 22.918Q-30.540 22.778-30.731 22.516Q-30.922 22.254-30.922 21.907M-30.282 21.915Q-30.282 22.188-30.040 22.344Q-29.797 22.500-29.512 22.500Q-29.293 22.500-29.061 22.442Q-28.829 22.383-28.667 22.245Q-28.504 22.106-28.504 21.883L-28.504 21.293Q-28.786 21.293-29.202 21.350Q-29.618 21.407-29.950 21.545Q-30.282 21.684-30.282 21.915M-26.844 23.668Q-26.844 23.368-26.696 23.106Q-26.547 22.844-26.297 22.684Q-26.481 22.430-26.481 22.110Q-26.481 21.825-26.329 21.563Q-26.579 21.239-26.579 20.821Q-26.579 20.461-26.387 20.165Q-26.196 19.868-25.877 19.700Q-25.559 19.532-25.196 19.532Q-24.989 19.532-24.786 19.592Q-24.583 19.653-24.418 19.754Q-24.036 19.493-23.563 19.493Q-23.325 19.493-23.143 19.624Q-22.961 19.754-22.961 19.981Q-22.961 20.129-23.063 20.235Q-23.165 20.340-23.321 20.340Q-23.454 20.340-23.549 20.262Q-23.645 20.184-23.676 20.059Q-23.821 20.067-24.020 20.157Q-23.817 20.469-23.817 20.821Q-23.817 21.102-23.928 21.334Q-24.040 21.567-24.235 21.745Q-24.430 21.922-24.682 22.020Q-24.934 22.118-25.196 22.118Q-25.583 22.118-25.915 21.930Q-25.926 21.930-25.936 22.002Q-25.946 22.075-25.946 22.110Q-25.946 22.231-25.879 22.338Q-25.813 22.446-25.692 22.485Q-25.676 22.481-25.663 22.479Q-25.649 22.477-25.626 22.477Q-25.610 22.477-25.579 22.485Q-25.547 22.493-25.540 22.493L-24.946 22.493Q-24.165 22.493-23.624 22.735Q-23.083 22.977-23.083 23.668Q-23.083 23.969-23.262 24.196Q-23.442 24.422-23.739 24.567Q-24.036 24.711-24.356 24.778Q-24.676 24.844-24.961 24.844Q-25.352 24.844-25.793 24.721Q-26.235 24.598-26.540 24.331Q-26.844 24.063-26.844 23.668M-26.305 23.661Q-26.305 23.879-26.065 24.020Q-25.825 24.161-25.501 24.227Q-25.176 24.293-24.961 24.293Q-24.747 24.293-24.422 24.227Q-24.098 24.161-23.858 24.020Q-23.618 23.879-23.618 23.661Q-23.618 23.372-23.842 23.233Q-24.067 23.094-24.348 23.061Q-24.629 23.028-24.977 23.028L-25.594 23.028Q-25.778 23.028-25.940 23.108Q-26.102 23.188-26.204 23.334Q-26.305 23.481-26.305 23.661M-25.196 21.563Q-24.895 21.563-24.676 21.344Q-24.458 21.125-24.458 20.821Q-24.458 20.665-24.512 20.534Q-24.567 20.403-24.672 20.297Q-24.778 20.192-24.909 20.137Q-25.040 20.082-25.196 20.082Q-25.501 20.082-25.719 20.301Q-25.938 20.520-25.938 20.821Q-25.938 21.118-25.715 21.340Q-25.493 21.563-25.196 21.563M-19.325 21.532L-21.766 21.532Q-21.711 21.809-21.514 22.032Q-21.317 22.254-21.040 22.377Q-20.762 22.500-20.477 22.500Q-20.004 22.500-19.782 22.211Q-19.774 22.200-19.717 22.094Q-19.661 21.989-19.612 21.946Q-19.563 21.903-19.469 21.891L-19.325 21.891Q-19.133 21.911-19.075 22.125L-19.075 22.180Q-19.141 22.481-19.372 22.678Q-19.602 22.875-19.915 22.967Q-20.227 23.059-20.532 23.059Q-21.016 23.059-21.456 22.831Q-21.895 22.602-22.163 22.202Q-22.430 21.801-22.430 21.309L-22.430 21.250Q-22.430 20.782-22.184 20.379Q-21.938 19.977-21.530 19.743Q-21.122 19.508-20.653 19.508Q-20.149 19.508-19.795 19.731Q-19.442 19.954-19.258 20.342Q-19.075 20.731-19.075 21.235L-19.075 21.293Q-19.133 21.508-19.325 21.532M-21.758 20.981L-19.731 20.981Q-19.778 20.571-20.016 20.319Q-20.254 20.067-20.653 20.067Q-21.047 20.067-21.354 20.329Q-21.661 20.590-21.758 20.981\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(80.173 37.253)\">\u003Cpath d=\"M-14.074 23.668Q-14.074 23.368-13.925 23.106Q-13.777 22.844-13.527 22.684Q-13.710 22.430-13.710 22.110Q-13.710 21.825-13.558 21.563Q-13.808 21.239-13.808 20.821Q-13.808 20.461-13.617 20.165Q-13.425 19.868-13.107 19.700Q-12.789 19.532-12.425 19.532Q-12.218 19.532-12.015 19.592Q-11.812 19.653-11.648 19.754Q-11.265 19.493-10.792 19.493Q-10.554 19.493-10.373 19.624Q-10.191 19.754-10.191 19.981Q-10.191 20.129-10.292 20.235Q-10.394 20.340-10.550 20.340Q-10.683 20.340-10.779 20.262Q-10.874 20.184-10.906 20.059Q-11.050 20.067-11.249 20.157Q-11.046 20.469-11.046 20.821Q-11.046 21.102-11.158 21.334Q-11.269 21.567-11.464 21.745Q-11.660 21.922-11.912 22.020Q-12.164 22.118-12.425 22.118Q-12.812 22.118-13.144 21.930Q-13.156 21.930-13.166 22.002Q-13.175 22.075-13.175 22.110Q-13.175 22.231-13.109 22.338Q-13.042 22.446-12.921 22.485Q-12.906 22.481-12.892 22.479Q-12.878 22.477-12.855 22.477Q-12.839 22.477-12.808 22.485Q-12.777 22.493-12.769 22.493L-12.175 22.493Q-11.394 22.493-10.853 22.735Q-10.312 22.977-10.312 23.668Q-10.312 23.969-10.492 24.196Q-10.671 24.422-10.968 24.567Q-11.265 24.711-11.585 24.778Q-11.906 24.844-12.191 24.844Q-12.582 24.844-13.023 24.721Q-13.464 24.598-13.769 24.331Q-14.074 24.063-14.074 23.668M-13.535 23.661Q-13.535 23.879-13.294 24.020Q-13.054 24.161-12.730 24.227Q-12.406 24.293-12.191 24.293Q-11.976 24.293-11.652 24.227Q-11.328 24.161-11.087 24.020Q-10.847 23.879-10.847 23.661Q-10.847 23.372-11.072 23.233Q-11.296 23.094-11.578 23.061Q-11.859 23.028-12.207 23.028L-12.824 23.028Q-13.007 23.028-13.169 23.108Q-13.332 23.188-13.433 23.334Q-13.535 23.481-13.535 23.661M-12.425 21.563Q-12.124 21.563-11.906 21.344Q-11.687 21.125-11.687 20.821Q-11.687 20.665-11.742 20.534Q-11.796 20.403-11.902 20.297Q-12.007 20.192-12.138 20.137Q-12.269 20.082-12.425 20.082Q-12.730 20.082-12.949 20.301Q-13.167 20.520-13.167 20.821Q-13.167 21.118-12.945 21.340Q-12.722 21.563-12.425 21.563M-9.843 22.782L-9.843 22.692Q-9.785 22.485-9.593 22.461L-8.882 22.461L-8.882 20.133L-9.593 20.133Q-9.789 20.110-9.843 19.891L-9.843 19.805Q-9.785 19.594-9.593 19.571L-8.492 19.571Q-8.292 19.590-8.242 19.805L-8.242 20.133Q-7.980 19.848-7.624 19.690Q-7.269 19.532-6.882 19.532Q-6.589 19.532-6.355 19.666Q-6.121 19.801-6.121 20.067Q-6.121 20.235-6.230 20.352Q-6.339 20.469-6.507 20.469Q-6.660 20.469-6.775 20.358Q-6.890 20.247-6.890 20.090Q-7.265 20.090-7.580 20.291Q-7.894 20.493-8.068 20.827Q-8.242 21.161-8.242 21.540L-8.242 22.461L-7.296 22.461Q-7.089 22.485-7.050 22.692L-7.050 22.782Q-7.089 22.997-7.296 23.020L-9.593 23.020Q-9.785 22.997-9.843 22.782M-3.699 23.059Q-4.171 23.059-4.556 22.815Q-4.941 22.571-5.164 22.161Q-5.386 21.750-5.386 21.293Q-5.386 20.950-5.261 20.627Q-5.136 20.305-4.906 20.051Q-4.675 19.797-4.369 19.653Q-4.062 19.508-3.699 19.508Q-3.335 19.508-3.023 19.655Q-2.710 19.801-2.488 20.047Q-2.265 20.293-2.138 20.614Q-2.011 20.934-2.011 21.293Q-2.011 21.750-2.236 22.163Q-2.460 22.575-2.845 22.817Q-3.230 23.059-3.699 23.059M-3.699 22.500Q-3.234 22.500-2.943 22.106Q-2.652 21.711-2.652 21.227Q-2.652 20.934-2.787 20.666Q-2.921 20.399-3.162 20.233Q-3.402 20.067-3.699 20.067Q-4.003 20.067-4.242 20.233Q-4.480 20.399-4.615 20.666Q-4.749 20.934-4.749 21.227Q-4.749 21.707-4.457 22.104Q-4.164 22.500-3.699 22.500M-0.566 22.797L-1.031 20.133L-1.191 20.133Q-1.398 20.110-1.437 19.891L-1.437 19.805Q-1.398 19.594-1.191 19.571L-0.023 19.571Q0.188 19.594 0.227 19.805L0.227 19.891Q0.188 20.110-0.023 20.133L-0.496 20.133Q-0.382 20.778-0.304 21.223Q-0.226 21.668-0.175 21.987Q-0.124 22.305-0.124 22.379Q-0.117 22.207 0.168 21.211Q0.208 21.098 0.305 21.024Q0.403 20.950 0.524 20.950L0.602 20.950Q0.723 20.950 0.821 21.024Q0.918 21.098 0.954 21.211Q1.063 21.594 1.145 21.918Q1.227 22.243 1.227 22.379Q1.239 22.090 1.586 20.133L1.114 20.133Q0.907 20.110 0.868 19.891L0.868 19.805Q0.907 19.594 1.114 19.571L2.290 19.571Q2.481 19.594 2.540 19.805L2.540 19.891Q2.485 20.110 2.290 20.133L2.122 20.133L1.657 22.797Q1.633 22.915 1.545 22.987Q1.458 23.059 1.336 23.059L1.196 23.059Q1.075 23.059 0.979 22.987Q0.883 22.915 0.840 22.797Q0.793 22.618 0.721 22.362Q0.649 22.106 0.606 21.897Q0.563 21.688 0.563 21.586Q0.555 21.868 0.274 22.797Q0.247 22.907 0.149 22.983Q0.051 23.059-0.070 23.059L-0.246 23.059Q-0.367 23.059-0.455 22.987Q-0.542 22.915-0.566 22.797M3.258 22.821L3.258 21.907Q3.286 21.700 3.497 21.676L3.665 21.676Q3.829 21.700 3.887 21.860Q4.090 22.500 4.817 22.500Q5.024 22.500 5.252 22.465Q5.481 22.430 5.649 22.315Q5.817 22.200 5.817 21.997Q5.817 21.786 5.594 21.672Q5.372 21.559 5.098 21.516L4.399 21.403Q3.258 21.192 3.258 20.469Q3.258 20.180 3.403 19.991Q3.547 19.801 3.788 19.694Q4.028 19.586 4.284 19.547Q4.540 19.508 4.817 19.508Q5.067 19.508 5.260 19.538Q5.454 19.567 5.618 19.645Q5.696 19.528 5.825 19.508L5.903 19.508Q6.001 19.520 6.063 19.582Q6.126 19.645 6.137 19.739L6.137 20.446Q6.126 20.540 6.063 20.606Q6.001 20.672 5.903 20.684L5.735 20.684Q5.641 20.672 5.575 20.606Q5.508 20.540 5.497 20.446Q5.497 20.067 4.801 20.067Q4.454 20.067 4.135 20.149Q3.817 20.231 3.817 20.477Q3.817 20.743 4.489 20.852L5.192 20.973Q5.676 21.055 6.026 21.303Q6.376 21.551 6.376 21.997Q6.376 22.387 6.139 22.629Q5.903 22.872 5.553 22.965Q5.204 23.059 4.817 23.059Q4.239 23.059 3.840 22.805Q3.770 22.930 3.721 22.987Q3.672 23.043 3.567 23.059L3.497 23.059Q3.282 23.036 3.258 22.821\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(80.173 37.253)\">\u003Cpath d=\"M11.594 21.907Q11.594 21.461 12.008 21.204Q12.422 20.946 12.963 20.846Q13.504 20.747 14.012 20.739Q14.012 20.524 13.877 20.372Q13.743 20.219 13.536 20.143Q13.329 20.067 13.118 20.067Q12.774 20.067 12.614 20.090L12.614 20.149Q12.614 20.317 12.495 20.432Q12.376 20.547 12.211 20.547Q12.036 20.547 11.920 20.424Q11.805 20.301 11.805 20.133Q11.805 19.727 12.186 19.618Q12.567 19.508 13.126 19.508Q13.395 19.508 13.663 19.586Q13.930 19.665 14.155 19.815Q14.379 19.965 14.516 20.186Q14.653 20.407 14.653 20.684L14.653 22.403Q14.653 22.461 15.180 22.461Q15.376 22.481 15.426 22.692L15.426 22.782Q15.376 22.997 15.180 23.020L15.036 23.020Q14.692 23.020 14.463 22.973Q14.235 22.926 14.090 22.739Q13.629 23.059 12.922 23.059Q12.586 23.059 12.282 22.918Q11.977 22.778 11.786 22.516Q11.594 22.254 11.594 21.907M12.235 21.915Q12.235 22.188 12.477 22.344Q12.719 22.500 13.004 22.500Q13.223 22.500 13.456 22.442Q13.688 22.383 13.850 22.245Q14.012 22.106 14.012 21.883L14.012 21.293Q13.731 21.293 13.315 21.350Q12.899 21.407 12.567 21.545Q12.235 21.684 12.235 21.915M16.020 22.821L16.020 21.907Q16.047 21.700 16.258 21.676L16.426 21.676Q16.590 21.700 16.649 21.860Q16.852 22.500 17.579 22.500Q17.786 22.500 18.014 22.465Q18.243 22.430 18.411 22.315Q18.579 22.200 18.579 21.997Q18.579 21.786 18.356 21.672Q18.133 21.559 17.860 21.516L17.161 21.403Q16.020 21.192 16.020 20.469Q16.020 20.180 16.165 19.991Q16.309 19.801 16.549 19.694Q16.790 19.586 17.045 19.547Q17.301 19.508 17.579 19.508Q17.829 19.508 18.022 19.538Q18.215 19.567 18.379 19.645Q18.458 19.528 18.586 19.508L18.665 19.508Q18.762 19.520 18.825 19.582Q18.887 19.645 18.899 19.739L18.899 20.446Q18.887 20.540 18.825 20.606Q18.762 20.672 18.665 20.684L18.497 20.684Q18.403 20.672 18.336 20.606Q18.270 20.540 18.258 20.446Q18.258 20.067 17.563 20.067Q17.215 20.067 16.897 20.149Q16.579 20.231 16.579 20.477Q16.579 20.743 17.251 20.852L17.954 20.973Q18.438 21.055 18.788 21.303Q19.137 21.551 19.137 21.997Q19.137 22.387 18.901 22.629Q18.665 22.872 18.315 22.965Q17.965 23.059 17.579 23.059Q17.001 23.059 16.602 22.805Q16.532 22.930 16.483 22.987Q16.434 23.043 16.329 23.059L16.258 23.059Q16.043 23.036 16.020 22.821\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(80.173 37.253)\">\u003Cpath d=\"M24.026 23.098Q23.670 23.098 23.401 22.918Q23.131 22.739 22.991 22.444Q22.850 22.149 22.850 21.790Q22.850 21.403 23.012 20.989Q23.174 20.575 23.458 20.237Q23.741 19.899 24.110 19.696Q24.479 19.493 24.881 19.493Q25.374 19.493 25.651 19.942L26.088 18.172Q26.131 18.008 26.131 17.957Q26.131 17.852 25.635 17.852Q25.538 17.821 25.538 17.723L25.561 17.622Q25.592 17.567 25.651 17.555L26.752 17.469Q26.795 17.469 26.834 17.500Q26.874 17.532 26.874 17.586L25.721 22.188Q25.682 22.434 25.682 22.485Q25.682 22.844 25.928 22.844Q26.073 22.844 26.174 22.737Q26.276 22.629 26.340 22.475Q26.405 22.321 26.454 22.131Q26.502 21.942 26.522 21.844Q26.549 21.774 26.612 21.774L26.713 21.774Q26.752 21.774 26.778 21.807Q26.803 21.840 26.803 21.868Q26.803 21.883 26.795 21.899Q26.682 22.391 26.483 22.745Q26.284 23.098 25.913 23.098Q25.631 23.098 25.405 22.956Q25.178 22.813 25.096 22.555Q24.874 22.797 24.600 22.948Q24.327 23.098 24.026 23.098M24.042 22.844Q24.252 22.844 24.461 22.731Q24.670 22.618 24.840 22.444Q25.010 22.270 25.139 22.075Q25.127 22.090 25.118 22.104Q25.108 22.118 25.096 22.133L25.530 20.411Q25.491 20.231 25.405 20.081Q25.319 19.930 25.182 19.838Q25.045 19.747 24.866 19.747Q24.530 19.747 24.258 20.028Q23.987 20.309 23.827 20.684Q23.702 21.004 23.604 21.424Q23.506 21.844 23.506 22.125Q23.506 22.415 23.639 22.629Q23.772 22.844 24.042 22.844\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(80.173 37.253)\">\u003Cpath d=\"M29.936 20.196L27.326 20.196L27.326 20.011Q27.332 19.988 27.352 19.962L28.503 18.907Q28.843 18.596 29.023 18.410Q29.204 18.224 29.349 17.964Q29.494 17.703 29.494 17.407Q29.494 17.134 29.368 16.919Q29.242 16.704 29.022 16.584Q28.802 16.464 28.527 16.464Q28.351 16.464 28.181 16.521Q28.011 16.578 27.879 16.685Q27.748 16.792 27.668 16.950Q27.756 16.950 27.834 16.994Q27.912 17.038 27.956 17.114Q27.999 17.190 27.999 17.287Q27.999 17.427 27.903 17.524Q27.806 17.621 27.663 17.621Q27.525 17.621 27.425 17.521Q27.326 17.422 27.326 17.287Q27.326 16.962 27.516 16.714Q27.707 16.467 28.010 16.336Q28.313 16.206 28.629 16.206Q29.010 16.206 29.353 16.341Q29.696 16.475 29.910 16.748Q30.124 17.020 30.124 17.407Q30.124 17.682 29.999 17.909Q29.874 18.136 29.694 18.308Q29.514 18.479 29.189 18.719Q28.864 18.960 28.779 19.027L28.023 19.631L28.556 19.631Q29.045 19.631 29.376 19.623Q29.707 19.616 29.722 19.601Q29.781 19.531 29.813 19.396Q29.845 19.261 29.877 19.050L30.124 19.050\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.180\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(80.173 37.253)\">\u003Cpath d=\"M31.886 24.426Q31.886 24.403 31.917 24.356Q32.210 24.094 32.376 23.727Q32.542 23.360 32.542 22.973L32.542 22.915Q32.414 23.020 32.246 23.020Q32.054 23.020 31.917 22.887Q31.781 22.754 31.781 22.555Q31.781 22.364 31.917 22.231Q32.054 22.098 32.246 22.098Q32.546 22.098 32.671 22.368Q32.796 22.637 32.796 22.973Q32.796 23.422 32.615 23.836Q32.433 24.250 32.093 24.547Q32.070 24.571 32.031 24.571Q31.984 24.571 31.935 24.526Q31.886 24.481 31.886 24.426\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(80.173 37.253)\">\u003Cpath d=\"M36.873 22.782L36.873 22.692Q36.924 22.485 37.119 22.461L38.158 22.461L38.158 20.133L37.186 20.133Q36.986 20.110 36.936 19.891L36.936 19.805Q36.986 19.594 37.186 19.571L38.553 19.571Q38.748 19.590 38.799 19.805L38.799 22.461L39.713 22.461Q39.908 22.485 39.959 22.692L39.959 22.782Q39.908 22.997 39.713 23.020L37.119 23.020Q36.924 22.997 36.873 22.782M37.904 18.594L37.904 18.540Q37.904 18.368 38.041 18.247Q38.178 18.125 38.354 18.125Q38.526 18.125 38.662 18.247Q38.799 18.368 38.799 18.540L38.799 18.594Q38.799 18.770 38.662 18.891Q38.526 19.012 38.354 19.012Q38.178 19.012 38.041 18.891Q37.904 18.770 37.904 18.594M40.572 22.782L40.572 22.692Q40.615 22.485 40.822 22.461L41.244 22.461L41.244 20.133L40.822 20.133Q40.615 20.110 40.572 19.891L40.572 19.805Q40.619 19.594 40.822 19.571L41.639 19.571Q41.834 19.594 41.885 19.805L41.885 19.891L41.877 19.915Q42.104 19.735 42.377 19.633Q42.651 19.532 42.944 19.532Q43.291 19.532 43.529 19.672Q43.768 19.813 43.883 20.071Q43.998 20.329 43.998 20.684L43.998 22.461L44.424 22.461Q44.631 22.485 44.670 22.692L44.670 22.782Q44.631 22.997 44.424 23.020L43.029 23.020Q42.834 22.997 42.783 22.782L42.783 22.692Q42.834 22.481 43.029 22.461L43.358 22.461L43.358 20.715Q43.358 20.407 43.268 20.249Q43.178 20.090 42.885 20.090Q42.615 20.090 42.387 20.221Q42.158 20.352 42.022 20.581Q41.885 20.809 41.885 21.075L41.885 22.461L42.311 22.461Q42.518 22.485 42.557 22.692L42.557 22.782Q42.518 22.997 42.311 23.020L40.822 23.020Q40.615 22.997 40.572 22.782M46.404 22.797L45.518 20.133L45.197 20.133Q44.998 20.110 44.947 19.891L44.947 19.805Q44.998 19.594 45.197 19.571L46.358 19.571Q46.553 19.590 46.604 19.805L46.604 19.891Q46.553 20.110 46.358 20.133L46.076 20.133L46.869 22.508L47.658 20.133L47.381 20.133Q47.182 20.110 47.131 19.891L47.131 19.805Q47.182 19.594 47.381 19.571L48.541 19.571Q48.736 19.594 48.787 19.805L48.787 19.891Q48.736 20.110 48.541 20.133L48.221 20.133L47.334 22.797Q47.291 22.911 47.190 22.985Q47.088 23.059 46.963 23.059L46.772 23.059Q46.654 23.059 46.551 22.987Q46.447 22.915 46.404 22.797M52.506 21.532L50.065 21.532Q50.119 21.809 50.317 22.032Q50.514 22.254 50.791 22.377Q51.069 22.500 51.354 22.500Q51.826 22.500 52.049 22.211Q52.057 22.200 52.113 22.094Q52.170 21.989 52.219 21.946Q52.268 21.903 52.361 21.891L52.506 21.891Q52.697 21.911 52.756 22.125L52.756 22.180Q52.690 22.481 52.459 22.678Q52.229 22.875 51.916 22.967Q51.604 23.059 51.299 23.059Q50.815 23.059 50.375 22.831Q49.936 22.602 49.668 22.202Q49.401 21.801 49.401 21.309L49.401 21.250Q49.401 20.782 49.647 20.379Q49.893 19.977 50.301 19.743Q50.709 19.508 51.178 19.508Q51.682 19.508 52.035 19.731Q52.389 19.954 52.572 20.342Q52.756 20.731 52.756 21.235L52.756 21.293Q52.697 21.508 52.506 21.532M50.072 20.981L52.100 20.981Q52.053 20.571 51.815 20.319Q51.576 20.067 51.178 20.067Q50.783 20.067 50.477 20.329Q50.170 20.590 50.072 20.981M53.463 22.782L53.463 22.692Q53.522 22.485 53.713 22.461L54.424 22.461L54.424 20.133L53.713 20.133Q53.518 20.110 53.463 19.891L53.463 19.805Q53.522 19.594 53.713 19.571L54.815 19.571Q55.014 19.590 55.065 19.805L55.065 20.133Q55.326 19.848 55.682 19.690Q56.037 19.532 56.424 19.532Q56.717 19.532 56.951 19.666Q57.186 19.801 57.186 20.067Q57.186 20.235 57.076 20.352Q56.967 20.469 56.799 20.469Q56.647 20.469 56.531 20.358Q56.416 20.247 56.416 20.090Q56.041 20.090 55.727 20.291Q55.412 20.493 55.238 20.827Q55.065 21.161 55.065 21.540L55.065 22.461L56.010 22.461Q56.217 22.485 56.256 22.692L56.256 22.782Q56.217 22.997 56.010 23.020L53.713 23.020Q53.522 22.997 53.463 22.782M58.072 22.821L58.072 21.907Q58.100 21.700 58.311 21.676L58.479 21.676Q58.643 21.700 58.701 21.860Q58.904 22.500 59.631 22.500Q59.838 22.500 60.067 22.465Q60.295 22.430 60.463 22.315Q60.631 22.200 60.631 21.997Q60.631 21.786 60.408 21.672Q60.186 21.559 59.912 21.516L59.213 21.403Q58.072 21.192 58.072 20.469Q58.072 20.180 58.217 19.991Q58.361 19.801 58.602 19.694Q58.842 19.586 59.098 19.547Q59.354 19.508 59.631 19.508Q59.881 19.508 60.074 19.538Q60.268 19.567 60.432 19.645Q60.510 19.528 60.639 19.508L60.717 19.508Q60.815 19.520 60.877 19.582Q60.940 19.645 60.951 19.739L60.951 20.446Q60.940 20.540 60.877 20.606Q60.815 20.672 60.717 20.684L60.549 20.684Q60.455 20.672 60.389 20.606Q60.322 20.540 60.311 20.446Q60.311 20.067 59.615 20.067Q59.268 20.067 58.949 20.149Q58.631 20.231 58.631 20.477Q58.631 20.743 59.303 20.852L60.006 20.973Q60.490 21.055 60.840 21.303Q61.190 21.551 61.190 21.997Q61.190 22.387 60.953 22.629Q60.717 22.872 60.367 22.965Q60.018 23.059 59.631 23.059Q59.053 23.059 58.654 22.805Q58.584 22.930 58.535 22.987Q58.486 23.043 58.381 23.059L58.311 23.059Q58.096 23.036 58.072 22.821M62.350 22.782L62.350 22.692Q62.401 22.485 62.596 22.461L63.635 22.461L63.635 20.133L62.662 20.133Q62.463 20.110 62.412 19.891L62.412 19.805Q62.463 19.594 62.662 19.571L64.029 19.571Q64.225 19.590 64.276 19.805L64.276 22.461L65.190 22.461Q65.385 22.485 65.436 22.692L65.436 22.782Q65.385 22.997 65.190 23.020L62.596 23.020Q62.401 22.997 62.350 22.782M63.381 18.594L63.381 18.540Q63.381 18.368 63.518 18.247Q63.654 18.125 63.830 18.125Q64.002 18.125 64.139 18.247Q64.276 18.368 64.276 18.540L64.276 18.594Q64.276 18.770 64.139 18.891Q64.002 19.012 63.830 19.012Q63.654 19.012 63.518 18.891Q63.381 18.770 63.381 18.594M68.100 23.059Q67.627 23.059 67.242 22.815Q66.858 22.571 66.635 22.161Q66.412 21.750 66.412 21.293Q66.412 20.950 66.537 20.627Q66.662 20.305 66.893 20.051Q67.123 19.797 67.430 19.653Q67.736 19.508 68.100 19.508Q68.463 19.508 68.776 19.655Q69.088 19.801 69.311 20.047Q69.533 20.293 69.660 20.614Q69.787 20.934 69.787 21.293Q69.787 21.750 69.563 22.163Q69.338 22.575 68.953 22.817Q68.569 23.059 68.100 23.059M68.100 22.500Q68.565 22.500 68.856 22.106Q69.147 21.711 69.147 21.227Q69.147 20.934 69.012 20.666Q68.877 20.399 68.637 20.233Q68.397 20.067 68.100 20.067Q67.795 20.067 67.557 20.233Q67.319 20.399 67.184 20.666Q67.049 20.934 67.049 21.227Q67.049 21.707 67.342 22.104Q67.635 22.500 68.100 22.500M70.295 22.782L70.295 22.692Q70.338 22.485 70.545 22.461L70.967 22.461L70.967 20.133L70.545 20.133Q70.338 20.110 70.295 19.891L70.295 19.805Q70.342 19.594 70.545 19.571L71.361 19.571Q71.557 19.594 71.608 19.805L71.608 19.891L71.600 19.915Q71.826 19.735 72.100 19.633Q72.373 19.532 72.666 19.532Q73.014 19.532 73.252 19.672Q73.490 19.813 73.606 20.071Q73.721 20.329 73.721 20.684L73.721 22.461L74.147 22.461Q74.354 22.485 74.393 22.692L74.393 22.782Q74.354 22.997 74.147 23.020L72.752 23.020Q72.557 22.997 72.506 22.782L72.506 22.692Q72.557 22.481 72.752 22.461L73.080 22.461L73.080 20.715Q73.080 20.407 72.990 20.249Q72.901 20.090 72.608 20.090Q72.338 20.090 72.110 20.221Q71.881 20.352 71.744 20.581Q71.608 20.809 71.608 21.075L71.608 22.461L72.033 22.461Q72.240 22.485 72.279 22.692L72.279 22.782Q72.240 22.997 72.033 23.020L70.545 23.020Q70.338 22.997 70.295 22.782\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(80.173 37.253)\">\u003Cpath d=\"M79.163 21.907Q79.163 21.461 79.577 21.204Q79.991 20.946 80.532 20.846Q81.073 20.747 81.581 20.739Q81.581 20.524 81.446 20.372Q81.312 20.219 81.105 20.143Q80.898 20.067 80.687 20.067Q80.343 20.067 80.183 20.090L80.183 20.149Q80.183 20.317 80.064 20.432Q79.945 20.547 79.780 20.547Q79.605 20.547 79.489 20.424Q79.374 20.301 79.374 20.133Q79.374 19.727 79.755 19.618Q80.136 19.508 80.695 19.508Q80.964 19.508 81.232 19.586Q81.499 19.665 81.724 19.815Q81.948 19.965 82.085 20.186Q82.222 20.407 82.222 20.684L82.222 22.403Q82.222 22.461 82.749 22.461Q82.945 22.481 82.995 22.692L82.995 22.782Q82.945 22.997 82.749 23.020L82.605 23.020Q82.261 23.020 82.032 22.973Q81.804 22.926 81.659 22.739Q81.198 23.059 80.491 23.059Q80.155 23.059 79.851 22.918Q79.546 22.778 79.355 22.516Q79.163 22.254 79.163 21.907M79.804 21.915Q79.804 22.188 80.046 22.344Q80.288 22.500 80.573 22.500Q80.792 22.500 81.025 22.442Q81.257 22.383 81.419 22.245Q81.581 22.106 81.581 21.883L81.581 21.293Q81.300 21.293 80.884 21.350Q80.468 21.407 80.136 21.545Q79.804 21.684 79.804 21.915M83.589 22.821L83.589 21.907Q83.616 21.700 83.827 21.676L83.995 21.676Q84.159 21.700 84.218 21.860Q84.421 22.500 85.148 22.500Q85.355 22.500 85.583 22.465Q85.812 22.430 85.980 22.315Q86.148 22.200 86.148 21.997Q86.148 21.786 85.925 21.672Q85.702 21.559 85.429 21.516L84.730 21.403Q83.589 21.192 83.589 20.469Q83.589 20.180 83.734 19.991Q83.878 19.801 84.118 19.694Q84.359 19.586 84.614 19.547Q84.870 19.508 85.148 19.508Q85.398 19.508 85.591 19.538Q85.784 19.567 85.948 19.645Q86.027 19.528 86.155 19.508L86.234 19.508Q86.331 19.520 86.394 19.582Q86.456 19.645 86.468 19.739L86.468 20.446Q86.456 20.540 86.394 20.606Q86.331 20.672 86.234 20.684L86.066 20.684Q85.972 20.672 85.905 20.606Q85.839 20.540 85.827 20.446Q85.827 20.067 85.132 20.067Q84.784 20.067 84.466 20.149Q84.148 20.231 84.148 20.477Q84.148 20.743 84.820 20.852L85.523 20.973Q86.007 21.055 86.357 21.303Q86.706 21.551 86.706 21.997Q86.706 22.387 86.470 22.629Q86.234 22.872 85.884 22.965Q85.534 23.059 85.148 23.059Q84.570 23.059 84.171 22.805Q84.101 22.930 84.052 22.987Q84.003 23.043 83.898 23.059L83.827 23.059Q83.612 23.036 83.589 22.821\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(80.173 37.253)\">\u003Cpath d=\"M91.594 23.098Q91.238 23.098 90.969 22.918Q90.699 22.739 90.559 22.444Q90.418 22.149 90.418 21.790Q90.418 21.403 90.580 20.989Q90.742 20.575 91.026 20.237Q91.309 19.899 91.678 19.696Q92.047 19.493 92.449 19.493Q92.942 19.493 93.219 19.942L93.656 18.172Q93.699 18.008 93.699 17.957Q93.699 17.852 93.203 17.852Q93.106 17.821 93.106 17.723L93.129 17.622Q93.160 17.567 93.219 17.555L94.320 17.469Q94.363 17.469 94.403 17.500Q94.442 17.532 94.442 17.586L93.289 22.188Q93.250 22.434 93.250 22.485Q93.250 22.844 93.496 22.844Q93.641 22.844 93.742 22.737Q93.844 22.629 93.908 22.475Q93.973 22.321 94.022 22.131Q94.070 21.942 94.090 21.844Q94.117 21.774 94.180 21.774L94.281 21.774Q94.320 21.774 94.346 21.807Q94.371 21.840 94.371 21.868Q94.371 21.883 94.363 21.899Q94.250 22.391 94.051 22.745Q93.852 23.098 93.481 23.098Q93.199 23.098 92.973 22.956Q92.746 22.813 92.664 22.555Q92.442 22.797 92.168 22.948Q91.895 23.098 91.594 23.098M91.610 22.844Q91.820 22.844 92.029 22.731Q92.238 22.618 92.408 22.444Q92.578 22.270 92.707 22.075Q92.695 22.090 92.686 22.104Q92.676 22.118 92.664 22.133L93.098 20.411Q93.059 20.231 92.973 20.081Q92.887 19.930 92.750 19.838Q92.613 19.747 92.434 19.747Q92.098 19.747 91.826 20.028Q91.555 20.309 91.395 20.684Q91.270 21.004 91.172 21.424Q91.074 21.844 91.074 22.125Q91.074 22.415 91.207 22.629Q91.340 22.844 91.610 22.844\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(80.173 37.253)\">\u003Cpath d=\"M95.237 19.745Q95.533 20.082 96.263 20.082Q96.521 20.082 96.701 19.954Q96.881 19.827 96.969 19.619Q97.057 19.411 97.057 19.153Q97.057 18.758 96.850 18.487Q96.644 18.216 96.257 18.216L95.791 18.216Q95.727 18.201 95.712 18.139L95.712 18.072Q95.727 18.016 95.791 17.999L96.193 17.975Q96.403 17.975 96.572 17.833Q96.740 17.691 96.833 17.477Q96.925 17.263 96.925 17.047Q96.925 16.759 96.740 16.594Q96.556 16.428 96.263 16.428Q96.002 16.428 95.778 16.496Q95.554 16.563 95.407 16.721Q95.536 16.739 95.615 16.828Q95.694 16.918 95.694 17.047Q95.694 17.184 95.599 17.279Q95.504 17.375 95.363 17.375Q95.229 17.375 95.132 17.278Q95.035 17.181 95.035 17.047Q95.035 16.759 95.226 16.568Q95.416 16.376 95.697 16.291Q95.979 16.206 96.263 16.206Q96.538 16.206 96.839 16.297Q97.139 16.387 97.347 16.576Q97.555 16.765 97.555 17.047Q97.555 17.416 97.309 17.688Q97.063 17.961 96.691 18.090Q97.110 18.183 97.427 18.466Q97.745 18.749 97.745 19.147Q97.745 19.510 97.526 19.776Q97.306 20.041 96.960 20.181Q96.614 20.322 96.263 20.322Q96.040 20.322 95.793 20.274Q95.545 20.225 95.325 20.115Q95.106 20.006 94.974 19.827Q94.842 19.648 94.842 19.393Q94.842 19.244 94.944 19.141Q95.047 19.039 95.196 19.039Q95.346 19.039 95.448 19.141Q95.551 19.244 95.551 19.393Q95.551 19.525 95.462 19.626Q95.372 19.727 95.237 19.745\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.180\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">The Hessian grows quadratically in \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> while the gradient grows only linearly, so at \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8141em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">9\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> it already holds \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8141em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">18\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> entries.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:218.816px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 164.112 176.357\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" d=\"M70.64 60.122 68.43 57.8l-2.2-2.323-2.19-2.322-2.179-2.322-2.166-2.322-2.15-2.323-2.133-2.322-2.113-2.322-2.09-2.323-2.064-2.322-2.031-2.322-1.995-2.322-1.95-2.323-1.9-2.322-1.834-2.322-1.76-2.323-1.667-2.322-1.557-2.322-1.42-2.323-1.26-2.322-1.064-2.322-.838-2.322-.58-2.323-.299-2.322V2.065l.295-2.323.577-2.322.837-2.322 1.063-2.322 1.257-2.323 1.42-2.322 1.555-2.322 1.668-2.323 1.758-2.322 1.835-2.322 1.898-2.322 1.95-2.323 1.995-2.322 2.031-2.322 2.063-2.323 2.09-2.322 2.113-2.322 2.133-2.323 2.15-2.322 2.165-2.322 2.179-2.322 2.19-2.323 2.2-2.322 2.21-2.322M-48.725 60.122l2.21-2.322 2.2-2.323 2.19-2.322 2.18-2.322 2.165-2.322 2.15-2.323 2.133-2.322 2.114-2.322 2.09-2.323 2.063-2.322 2.032-2.322 1.994-2.322 1.95-2.323 1.9-2.322 1.835-2.322 1.76-2.323 1.667-2.322 1.556-2.322 1.421-2.323 1.259-2.322 1.064-2.322.838-2.322.581-2.323.298-2.322.001-2.322-.295-2.323-.578-2.322-.836-2.322-1.064-2.322-1.257-2.323-1.42-2.322-1.555-2.322-1.667-2.323-1.759-2.322-1.835-2.322-1.897-2.322-1.95-2.323-1.996-2.322-2.031-2.322-2.063-2.323-2.09-2.322-2.112-2.322-2.133-2.323-2.15-2.322-2.166-2.322-2.178-2.322-2.19-2.323-2.201-2.322-2.21-2.322M-45.948-56.466l2.322 2.21 2.323 2.2 2.322 2.19 2.322 2.18 2.323 2.165 2.322 2.15 2.322 2.133 2.323 2.114 2.322 2.09 2.322 2.063 2.322 2.032 2.323 1.994 2.322 1.95 2.322 1.9 2.323 1.834 2.322 1.76 2.322 1.668 2.322 1.556 2.323 1.421 2.322 1.259 2.322 1.064 2.323.838 2.322.581 2.322.298h2.322l2.323-.294 2.322-.578 2.322-.836 2.323-1.064 2.322-1.257 2.322-1.42 2.323-1.555 2.322-1.667 2.322-1.759 2.322-1.835 2.323-1.898 2.322-1.95 2.322-1.995 2.323-2.031 2.322-2.063 2.322-2.09 2.322-2.112 2.323-2.133 2.322-2.15 2.322-2.166 2.323-2.179 2.322-2.19 2.322-2.2 2.322-2.21M-45.948 62.9l2.322-2.21 2.323-2.201 2.322-2.19 2.322-2.18 2.323-2.165 2.322-2.15 2.322-2.133 2.323-2.114 2.322-2.09 2.322-2.063 2.322-2.032 2.323-1.994 2.322-1.95 2.322-1.9 2.323-1.834 2.322-1.76 2.322-1.668 2.322-1.556 2.323-1.421L.498 24.03l2.322-1.064 2.323-.838 2.322-.581 2.322-.298h2.322l2.323.294 2.322.578 2.322.837L21.4 24.02l2.322 1.257 2.322 1.42 2.323 1.555 2.322 1.667 2.322 1.759 2.322 1.835 2.323 1.898 2.322 1.95 2.322 1.995 2.323 2.031 2.322 2.063 2.322 2.09 2.322 2.113 2.323 2.133 2.322 2.15 2.322 2.165 2.323 2.179 2.322 2.19 2.322 2.2 2.322 2.21\"\u002F>\u003Cpath fill=\"none\" d=\"M75.213 60.122 73.165 57.8l-2.027-2.323-2.005-2.322-1.981-2.322-1.954-2.322-1.923-2.323-1.89-2.322-1.853-2.322-1.81-2.323-1.764-2.322-1.71-2.322-1.653-2.322-1.584-2.323-1.51-2.322-1.427-2.322-1.332-2.323-1.227-2.322-1.111-2.322-.983-2.323-.842-2.322-.69-2.322-.529-2.322-.358-2.323-.18-2.322V2.065L41-.258l.356-2.322.526-2.322.69-2.322.841-2.323.981-2.322 1.11-2.322 1.227-2.323 1.331-2.322 1.426-2.322 1.509-2.322 1.584-2.323 1.651-2.322 1.71-2.322 1.764-2.323 1.81-2.322 1.853-2.322 1.89-2.323 1.923-2.322 1.953-2.322 1.981-2.322 2.005-2.323 2.027-2.322 2.047-2.322M-53.298 60.122l2.048-2.322 2.027-2.323 2.005-2.322 1.981-2.322 1.954-2.322 1.924-2.323 1.89-2.322 1.852-2.322 1.811-2.323 1.764-2.322 1.71-2.322 1.652-2.322 1.585-2.323 1.51-2.322 1.426-2.322 1.332-2.323 1.228-2.322 1.11-2.322.983-2.323.842-2.322.69-2.322.53-2.322.357-2.323.18-2.322V2.065l-.178-2.323-.356-2.322-.527-2.322-.69-2.322-.84-2.323-.982-2.322-1.11-2.322-1.227-2.323-1.33-2.322-1.427-2.322-1.509-2.322-1.584-2.323-1.651-2.322-1.71-2.322-1.764-2.323-1.81-2.322-1.852-2.322-1.89-2.323-1.923-2.322-1.954-2.322-1.98-2.322-2.006-2.323-2.027-2.322-2.047-2.322M-45.948-61.039l2.322 2.048 2.323 2.027 2.322 2.005 2.322 1.981 2.323 1.954 2.322 1.924 2.322 1.89 2.323 1.852 2.322 1.81 2.322 1.764 2.322 1.711 2.323 1.652 2.322 1.585 2.322 1.51 2.323 1.426 2.322 1.332 2.322 1.228 2.322 1.11 2.323.983 2.322.842 2.322.69 2.323.529 2.322.358 2.322.18h2.322l2.323-.179 2.322-.356 2.322-.526 2.323-.69 2.322-.841 2.322-.981 2.323-1.11 2.322-1.227 2.322-1.33 2.322-1.427 2.323-1.509 2.322-1.584 2.322-1.651 2.323-1.71 2.322-1.764 2.322-1.81 2.322-1.852 2.323-1.89 2.322-1.923 2.322-1.954 2.323-1.98 2.322-2.006 2.322-2.027 2.322-2.047M-45.948 67.472l2.322-2.048 2.323-2.027 2.322-2.005 2.322-1.981 2.323-1.954 2.322-1.924 2.322-1.89 2.323-1.852 2.322-1.81 2.322-1.764 2.322-1.71 2.323-1.653 2.322-1.585 2.322-1.51 2.323-1.426L-8.791 39l2.322-1.228 2.322-1.11 2.323-.983 2.322-.842 2.322-.69 2.323-.529 2.322-.358 2.322-.18h2.322l2.323.179 2.322.356 2.322.526 2.323.69 2.322.841 2.322.981 2.323 1.11 2.322 1.227 2.322 1.331 2.322 1.426 2.323 1.509 2.322 1.584 2.322 1.651 2.323 1.71 2.322 1.764 2.322 1.81 2.322 1.852 2.323 1.89 2.322 1.924 2.322 1.953 2.323 1.98 2.322 2.006 2.322 2.027 2.322 2.047\"\u002F>\u003Cpath fill=\"none\" d=\"M80.652 60.122 78.77 57.8l-1.856-2.323-1.824-2.322-1.791-2.322-1.756-2.322-1.715-2.323-1.674-2.322-1.626-2.322-1.575-2.323-1.52-2.322-1.46-2.322-1.394-2.322-1.322-2.323-1.245-2.322-1.16-2.322-1.071-2.323-.974-2.322-.87-2.322-.76-2.323-.644-2.322-.523-2.322-.396-2.322-.267-2.323-.135-2.322V2.065l.133-2.323.266-2.322.396-2.322.521-2.322.643-2.323.76-2.322.868-2.322.974-2.323 1.07-2.322 1.16-2.322 1.244-2.322 1.321-2.323 1.394-2.322 1.458-2.322 1.52-2.323 1.575-2.322 1.626-2.322 1.673-2.323 1.716-2.322 1.754-2.322 1.792-2.322 1.824-2.323 1.854-2.322 1.883-2.322M-58.737 60.122l1.883-2.322L-55 55.477l1.824-2.322 1.792-2.322 1.755-2.322 1.716-2.323 1.673-2.322 1.626-2.322 1.576-2.323 1.52-2.322 1.46-2.322 1.393-2.322 1.322-2.323 1.245-2.322 1.161-2.322 1.07-2.323.974-2.322.87-2.322.76-2.323.645-2.322.522-2.322.397-2.322.266-2.323.135-2.322V2.065l-.133-2.323-.266-2.322-.395-2.322-.522-2.322-.643-2.323-.76-2.322-.868-2.322-.973-2.323-1.07-2.322-1.16-2.322-1.245-2.322-1.32-2.323-1.394-2.322-1.46-2.322-1.519-2.323-1.575-2.322-1.626-2.322-1.672-2.323-1.716-2.322-1.755-2.322-1.791-2.322-1.824-2.323-1.855-2.322-1.882-2.322M-45.948-66.478l2.322 1.883 2.323 1.855 2.322 1.824 2.322 1.791 2.323 1.756 2.322 1.716 2.322 1.673 2.323 1.626 2.322 1.576 2.322 1.52 2.322 1.46 2.323 1.393 2.322 1.322 2.322 1.245 2.323 1.16 2.322 1.072 2.322.973 2.322.87 2.323.76 2.322.644 2.322.523 2.323.397 2.322.266 2.322.135h2.322l2.323-.133 2.322-.266 2.322-.395 2.323-.522 2.322-.643 2.322-.76 2.323-.868 2.322-.974 2.322-1.07 2.322-1.16 2.323-1.244 2.322-1.32 2.322-1.394 2.323-1.46 2.322-1.519 2.322-1.575 2.322-1.626 2.323-1.673 2.322-1.715 2.322-1.755 2.323-1.792 2.322-1.823 2.322-1.855 2.322-1.883M-45.948 72.91l2.322-1.882 2.323-1.855 2.322-1.824 2.322-1.791 2.323-1.756 2.322-1.715 2.322-1.674 2.323-1.626 2.322-1.576 2.322-1.52 2.322-1.46 2.323-1.393 2.322-1.322 2.322-1.245 2.323-1.16 2.322-1.071 2.322-.974 2.322-.87 2.323-.76 2.322-.644 2.322-.523 2.323-.397 2.322-.266 2.322-.135h2.322l2.323.133 2.322.266 2.322.395 2.323.522 2.322.643 2.322.76 2.323.868 2.322.974 2.322 1.07 2.322 1.16 2.323 1.244 2.322 1.32 2.322 1.395 2.323 1.458 2.322 1.52 2.322 1.575 2.322 1.626 2.323 1.673 2.322 1.715 2.322 1.755 2.323 1.792 2.322 1.824 2.322 1.854 2.322 1.883\"\u002F>\u003Cpath stroke=\"none\" d=\"M13.558 3.217a2.6 2.6 0 1 0-5.2 0 2.6 2.6 0 0 0 5.2 0m-2.6 0\"\u002F>\u003Cg transform=\"translate(6.947 -5.81)\">\u003Cpath d=\"M11.544 3.018L11.544 2.104Q11.571 1.897 11.782 1.873L11.950 1.873Q12.114 1.897 12.173 2.057Q12.376 2.697 13.103 2.697Q13.310 2.697 13.538 2.662Q13.767 2.627 13.935 2.512Q14.103 2.397 14.103 2.194Q14.103 1.983 13.880 1.869Q13.657 1.756 13.384 1.713L12.685 1.600Q11.544 1.389 11.544 0.666Q11.544 0.377 11.688 0.188Q11.833-0.002 12.073-0.109Q12.313-0.217 12.569-0.256Q12.825-0.295 13.103-0.295Q13.353-0.295 13.546-0.265Q13.739-0.236 13.903-0.158Q13.981-0.275 14.110-0.295L14.188-0.295Q14.286-0.283 14.349-0.220Q14.411-0.158 14.423-0.064L14.423 0.643Q14.411 0.737 14.349 0.803Q14.286 0.869 14.188 0.881L14.021 0.881Q13.927 0.869 13.860 0.803Q13.794 0.737 13.782 0.643Q13.782 0.264 13.087 0.264Q12.739 0.264 12.421 0.346Q12.103 0.428 12.103 0.674Q12.103 0.940 12.774 1.049L13.478 1.170Q13.962 1.252 14.312 1.500Q14.661 1.748 14.661 2.194Q14.661 2.584 14.425 2.826Q14.188 3.069 13.839 3.162Q13.489 3.256 13.103 3.256Q12.524 3.256 12.126 3.002Q12.056 3.127 12.007 3.184Q11.958 3.240 11.853 3.256L11.782 3.256Q11.567 3.233 11.544 3.018M15.610 2.104Q15.610 1.658 16.024 1.401Q16.438 1.143 16.979 1.043Q17.520 0.944 18.028 0.936Q18.028 0.721 17.894 0.569Q17.759 0.416 17.552 0.340Q17.345 0.264 17.134 0.264Q16.790 0.264 16.630 0.287L16.630 0.346Q16.630 0.514 16.511 0.629Q16.392 0.744 16.228 0.744Q16.052 0.744 15.937 0.621Q15.821 0.498 15.821 0.330Q15.821-0.076 16.202-0.185Q16.583-0.295 17.142-0.295Q17.411-0.295 17.679-0.217Q17.946-0.138 18.171 0.012Q18.395 0.162 18.532 0.383Q18.669 0.604 18.669 0.881L18.669 2.600Q18.669 2.658 19.196 2.658Q19.392 2.678 19.442 2.889L19.442 2.979Q19.392 3.194 19.196 3.217L19.052 3.217Q18.708 3.217 18.479 3.170Q18.251 3.123 18.106 2.936Q17.645 3.256 16.938 3.256Q16.603 3.256 16.298 3.115Q15.993 2.975 15.802 2.713Q15.610 2.451 15.610 2.104M16.251 2.112Q16.251 2.385 16.493 2.541Q16.735 2.697 17.020 2.697Q17.239 2.697 17.472 2.639Q17.704 2.580 17.866 2.442Q18.028 2.303 18.028 2.080L18.028 1.490Q17.747 1.490 17.331 1.547Q16.915 1.604 16.583 1.742Q16.251 1.881 16.251 2.112M21.313 3.256Q20.849 3.256 20.483 3.006Q20.118 2.756 19.913 2.352Q19.708 1.947 19.708 1.490Q19.708 1.147 19.833 0.826Q19.958 0.506 20.190 0.256Q20.423 0.006 20.728-0.133Q21.032-0.271 21.388-0.271Q21.899-0.271 22.306 0.049L22.306-1.111L21.884-1.111Q21.673-1.135 21.634-1.349L21.634-1.439Q21.673-1.646 21.884-1.670L22.696-1.670Q22.895-1.646 22.946-1.439L22.946 2.658L23.372 2.658Q23.579 2.682 23.618 2.889L23.618 2.979Q23.579 3.194 23.372 3.217L22.556 3.217Q22.356 3.194 22.306 2.979L22.306 2.850Q22.110 3.041 21.849 3.149Q21.587 3.256 21.313 3.256M21.353 2.697Q21.712 2.697 21.964 2.428Q22.216 2.158 22.306 1.783L22.306 0.951Q22.247 0.764 22.120 0.613Q21.993 0.463 21.815 0.375Q21.638 0.287 21.442 0.287Q21.134 0.287 20.884 0.457Q20.634 0.627 20.489 0.912Q20.345 1.197 20.345 1.498Q20.345 1.947 20.630 2.322Q20.915 2.697 21.353 2.697M25.560 3.256Q25.095 3.256 24.729 3.006Q24.364 2.756 24.159 2.352Q23.954 1.947 23.954 1.490Q23.954 1.147 24.079 0.826Q24.204 0.506 24.437 0.256Q24.669 0.006 24.974-0.133Q25.278-0.271 25.634-0.271Q26.145-0.271 26.552 0.049L26.552-1.111L26.130-1.111Q25.919-1.135 25.880-1.349L25.880-1.439Q25.919-1.646 26.130-1.670L26.942-1.670Q27.142-1.646 27.192-1.439L27.192 2.658L27.618 2.658Q27.825 2.682 27.864 2.889L27.864 2.979Q27.825 3.194 27.618 3.217L26.802 3.217Q26.603 3.194 26.552 2.979L26.552 2.850Q26.356 3.041 26.095 3.149Q25.833 3.256 25.560 3.256M25.599 2.697Q25.958 2.697 26.210 2.428Q26.462 2.158 26.552 1.783L26.552 0.951Q26.493 0.764 26.366 0.613Q26.239 0.463 26.062 0.375Q25.884 0.287 25.688 0.287Q25.380 0.287 25.130 0.457Q24.880 0.627 24.735 0.912Q24.591 1.197 24.591 1.498Q24.591 1.947 24.876 2.322Q25.161 2.697 25.599 2.697M28.392 2.979L28.392 2.889Q28.442 2.682 28.638 2.658L29.743 2.658L29.743-1.111L28.638-1.111Q28.442-1.135 28.392-1.349L28.392-1.439Q28.442-1.646 28.638-1.670L30.134-1.670Q30.325-1.646 30.384-1.439L30.384 2.658L31.485 2.658Q31.685 2.682 31.735 2.889L31.735 2.979Q31.685 3.194 31.485 3.217L28.638 3.217Q28.442 3.194 28.392 2.979M35.700 1.729L33.259 1.729Q33.313 2.006 33.511 2.229Q33.708 2.451 33.985 2.574Q34.263 2.697 34.548 2.697Q35.020 2.697 35.243 2.408Q35.251 2.397 35.308 2.291Q35.364 2.186 35.413 2.143Q35.462 2.100 35.556 2.088L35.700 2.088Q35.892 2.108 35.950 2.322L35.950 2.377Q35.884 2.678 35.653 2.875Q35.423 3.072 35.110 3.164Q34.798 3.256 34.493 3.256Q34.009 3.256 33.569 3.028Q33.130 2.799 32.862 2.399Q32.595 1.998 32.595 1.506L32.595 1.447Q32.595 0.979 32.841 0.576Q33.087 0.174 33.495-0.060Q33.903-0.295 34.372-0.295Q34.876-0.295 35.229-0.072Q35.583 0.151 35.767 0.539Q35.950 0.928 35.950 1.432L35.950 1.490Q35.892 1.705 35.700 1.729M33.267 1.178L35.294 1.178Q35.247 0.768 35.009 0.516Q34.770 0.264 34.372 0.264Q33.978 0.264 33.671 0.526Q33.364 0.787 33.267 1.178\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-accent)\" stroke=\"none\" d=\"M13.358 65.812a2.4 2.4 0 1 0-4.8 0 2.4 2.4 0 0 0 4.8 0m-2.4 0\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cg transform=\"translate(7.8 65.04)\">\u003Cpath d=\"M11.028 2.979L11.028 2.889Q11.071 2.682 11.278 2.658L11.700 2.658L11.700-1.111L11.278-1.111Q11.071-1.135 11.028-1.349L11.028-1.439Q11.071-1.646 11.278-1.670L12.095-1.670Q12.290-1.646 12.341-1.439L12.341 0.112Q12.802-0.271 13.399-0.271Q13.747-0.271 13.985-0.131Q14.224 0.010 14.339 0.268Q14.454 0.526 14.454 0.881L14.454 2.658L14.880 2.658Q15.087 2.682 15.126 2.889L15.126 2.979Q15.087 3.194 14.880 3.217L13.485 3.217Q13.290 3.194 13.239 2.979L13.239 2.889Q13.290 2.678 13.485 2.658L13.813 2.658L13.813 0.912Q13.813 0.604 13.724 0.446Q13.634 0.287 13.341 0.287Q13.071 0.287 12.843 0.418Q12.614 0.549 12.478 0.778Q12.341 1.006 12.341 1.272L12.341 2.658L12.767 2.658Q12.974 2.682 13.013 2.889L13.013 2.979Q12.974 3.194 12.767 3.217L11.278 3.217Q11.071 3.194 11.028 2.979M18.716 1.729L16.274 1.729Q16.329 2.006 16.526 2.229Q16.724 2.451 17.001 2.574Q17.278 2.697 17.563 2.697Q18.036 2.697 18.259 2.408Q18.267 2.397 18.323 2.291Q18.380 2.186 18.429 2.143Q18.478 2.100 18.571 2.088L18.716 2.088Q18.907 2.108 18.966 2.322L18.966 2.377Q18.899 2.678 18.669 2.875Q18.438 3.072 18.126 3.164Q17.813 3.256 17.509 3.256Q17.024 3.256 16.585 3.028Q16.145 2.799 15.878 2.399Q15.610 1.998 15.610 1.506L15.610 1.447Q15.610 0.979 15.856 0.576Q16.103 0.174 16.511-0.060Q16.919-0.295 17.388-0.295Q17.892-0.295 18.245-0.072Q18.599 0.151 18.782 0.539Q18.966 0.928 18.966 1.432L18.966 1.490Q18.907 1.705 18.716 1.729M16.282 1.178L18.310 1.178Q18.263 0.768 18.024 0.516Q17.786 0.264 17.388 0.264Q16.993 0.264 16.687 0.526Q16.380 0.787 16.282 1.178M19.673 2.979L19.673 2.889Q19.731 2.682 19.923 2.658L20.634 2.658L20.634 0.330L19.923 0.330Q19.728 0.307 19.673 0.088L19.673 0.002Q19.731-0.209 19.923-0.232L21.024-0.232Q21.224-0.213 21.274 0.002L21.274 0.330Q21.536 0.045 21.892-0.113Q22.247-0.271 22.634-0.271Q22.927-0.271 23.161-0.137Q23.395-0.002 23.395 0.264Q23.395 0.432 23.286 0.549Q23.177 0.666 23.009 0.666Q22.856 0.666 22.741 0.555Q22.626 0.444 22.626 0.287Q22.251 0.287 21.937 0.488Q21.622 0.690 21.448 1.024Q21.274 1.358 21.274 1.737L21.274 2.658L22.220 2.658Q22.427 2.682 22.466 2.889L22.466 2.979Q22.427 3.194 22.220 3.217L19.923 3.217Q19.731 3.194 19.673 2.979M27.208 1.729L24.767 1.729Q24.821 2.006 25.019 2.229Q25.216 2.451 25.493 2.574Q25.770 2.697 26.056 2.697Q26.528 2.697 26.751 2.408Q26.759 2.397 26.815 2.291Q26.872 2.186 26.921 2.143Q26.970 2.100 27.063 2.088L27.208 2.088Q27.399 2.108 27.458 2.322L27.458 2.377Q27.392 2.678 27.161 2.875Q26.931 3.072 26.618 3.164Q26.306 3.256 26.001 3.256Q25.517 3.256 25.077 3.028Q24.638 2.799 24.370 2.399Q24.103 1.998 24.103 1.506L24.103 1.447Q24.103 0.979 24.349 0.576Q24.595 0.174 25.003-0.060Q25.411-0.295 25.880-0.295Q26.384-0.295 26.737-0.072Q27.091 0.151 27.274 0.539Q27.458 0.928 27.458 1.432L27.458 1.490Q27.399 1.705 27.208 1.729M24.774 1.178L26.802 1.178Q26.755 0.768 26.517 0.516Q26.278 0.264 25.880 0.264Q25.485 0.264 25.179 0.526Q24.872 0.787 24.774 1.178\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"M10.958 65.812v28.099\"\u002F>\u003Cpath stroke=\"none\" d=\"m10.958 97.11 2.56-5.12-2.56 1.92-2.56-1.92\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\">\u003Cg fill=\"var(--tk-good)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(7.8 89.76)\">\u003Cpath d=\"M11.196 3.865Q11.196 3.565 11.345 3.303Q11.493 3.041 11.743 2.881Q11.560 2.627 11.560 2.307Q11.560 2.022 11.712 1.760Q11.462 1.436 11.462 1.018Q11.462 0.658 11.653 0.362Q11.845 0.065 12.163-0.103Q12.481-0.271 12.845-0.271Q13.052-0.271 13.255-0.211Q13.458-0.150 13.622-0.049Q14.005-0.310 14.478-0.310Q14.716-0.310 14.897-0.179Q15.079-0.049 15.079 0.178Q15.079 0.326 14.978 0.432Q14.876 0.537 14.720 0.537Q14.587 0.537 14.491 0.459Q14.396 0.381 14.364 0.256Q14.220 0.264 14.021 0.354Q14.224 0.666 14.224 1.018Q14.224 1.299 14.112 1.531Q14.001 1.764 13.806 1.942Q13.610 2.119 13.358 2.217Q13.106 2.315 12.845 2.315Q12.458 2.315 12.126 2.127Q12.114 2.127 12.104 2.199Q12.095 2.272 12.095 2.307Q12.095 2.428 12.161 2.535Q12.228 2.643 12.349 2.682Q12.364 2.678 12.378 2.676Q12.392 2.674 12.415 2.674Q12.431 2.674 12.462 2.682Q12.493 2.690 12.501 2.690L13.095 2.690Q13.876 2.690 14.417 2.932Q14.958 3.174 14.958 3.865Q14.958 4.166 14.778 4.393Q14.599 4.619 14.302 4.764Q14.005 4.908 13.685 4.975Q13.364 5.041 13.079 5.041Q12.688 5.041 12.247 4.918Q11.806 4.795 11.501 4.528Q11.196 4.260 11.196 3.865M11.735 3.858Q11.735 4.076 11.976 4.217Q12.216 4.358 12.540 4.424Q12.864 4.490 13.079 4.490Q13.294 4.490 13.618 4.424Q13.942 4.358 14.183 4.217Q14.423 4.076 14.423 3.858Q14.423 3.569 14.198 3.430Q13.974 3.291 13.692 3.258Q13.411 3.225 13.063 3.225L12.446 3.225Q12.263 3.225 12.101 3.305Q11.938 3.385 11.837 3.531Q11.735 3.678 11.735 3.858M12.845 1.760Q13.146 1.760 13.364 1.541Q13.583 1.322 13.583 1.018Q13.583 0.862 13.528 0.731Q13.474 0.600 13.368 0.494Q13.263 0.389 13.132 0.334Q13.001 0.280 12.845 0.280Q12.540 0.280 12.321 0.498Q12.103 0.717 12.103 1.018Q12.103 1.315 12.325 1.537Q12.548 1.760 12.845 1.760M15.427 2.979L15.427 2.889Q15.485 2.682 15.677 2.658L16.388 2.658L16.388 0.330L15.677 0.330Q15.481 0.307 15.427 0.088L15.427 0.002Q15.485-0.209 15.677-0.232L16.778-0.232Q16.978-0.213 17.028 0.002L17.028 0.330Q17.290 0.045 17.645-0.113Q18.001-0.271 18.388-0.271Q18.681-0.271 18.915-0.137Q19.149-0.002 19.149 0.264Q19.149 0.432 19.040 0.549Q18.931 0.666 18.763 0.666Q18.610 0.666 18.495 0.555Q18.380 0.444 18.380 0.287Q18.005 0.287 17.690 0.488Q17.376 0.690 17.202 1.024Q17.028 1.358 17.028 1.737L17.028 2.658L17.974 2.658Q18.181 2.682 18.220 2.889L18.220 2.979Q18.181 3.194 17.974 3.217L15.677 3.217Q15.485 3.194 15.427 2.979M19.856 2.104Q19.856 1.658 20.270 1.401Q20.685 1.143 21.226 1.043Q21.767 0.944 22.274 0.936Q22.274 0.721 22.140 0.569Q22.005 0.416 21.798 0.340Q21.591 0.264 21.380 0.264Q21.036 0.264 20.876 0.287L20.876 0.346Q20.876 0.514 20.757 0.629Q20.638 0.744 20.474 0.744Q20.298 0.744 20.183 0.621Q20.067 0.498 20.067 0.330Q20.067-0.076 20.448-0.185Q20.829-0.295 21.388-0.295Q21.657-0.295 21.925-0.217Q22.192-0.138 22.417 0.012Q22.642 0.162 22.778 0.383Q22.915 0.604 22.915 0.881L22.915 2.600Q22.915 2.658 23.442 2.658Q23.638 2.678 23.688 2.889L23.688 2.979Q23.638 3.194 23.442 3.217L23.298 3.217Q22.954 3.217 22.726 3.170Q22.497 3.123 22.353 2.936Q21.892 3.256 21.185 3.256Q20.849 3.256 20.544 3.115Q20.239 2.975 20.048 2.713Q19.856 2.451 19.856 2.104M20.497 2.112Q20.497 2.385 20.739 2.541Q20.981 2.697 21.267 2.697Q21.485 2.697 21.718 2.639Q21.950 2.580 22.112 2.442Q22.274 2.303 22.274 2.080L22.274 1.490Q21.993 1.490 21.577 1.547Q21.161 1.604 20.829 1.742Q20.497 1.881 20.497 2.112M25.560 3.256Q25.095 3.256 24.729 3.006Q24.364 2.756 24.159 2.352Q23.954 1.947 23.954 1.490Q23.954 1.147 24.079 0.826Q24.204 0.506 24.437 0.256Q24.669 0.006 24.974-0.133Q25.278-0.271 25.634-0.271Q26.145-0.271 26.552 0.049L26.552-1.111L26.130-1.111Q25.919-1.135 25.880-1.349L25.880-1.439Q25.919-1.646 26.130-1.670L26.942-1.670Q27.142-1.646 27.192-1.439L27.192 2.658L27.618 2.658Q27.825 2.682 27.864 2.889L27.864 2.979Q27.825 3.194 27.618 3.217L26.802 3.217Q26.603 3.194 26.552 2.979L26.552 2.850Q26.356 3.041 26.095 3.149Q25.833 3.256 25.560 3.256M25.599 2.697Q25.958 2.697 26.210 2.428Q26.462 2.158 26.552 1.783L26.552 0.951Q26.493 0.764 26.366 0.613Q26.239 0.463 26.062 0.375Q25.884 0.287 25.688 0.287Q25.380 0.287 25.130 0.457Q24.880 0.627 24.735 0.912Q24.591 1.197 24.591 1.498Q24.591 1.947 24.876 2.322Q25.161 2.697 25.599 2.697M28.560 2.979L28.560 2.889Q28.610 2.682 28.806 2.658L29.845 2.658L29.845 0.330L28.872 0.330Q28.673 0.307 28.622 0.088L28.622 0.002Q28.673-0.209 28.872-0.232L30.239-0.232Q30.435-0.213 30.485 0.002L30.485 2.658L31.399 2.658Q31.595 2.682 31.645 2.889L31.645 2.979Q31.595 3.194 31.399 3.217L28.806 3.217Q28.610 3.194 28.560 2.979M29.591-1.209L29.591-1.263Q29.591-1.435 29.728-1.556Q29.864-1.678 30.040-1.678Q30.212-1.678 30.349-1.556Q30.485-1.435 30.485-1.263L30.485-1.209Q30.485-1.033 30.349-0.912Q30.212-0.791 30.040-0.791Q29.864-0.791 29.728-0.912Q29.591-1.033 29.591-1.209M35.700 1.729L33.259 1.729Q33.313 2.006 33.511 2.229Q33.708 2.451 33.985 2.574Q34.263 2.697 34.548 2.697Q35.020 2.697 35.243 2.408Q35.251 2.397 35.308 2.291Q35.364 2.186 35.413 2.143Q35.462 2.100 35.556 2.088L35.700 2.088Q35.892 2.108 35.950 2.322L35.950 2.377Q35.884 2.678 35.653 2.875Q35.423 3.072 35.110 3.164Q34.798 3.256 34.493 3.256Q34.009 3.256 33.569 3.028Q33.130 2.799 32.862 2.399Q32.595 1.998 32.595 1.506L32.595 1.447Q32.595 0.979 32.841 0.576Q33.087 0.174 33.495-0.060Q33.903-0.295 34.372-0.295Q34.876-0.295 35.229-0.072Q35.583 0.151 35.767 0.539Q35.950 0.928 35.950 1.432L35.950 1.490Q35.892 1.705 35.700 1.729M33.267 1.178L35.294 1.178Q35.247 0.768 35.009 0.516Q34.770 0.264 34.372 0.264Q33.978 0.264 33.671 0.526Q33.364 0.787 33.267 1.178M36.505 2.979L36.505 2.889Q36.548 2.682 36.755 2.658L37.177 2.658L37.177 0.330L36.755 0.330Q36.548 0.307 36.505 0.088L36.505 0.002Q36.552-0.209 36.755-0.232L37.571-0.232Q37.767-0.209 37.817 0.002L37.817 0.088L37.810 0.112Q38.036-0.068 38.310-0.170Q38.583-0.271 38.876-0.271Q39.224-0.271 39.462-0.131Q39.700 0.010 39.815 0.268Q39.931 0.526 39.931 0.881L39.931 2.658L40.356 2.658Q40.563 2.682 40.603 2.889L40.603 2.979Q40.563 3.194 40.356 3.217L38.962 3.217Q38.767 3.194 38.716 2.979L38.716 2.889Q38.767 2.678 38.962 2.658L39.290 2.658L39.290 0.912Q39.290 0.604 39.200 0.446Q39.110 0.287 38.817 0.287Q38.548 0.287 38.319 0.418Q38.091 0.549 37.954 0.778Q37.817 1.006 37.817 1.272L37.817 2.658L38.243 2.658Q38.450 2.682 38.489 2.889L38.489 2.979Q38.450 3.194 38.243 3.217L36.755 3.217Q36.548 3.194 36.505 2.979M41.880 2.112L41.880 0.330L41.130 0.330Q40.931 0.307 40.880 0.088L40.880 0.002Q40.931-0.209 41.130-0.232L41.880-0.232L41.880-0.982Q41.931-1.189 42.130-1.217L42.274-1.217Q42.470-1.189 42.520-0.982L42.520-0.232L43.880-0.232Q44.071-0.213 44.130 0.002L44.130 0.088Q44.075 0.307 43.880 0.330L42.520 0.330L42.520 2.080Q42.520 2.697 43.095 2.697Q43.345 2.697 43.509 2.512Q43.673 2.326 43.673 2.080Q43.673 1.987 43.745 1.916Q43.817 1.846 43.919 1.834L44.063 1.834Q44.263 1.858 44.313 2.065L44.313 2.112Q44.313 2.436 44.130 2.699Q43.946 2.963 43.653 3.110Q43.360 3.256 43.040 3.256Q42.528 3.256 42.204 2.946Q41.880 2.635 41.880 2.112\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(7.8 89.76)\">\u003Cpath d=\"M49.657 2.979L49.657 2.889Q49.708 2.682 49.903 2.658L51.009 2.658L51.009-1.111L49.903-1.111Q49.708-1.135 49.657-1.349L49.657-1.439Q49.708-1.646 49.903-1.670L51.399-1.670Q51.591-1.646 51.649-1.439L51.649 2.658L52.751 2.658Q52.950 2.682 53.001 2.889L53.001 2.979Q52.950 3.194 52.751 3.217L49.903 3.217Q49.708 3.194 49.657 2.979M56.966 1.729L54.524 1.729Q54.579 2.006 54.776 2.229Q54.974 2.451 55.251 2.574Q55.528 2.697 55.813 2.697Q56.286 2.697 56.509 2.408Q56.517 2.397 56.573 2.291Q56.630 2.186 56.679 2.143Q56.728 2.100 56.821 2.088L56.966 2.088Q57.157 2.108 57.216 2.322L57.216 2.377Q57.149 2.678 56.919 2.875Q56.688 3.072 56.376 3.164Q56.063 3.256 55.759 3.256Q55.274 3.256 54.835 3.028Q54.395 2.799 54.128 2.399Q53.860 1.998 53.860 1.506L53.860 1.447Q53.860 0.979 54.106 0.576Q54.353 0.174 54.761-0.060Q55.169-0.295 55.638-0.295Q56.142-0.295 56.495-0.072Q56.849 0.151 57.032 0.539Q57.216 0.928 57.216 1.432L57.216 1.490Q57.157 1.705 56.966 1.729M54.532 1.178L56.560 1.178Q56.513 0.768 56.274 0.516Q56.036 0.264 55.638 0.264Q55.243 0.264 54.937 0.526Q54.630 0.787 54.532 1.178M58.106 2.104Q58.106 1.658 58.520 1.401Q58.935 1.143 59.476 1.043Q60.017 0.944 60.524 0.936Q60.524 0.721 60.390 0.569Q60.255 0.416 60.048 0.340Q59.841 0.264 59.630 0.264Q59.286 0.264 59.126 0.287L59.126 0.346Q59.126 0.514 59.007 0.629Q58.888 0.744 58.724 0.744Q58.548 0.744 58.433 0.621Q58.317 0.498 58.317 0.330Q58.317-0.076 58.698-0.185Q59.079-0.295 59.638-0.295Q59.907-0.295 60.175-0.217Q60.442-0.138 60.667 0.012Q60.892 0.162 61.028 0.383Q61.165 0.604 61.165 0.881L61.165 2.600Q61.165 2.658 61.692 2.658Q61.888 2.678 61.938 2.889L61.938 2.979Q61.888 3.194 61.692 3.217L61.548 3.217Q61.204 3.217 60.976 3.170Q60.747 3.123 60.603 2.936Q60.142 3.256 59.435 3.256Q59.099 3.256 58.794 3.115Q58.489 2.975 58.298 2.713Q58.106 2.451 58.106 2.104M58.747 2.112Q58.747 2.385 58.989 2.541Q59.231 2.697 59.517 2.697Q59.735 2.697 59.968 2.639Q60.200 2.580 60.362 2.442Q60.524 2.303 60.524 2.080L60.524 1.490Q60.243 1.490 59.827 1.547Q59.411 1.604 59.079 1.742Q58.747 1.881 58.747 2.112M63.603 2.994L62.716 0.330L62.395 0.330Q62.196 0.307 62.145 0.088L62.145 0.002Q62.196-0.209 62.395-0.232L63.556-0.232Q63.751-0.213 63.802 0.002L63.802 0.088Q63.751 0.307 63.556 0.330L63.274 0.330L64.067 2.705L64.856 0.330L64.579 0.330Q64.380 0.307 64.329 0.088L64.329 0.002Q64.380-0.209 64.579-0.232L65.739-0.232Q65.935-0.209 65.985 0.002L65.985 0.088Q65.935 0.307 65.739 0.330L65.419 0.330L64.532 2.994Q64.489 3.108 64.388 3.182Q64.286 3.256 64.161 3.256L63.970 3.256Q63.853 3.256 63.749 3.184Q63.645 3.112 63.603 2.994M69.704 1.729L67.263 1.729Q67.317 2.006 67.515 2.229Q67.712 2.451 67.989 2.574Q68.267 2.697 68.552 2.697Q69.024 2.697 69.247 2.408Q69.255 2.397 69.312 2.291Q69.368 2.186 69.417 2.143Q69.466 2.100 69.560 2.088L69.704 2.088Q69.895 2.108 69.954 2.322L69.954 2.377Q69.888 2.678 69.657 2.875Q69.427 3.072 69.114 3.164Q68.802 3.256 68.497 3.256Q68.013 3.256 67.573 3.028Q67.134 2.799 66.866 2.399Q66.599 1.998 66.599 1.506L66.599 1.447Q66.599 0.979 66.845 0.576Q67.091 0.174 67.499-0.060Q67.907-0.295 68.376-0.295Q68.880-0.295 69.233-0.072Q69.587 0.151 69.770 0.539Q69.954 0.928 69.954 1.432L69.954 1.490Q69.895 1.705 69.704 1.729M67.270 1.178L69.298 1.178Q69.251 0.768 69.013 0.516Q68.774 0.264 68.376 0.264Q67.981 0.264 67.675 0.526Q67.368 0.787 67.270 1.178M71.024 3.018L71.024 2.104Q71.052 1.897 71.263 1.873L71.431 1.873Q71.595 1.897 71.653 2.057Q71.856 2.697 72.583 2.697Q72.790 2.697 73.019 2.662Q73.247 2.627 73.415 2.512Q73.583 2.397 73.583 2.194Q73.583 1.983 73.360 1.869Q73.138 1.756 72.864 1.713L72.165 1.600Q71.024 1.389 71.024 0.666Q71.024 0.377 71.169 0.188Q71.313-0.002 71.554-0.109Q71.794-0.217 72.050-0.256Q72.306-0.295 72.583-0.295Q72.833-0.295 73.026-0.265Q73.220-0.236 73.384-0.158Q73.462-0.275 73.591-0.295L73.669-0.295Q73.767-0.283 73.829-0.220Q73.892-0.158 73.903-0.064L73.903 0.643Q73.892 0.737 73.829 0.803Q73.767 0.869 73.669 0.881L73.501 0.881Q73.407 0.869 73.341 0.803Q73.274 0.737 73.263 0.643Q73.263 0.264 72.567 0.264Q72.220 0.264 71.901 0.346Q71.583 0.428 71.583 0.674Q71.583 0.940 72.255 1.049L72.958 1.170Q73.442 1.252 73.792 1.500Q74.142 1.748 74.142 2.194Q74.142 2.584 73.905 2.826Q73.669 3.069 73.319 3.162Q72.970 3.256 72.583 3.256Q72.005 3.256 71.606 3.002Q71.536 3.127 71.487 3.184Q71.438 3.240 71.333 3.256L71.263 3.256Q71.048 3.233 71.024 3.018\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"M10.958 65.812V22.066\"\u002F>\u003Cpath stroke=\"none\" d=\"m10.958 18.866-2.56 5.12 2.56-1.92 2.56 1.92\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(-34 -67.265)\">\u003Cpath d=\"M11.157 2.979L11.157 2.889Q11.208 2.682 11.407 2.658L11.677 2.658L11.677-1.111L11.407-1.111Q11.208-1.135 11.157-1.349L11.157-1.439Q11.208-1.646 11.407-1.670L12.188-1.670Q12.317-1.670 12.423-1.595Q12.528-1.521 12.567-1.408Q12.845-0.619 13.005-0.170Q13.165 0.280 13.419 0.998Q13.673 1.717 13.812 2.127Q13.950 2.537 13.950 2.576L13.950-1.111L13.677-1.111Q13.481-1.135 13.431-1.349L13.431-1.439Q13.481-1.646 13.677-1.670L14.751-1.670Q14.946-1.646 14.997-1.439L14.997-1.349Q14.946-1.135 14.751-1.111L14.478-1.111L14.478 2.979Q14.427 3.194 14.231 3.217L13.966 3.217Q13.841 3.217 13.737 3.143Q13.634 3.069 13.591 2.951Q13.294 2.104 12.921 1.055Q12.548 0.006 12.378-0.482Q12.208-0.970 12.204-1.029L12.204 2.658L12.478 2.658Q12.669 2.682 12.728 2.889L12.728 2.979Q12.669 3.194 12.478 3.217L11.407 3.217Q11.208 3.194 11.157 2.979M18.716 1.729L16.274 1.729Q16.329 2.006 16.526 2.229Q16.724 2.451 17.001 2.574Q17.278 2.697 17.563 2.697Q18.036 2.697 18.259 2.408Q18.267 2.397 18.323 2.291Q18.380 2.186 18.429 2.143Q18.478 2.100 18.571 2.088L18.716 2.088Q18.907 2.108 18.966 2.322L18.966 2.377Q18.899 2.678 18.669 2.875Q18.438 3.072 18.126 3.164Q17.813 3.256 17.509 3.256Q17.024 3.256 16.585 3.028Q16.145 2.799 15.878 2.399Q15.610 1.998 15.610 1.506L15.610 1.447Q15.610 0.979 15.856 0.576Q16.103 0.174 16.511-0.060Q16.919-0.295 17.388-0.295Q17.892-0.295 18.245-0.072Q18.599 0.151 18.782 0.539Q18.966 0.928 18.966 1.432L18.966 1.490Q18.907 1.705 18.716 1.729M16.282 1.178L18.310 1.178Q18.263 0.768 18.024 0.516Q17.786 0.264 17.388 0.264Q16.993 0.264 16.687 0.526Q16.380 0.787 16.282 1.178M20.458 2.994L19.993 0.330L19.833 0.330Q19.626 0.307 19.587 0.088L19.587 0.002Q19.626-0.209 19.833-0.232L21.001-0.232Q21.212-0.209 21.251 0.002L21.251 0.088Q21.212 0.307 21.001 0.330L20.528 0.330Q20.642 0.975 20.720 1.420Q20.798 1.865 20.849 2.184Q20.899 2.502 20.899 2.576Q20.907 2.405 21.192 1.408Q21.231 1.295 21.329 1.221Q21.427 1.147 21.548 1.147L21.626 1.147Q21.747 1.147 21.845 1.221Q21.942 1.295 21.978 1.408Q22.087 1.791 22.169 2.115Q22.251 2.440 22.251 2.576Q22.263 2.287 22.610 0.330L22.138 0.330Q21.931 0.307 21.892 0.088L21.892 0.002Q21.931-0.209 22.138-0.232L23.313-0.232Q23.505-0.209 23.563 0.002L23.563 0.088Q23.509 0.307 23.313 0.330L23.145 0.330L22.681 2.994Q22.657 3.112 22.569 3.184Q22.481 3.256 22.360 3.256L22.220 3.256Q22.099 3.256 22.003 3.184Q21.907 3.112 21.864 2.994Q21.817 2.815 21.745 2.559Q21.673 2.303 21.630 2.094Q21.587 1.885 21.587 1.783Q21.579 2.065 21.298 2.994Q21.270 3.104 21.173 3.180Q21.075 3.256 20.954 3.256L20.778 3.256Q20.657 3.256 20.569 3.184Q20.481 3.112 20.458 2.994M24.895 2.112L24.895 0.330L24.145 0.330Q23.946 0.307 23.895 0.088L23.895 0.002Q23.946-0.209 24.145-0.232L24.895-0.232L24.895-0.982Q24.946-1.189 25.145-1.217L25.290-1.217Q25.485-1.189 25.536-0.982L25.536-0.232L26.895-0.232Q27.087-0.213 27.145 0.002L27.145 0.088Q27.091 0.307 26.895 0.330L25.536 0.330L25.536 2.080Q25.536 2.697 26.110 2.697Q26.360 2.697 26.524 2.512Q26.688 2.326 26.688 2.080Q26.688 1.987 26.761 1.916Q26.833 1.846 26.935 1.834L27.079 1.834Q27.278 1.858 27.329 2.065L27.329 2.112Q27.329 2.436 27.145 2.699Q26.962 2.963 26.669 3.110Q26.376 3.256 26.056 3.256Q25.544 3.256 25.220 2.946Q24.895 2.635 24.895 2.112M30.063 3.256Q29.591 3.256 29.206 3.012Q28.821 2.768 28.599 2.358Q28.376 1.947 28.376 1.490Q28.376 1.147 28.501 0.824Q28.626 0.502 28.856 0.248Q29.087-0.006 29.394-0.150Q29.700-0.295 30.063-0.295Q30.427-0.295 30.739-0.148Q31.052-0.002 31.274 0.244Q31.497 0.490 31.624 0.811Q31.751 1.131 31.751 1.490Q31.751 1.947 31.526 2.360Q31.302 2.772 30.917 3.014Q30.532 3.256 30.063 3.256M30.063 2.697Q30.528 2.697 30.819 2.303Q31.110 1.908 31.110 1.424Q31.110 1.131 30.976 0.863Q30.841 0.596 30.601 0.430Q30.360 0.264 30.063 0.264Q29.759 0.264 29.520 0.430Q29.282 0.596 29.147 0.863Q29.013 1.131 29.013 1.424Q29.013 1.905 29.306 2.301Q29.599 2.697 30.063 2.697M32.259 2.979L32.259 2.889Q32.302 2.682 32.509 2.658L32.931 2.658L32.931 0.330L32.509 0.330Q32.302 0.307 32.259 0.088L32.259 0.002Q32.306-0.209 32.509-0.232L33.325-0.232Q33.520-0.209 33.571 0.002L33.571 0.088L33.563 0.112Q33.790-0.068 34.063-0.170Q34.337-0.271 34.630-0.271Q34.978-0.271 35.216-0.131Q35.454 0.010 35.569 0.268Q35.685 0.526 35.685 0.881L35.685 2.658L36.110 2.658Q36.317 2.682 36.356 2.889L36.356 2.979Q36.317 3.194 36.110 3.217L34.716 3.217Q34.520 3.194 34.470 2.979L34.470 2.889Q34.520 2.678 34.716 2.658L35.044 2.658L35.044 0.912Q35.044 0.604 34.954 0.446Q34.864 0.287 34.571 0.287Q34.302 0.287 34.073 0.418Q33.845 0.549 33.708 0.778Q33.571 1.006 33.571 1.272L33.571 2.658L33.997 2.658Q34.204 2.682 34.243 2.889L34.243 2.979Q34.204 3.194 33.997 3.217L32.509 3.217Q32.302 3.194 32.259 2.979\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-34 -67.265)\">\u003Cpath d=\"M41.267 1.490Q41.267 1.010 41.511 0.596Q41.755 0.182 42.171-0.056Q42.587-0.295 43.067-0.295Q43.622-0.295 44.001-0.185Q44.380-0.076 44.380 0.330Q44.380 0.498 44.267 0.621Q44.153 0.744 43.981 0.744Q43.810 0.744 43.690 0.629Q43.571 0.514 43.571 0.346L43.571 0.287Q43.411 0.264 43.075 0.264Q42.747 0.264 42.479 0.434Q42.212 0.604 42.060 0.887Q41.907 1.170 41.907 1.490Q41.907 1.811 42.079 2.092Q42.251 2.373 42.536 2.535Q42.821 2.697 43.149 2.697Q43.462 2.697 43.589 2.594Q43.716 2.490 43.833 2.301Q43.950 2.112 44.067 2.096L44.235 2.096Q44.341 2.108 44.405 2.176Q44.470 2.244 44.470 2.346Q44.470 2.393 44.450 2.432Q44.341 2.725 44.138 2.905Q43.935 3.084 43.659 3.170Q43.384 3.256 43.067 3.256Q42.583 3.256 42.167 3.018Q41.751 2.780 41.509 2.377Q41.267 1.975 41.267 1.490M45.403 2.979L45.403 2.889Q45.454 2.682 45.649 2.658L46.755 2.658L46.755-1.111L45.649-1.111Q45.454-1.135 45.403-1.349L45.403-1.439Q45.454-1.646 45.649-1.670L47.145-1.670Q47.337-1.646 47.395-1.439L47.395 2.658L48.497 2.658Q48.696 2.682 48.747 2.889L48.747 2.979Q48.696 3.194 48.497 3.217L45.649 3.217Q45.454 3.194 45.403 2.979M49.817 2.979L49.817 2.889Q49.868 2.682 50.063 2.658L51.103 2.658L51.103 0.330L50.130 0.330Q49.931 0.307 49.880 0.088L49.880 0.002Q49.931-0.209 50.130-0.232L51.497-0.232Q51.692-0.213 51.743 0.002L51.743 2.658L52.657 2.658Q52.853 2.682 52.903 2.889L52.903 2.979Q52.853 3.194 52.657 3.217L50.063 3.217Q49.868 3.194 49.817 2.979M50.849-1.209L50.849-1.263Q50.849-1.435 50.985-1.556Q51.122-1.678 51.298-1.678Q51.470-1.678 51.606-1.556Q51.743-1.435 51.743-1.263L51.743-1.209Q51.743-1.033 51.606-0.912Q51.470-0.791 51.298-0.791Q51.122-0.791 50.985-0.912Q50.849-1.033 50.849-1.209M53.392 2.979L53.392 2.889Q53.442 2.678 53.638 2.658L53.837 2.658L53.837 0.330L53.638 0.330Q53.431 0.307 53.392 0.088L53.392 0.002Q53.442-0.213 53.638-0.232L54.118-0.232Q54.310-0.209 54.368 0.002Q54.688-0.271 55.103-0.271Q55.294-0.271 55.462-0.162Q55.630-0.053 55.704 0.119Q55.880-0.072 56.103-0.172Q56.325-0.271 56.567-0.271Q56.985-0.271 57.140 0.071Q57.294 0.412 57.294 0.881L57.294 2.658L57.493 2.658Q57.704 2.682 57.743 2.889L57.743 2.979Q57.692 3.194 57.493 3.217L56.677 3.217Q56.481 3.194 56.431 2.979L56.431 2.889Q56.481 2.682 56.677 2.658L56.767 2.658L56.767 0.912Q56.767 0.287 56.517 0.287Q56.185 0.287 56.007 0.598Q55.829 0.908 55.829 1.272L55.829 2.658L56.032 2.658Q56.239 2.682 56.278 2.889L56.278 2.979Q56.228 3.194 56.032 3.217L55.216 3.217Q55.017 3.194 54.966 2.979L54.966 2.889Q55.017 2.682 55.216 2.658L55.302 2.658L55.302 0.912Q55.302 0.287 55.056 0.287Q54.724 0.287 54.546 0.600Q54.368 0.912 54.368 1.272L54.368 2.658L54.567 2.658Q54.774 2.682 54.813 2.889L54.813 2.979Q54.763 3.197 54.567 3.217L53.638 3.217Q53.431 3.194 53.392 2.979M58.435 2.979L58.435-1.111L58.013-1.111Q57.806-1.135 57.763-1.349L57.763-1.439Q57.806-1.646 58.013-1.670L58.829-1.670Q59.024-1.646 59.075-1.439L59.075 0.072Q59.286-0.095 59.550-0.183Q59.813-0.271 60.083-0.271Q60.423-0.271 60.720-0.127Q61.017 0.018 61.231 0.270Q61.446 0.522 61.562 0.834Q61.677 1.147 61.677 1.490Q61.677 1.955 61.450 2.365Q61.224 2.776 60.837 3.016Q60.450 3.256 59.981 3.256Q59.462 3.256 59.075 2.889L59.075 2.979Q59.024 3.197 58.829 3.217L58.685 3.217Q58.493 3.194 58.435 2.979M59.938 2.697Q60.247 2.697 60.497 2.528Q60.747 2.358 60.892 2.076Q61.036 1.795 61.036 1.490Q61.036 1.201 60.911 0.922Q60.786 0.643 60.554 0.465Q60.321 0.287 60.020 0.287Q59.700 0.287 59.438 0.473Q59.177 0.658 59.075 0.959L59.075 1.811Q59.165 2.178 59.386 2.438Q59.606 2.697 59.938 2.697M62.524 3.018L62.524 2.104Q62.552 1.897 62.763 1.873L62.931 1.873Q63.095 1.897 63.153 2.057Q63.356 2.697 64.083 2.697Q64.290 2.697 64.519 2.662Q64.747 2.627 64.915 2.512Q65.083 2.397 65.083 2.194Q65.083 1.983 64.860 1.869Q64.638 1.756 64.364 1.713L63.665 1.600Q62.524 1.389 62.524 0.666Q62.524 0.377 62.669 0.188Q62.813-0.002 63.054-0.109Q63.294-0.217 63.550-0.256Q63.806-0.295 64.083-0.295Q64.333-0.295 64.526-0.265Q64.720-0.236 64.884-0.158Q64.962-0.275 65.091-0.295L65.169-0.295Q65.267-0.283 65.329-0.220Q65.392-0.158 65.403-0.064L65.403 0.643Q65.392 0.737 65.329 0.803Q65.267 0.869 65.169 0.881L65.001 0.881Q64.907 0.869 64.841 0.803Q64.774 0.737 64.763 0.643Q64.763 0.264 64.067 0.264Q63.720 0.264 63.401 0.346Q63.083 0.428 63.083 0.674Q63.083 0.940 63.755 1.049L64.458 1.170Q64.942 1.252 65.292 1.500Q65.642 1.748 65.642 2.194Q65.642 2.584 65.405 2.826Q65.169 3.069 64.819 3.162Q64.470 3.256 64.083 3.256Q63.505 3.256 63.106 3.002Q63.036 3.127 62.987 3.184Q62.938 3.240 62.833 3.256L62.763 3.256Q62.548 3.233 62.524 3.018\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-34 -67.265)\">\u003Cpath d=\"M71.076 2.979L71.076 2.889Q71.127 2.682 71.322 2.658L72.361 2.658L72.361 0.330L71.389 0.330Q71.189 0.307 71.139 0.088L71.139 0.002Q71.189-0.209 71.389-0.232L72.756-0.232Q72.951-0.213 73.002 0.002L73.002 2.658L73.916 2.658Q74.111 2.682 74.162 2.889L74.162 2.979Q74.111 3.194 73.916 3.217L71.322 3.217Q71.127 3.194 71.076 2.979M72.107-1.209L72.107-1.263Q72.107-1.435 72.244-1.556Q72.381-1.678 72.557-1.678Q72.729-1.678 72.865-1.556Q73.002-1.435 73.002-1.263L73.002-1.209Q73.002-1.033 72.865-0.912Q72.729-0.791 72.557-0.791Q72.381-0.791 72.244-0.912Q72.107-1.033 72.107-1.209M74.775 2.979L74.775 2.889Q74.818 2.682 75.025 2.658L75.447 2.658L75.447 0.330L75.025 0.330Q74.818 0.307 74.775 0.088L74.775 0.002Q74.822-0.209 75.025-0.232L75.842-0.232Q76.037-0.209 76.088 0.002L76.088 0.088L76.080 0.112Q76.307-0.068 76.580-0.170Q76.854-0.271 77.147-0.271Q77.494-0.271 77.732-0.131Q77.971 0.010 78.086 0.268Q78.201 0.526 78.201 0.881L78.201 2.658L78.627 2.658Q78.834 2.682 78.873 2.889L78.873 2.979Q78.834 3.194 78.627 3.217L77.232 3.217Q77.037 3.194 76.986 2.979L76.986 2.889Q77.037 2.678 77.232 2.658L77.561 2.658L77.561 0.912Q77.561 0.604 77.471 0.446Q77.381 0.287 77.088 0.287Q76.818 0.287 76.590 0.418Q76.361 0.549 76.225 0.778Q76.088 1.006 76.088 1.272L76.088 2.658L76.514 2.658Q76.721 2.682 76.760 2.889L76.760 2.979Q76.721 3.194 76.514 3.217L75.025 3.217Q74.818 3.194 74.775 2.979\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">Near a saddle, the negative-curvature axis flips the sign of its Newton component, so the Newton step (red) climbs back toward the saddle.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:410.992px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 308.244 150.388\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" d=\"M113.26-.41c0-5.499-12.42-9.957-27.742-9.957-15.32 0-27.74 4.458-27.74 9.958s12.42 9.958 27.74 9.958c15.322 0 27.742-4.458 27.742-9.958Zm-27.742 0\"\u002F>\u003Cpath fill=\"none\" d=\"M141.001-.41c0-11-24.84-19.916-55.483-19.916S30.036-11.409 30.036-.409s24.84 19.917 55.482 19.917c30.643 0 55.483-8.917 55.483-19.917Zm-55.483 0\"\u002F>\u003Cpath fill=\"none\" d=\"M168.742-.41c0-16.499-37.26-29.874-83.224-29.874C39.555-30.284 2.295-16.91 2.295-.41s37.26 29.875 83.223 29.875S168.742 16.09 168.742-.41Zm-83.224 0\"\u002F>\u003Cpath fill=\"none\" d=\"M196.484-.41c0-21.999-49.68-39.833-110.966-39.833S-25.447-22.409-25.447-.409s49.68 39.833 110.965 39.833S196.484 21.591 196.484-.409Zm-110.966 0\"\u002F>\u003Cpath fill=\"none\" d=\"M229.774-.41c0-28.599-64.585-51.783-144.256-51.783-79.67 0-144.255 23.184-144.255 51.784S5.848 51.375 85.518 51.375c79.671 0 144.256-23.184 144.256-51.784Zm-144.256 0\"\u002F>\u003Cpath fill=\"var(--tk-accent)\" stroke=\"none\" d=\"M87.918-.41a2.4 2.4 0 1 0-4.8 0 2.4 2.4 0 0 0 4.8 0m-2.4 0\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cg transform=\"translate(30.65 62.482)\">\u003Cpath d=\"M85.463-0.647L85.463-0.737Q85.514-0.948 85.709-0.968L85.909-0.968L85.909-3.296L85.709-3.296Q85.502-3.319 85.463-3.538L85.463-3.624Q85.514-3.839 85.709-3.858L86.190-3.858Q86.381-3.835 86.440-3.624Q86.760-3.897 87.174-3.897Q87.366-3.897 87.534-3.788Q87.702-3.679 87.776-3.507Q87.952-3.698 88.174-3.798Q88.397-3.897 88.639-3.897Q89.057-3.897 89.211-3.555Q89.366-3.214 89.366-2.745L89.366-0.968L89.565-0.968Q89.776-0.944 89.815-0.737L89.815-0.647Q89.764-0.432 89.565-0.409L88.748-0.409Q88.553-0.432 88.502-0.647L88.502-0.737Q88.553-0.944 88.748-0.968L88.838-0.968L88.838-2.714Q88.838-3.339 88.588-3.339Q88.256-3.339 88.079-3.028Q87.901-2.718 87.901-2.354L87.901-0.968L88.104-0.968Q88.311-0.944 88.350-0.737L88.350-0.647Q88.299-0.432 88.104-0.409L87.288-0.409Q87.088-0.432 87.038-0.647L87.038-0.737Q87.088-0.944 87.288-0.968L87.373-0.968L87.373-2.714Q87.373-3.339 87.127-3.339Q86.795-3.339 86.618-3.026Q86.440-2.714 86.440-2.354L86.440-0.968L86.639-0.968Q86.846-0.944 86.885-0.737L86.885-0.647Q86.834-0.429 86.639-0.409L85.709-0.409Q85.502-0.432 85.463-0.647M90.381-0.647L90.381-0.737Q90.432-0.944 90.627-0.968L91.666-0.968L91.666-3.296L90.694-3.296Q90.495-3.319 90.444-3.538L90.444-3.624Q90.495-3.835 90.694-3.858L92.061-3.858Q92.256-3.839 92.307-3.624L92.307-0.968L93.221-0.968Q93.416-0.944 93.467-0.737L93.467-0.647Q93.416-0.432 93.221-0.409L90.627-0.409Q90.432-0.432 90.381-0.647M91.413-4.835L91.413-4.889Q91.413-5.061 91.549-5.182Q91.686-5.304 91.862-5.304Q92.034-5.304 92.170-5.182Q92.307-5.061 92.307-4.889L92.307-4.835Q92.307-4.659 92.170-4.538Q92.034-4.417 91.862-4.417Q91.686-4.417 91.549-4.538Q91.413-4.659 91.413-4.835M94.081-0.647L94.081-0.737Q94.123-0.944 94.331-0.968L94.752-0.968L94.752-3.296L94.331-3.296Q94.123-3.319 94.081-3.538L94.081-3.624Q94.127-3.835 94.331-3.858L95.147-3.858Q95.342-3.835 95.393-3.624L95.393-3.538L95.385-3.514Q95.612-3.694 95.885-3.796Q96.159-3.897 96.452-3.897Q96.799-3.897 97.038-3.757Q97.276-3.616 97.391-3.358Q97.506-3.100 97.506-2.745L97.506-0.968L97.932-0.968Q98.139-0.944 98.178-0.737L98.178-0.647Q98.139-0.432 97.932-0.409L96.538-0.409Q96.342-0.432 96.291-0.647L96.291-0.737Q96.342-0.948 96.538-0.968L96.866-0.968L96.866-2.714Q96.866-3.022 96.776-3.180Q96.686-3.339 96.393-3.339Q96.123-3.339 95.895-3.208Q95.666-3.077 95.530-2.848Q95.393-2.620 95.393-2.354L95.393-0.968L95.819-0.968Q96.026-0.944 96.065-0.737L96.065-0.647Q96.026-0.432 95.819-0.409L94.331-0.409Q94.123-0.432 94.081-0.647M98.873-0.647L98.873-0.737Q98.924-0.944 99.120-0.968L100.159-0.968L100.159-3.296L99.186-3.296Q98.987-3.319 98.936-3.538L98.936-3.624Q98.987-3.835 99.186-3.858L100.553-3.858Q100.748-3.839 100.799-3.624L100.799-0.968L101.713-0.968Q101.909-0.944 101.959-0.737L101.959-0.647Q101.909-0.432 101.713-0.409L99.120-0.409Q98.924-0.432 98.873-0.647M99.905-4.835L99.905-4.889Q99.905-5.061 100.041-5.182Q100.178-5.304 100.354-5.304Q100.526-5.304 100.663-5.182Q100.799-5.061 100.799-4.889L100.799-4.835Q100.799-4.659 100.663-4.538Q100.526-4.417 100.354-4.417Q100.178-4.417 100.041-4.538Q99.905-4.659 99.905-4.835M102.448-0.647L102.448-0.737Q102.498-0.948 102.694-0.968L102.893-0.968L102.893-3.296L102.694-3.296Q102.487-3.319 102.448-3.538L102.448-3.624Q102.498-3.839 102.694-3.858L103.174-3.858Q103.366-3.835 103.424-3.624Q103.745-3.897 104.159-3.897Q104.350-3.897 104.518-3.788Q104.686-3.679 104.760-3.507Q104.936-3.698 105.159-3.798Q105.381-3.897 105.623-3.897Q106.041-3.897 106.196-3.555Q106.350-3.214 106.350-2.745L106.350-0.968L106.549-0.968Q106.760-0.944 106.799-0.737L106.799-0.647Q106.748-0.432 106.549-0.409L105.733-0.409Q105.538-0.432 105.487-0.647L105.487-0.737Q105.538-0.944 105.733-0.968L105.823-0.968L105.823-2.714Q105.823-3.339 105.573-3.339Q105.241-3.339 105.063-3.028Q104.885-2.718 104.885-2.354L104.885-0.968L105.088-0.968Q105.295-0.944 105.334-0.737L105.334-0.647Q105.284-0.432 105.088-0.409L104.272-0.409Q104.073-0.432 104.022-0.647L104.022-0.737Q104.073-0.944 104.272-0.968L104.358-0.968L104.358-2.714Q104.358-3.339 104.112-3.339Q103.780-3.339 103.602-3.026Q103.424-2.714 103.424-2.354L103.424-0.968L103.623-0.968Q103.831-0.944 103.870-0.737L103.870-0.647Q103.819-0.429 103.623-0.409L102.694-0.409Q102.487-0.432 102.448-0.647M107.491-1.264L107.491-3.296L107.069-3.296Q106.862-3.319 106.819-3.538L106.819-3.624Q106.866-3.835 107.069-3.858L107.885-3.858Q108.081-3.835 108.131-3.624L108.131-1.296Q108.131-1.061 108.301-0.995Q108.471-0.929 108.756-0.929Q108.963-0.929 109.159-1.005Q109.354-1.081 109.479-1.231Q109.604-1.382 109.604-1.593L109.604-3.296L109.182-3.296Q108.971-3.319 108.932-3.538L108.932-3.624Q108.971-3.835 109.182-3.858L109.995-3.858Q110.194-3.835 110.245-3.624L110.245-0.968L110.670-0.968Q110.877-0.944 110.916-0.737L110.916-0.647Q110.877-0.432 110.670-0.409L109.854-0.409Q109.655-0.432 109.604-0.632Q109.202-0.370 108.694-0.370Q108.459-0.370 108.245-0.411Q108.030-0.452 107.864-0.554Q107.698-0.655 107.594-0.833Q107.491-1.011 107.491-1.264M110.940-0.647L110.940-0.737Q110.991-0.948 111.186-0.968L111.385-0.968L111.385-3.296L111.186-3.296Q110.979-3.319 110.940-3.538L110.940-3.624Q110.991-3.839 111.186-3.858L111.666-3.858Q111.858-3.835 111.916-3.624Q112.237-3.897 112.651-3.897Q112.842-3.897 113.010-3.788Q113.178-3.679 113.252-3.507Q113.428-3.698 113.651-3.798Q113.873-3.897 114.116-3.897Q114.534-3.897 114.688-3.555Q114.842-3.214 114.842-2.745L114.842-0.968L115.041-0.968Q115.252-0.944 115.291-0.737L115.291-0.647Q115.241-0.432 115.041-0.409L114.225-0.409Q114.030-0.432 113.979-0.647L113.979-0.737Q114.030-0.944 114.225-0.968L114.315-0.968L114.315-2.714Q114.315-3.339 114.065-3.339Q113.733-3.339 113.555-3.028Q113.377-2.718 113.377-2.354L113.377-0.968L113.581-0.968Q113.788-0.944 113.827-0.737L113.827-0.647Q113.776-0.432 113.581-0.409L112.764-0.409Q112.565-0.432 112.514-0.647L112.514-0.737Q112.565-0.944 112.764-0.968L112.850-0.968L112.850-2.714Q112.850-3.339 112.604-3.339Q112.272-3.339 112.094-3.026Q111.916-2.714 111.916-2.354L111.916-0.968L112.116-0.968Q112.323-0.944 112.362-0.737L112.362-0.647Q112.311-0.429 112.116-0.409L111.186-0.409Q110.979-0.432 110.940-0.647\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath stroke=\"none\" d=\"M-14.712-41.666a2.2 2.2 0 1 0-4.4 0 2.2 2.2 0 0 0 4.4 0m-2.2 0\"\u002F>\u003Cg transform=\"translate(-113.055 -49.058)\">\u003Cpath d=\"M86.104-0.608L86.104-1.522Q86.131-1.729 86.342-1.753L86.510-1.753Q86.674-1.729 86.733-1.569Q86.936-0.929 87.663-0.929Q87.870-0.929 88.098-0.964Q88.327-0.999 88.495-1.114Q88.663-1.229 88.663-1.432Q88.663-1.643 88.440-1.757Q88.217-1.870 87.944-1.913L87.245-2.026Q86.104-2.237 86.104-2.960Q86.104-3.249 86.248-3.438Q86.393-3.628 86.633-3.735Q86.873-3.843 87.129-3.882Q87.385-3.921 87.663-3.921Q87.913-3.921 88.106-3.891Q88.299-3.862 88.463-3.784Q88.541-3.901 88.670-3.921L88.748-3.921Q88.846-3.909 88.909-3.846Q88.971-3.784 88.983-3.690L88.983-2.983Q88.971-2.889 88.909-2.823Q88.846-2.757 88.748-2.745L88.581-2.745Q88.487-2.757 88.420-2.823Q88.354-2.889 88.342-2.983Q88.342-3.362 87.647-3.362Q87.299-3.362 86.981-3.280Q86.663-3.198 86.663-2.952Q86.663-2.686 87.334-2.577L88.038-2.456Q88.522-2.374 88.872-2.126Q89.221-1.878 89.221-1.432Q89.221-1.042 88.985-0.800Q88.748-0.557 88.399-0.464Q88.049-0.370 87.663-0.370Q87.084-0.370 86.686-0.624Q86.616-0.499 86.567-0.442Q86.518-0.386 86.413-0.370L86.342-0.370Q86.127-0.393 86.104-0.608M90.963-1.514L90.963-3.296L90.213-3.296Q90.014-3.319 89.963-3.538L89.963-3.624Q90.014-3.835 90.213-3.858L90.963-3.858L90.963-4.608Q91.014-4.815 91.213-4.843L91.358-4.843Q91.553-4.815 91.604-4.608L91.604-3.858L92.963-3.858Q93.155-3.839 93.213-3.624L93.213-3.538Q93.159-3.319 92.963-3.296L91.604-3.296L91.604-1.546Q91.604-0.929 92.178-0.929Q92.428-0.929 92.592-1.114Q92.756-1.300 92.756-1.546Q92.756-1.639 92.829-1.710Q92.901-1.780 93.002-1.792L93.147-1.792Q93.346-1.768 93.397-1.561L93.397-1.514Q93.397-1.190 93.213-0.927Q93.030-0.663 92.737-0.516Q92.444-0.370 92.123-0.370Q91.612-0.370 91.288-0.680Q90.963-0.991 90.963-1.514M94.416-1.522Q94.416-1.968 94.831-2.225Q95.245-2.483 95.786-2.583Q96.327-2.682 96.834-2.690Q96.834-2.905 96.700-3.057Q96.565-3.210 96.358-3.286Q96.151-3.362 95.940-3.362Q95.596-3.362 95.436-3.339L95.436-3.280Q95.436-3.112 95.317-2.997Q95.198-2.882 95.034-2.882Q94.858-2.882 94.743-3.005Q94.627-3.128 94.627-3.296Q94.627-3.702 95.008-3.811Q95.389-3.921 95.948-3.921Q96.217-3.921 96.485-3.843Q96.752-3.764 96.977-3.614Q97.202-3.464 97.338-3.243Q97.475-3.022 97.475-2.745L97.475-1.026Q97.475-0.968 98.002-0.968Q98.198-0.948 98.248-0.737L98.248-0.647Q98.198-0.432 98.002-0.409L97.858-0.409Q97.514-0.409 97.286-0.456Q97.057-0.503 96.913-0.690Q96.452-0.370 95.745-0.370Q95.409-0.370 95.104-0.511Q94.799-0.651 94.608-0.913Q94.416-1.175 94.416-1.522M95.057-1.514Q95.057-1.241 95.299-1.085Q95.541-0.929 95.827-0.929Q96.045-0.929 96.278-0.987Q96.510-1.046 96.672-1.184Q96.834-1.323 96.834-1.546L96.834-2.136Q96.553-2.136 96.137-2.079Q95.721-2.022 95.389-1.884Q95.057-1.745 95.057-1.514M98.479-0.647L98.479-0.737Q98.538-0.944 98.729-0.968L99.440-0.968L99.440-3.296L98.729-3.296Q98.534-3.319 98.479-3.538L98.479-3.624Q98.538-3.835 98.729-3.858L99.831-3.858Q100.030-3.839 100.081-3.624L100.081-3.296Q100.342-3.581 100.698-3.739Q101.053-3.897 101.440-3.897Q101.733-3.897 101.967-3.763Q102.202-3.628 102.202-3.362Q102.202-3.194 102.092-3.077Q101.983-2.960 101.815-2.960Q101.663-2.960 101.547-3.071Q101.432-3.182 101.432-3.339Q101.057-3.339 100.743-3.138Q100.428-2.936 100.254-2.602Q100.081-2.268 100.081-1.889L100.081-0.968L101.026-0.968Q101.233-0.944 101.272-0.737L101.272-0.647Q101.233-0.432 101.026-0.409L98.729-0.409Q98.538-0.432 98.479-0.647M103.702-1.514L103.702-3.296L102.952-3.296Q102.752-3.319 102.702-3.538L102.702-3.624Q102.752-3.835 102.952-3.858L103.702-3.858L103.702-4.608Q103.752-4.815 103.952-4.843L104.096-4.843Q104.291-4.815 104.342-4.608L104.342-3.858L105.702-3.858Q105.893-3.839 105.952-3.624L105.952-3.538Q105.897-3.319 105.702-3.296L104.342-3.296L104.342-1.546Q104.342-0.929 104.916-0.929Q105.166-0.929 105.331-1.114Q105.495-1.300 105.495-1.546Q105.495-1.639 105.567-1.710Q105.639-1.780 105.741-1.792L105.885-1.792Q106.084-1.768 106.135-1.561L106.135-1.514Q106.135-1.190 105.952-0.927Q105.768-0.663 105.475-0.516Q105.182-0.370 104.862-0.370Q104.350-0.370 104.026-0.680Q103.702-0.991 103.702-1.514\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"m-16.912-41.666 35.245 54.223\"\u002F>\u003Cpath stroke=\"none\" d=\"m20.077 15.24-.644-5.688-1.1 3.005-3.193-.215\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"m20.077 15.24 22.405-33.476\"\u002F>\u003Cpath stroke=\"none\" d=\"m44.262-20.895-4.975 2.83 3.195-.171 1.06 3.02\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"M44.262-20.895 59.822 8.15\"\u002F>\u003Cpath stroke=\"none\" d=\"m61.333 10.972-.16-5.722-1.35 2.901-3.164-.483\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"m61.333 10.972 10.56-17.19\"\u002F>\u003Cpath stroke=\"none\" d=\"m73.568-8.945-4.86 3.022 3.186-.296 1.176 2.976\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(-79.525 -63.639)\">\u003Cpath d=\"M86.104-0.608L86.104-1.522Q86.131-1.729 86.342-1.753L86.510-1.753Q86.674-1.729 86.733-1.569Q86.936-0.929 87.663-0.929Q87.870-0.929 88.098-0.964Q88.327-0.999 88.495-1.114Q88.663-1.229 88.663-1.432Q88.663-1.643 88.440-1.757Q88.217-1.870 87.944-1.913L87.245-2.026Q86.104-2.237 86.104-2.960Q86.104-3.249 86.248-3.438Q86.393-3.628 86.633-3.735Q86.873-3.843 87.129-3.882Q87.385-3.921 87.663-3.921Q87.913-3.921 88.106-3.891Q88.299-3.862 88.463-3.784Q88.541-3.901 88.670-3.921L88.748-3.921Q88.846-3.909 88.909-3.846Q88.971-3.784 88.983-3.690L88.983-2.983Q88.971-2.889 88.909-2.823Q88.846-2.757 88.748-2.745L88.581-2.745Q88.487-2.757 88.420-2.823Q88.354-2.889 88.342-2.983Q88.342-3.362 87.647-3.362Q87.299-3.362 86.981-3.280Q86.663-3.198 86.663-2.952Q86.663-2.686 87.334-2.577L88.038-2.456Q88.522-2.374 88.872-2.126Q89.221-1.878 89.221-1.432Q89.221-1.042 88.985-0.800Q88.748-0.557 88.399-0.464Q88.049-0.370 87.663-0.370Q87.084-0.370 86.686-0.624Q86.616-0.499 86.567-0.442Q86.518-0.386 86.413-0.370L86.342-0.370Q86.127-0.393 86.104-0.608M90.963-1.514L90.963-3.296L90.213-3.296Q90.014-3.319 89.963-3.538L89.963-3.624Q90.014-3.835 90.213-3.858L90.963-3.858L90.963-4.608Q91.014-4.815 91.213-4.843L91.358-4.843Q91.553-4.815 91.604-4.608L91.604-3.858L92.963-3.858Q93.155-3.839 93.213-3.624L93.213-3.538Q93.159-3.319 92.963-3.296L91.604-3.296L91.604-1.546Q91.604-0.929 92.178-0.929Q92.428-0.929 92.592-1.114Q92.756-1.300 92.756-1.546Q92.756-1.639 92.829-1.710Q92.901-1.780 93.002-1.792L93.147-1.792Q93.346-1.768 93.397-1.561L93.397-1.514Q93.397-1.190 93.213-0.927Q93.030-0.663 92.737-0.516Q92.444-0.370 92.123-0.370Q91.612-0.370 91.288-0.680Q90.963-0.991 90.963-1.514M97.522-1.897L95.081-1.897Q95.135-1.620 95.332-1.397Q95.530-1.175 95.807-1.052Q96.084-0.929 96.370-0.929Q96.842-0.929 97.065-1.218Q97.073-1.229 97.129-1.335Q97.186-1.440 97.235-1.483Q97.284-1.526 97.377-1.538L97.522-1.538Q97.713-1.518 97.772-1.304L97.772-1.249Q97.706-0.948 97.475-0.751Q97.245-0.554 96.932-0.462Q96.620-0.370 96.315-0.370Q95.831-0.370 95.391-0.598Q94.952-0.827 94.684-1.227Q94.416-1.628 94.416-2.120L94.416-2.179Q94.416-2.647 94.663-3.050Q94.909-3.452 95.317-3.686Q95.725-3.921 96.194-3.921Q96.698-3.921 97.051-3.698Q97.405-3.475 97.588-3.087Q97.772-2.698 97.772-2.194L97.772-2.136Q97.713-1.921 97.522-1.897M95.088-2.448L97.116-2.448Q97.069-2.858 96.831-3.110Q96.592-3.362 96.194-3.362Q95.799-3.362 95.493-3.100Q95.186-2.839 95.088-2.448M101.768-1.897L99.327-1.897Q99.381-1.620 99.579-1.397Q99.776-1.175 100.053-1.052Q100.331-0.929 100.616-0.929Q101.088-0.929 101.311-1.218Q101.319-1.229 101.375-1.335Q101.432-1.440 101.481-1.483Q101.530-1.526 101.623-1.538L101.768-1.538Q101.959-1.518 102.018-1.304L102.018-1.249Q101.952-0.948 101.721-0.751Q101.491-0.554 101.178-0.462Q100.866-0.370 100.561-0.370Q100.077-0.370 99.637-0.598Q99.198-0.827 98.930-1.227Q98.663-1.628 98.663-2.120L98.663-2.179Q98.663-2.647 98.909-3.050Q99.155-3.452 99.563-3.686Q99.971-3.921 100.440-3.921Q100.944-3.921 101.297-3.698Q101.651-3.475 101.834-3.087Q102.018-2.698 102.018-2.194L102.018-2.136Q101.959-1.921 101.768-1.897M99.334-2.448L101.362-2.448Q101.315-2.858 101.077-3.110Q100.838-3.362 100.440-3.362Q100.045-3.362 99.739-3.100Q99.432-2.839 99.334-2.448M102.573 1.134L102.573 1.048Q102.616 0.829 102.823 0.806L103.245 0.806L103.245-3.296L102.823-3.296Q102.616-3.319 102.573-3.538L102.573-3.624Q102.620-3.835 102.823-3.858L103.639-3.858Q103.834-3.839 103.885-3.624L103.885-3.554Q104.096-3.721 104.360-3.809Q104.623-3.897 104.893-3.897Q105.233-3.897 105.530-3.753Q105.827-3.608 106.041-3.356Q106.256-3.104 106.372-2.792Q106.487-2.479 106.487-2.136Q106.487-1.671 106.260-1.261Q106.034-0.850 105.647-0.610Q105.260-0.370 104.791-0.370Q104.272-0.370 103.885-0.737L103.885 0.806L104.311 0.806Q104.518 0.829 104.557 1.048L104.557 1.134Q104.518 1.345 104.311 1.368L102.823 1.368Q102.620 1.345 102.573 1.134M104.748-0.929Q105.057-0.929 105.307-1.098Q105.557-1.268 105.702-1.550Q105.846-1.831 105.846-2.136Q105.846-2.425 105.721-2.704Q105.596-2.983 105.364-3.161Q105.131-3.339 104.831-3.339Q104.510-3.339 104.248-3.153Q103.987-2.968 103.885-2.667L103.885-1.815Q103.975-1.448 104.196-1.188Q104.416-0.929 104.748-0.929M110.260-1.897L107.819-1.897Q107.873-1.620 108.071-1.397Q108.268-1.175 108.545-1.052Q108.823-0.929 109.108-0.929Q109.581-0.929 109.803-1.218Q109.811-1.229 109.868-1.335Q109.924-1.440 109.973-1.483Q110.022-1.526 110.116-1.538L110.260-1.538Q110.452-1.518 110.510-1.304L110.510-1.249Q110.444-0.948 110.213-0.751Q109.983-0.554 109.670-0.462Q109.358-0.370 109.053-0.370Q108.569-0.370 108.129-0.598Q107.690-0.827 107.422-1.227Q107.155-1.628 107.155-2.120L107.155-2.179Q107.155-2.647 107.401-3.050Q107.647-3.452 108.055-3.686Q108.463-3.921 108.932-3.921Q109.436-3.921 109.789-3.698Q110.143-3.475 110.327-3.087Q110.510-2.698 110.510-2.194L110.510-2.136Q110.452-1.921 110.260-1.897M107.827-2.448L109.854-2.448Q109.807-2.858 109.569-3.110Q109.331-3.362 108.932-3.362Q108.538-3.362 108.231-3.100Q107.924-2.839 107.827-2.448M111.581-0.608L111.581-1.522Q111.608-1.729 111.819-1.753L111.987-1.753Q112.151-1.729 112.209-1.569Q112.413-0.929 113.139-0.929Q113.346-0.929 113.575-0.964Q113.803-0.999 113.971-1.114Q114.139-1.229 114.139-1.432Q114.139-1.643 113.916-1.757Q113.694-1.870 113.420-1.913L112.721-2.026Q111.581-2.237 111.581-2.960Q111.581-3.249 111.725-3.438Q111.870-3.628 112.110-3.735Q112.350-3.843 112.606-3.882Q112.862-3.921 113.139-3.921Q113.389-3.921 113.582-3.891Q113.776-3.862 113.940-3.784Q114.018-3.901 114.147-3.921L114.225-3.921Q114.323-3.909 114.385-3.846Q114.448-3.784 114.459-3.690L114.459-2.983Q114.448-2.889 114.385-2.823Q114.323-2.757 114.225-2.745L114.057-2.745Q113.963-2.757 113.897-2.823Q113.831-2.889 113.819-2.983Q113.819-3.362 113.123-3.362Q112.776-3.362 112.457-3.280Q112.139-3.198 112.139-2.952Q112.139-2.686 112.811-2.577L113.514-2.456Q113.998-2.374 114.348-2.126Q114.698-1.878 114.698-1.432Q114.698-1.042 114.461-0.800Q114.225-0.557 113.875-0.464Q113.526-0.370 113.139-0.370Q112.561-0.370 112.163-0.624Q112.092-0.499 112.043-0.442Q111.995-0.386 111.889-0.370L111.819-0.370Q111.604-0.393 111.581-0.608M116.440-1.514L116.440-3.296L115.690-3.296Q115.491-3.319 115.440-3.538L115.440-3.624Q115.491-3.835 115.690-3.858L116.440-3.858L116.440-4.608Q116.491-4.815 116.690-4.843L116.834-4.843Q117.030-4.815 117.081-4.608L117.081-3.858L118.440-3.858Q118.631-3.839 118.690-3.624L118.690-3.538Q118.635-3.319 118.440-3.296L117.081-3.296L117.081-1.546Q117.081-0.929 117.655-0.929Q117.905-0.929 118.069-1.114Q118.233-1.300 118.233-1.546Q118.233-1.639 118.305-1.710Q118.377-1.780 118.479-1.792L118.623-1.792Q118.823-1.768 118.873-1.561L118.873-1.514Q118.873-1.190 118.690-0.927Q118.506-0.663 118.213-0.516Q117.920-0.370 117.600-0.370Q117.088-0.370 116.764-0.680Q116.440-0.991 116.440-1.514\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-79.525 -63.639)\">\u003Cpath d=\"M125.632-0.370Q125.167-0.370 124.802-0.620Q124.437-0.870 124.232-1.274Q124.027-1.679 124.027-2.136Q124.027-2.479 124.152-2.800Q124.277-3.120 124.509-3.370Q124.742-3.620 125.046-3.759Q125.351-3.897 125.707-3.897Q126.218-3.897 126.624-3.577L126.624-4.737L126.203-4.737Q125.992-4.761 125.953-4.975L125.953-5.065Q125.992-5.272 126.203-5.296L127.015-5.296Q127.214-5.272 127.265-5.065L127.265-0.968L127.691-0.968Q127.898-0.944 127.937-0.737L127.937-0.647Q127.898-0.432 127.691-0.409L126.874-0.409Q126.675-0.432 126.624-0.647L126.624-0.776Q126.429-0.585 126.167-0.477Q125.906-0.370 125.632-0.370M125.671-0.929Q126.031-0.929 126.283-1.198Q126.535-1.468 126.624-1.843L126.624-2.675Q126.566-2.862 126.439-3.013Q126.312-3.163 126.134-3.251Q125.957-3.339 125.761-3.339Q125.453-3.339 125.203-3.169Q124.953-2.999 124.808-2.714Q124.664-2.429 124.664-2.128Q124.664-1.679 124.949-1.304Q125.234-0.929 125.671-0.929M131.527-1.897L129.085-1.897Q129.140-1.620 129.337-1.397Q129.535-1.175 129.812-1.052Q130.089-0.929 130.374-0.929Q130.847-0.929 131.070-1.218Q131.078-1.229 131.134-1.335Q131.191-1.440 131.240-1.483Q131.289-1.526 131.382-1.538L131.527-1.538Q131.718-1.518 131.777-1.304L131.777-1.249Q131.710-0.948 131.480-0.751Q131.249-0.554 130.937-0.462Q130.624-0.370 130.320-0.370Q129.835-0.370 129.396-0.598Q128.957-0.827 128.689-1.227Q128.421-1.628 128.421-2.120L128.421-2.179Q128.421-2.647 128.667-3.050Q128.914-3.452 129.322-3.686Q129.730-3.921 130.199-3.921Q130.703-3.921 131.056-3.698Q131.410-3.475 131.593-3.087Q131.777-2.698 131.777-2.194L131.777-2.136Q131.718-1.921 131.527-1.897M129.093-2.448L131.121-2.448Q131.074-2.858 130.835-3.110Q130.597-3.362 130.199-3.362Q129.804-3.362 129.498-3.100Q129.191-2.839 129.093-2.448M132.847-0.608L132.847-1.522Q132.874-1.729 133.085-1.753L133.253-1.753Q133.417-1.729 133.476-1.569Q133.679-0.929 134.406-0.929Q134.613-0.929 134.841-0.964Q135.070-0.999 135.238-1.114Q135.406-1.229 135.406-1.432Q135.406-1.643 135.183-1.757Q134.960-1.870 134.687-1.913L133.988-2.026Q132.847-2.237 132.847-2.960Q132.847-3.249 132.992-3.438Q133.136-3.628 133.376-3.735Q133.617-3.843 133.873-3.882Q134.128-3.921 134.406-3.921Q134.656-3.921 134.849-3.891Q135.042-3.862 135.207-3.784Q135.285-3.901 135.414-3.921L135.492-3.921Q135.589-3.909 135.652-3.846Q135.714-3.784 135.726-3.690L135.726-2.983Q135.714-2.889 135.652-2.823Q135.589-2.757 135.492-2.745L135.324-2.745Q135.230-2.757 135.164-2.823Q135.097-2.889 135.085-2.983Q135.085-3.362 134.390-3.362Q134.042-3.362 133.724-3.280Q133.406-3.198 133.406-2.952Q133.406-2.686 134.078-2.577L134.781-2.456Q135.265-2.374 135.615-2.126Q135.964-1.878 135.964-1.432Q135.964-1.042 135.728-0.800Q135.492-0.557 135.142-0.464Q134.792-0.370 134.406-0.370Q133.828-0.370 133.429-0.624Q133.359-0.499 133.310-0.442Q133.261-0.386 133.156-0.370L133.085-0.370Q132.871-0.393 132.847-0.608M137.066-2.136Q137.066-2.616 137.310-3.030Q137.554-3.444 137.970-3.682Q138.386-3.921 138.867-3.921Q139.421-3.921 139.800-3.811Q140.179-3.702 140.179-3.296Q140.179-3.128 140.066-3.005Q139.953-2.882 139.781-2.882Q139.609-2.882 139.490-2.997Q139.371-3.112 139.371-3.280L139.371-3.339Q139.210-3.362 138.874-3.362Q138.546-3.362 138.279-3.192Q138.011-3.022 137.859-2.739Q137.707-2.456 137.707-2.136Q137.707-1.815 137.878-1.534Q138.050-1.253 138.335-1.091Q138.621-0.929 138.949-0.929Q139.261-0.929 139.388-1.032Q139.515-1.136 139.632-1.325Q139.749-1.514 139.867-1.530L140.035-1.530Q140.140-1.518 140.205-1.450Q140.269-1.382 140.269-1.280Q140.269-1.233 140.249-1.194Q140.140-0.901 139.937-0.722Q139.734-0.542 139.458-0.456Q139.183-0.370 138.867-0.370Q138.382-0.370 137.966-0.608Q137.550-0.847 137.308-1.249Q137.066-1.651 137.066-2.136M144.265-1.897L141.824-1.897Q141.878-1.620 142.076-1.397Q142.273-1.175 142.550-1.052Q142.828-0.929 143.113-0.929Q143.585-0.929 143.808-1.218Q143.816-1.229 143.873-1.335Q143.929-1.440 143.978-1.483Q144.027-1.526 144.121-1.538L144.265-1.538Q144.457-1.518 144.515-1.304L144.515-1.249Q144.449-0.948 144.218-0.751Q143.988-0.554 143.675-0.462Q143.363-0.370 143.058-0.370Q142.574-0.370 142.134-0.598Q141.695-0.827 141.427-1.227Q141.160-1.628 141.160-2.120L141.160-2.179Q141.160-2.647 141.406-3.050Q141.652-3.452 142.060-3.686Q142.468-3.921 142.937-3.921Q143.441-3.921 143.794-3.698Q144.148-3.475 144.332-3.087Q144.515-2.698 144.515-2.194L144.515-2.136Q144.457-1.921 144.265-1.897M141.832-2.448L143.859-2.448Q143.812-2.858 143.574-3.110Q143.335-3.362 142.937-3.362Q142.542-3.362 142.236-3.100Q141.929-2.839 141.832-2.448M145.070-0.647L145.070-0.737Q145.113-0.944 145.320-0.968L145.742-0.968L145.742-3.296L145.320-3.296Q145.113-3.319 145.070-3.538L145.070-3.624Q145.117-3.835 145.320-3.858L146.136-3.858Q146.332-3.835 146.382-3.624L146.382-3.538L146.374-3.514Q146.601-3.694 146.874-3.796Q147.148-3.897 147.441-3.897Q147.789-3.897 148.027-3.757Q148.265-3.616 148.380-3.358Q148.496-3.100 148.496-2.745L148.496-0.968L148.921-0.968Q149.128-0.944 149.167-0.737L149.167-0.647Q149.128-0.432 148.921-0.409L147.527-0.409Q147.332-0.432 147.281-0.647L147.281-0.737Q147.332-0.948 147.527-0.968L147.855-0.968L147.855-2.714Q147.855-3.022 147.765-3.180Q147.675-3.339 147.382-3.339Q147.113-3.339 146.884-3.208Q146.656-3.077 146.519-2.848Q146.382-2.620 146.382-2.354L146.382-0.968L146.808-0.968Q147.015-0.944 147.054-0.737L147.054-0.647Q147.015-0.432 146.808-0.409L145.320-0.409Q145.113-0.432 145.070-0.647M150.445-1.514L150.445-3.296L149.695-3.296Q149.496-3.319 149.445-3.538L149.445-3.624Q149.496-3.835 149.695-3.858L150.445-3.858L150.445-4.608Q150.496-4.815 150.695-4.843L150.839-4.843Q151.035-4.815 151.085-4.608L151.085-3.858L152.445-3.858Q152.636-3.839 152.695-3.624L152.695-3.538Q152.640-3.319 152.445-3.296L151.085-3.296L151.085-1.546Q151.085-0.929 151.660-0.929Q151.910-0.929 152.074-1.114Q152.238-1.300 152.238-1.546Q152.238-1.639 152.310-1.710Q152.382-1.780 152.484-1.792L152.628-1.792Q152.828-1.768 152.878-1.561L152.878-1.514Q152.878-1.190 152.695-0.927Q152.511-0.663 152.218-0.516Q151.925-0.370 151.605-0.370Q151.093-0.370 150.769-0.680Q150.445-0.991 150.445-1.514\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"m-16.912-41.666 55.046 77.064\"\u002F>\u003Cpath stroke=\"none\" d=\"m39.994 38.002-.893-5.654-.967 3.05-3.199-.074\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"M39.994 38.002 83.073 1.654\"\u002F>\u003Cpath stroke=\"none\" d=\"M85.518-.41 79.955.937l3.118.718.183 3.195\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\">\u003Cg fill=\"var(--tk-good)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(-107.15 67.417)\">\u003Cpath d=\"M86.077-2.136Q86.077-2.616 86.321-3.030Q86.565-3.444 86.981-3.682Q87.397-3.921 87.877-3.921Q88.432-3.921 88.811-3.811Q89.190-3.702 89.190-3.296Q89.190-3.128 89.077-3.005Q88.963-2.882 88.791-2.882Q88.620-2.882 88.500-2.997Q88.381-3.112 88.381-3.280L88.381-3.339Q88.221-3.362 87.885-3.362Q87.557-3.362 87.289-3.192Q87.022-3.022 86.870-2.739Q86.717-2.456 86.717-2.136Q86.717-1.815 86.889-1.534Q87.061-1.253 87.346-1.091Q87.631-0.929 87.959-0.929Q88.272-0.929 88.399-1.032Q88.526-1.136 88.643-1.325Q88.760-1.514 88.877-1.530L89.045-1.530Q89.151-1.518 89.215-1.450Q89.280-1.382 89.280-1.280Q89.280-1.233 89.260-1.194Q89.151-0.901 88.948-0.722Q88.745-0.542 88.469-0.456Q88.194-0.370 87.877-0.370Q87.393-0.370 86.977-0.608Q86.561-0.847 86.319-1.249Q86.077-1.651 86.077-2.136M91.885-0.370Q91.413-0.370 91.028-0.614Q90.643-0.858 90.420-1.268Q90.198-1.679 90.198-2.136Q90.198-2.479 90.323-2.802Q90.448-3.124 90.678-3.378Q90.909-3.632 91.215-3.776Q91.522-3.921 91.885-3.921Q92.248-3.921 92.561-3.774Q92.873-3.628 93.096-3.382Q93.319-3.136 93.446-2.815Q93.573-2.495 93.573-2.136Q93.573-1.679 93.348-1.266Q93.123-0.854 92.739-0.612Q92.354-0.370 91.885-0.370M91.885-0.929Q92.350-0.929 92.641-1.323Q92.932-1.718 92.932-2.202Q92.932-2.495 92.797-2.763Q92.663-3.030 92.422-3.196Q92.182-3.362 91.885-3.362Q91.581-3.362 91.342-3.196Q91.104-3.030 90.969-2.763Q90.834-2.495 90.834-2.202Q90.834-1.722 91.127-1.325Q91.420-0.929 91.885-0.929M94.081-0.647L94.081-0.737Q94.123-0.944 94.331-0.968L94.752-0.968L94.752-3.296L94.331-3.296Q94.123-3.319 94.081-3.538L94.081-3.624Q94.127-3.835 94.331-3.858L95.147-3.858Q95.342-3.835 95.393-3.624L95.393-3.538L95.385-3.514Q95.612-3.694 95.885-3.796Q96.159-3.897 96.452-3.897Q96.799-3.897 97.038-3.757Q97.276-3.616 97.391-3.358Q97.506-3.100 97.506-2.745L97.506-0.968L97.932-0.968Q98.139-0.944 98.178-0.737L98.178-0.647Q98.139-0.432 97.932-0.409L96.538-0.409Q96.342-0.432 96.291-0.647L96.291-0.737Q96.342-0.948 96.538-0.968L96.866-0.968L96.866-2.714Q96.866-3.022 96.776-3.180Q96.686-3.339 96.393-3.339Q96.123-3.339 95.895-3.208Q95.666-3.077 95.530-2.848Q95.393-2.620 95.393-2.354L95.393-0.968L95.819-0.968Q96.026-0.944 96.065-0.737L96.065-0.647Q96.026-0.432 95.819-0.409L94.331-0.409Q94.123-0.432 94.081-0.647M98.639 0.775L98.639 0.720Q98.639 0.564 98.764 0.454Q98.889 0.345 99.049 0.345Q99.202 0.345 99.325 0.456Q99.448 0.568 99.448 0.720L99.448 0.775Q99.448 0.798 99.446 0.806Q99.444 0.814 99.440 0.821Q99.600 0.849 99.913 0.849Q100.264 0.849 100.448 0.554Q100.631 0.259 100.631-0.104L100.631-3.296L99.530-3.296Q99.331-3.319 99.280-3.538L99.280-3.624Q99.331-3.835 99.530-3.858L101.026-3.858Q101.221-3.835 101.272-3.624L101.272-0.050Q101.272 0.321 101.090 0.661Q100.909 1.001 100.592 1.204Q100.276 1.407 99.905 1.407Q99.659 1.407 99.471 1.396Q99.284 1.384 99.086 1.321Q98.889 1.259 98.764 1.128Q98.639 0.997 98.639 0.775M100.377-4.835L100.377-4.889Q100.377-5.061 100.514-5.182Q100.651-5.304 100.823-5.304Q100.998-5.304 101.135-5.182Q101.272-5.061 101.272-4.889L101.272-4.835Q101.272-4.659 101.135-4.538Q100.998-4.417 100.823-4.417Q100.651-4.417 100.514-4.538Q100.377-4.659 100.377-4.835M103.245-1.264L103.245-3.296L102.823-3.296Q102.616-3.319 102.573-3.538L102.573-3.624Q102.620-3.835 102.823-3.858L103.639-3.858Q103.834-3.835 103.885-3.624L103.885-1.296Q103.885-1.061 104.055-0.995Q104.225-0.929 104.510-0.929Q104.717-0.929 104.913-1.005Q105.108-1.081 105.233-1.231Q105.358-1.382 105.358-1.593L105.358-3.296L104.936-3.296Q104.725-3.319 104.686-3.538L104.686-3.624Q104.725-3.835 104.936-3.858L105.748-3.858Q105.948-3.835 105.998-3.624L105.998-0.968L106.424-0.968Q106.631-0.944 106.670-0.737L106.670-0.647Q106.631-0.432 106.424-0.409L105.608-0.409Q105.409-0.432 105.358-0.632Q104.956-0.370 104.448-0.370Q104.213-0.370 103.998-0.411Q103.784-0.452 103.618-0.554Q103.452-0.655 103.348-0.833Q103.245-1.011 103.245-1.264M106.987 0.239Q106.987-0.061 107.135-0.323Q107.284-0.585 107.534-0.745Q107.350-0.999 107.350-1.319Q107.350-1.604 107.502-1.866Q107.252-2.190 107.252-2.608Q107.252-2.968 107.444-3.264Q107.635-3.561 107.954-3.729Q108.272-3.897 108.635-3.897Q108.842-3.897 109.045-3.837Q109.248-3.776 109.413-3.675Q109.795-3.936 110.268-3.936Q110.506-3.936 110.688-3.805Q110.870-3.675 110.870-3.448Q110.870-3.300 110.768-3.194Q110.666-3.089 110.510-3.089Q110.377-3.089 110.282-3.167Q110.186-3.245 110.155-3.370Q110.010-3.362 109.811-3.272Q110.014-2.960 110.014-2.608Q110.014-2.327 109.903-2.095Q109.791-1.862 109.596-1.684Q109.401-1.507 109.149-1.409Q108.897-1.311 108.635-1.311Q108.248-1.311 107.916-1.499Q107.905-1.499 107.895-1.427Q107.885-1.354 107.885-1.319Q107.885-1.198 107.952-1.091Q108.018-0.983 108.139-0.944Q108.155-0.948 108.168-0.950Q108.182-0.952 108.206-0.952Q108.221-0.952 108.252-0.944Q108.284-0.936 108.291-0.936L108.885-0.936Q109.666-0.936 110.207-0.694Q110.748-0.452 110.748 0.239Q110.748 0.540 110.569 0.767Q110.389 0.993 110.092 1.138Q109.795 1.282 109.475 1.349Q109.155 1.415 108.870 1.415Q108.479 1.415 108.038 1.292Q107.596 1.169 107.291 0.902Q106.987 0.634 106.987 0.239M107.526 0.232Q107.526 0.450 107.766 0.591Q108.006 0.732 108.331 0.798Q108.655 0.864 108.870 0.864Q109.084 0.864 109.409 0.798Q109.733 0.732 109.973 0.591Q110.213 0.450 110.213 0.232Q110.213-0.057 109.989-0.196Q109.764-0.335 109.483-0.368Q109.202-0.401 108.854-0.401L108.237-0.401Q108.053-0.401 107.891-0.321Q107.729-0.241 107.627-0.095Q107.526 0.052 107.526 0.232M108.635-1.866Q108.936-1.866 109.155-2.085Q109.373-2.304 109.373-2.608Q109.373-2.764 109.319-2.895Q109.264-3.026 109.159-3.132Q109.053-3.237 108.922-3.292Q108.791-3.346 108.635-3.346Q108.331-3.346 108.112-3.128Q107.893-2.909 107.893-2.608Q107.893-2.311 108.116-2.089Q108.338-1.866 108.635-1.866M111.401-1.522Q111.401-1.968 111.815-2.225Q112.229-2.483 112.770-2.583Q113.311-2.682 113.819-2.690Q113.819-2.905 113.684-3.057Q113.549-3.210 113.342-3.286Q113.135-3.362 112.924-3.362Q112.581-3.362 112.420-3.339L112.420-3.280Q112.420-3.112 112.301-2.997Q112.182-2.882 112.018-2.882Q111.842-2.882 111.727-3.005Q111.612-3.128 111.612-3.296Q111.612-3.702 111.993-3.811Q112.373-3.921 112.932-3.921Q113.202-3.921 113.469-3.843Q113.737-3.764 113.961-3.614Q114.186-3.464 114.323-3.243Q114.459-3.022 114.459-2.745L114.459-1.026Q114.459-0.968 114.987-0.968Q115.182-0.948 115.233-0.737L115.233-0.647Q115.182-0.432 114.987-0.409L114.842-0.409Q114.498-0.409 114.270-0.456Q114.041-0.503 113.897-0.690Q113.436-0.370 112.729-0.370Q112.393-0.370 112.088-0.511Q111.784-0.651 111.592-0.913Q111.401-1.175 111.401-1.522M112.041-1.514Q112.041-1.241 112.284-1.085Q112.526-0.929 112.811-0.929Q113.030-0.929 113.262-0.987Q113.495-1.046 113.657-1.184Q113.819-1.323 113.819-1.546L113.819-2.136Q113.538-2.136 113.122-2.079Q112.706-2.022 112.373-1.884Q112.041-1.745 112.041-1.514M116.440-1.514L116.440-3.296L115.690-3.296Q115.491-3.319 115.440-3.538L115.440-3.624Q115.491-3.835 115.690-3.858L116.440-3.858L116.440-4.608Q116.491-4.815 116.690-4.843L116.834-4.843Q117.030-4.815 117.081-4.608L117.081-3.858L118.440-3.858Q118.631-3.839 118.690-3.624L118.690-3.538Q118.635-3.319 118.440-3.296L117.081-3.296L117.081-1.546Q117.081-0.929 117.655-0.929Q117.905-0.929 118.069-1.114Q118.233-1.300 118.233-1.546Q118.233-1.639 118.305-1.710Q118.377-1.780 118.479-1.792L118.623-1.792Q118.823-1.768 118.873-1.561L118.873-1.514Q118.873-1.190 118.690-0.927Q118.506-0.663 118.213-0.516Q117.920-0.370 117.600-0.370Q117.088-0.370 116.764-0.680Q116.440-0.991 116.440-1.514M122.998-1.897L120.557-1.897Q120.612-1.620 120.809-1.397Q121.006-1.175 121.284-1.052Q121.561-0.929 121.846-0.929Q122.319-0.929 122.541-1.218Q122.549-1.229 122.606-1.335Q122.663-1.440 122.711-1.483Q122.760-1.526 122.854-1.538L122.998-1.538Q123.190-1.518 123.248-1.304L123.248-1.249Q123.182-0.948 122.952-0.751Q122.721-0.554 122.409-0.462Q122.096-0.370 121.791-0.370Q121.307-0.370 120.868-0.598Q120.428-0.827 120.161-1.227Q119.893-1.628 119.893-2.120L119.893-2.179Q119.893-2.647 120.139-3.050Q120.385-3.452 120.793-3.686Q121.202-3.921 121.670-3.921Q122.174-3.921 122.528-3.698Q122.881-3.475 123.065-3.087Q123.248-2.698 123.248-2.194L123.248-2.136Q123.190-1.921 122.998-1.897M120.565-2.448L122.592-2.448Q122.545-2.858 122.307-3.110Q122.069-3.362 121.670-3.362Q121.276-3.362 120.969-3.100Q120.663-2.839 120.565-2.448\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-107.15 67.417)\">\u003Cpath d=\"M128.257 0.239Q128.257-0.061 128.406-0.323Q128.554-0.585 128.804-0.745Q128.621-0.999 128.621-1.319Q128.621-1.604 128.773-1.866Q128.523-2.190 128.523-2.608Q128.523-2.968 128.714-3.264Q128.906-3.561 129.224-3.729Q129.542-3.897 129.906-3.897Q130.113-3.897 130.316-3.837Q130.519-3.776 130.683-3.675Q131.066-3.936 131.539-3.936Q131.777-3.936 131.958-3.805Q132.140-3.675 132.140-3.448Q132.140-3.300 132.039-3.194Q131.937-3.089 131.781-3.089Q131.648-3.089 131.552-3.167Q131.457-3.245 131.425-3.370Q131.281-3.362 131.082-3.272Q131.285-2.960 131.285-2.608Q131.285-2.327 131.173-2.095Q131.062-1.862 130.867-1.684Q130.671-1.507 130.419-1.409Q130.167-1.311 129.906-1.311Q129.519-1.311 129.187-1.499Q129.175-1.499 129.165-1.427Q129.156-1.354 129.156-1.319Q129.156-1.198 129.222-1.091Q129.289-0.983 129.410-0.944Q129.425-0.948 129.439-0.950Q129.453-0.952 129.476-0.952Q129.492-0.952 129.523-0.944Q129.554-0.936 129.562-0.936L130.156-0.936Q130.937-0.936 131.478-0.694Q132.019-0.452 132.019 0.239Q132.019 0.540 131.839 0.767Q131.660 0.993 131.363 1.138Q131.066 1.282 130.746 1.349Q130.425 1.415 130.140 1.415Q129.749 1.415 129.308 1.292Q128.867 1.169 128.562 0.902Q128.257 0.634 128.257 0.239M128.796 0.232Q128.796 0.450 129.037 0.591Q129.277 0.732 129.601 0.798Q129.925 0.864 130.140 0.864Q130.355 0.864 130.679 0.798Q131.003 0.732 131.244 0.591Q131.484 0.450 131.484 0.232Q131.484-0.057 131.259-0.196Q131.035-0.335 130.753-0.368Q130.472-0.401 130.124-0.401L129.507-0.401Q129.324-0.401 129.162-0.321Q128.999-0.241 128.898-0.095Q128.796 0.052 128.796 0.232M129.906-1.866Q130.207-1.866 130.425-2.085Q130.644-2.304 130.644-2.608Q130.644-2.764 130.589-2.895Q130.535-3.026 130.429-3.132Q130.324-3.237 130.193-3.292Q130.062-3.346 129.906-3.346Q129.601-3.346 129.382-3.128Q129.164-2.909 129.164-2.608Q129.164-2.311 129.386-2.089Q129.609-1.866 129.906-1.866M132.488-0.647L132.488-0.737Q132.546-0.944 132.738-0.968L133.449-0.968L133.449-3.296L132.738-3.296Q132.542-3.319 132.488-3.538L132.488-3.624Q132.546-3.835 132.738-3.858L133.839-3.858Q134.039-3.839 134.089-3.624L134.089-3.296Q134.351-3.581 134.707-3.739Q135.062-3.897 135.449-3.897Q135.742-3.897 135.976-3.763Q136.210-3.628 136.210-3.362Q136.210-3.194 136.101-3.077Q135.992-2.960 135.824-2.960Q135.671-2.960 135.556-3.071Q135.441-3.182 135.441-3.339Q135.066-3.339 134.751-3.138Q134.437-2.936 134.263-2.602Q134.089-2.268 134.089-1.889L134.089-0.968L135.035-0.968Q135.242-0.944 135.281-0.737L135.281-0.647Q135.242-0.432 135.035-0.409L132.738-0.409Q132.546-0.432 132.488-0.647M136.917-1.522Q136.917-1.968 137.332-2.225Q137.746-2.483 138.287-2.583Q138.828-2.682 139.335-2.690Q139.335-2.905 139.201-3.057Q139.066-3.210 138.859-3.286Q138.652-3.362 138.441-3.362Q138.097-3.362 137.937-3.339L137.937-3.280Q137.937-3.112 137.818-2.997Q137.699-2.882 137.535-2.882Q137.359-2.882 137.244-3.005Q137.128-3.128 137.128-3.296Q137.128-3.702 137.509-3.811Q137.890-3.921 138.449-3.921Q138.718-3.921 138.986-3.843Q139.253-3.764 139.478-3.614Q139.703-3.464 139.839-3.243Q139.976-3.022 139.976-2.745L139.976-1.026Q139.976-0.968 140.503-0.968Q140.699-0.948 140.749-0.737L140.749-0.647Q140.699-0.432 140.503-0.409L140.359-0.409Q140.015-0.409 139.787-0.456Q139.558-0.503 139.414-0.690Q138.953-0.370 138.246-0.370Q137.910-0.370 137.605-0.511Q137.300-0.651 137.109-0.913Q136.917-1.175 136.917-1.522M137.558-1.514Q137.558-1.241 137.800-1.085Q138.042-0.929 138.328-0.929Q138.546-0.929 138.779-0.987Q139.011-1.046 139.173-1.184Q139.335-1.323 139.335-1.546L139.335-2.136Q139.054-2.136 138.638-2.079Q138.222-2.022 137.890-1.884Q137.558-1.745 137.558-1.514M142.621-0.370Q142.156-0.370 141.790-0.620Q141.425-0.870 141.220-1.274Q141.015-1.679 141.015-2.136Q141.015-2.479 141.140-2.800Q141.265-3.120 141.498-3.370Q141.730-3.620 142.035-3.759Q142.339-3.897 142.695-3.897Q143.207-3.897 143.613-3.577L143.613-4.737L143.191-4.737Q142.980-4.761 142.941-4.975L142.941-5.065Q142.980-5.272 143.191-5.296L144.003-5.296Q144.203-5.272 144.253-5.065L144.253-0.968L144.679-0.968Q144.886-0.944 144.925-0.737L144.925-0.647Q144.886-0.432 144.679-0.409L143.863-0.409Q143.664-0.432 143.613-0.647L143.613-0.776Q143.417-0.585 143.156-0.477Q142.894-0.370 142.621-0.370M142.660-0.929Q143.019-0.929 143.271-1.198Q143.523-1.468 143.613-1.843L143.613-2.675Q143.554-2.862 143.427-3.013Q143.300-3.163 143.123-3.251Q142.945-3.339 142.749-3.339Q142.441-3.339 142.191-3.169Q141.941-2.999 141.796-2.714Q141.652-2.429 141.652-2.128Q141.652-1.679 141.937-1.304Q142.222-0.929 142.660-0.929M145.621-0.647L145.621-0.737Q145.671-0.944 145.867-0.968L146.906-0.968L146.906-3.296L145.933-3.296Q145.734-3.319 145.683-3.538L145.683-3.624Q145.734-3.835 145.933-3.858L147.300-3.858Q147.496-3.839 147.546-3.624L147.546-0.968L148.460-0.968Q148.656-0.944 148.707-0.737L148.707-0.647Q148.656-0.432 148.460-0.409L145.867-0.409Q145.671-0.432 145.621-0.647M146.652-4.835L146.652-4.889Q146.652-5.061 146.789-5.182Q146.925-5.304 147.101-5.304Q147.273-5.304 147.410-5.182Q147.546-5.061 147.546-4.889L147.546-4.835Q147.546-4.659 147.410-4.538Q147.273-4.417 147.101-4.417Q146.925-4.417 146.789-4.538Q146.652-4.659 146.652-4.835M152.761-1.897L150.320-1.897Q150.374-1.620 150.572-1.397Q150.769-1.175 151.046-1.052Q151.324-0.929 151.609-0.929Q152.082-0.929 152.304-1.218Q152.312-1.229 152.369-1.335Q152.425-1.440 152.474-1.483Q152.523-1.526 152.617-1.538L152.761-1.538Q152.953-1.518 153.011-1.304L153.011-1.249Q152.945-0.948 152.714-0.751Q152.484-0.554 152.171-0.462Q151.859-0.370 151.554-0.370Q151.070-0.370 150.630-0.598Q150.191-0.827 149.923-1.227Q149.656-1.628 149.656-2.120L149.656-2.179Q149.656-2.647 149.902-3.050Q150.148-3.452 150.556-3.686Q150.964-3.921 151.433-3.921Q151.937-3.921 152.290-3.698Q152.644-3.475 152.828-3.087Q153.011-2.698 153.011-2.194L153.011-2.136Q152.953-1.921 152.761-1.897M150.328-2.448L152.355-2.448Q152.308-2.858 152.070-3.110Q151.832-3.362 151.433-3.362Q151.039-3.362 150.732-3.100Q150.425-2.839 150.328-2.448M153.566-0.647L153.566-0.737Q153.609-0.944 153.816-0.968L154.238-0.968L154.238-3.296L153.816-3.296Q153.609-3.319 153.566-3.538L153.566-3.624Q153.613-3.835 153.816-3.858L154.632-3.858Q154.828-3.835 154.878-3.624L154.878-3.538L154.871-3.514Q155.097-3.694 155.371-3.796Q155.644-3.897 155.937-3.897Q156.285-3.897 156.523-3.757Q156.761-3.616 156.876-3.358Q156.992-3.100 156.992-2.745L156.992-0.968L157.417-0.968Q157.624-0.944 157.664-0.737L157.664-0.647Q157.624-0.432 157.417-0.409L156.023-0.409Q155.828-0.432 155.777-0.647L155.777-0.737Q155.828-0.948 156.023-0.968L156.351-0.968L156.351-2.714Q156.351-3.022 156.261-3.180Q156.171-3.339 155.878-3.339Q155.609-3.339 155.380-3.208Q155.152-3.077 155.015-2.848Q154.878-2.620 154.878-2.354L154.878-0.968L155.304-0.968Q155.511-0.944 155.550-0.737L155.550-0.647Q155.511-0.432 155.304-0.409L153.816-0.409Q153.609-0.432 153.566-0.647M158.941-1.514L158.941-3.296L158.191-3.296Q157.992-3.319 157.941-3.538L157.941-3.624Q157.992-3.835 158.191-3.858L158.941-3.858L158.941-4.608Q158.992-4.815 159.191-4.843L159.335-4.843Q159.531-4.815 159.582-4.608L159.582-3.858L160.941-3.858Q161.132-3.839 161.191-3.624L161.191-3.538Q161.136-3.319 160.941-3.296L159.582-3.296L159.582-1.546Q159.582-0.929 160.156-0.929Q160.406-0.929 160.570-1.114Q160.734-1.300 160.734-1.546Q160.734-1.639 160.806-1.710Q160.878-1.780 160.980-1.792L161.124-1.792Q161.324-1.768 161.374-1.561L161.374-1.514Q161.374-1.190 161.191-0.927Q161.007-0.663 160.714-0.516Q160.421-0.370 160.101-0.370Q159.589-0.370 159.265-0.680Q158.941-0.991 158.941-1.514M162.574-0.608L162.574-1.522Q162.601-1.729 162.812-1.753L162.980-1.753Q163.144-1.729 163.203-1.569Q163.406-0.929 164.132-0.929Q164.339-0.929 164.568-0.964Q164.796-0.999 164.964-1.114Q165.132-1.229 165.132-1.432Q165.132-1.643 164.910-1.757Q164.687-1.870 164.414-1.913L163.714-2.026Q162.574-2.237 162.574-2.960Q162.574-3.249 162.718-3.438Q162.863-3.628 163.103-3.735Q163.343-3.843 163.599-3.882Q163.855-3.921 164.132-3.921Q164.382-3.921 164.576-3.891Q164.769-3.862 164.933-3.784Q165.011-3.901 165.140-3.921L165.218-3.921Q165.316-3.909 165.378-3.846Q165.441-3.784 165.453-3.690L165.453-2.983Q165.441-2.889 165.378-2.823Q165.316-2.757 165.218-2.745L165.050-2.745Q164.957-2.757 164.890-2.823Q164.824-2.889 164.812-2.983Q164.812-3.362 164.117-3.362Q163.769-3.362 163.451-3.280Q163.132-3.198 163.132-2.952Q163.132-2.686 163.804-2.577L164.507-2.456Q164.992-2.374 165.341-2.126Q165.691-1.878 165.691-1.432Q165.691-1.042 165.455-0.800Q165.218-0.557 164.869-0.464Q164.519-0.370 164.132-0.370Q163.554-0.370 163.156-0.624Q163.085-0.499 163.037-0.442Q162.988-0.386 162.882-0.370L162.812-0.370Q162.597-0.393 162.574-0.608\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-107.15 67.417)\">\u003Cpath d=\"M173.633 0.232Q173.102-0.077 172.702-0.565Q172.301-1.054 172.090-1.647Q171.879-2.241 171.879-2.858Q171.879-3.475 172.088-4.063Q172.297-4.651 172.694-5.132Q173.090-5.612 173.625-5.936Q173.696-5.960 173.727-5.960L173.817-5.960Q173.915-5.948 173.981-5.882Q174.047-5.815 174.047-5.714Q174.047-5.569 173.946-5.507Q173.497-5.218 173.170-4.802Q172.844-4.386 172.682-3.893Q172.520-3.401 172.520-2.858Q172.520-2.452 172.616-2.065Q172.711-1.679 172.889-1.343Q173.067-1.007 173.327-0.723Q173.586-0.440 173.934-0.210Q174.047-0.132 174.047 0.005Q174.047 0.103 173.981 0.173Q173.915 0.243 173.817 0.255L173.727 0.255Q173.668 0.255 173.633 0.232M175.192-0.647L175.192-0.722Q175.223-0.889 175.325-0.936L176.614-2.003Q176.946-2.280 177.127-2.432Q177.309-2.585 177.504-2.805Q177.700-3.026 177.821-3.276Q177.942-3.526 177.942-3.792Q177.942-4.116 177.766-4.350Q177.590-4.585 177.311-4.700Q177.032-4.815 176.711-4.815Q176.454-4.815 176.225-4.692Q175.997-4.569 175.895-4.354Q175.997-4.221 175.997-4.073Q175.997-3.913 175.877-3.790Q175.758-3.667 175.598-3.667Q175.422-3.667 175.307-3.792Q175.192-3.917 175.192-4.089Q175.192-4.382 175.327-4.624Q175.461-4.866 175.700-5.038Q175.938-5.210 176.206-5.294Q176.473-5.378 176.766-5.378Q177.247-5.378 177.663-5.188Q178.079-4.999 178.331-4.638Q178.583-4.276 178.583-3.792Q178.583-3.448 178.450-3.145Q178.317-2.843 178.092-2.583Q177.868-2.323 177.559-2.061Q177.250-1.800 177.040-1.624L176.231-0.968L177.942-0.968L177.942-1.112Q177.993-1.323 178.192-1.346L178.333-1.346Q178.532-1.327 178.583-1.112L178.583-0.647Q178.532-0.432 178.333-0.409L175.438-0.409Q175.243-0.429 175.192-0.647\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-107.15 67.417)\">\u003Cpath d=\"M183.856-0.608L183.856-1.522Q183.883-1.729 184.094-1.753L184.262-1.753Q184.426-1.729 184.485-1.569Q184.688-0.929 185.415-0.929Q185.622-0.929 185.850-0.964Q186.079-0.999 186.247-1.114Q186.415-1.229 186.415-1.432Q186.415-1.643 186.192-1.757Q185.969-1.870 185.696-1.913L184.997-2.026Q183.856-2.237 183.856-2.960Q183.856-3.249 184-3.438Q184.145-3.628 184.385-3.735Q184.625-3.843 184.881-3.882Q185.137-3.921 185.415-3.921Q185.665-3.921 185.858-3.891Q186.051-3.862 186.215-3.784Q186.293-3.901 186.422-3.921L186.500-3.921Q186.598-3.909 186.661-3.846Q186.723-3.784 186.735-3.690L186.735-2.983Q186.723-2.889 186.661-2.823Q186.598-2.757 186.500-2.745L186.333-2.745Q186.239-2.757 186.172-2.823Q186.106-2.889 186.094-2.983Q186.094-3.362 185.399-3.362Q185.051-3.362 184.733-3.280Q184.415-3.198 184.415-2.952Q184.415-2.686 185.086-2.577L185.790-2.456Q186.274-2.374 186.624-2.126Q186.973-1.878 186.973-1.432Q186.973-1.042 186.737-0.800Q186.500-0.557 186.151-0.464Q185.801-0.370 185.415-0.370Q184.836-0.370 184.438-0.624Q184.368-0.499 184.319-0.442Q184.270-0.386 184.165-0.370L184.094-0.370Q183.879-0.393 183.856-0.608M188.715-1.514L188.715-3.296L187.965-3.296Q187.766-3.319 187.715-3.538L187.715-3.624Q187.766-3.835 187.965-3.858L188.715-3.858L188.715-4.608Q188.766-4.815 188.965-4.843L189.110-4.843Q189.305-4.815 189.356-4.608L189.356-3.858L190.715-3.858Q190.907-3.839 190.965-3.624L190.965-3.538Q190.911-3.319 190.715-3.296L189.356-3.296L189.356-1.546Q189.356-0.929 189.930-0.929Q190.180-0.929 190.344-1.114Q190.508-1.300 190.508-1.546Q190.508-1.639 190.581-1.710Q190.653-1.780 190.754-1.792L190.899-1.792Q191.098-1.768 191.149-1.561L191.149-1.514Q191.149-1.190 190.965-0.927Q190.782-0.663 190.489-0.516Q190.196-0.370 189.875-0.370Q189.364-0.370 189.040-0.680Q188.715-0.991 188.715-1.514M195.274-1.897L192.833-1.897Q192.887-1.620 193.084-1.397Q193.282-1.175 193.559-1.052Q193.836-0.929 194.122-0.929Q194.594-0.929 194.817-1.218Q194.825-1.229 194.881-1.335Q194.938-1.440 194.987-1.483Q195.036-1.526 195.129-1.538L195.274-1.538Q195.465-1.518 195.524-1.304L195.524-1.249Q195.458-0.948 195.227-0.751Q194.997-0.554 194.684-0.462Q194.372-0.370 194.067-0.370Q193.583-0.370 193.143-0.598Q192.704-0.827 192.436-1.227Q192.168-1.628 192.168-2.120L192.168-2.179Q192.168-2.647 192.415-3.050Q192.661-3.452 193.069-3.686Q193.477-3.921 193.946-3.921Q194.450-3.921 194.803-3.698Q195.157-3.475 195.340-3.087Q195.524-2.698 195.524-2.194L195.524-2.136Q195.465-1.921 195.274-1.897M192.840-2.448L194.868-2.448Q194.821-2.858 194.583-3.110Q194.344-3.362 193.946-3.362Q193.551-3.362 193.245-3.100Q192.938-2.839 192.840-2.448M196.079 1.134L196.079 1.048Q196.122 0.829 196.329 0.806L196.750 0.806L196.750-3.296L196.329-3.296Q196.122-3.319 196.079-3.538L196.079-3.624Q196.125-3.835 196.329-3.858L197.145-3.858Q197.340-3.839 197.391-3.624L197.391-3.554Q197.602-3.721 197.866-3.809Q198.129-3.897 198.399-3.897Q198.739-3.897 199.036-3.753Q199.333-3.608 199.547-3.356Q199.762-3.104 199.877-2.792Q199.993-2.479 199.993-2.136Q199.993-1.671 199.766-1.261Q199.540-0.850 199.153-0.610Q198.766-0.370 198.297-0.370Q197.778-0.370 197.391-0.737L197.391 0.806L197.817 0.806Q198.024 0.829 198.063 1.048L198.063 1.134Q198.024 1.345 197.817 1.368L196.329 1.368Q196.125 1.345 196.079 1.134M198.254-0.929Q198.563-0.929 198.813-1.098Q199.063-1.268 199.208-1.550Q199.352-1.831 199.352-2.136Q199.352-2.425 199.227-2.704Q199.102-2.983 198.870-3.161Q198.637-3.339 198.336-3.339Q198.016-3.339 197.754-3.153Q197.493-2.968 197.391-2.667L197.391-1.815Q197.481-1.448 197.702-1.188Q197.922-0.929 198.254-0.929M200.840-0.608L200.840-1.522Q200.868-1.729 201.079-1.753L201.247-1.753Q201.411-1.729 201.469-1.569Q201.672-0.929 202.399-0.929Q202.606-0.929 202.834-0.964Q203.063-0.999 203.231-1.114Q203.399-1.229 203.399-1.432Q203.399-1.643 203.176-1.757Q202.954-1.870 202.680-1.913L201.981-2.026Q200.840-2.237 200.840-2.960Q200.840-3.249 200.985-3.438Q201.129-3.628 201.370-3.735Q201.610-3.843 201.866-3.882Q202.122-3.921 202.399-3.921Q202.649-3.921 202.842-3.891Q203.036-3.862 203.200-3.784Q203.278-3.901 203.407-3.921L203.485-3.921Q203.583-3.909 203.645-3.846Q203.708-3.784 203.719-3.690L203.719-2.983Q203.708-2.889 203.645-2.823Q203.583-2.757 203.485-2.745L203.317-2.745Q203.223-2.757 203.157-2.823Q203.090-2.889 203.079-2.983Q203.079-3.362 202.383-3.362Q202.036-3.362 201.717-3.280Q201.399-3.198 201.399-2.952Q201.399-2.686 202.071-2.577L202.774-2.456Q203.258-2.374 203.608-2.126Q203.958-1.878 203.958-1.432Q203.958-1.042 203.721-0.800Q203.485-0.557 203.135-0.464Q202.786-0.370 202.399-0.370Q201.821-0.370 201.422-0.624Q201.352-0.499 201.303-0.442Q201.254-0.386 201.149-0.370L201.079-0.370Q200.864-0.393 200.840-0.608M205.540 0.255L205.454 0.255Q205.348 0.243 205.280 0.171Q205.211 0.099 205.211 0.005Q205.211-0.136 205.317-0.202Q205.653-0.417 205.922-0.706Q206.192-0.995 206.375-1.343Q206.559-1.690 206.649-2.069Q206.739-2.448 206.739-2.858Q206.739-3.397 206.575-3.889Q206.411-4.382 206.092-4.796Q205.774-5.210 205.333-5.499Q205.211-5.581 205.211-5.714Q205.211-5.811 205.280-5.880Q205.348-5.948 205.454-5.960L205.540-5.960Q205.594-5.960 205.629-5.936Q206.016-5.714 206.356-5.366Q206.696-5.018 206.916-4.624Q207.137-4.229 207.258-3.784Q207.379-3.339 207.379-2.858Q207.379-2.374 207.258-1.923Q207.137-1.471 206.913-1.075Q206.688-0.679 206.364-0.345Q206.040-0.011 205.637 0.232Q205.567 0.255 205.540 0.255\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">Steepest descent (red) takes orthogonal steps that zig-zag across the stretched valley; conjugate gradients (green) bends each direction to be \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0813em;\">H\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>-conjugate to the last and reaches the minimum without retracing.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:420.648px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 315.486 228.537\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-55.64 124.687H189.3\"\u002F>\u003Cpath stroke=\"none\" d=\"m191.9 124.687-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003Cg transform=\"translate(242.736 1.325)\">\u003Cpath d=\"M-47.034 126.230L-47.034 126.144Q-46.991 125.925-46.784 125.902L-46.362 125.902L-46.362 121.800L-46.784 121.800Q-46.991 121.777-47.034 121.558L-47.034 121.472Q-46.987 121.261-46.784 121.238L-45.967 121.238Q-45.772 121.257-45.721 121.472L-45.721 121.542Q-45.510 121.374-45.247 121.287Q-44.983 121.199-44.713 121.199Q-44.374 121.199-44.077 121.343Q-43.780 121.488-43.565 121.740Q-43.350 121.992-43.235 122.304Q-43.120 122.617-43.120 122.960Q-43.120 123.425-43.346 123.835Q-43.573 124.246-43.959 124.486Q-44.346 124.726-44.815 124.726Q-45.334 124.726-45.721 124.359L-45.721 125.902L-45.295 125.902Q-45.088 125.925-45.049 126.144L-45.049 126.230Q-45.088 126.441-45.295 126.464L-46.784 126.464Q-46.987 126.441-47.034 126.230M-44.858 124.167Q-44.549 124.167-44.299 123.998Q-44.049 123.828-43.905 123.546Q-43.760 123.265-43.760 122.960Q-43.760 122.671-43.885 122.392Q-44.010 122.113-44.243 121.935Q-44.475 121.757-44.776 121.757Q-45.096 121.757-45.358 121.943Q-45.620 122.128-45.721 122.429L-45.721 123.281Q-45.631 123.648-45.411 123.908Q-45.190 124.167-44.858 124.167M-42.452 123.574Q-42.452 123.128-42.038 122.871Q-41.624 122.613-41.083 122.513Q-40.541 122.414-40.034 122.406Q-40.034 122.191-40.168 122.039Q-40.303 121.886-40.510 121.810Q-40.717 121.734-40.928 121.734Q-41.272 121.734-41.432 121.757L-41.432 121.816Q-41.432 121.984-41.551 122.099Q-41.670 122.214-41.834 122.214Q-42.010 122.214-42.125 122.091Q-42.241 121.968-42.241 121.800Q-42.241 121.394-41.860 121.285Q-41.479 121.175-40.920 121.175Q-40.651 121.175-40.383 121.253Q-40.116 121.332-39.891 121.482Q-39.666 121.632-39.530 121.853Q-39.393 122.074-39.393 122.351L-39.393 124.070Q-39.393 124.128-38.866 124.128Q-38.670 124.148-38.620 124.359L-38.620 124.449Q-38.670 124.664-38.866 124.687L-39.010 124.687Q-39.354 124.687-39.583 124.640Q-39.811 124.593-39.956 124.406Q-40.416 124.726-41.124 124.726Q-41.459 124.726-41.764 124.585Q-42.069 124.445-42.260 124.183Q-42.452 123.921-42.452 123.574M-41.811 123.582Q-41.811 123.855-41.569 124.011Q-41.327 124.167-41.041 124.167Q-40.823 124.167-40.590 124.109Q-40.358 124.050-40.196 123.912Q-40.034 123.773-40.034 123.550L-40.034 122.960Q-40.315 122.960-40.731 123.017Q-41.147 123.074-41.479 123.212Q-41.811 123.351-41.811 123.582M-38.389 124.449L-38.389 124.359Q-38.331 124.152-38.139 124.128L-37.428 124.128L-37.428 121.800L-38.139 121.800Q-38.334 121.777-38.389 121.558L-38.389 121.472Q-38.331 121.261-38.139 121.238L-37.038 121.238Q-36.838 121.257-36.788 121.472L-36.788 121.800Q-36.526 121.515-36.170 121.357Q-35.815 121.199-35.428 121.199Q-35.135 121.199-34.901 121.333Q-34.666 121.468-34.666 121.734Q-34.666 121.902-34.776 122.019Q-34.885 122.136-35.053 122.136Q-35.206 122.136-35.321 122.025Q-35.436 121.914-35.436 121.757Q-35.811 121.757-36.125 121.958Q-36.440 122.160-36.614 122.494Q-36.788 122.828-36.788 123.207L-36.788 124.128L-35.842 124.128Q-35.635 124.152-35.596 124.359L-35.596 124.449Q-35.635 124.664-35.842 124.687L-38.139 124.687Q-38.331 124.664-38.389 124.449M-33.959 123.574Q-33.959 123.128-33.545 122.871Q-33.131 122.613-32.590 122.513Q-32.049 122.414-31.541 122.406Q-31.541 122.191-31.676 122.039Q-31.811 121.886-32.018 121.810Q-32.225 121.734-32.436 121.734Q-32.780 121.734-32.940 121.757L-32.940 121.816Q-32.940 121.984-33.059 122.099Q-33.178 122.214-33.342 122.214Q-33.518 122.214-33.633 122.091Q-33.749 121.968-33.749 121.800Q-33.749 121.394-33.368 121.285Q-32.987 121.175-32.428 121.175Q-32.159 121.175-31.891 121.253Q-31.624 121.332-31.399 121.482Q-31.174 121.632-31.038 121.853Q-30.901 122.074-30.901 122.351L-30.901 124.070Q-30.901 124.128-30.374 124.128Q-30.178 124.148-30.127 124.359L-30.127 124.449Q-30.178 124.664-30.374 124.687L-30.518 124.687Q-30.862 124.687-31.090 124.640Q-31.319 124.593-31.463 124.406Q-31.924 124.726-32.631 124.726Q-32.967 124.726-33.272 124.585Q-33.577 124.445-33.768 124.183Q-33.959 123.921-33.959 123.574M-33.319 123.582Q-33.319 123.855-33.077 124.011Q-32.834 124.167-32.549 124.167Q-32.331 124.167-32.098 124.109Q-31.866 124.050-31.704 123.912Q-31.541 123.773-31.541 123.550L-31.541 122.960Q-31.823 122.960-32.239 123.017Q-32.655 123.074-32.987 123.212Q-33.319 123.351-33.319 123.582M-30.174 124.449L-30.174 124.359Q-30.124 124.148-29.928 124.128L-29.729 124.128L-29.729 121.800L-29.928 121.800Q-30.135 121.777-30.174 121.558L-30.174 121.472Q-30.124 121.257-29.928 121.238L-29.448 121.238Q-29.256 121.261-29.198 121.472Q-28.877 121.199-28.463 121.199Q-28.272 121.199-28.104 121.308Q-27.936 121.417-27.862 121.589Q-27.686 121.398-27.463 121.298Q-27.241 121.199-26.999 121.199Q-26.581 121.199-26.426 121.541Q-26.272 121.882-26.272 122.351L-26.272 124.128L-26.073 124.128Q-25.862 124.152-25.823 124.359L-25.823 124.449Q-25.874 124.664-26.073 124.687L-26.889 124.687Q-27.084 124.664-27.135 124.449L-27.135 124.359Q-27.084 124.152-26.889 124.128L-26.799 124.128L-26.799 122.382Q-26.799 121.757-27.049 121.757Q-27.381 121.757-27.559 122.068Q-27.737 122.378-27.737 122.742L-27.737 124.128L-27.534 124.128Q-27.327 124.152-27.288 124.359L-27.288 124.449Q-27.338 124.664-27.534 124.687L-28.350 124.687Q-28.549 124.664-28.600 124.449L-28.600 124.359Q-28.549 124.152-28.350 124.128L-28.264 124.128L-28.264 122.382Q-28.264 121.757-28.510 121.757Q-28.842 121.757-29.020 122.070Q-29.198 122.382-29.198 122.742L-29.198 124.128L-28.999 124.128Q-28.791 124.152-28.752 124.359L-28.752 124.449Q-28.803 124.667-28.999 124.687L-29.928 124.687Q-30.135 124.664-30.174 124.449M-22.362 123.199L-24.803 123.199Q-24.749 123.476-24.551 123.699Q-24.354 123.921-24.077 124.044Q-23.799 124.167-23.514 124.167Q-23.041 124.167-22.819 123.878Q-22.811 123.867-22.754 123.761Q-22.698 123.656-22.649 123.613Q-22.600 123.570-22.506 123.558L-22.362 123.558Q-22.170 123.578-22.112 123.792L-22.112 123.847Q-22.178 124.148-22.409 124.345Q-22.639 124.542-22.952 124.634Q-23.264 124.726-23.569 124.726Q-24.053 124.726-24.493 124.498Q-24.932 124.269-25.200 123.869Q-25.467 123.468-25.467 122.976L-25.467 122.917Q-25.467 122.449-25.221 122.046Q-24.975 121.644-24.567 121.410Q-24.159 121.175-23.690 121.175Q-23.186 121.175-22.833 121.398Q-22.479 121.621-22.295 122.009Q-22.112 122.398-22.112 122.902L-22.112 122.960Q-22.170 123.175-22.362 123.199M-24.795 122.648L-22.768 122.648Q-22.815 122.238-23.053 121.986Q-23.291 121.734-23.690 121.734Q-24.084 121.734-24.391 121.996Q-24.698 122.257-24.795 122.648M-20.428 123.582L-20.428 121.800L-21.178 121.800Q-21.377 121.777-21.428 121.558L-21.428 121.472Q-21.377 121.261-21.178 121.238L-20.428 121.238L-20.428 120.488Q-20.377 120.281-20.178 120.253L-20.034 120.253Q-19.838 120.281-19.788 120.488L-19.788 121.238L-18.428 121.238Q-18.237 121.257-18.178 121.472L-18.178 121.558Q-18.233 121.777-18.428 121.800L-19.788 121.800L-19.788 123.550Q-19.788 124.167-19.213 124.167Q-18.963 124.167-18.799 123.982Q-18.635 123.796-18.635 123.550Q-18.635 123.457-18.563 123.386Q-18.491 123.316-18.389 123.304L-18.245 123.304Q-18.045 123.328-17.995 123.535L-17.995 123.582Q-17.995 123.906-18.178 124.169Q-18.362 124.433-18.655 124.580Q-18.948 124.726-19.268 124.726Q-19.780 124.726-20.104 124.416Q-20.428 124.105-20.428 123.582M-13.870 123.199L-16.311 123.199Q-16.256 123.476-16.059 123.699Q-15.862 123.921-15.584 124.044Q-15.307 124.167-15.022 124.167Q-14.549 124.167-14.327 123.878Q-14.319 123.867-14.262 123.761Q-14.206 123.656-14.157 123.613Q-14.108 123.570-14.014 123.558L-13.870 123.558Q-13.678 123.578-13.620 123.792L-13.620 123.847Q-13.686 124.148-13.916 124.345Q-14.147 124.542-14.459 124.634Q-14.772 124.726-15.077 124.726Q-15.561 124.726-16 124.498Q-16.440 124.269-16.708 123.869Q-16.975 123.468-16.975 122.976L-16.975 122.917Q-16.975 122.449-16.729 122.046Q-16.483 121.644-16.075 121.410Q-15.666 121.175-15.198 121.175Q-14.694 121.175-14.340 121.398Q-13.987 121.621-13.803 122.009Q-13.620 122.398-13.620 122.902L-13.620 122.960Q-13.678 123.175-13.870 123.199M-16.303 122.648L-14.276 122.648Q-14.323 122.238-14.561 121.986Q-14.799 121.734-15.198 121.734Q-15.592 121.734-15.899 121.996Q-16.206 122.257-16.303 122.648M-12.913 124.449L-12.913 124.359Q-12.854 124.152-12.663 124.128L-11.952 124.128L-11.952 121.800L-12.663 121.800Q-12.858 121.777-12.913 121.558L-12.913 121.472Q-12.854 121.261-12.663 121.238L-11.561 121.238Q-11.362 121.257-11.311 121.472L-11.311 121.800Q-11.049 121.515-10.694 121.357Q-10.338 121.199-9.952 121.199Q-9.659 121.199-9.424 121.333Q-9.190 121.468-9.190 121.734Q-9.190 121.902-9.299 122.019Q-9.409 122.136-9.577 122.136Q-9.729 122.136-9.844 122.025Q-9.959 121.914-9.959 121.757Q-10.334 121.757-10.649 121.958Q-10.963 122.160-11.137 122.494Q-11.311 122.828-11.311 123.207L-11.311 124.128L-10.366 124.128Q-10.159 124.152-10.120 124.359L-10.120 124.449Q-10.159 124.664-10.366 124.687L-12.663 124.687Q-12.854 124.664-12.913 124.449\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-47.104 133.223V24.857\"\u002F>\u003Cpath stroke=\"none\" d=\"m-47.104 22.257-2.08 4.16 2.08-1.56 2.08 1.56\"\u002F>\u003Cg transform=\"translate(-8.5 -106.163)\">\u003Cpath d=\"M-46.655 124.449L-46.655 124.359Q-46.604 124.152-46.409 124.128L-45.303 124.128L-45.303 120.359L-46.409 120.359Q-46.604 120.335-46.655 120.121L-46.655 120.031Q-46.604 119.824-46.409 119.800L-44.913 119.800Q-44.721 119.824-44.663 120.031L-44.663 124.128L-43.561 124.128Q-43.362 124.152-43.311 124.359L-43.311 124.449Q-43.362 124.664-43.561 124.687L-46.409 124.687Q-46.604 124.664-46.655 124.449M-40.737 124.726Q-41.209 124.726-41.594 124.482Q-41.979 124.238-42.202 123.828Q-42.424 123.417-42.424 122.960Q-42.424 122.617-42.299 122.294Q-42.174 121.972-41.944 121.718Q-41.713 121.464-41.407 121.320Q-41.100 121.175-40.737 121.175Q-40.374 121.175-40.061 121.322Q-39.749 121.468-39.526 121.714Q-39.303 121.960-39.176 122.281Q-39.049 122.601-39.049 122.960Q-39.049 123.417-39.274 123.830Q-39.499 124.242-39.883 124.484Q-40.268 124.726-40.737 124.726M-40.737 124.167Q-40.272 124.167-39.981 123.773Q-39.690 123.378-39.690 122.894Q-39.690 122.601-39.825 122.333Q-39.959 122.066-40.200 121.900Q-40.440 121.734-40.737 121.734Q-41.041 121.734-41.280 121.900Q-41.518 122.066-41.653 122.333Q-41.788 122.601-41.788 122.894Q-41.788 123.374-41.495 123.771Q-41.202 124.167-40.737 124.167M-38.026 124.488L-38.026 123.574Q-37.999 123.367-37.788 123.343L-37.620 123.343Q-37.456 123.367-37.397 123.527Q-37.194 124.167-36.467 124.167Q-36.260 124.167-36.032 124.132Q-35.803 124.097-35.635 123.982Q-35.467 123.867-35.467 123.664Q-35.467 123.453-35.690 123.339Q-35.913 123.226-36.186 123.183L-36.885 123.070Q-38.026 122.859-38.026 122.136Q-38.026 121.847-37.881 121.658Q-37.737 121.468-37.497 121.361Q-37.256 121.253-37 121.214Q-36.745 121.175-36.467 121.175Q-36.217 121.175-36.024 121.205Q-35.831 121.234-35.666 121.312Q-35.588 121.195-35.459 121.175L-35.381 121.175Q-35.284 121.187-35.221 121.249Q-35.159 121.312-35.147 121.406L-35.147 122.113Q-35.159 122.207-35.221 122.273Q-35.284 122.339-35.381 122.351L-35.549 122.351Q-35.643 122.339-35.709 122.273Q-35.776 122.207-35.788 122.113Q-35.788 121.734-36.483 121.734Q-36.831 121.734-37.149 121.816Q-37.467 121.898-37.467 122.144Q-37.467 122.410-36.795 122.519L-36.092 122.640Q-35.608 122.722-35.258 122.970Q-34.909 123.218-34.909 123.664Q-34.909 124.054-35.145 124.296Q-35.381 124.539-35.731 124.632Q-36.081 124.726-36.467 124.726Q-37.045 124.726-37.444 124.472Q-37.514 124.597-37.563 124.654Q-37.612 124.710-37.717 124.726L-37.788 124.726Q-38.002 124.703-38.026 124.488M-33.780 124.488L-33.780 123.574Q-33.752 123.367-33.541 123.343L-33.374 123.343Q-33.209 123.367-33.151 123.527Q-32.948 124.167-32.221 124.167Q-32.014 124.167-31.786 124.132Q-31.557 124.097-31.389 123.982Q-31.221 123.867-31.221 123.664Q-31.221 123.453-31.444 123.339Q-31.666 123.226-31.940 123.183L-32.639 123.070Q-33.780 122.859-33.780 122.136Q-33.780 121.847-33.635 121.658Q-33.491 121.468-33.250 121.361Q-33.010 121.253-32.754 121.214Q-32.499 121.175-32.221 121.175Q-31.971 121.175-31.778 121.205Q-31.584 121.234-31.420 121.312Q-31.342 121.195-31.213 121.175L-31.135 121.175Q-31.038 121.187-30.975 121.249Q-30.913 121.312-30.901 121.406L-30.901 122.113Q-30.913 122.207-30.975 122.273Q-31.038 122.339-31.135 122.351L-31.303 122.351Q-31.397 122.339-31.463 122.273Q-31.530 122.207-31.541 122.113Q-31.541 121.734-32.237 121.734Q-32.584 121.734-32.903 121.816Q-33.221 121.898-33.221 122.144Q-33.221 122.410-32.549 122.519L-31.846 122.640Q-31.362 122.722-31.012 122.970Q-30.663 123.218-30.663 123.664Q-30.663 124.054-30.899 124.296Q-31.135 124.539-31.485 124.632Q-31.834 124.726-32.221 124.726Q-32.799 124.726-33.198 124.472Q-33.268 124.597-33.317 124.654Q-33.366 124.710-33.471 124.726L-33.541 124.726Q-33.756 124.703-33.780 124.488\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-accent)\" d=\"m-32.877-71.67 1.745 4.662 1.745 4.602 1.746 4.54 1.745 4.483 1.745 4.42 1.746 4.363 1.745 4.302 1.745 4.24 1.745 4.183 1.746 4.121 1.745 4.063 1.745 4.002 1.746 3.941 1.745 3.883 1.745 3.82 1.746 3.764L-3.207-.58l1.745 3.642L.283 6.645l1.746 3.522 1.745 3.463 1.745 3.403 1.746 3.341 1.745 3.283 1.745 3.222 1.746 3.163 1.745 3.103 1.745 3.042 1.745 2.983 1.746 2.922 1.745 2.864 1.745 2.802 1.746 2.743 1.745 2.684 1.745 2.622 1.746 2.564 1.745 2.502 1.745 2.443L35.19 65.7l1.746 2.323 1.745 2.263 1.745 2.203 1.746 2.144 1.745 2.084 1.745 2.023 1.746 1.964 1.745 1.903 1.745 1.844 1.746 1.784 1.745 1.723 1.745 1.664 1.745 1.604 1.746 1.543 1.745 1.485 1.745 1.423 1.746 1.365 1.745 1.303 1.745 1.245 1.746 1.184 1.745 1.124 1.745 1.065 1.745 1.003 1.746.945 1.745.884 1.745.825 1.746.764 1.745.705 1.745.644 1.746.585 1.745.525 1.745.465 1.745.404 1.746.345 1.745.285 1.745.225 1.746.165 1.745.105 1.745.045 1.746-.015 1.745-.075 1.745-.135 1.745-.194 1.746-.255 1.745-.314 1.745-.375 1.746-.435 1.745-.494 1.745-.555 1.746-.614 1.745-.674 1.745-.735 1.745-.794 1.746-.854 1.745-.914 1.745-.974 1.746-1.034 1.745-1.094 1.745-1.154 1.746-1.214 1.745-1.274 1.745-1.334 1.745-1.393 1.746-1.454 1.745-1.514 1.745-1.573 1.746-1.634 1.745-1.693 1.745-1.754 1.746-1.813 1.745-1.874 1.745-1.933 1.745-1.993 1.746-2.054 1.745-2.112 1.745-2.174 1.746-2.232 1.745-2.294 1.745-2.353 1.746-2.412\" style=\"stroke-width:1.2\"\u002F>\u003Cpath stroke=\"none\" d=\"M3.666 37.622a2.4 2.4 0 1 0-4.8 0 2.4 2.4 0 0 0 4.8 0m-2.4 0\"\u002F>\u003Cg stroke=\"none\" font-size=\"8\">\u003Cg transform=\"translate(56.17 -75.482)\">\u003Cpath d=\"M-47.034 124.449L-47.034 124.359Q-46.991 124.152-46.784 124.128L-46.362 124.128L-46.362 120.359L-46.784 120.359Q-46.991 120.335-47.034 120.121L-47.034 120.031Q-46.991 119.824-46.784 119.800L-45.967 119.800Q-45.772 119.824-45.721 120.031L-45.721 121.582Q-45.260 121.199-44.663 121.199Q-44.315 121.199-44.077 121.339Q-43.838 121.480-43.723 121.738Q-43.608 121.996-43.608 122.351L-43.608 124.128L-43.182 124.128Q-42.975 124.152-42.936 124.359L-42.936 124.449Q-42.975 124.664-43.182 124.687L-44.577 124.687Q-44.772 124.664-44.823 124.449L-44.823 124.359Q-44.772 124.148-44.577 124.128L-44.249 124.128L-44.249 122.382Q-44.249 122.074-44.338 121.916Q-44.428 121.757-44.721 121.757Q-44.991 121.757-45.219 121.888Q-45.448 122.019-45.584 122.248Q-45.721 122.476-45.721 122.742L-45.721 124.128L-45.295 124.128Q-45.088 124.152-45.049 124.359L-45.049 124.449Q-45.088 124.664-45.295 124.687L-46.784 124.687Q-46.991 124.664-47.034 124.449M-39.346 123.199L-41.788 123.199Q-41.733 123.476-41.536 123.699Q-41.338 123.921-41.061 124.044Q-40.784 124.167-40.499 124.167Q-40.026 124.167-39.803 123.878Q-39.795 123.867-39.739 123.761Q-39.682 123.656-39.633 123.613Q-39.584 123.570-39.491 123.558L-39.346 123.558Q-39.155 123.578-39.096 123.792L-39.096 123.847Q-39.163 124.148-39.393 124.345Q-39.624 124.542-39.936 124.634Q-40.249 124.726-40.553 124.726Q-41.038 124.726-41.477 124.498Q-41.916 124.269-42.184 123.869Q-42.452 123.468-42.452 122.976L-42.452 122.917Q-42.452 122.449-42.206 122.046Q-41.959 121.644-41.551 121.410Q-41.143 121.175-40.674 121.175Q-40.170 121.175-39.817 121.398Q-39.463 121.621-39.280 122.009Q-39.096 122.398-39.096 122.902L-39.096 122.960Q-39.155 123.175-39.346 123.199M-41.780 122.648L-39.752 122.648Q-39.799 122.238-40.038 121.986Q-40.276 121.734-40.674 121.734Q-41.069 121.734-41.375 121.996Q-41.682 122.257-41.780 122.648M-38.389 124.449L-38.389 124.359Q-38.331 124.152-38.139 124.128L-37.428 124.128L-37.428 121.800L-38.139 121.800Q-38.334 121.777-38.389 121.558L-38.389 121.472Q-38.331 121.261-38.139 121.238L-37.038 121.238Q-36.838 121.257-36.788 121.472L-36.788 121.800Q-36.526 121.515-36.170 121.357Q-35.815 121.199-35.428 121.199Q-35.135 121.199-34.901 121.333Q-34.666 121.468-34.666 121.734Q-34.666 121.902-34.776 122.019Q-34.885 122.136-35.053 122.136Q-35.206 122.136-35.321 122.025Q-35.436 121.914-35.436 121.757Q-35.811 121.757-36.125 121.958Q-36.440 122.160-36.614 122.494Q-36.788 122.828-36.788 123.207L-36.788 124.128L-35.842 124.128Q-35.635 124.152-35.596 124.359L-35.596 124.449Q-35.635 124.664-35.842 124.687L-38.139 124.687Q-38.331 124.664-38.389 124.449M-30.854 123.199L-33.295 123.199Q-33.241 123.476-33.043 123.699Q-32.846 123.921-32.569 124.044Q-32.291 124.167-32.006 124.167Q-31.534 124.167-31.311 123.878Q-31.303 123.867-31.247 123.761Q-31.190 123.656-31.141 123.613Q-31.092 123.570-30.999 123.558L-30.854 123.558Q-30.663 123.578-30.604 123.792L-30.604 123.847Q-30.670 124.148-30.901 124.345Q-31.131 124.542-31.444 124.634Q-31.756 124.726-32.061 124.726Q-32.545 124.726-32.985 124.498Q-33.424 124.269-33.692 123.869Q-33.959 123.468-33.959 122.976L-33.959 122.917Q-33.959 122.449-33.713 122.046Q-33.467 121.644-33.059 121.410Q-32.651 121.175-32.182 121.175Q-31.678 121.175-31.325 121.398Q-30.971 121.621-30.788 122.009Q-30.604 122.398-30.604 122.902L-30.604 122.960Q-30.663 123.175-30.854 123.199M-33.288 122.648L-31.260 122.648Q-31.307 122.238-31.545 121.986Q-31.784 121.734-32.182 121.734Q-32.577 121.734-32.883 121.996Q-33.190 122.257-33.288 122.648\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(56.17 -75.482)\">\u003Cpath d=\"M-25.447 123.574Q-25.447 123.128-25.033 122.871Q-24.619 122.613-24.078 122.513Q-23.537 122.414-23.029 122.406Q-23.029 122.191-23.164 122.039Q-23.298 121.886-23.505 121.810Q-23.712 121.734-23.923 121.734Q-24.267 121.734-24.427 121.757L-24.427 121.816Q-24.427 121.984-24.546 122.099Q-24.666 122.214-24.830 122.214Q-25.005 122.214-25.121 122.091Q-25.236 121.968-25.236 121.800Q-25.236 121.394-24.855 121.285Q-24.474 121.175-23.916 121.175Q-23.646 121.175-23.378 121.253Q-23.111 121.332-22.886 121.482Q-22.662 121.632-22.525 121.853Q-22.388 122.074-22.388 122.351L-22.388 124.070Q-22.388 124.128-21.861 124.128Q-21.666 124.148-21.615 124.359L-21.615 124.449Q-21.666 124.664-21.861 124.687L-22.005 124.687Q-22.349 124.687-22.578 124.640Q-22.806 124.593-22.951 124.406Q-23.412 124.726-24.119 124.726Q-24.455 124.726-24.759 124.585Q-25.064 124.445-25.255 124.183Q-25.447 123.921-25.447 123.574M-24.806 123.582Q-24.806 123.855-24.564 124.011Q-24.322 124.167-24.037 124.167Q-23.818 124.167-23.585 124.109Q-23.353 124.050-23.191 123.912Q-23.029 123.773-23.029 123.550L-23.029 122.960Q-23.310 122.960-23.726 123.017Q-24.142 123.074-24.474 123.212Q-24.806 123.351-24.806 123.582M-20.408 123.582L-20.408 121.800L-21.158 121.800Q-21.357 121.777-21.408 121.558L-21.408 121.472Q-21.357 121.261-21.158 121.238L-20.408 121.238L-20.408 120.488Q-20.357 120.281-20.158 120.253L-20.013 120.253Q-19.818 120.281-19.767 120.488L-19.767 121.238L-18.408 121.238Q-18.216 121.257-18.158 121.472L-18.158 121.558Q-18.212 121.777-18.408 121.800L-19.767 121.800L-19.767 123.550Q-19.767 124.167-19.193 124.167Q-18.943 124.167-18.779 123.982Q-18.615 123.796-18.615 123.550Q-18.615 123.457-18.542 123.386Q-18.470 123.316-18.369 123.304L-18.224 123.304Q-18.025 123.328-17.974 123.535L-17.974 123.582Q-17.974 123.906-18.158 124.169Q-18.341 124.433-18.634 124.580Q-18.927 124.726-19.248 124.726Q-19.759 124.726-20.083 124.416Q-20.408 124.105-20.408 123.582\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(56.17 -75.482)\">\u003Cpath d=\"M-13.969 124.015Q-13.969 123.906-13.946 123.800L-13.375 121.535L-14.223 121.535Q-14.329 121.507-14.329 121.406L-14.305 121.304Q-14.286 121.253-14.207 121.238L-13.305 121.238L-12.985 119.976Q-12.957 119.847-12.852 119.767Q-12.747 119.687-12.618 119.687Q-12.508 119.687-12.430 119.761Q-12.352 119.835-12.352 119.945Q-12.352 119.992-12.360 120.015L-12.665 121.238L-11.817 121.238Q-11.719 121.269-11.719 121.359L-11.743 121.464Q-11.750 121.515-11.832 121.535L-12.735 121.535L-13.321 123.855Q-13.360 124.101-13.360 124.152Q-13.360 124.511-13.114 124.511Q-12.750 124.511-12.473 124.195Q-12.196 123.878-12.047 123.488Q-12.016 123.441-11.969 123.441L-11.864 123.441Q-11.825 123.441-11.801 123.470Q-11.778 123.499-11.778 123.535Q-11.778 123.550-11.786 123.566Q-11.969 124.050-12.323 124.408Q-12.676 124.765-13.129 124.765Q-13.473 124.765-13.721 124.558Q-13.969 124.351-13.969 124.015\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"m-27.187 21.688 56.906 60.32\" style=\"stroke-width:.8\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(18.62 -110.586)\">\u003Cpath d=\"M-46.518 124.488L-46.518 123.574Q-46.491 123.367-46.280 123.343L-46.112 123.343Q-45.948 123.367-45.889 123.527Q-45.686 124.167-44.959 124.167Q-44.752 124.167-44.524 124.132Q-44.295 124.097-44.127 123.982Q-43.959 123.867-43.959 123.664Q-43.959 123.453-44.182 123.339Q-44.405 123.226-44.678 123.183L-45.377 123.070Q-46.518 122.859-46.518 122.136Q-46.518 121.847-46.374 121.658Q-46.229 121.468-45.989 121.361Q-45.749 121.253-45.493 121.214Q-45.237 121.175-44.959 121.175Q-44.709 121.175-44.516 121.205Q-44.323 121.234-44.159 121.312Q-44.081 121.195-43.952 121.175L-43.874 121.175Q-43.776 121.187-43.713 121.249Q-43.651 121.312-43.639 121.406L-43.639 122.113Q-43.651 122.207-43.713 122.273Q-43.776 122.339-43.874 122.351L-44.041 122.351Q-44.135 122.339-44.202 122.273Q-44.268 122.207-44.280 122.113Q-44.280 121.734-44.975 121.734Q-45.323 121.734-45.641 121.816Q-45.959 121.898-45.959 122.144Q-45.959 122.410-45.288 122.519L-44.584 122.640Q-44.100 122.722-43.750 122.970Q-43.401 123.218-43.401 123.664Q-43.401 124.054-43.637 124.296Q-43.874 124.539-44.223 124.632Q-44.573 124.726-44.959 124.726Q-45.538 124.726-45.936 124.472Q-46.006 124.597-46.055 124.654Q-46.104 124.710-46.209 124.726L-46.280 124.726Q-46.495 124.703-46.518 124.488M-41.659 123.582L-41.659 121.800L-42.409 121.800Q-42.608 121.777-42.659 121.558L-42.659 121.472Q-42.608 121.261-42.409 121.238L-41.659 121.238L-41.659 120.488Q-41.608 120.281-41.409 120.253L-41.264 120.253Q-41.069 120.281-41.018 120.488L-41.018 121.238L-39.659 121.238Q-39.467 121.257-39.409 121.472L-39.409 121.558Q-39.463 121.777-39.659 121.800L-41.018 121.800L-41.018 123.550Q-41.018 124.167-40.444 124.167Q-40.194 124.167-40.030 123.982Q-39.866 123.796-39.866 123.550Q-39.866 123.457-39.793 123.386Q-39.721 123.316-39.620 123.304L-39.475 123.304Q-39.276 123.328-39.225 123.535L-39.225 123.582Q-39.225 123.906-39.409 124.169Q-39.592 124.433-39.885 124.580Q-40.178 124.726-40.499 124.726Q-41.010 124.726-41.334 124.416Q-41.659 124.105-41.659 123.582M-35.100 123.199L-37.541 123.199Q-37.487 123.476-37.290 123.699Q-37.092 123.921-36.815 124.044Q-36.538 124.167-36.252 124.167Q-35.780 124.167-35.557 123.878Q-35.549 123.867-35.493 123.761Q-35.436 123.656-35.387 123.613Q-35.338 123.570-35.245 123.558L-35.100 123.558Q-34.909 123.578-34.850 123.792L-34.850 123.847Q-34.916 124.148-35.147 124.345Q-35.377 124.542-35.690 124.634Q-36.002 124.726-36.307 124.726Q-36.791 124.726-37.231 124.498Q-37.670 124.269-37.938 123.869Q-38.206 123.468-38.206 122.976L-38.206 122.917Q-38.206 122.449-37.959 122.046Q-37.713 121.644-37.305 121.410Q-36.897 121.175-36.428 121.175Q-35.924 121.175-35.571 121.398Q-35.217 121.621-35.034 122.009Q-34.850 122.398-34.850 122.902L-34.850 122.960Q-34.909 123.175-35.100 123.199M-37.534 122.648L-35.506 122.648Q-35.553 122.238-35.791 121.986Q-36.030 121.734-36.428 121.734Q-36.823 121.734-37.129 121.996Q-37.436 122.257-37.534 122.648M-30.854 123.199L-33.295 123.199Q-33.241 123.476-33.043 123.699Q-32.846 123.921-32.569 124.044Q-32.291 124.167-32.006 124.167Q-31.534 124.167-31.311 123.878Q-31.303 123.867-31.247 123.761Q-31.190 123.656-31.141 123.613Q-31.092 123.570-30.999 123.558L-30.854 123.558Q-30.663 123.578-30.604 123.792L-30.604 123.847Q-30.670 124.148-30.901 124.345Q-31.131 124.542-31.444 124.634Q-31.756 124.726-32.061 124.726Q-32.545 124.726-32.985 124.498Q-33.424 124.269-33.692 123.869Q-33.959 123.468-33.959 122.976L-33.959 122.917Q-33.959 122.449-33.713 122.046Q-33.467 121.644-33.059 121.410Q-32.651 121.175-32.182 121.175Q-31.678 121.175-31.325 121.398Q-30.971 121.621-30.788 122.009Q-30.604 122.398-30.604 122.902L-30.604 122.960Q-30.663 123.175-30.854 123.199M-33.288 122.648L-31.260 122.648Q-31.307 122.238-31.545 121.986Q-31.784 121.734-32.182 121.734Q-32.577 121.734-32.883 121.996Q-33.190 122.257-33.288 122.648M-30.049 126.230L-30.049 126.144Q-30.006 125.925-29.799 125.902L-29.377 125.902L-29.377 121.800L-29.799 121.800Q-30.006 121.777-30.049 121.558L-30.049 121.472Q-30.002 121.261-29.799 121.238L-28.983 121.238Q-28.788 121.257-28.737 121.472L-28.737 121.542Q-28.526 121.374-28.262 121.287Q-27.999 121.199-27.729 121.199Q-27.389 121.199-27.092 121.343Q-26.795 121.488-26.581 121.740Q-26.366 121.992-26.250 122.304Q-26.135 122.617-26.135 122.960Q-26.135 123.425-26.362 123.835Q-26.588 124.246-26.975 124.486Q-27.362 124.726-27.831 124.726Q-28.350 124.726-28.737 124.359L-28.737 125.902L-28.311 125.902Q-28.104 125.925-28.065 126.144L-28.065 126.230Q-28.104 126.441-28.311 126.464L-29.799 126.464Q-30.002 126.441-30.049 126.230M-27.874 124.167Q-27.565 124.167-27.315 123.998Q-27.065 123.828-26.920 123.546Q-26.776 123.265-26.776 122.960Q-26.776 122.671-26.901 122.392Q-27.026 122.113-27.258 121.935Q-27.491 121.757-27.791 121.757Q-28.112 121.757-28.374 121.943Q-28.635 122.128-28.737 122.429L-28.737 123.281Q-28.647 123.648-28.426 123.908Q-28.206 124.167-27.874 124.167\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(18.62 -110.586)\">\u003Cpath d=\"M-21.365 125.335Q-21.365 125.035-21.216 124.773Q-21.068 124.511-20.818 124.351Q-21.001 124.097-21.001 123.777Q-21.001 123.492-20.849 123.230Q-21.099 122.906-21.099 122.488Q-21.099 122.128-20.908 121.832Q-20.716 121.535-20.398 121.367Q-20.080 121.199-19.716 121.199Q-19.509 121.199-19.306 121.259Q-19.103 121.320-18.939 121.421Q-18.556 121.160-18.083 121.160Q-17.845 121.160-17.664 121.291Q-17.482 121.421-17.482 121.648Q-17.482 121.796-17.583 121.902Q-17.685 122.007-17.841 122.007Q-17.974 122.007-18.070 121.929Q-18.166 121.851-18.197 121.726Q-18.341 121.734-18.541 121.824Q-18.337 122.136-18.337 122.488Q-18.337 122.769-18.449 123.001Q-18.560 123.234-18.755 123.412Q-18.951 123.589-19.203 123.687Q-19.455 123.785-19.716 123.785Q-20.103 123.785-20.435 123.597Q-20.447 123.597-20.457 123.669Q-20.466 123.742-20.466 123.777Q-20.466 123.898-20.400 124.005Q-20.333 124.113-20.212 124.152Q-20.197 124.148-20.183 124.146Q-20.169 124.144-20.146 124.144Q-20.130 124.144-20.099 124.152Q-20.068 124.160-20.060 124.160L-19.466 124.160Q-18.685 124.160-18.144 124.402Q-17.603 124.644-17.603 125.335Q-17.603 125.636-17.783 125.863Q-17.962 126.089-18.259 126.234Q-18.556 126.378-18.876 126.445Q-19.197 126.511-19.482 126.511Q-19.873 126.511-20.314 126.388Q-20.755 126.265-21.060 125.998Q-21.365 125.730-21.365 125.335M-20.826 125.328Q-20.826 125.546-20.585 125.687Q-20.345 125.828-20.021 125.894Q-19.697 125.960-19.482 125.960Q-19.267 125.960-18.943 125.894Q-18.619 125.828-18.378 125.687Q-18.138 125.546-18.138 125.328Q-18.138 125.039-18.363 124.900Q-18.587 124.761-18.869 124.728Q-19.150 124.695-19.498 124.695L-20.115 124.695Q-20.298 124.695-20.460 124.775Q-20.623 124.855-20.724 125.001Q-20.826 125.148-20.826 125.328M-19.716 123.230Q-19.416 123.230-19.197 123.011Q-18.978 122.792-18.978 122.488Q-18.978 122.332-19.033 122.201Q-19.087 122.070-19.193 121.964Q-19.298 121.859-19.429 121.804Q-19.560 121.749-19.716 121.749Q-20.021 121.749-20.240 121.968Q-20.458 122.187-20.458 122.488Q-20.458 122.785-20.236 123.007Q-20.013 123.230-19.716 123.230M-17.134 124.449L-17.134 124.359Q-17.076 124.152-16.884 124.128L-16.173 124.128L-16.173 121.800L-16.884 121.800Q-17.080 121.777-17.134 121.558L-17.134 121.472Q-17.076 121.261-16.884 121.238L-15.783 121.238Q-15.583 121.257-15.533 121.472L-15.533 121.800Q-15.271 121.515-14.916 121.357Q-14.560 121.199-14.173 121.199Q-13.880 121.199-13.646 121.333Q-13.412 121.468-13.412 121.734Q-13.412 121.902-13.521 122.019Q-13.630 122.136-13.798 122.136Q-13.951 122.136-14.066 122.025Q-14.181 121.914-14.181 121.757Q-14.556 121.757-14.871 121.958Q-15.185 122.160-15.359 122.494Q-15.533 122.828-15.533 123.207L-15.533 124.128L-14.587 124.128Q-14.380 124.152-14.341 124.359L-14.341 124.449Q-14.380 124.664-14.587 124.687L-16.884 124.687Q-17.076 124.664-17.134 124.449M-12.705 123.574Q-12.705 123.128-12.291 122.871Q-11.876 122.613-11.335 122.513Q-10.794 122.414-10.287 122.406Q-10.287 122.191-10.421 122.039Q-10.556 121.886-10.763 121.810Q-10.970 121.734-11.181 121.734Q-11.525 121.734-11.685 121.757L-11.685 121.816Q-11.685 121.984-11.804 122.099Q-11.923 122.214-12.087 122.214Q-12.263 122.214-12.378 122.091Q-12.494 121.968-12.494 121.800Q-12.494 121.394-12.113 121.285Q-11.732 121.175-11.173 121.175Q-10.904 121.175-10.636 121.253Q-10.369 121.332-10.144 121.482Q-9.919 121.632-9.783 121.853Q-9.646 122.074-9.646 122.351L-9.646 124.070Q-9.646 124.128-9.119 124.128Q-8.923 124.148-8.873 124.359L-8.873 124.449Q-8.923 124.664-9.119 124.687L-9.263 124.687Q-9.607 124.687-9.835 124.640Q-10.064 124.593-10.208 124.406Q-10.669 124.726-11.376 124.726Q-11.712 124.726-12.017 124.585Q-12.322 124.445-12.513 124.183Q-12.705 123.921-12.705 123.574M-12.064 123.582Q-12.064 123.855-11.822 124.011Q-11.580 124.167-11.294 124.167Q-11.076 124.167-10.843 124.109Q-10.611 124.050-10.449 123.912Q-10.287 123.773-10.287 123.550L-10.287 122.960Q-10.568 122.960-10.984 123.017Q-11.400 123.074-11.732 123.212Q-12.064 123.351-12.064 123.582M-7.001 124.726Q-7.466 124.726-7.832 124.476Q-8.197 124.226-8.402 123.822Q-8.607 123.417-8.607 122.960Q-8.607 122.617-8.482 122.296Q-8.357 121.976-8.124 121.726Q-7.892 121.476-7.587 121.337Q-7.283 121.199-6.927 121.199Q-6.416 121.199-6.009 121.519L-6.009 120.359L-6.431 120.359Q-6.642 120.335-6.681 120.121L-6.681 120.031Q-6.642 119.824-6.431 119.800L-5.619 119.800Q-5.419 119.824-5.369 120.031L-5.369 124.128L-4.943 124.128Q-4.736 124.152-4.697 124.359L-4.697 124.449Q-4.736 124.664-4.943 124.687L-5.759 124.687Q-5.958 124.664-6.009 124.449L-6.009 124.320Q-6.205 124.511-6.466 124.619Q-6.728 124.726-7.001 124.726M-6.962 124.167Q-6.603 124.167-6.351 123.898Q-6.099 123.628-6.009 123.253L-6.009 122.421Q-6.068 122.234-6.195 122.083Q-6.322 121.933-6.499 121.845Q-6.677 121.757-6.873 121.757Q-7.181 121.757-7.431 121.927Q-7.681 122.097-7.826 122.382Q-7.970 122.667-7.970 122.968Q-7.970 123.417-7.685 123.792Q-7.400 124.167-6.962 124.167M-4.001 124.449L-4.001 124.359Q-3.951 124.152-3.755 124.128L-2.716 124.128L-2.716 121.800L-3.689 121.800Q-3.888 121.777-3.939 121.558L-3.939 121.472Q-3.888 121.261-3.689 121.238L-2.322 121.238Q-2.126 121.257-2.076 121.472L-2.076 124.128L-1.162 124.128Q-0.966 124.152-0.916 124.359L-0.916 124.449Q-0.966 124.664-1.162 124.687L-3.755 124.687Q-3.951 124.664-4.001 124.449M-2.970 120.261L-2.970 120.207Q-2.970 120.035-2.833 119.914Q-2.697 119.792-2.521 119.792Q-2.349 119.792-2.212 119.914Q-2.076 120.035-2.076 120.207L-2.076 120.261Q-2.076 120.437-2.212 120.558Q-2.349 120.679-2.521 120.679Q-2.697 120.679-2.833 120.558Q-2.970 120.437-2.970 120.261M3.139 123.199L0.698 123.199Q0.752 123.476 0.950 123.699Q1.147 123.921 1.424 124.044Q1.702 124.167 1.987 124.167Q2.459 124.167 2.682 123.878Q2.690 123.867 2.747 123.761Q2.803 123.656 2.852 123.613Q2.901 123.570 2.995 123.558L3.139 123.558Q3.331 123.578 3.389 123.792L3.389 123.847Q3.323 124.148 3.092 124.345Q2.862 124.542 2.549 124.634Q2.237 124.726 1.932 124.726Q1.448 124.726 1.008 124.498Q0.569 124.269 0.301 123.869Q0.034 123.468 0.034 122.976L0.034 122.917Q0.034 122.449 0.280 122.046Q0.526 121.644 0.934 121.410Q1.342 121.175 1.811 121.175Q2.315 121.175 2.668 121.398Q3.022 121.621 3.206 122.009Q3.389 122.398 3.389 122.902L3.389 122.960Q3.331 123.175 3.139 123.199M0.706 122.648L2.733 122.648Q2.686 122.238 2.448 121.986Q2.209 121.734 1.811 121.734Q1.417 121.734 1.110 121.996Q0.803 122.257 0.706 122.648M3.944 124.449L3.944 124.359Q3.987 124.152 4.194 124.128L4.616 124.128L4.616 121.800L4.194 121.800Q3.987 121.777 3.944 121.558L3.944 121.472Q3.991 121.261 4.194 121.238L5.010 121.238Q5.206 121.261 5.256 121.472L5.256 121.558L5.249 121.582Q5.475 121.402 5.749 121.300Q6.022 121.199 6.315 121.199Q6.663 121.199 6.901 121.339Q7.139 121.480 7.254 121.738Q7.370 121.996 7.370 122.351L7.370 124.128L7.795 124.128Q8.002 124.152 8.042 124.359L8.042 124.449Q8.002 124.664 7.795 124.687L6.401 124.687Q6.206 124.664 6.155 124.449L6.155 124.359Q6.206 124.148 6.401 124.128L6.729 124.128L6.729 122.382Q6.729 122.074 6.639 121.916Q6.549 121.757 6.256 121.757Q5.987 121.757 5.758 121.888Q5.530 122.019 5.393 122.248Q5.256 122.476 5.256 122.742L5.256 124.128L5.682 124.128Q5.889 124.152 5.928 124.359L5.928 124.449Q5.889 124.664 5.682 124.687L4.194 124.687Q3.987 124.664 3.944 124.449M9.319 123.582L9.319 121.800L8.569 121.800Q8.370 121.777 8.319 121.558L8.319 121.472Q8.370 121.261 8.569 121.238L9.319 121.238L9.319 120.488Q9.370 120.281 9.569 120.253L9.713 120.253Q9.909 120.281 9.959 120.488L9.959 121.238L11.319 121.238Q11.510 121.257 11.569 121.472L11.569 121.558Q11.514 121.777 11.319 121.800L9.959 121.800L9.959 123.550Q9.959 124.167 10.534 124.167Q10.784 124.167 10.948 123.982Q11.112 123.796 11.112 123.550Q11.112 123.457 11.184 123.386Q11.256 123.316 11.358 123.304L11.502 123.304Q11.702 123.328 11.752 123.535L11.752 123.582Q11.752 123.906 11.569 124.169Q11.385 124.433 11.092 124.580Q10.799 124.726 10.479 124.726Q9.967 124.726 9.643 124.416Q9.319 124.105 9.319 123.582\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath stroke=\"none\" d=\"M69.107 97.942a2.4 2.4 0 1 0-4.8 0 2.4 2.4 0 0 0 4.8 0m-2.4 0\"\u002F>\u003Cg stroke=\"none\" font-size=\"8\">\u003Cg transform=\"translate(76.327 -57.004)\">\u003Cpath d=\"M-47.034 124.449L-47.034 124.359Q-46.991 124.152-46.784 124.128L-46.362 124.128L-46.362 120.359L-46.784 120.359Q-46.991 120.335-47.034 120.121L-47.034 120.031Q-46.991 119.824-46.784 119.800L-45.967 119.800Q-45.772 119.824-45.721 120.031L-45.721 121.582Q-45.260 121.199-44.663 121.199Q-44.315 121.199-44.077 121.339Q-43.838 121.480-43.723 121.738Q-43.608 121.996-43.608 122.351L-43.608 124.128L-43.182 124.128Q-42.975 124.152-42.936 124.359L-42.936 124.449Q-42.975 124.664-43.182 124.687L-44.577 124.687Q-44.772 124.664-44.823 124.449L-44.823 124.359Q-44.772 124.148-44.577 124.128L-44.249 124.128L-44.249 122.382Q-44.249 122.074-44.338 121.916Q-44.428 121.757-44.721 121.757Q-44.991 121.757-45.219 121.888Q-45.448 122.019-45.584 122.248Q-45.721 122.476-45.721 122.742L-45.721 124.128L-45.295 124.128Q-45.088 124.152-45.049 124.359L-45.049 124.449Q-45.088 124.664-45.295 124.687L-46.784 124.687Q-46.991 124.664-47.034 124.449M-39.346 123.199L-41.788 123.199Q-41.733 123.476-41.536 123.699Q-41.338 123.921-41.061 124.044Q-40.784 124.167-40.499 124.167Q-40.026 124.167-39.803 123.878Q-39.795 123.867-39.739 123.761Q-39.682 123.656-39.633 123.613Q-39.584 123.570-39.491 123.558L-39.346 123.558Q-39.155 123.578-39.096 123.792L-39.096 123.847Q-39.163 124.148-39.393 124.345Q-39.624 124.542-39.936 124.634Q-40.249 124.726-40.553 124.726Q-41.038 124.726-41.477 124.498Q-41.916 124.269-42.184 123.869Q-42.452 123.468-42.452 122.976L-42.452 122.917Q-42.452 122.449-42.206 122.046Q-41.959 121.644-41.551 121.410Q-41.143 121.175-40.674 121.175Q-40.170 121.175-39.817 121.398Q-39.463 121.621-39.280 122.009Q-39.096 122.398-39.096 122.902L-39.096 122.960Q-39.155 123.175-39.346 123.199M-41.780 122.648L-39.752 122.648Q-39.799 122.238-40.038 121.986Q-40.276 121.734-40.674 121.734Q-41.069 121.734-41.375 121.996Q-41.682 122.257-41.780 122.648M-38.389 124.449L-38.389 124.359Q-38.331 124.152-38.139 124.128L-37.428 124.128L-37.428 121.800L-38.139 121.800Q-38.334 121.777-38.389 121.558L-38.389 121.472Q-38.331 121.261-38.139 121.238L-37.038 121.238Q-36.838 121.257-36.788 121.472L-36.788 121.800Q-36.526 121.515-36.170 121.357Q-35.815 121.199-35.428 121.199Q-35.135 121.199-34.901 121.333Q-34.666 121.468-34.666 121.734Q-34.666 121.902-34.776 122.019Q-34.885 122.136-35.053 122.136Q-35.206 122.136-35.321 122.025Q-35.436 121.914-35.436 121.757Q-35.811 121.757-36.125 121.958Q-36.440 122.160-36.614 122.494Q-36.788 122.828-36.788 123.207L-36.788 124.128L-35.842 124.128Q-35.635 124.152-35.596 124.359L-35.596 124.449Q-35.635 124.664-35.842 124.687L-38.139 124.687Q-38.331 124.664-38.389 124.449M-30.854 123.199L-33.295 123.199Q-33.241 123.476-33.043 123.699Q-32.846 123.921-32.569 124.044Q-32.291 124.167-32.006 124.167Q-31.534 124.167-31.311 123.878Q-31.303 123.867-31.247 123.761Q-31.190 123.656-31.141 123.613Q-31.092 123.570-30.999 123.558L-30.854 123.558Q-30.663 123.578-30.604 123.792L-30.604 123.847Q-30.670 124.148-30.901 124.345Q-31.131 124.542-31.444 124.634Q-31.756 124.726-32.061 124.726Q-32.545 124.726-32.985 124.498Q-33.424 124.269-33.692 123.869Q-33.959 123.468-33.959 122.976L-33.959 122.917Q-33.959 122.449-33.713 122.046Q-33.467 121.644-33.059 121.410Q-32.651 121.175-32.182 121.175Q-31.678 121.175-31.325 121.398Q-30.971 121.621-30.788 122.009Q-30.604 122.398-30.604 122.902L-30.604 122.960Q-30.663 123.175-30.854 123.199M-33.288 122.648L-31.260 122.648Q-31.307 122.238-31.545 121.986Q-31.784 121.734-32.182 121.734Q-32.577 121.734-32.883 121.996Q-33.190 122.257-33.288 122.648\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(76.327 -57.004)\">\u003Cpath d=\"M-25.447 123.574Q-25.447 123.128-25.033 122.871Q-24.619 122.613-24.078 122.513Q-23.537 122.414-23.029 122.406Q-23.029 122.191-23.164 122.039Q-23.298 121.886-23.505 121.810Q-23.712 121.734-23.923 121.734Q-24.267 121.734-24.427 121.757L-24.427 121.816Q-24.427 121.984-24.546 122.099Q-24.666 122.214-24.830 122.214Q-25.005 122.214-25.121 122.091Q-25.236 121.968-25.236 121.800Q-25.236 121.394-24.855 121.285Q-24.474 121.175-23.916 121.175Q-23.646 121.175-23.378 121.253Q-23.111 121.332-22.886 121.482Q-22.662 121.632-22.525 121.853Q-22.388 122.074-22.388 122.351L-22.388 124.070Q-22.388 124.128-21.861 124.128Q-21.666 124.148-21.615 124.359L-21.615 124.449Q-21.666 124.664-21.861 124.687L-22.005 124.687Q-22.349 124.687-22.578 124.640Q-22.806 124.593-22.951 124.406Q-23.412 124.726-24.119 124.726Q-24.455 124.726-24.759 124.585Q-25.064 124.445-25.255 124.183Q-25.447 123.921-25.447 123.574M-24.806 123.582Q-24.806 123.855-24.564 124.011Q-24.322 124.167-24.037 124.167Q-23.818 124.167-23.585 124.109Q-23.353 124.050-23.191 123.912Q-23.029 123.773-23.029 123.550L-23.029 122.960Q-23.310 122.960-23.726 123.017Q-24.142 123.074-24.474 123.212Q-24.806 123.351-24.806 123.582M-20.408 123.582L-20.408 121.800L-21.158 121.800Q-21.357 121.777-21.408 121.558L-21.408 121.472Q-21.357 121.261-21.158 121.238L-20.408 121.238L-20.408 120.488Q-20.357 120.281-20.158 120.253L-20.013 120.253Q-19.818 120.281-19.767 120.488L-19.767 121.238L-18.408 121.238Q-18.216 121.257-18.158 121.472L-18.158 121.558Q-18.212 121.777-18.408 121.800L-19.767 121.800L-19.767 123.550Q-19.767 124.167-19.193 124.167Q-18.943 124.167-18.779 123.982Q-18.615 123.796-18.615 123.550Q-18.615 123.457-18.542 123.386Q-18.470 123.316-18.369 123.304L-18.224 123.304Q-18.025 123.328-17.974 123.535L-17.974 123.582Q-17.974 123.906-18.158 124.169Q-18.341 124.433-18.634 124.580Q-18.927 124.726-19.248 124.726Q-19.759 124.726-20.083 124.416Q-20.408 124.105-20.408 123.582\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(76.327 -57.004)\">\u003Cpath d=\"M-13.969 124.015Q-13.969 123.906-13.946 123.800L-13.375 121.535L-14.223 121.535Q-14.329 121.507-14.329 121.406L-14.305 121.304Q-14.286 121.253-14.207 121.238L-13.305 121.238L-12.985 119.976Q-12.957 119.847-12.852 119.767Q-12.747 119.687-12.618 119.687Q-12.508 119.687-12.430 119.761Q-12.352 119.835-12.352 119.945Q-12.352 119.992-12.360 120.015L-12.665 121.238L-11.817 121.238Q-11.719 121.269-11.719 121.359L-11.743 121.464Q-11.750 121.515-11.832 121.535L-12.735 121.535L-13.321 123.855Q-13.360 124.101-13.360 124.152Q-13.360 124.511-13.114 124.511Q-12.750 124.511-12.473 124.195Q-12.196 123.878-12.047 123.488Q-12.016 123.441-11.969 123.441L-11.864 123.441Q-11.825 123.441-11.801 123.470Q-11.778 123.499-11.778 123.535Q-11.778 123.550-11.786 123.566Q-11.969 124.050-12.323 124.408Q-12.676 124.765-13.129 124.765Q-13.473 124.765-13.721 124.558Q-13.969 124.351-13.969 124.015\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(76.327 -57.004)\">\u003Cpath d=\"M-8.329 122.871L-10.802 122.871Q-10.880 122.859-10.929 122.810Q-10.977 122.761-10.977 122.687Q-10.977 122.613-10.929 122.564Q-10.880 122.515-10.802 122.503L-8.329 122.503L-8.329 120.023Q-8.302 119.855-8.145 119.855Q-8.071 119.855-8.022 119.904Q-7.973 119.953-7.962 120.023L-7.962 122.503L-5.489 122.503Q-5.321 122.535-5.321 122.687Q-5.321 122.839-5.489 122.871L-7.962 122.871L-7.962 125.351Q-7.973 125.421-8.022 125.470Q-8.071 125.519-8.145 125.519Q-8.302 125.519-8.329 125.351\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(76.327 -57.004)\">\u003Cpath d=\"M-1.245 124.687L-4.038 124.687L-4.038 124.390Q-2.976 124.390-2.976 124.128L-2.976 119.960Q-3.405 120.175-4.085 120.175L-4.085 119.878Q-3.066 119.878-2.550 119.367L-2.405 119.367Q-2.331 119.386-2.312 119.464L-2.312 124.128Q-2.312 124.390-1.245 124.390\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-good)\" d=\"M38.255 77.456 95.16 110.46\" style=\"stroke-width:.8\"\u002F>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\">\u003Cg fill=\"var(--tk-good)\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(122.49 -36.856)\">\u003Cpath d=\"M-46.866 125.335Q-46.866 125.035-46.717 124.773Q-46.569 124.511-46.319 124.351Q-46.502 124.097-46.502 123.777Q-46.502 123.492-46.350 123.230Q-46.600 122.906-46.600 122.488Q-46.600 122.128-46.409 121.832Q-46.217 121.535-45.899 121.367Q-45.581 121.199-45.217 121.199Q-45.010 121.199-44.807 121.259Q-44.604 121.320-44.440 121.421Q-44.057 121.160-43.584 121.160Q-43.346 121.160-43.165 121.291Q-42.983 121.421-42.983 121.648Q-42.983 121.796-43.084 121.902Q-43.186 122.007-43.342 122.007Q-43.475 122.007-43.571 121.929Q-43.666 121.851-43.698 121.726Q-43.842 121.734-44.041 121.824Q-43.838 122.136-43.838 122.488Q-43.838 122.769-43.950 123.001Q-44.061 123.234-44.256 123.412Q-44.452 123.589-44.704 123.687Q-44.956 123.785-45.217 123.785Q-45.604 123.785-45.936 123.597Q-45.948 123.597-45.958 123.669Q-45.967 123.742-45.967 123.777Q-45.967 123.898-45.901 124.005Q-45.834 124.113-45.713 124.152Q-45.698 124.148-45.684 124.146Q-45.670 124.144-45.647 124.144Q-45.631 124.144-45.600 124.152Q-45.569 124.160-45.561 124.160L-44.967 124.160Q-44.186 124.160-43.645 124.402Q-43.104 124.644-43.104 125.335Q-43.104 125.636-43.284 125.863Q-43.463 126.089-43.760 126.234Q-44.057 126.378-44.377 126.445Q-44.698 126.511-44.983 126.511Q-45.374 126.511-45.815 126.388Q-46.256 126.265-46.561 125.998Q-46.866 125.730-46.866 125.335M-46.327 125.328Q-46.327 125.546-46.086 125.687Q-45.846 125.828-45.522 125.894Q-45.198 125.960-44.983 125.960Q-44.768 125.960-44.444 125.894Q-44.120 125.828-43.879 125.687Q-43.639 125.546-43.639 125.328Q-43.639 125.039-43.864 124.900Q-44.088 124.761-44.370 124.728Q-44.651 124.695-44.999 124.695L-45.616 124.695Q-45.799 124.695-45.961 124.775Q-46.124 124.855-46.225 125.001Q-46.327 125.148-46.327 125.328M-45.217 123.230Q-44.916 123.230-44.698 123.011Q-44.479 122.792-44.479 122.488Q-44.479 122.332-44.534 122.201Q-44.588 122.070-44.694 121.964Q-44.799 121.859-44.930 121.804Q-45.061 121.749-45.217 121.749Q-45.522 121.749-45.741 121.968Q-45.959 122.187-45.959 122.488Q-45.959 122.785-45.737 123.007Q-45.514 123.230-45.217 123.230M-39.346 123.199L-41.788 123.199Q-41.733 123.476-41.536 123.699Q-41.338 123.921-41.061 124.044Q-40.784 124.167-40.499 124.167Q-40.026 124.167-39.803 123.878Q-39.795 123.867-39.739 123.761Q-39.682 123.656-39.633 123.613Q-39.584 123.570-39.491 123.558L-39.346 123.558Q-39.155 123.578-39.096 123.792L-39.096 123.847Q-39.163 124.148-39.393 124.345Q-39.624 124.542-39.936 124.634Q-40.249 124.726-40.553 124.726Q-41.038 124.726-41.477 124.498Q-41.916 124.269-42.184 123.869Q-42.452 123.468-42.452 122.976L-42.452 122.917Q-42.452 122.449-42.206 122.046Q-41.959 121.644-41.551 121.410Q-41.143 121.175-40.674 121.175Q-40.170 121.175-39.817 121.398Q-39.463 121.621-39.280 122.009Q-39.096 122.398-39.096 122.902L-39.096 122.960Q-39.155 123.175-39.346 123.199M-41.780 122.648L-39.752 122.648Q-39.799 122.238-40.038 121.986Q-40.276 121.734-40.674 121.734Q-41.069 121.734-41.375 121.996Q-41.682 122.257-41.780 122.648M-38.541 124.449L-38.541 124.359Q-38.499 124.152-38.291 124.128L-37.870 124.128L-37.870 121.800L-38.291 121.800Q-38.499 121.777-38.541 121.558L-38.541 121.472Q-38.495 121.261-38.291 121.238L-37.475 121.238Q-37.280 121.261-37.229 121.472L-37.229 121.558L-37.237 121.582Q-37.010 121.402-36.737 121.300Q-36.463 121.199-36.170 121.199Q-35.823 121.199-35.584 121.339Q-35.346 121.480-35.231 121.738Q-35.116 121.996-35.116 122.351L-35.116 124.128L-34.690 124.128Q-34.483 124.152-34.444 124.359L-34.444 124.449Q-34.483 124.664-34.690 124.687L-36.084 124.687Q-36.280 124.664-36.331 124.449L-36.331 124.359Q-36.280 124.148-36.084 124.128L-35.756 124.128L-35.756 122.382Q-35.756 122.074-35.846 121.916Q-35.936 121.757-36.229 121.757Q-36.499 121.757-36.727 121.888Q-36.956 122.019-37.092 122.248Q-37.229 122.476-37.229 122.742L-37.229 124.128L-36.803 124.128Q-36.596 124.152-36.557 124.359L-36.557 124.449Q-36.596 124.664-36.803 124.687L-38.291 124.687Q-38.499 124.664-38.541 124.449M-33.166 123.582L-33.166 121.800L-33.916 121.800Q-34.116 121.777-34.166 121.558L-34.166 121.472Q-34.116 121.261-33.916 121.238L-33.166 121.238L-33.166 120.488Q-33.116 120.281-32.916 120.253L-32.772 120.253Q-32.577 120.281-32.526 120.488L-32.526 121.238L-31.166 121.238Q-30.975 121.257-30.916 121.472L-30.916 121.558Q-30.971 121.777-31.166 121.800L-32.526 121.800L-32.526 123.550Q-32.526 124.167-31.952 124.167Q-31.702 124.167-31.538 123.982Q-31.374 123.796-31.374 123.550Q-31.374 123.457-31.301 123.386Q-31.229 123.316-31.127 123.304L-30.983 123.304Q-30.784 123.328-30.733 123.535L-30.733 123.582Q-30.733 123.906-30.916 124.169Q-31.100 124.433-31.393 124.580Q-31.686 124.726-32.006 124.726Q-32.518 124.726-32.842 124.416Q-33.166 124.105-33.166 123.582M-29.670 124.449L-29.670 124.359Q-29.620 124.152-29.424 124.128L-28.319 124.128L-28.319 120.359L-29.424 120.359Q-29.620 120.335-29.670 120.121L-29.670 120.031Q-29.620 119.824-29.424 119.800L-27.928 119.800Q-27.737 119.824-27.678 120.031L-27.678 124.128L-26.577 124.128Q-26.377 124.152-26.327 124.359L-26.327 124.449Q-26.377 124.664-26.577 124.687L-29.424 124.687Q-29.620 124.664-29.670 124.449M-22.362 123.199L-24.803 123.199Q-24.749 123.476-24.551 123.699Q-24.354 123.921-24.077 124.044Q-23.799 124.167-23.514 124.167Q-23.041 124.167-22.819 123.878Q-22.811 123.867-22.754 123.761Q-22.698 123.656-22.649 123.613Q-22.600 123.570-22.506 123.558L-22.362 123.558Q-22.170 123.578-22.112 123.792L-22.112 123.847Q-22.178 124.148-22.409 124.345Q-22.639 124.542-22.952 124.634Q-23.264 124.726-23.569 124.726Q-24.053 124.726-24.493 124.498Q-24.932 124.269-25.200 123.869Q-25.467 123.468-25.467 122.976L-25.467 122.917Q-25.467 122.449-25.221 122.046Q-24.975 121.644-24.567 121.410Q-24.159 121.175-23.690 121.175Q-23.186 121.175-22.833 121.398Q-22.479 121.621-22.295 122.009Q-22.112 122.398-22.112 122.902L-22.112 122.960Q-22.170 123.175-22.362 123.199M-24.795 122.648L-22.768 122.648Q-22.815 122.238-23.053 121.986Q-23.291 121.734-23.690 121.734Q-24.084 121.734-24.391 121.996Q-24.698 122.257-24.795 122.648M-21.405 124.449L-21.405 124.359Q-21.346 124.152-21.155 124.128L-20.444 124.128L-20.444 121.800L-21.155 121.800Q-21.350 121.777-21.405 121.558L-21.405 121.472Q-21.346 121.261-21.155 121.238L-20.053 121.238Q-19.854 121.257-19.803 121.472L-19.803 121.800Q-19.541 121.515-19.186 121.357Q-18.831 121.199-18.444 121.199Q-18.151 121.199-17.916 121.333Q-17.682 121.468-17.682 121.734Q-17.682 121.902-17.791 122.019Q-17.901 122.136-18.069 122.136Q-18.221 122.136-18.336 122.025Q-18.452 121.914-18.452 121.757Q-18.827 121.757-19.141 121.958Q-19.456 122.160-19.629 122.494Q-19.803 122.828-19.803 123.207L-19.803 124.128L-18.858 124.128Q-18.651 124.152-18.612 124.359L-18.612 124.449Q-18.651 124.664-18.858 124.687L-21.155 124.687Q-21.346 124.664-21.405 124.449\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(122.49 -36.856)\">\u003Cpath d=\"M-12.865 125.335Q-12.865 125.035-12.716 124.773Q-12.568 124.511-12.318 124.351Q-12.501 124.097-12.501 123.777Q-12.501 123.492-12.349 123.230Q-12.599 122.906-12.599 122.488Q-12.599 122.128-12.408 121.832Q-12.216 121.535-11.898 121.367Q-11.580 121.199-11.216 121.199Q-11.009 121.199-10.806 121.259Q-10.603 121.320-10.439 121.421Q-10.056 121.160-9.583 121.160Q-9.345 121.160-9.164 121.291Q-8.982 121.421-8.982 121.648Q-8.982 121.796-9.083 121.902Q-9.185 122.007-9.341 122.007Q-9.474 122.007-9.570 121.929Q-9.665 121.851-9.697 121.726Q-9.841 121.734-10.040 121.824Q-9.837 122.136-9.837 122.488Q-9.837 122.769-9.949 123.001Q-10.060 123.234-10.255 123.412Q-10.451 123.589-10.703 123.687Q-10.955 123.785-11.216 123.785Q-11.603 123.785-11.935 123.597Q-11.947 123.597-11.957 123.669Q-11.966 123.742-11.966 123.777Q-11.966 123.898-11.900 124.005Q-11.833 124.113-11.712 124.152Q-11.697 124.148-11.683 124.146Q-11.669 124.144-11.646 124.144Q-11.630 124.144-11.599 124.152Q-11.568 124.160-11.560 124.160L-10.966 124.160Q-10.185 124.160-9.644 124.402Q-9.103 124.644-9.103 125.335Q-9.103 125.636-9.283 125.863Q-9.462 126.089-9.759 126.234Q-10.056 126.378-10.376 126.445Q-10.697 126.511-10.982 126.511Q-11.373 126.511-11.814 126.388Q-12.255 126.265-12.560 125.998Q-12.865 125.730-12.865 125.335M-12.326 125.328Q-12.326 125.546-12.085 125.687Q-11.845 125.828-11.521 125.894Q-11.197 125.960-10.982 125.960Q-10.767 125.960-10.443 125.894Q-10.119 125.828-9.878 125.687Q-9.638 125.546-9.638 125.328Q-9.638 125.039-9.863 124.900Q-10.087 124.761-10.369 124.728Q-10.650 124.695-10.998 124.695L-11.615 124.695Q-11.798 124.695-11.960 124.775Q-12.123 124.855-12.224 125.001Q-12.326 125.148-12.326 125.328M-11.216 123.230Q-10.915 123.230-10.697 123.011Q-10.478 122.792-10.478 122.488Q-10.478 122.332-10.533 122.201Q-10.587 122.070-10.693 121.964Q-10.798 121.859-10.929 121.804Q-11.060 121.749-11.216 121.749Q-11.521 121.749-11.740 121.968Q-11.958 122.187-11.958 122.488Q-11.958 122.785-11.736 123.007Q-11.513 123.230-11.216 123.230M-8.634 124.449L-8.634 124.359Q-8.576 124.152-8.384 124.128L-7.673 124.128L-7.673 121.800L-8.384 121.800Q-8.580 121.777-8.634 121.558L-8.634 121.472Q-8.576 121.261-8.384 121.238L-7.283 121.238Q-7.083 121.257-7.033 121.472L-7.033 121.800Q-6.771 121.515-6.415 121.357Q-6.060 121.199-5.673 121.199Q-5.380 121.199-5.146 121.333Q-4.912 121.468-4.912 121.734Q-4.912 121.902-5.021 122.019Q-5.130 122.136-5.298 122.136Q-5.451 122.136-5.566 122.025Q-5.681 121.914-5.681 121.757Q-6.056 121.757-6.371 121.958Q-6.685 122.160-6.859 122.494Q-7.033 122.828-7.033 123.207L-7.033 124.128L-6.087 124.128Q-5.880 124.152-5.841 124.359L-5.841 124.449Q-5.880 124.664-6.087 124.687L-8.384 124.687Q-8.576 124.664-8.634 124.449M-4.205 123.574Q-4.205 123.128-3.790 122.871Q-3.376 122.613-2.835 122.513Q-2.294 122.414-1.787 122.406Q-1.787 122.191-1.921 122.039Q-2.056 121.886-2.263 121.810Q-2.470 121.734-2.681 121.734Q-3.025 121.734-3.185 121.757L-3.185 121.816Q-3.185 121.984-3.304 122.099Q-3.423 122.214-3.587 122.214Q-3.763 122.214-3.878 122.091Q-3.994 121.968-3.994 121.800Q-3.994 121.394-3.613 121.285Q-3.232 121.175-2.673 121.175Q-2.404 121.175-2.136 121.253Q-1.869 121.332-1.644 121.482Q-1.419 121.632-1.283 121.853Q-1.146 122.074-1.146 122.351L-1.146 124.070Q-1.146 124.128-0.619 124.128Q-0.423 124.148-0.373 124.359L-0.373 124.449Q-0.423 124.664-0.619 124.687L-0.763 124.687Q-1.107 124.687-1.335 124.640Q-1.564 124.593-1.708 124.406Q-2.169 124.726-2.876 124.726Q-3.212 124.726-3.517 124.585Q-3.822 124.445-4.013 124.183Q-4.205 123.921-4.205 123.574M-3.564 123.582Q-3.564 123.855-3.322 124.011Q-3.080 124.167-2.794 124.167Q-2.576 124.167-2.343 124.109Q-2.111 124.050-1.949 123.912Q-1.787 123.773-1.787 123.550L-1.787 122.960Q-2.068 122.960-2.484 123.017Q-2.900 123.074-3.232 123.212Q-3.564 123.351-3.564 123.582M1.499 124.726Q1.034 124.726 0.668 124.476Q0.303 124.226 0.098 123.822Q-0.107 123.417-0.107 122.960Q-0.107 122.617 0.018 122.296Q0.143 121.976 0.376 121.726Q0.608 121.476 0.913 121.337Q1.217 121.199 1.573 121.199Q2.085 121.199 2.491 121.519L2.491 120.359L2.069 120.359Q1.858 120.335 1.819 120.121L1.819 120.031Q1.858 119.824 2.069 119.800L2.881 119.800Q3.081 119.824 3.131 120.031L3.131 124.128L3.557 124.128Q3.764 124.152 3.803 124.359L3.803 124.449Q3.764 124.664 3.557 124.687L2.741 124.687Q2.542 124.664 2.491 124.449L2.491 124.320Q2.295 124.511 2.034 124.619Q1.772 124.726 1.499 124.726M1.538 124.167Q1.897 124.167 2.149 123.898Q2.401 123.628 2.491 123.253L2.491 122.421Q2.432 122.234 2.305 122.083Q2.178 121.933 2.001 121.845Q1.823 121.757 1.627 121.757Q1.319 121.757 1.069 121.927Q0.819 122.097 0.674 122.382Q0.530 122.667 0.530 122.968Q0.530 123.417 0.815 123.792Q1.100 124.167 1.538 124.167M4.499 124.449L4.499 124.359Q4.549 124.152 4.745 124.128L5.784 124.128L5.784 121.800L4.811 121.800Q4.612 121.777 4.561 121.558L4.561 121.472Q4.612 121.261 4.811 121.238L6.178 121.238Q6.374 121.257 6.424 121.472L6.424 124.128L7.338 124.128Q7.534 124.152 7.585 124.359L7.585 124.449Q7.534 124.664 7.338 124.687L4.745 124.687Q4.549 124.664 4.499 124.449M5.530 120.261L5.530 120.207Q5.530 120.035 5.667 119.914Q5.803 119.792 5.979 119.792Q6.151 119.792 6.288 119.914Q6.424 120.035 6.424 120.207L6.424 120.261Q6.424 120.437 6.288 120.558Q6.151 120.679 5.979 120.679Q5.803 120.679 5.667 120.558Q5.530 120.437 5.530 120.261M11.639 123.199L9.198 123.199Q9.252 123.476 9.450 123.699Q9.647 123.921 9.924 124.044Q10.202 124.167 10.487 124.167Q10.960 124.167 11.182 123.878Q11.190 123.867 11.247 123.761Q11.303 123.656 11.352 123.613Q11.401 123.570 11.495 123.558L11.639 123.558Q11.831 123.578 11.889 123.792L11.889 123.847Q11.823 124.148 11.592 124.345Q11.362 124.542 11.049 124.634Q10.737 124.726 10.432 124.726Q9.948 124.726 9.508 124.498Q9.069 124.269 8.801 123.869Q8.534 123.468 8.534 122.976L8.534 122.917Q8.534 122.449 8.780 122.046Q9.026 121.644 9.434 121.410Q9.842 121.175 10.311 121.175Q10.815 121.175 11.168 121.398Q11.522 121.621 11.706 122.009Q11.889 122.398 11.889 122.902L11.889 122.960Q11.831 123.175 11.639 123.199M9.206 122.648L11.233 122.648Q11.186 122.238 10.948 121.986Q10.710 121.734 10.311 121.734Q9.917 121.734 9.610 121.996Q9.303 122.257 9.206 122.648M12.444 124.449L12.444 124.359Q12.487 124.152 12.694 124.128L13.116 124.128L13.116 121.800L12.694 121.800Q12.487 121.777 12.444 121.558L12.444 121.472Q12.491 121.261 12.694 121.238L13.510 121.238Q13.706 121.261 13.756 121.472L13.756 121.558L13.749 121.582Q13.975 121.402 14.249 121.300Q14.522 121.199 14.815 121.199Q15.163 121.199 15.401 121.339Q15.639 121.480 15.754 121.738Q15.870 121.996 15.870 122.351L15.870 124.128L16.295 124.128Q16.502 124.152 16.542 124.359L16.542 124.449Q16.502 124.664 16.295 124.687L14.901 124.687Q14.706 124.664 14.655 124.449L14.655 124.359Q14.706 124.148 14.901 124.128L15.229 124.128L15.229 122.382Q15.229 122.074 15.139 121.916Q15.049 121.757 14.756 121.757Q14.487 121.757 14.258 121.888Q14.030 122.019 13.893 122.248Q13.756 122.476 13.756 122.742L13.756 124.128L14.182 124.128Q14.389 124.152 14.428 124.359L14.428 124.449Q14.389 124.664 14.182 124.687L12.694 124.687Q12.487 124.664 12.444 124.449M17.819 123.582L17.819 121.800L17.069 121.800Q16.870 121.777 16.819 121.558L16.819 121.472Q16.870 121.261 17.069 121.238L17.819 121.238L17.819 120.488Q17.870 120.281 18.069 120.253L18.213 120.253Q18.409 120.281 18.459 120.488L18.459 121.238L19.819 121.238Q20.010 121.257 20.069 121.472L20.069 121.558Q20.014 121.777 19.819 121.800L18.459 121.800L18.459 123.550Q18.459 124.167 19.034 124.167Q19.284 124.167 19.448 123.982Q19.612 123.796 19.612 123.550Q19.612 123.457 19.684 123.386Q19.756 123.316 19.858 123.304L20.002 123.304Q20.202 123.328 20.252 123.535L20.252 123.582Q20.252 123.906 20.069 124.169Q19.885 124.433 19.592 124.580Q19.299 124.726 18.979 124.726Q18.467 124.726 18.143 124.416Q17.819 124.105 17.819 123.582\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M3.866 134.646h60.241\"\u002F>\u003Cpath stroke=\"none\" d=\"m1.266 134.646 4.16 2.08-1.56-2.08 1.56-2.08M66.707 134.646l-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"none\" font-size=\"8\">\u003Cg transform=\"translate(29.541 20.47)\">\u003Cpath d=\"M-46.518 124.488L-46.518 123.574Q-46.491 123.367-46.280 123.343L-46.112 123.343Q-45.948 123.367-45.889 123.527Q-45.686 124.167-44.959 124.167Q-44.752 124.167-44.524 124.132Q-44.295 124.097-44.127 123.982Q-43.959 123.867-43.959 123.664Q-43.959 123.453-44.182 123.339Q-44.405 123.226-44.678 123.183L-45.377 123.070Q-46.518 122.859-46.518 122.136Q-46.518 121.847-46.374 121.658Q-46.229 121.468-45.989 121.361Q-45.749 121.253-45.493 121.214Q-45.237 121.175-44.959 121.175Q-44.709 121.175-44.516 121.205Q-44.323 121.234-44.159 121.312Q-44.081 121.195-43.952 121.175L-43.874 121.175Q-43.776 121.187-43.713 121.249Q-43.651 121.312-43.639 121.406L-43.639 122.113Q-43.651 122.207-43.713 122.273Q-43.776 122.339-43.874 122.351L-44.041 122.351Q-44.135 122.339-44.202 122.273Q-44.268 122.207-44.280 122.113Q-44.280 121.734-44.975 121.734Q-45.323 121.734-45.641 121.816Q-45.959 121.898-45.959 122.144Q-45.959 122.410-45.288 122.519L-44.584 122.640Q-44.100 122.722-43.750 122.970Q-43.401 123.218-43.401 123.664Q-43.401 124.054-43.637 124.296Q-43.874 124.539-44.223 124.632Q-44.573 124.726-44.959 124.726Q-45.538 124.726-45.936 124.472Q-46.006 124.597-46.055 124.654Q-46.104 124.710-46.209 124.726L-46.280 124.726Q-46.495 124.703-46.518 124.488M-41.659 123.582L-41.659 121.800L-42.409 121.800Q-42.608 121.777-42.659 121.558L-42.659 121.472Q-42.608 121.261-42.409 121.238L-41.659 121.238L-41.659 120.488Q-41.608 120.281-41.409 120.253L-41.264 120.253Q-41.069 120.281-41.018 120.488L-41.018 121.238L-39.659 121.238Q-39.467 121.257-39.409 121.472L-39.409 121.558Q-39.463 121.777-39.659 121.800L-41.018 121.800L-41.018 123.550Q-41.018 124.167-40.444 124.167Q-40.194 124.167-40.030 123.982Q-39.866 123.796-39.866 123.550Q-39.866 123.457-39.793 123.386Q-39.721 123.316-39.620 123.304L-39.475 123.304Q-39.276 123.328-39.225 123.535L-39.225 123.582Q-39.225 123.906-39.409 124.169Q-39.592 124.433-39.885 124.580Q-40.178 124.726-40.499 124.726Q-41.010 124.726-41.334 124.416Q-41.659 124.105-41.659 123.582M-35.100 123.199L-37.541 123.199Q-37.487 123.476-37.290 123.699Q-37.092 123.921-36.815 124.044Q-36.538 124.167-36.252 124.167Q-35.780 124.167-35.557 123.878Q-35.549 123.867-35.493 123.761Q-35.436 123.656-35.387 123.613Q-35.338 123.570-35.245 123.558L-35.100 123.558Q-34.909 123.578-34.850 123.792L-34.850 123.847Q-34.916 124.148-35.147 124.345Q-35.377 124.542-35.690 124.634Q-36.002 124.726-36.307 124.726Q-36.791 124.726-37.231 124.498Q-37.670 124.269-37.938 123.869Q-38.206 123.468-38.206 122.976L-38.206 122.917Q-38.206 122.449-37.959 122.046Q-37.713 121.644-37.305 121.410Q-36.897 121.175-36.428 121.175Q-35.924 121.175-35.571 121.398Q-35.217 121.621-35.034 122.009Q-34.850 122.398-34.850 122.902L-34.850 122.960Q-34.909 123.175-35.100 123.199M-37.534 122.648L-35.506 122.648Q-35.553 122.238-35.791 121.986Q-36.030 121.734-36.428 121.734Q-36.823 121.734-37.129 121.996Q-37.436 122.257-37.534 122.648M-34.295 126.230L-34.295 126.144Q-34.252 125.925-34.045 125.902L-33.624 125.902L-33.624 121.800L-34.045 121.800Q-34.252 121.777-34.295 121.558L-34.295 121.472Q-34.249 121.261-34.045 121.238L-33.229 121.238Q-33.034 121.257-32.983 121.472L-32.983 121.542Q-32.772 121.374-32.508 121.287Q-32.245 121.199-31.975 121.199Q-31.635 121.199-31.338 121.343Q-31.041 121.488-30.827 121.740Q-30.612 121.992-30.497 122.304Q-30.381 122.617-30.381 122.960Q-30.381 123.425-30.608 123.835Q-30.834 124.246-31.221 124.486Q-31.608 124.726-32.077 124.726Q-32.596 124.726-32.983 124.359L-32.983 125.902L-32.557 125.902Q-32.350 125.925-32.311 126.144L-32.311 126.230Q-32.350 126.441-32.557 126.464L-34.045 126.464Q-34.249 126.441-34.295 126.230M-32.120 124.167Q-31.811 124.167-31.561 123.998Q-31.311 123.828-31.166 123.546Q-31.022 123.265-31.022 122.960Q-31.022 122.671-31.147 122.392Q-31.272 122.113-31.504 121.935Q-31.737 121.757-32.038 121.757Q-32.358 121.757-32.620 121.943Q-32.881 122.128-32.983 122.429L-32.983 123.281Q-32.893 123.648-32.672 123.908Q-32.452 124.167-32.120 124.167\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(29.541 20.47)\">\u003Cpath d=\"M-26.477 124.207Q-26.262 124.511-25.598 124.511Q-25.168 124.511-24.811 124.310Q-24.454 124.109-24.454 123.710Q-24.454 123.507-24.614 123.384Q-24.774 123.261-24.981 123.222L-25.446 123.136Q-25.762 123.066-25.977 122.859Q-26.192 122.652-26.192 122.343Q-26.192 121.980-25.987 121.708Q-25.782 121.437-25.452 121.298Q-25.122 121.160-24.766 121.160Q-24.379 121.160-24.069 121.328Q-23.758 121.496-23.758 121.847Q-23.758 122.039-23.864 122.179Q-23.969 122.320-24.157 122.320Q-24.266 122.320-24.348 122.248Q-24.430 122.175-24.430 122.062Q-24.430 121.917-24.332 121.808Q-24.235 121.699-24.094 121.679Q-24.184 121.535-24.375 121.474Q-24.567 121.414-24.782 121.414Q-25.114 121.414-25.387 121.585Q-25.661 121.757-25.661 122.070Q-25.661 122.226-25.549 122.320Q-25.438 122.414-25.262 122.457L-24.805 122.542Q-24.430 122.613-24.178 122.845Q-23.926 123.078-23.926 123.441Q-23.926 123.714-24.071 123.982Q-24.215 124.249-24.454 124.429Q-24.930 124.765-25.614 124.765Q-25.891 124.765-26.172 124.693Q-26.454 124.621-26.645 124.447Q-26.836 124.273-26.836 123.992Q-26.836 123.769-26.707 123.605Q-26.579 123.441-26.360 123.441Q-26.215 123.441-26.127 123.523Q-26.040 123.605-26.040 123.742Q-26.040 123.921-26.170 124.064Q-26.301 124.207-26.477 124.207\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(29.541 20.47)\">\u003Cpath d=\"M-17.393 125.328Q-17.924 125.019-18.324 124.531Q-18.725 124.042-18.936 123.449Q-19.147 122.855-19.147 122.238Q-19.147 121.621-18.938 121.033Q-18.729 120.445-18.332 119.964Q-17.936 119.484-17.401 119.160Q-17.330 119.136-17.299 119.136L-17.209 119.136Q-17.111 119.148-17.045 119.214Q-16.979 119.281-16.979 119.382Q-16.979 119.527-17.080 119.589Q-17.529 119.878-17.856 120.294Q-18.182 120.710-18.344 121.203Q-18.506 121.695-18.506 122.238Q-18.506 122.644-18.410 123.031Q-18.315 123.417-18.137 123.753Q-17.959 124.089-17.699 124.373Q-17.440 124.656-17.092 124.886Q-16.979 124.964-16.979 125.101Q-16.979 125.199-17.045 125.269Q-17.111 125.339-17.209 125.351L-17.299 125.351Q-17.358 125.351-17.393 125.328M-15.701 122.960Q-15.701 122.480-15.457 122.066Q-15.213 121.652-14.797 121.414Q-14.381 121.175-13.901 121.175Q-13.346 121.175-12.967 121.285Q-12.588 121.394-12.588 121.800Q-12.588 121.968-12.701 122.091Q-12.815 122.214-12.986 122.214Q-13.158 122.214-13.277 122.099Q-13.397 121.984-13.397 121.816L-13.397 121.757Q-13.557 121.734-13.893 121.734Q-14.221 121.734-14.488 121.904Q-14.756 122.074-14.908 122.357Q-15.061 122.640-15.061 122.960Q-15.061 123.281-14.889 123.562Q-14.717 123.843-14.432 124.005Q-14.147 124.167-13.819 124.167Q-13.506 124.167-13.379 124.064Q-13.252 123.960-13.135 123.771Q-13.018 123.582-12.901 123.566L-12.733 123.566Q-12.627 123.578-12.563 123.646Q-12.498 123.714-12.498 123.816Q-12.498 123.863-12.518 123.902Q-12.627 124.195-12.830 124.374Q-13.033 124.554-13.309 124.640Q-13.584 124.726-13.901 124.726Q-14.385 124.726-14.801 124.488Q-15.217 124.249-15.459 123.847Q-15.701 123.445-15.701 122.960M-11.944 124.449L-11.944 124.359Q-11.901 124.152-11.694 124.128L-11.272 124.128L-11.272 120.359L-11.694 120.359Q-11.901 120.335-11.944 120.121L-11.944 120.031Q-11.901 119.824-11.694 119.800L-10.877 119.800Q-10.682 119.824-10.631 120.031L-10.631 121.582Q-10.170 121.199-9.572 121.199Q-9.225 121.199-8.986 121.339Q-8.748 121.480-8.633 121.738Q-8.518 121.996-8.518 122.351L-8.518 124.128L-8.092 124.128Q-7.885 124.152-7.846 124.359L-7.846 124.449Q-7.885 124.664-8.092 124.687L-9.486 124.687Q-9.682 124.664-9.733 124.449L-9.733 124.359Q-9.682 124.148-9.486 124.128L-9.158 124.128L-9.158 122.382Q-9.158 122.074-9.248 121.916Q-9.338 121.757-9.631 121.757Q-9.901 121.757-10.129 121.888Q-10.358 122.019-10.494 122.248Q-10.631 122.476-10.631 122.742L-10.631 124.128L-10.205 124.128Q-9.998 124.152-9.959 124.359L-9.959 124.449Q-9.998 124.664-10.205 124.687L-11.694 124.687Q-11.901 124.664-11.944 124.449M-7.361 123.574Q-7.361 123.128-6.947 122.871Q-6.533 122.613-5.992 122.513Q-5.451 122.414-4.944 122.406Q-4.944 122.191-5.078 122.039Q-5.213 121.886-5.420 121.810Q-5.627 121.734-5.838 121.734Q-6.182 121.734-6.342 121.757L-6.342 121.816Q-6.342 121.984-6.461 122.099Q-6.580 122.214-6.744 122.214Q-6.920 122.214-7.035 122.091Q-7.151 121.968-7.151 121.800Q-7.151 121.394-6.770 121.285Q-6.389 121.175-5.830 121.175Q-5.561 121.175-5.293 121.253Q-5.026 121.332-4.801 121.482Q-4.576 121.632-4.440 121.853Q-4.303 122.074-4.303 122.351L-4.303 124.070Q-4.303 124.128-3.776 124.128Q-3.580 124.148-3.529 124.359L-3.529 124.449Q-3.580 124.664-3.776 124.687L-3.920 124.687Q-4.264 124.687-4.492 124.640Q-4.721 124.593-4.865 124.406Q-5.326 124.726-6.033 124.726Q-6.369 124.726-6.674 124.585Q-6.979 124.445-7.170 124.183Q-7.361 123.921-7.361 123.574M-6.721 123.582Q-6.721 123.855-6.479 124.011Q-6.236 124.167-5.951 124.167Q-5.733 124.167-5.500 124.109Q-5.268 124.050-5.106 123.912Q-4.944 123.773-4.944 123.550L-4.944 122.960Q-5.225 122.960-5.641 123.017Q-6.057 123.074-6.389 123.212Q-6.721 123.351-6.721 123.582M-3.451 124.449L-3.451 124.359Q-3.408 124.152-3.201 124.128L-2.779 124.128L-2.779 121.800L-3.201 121.800Q-3.408 121.777-3.451 121.558L-3.451 121.472Q-3.404 121.261-3.201 121.238L-2.385 121.238Q-2.190 121.261-2.139 121.472L-2.139 121.558L-2.147 121.582Q-1.920 121.402-1.647 121.300Q-1.373 121.199-1.080 121.199Q-0.733 121.199-0.494 121.339Q-0.256 121.480-0.141 121.738Q-0.026 121.996-0.026 122.351L-0.026 124.128L0.400 124.128Q0.607 124.152 0.646 124.359L0.646 124.449Q0.607 124.664 0.400 124.687L-0.994 124.687Q-1.190 124.664-1.240 124.449L-1.240 124.359Q-1.190 124.148-0.994 124.128L-0.666 124.128L-0.666 122.382Q-0.666 122.074-0.756 121.916Q-0.846 121.757-1.139 121.757Q-1.408 121.757-1.637 121.888Q-1.865 122.019-2.002 122.248Q-2.139 122.476-2.139 122.742L-2.139 124.128L-1.713 124.128Q-1.506 124.152-1.467 124.359L-1.467 124.449Q-1.506 124.664-1.713 124.687L-3.201 124.687Q-3.408 124.664-3.451 124.449M0.963 125.335Q0.963 125.035 1.111 124.773Q1.260 124.511 1.510 124.351Q1.326 124.097 1.326 123.777Q1.326 123.492 1.478 123.230Q1.228 122.906 1.228 122.488Q1.228 122.128 1.420 121.832Q1.611 121.535 1.930 121.367Q2.248 121.199 2.611 121.199Q2.818 121.199 3.021 121.259Q3.224 121.320 3.389 121.421Q3.771 121.160 4.244 121.160Q4.482 121.160 4.664 121.291Q4.846 121.421 4.846 121.648Q4.846 121.796 4.744 121.902Q4.642 122.007 4.486 122.007Q4.353 122.007 4.258 121.929Q4.162 121.851 4.131 121.726Q3.986 121.734 3.787 121.824Q3.990 122.136 3.990 122.488Q3.990 122.769 3.879 123.001Q3.767 123.234 3.572 123.412Q3.377 123.589 3.125 123.687Q2.873 123.785 2.611 123.785Q2.224 123.785 1.892 123.597Q1.881 123.597 1.871 123.669Q1.861 123.742 1.861 123.777Q1.861 123.898 1.928 124.005Q1.994 124.113 2.115 124.152Q2.131 124.148 2.144 124.146Q2.158 124.144 2.181 124.144Q2.197 124.144 2.228 124.152Q2.260 124.160 2.267 124.160L2.861 124.160Q3.642 124.160 4.183 124.402Q4.724 124.644 4.724 125.335Q4.724 125.636 4.545 125.863Q4.365 126.089 4.068 126.234Q3.771 126.378 3.451 126.445Q3.131 126.511 2.846 126.511Q2.455 126.511 2.014 126.388Q1.572 126.265 1.267 125.998Q0.963 125.730 0.963 125.335M1.502 125.328Q1.502 125.546 1.742 125.687Q1.982 125.828 2.306 125.894Q2.631 125.960 2.846 125.960Q3.060 125.960 3.385 125.894Q3.709 125.828 3.949 125.687Q4.189 125.546 4.189 125.328Q4.189 125.039 3.965 124.900Q3.740 124.761 3.459 124.728Q3.178 124.695 2.830 124.695L2.213 124.695Q2.029 124.695 1.867 124.775Q1.705 124.855 1.603 125.001Q1.502 125.148 1.502 125.328M2.611 123.230Q2.912 123.230 3.131 123.011Q3.349 122.792 3.349 122.488Q3.349 122.332 3.295 122.201Q3.240 122.070 3.135 121.964Q3.029 121.859 2.898 121.804Q2.767 121.749 2.611 121.749Q2.306 121.749 2.088 121.968Q1.869 122.187 1.869 122.488Q1.869 122.785 2.092 123.007Q2.314 123.230 2.611 123.230M8.482 123.199L6.041 123.199Q6.096 123.476 6.293 123.699Q6.490 123.921 6.767 124.044Q7.045 124.167 7.330 124.167Q7.803 124.167 8.025 123.878Q8.033 123.867 8.090 123.761Q8.146 123.656 8.195 123.613Q8.244 123.570 8.338 123.558L8.482 123.558Q8.674 123.578 8.732 123.792L8.732 123.847Q8.666 124.148 8.435 124.345Q8.205 124.542 7.892 124.634Q7.580 124.726 7.275 124.726Q6.791 124.726 6.351 124.498Q5.912 124.269 5.644 123.869Q5.377 123.468 5.377 122.976L5.377 122.917Q5.377 122.449 5.623 122.046Q5.869 121.644 6.277 121.410Q6.685 121.175 7.154 121.175Q7.658 121.175 8.012 121.398Q8.365 121.621 8.549 122.009Q8.732 122.398 8.732 122.902L8.732 122.960Q8.674 123.175 8.482 123.199M6.049 122.648L8.076 122.648Q8.029 122.238 7.791 121.986Q7.553 121.734 7.154 121.734Q6.760 121.734 6.453 121.996Q6.146 122.257 6.049 122.648\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(29.541 20.47)\">\u003Cpath d=\"M14.111 124.449L14.111 124.359Q14.162 124.152 14.357 124.128L15.396 124.128L15.396 121.800L14.424 121.800Q14.224 121.777 14.174 121.558L14.174 121.472Q14.224 121.261 14.424 121.238L15.791 121.238Q15.986 121.257 16.037 121.472L16.037 124.128L16.951 124.128Q17.146 124.152 17.197 124.359L17.197 124.449Q17.146 124.664 16.951 124.687L14.357 124.687Q14.162 124.664 14.111 124.449M15.142 120.261L15.142 120.207Q15.142 120.035 15.279 119.914Q15.416 119.792 15.592 119.792Q15.764 119.792 15.900 119.914Q16.037 120.035 16.037 120.207L16.037 120.261Q16.037 120.437 15.900 120.558Q15.764 120.679 15.592 120.679Q15.416 120.679 15.279 120.558Q15.142 120.437 15.142 120.261M17.810 124.449L17.810 124.359Q17.853 124.152 18.060 124.128L18.482 124.128L18.482 121.800L18.060 121.800Q17.853 121.777 17.810 121.558L17.810 121.472Q17.857 121.261 18.060 121.238L18.877 121.238Q19.072 121.261 19.123 121.472L19.123 121.558L19.115 121.582Q19.342 121.402 19.615 121.300Q19.889 121.199 20.181 121.199Q20.529 121.199 20.767 121.339Q21.006 121.480 21.121 121.738Q21.236 121.996 21.236 122.351L21.236 124.128L21.662 124.128Q21.869 124.152 21.908 124.359L21.908 124.449Q21.869 124.664 21.662 124.687L20.267 124.687Q20.072 124.664 20.021 124.449L20.021 124.359Q20.072 124.148 20.267 124.128L20.596 124.128L20.596 122.382Q20.596 122.074 20.506 121.916Q20.416 121.757 20.123 121.757Q19.853 121.757 19.625 121.888Q19.396 122.019 19.260 122.248Q19.123 122.476 19.123 122.742L19.123 124.128L19.549 124.128Q19.756 124.152 19.795 124.359L19.795 124.449Q19.756 124.664 19.549 124.687L18.060 124.687Q17.853 124.664 17.810 124.449\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(29.541 20.47)\">\u003Cpath d=\"M27.253 124.464L26.788 121.800L26.628 121.800Q26.421 121.777 26.382 121.558L26.382 121.472Q26.421 121.261 26.628 121.238L27.796 121.238Q28.007 121.261 28.046 121.472L28.046 121.558Q28.007 121.777 27.796 121.800L27.323 121.800Q27.436 122.445 27.515 122.890Q27.593 123.335 27.643 123.654Q27.694 123.972 27.694 124.046Q27.702 123.874 27.987 122.878Q28.026 122.765 28.124 122.691Q28.222 122.617 28.343 122.617L28.421 122.617Q28.542 122.617 28.640 122.691Q28.737 122.765 28.772 122.878Q28.882 123.261 28.964 123.585Q29.046 123.910 29.046 124.046Q29.058 123.757 29.405 121.800L28.933 121.800Q28.725 121.777 28.686 121.558L28.686 121.472Q28.725 121.261 28.933 121.238L30.108 121.238Q30.300 121.261 30.358 121.472L30.358 121.558Q30.304 121.777 30.108 121.800L29.940 121.800L29.475 124.464Q29.452 124.582 29.364 124.654Q29.276 124.726 29.155 124.726L29.015 124.726Q28.893 124.726 28.798 124.654Q28.702 124.582 28.659 124.464Q28.612 124.285 28.540 124.029Q28.468 123.773 28.425 123.564Q28.382 123.355 28.382 123.253Q28.374 123.535 28.093 124.464Q28.065 124.574 27.968 124.650Q27.870 124.726 27.749 124.726L27.573 124.726Q27.452 124.726 27.364 124.654Q27.276 124.582 27.253 124.464M34.003 123.199L31.561 123.199Q31.616 123.476 31.813 123.699Q32.011 123.921 32.288 124.044Q32.565 124.167 32.850 124.167Q33.323 124.167 33.546 123.878Q33.554 123.867 33.610 123.761Q33.667 123.656 33.716 123.613Q33.765 123.570 33.858 123.558L34.003 123.558Q34.194 123.578 34.253 123.792L34.253 123.847Q34.186 124.148 33.956 124.345Q33.725 124.542 33.413 124.634Q33.100 124.726 32.796 124.726Q32.311 124.726 31.872 124.498Q31.433 124.269 31.165 123.869Q30.897 123.468 30.897 122.976L30.897 122.917Q30.897 122.449 31.143 122.046Q31.390 121.644 31.798 121.410Q32.206 121.175 32.675 121.175Q33.179 121.175 33.532 121.398Q33.886 121.621 34.069 122.009Q34.253 122.398 34.253 122.902L34.253 122.960Q34.194 123.175 34.003 123.199M31.569 122.648L33.597 122.648Q33.550 122.238 33.311 121.986Q33.073 121.734 32.675 121.734Q32.280 121.734 31.974 121.996Q31.667 122.257 31.569 122.648M35.354 124.449L35.354 124.359Q35.405 124.152 35.600 124.128L36.640 124.128L36.640 121.800L35.667 121.800Q35.468 121.777 35.417 121.558L35.417 121.472Q35.468 121.261 35.667 121.238L37.034 121.238Q37.229 121.257 37.280 121.472L37.280 124.128L38.194 124.128Q38.390 124.152 38.440 124.359L38.440 124.449Q38.390 124.664 38.194 124.687L35.600 124.687Q35.405 124.664 35.354 124.449M36.386 120.261L36.386 120.207Q36.386 120.035 36.522 119.914Q36.659 119.792 36.835 119.792Q37.007 119.792 37.143 119.914Q37.280 120.035 37.280 120.207L37.280 120.261Q37.280 120.437 37.143 120.558Q37.007 120.679 36.835 120.679Q36.659 120.679 36.522 120.558Q36.386 120.437 36.386 120.261M39.222 125.335Q39.222 125.035 39.370 124.773Q39.518 124.511 39.768 124.351Q39.585 124.097 39.585 123.777Q39.585 123.492 39.737 123.230Q39.487 122.906 39.487 122.488Q39.487 122.128 39.679 121.832Q39.870 121.535 40.188 121.367Q40.507 121.199 40.870 121.199Q41.077 121.199 41.280 121.259Q41.483 121.320 41.647 121.421Q42.030 121.160 42.503 121.160Q42.741 121.160 42.923 121.291Q43.104 121.421 43.104 121.648Q43.104 121.796 43.003 121.902Q42.901 122.007 42.745 122.007Q42.612 122.007 42.516 121.929Q42.421 121.851 42.390 121.726Q42.245 121.734 42.046 121.824Q42.249 122.136 42.249 122.488Q42.249 122.769 42.138 123.001Q42.026 123.234 41.831 123.412Q41.636 123.589 41.384 123.687Q41.132 123.785 40.870 123.785Q40.483 123.785 40.151 123.597Q40.140 123.597 40.130 123.669Q40.120 123.742 40.120 123.777Q40.120 123.898 40.186 124.005Q40.253 124.113 40.374 124.152Q40.390 124.148 40.403 124.146Q40.417 124.144 40.440 124.144Q40.456 124.144 40.487 124.152Q40.518 124.160 40.526 124.160L41.120 124.160Q41.901 124.160 42.442 124.402Q42.983 124.644 42.983 125.335Q42.983 125.636 42.804 125.863Q42.624 126.089 42.327 126.234Q42.030 126.378 41.710 126.445Q41.390 126.511 41.104 126.511Q40.714 126.511 40.272 126.388Q39.831 126.265 39.526 125.998Q39.222 125.730 39.222 125.335M39.761 125.328Q39.761 125.546 40.001 125.687Q40.241 125.828 40.565 125.894Q40.890 125.960 41.104 125.960Q41.319 125.960 41.643 125.894Q41.968 125.828 42.208 125.687Q42.448 125.546 42.448 125.328Q42.448 125.039 42.224 124.900Q41.999 124.761 41.718 124.728Q41.436 124.695 41.089 124.695L40.472 124.695Q40.288 124.695 40.126 124.775Q39.964 124.855 39.862 125.001Q39.761 125.148 39.761 125.328M40.870 123.230Q41.171 123.230 41.390 123.011Q41.608 122.792 41.608 122.488Q41.608 122.332 41.554 122.201Q41.499 122.070 41.393 121.964Q41.288 121.859 41.157 121.804Q41.026 121.749 40.870 121.749Q40.565 121.749 40.347 121.968Q40.128 122.187 40.128 122.488Q40.128 122.785 40.350 123.007Q40.573 123.230 40.870 123.230M43.300 124.449L43.300 124.359Q43.343 124.152 43.550 124.128L43.972 124.128L43.972 120.359L43.550 120.359Q43.343 120.335 43.300 120.121L43.300 120.031Q43.343 119.824 43.550 119.800L44.366 119.800Q44.561 119.824 44.612 120.031L44.612 121.582Q45.073 121.199 45.671 121.199Q46.018 121.199 46.257 121.339Q46.495 121.480 46.610 121.738Q46.725 121.996 46.725 122.351L46.725 124.128L47.151 124.128Q47.358 124.152 47.397 124.359L47.397 124.449Q47.358 124.664 47.151 124.687L45.757 124.687Q45.561 124.664 45.511 124.449L45.511 124.359Q45.561 124.148 45.757 124.128L46.085 124.128L46.085 122.382Q46.085 122.074 45.995 121.916Q45.905 121.757 45.612 121.757Q45.343 121.757 45.114 121.888Q44.886 122.019 44.749 122.248Q44.612 122.476 44.612 122.742L44.612 124.128L45.038 124.128Q45.245 124.152 45.284 124.359L45.284 124.449Q45.245 124.664 45.038 124.687L43.550 124.687Q43.343 124.664 43.300 124.449M48.675 123.582L48.675 121.800L47.925 121.800Q47.725 121.777 47.675 121.558L47.675 121.472Q47.725 121.261 47.925 121.238L48.675 121.238L48.675 120.488Q48.725 120.281 48.925 120.253L49.069 120.253Q49.265 120.281 49.315 120.488L49.315 121.238L50.675 121.238Q50.866 121.257 50.925 121.472L50.925 121.558Q50.870 121.777 50.675 121.800L49.315 121.800L49.315 123.550Q49.315 124.167 49.890 124.167Q50.140 124.167 50.304 123.982Q50.468 123.796 50.468 123.550Q50.468 123.457 50.540 123.386Q50.612 123.316 50.714 123.304L50.858 123.304Q51.058 123.328 51.108 123.535L51.108 123.582Q51.108 123.906 50.925 124.169Q50.741 124.433 50.448 124.580Q50.155 124.726 49.835 124.726Q49.323 124.726 48.999 124.416Q48.675 124.105 48.675 123.582M52.761 125.351L52.675 125.351Q52.569 125.339 52.501 125.267Q52.433 125.195 52.433 125.101Q52.433 124.960 52.538 124.894Q52.874 124.679 53.143 124.390Q53.413 124.101 53.597 123.753Q53.780 123.406 53.870 123.027Q53.960 122.648 53.960 122.238Q53.960 121.699 53.796 121.207Q53.632 120.714 53.313 120.300Q52.995 119.886 52.554 119.597Q52.433 119.515 52.433 119.382Q52.433 119.285 52.501 119.216Q52.569 119.148 52.675 119.136L52.761 119.136Q52.815 119.136 52.850 119.160Q53.237 119.382 53.577 119.730Q53.917 120.078 54.138 120.472Q54.358 120.867 54.479 121.312Q54.600 121.757 54.600 122.238Q54.600 122.722 54.479 123.173Q54.358 123.624 54.134 124.021Q53.909 124.417 53.585 124.751Q53.261 125.085 52.858 125.328Q52.788 125.351 52.761 125.351\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(151.487 -75.267)\">\u003Cpath d=\"M-46.518 124.488L-46.518 123.574Q-46.491 123.367-46.280 123.343L-46.112 123.343Q-45.948 123.367-45.889 123.527Q-45.686 124.167-44.959 124.167Q-44.752 124.167-44.524 124.132Q-44.295 124.097-44.127 123.982Q-43.959 123.867-43.959 123.664Q-43.959 123.453-44.182 123.339Q-44.405 123.226-44.678 123.183L-45.377 123.070Q-46.518 122.859-46.518 122.136Q-46.518 121.847-46.374 121.658Q-46.229 121.468-45.989 121.361Q-45.749 121.253-45.493 121.214Q-45.237 121.175-44.959 121.175Q-44.709 121.175-44.516 121.205Q-44.323 121.234-44.159 121.312Q-44.081 121.195-43.952 121.175L-43.874 121.175Q-43.776 121.187-43.713 121.249Q-43.651 121.312-43.639 121.406L-43.639 122.113Q-43.651 122.207-43.713 122.273Q-43.776 122.339-43.874 122.351L-44.041 122.351Q-44.135 122.339-44.202 122.273Q-44.268 122.207-44.280 122.113Q-44.280 121.734-44.975 121.734Q-45.323 121.734-45.641 121.816Q-45.959 121.898-45.959 122.144Q-45.959 122.410-45.288 122.519L-44.584 122.640Q-44.100 122.722-43.750 122.970Q-43.401 123.218-43.401 123.664Q-43.401 124.054-43.637 124.296Q-43.874 124.539-44.223 124.632Q-44.573 124.726-44.959 124.726Q-45.538 124.726-45.936 124.472Q-46.006 124.597-46.055 124.654Q-46.104 124.710-46.209 124.726L-46.280 124.726Q-46.495 124.703-46.518 124.488M-42.409 124.449L-42.409 124.359Q-42.358 124.152-42.163 124.128L-41.057 124.128L-41.057 120.359L-42.163 120.359Q-42.358 120.335-42.409 120.121L-42.409 120.031Q-42.358 119.824-42.163 119.800L-40.666 119.800Q-40.475 119.824-40.416 120.031L-40.416 124.128L-39.315 124.128Q-39.116 124.152-39.065 124.359L-39.065 124.449Q-39.116 124.664-39.315 124.687L-42.163 124.687Q-42.358 124.664-42.409 124.449M-36.491 124.726Q-36.963 124.726-37.348 124.482Q-37.733 124.238-37.956 123.828Q-38.178 123.417-38.178 122.960Q-38.178 122.617-38.053 122.294Q-37.928 121.972-37.698 121.718Q-37.467 121.464-37.161 121.320Q-36.854 121.175-36.491 121.175Q-36.127 121.175-35.815 121.322Q-35.502 121.468-35.280 121.714Q-35.057 121.960-34.930 122.281Q-34.803 122.601-34.803 122.960Q-34.803 123.417-35.028 123.830Q-35.252 124.242-35.637 124.484Q-36.022 124.726-36.491 124.726M-36.491 124.167Q-36.026 124.167-35.735 123.773Q-35.444 123.378-35.444 122.894Q-35.444 122.601-35.579 122.333Q-35.713 122.066-35.954 121.900Q-36.194 121.734-36.491 121.734Q-36.795 121.734-37.034 121.900Q-37.272 122.066-37.407 122.333Q-37.541 122.601-37.541 122.894Q-37.541 123.374-37.249 123.771Q-36.956 124.167-36.491 124.167M-34.295 126.230L-34.295 126.144Q-34.252 125.925-34.045 125.902L-33.624 125.902L-33.624 121.800L-34.045 121.800Q-34.252 121.777-34.295 121.558L-34.295 121.472Q-34.249 121.261-34.045 121.238L-33.229 121.238Q-33.034 121.257-32.983 121.472L-32.983 121.542Q-32.772 121.374-32.508 121.287Q-32.245 121.199-31.975 121.199Q-31.635 121.199-31.338 121.343Q-31.041 121.488-30.827 121.740Q-30.612 121.992-30.497 122.304Q-30.381 122.617-30.381 122.960Q-30.381 123.425-30.608 123.835Q-30.834 124.246-31.221 124.486Q-31.608 124.726-32.077 124.726Q-32.596 124.726-32.983 124.359L-32.983 125.902L-32.557 125.902Q-32.350 125.925-32.311 126.144L-32.311 126.230Q-32.350 126.441-32.557 126.464L-34.045 126.464Q-34.249 126.441-34.295 126.230M-32.120 124.167Q-31.811 124.167-31.561 123.998Q-31.311 123.828-31.166 123.546Q-31.022 123.265-31.022 122.960Q-31.022 122.671-31.147 122.392Q-31.272 122.113-31.504 121.935Q-31.737 121.757-32.038 121.757Q-32.358 121.757-32.620 121.943Q-32.881 122.128-32.983 122.429L-32.983 123.281Q-32.893 123.648-32.672 123.908Q-32.452 124.167-32.120 124.167M-26.608 123.199L-29.049 123.199Q-28.995 123.476-28.797 123.699Q-28.600 123.921-28.323 124.044Q-28.045 124.167-27.760 124.167Q-27.288 124.167-27.065 123.878Q-27.057 123.867-27 123.761Q-26.944 123.656-26.895 123.613Q-26.846 123.570-26.752 123.558L-26.608 123.558Q-26.416 123.578-26.358 123.792L-26.358 123.847Q-26.424 124.148-26.655 124.345Q-26.885 124.542-27.198 124.634Q-27.510 124.726-27.815 124.726Q-28.299 124.726-28.739 124.498Q-29.178 124.269-29.446 123.869Q-29.713 123.468-29.713 122.976L-29.713 122.917Q-29.713 122.449-29.467 122.046Q-29.221 121.644-28.813 121.410Q-28.405 121.175-27.936 121.175Q-27.432 121.175-27.079 121.398Q-26.725 121.621-26.541 122.009Q-26.358 122.398-26.358 122.902L-26.358 122.960Q-26.416 123.175-26.608 123.199M-29.041 122.648L-27.014 122.648Q-27.061 122.238-27.299 121.986Q-27.538 121.734-27.936 121.734Q-28.331 121.734-28.637 121.996Q-28.944 122.257-29.041 122.648\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(151.487 -75.267)\">\u003Cpath d=\"M-21.044 122.960Q-21.044 122.480-20.800 122.066Q-20.556 121.652-20.140 121.414Q-19.724 121.175-19.244 121.175Q-18.689 121.175-18.310 121.285Q-17.931 121.394-17.931 121.800Q-17.931 121.968-18.044 122.091Q-18.158 122.214-18.330 122.214Q-18.501 122.214-18.621 122.099Q-18.740 121.984-18.740 121.816L-18.740 121.757Q-18.900 121.734-19.236 121.734Q-19.564 121.734-19.832 121.904Q-20.099 122.074-20.251 122.357Q-20.404 122.640-20.404 122.960Q-20.404 123.281-20.232 123.562Q-20.060 123.843-19.775 124.005Q-19.490 124.167-19.162 124.167Q-18.849 124.167-18.722 124.064Q-18.595 123.960-18.478 123.771Q-18.361 123.582-18.244 123.566L-18.076 123.566Q-17.970 123.578-17.906 123.646Q-17.841 123.714-17.841 123.816Q-17.841 123.863-17.861 123.902Q-17.970 124.195-18.173 124.374Q-18.376 124.554-18.652 124.640Q-18.927 124.726-19.244 124.726Q-19.728 124.726-20.144 124.488Q-20.560 124.249-20.802 123.847Q-21.044 123.445-21.044 122.960M-17.287 124.449L-17.287 124.359Q-17.244 124.152-17.037 124.128L-16.615 124.128L-16.615 120.359L-17.037 120.359Q-17.244 120.335-17.287 120.121L-17.287 120.031Q-17.244 119.824-17.037 119.800L-16.220 119.800Q-16.025 119.824-15.974 120.031L-15.974 121.582Q-15.513 121.199-14.916 121.199Q-14.568 121.199-14.330 121.339Q-14.091 121.480-13.976 121.738Q-13.861 121.996-13.861 122.351L-13.861 124.128L-13.435 124.128Q-13.228 124.152-13.189 124.359L-13.189 124.449Q-13.228 124.664-13.435 124.687L-14.830 124.687Q-15.025 124.664-15.076 124.449L-15.076 124.359Q-15.025 124.148-14.830 124.128L-14.501 124.128L-14.501 122.382Q-14.501 122.074-14.591 121.916Q-14.681 121.757-14.974 121.757Q-15.244 121.757-15.472 121.888Q-15.701 122.019-15.837 122.248Q-15.974 122.476-15.974 122.742L-15.974 124.128L-15.548 124.128Q-15.341 124.152-15.302 124.359L-15.302 124.449Q-15.341 124.664-15.548 124.687L-17.037 124.687Q-17.244 124.664-17.287 124.449M-12.705 123.574Q-12.705 123.128-12.291 122.871Q-11.876 122.613-11.335 122.513Q-10.794 122.414-10.287 122.406Q-10.287 122.191-10.421 122.039Q-10.556 121.886-10.763 121.810Q-10.970 121.734-11.181 121.734Q-11.525 121.734-11.685 121.757L-11.685 121.816Q-11.685 121.984-11.804 122.099Q-11.923 122.214-12.087 122.214Q-12.263 122.214-12.378 122.091Q-12.494 121.968-12.494 121.800Q-12.494 121.394-12.113 121.285Q-11.732 121.175-11.173 121.175Q-10.904 121.175-10.636 121.253Q-10.369 121.332-10.144 121.482Q-9.919 121.632-9.783 121.853Q-9.646 122.074-9.646 122.351L-9.646 124.070Q-9.646 124.128-9.119 124.128Q-8.923 124.148-8.873 124.359L-8.873 124.449Q-8.923 124.664-9.119 124.687L-9.263 124.687Q-9.607 124.687-9.835 124.640Q-10.064 124.593-10.208 124.406Q-10.669 124.726-11.376 124.726Q-11.712 124.726-12.017 124.585Q-12.322 124.445-12.513 124.183Q-12.705 123.921-12.705 123.574M-12.064 123.582Q-12.064 123.855-11.822 124.011Q-11.580 124.167-11.294 124.167Q-11.076 124.167-10.843 124.109Q-10.611 124.050-10.449 123.912Q-10.287 123.773-10.287 123.550L-10.287 122.960Q-10.568 122.960-10.984 123.017Q-11.400 123.074-11.732 123.212Q-12.064 123.351-12.064 123.582M-8.794 124.449L-8.794 124.359Q-8.751 124.152-8.544 124.128L-8.123 124.128L-8.123 121.800L-8.544 121.800Q-8.751 121.777-8.794 121.558L-8.794 121.472Q-8.748 121.261-8.544 121.238L-7.728 121.238Q-7.533 121.261-7.482 121.472L-7.482 121.558L-7.490 121.582Q-7.263 121.402-6.990 121.300Q-6.716 121.199-6.423 121.199Q-6.076 121.199-5.837 121.339Q-5.599 121.480-5.484 121.738Q-5.369 121.996-5.369 122.351L-5.369 124.128L-4.943 124.128Q-4.736 124.152-4.697 124.359L-4.697 124.449Q-4.736 124.664-4.943 124.687L-6.337 124.687Q-6.533 124.664-6.583 124.449L-6.583 124.359Q-6.533 124.148-6.337 124.128L-6.009 124.128L-6.009 122.382Q-6.009 122.074-6.099 121.916Q-6.189 121.757-6.482 121.757Q-6.751 121.757-6.980 121.888Q-7.208 122.019-7.345 122.248Q-7.482 122.476-7.482 122.742L-7.482 124.128L-7.056 124.128Q-6.849 124.152-6.810 124.359L-6.810 124.449Q-6.849 124.664-7.056 124.687L-8.544 124.687Q-8.751 124.664-8.794 124.449M-4.380 125.335Q-4.380 125.035-4.232 124.773Q-4.083 124.511-3.833 124.351Q-4.017 124.097-4.017 123.777Q-4.017 123.492-3.865 123.230Q-4.115 122.906-4.115 122.488Q-4.115 122.128-3.923 121.832Q-3.732 121.535-3.414 121.367Q-3.095 121.199-2.732 121.199Q-2.525 121.199-2.322 121.259Q-2.119 121.320-1.955 121.421Q-1.572 121.160-1.099 121.160Q-0.861 121.160-0.679 121.291Q-0.498 121.421-0.498 121.648Q-0.498 121.796-0.599 121.902Q-0.701 122.007-0.857 122.007Q-0.990 122.007-1.085 121.929Q-1.181 121.851-1.212 121.726Q-1.357 121.734-1.556 121.824Q-1.353 122.136-1.353 122.488Q-1.353 122.769-1.464 123.001Q-1.576 123.234-1.771 123.412Q-1.966 123.589-2.218 123.687Q-2.470 123.785-2.732 123.785Q-3.119 123.785-3.451 123.597Q-3.462 123.597-3.472 123.669Q-3.482 123.742-3.482 123.777Q-3.482 123.898-3.416 124.005Q-3.349 124.113-3.228 124.152Q-3.212 124.148-3.199 124.146Q-3.185 124.144-3.162 124.144Q-3.146 124.144-3.115 124.152Q-3.083 124.160-3.076 124.160L-2.482 124.160Q-1.701 124.160-1.160 124.402Q-0.619 124.644-0.619 125.335Q-0.619 125.636-0.798 125.863Q-0.978 126.089-1.275 126.234Q-1.572 126.378-1.892 126.445Q-2.212 126.511-2.498 126.511Q-2.888 126.511-3.330 126.388Q-3.771 126.265-4.076 125.998Q-4.380 125.730-4.380 125.335M-3.841 125.328Q-3.841 125.546-3.601 125.687Q-3.361 125.828-3.037 125.894Q-2.712 125.960-2.498 125.960Q-2.283 125.960-1.958 125.894Q-1.634 125.828-1.394 125.687Q-1.154 125.546-1.154 125.328Q-1.154 125.039-1.378 124.900Q-1.603 124.761-1.884 124.728Q-2.166 124.695-2.513 124.695L-3.130 124.695Q-3.314 124.695-3.476 124.775Q-3.638 124.855-3.740 125.001Q-3.841 125.148-3.841 125.328M-2.732 123.230Q-2.431 123.230-2.212 123.011Q-1.994 122.792-1.994 122.488Q-1.994 122.332-2.048 122.201Q-2.103 122.070-2.208 121.964Q-2.314 121.859-2.445 121.804Q-2.576 121.749-2.732 121.749Q-3.037 121.749-3.255 121.968Q-3.474 122.187-3.474 122.488Q-3.474 122.785-3.251 123.007Q-3.029 123.230-2.732 123.230M3.139 123.199L0.698 123.199Q0.752 123.476 0.950 123.699Q1.147 123.921 1.424 124.044Q1.702 124.167 1.987 124.167Q2.459 124.167 2.682 123.878Q2.690 123.867 2.747 123.761Q2.803 123.656 2.852 123.613Q2.901 123.570 2.995 123.558L3.139 123.558Q3.331 123.578 3.389 123.792L3.389 123.847Q3.323 124.148 3.092 124.345Q2.862 124.542 2.549 124.634Q2.237 124.726 1.932 124.726Q1.448 124.726 1.008 124.498Q0.569 124.269 0.301 123.869Q0.034 123.468 0.034 122.976L0.034 122.917Q0.034 122.449 0.280 122.046Q0.526 121.644 0.934 121.410Q1.342 121.175 1.811 121.175Q2.315 121.175 2.668 121.398Q3.022 121.621 3.206 122.009Q3.389 122.398 3.389 122.902L3.389 122.960Q3.331 123.175 3.139 123.199M0.706 122.648L2.733 122.648Q2.686 122.238 2.448 121.986Q2.209 121.734 1.811 121.734Q1.417 121.734 1.110 121.996Q0.803 122.257 0.706 122.648\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(151.487 -75.267)\">\u003Cpath d=\"M8.385 125.335Q8.385 125.035 8.534 124.773Q8.682 124.511 8.932 124.351Q8.749 124.097 8.749 123.777Q8.749 123.492 8.901 123.230Q8.651 122.906 8.651 122.488Q8.651 122.128 8.842 121.832Q9.034 121.535 9.352 121.367Q9.670 121.199 10.034 121.199Q10.241 121.199 10.444 121.259Q10.647 121.320 10.811 121.421Q11.194 121.160 11.667 121.160Q11.905 121.160 12.086 121.291Q12.268 121.421 12.268 121.648Q12.268 121.796 12.167 121.902Q12.065 122.007 11.909 122.007Q11.776 122.007 11.680 121.929Q11.585 121.851 11.553 121.726Q11.409 121.734 11.210 121.824Q11.413 122.136 11.413 122.488Q11.413 122.769 11.301 123.001Q11.190 123.234 10.995 123.412Q10.799 123.589 10.547 123.687Q10.295 123.785 10.034 123.785Q9.647 123.785 9.315 123.597Q9.303 123.597 9.293 123.669Q9.284 123.742 9.284 123.777Q9.284 123.898 9.350 124.005Q9.417 124.113 9.538 124.152Q9.553 124.148 9.567 124.146Q9.581 124.144 9.604 124.144Q9.620 124.144 9.651 124.152Q9.682 124.160 9.690 124.160L10.284 124.160Q11.065 124.160 11.606 124.402Q12.147 124.644 12.147 125.335Q12.147 125.636 11.967 125.863Q11.788 126.089 11.491 126.234Q11.194 126.378 10.874 126.445Q10.553 126.511 10.268 126.511Q9.877 126.511 9.436 126.388Q8.995 126.265 8.690 125.998Q8.385 125.730 8.385 125.335M8.924 125.328Q8.924 125.546 9.165 125.687Q9.405 125.828 9.729 125.894Q10.053 125.960 10.268 125.960Q10.483 125.960 10.807 125.894Q11.131 125.828 11.372 125.687Q11.612 125.546 11.612 125.328Q11.612 125.039 11.387 124.900Q11.163 124.761 10.881 124.728Q10.600 124.695 10.252 124.695L9.635 124.695Q9.452 124.695 9.290 124.775Q9.127 124.855 9.026 125.001Q8.924 125.148 8.924 125.328M10.034 123.230Q10.335 123.230 10.553 123.011Q10.772 122.792 10.772 122.488Q10.772 122.332 10.717 122.201Q10.663 122.070 10.557 121.964Q10.452 121.859 10.321 121.804Q10.190 121.749 10.034 121.749Q9.729 121.749 9.510 121.968Q9.292 122.187 9.292 122.488Q9.292 122.785 9.514 123.007Q9.737 123.230 10.034 123.230M13.010 124.449L13.010 124.359Q13.061 124.152 13.256 124.128L14.295 124.128L14.295 121.800L13.323 121.800Q13.124 121.777 13.073 121.558L13.073 121.472Q13.124 121.261 13.323 121.238L14.690 121.238Q14.885 121.257 14.936 121.472L14.936 124.128L15.850 124.128Q16.045 124.152 16.096 124.359L16.096 124.449Q16.045 124.664 15.850 124.687L13.256 124.687Q13.061 124.664 13.010 124.449M14.042 120.261L14.042 120.207Q14.042 120.035 14.178 119.914Q14.315 119.792 14.491 119.792Q14.663 119.792 14.799 119.914Q14.936 120.035 14.936 120.207L14.936 120.261Q14.936 120.437 14.799 120.558Q14.663 120.679 14.491 120.679Q14.315 120.679 14.178 120.558Q14.042 120.437 14.042 120.261M18.295 124.464L17.409 121.800L17.088 121.800Q16.889 121.777 16.838 121.558L16.838 121.472Q16.889 121.261 17.088 121.238L18.249 121.238Q18.444 121.257 18.495 121.472L18.495 121.558Q18.444 121.777 18.249 121.800L17.967 121.800L18.760 124.175L19.549 121.800L19.272 121.800Q19.073 121.777 19.022 121.558L19.022 121.472Q19.073 121.261 19.272 121.238L20.432 121.238Q20.627 121.261 20.678 121.472L20.678 121.558Q20.627 121.777 20.432 121.800L20.112 121.800L19.225 124.464Q19.182 124.578 19.081 124.652Q18.979 124.726 18.854 124.726L18.663 124.726Q18.545 124.726 18.442 124.654Q18.338 124.582 18.295 124.464M24.397 123.199L21.956 123.199Q22.010 123.476 22.208 123.699Q22.405 123.921 22.682 124.044Q22.959 124.167 23.245 124.167Q23.717 124.167 23.940 123.878Q23.948 123.867 24.004 123.761Q24.061 123.656 24.110 123.613Q24.159 123.570 24.252 123.558L24.397 123.558Q24.588 123.578 24.647 123.792L24.647 123.847Q24.581 124.148 24.350 124.345Q24.120 124.542 23.807 124.634Q23.495 124.726 23.190 124.726Q22.706 124.726 22.266 124.498Q21.827 124.269 21.559 123.869Q21.292 123.468 21.292 122.976L21.292 122.917Q21.292 122.449 21.538 122.046Q21.784 121.644 22.192 121.410Q22.600 121.175 23.069 121.175Q23.573 121.175 23.926 121.398Q24.280 121.621 24.463 122.009Q24.647 122.398 24.647 122.902L24.647 122.960Q24.588 123.175 24.397 123.199M21.963 122.648L23.991 122.648Q23.944 122.238 23.706 121.986Q23.467 121.734 23.069 121.734Q22.674 121.734 22.368 121.996Q22.061 122.257 21.963 122.648M25.717 124.488L25.717 123.574Q25.745 123.367 25.956 123.343L26.124 123.343Q26.288 123.367 26.346 123.527Q26.549 124.167 27.276 124.167Q27.483 124.167 27.711 124.132Q27.940 124.097 28.108 123.982Q28.276 123.867 28.276 123.664Q28.276 123.453 28.053 123.339Q27.831 123.226 27.557 123.183L26.858 123.070Q25.717 122.859 25.717 122.136Q25.717 121.847 25.862 121.658Q26.006 121.468 26.247 121.361Q26.487 121.253 26.743 121.214Q26.999 121.175 27.276 121.175Q27.526 121.175 27.719 121.205Q27.913 121.234 28.077 121.312Q28.155 121.195 28.284 121.175L28.362 121.175Q28.459 121.187 28.522 121.249Q28.584 121.312 28.596 121.406L28.596 122.113Q28.584 122.207 28.522 122.273Q28.459 122.339 28.362 122.351L28.194 122.351Q28.100 122.339 28.034 122.273Q27.967 122.207 27.956 122.113Q27.956 121.734 27.260 121.734Q26.913 121.734 26.594 121.816Q26.276 121.898 26.276 122.144Q26.276 122.410 26.948 122.519L27.651 122.640Q28.135 122.722 28.485 122.970Q28.834 123.218 28.834 123.664Q28.834 124.054 28.598 124.296Q28.362 124.539 28.012 124.632Q27.663 124.726 27.276 124.726Q26.698 124.726 26.299 124.472Q26.229 124.597 26.180 124.654Q26.131 124.710 26.026 124.726L25.956 124.726Q25.741 124.703 25.717 124.488\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(151.487 -75.267)\">\u003Cpath d=\"M34.207 122.960Q34.207 122.480 34.451 122.066Q34.695 121.652 35.111 121.414Q35.527 121.175 36.007 121.175Q36.562 121.175 36.941 121.285Q37.320 121.394 37.320 121.800Q37.320 121.968 37.207 122.091Q37.093 122.214 36.921 122.214Q36.750 122.214 36.630 122.099Q36.511 121.984 36.511 121.816L36.511 121.757Q36.351 121.734 36.015 121.734Q35.687 121.734 35.419 121.904Q35.152 122.074 35 122.357Q34.847 122.640 34.847 122.960Q34.847 123.281 35.019 123.562Q35.191 123.843 35.476 124.005Q35.761 124.167 36.089 124.167Q36.402 124.167 36.529 124.064Q36.656 123.960 36.773 123.771Q36.890 123.582 37.007 123.566L37.175 123.566Q37.281 123.578 37.345 123.646Q37.410 123.714 37.410 123.816Q37.410 123.863 37.390 123.902Q37.281 124.195 37.078 124.374Q36.875 124.554 36.599 124.640Q36.324 124.726 36.007 124.726Q35.523 124.726 35.107 124.488Q34.691 124.249 34.449 123.847Q34.207 123.445 34.207 122.960M38.636 123.832L38.636 121.800L38.214 121.800Q38.007 121.777 37.964 121.558L37.964 121.472Q38.011 121.261 38.214 121.238L39.031 121.238Q39.226 121.261 39.277 121.472L39.277 123.800Q39.277 124.035 39.447 124.101Q39.617 124.167 39.902 124.167Q40.109 124.167 40.304 124.091Q40.500 124.015 40.625 123.865Q40.750 123.714 40.750 123.503L40.750 121.800L40.328 121.800Q40.117 121.777 40.078 121.558L40.078 121.472Q40.117 121.261 40.328 121.238L41.140 121.238Q41.339 121.261 41.390 121.472L41.390 124.128L41.816 124.128Q42.023 124.152 42.062 124.359L42.062 124.449Q42.023 124.664 41.816 124.687L41 124.687Q40.800 124.664 40.750 124.464Q40.347 124.726 39.839 124.726Q39.605 124.726 39.390 124.685Q39.175 124.644 39.009 124.542Q38.843 124.441 38.740 124.263Q38.636 124.085 38.636 123.832M42.363 124.449L42.363 124.359Q42.421 124.152 42.613 124.128L43.324 124.128L43.324 121.800L42.613 121.800Q42.418 121.777 42.363 121.558L42.363 121.472Q42.421 121.261 42.613 121.238L43.714 121.238Q43.914 121.257 43.964 121.472L43.964 121.800Q44.226 121.515 44.582 121.357Q44.937 121.199 45.324 121.199Q45.617 121.199 45.851 121.333Q46.086 121.468 46.086 121.734Q46.086 121.902 45.976 122.019Q45.867 122.136 45.699 122.136Q45.546 122.136 45.431 122.025Q45.316 121.914 45.316 121.757Q44.941 121.757 44.627 121.958Q44.312 122.160 44.138 122.494Q43.964 122.828 43.964 123.207L43.964 124.128L44.910 124.128Q45.117 124.152 45.156 124.359L45.156 124.449Q45.117 124.664 44.910 124.687L42.613 124.687Q42.421 124.664 42.363 124.449M48.043 124.464L47.156 121.800L46.836 121.800Q46.636 121.777 46.586 121.558L46.586 121.472Q46.636 121.261 46.836 121.238L47.996 121.238Q48.191 121.257 48.242 121.472L48.242 121.558Q48.191 121.777 47.996 121.800L47.714 121.800L48.507 124.175L49.296 121.800L49.019 121.800Q48.820 121.777 48.769 121.558L48.769 121.472Q48.820 121.261 49.019 121.238L50.179 121.238Q50.375 121.261 50.425 121.472L50.425 121.558Q50.375 121.777 50.179 121.800L49.859 121.800L48.972 124.464Q48.929 124.578 48.828 124.652Q48.726 124.726 48.601 124.726L48.410 124.726Q48.293 124.726 48.189 124.654Q48.086 124.582 48.043 124.464M51.039 123.574Q51.039 123.128 51.453 122.871Q51.867 122.613 52.408 122.513Q52.949 122.414 53.457 122.406Q53.457 122.191 53.322 122.039Q53.187 121.886 52.980 121.810Q52.773 121.734 52.562 121.734Q52.218 121.734 52.058 121.757L52.058 121.816Q52.058 121.984 51.939 122.099Q51.820 122.214 51.656 122.214Q51.480 122.214 51.365 122.091Q51.250 121.968 51.250 121.800Q51.250 121.394 51.630 121.285Q52.011 121.175 52.570 121.175Q52.839 121.175 53.107 121.253Q53.375 121.332 53.599 121.482Q53.824 121.632 53.961 121.853Q54.097 122.074 54.097 122.351L54.097 124.070Q54.097 124.128 54.625 124.128Q54.820 124.148 54.871 124.359L54.871 124.449Q54.820 124.664 54.625 124.687L54.480 124.687Q54.136 124.687 53.908 124.640Q53.679 124.593 53.535 124.406Q53.074 124.726 52.367 124.726Q52.031 124.726 51.726 124.585Q51.421 124.445 51.230 124.183Q51.039 123.921 51.039 123.574M51.679 123.582Q51.679 123.855 51.921 124.011Q52.164 124.167 52.449 124.167Q52.668 124.167 52.900 124.109Q53.132 124.050 53.294 123.912Q53.457 123.773 53.457 123.550L53.457 122.960Q53.175 122.960 52.759 123.017Q52.343 123.074 52.011 123.212Q51.679 123.351 51.679 123.582M56.078 123.582L56.078 121.800L55.328 121.800Q55.128 121.777 55.078 121.558L55.078 121.472Q55.128 121.261 55.328 121.238L56.078 121.238L56.078 120.488Q56.128 120.281 56.328 120.253L56.472 120.253Q56.668 120.281 56.718 120.488L56.718 121.238L58.078 121.238Q58.269 121.257 58.328 121.472L58.328 121.558Q58.273 121.777 58.078 121.800L56.718 121.800L56.718 123.550Q56.718 124.167 57.293 124.167Q57.543 124.167 57.707 123.982Q57.871 123.796 57.871 123.550Q57.871 123.457 57.943 123.386Q58.015 123.316 58.117 123.304L58.261 123.304Q58.461 123.328 58.511 123.535L58.511 123.582Q58.511 123.906 58.328 124.169Q58.144 124.433 57.851 124.580Q57.558 124.726 57.238 124.726Q56.726 124.726 56.402 124.416Q56.078 124.105 56.078 123.582M59.867 123.832L59.867 121.800L59.445 121.800Q59.238 121.777 59.195 121.558L59.195 121.472Q59.242 121.261 59.445 121.238L60.261 121.238Q60.457 121.261 60.507 121.472L60.507 123.800Q60.507 124.035 60.677 124.101Q60.847 124.167 61.132 124.167Q61.339 124.167 61.535 124.091Q61.730 124.015 61.855 123.865Q61.980 123.714 61.980 123.503L61.980 121.800L61.558 121.800Q61.347 121.777 61.308 121.558L61.308 121.472Q61.347 121.261 61.558 121.238L62.371 121.238Q62.570 121.261 62.621 121.472L62.621 124.128L63.046 124.128Q63.253 124.152 63.293 124.359L63.293 124.449Q63.253 124.664 63.046 124.687L62.230 124.687Q62.031 124.664 61.980 124.464Q61.578 124.726 61.070 124.726Q60.836 124.726 60.621 124.685Q60.406 124.644 60.240 124.542Q60.074 124.441 59.970 124.263Q59.867 124.085 59.867 123.832M63.593 124.449L63.593 124.359Q63.652 124.152 63.843 124.128L64.554 124.128L64.554 121.800L63.843 121.800Q63.648 121.777 63.593 121.558L63.593 121.472Q63.652 121.261 63.843 121.238L64.945 121.238Q65.144 121.257 65.195 121.472L65.195 121.800Q65.457 121.515 65.812 121.357Q66.168 121.199 66.554 121.199Q66.847 121.199 67.082 121.333Q67.316 121.468 67.316 121.734Q67.316 121.902 67.207 122.019Q67.097 122.136 66.929 122.136Q66.777 122.136 66.662 122.025Q66.546 121.914 66.546 121.757Q66.171 121.757 65.857 121.958Q65.543 122.160 65.369 122.494Q65.195 122.828 65.195 123.207L65.195 124.128L66.140 124.128Q66.347 124.152 66.386 124.359L66.386 124.449Q66.347 124.664 66.140 124.687L63.843 124.687Q63.652 124.664 63.593 124.449M71.128 123.199L68.687 123.199Q68.742 123.476 68.939 123.699Q69.136 123.921 69.414 124.044Q69.691 124.167 69.976 124.167Q70.449 124.167 70.671 123.878Q70.679 123.867 70.736 123.761Q70.793 123.656 70.841 123.613Q70.890 123.570 70.984 123.558L71.128 123.558Q71.320 123.578 71.378 123.792L71.378 123.847Q71.312 124.148 71.082 124.345Q70.851 124.542 70.539 124.634Q70.226 124.726 69.921 124.726Q69.437 124.726 68.998 124.498Q68.558 124.269 68.291 123.869Q68.023 123.468 68.023 122.976L68.023 122.917Q68.023 122.449 68.269 122.046Q68.515 121.644 68.923 121.410Q69.332 121.175 69.800 121.175Q70.304 121.175 70.658 121.398Q71.011 121.621 71.195 122.009Q71.378 122.398 71.378 122.902L71.378 122.960Q71.320 123.175 71.128 123.199M68.695 122.648L70.722 122.648Q70.675 122.238 70.437 121.986Q70.199 121.734 69.800 121.734Q69.406 121.734 69.099 121.996Q68.793 122.257 68.695 122.648\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">BFGS reads curvature off successive gradients: the secant relation \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6776em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">≈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0813em;\">H\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> feeds a rank-two update of the inverse-Hessian estimate.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:595.071px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 446.303 91.466\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath d=\"M-65.203.097h56.905v-22.833h-56.905Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(-14.875 6.306)\">\u003Cpath d=\"M-36.431-21.057L-36.431-21.147Q-36.380-21.354-36.185-21.378L-35.302-21.378L-35.302-23.706L-36.157-23.706Q-36.356-23.729-36.407-23.948L-36.407-24.034Q-36.356-24.245-36.157-24.268L-35.302-24.268L-35.302-24.721Q-35.302-25.186-34.896-25.467Q-34.489-25.749-34.009-25.749Q-33.696-25.749-33.452-25.628Q-33.208-25.506-33.208-25.225Q-33.208-25.061-33.317-24.944Q-33.427-24.827-33.591-24.827Q-33.739-24.827-33.860-24.932Q-33.981-25.038-33.981-25.186L-34.063-25.186Q-34.216-25.186-34.353-25.126Q-34.489-25.065-34.575-24.950Q-34.661-24.835-34.661-24.690L-34.661-24.268L-33.630-24.268Q-33.435-24.249-33.384-24.034L-33.384-23.948Q-33.435-23.729-33.630-23.706L-34.661-23.706L-34.661-21.378L-33.782-21.378Q-33.587-21.354-33.536-21.147L-33.536-21.057Q-33.587-20.842-33.782-20.819L-36.185-20.819Q-36.380-20.842-36.431-21.057M-30.384-20.780Q-30.856-20.780-31.241-21.024Q-31.626-21.268-31.849-21.678Q-32.071-22.089-32.071-22.546Q-32.071-22.889-31.946-23.212Q-31.821-23.534-31.591-23.788Q-31.360-24.042-31.054-24.186Q-30.747-24.331-30.384-24.331Q-30.021-24.331-29.708-24.184Q-29.396-24.038-29.173-23.792Q-28.950-23.546-28.823-23.225Q-28.696-22.905-28.696-22.546Q-28.696-22.089-28.921-21.676Q-29.146-21.264-29.530-21.022Q-29.915-20.780-30.384-20.780M-30.384-21.339Q-29.919-21.339-29.628-21.733Q-29.337-22.128-29.337-22.612Q-29.337-22.905-29.472-23.173Q-29.606-23.440-29.847-23.606Q-30.087-23.772-30.384-23.772Q-30.688-23.772-30.927-23.606Q-31.165-23.440-31.300-23.173Q-31.435-22.905-31.435-22.612Q-31.435-22.131-31.142-21.735Q-30.849-21.339-30.384-21.339M-28.036-21.057L-28.036-21.147Q-27.978-21.354-27.786-21.378L-27.075-21.378L-27.075-23.706L-27.786-23.706Q-27.981-23.729-28.036-23.948L-28.036-24.034Q-27.978-24.245-27.786-24.268L-26.685-24.268Q-26.485-24.249-26.435-24.034L-26.435-23.706Q-26.173-23.991-25.817-24.149Q-25.462-24.307-25.075-24.307Q-24.782-24.307-24.548-24.173Q-24.313-24.038-24.313-23.772Q-24.313-23.604-24.423-23.487Q-24.532-23.370-24.700-23.370Q-24.853-23.370-24.968-23.481Q-25.083-23.592-25.083-23.749Q-25.458-23.749-25.772-23.548Q-26.087-23.346-26.261-23.012Q-26.435-22.678-26.435-22.299L-26.435-21.378L-25.489-21.378Q-25.282-21.354-25.243-21.147L-25.243-21.057Q-25.282-20.842-25.489-20.819L-27.786-20.819Q-27.978-20.842-28.036-21.057M-23.005-21.042L-23.470-23.706L-23.630-23.706Q-23.837-23.729-23.876-23.948L-23.876-24.034Q-23.837-24.245-23.630-24.268L-22.462-24.268Q-22.251-24.245-22.212-24.034L-22.212-23.948Q-22.251-23.729-22.462-23.706L-22.935-23.706Q-22.821-23.061-22.743-22.616Q-22.665-22.171-22.614-21.852Q-22.563-21.534-22.563-21.460Q-22.556-21.631-22.271-22.628Q-22.231-22.741-22.134-22.815Q-22.036-22.889-21.915-22.889L-21.837-22.889Q-21.716-22.889-21.618-22.815Q-21.521-22.741-21.485-22.628Q-21.376-22.245-21.294-21.921Q-21.212-21.596-21.212-21.460Q-21.200-21.749-20.853-23.706L-21.325-23.706Q-21.532-23.729-21.571-23.948L-21.571-24.034Q-21.532-24.245-21.325-24.268L-20.149-24.268Q-19.958-24.245-19.899-24.034L-19.899-23.948Q-19.954-23.729-20.149-23.706L-20.317-23.706L-20.782-21.042Q-20.806-20.924-20.894-20.852Q-20.981-20.780-21.103-20.780L-21.243-20.780Q-21.364-20.780-21.460-20.852Q-21.556-20.924-21.599-21.042Q-21.646-21.221-21.718-21.477Q-21.790-21.733-21.833-21.942Q-21.876-22.151-21.876-22.253Q-21.884-21.971-22.165-21.042Q-22.192-20.932-22.290-20.856Q-22.388-20.780-22.509-20.780L-22.685-20.780Q-22.806-20.780-22.894-20.852Q-22.981-20.924-23.005-21.042M-19.360-21.932Q-19.360-22.378-18.946-22.635Q-18.532-22.893-17.991-22.993Q-17.450-23.092-16.942-23.100Q-16.942-23.315-17.077-23.467Q-17.212-23.620-17.419-23.696Q-17.626-23.772-17.837-23.772Q-18.181-23.772-18.341-23.749L-18.341-23.690Q-18.341-23.522-18.460-23.407Q-18.579-23.292-18.743-23.292Q-18.919-23.292-19.034-23.415Q-19.149-23.538-19.149-23.706Q-19.149-24.112-18.769-24.221Q-18.388-24.331-17.829-24.331Q-17.560-24.331-17.292-24.253Q-17.024-24.174-16.800-24.024Q-16.575-23.874-16.438-23.653Q-16.302-23.432-16.302-23.155L-16.302-21.436Q-16.302-21.378-15.774-21.378Q-15.579-21.358-15.528-21.147L-15.528-21.057Q-15.579-20.842-15.774-20.819L-15.919-20.819Q-16.263-20.819-16.491-20.866Q-16.720-20.913-16.864-21.100Q-17.325-20.780-18.032-20.780Q-18.368-20.780-18.673-20.921Q-18.978-21.061-19.169-21.323Q-19.360-21.585-19.360-21.932M-18.720-21.924Q-18.720-21.651-18.478-21.495Q-18.235-21.339-17.950-21.339Q-17.731-21.339-17.499-21.397Q-17.267-21.456-17.105-21.594Q-16.942-21.733-16.942-21.956L-16.942-22.546Q-17.224-22.546-17.640-22.489Q-18.056-22.432-18.388-22.294Q-18.720-22.155-18.720-21.924M-15.298-21.057L-15.298-21.147Q-15.239-21.354-15.048-21.378L-14.337-21.378L-14.337-23.706L-15.048-23.706Q-15.243-23.729-15.298-23.948L-15.298-24.034Q-15.239-24.245-15.048-24.268L-13.946-24.268Q-13.747-24.249-13.696-24.034L-13.696-23.706Q-13.435-23.991-13.079-24.149Q-12.724-24.307-12.337-24.307Q-12.044-24.307-11.810-24.173Q-11.575-24.038-11.575-23.772Q-11.575-23.604-11.685-23.487Q-11.794-23.370-11.962-23.370Q-12.114-23.370-12.230-23.481Q-12.345-23.592-12.345-23.749Q-12.720-23.749-13.034-23.548Q-13.349-23.346-13.522-23.012Q-13.696-22.678-13.696-22.299L-13.696-21.378L-12.751-21.378Q-12.544-21.354-12.505-21.147L-12.505-21.057Q-12.544-20.842-12.751-20.819L-15.048-20.819Q-15.239-20.842-15.298-21.057M-9.411-20.780Q-9.876-20.780-10.241-21.030Q-10.606-21.280-10.812-21.684Q-11.017-22.089-11.017-22.546Q-11.017-22.889-10.892-23.210Q-10.767-23.530-10.534-23.780Q-10.302-24.030-9.997-24.169Q-9.692-24.307-9.337-24.307Q-8.825-24.307-8.419-23.987L-8.419-25.147L-8.841-25.147Q-9.052-25.171-9.091-25.385L-9.091-25.475Q-9.052-25.682-8.841-25.706L-8.028-25.706Q-7.829-25.682-7.778-25.475L-7.778-21.378L-7.353-21.378Q-7.146-21.354-7.106-21.147L-7.106-21.057Q-7.146-20.842-7.353-20.819L-8.169-20.819Q-8.368-20.842-8.419-21.057L-8.419-21.186Q-8.614-20.995-8.876-20.887Q-9.138-20.780-9.411-20.780M-9.372-21.339Q-9.013-21.339-8.761-21.608Q-8.509-21.878-8.419-22.253L-8.419-23.085Q-8.478-23.272-8.605-23.423Q-8.731-23.573-8.909-23.661Q-9.087-23.749-9.282-23.749Q-9.591-23.749-9.841-23.579Q-10.091-23.409-10.235-23.124Q-10.380-22.839-10.380-22.538Q-10.380-22.089-10.095-21.714Q-9.810-21.339-9.372-21.339\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-14.875 6.306)\">\u003Cpath d=\"M-30.306-9.776L-30.306-9.862Q-30.263-10.081-30.056-10.104L-29.634-10.104L-29.634-14.206L-30.056-14.206Q-30.263-14.229-30.306-14.448L-30.306-14.534Q-30.259-14.745-30.056-14.768L-29.239-14.768Q-29.044-14.749-28.993-14.534L-28.993-14.464Q-28.782-14.632-28.519-14.719Q-28.255-14.807-27.985-14.807Q-27.646-14.807-27.349-14.663Q-27.052-14.518-26.837-14.266Q-26.622-14.014-26.507-13.702Q-26.392-13.389-26.392-13.046Q-26.392-12.581-26.618-12.171Q-26.845-11.760-27.231-11.520Q-27.618-11.280-28.087-11.280Q-28.606-11.280-28.993-11.647L-28.993-10.104L-28.567-10.104Q-28.360-10.081-28.321-9.862L-28.321-9.776Q-28.360-9.565-28.567-9.542L-30.056-9.542Q-30.259-9.565-30.306-9.776M-28.130-11.839Q-27.821-11.839-27.571-12.008Q-27.321-12.178-27.177-12.460Q-27.032-12.741-27.032-13.046Q-27.032-13.335-27.157-13.614Q-27.282-13.893-27.515-14.071Q-27.747-14.249-28.048-14.249Q-28.368-14.249-28.630-14.063Q-28.892-13.878-28.993-13.577L-28.993-12.725Q-28.903-12.358-28.683-12.098Q-28.462-11.839-28.130-11.839M-25.724-12.432Q-25.724-12.878-25.310-13.135Q-24.896-13.393-24.355-13.493Q-23.814-13.592-23.306-13.600Q-23.306-13.815-23.440-13.967Q-23.575-14.120-23.782-14.196Q-23.989-14.272-24.200-14.272Q-24.544-14.272-24.704-14.249L-24.704-14.190Q-24.704-14.022-24.823-13.907Q-24.942-13.792-25.106-13.792Q-25.282-13.792-25.397-13.915Q-25.513-14.038-25.513-14.206Q-25.513-14.612-25.132-14.721Q-24.751-14.831-24.192-14.831Q-23.923-14.831-23.655-14.753Q-23.388-14.674-23.163-14.524Q-22.939-14.374-22.802-14.153Q-22.665-13.932-22.665-13.655L-22.665-11.936Q-22.665-11.878-22.138-11.878Q-21.942-11.858-21.892-11.647L-21.892-11.557Q-21.942-11.342-22.138-11.319L-22.282-11.319Q-22.626-11.319-22.855-11.366Q-23.083-11.413-23.228-11.600Q-23.689-11.280-24.396-11.280Q-24.731-11.280-25.036-11.421Q-25.341-11.561-25.532-11.823Q-25.724-12.085-25.724-12.432M-25.083-12.424Q-25.083-12.151-24.841-11.995Q-24.599-11.839-24.314-11.839Q-24.095-11.839-23.862-11.897Q-23.630-11.956-23.468-12.094Q-23.306-12.233-23.306-12.456L-23.306-13.046Q-23.587-13.046-24.003-12.989Q-24.419-12.932-24.751-12.794Q-25.083-12.655-25.083-12.424M-21.298-11.518L-21.298-12.432Q-21.271-12.639-21.060-12.663L-20.892-12.663Q-20.728-12.639-20.669-12.479Q-20.466-11.839-19.739-11.839Q-19.532-11.839-19.304-11.874Q-19.075-11.909-18.907-12.024Q-18.739-12.139-18.739-12.342Q-18.739-12.553-18.962-12.667Q-19.185-12.780-19.458-12.823L-20.157-12.936Q-21.298-13.147-21.298-13.870Q-21.298-14.159-21.153-14.348Q-21.009-14.538-20.769-14.645Q-20.528-14.753-20.272-14.792Q-20.017-14.831-19.739-14.831Q-19.489-14.831-19.296-14.801Q-19.103-14.772-18.939-14.694Q-18.860-14.811-18.731-14.831L-18.653-14.831Q-18.556-14.819-18.493-14.757Q-18.431-14.694-18.419-14.600L-18.419-13.893Q-18.431-13.799-18.493-13.733Q-18.556-13.667-18.653-13.655L-18.821-13.655Q-18.915-13.667-18.981-13.733Q-19.048-13.799-19.060-13.893Q-19.060-14.272-19.755-14.272Q-20.103-14.272-20.421-14.190Q-20.739-14.108-20.739-13.862Q-20.739-13.596-20.067-13.487L-19.364-13.366Q-18.880-13.284-18.530-13.036Q-18.181-12.788-18.181-12.342Q-18.181-11.952-18.417-11.710Q-18.653-11.467-19.003-11.374Q-19.353-11.280-19.739-11.280Q-20.317-11.280-20.716-11.534Q-20.786-11.409-20.835-11.352Q-20.884-11.296-20.989-11.280L-21.060-11.280Q-21.274-11.303-21.298-11.518M-17.052-11.518L-17.052-12.432Q-17.024-12.639-16.814-12.663L-16.646-12.663Q-16.481-12.639-16.423-12.479Q-16.220-11.839-15.493-11.839Q-15.286-11.839-15.058-11.874Q-14.829-11.909-14.661-12.024Q-14.493-12.139-14.493-12.342Q-14.493-12.553-14.716-12.667Q-14.939-12.780-15.212-12.823L-15.911-12.936Q-17.052-13.147-17.052-13.870Q-17.052-14.159-16.907-14.348Q-16.763-14.538-16.522-14.645Q-16.282-14.753-16.026-14.792Q-15.771-14.831-15.493-14.831Q-15.243-14.831-15.050-14.801Q-14.856-14.772-14.692-14.694Q-14.614-14.811-14.485-14.831L-14.407-14.831Q-14.310-14.819-14.247-14.757Q-14.185-14.694-14.173-14.600L-14.173-13.893Q-14.185-13.799-14.247-13.733Q-14.310-13.667-14.407-13.655L-14.575-13.655Q-14.669-13.667-14.735-13.733Q-14.802-13.799-14.814-13.893Q-14.814-14.272-15.509-14.272Q-15.856-14.272-16.175-14.190Q-16.493-14.108-16.493-13.862Q-16.493-13.596-15.821-13.487L-15.118-13.366Q-14.634-13.284-14.284-13.036Q-13.935-12.788-13.935-12.342Q-13.935-11.952-14.171-11.710Q-14.407-11.467-14.757-11.374Q-15.106-11.280-15.493-11.280Q-16.071-11.280-16.470-11.534Q-16.540-11.409-16.589-11.352Q-16.638-11.296-16.743-11.280L-16.814-11.280Q-17.028-11.303-17.052-11.518\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath d=\"M29.49.097h56.906v-22.833H29.491Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(77.694 6.306)\">\u003Cpath d=\"M-36.009-21.057L-36.009-25.147L-36.431-25.147Q-36.638-25.171-36.681-25.385L-36.681-25.475Q-36.638-25.682-36.431-25.706L-35.614-25.706Q-35.419-25.682-35.368-25.475L-35.368-23.964Q-35.157-24.131-34.894-24.219Q-34.630-24.307-34.360-24.307Q-34.021-24.307-33.724-24.163Q-33.427-24.018-33.212-23.766Q-32.997-23.514-32.882-23.202Q-32.767-22.889-32.767-22.546Q-32.767-22.081-32.993-21.671Q-33.220-21.260-33.606-21.020Q-33.993-20.780-34.462-20.780Q-34.981-20.780-35.368-21.147L-35.368-21.057Q-35.419-20.839-35.614-20.819L-35.759-20.819Q-35.950-20.842-36.009-21.057M-34.505-21.339Q-34.196-21.339-33.946-21.508Q-33.696-21.678-33.552-21.960Q-33.407-22.241-33.407-22.546Q-33.407-22.835-33.532-23.114Q-33.657-23.393-33.890-23.571Q-34.122-23.749-34.423-23.749Q-34.743-23.749-35.005-23.563Q-35.267-23.378-35.368-23.077L-35.368-22.225Q-35.278-21.858-35.058-21.598Q-34.837-21.339-34.505-21.339M-32.099-21.932Q-32.099-22.378-31.685-22.635Q-31.271-22.893-30.730-22.993Q-30.188-23.092-29.681-23.100Q-29.681-23.315-29.815-23.467Q-29.950-23.620-30.157-23.696Q-30.364-23.772-30.575-23.772Q-30.919-23.772-31.079-23.749L-31.079-23.690Q-31.079-23.522-31.198-23.407Q-31.317-23.292-31.481-23.292Q-31.657-23.292-31.772-23.415Q-31.888-23.538-31.888-23.706Q-31.888-24.112-31.507-24.221Q-31.126-24.331-30.567-24.331Q-30.298-24.331-30.030-24.253Q-29.763-24.174-29.538-24.024Q-29.313-23.874-29.177-23.653Q-29.040-23.432-29.040-23.155L-29.040-21.436Q-29.040-21.378-28.513-21.378Q-28.317-21.358-28.267-21.147L-28.267-21.057Q-28.317-20.842-28.513-20.819L-28.657-20.819Q-29.001-20.819-29.230-20.866Q-29.458-20.913-29.603-21.100Q-30.063-20.780-30.771-20.780Q-31.106-20.780-31.411-20.921Q-31.716-21.061-31.907-21.323Q-32.099-21.585-32.099-21.932M-31.458-21.924Q-31.458-21.651-31.216-21.495Q-30.974-21.339-30.688-21.339Q-30.470-21.339-30.237-21.397Q-30.005-21.456-29.843-21.594Q-29.681-21.733-29.681-21.956L-29.681-22.546Q-29.962-22.546-30.378-22.489Q-30.794-22.432-31.126-22.294Q-31.458-22.155-31.458-21.924M-27.700-22.546Q-27.700-23.026-27.456-23.440Q-27.212-23.854-26.796-24.092Q-26.380-24.331-25.899-24.331Q-25.345-24.331-24.966-24.221Q-24.587-24.112-24.587-23.706Q-24.587-23.538-24.700-23.415Q-24.813-23.292-24.985-23.292Q-25.157-23.292-25.276-23.407Q-25.396-23.522-25.396-23.690L-25.396-23.749Q-25.556-23.772-25.892-23.772Q-26.220-23.772-26.487-23.602Q-26.755-23.432-26.907-23.149Q-27.060-22.866-27.060-22.546Q-27.060-22.225-26.888-21.944Q-26.716-21.663-26.431-21.501Q-26.146-21.339-25.817-21.339Q-25.505-21.339-25.378-21.442Q-25.251-21.546-25.134-21.735Q-25.017-21.924-24.899-21.940L-24.731-21.940Q-24.626-21.928-24.562-21.860Q-24.497-21.792-24.497-21.690Q-24.497-21.643-24.517-21.604Q-24.626-21.311-24.829-21.131Q-25.032-20.952-25.308-20.866Q-25.583-20.780-25.899-20.780Q-26.384-20.780-26.800-21.018Q-27.216-21.256-27.458-21.659Q-27.700-22.061-27.700-22.546M-23.868-21.057L-23.868-21.147Q-23.817-21.354-23.622-21.378L-23.157-21.378L-23.157-25.147L-23.622-25.147Q-23.817-25.171-23.868-25.385L-23.868-25.475Q-23.817-25.682-23.622-25.706L-22.876-25.706Q-22.681-25.686-22.630-25.475L-22.630-22.651L-21.485-23.706L-21.782-23.706Q-21.989-23.729-22.028-23.948L-22.028-24.034Q-21.989-24.245-21.782-24.268L-20.333-24.268Q-20.130-24.245-20.083-24.034L-20.083-23.948Q-20.126-23.729-20.333-23.706L-20.700-23.706L-21.646-22.842L-20.493-21.378L-20.157-21.378Q-19.950-21.354-19.907-21.147L-19.907-21.057Q-19.950-20.842-20.157-20.819L-21.325-20.819Q-21.521-20.842-21.571-21.057L-21.571-21.147Q-21.521-21.354-21.325-21.378L-21.165-21.378L-22.028-22.483L-22.630-21.932L-22.630-21.378L-22.165-21.378Q-21.974-21.354-21.915-21.147L-21.915-21.057Q-21.974-20.842-22.165-20.819L-23.622-20.819Q-23.817-20.842-23.868-21.057M-18.759-21.042L-19.224-23.706L-19.384-23.706Q-19.591-23.729-19.630-23.948L-19.630-24.034Q-19.591-24.245-19.384-24.268L-18.216-24.268Q-18.005-24.245-17.966-24.034L-17.966-23.948Q-18.005-23.729-18.216-23.706L-18.688-23.706Q-18.575-23.061-18.497-22.616Q-18.419-22.171-18.368-21.852Q-18.317-21.534-18.317-21.460Q-18.310-21.631-18.024-22.628Q-17.985-22.741-17.888-22.815Q-17.790-22.889-17.669-22.889L-17.591-22.889Q-17.470-22.889-17.372-22.815Q-17.274-22.741-17.239-22.628Q-17.130-22.245-17.048-21.921Q-16.966-21.596-16.966-21.460Q-16.954-21.749-16.606-23.706L-17.079-23.706Q-17.286-23.729-17.325-23.948L-17.325-24.034Q-17.286-24.245-17.079-24.268L-15.903-24.268Q-15.712-24.245-15.653-24.034L-15.653-23.948Q-15.708-23.729-15.903-23.706L-16.071-23.706L-16.536-21.042Q-16.560-20.924-16.647-20.852Q-16.735-20.780-16.856-20.780L-16.997-20.780Q-17.118-20.780-17.214-20.852Q-17.310-20.924-17.353-21.042Q-17.399-21.221-17.472-21.477Q-17.544-21.733-17.587-21.942Q-17.630-22.151-17.630-22.253Q-17.638-21.971-17.919-21.042Q-17.946-20.932-18.044-20.856Q-18.142-20.780-18.263-20.780L-18.438-20.780Q-18.560-20.780-18.647-20.852Q-18.735-20.924-18.759-21.042M-15.114-21.932Q-15.114-22.378-14.700-22.635Q-14.286-22.893-13.745-22.993Q-13.204-23.092-12.696-23.100Q-12.696-23.315-12.831-23.467Q-12.966-23.620-13.173-23.696Q-13.380-23.772-13.591-23.772Q-13.935-23.772-14.095-23.749L-14.095-23.690Q-14.095-23.522-14.214-23.407Q-14.333-23.292-14.497-23.292Q-14.673-23.292-14.788-23.415Q-14.903-23.538-14.903-23.706Q-14.903-24.112-14.522-24.221Q-14.142-24.331-13.583-24.331Q-13.313-24.331-13.046-24.253Q-12.778-24.174-12.554-24.024Q-12.329-23.874-12.192-23.653Q-12.056-23.432-12.056-23.155L-12.056-21.436Q-12.056-21.378-11.528-21.378Q-11.333-21.358-11.282-21.147L-11.282-21.057Q-11.333-20.842-11.528-20.819L-11.673-20.819Q-12.017-20.819-12.245-20.866Q-12.474-20.913-12.618-21.100Q-13.079-20.780-13.786-20.780Q-14.122-20.780-14.427-20.921Q-14.731-21.061-14.923-21.323Q-15.114-21.585-15.114-21.932M-14.474-21.924Q-14.474-21.651-14.231-21.495Q-13.989-21.339-13.704-21.339Q-13.485-21.339-13.253-21.397Q-13.021-21.456-12.858-21.594Q-12.696-21.733-12.696-21.956L-12.696-22.546Q-12.978-22.546-13.394-22.489Q-13.810-22.432-14.142-22.294Q-14.474-22.155-14.474-21.924M-11.052-21.057L-11.052-21.147Q-10.993-21.354-10.802-21.378L-10.091-21.378L-10.091-23.706L-10.802-23.706Q-10.997-23.729-11.052-23.948L-11.052-24.034Q-10.993-24.245-10.802-24.268L-9.700-24.268Q-9.501-24.249-9.450-24.034L-9.450-23.706Q-9.188-23.991-8.833-24.149Q-8.478-24.307-8.091-24.307Q-7.798-24.307-7.563-24.173Q-7.329-24.038-7.329-23.772Q-7.329-23.604-7.438-23.487Q-7.548-23.370-7.716-23.370Q-7.868-23.370-7.983-23.481Q-8.099-23.592-8.099-23.749Q-8.474-23.749-8.788-23.548Q-9.103-23.346-9.276-23.012Q-9.450-22.678-9.450-22.299L-9.450-21.378L-8.505-21.378Q-8.298-21.354-8.259-21.147L-8.259-21.057Q-8.298-20.842-8.505-20.819L-10.802-20.819Q-10.993-20.842-11.052-21.057M-5.165-20.780Q-5.630-20.780-5.995-21.030Q-6.360-21.280-6.565-21.684Q-6.771-22.089-6.771-22.546Q-6.771-22.889-6.646-23.210Q-6.521-23.530-6.288-23.780Q-6.056-24.030-5.751-24.169Q-5.446-24.307-5.091-24.307Q-4.579-24.307-4.173-23.987L-4.173-25.147L-4.595-25.147Q-4.806-25.171-4.845-25.385L-4.845-25.475Q-4.806-25.682-4.595-25.706L-3.782-25.706Q-3.583-25.682-3.532-25.475L-3.532-21.378L-3.106-21.378Q-2.899-21.354-2.860-21.147L-2.860-21.057Q-2.899-20.842-3.106-20.819L-3.923-20.819Q-4.122-20.842-4.173-21.057L-4.173-21.186Q-4.368-20.995-4.630-20.887Q-4.892-20.780-5.165-20.780M-5.126-21.339Q-4.767-21.339-4.515-21.608Q-4.263-21.878-4.173-22.253L-4.173-23.085Q-4.231-23.272-4.358-23.423Q-4.485-23.573-4.663-23.661Q-4.841-23.749-5.036-23.749Q-5.345-23.749-5.595-23.579Q-5.845-23.409-5.989-23.124Q-6.134-22.839-6.134-22.538Q-6.134-22.089-5.849-21.714Q-5.563-21.339-5.126-21.339\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(77.694 6.306)\">\u003Cpath d=\"M-28.180-9.776L-28.180-9.862Q-28.137-10.081-27.930-10.104L-27.508-10.104L-27.508-14.206L-27.930-14.206Q-28.137-14.229-28.180-14.448L-28.180-14.534Q-28.133-14.745-27.930-14.768L-27.113-14.768Q-26.918-14.749-26.867-14.534L-26.867-14.464Q-26.656-14.632-26.393-14.719Q-26.129-14.807-25.859-14.807Q-25.520-14.807-25.223-14.663Q-24.926-14.518-24.711-14.266Q-24.496-14.014-24.381-13.702Q-24.266-13.389-24.266-13.046Q-24.266-12.581-24.492-12.171Q-24.719-11.760-25.105-11.520Q-25.492-11.280-25.961-11.280Q-26.480-11.280-26.867-11.647L-26.867-10.104L-26.441-10.104Q-26.234-10.081-26.195-9.862L-26.195-9.776Q-26.234-9.565-26.441-9.542L-27.930-9.542Q-28.133-9.565-28.180-9.776M-26.004-11.839Q-25.695-11.839-25.445-12.008Q-25.195-12.178-25.051-12.460Q-24.906-12.741-24.906-13.046Q-24.906-13.335-25.031-13.614Q-25.156-13.893-25.389-14.071Q-25.621-14.249-25.922-14.249Q-26.242-14.249-26.504-14.063Q-26.766-13.878-26.867-13.577L-26.867-12.725Q-26.777-12.358-26.557-12.098Q-26.336-11.839-26.004-11.839M-23.598-12.432Q-23.598-12.878-23.184-13.135Q-22.770-13.393-22.229-13.493Q-21.687-13.592-21.180-13.600Q-21.180-13.815-21.314-13.967Q-21.449-14.120-21.656-14.196Q-21.863-14.272-22.074-14.272Q-22.418-14.272-22.578-14.249L-22.578-14.190Q-22.578-14.022-22.697-13.907Q-22.816-13.792-22.980-13.792Q-23.156-13.792-23.271-13.915Q-23.387-14.038-23.387-14.206Q-23.387-14.612-23.006-14.721Q-22.625-14.831-22.066-14.831Q-21.797-14.831-21.529-14.753Q-21.262-14.674-21.037-14.524Q-20.812-14.374-20.676-14.153Q-20.539-13.932-20.539-13.655L-20.539-11.936Q-20.539-11.878-20.012-11.878Q-19.816-11.858-19.766-11.647L-19.766-11.557Q-19.816-11.342-20.012-11.319L-20.156-11.319Q-20.500-11.319-20.729-11.366Q-20.957-11.413-21.102-11.600Q-21.562-11.280-22.270-11.280Q-22.605-11.280-22.910-11.421Q-23.215-11.561-23.406-11.823Q-23.598-12.085-23.598-12.432M-22.957-12.424Q-22.957-12.151-22.715-11.995Q-22.473-11.839-22.187-11.839Q-21.969-11.839-21.736-11.897Q-21.504-11.956-21.342-12.094Q-21.180-12.233-21.180-12.456L-21.180-13.046Q-21.461-13.046-21.877-12.989Q-22.293-12.932-22.625-12.794Q-22.957-12.655-22.957-12.424M-19.172-11.518L-19.172-12.432Q-19.145-12.639-18.934-12.663L-18.766-12.663Q-18.602-12.639-18.543-12.479Q-18.340-11.839-17.613-11.839Q-17.406-11.839-17.178-11.874Q-16.949-11.909-16.781-12.024Q-16.613-12.139-16.613-12.342Q-16.613-12.553-16.836-12.667Q-17.059-12.780-17.332-12.823L-18.031-12.936Q-19.172-13.147-19.172-13.870Q-19.172-14.159-19.027-14.348Q-18.883-14.538-18.643-14.645Q-18.402-14.753-18.146-14.792Q-17.891-14.831-17.613-14.831Q-17.363-14.831-17.170-14.801Q-16.977-14.772-16.812-14.694Q-16.734-14.811-16.605-14.831L-16.527-14.831Q-16.430-14.819-16.367-14.757Q-16.305-14.694-16.293-14.600L-16.293-13.893Q-16.305-13.799-16.367-13.733Q-16.430-13.667-16.527-13.655L-16.695-13.655Q-16.789-13.667-16.855-13.733Q-16.922-13.799-16.934-13.893Q-16.934-14.272-17.629-14.272Q-17.977-14.272-18.295-14.190Q-18.613-14.108-18.613-13.862Q-18.613-13.596-17.941-13.487L-17.238-13.366Q-16.754-13.284-16.404-13.036Q-16.055-12.788-16.055-12.342Q-16.055-11.952-16.291-11.710Q-16.527-11.467-16.877-11.374Q-17.227-11.280-17.613-11.280Q-18.191-11.280-18.590-11.534Q-18.660-11.409-18.709-11.352Q-18.758-11.296-18.863-11.280L-18.934-11.280Q-19.148-11.303-19.172-11.518M-14.926-11.518L-14.926-12.432Q-14.898-12.639-14.687-12.663L-14.520-12.663Q-14.355-12.639-14.297-12.479Q-14.094-11.839-13.367-11.839Q-13.160-11.839-12.932-11.874Q-12.703-11.909-12.535-12.024Q-12.367-12.139-12.367-12.342Q-12.367-12.553-12.590-12.667Q-12.812-12.780-13.086-12.823L-13.785-12.936Q-14.926-13.147-14.926-13.870Q-14.926-14.159-14.781-14.348Q-14.637-14.538-14.396-14.645Q-14.156-14.753-13.900-14.792Q-13.645-14.831-13.367-14.831Q-13.117-14.831-12.924-14.801Q-12.730-14.772-12.566-14.694Q-12.488-14.811-12.359-14.831L-12.281-14.831Q-12.184-14.819-12.121-14.757Q-12.059-14.694-12.047-14.600L-12.047-13.893Q-12.059-13.799-12.121-13.733Q-12.184-13.667-12.281-13.655L-12.449-13.655Q-12.543-13.667-12.609-13.733Q-12.676-13.799-12.687-13.893Q-12.687-14.272-13.383-14.272Q-13.730-14.272-14.049-14.190Q-14.367-14.108-14.367-13.862Q-14.367-13.596-13.695-13.487L-12.992-13.366Q-12.508-13.284-12.158-13.036Q-11.809-12.788-11.809-12.342Q-11.809-11.952-12.045-11.710Q-12.281-11.467-12.631-11.374Q-12.980-11.280-13.367-11.280Q-13.945-11.280-14.344-11.534Q-14.414-11.409-14.463-11.352Q-14.512-11.296-14.617-11.280L-14.687-11.280Q-14.902-11.303-14.926-11.518\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-neutral)\" style=\"stroke-width:.8\">\u003Cpath d=\"M124.185.062h62.596V-22.7h-62.596Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-size=\"8\">\u003Cg transform=\"translate(178.174 1.667)\">\u003Cpath d=\"M-34.888-11.280Q-35.353-11.280-35.718-11.530Q-36.083-11.780-36.288-12.184Q-36.493-12.589-36.493-13.046Q-36.493-13.389-36.368-13.710Q-36.243-14.030-36.011-14.280Q-35.778-14.530-35.474-14.669Q-35.169-14.807-34.813-14.807Q-34.302-14.807-33.896-14.487L-33.896-15.647L-34.317-15.647Q-34.528-15.671-34.567-15.885L-34.567-15.975Q-34.528-16.182-34.317-16.206L-33.505-16.206Q-33.306-16.182-33.255-15.975L-33.255-11.878L-32.829-11.878Q-32.622-11.854-32.583-11.647L-32.583-11.557Q-32.622-11.342-32.829-11.319L-33.646-11.319Q-33.845-11.342-33.896-11.557L-33.896-11.686Q-34.091-11.495-34.353-11.387Q-34.614-11.280-34.888-11.280M-34.849-11.839Q-34.489-11.839-34.237-12.108Q-33.985-12.378-33.896-12.753L-33.896-13.585Q-33.954-13.772-34.081-13.923Q-34.208-14.073-34.386-14.161Q-34.563-14.249-34.759-14.249Q-35.067-14.249-35.317-14.079Q-35.567-13.909-35.712-13.624Q-35.856-13.339-35.856-13.038Q-35.856-12.589-35.571-12.214Q-35.286-11.839-34.849-11.839M-30.384-11.280Q-30.856-11.280-31.241-11.524Q-31.626-11.768-31.849-12.178Q-32.071-12.589-32.071-13.046Q-32.071-13.389-31.946-13.712Q-31.821-14.034-31.591-14.288Q-31.360-14.542-31.054-14.686Q-30.747-14.831-30.384-14.831Q-30.021-14.831-29.708-14.684Q-29.396-14.538-29.173-14.292Q-28.950-14.046-28.823-13.725Q-28.696-13.405-28.696-13.046Q-28.696-12.589-28.921-12.176Q-29.146-11.764-29.530-11.522Q-29.915-11.280-30.384-11.280M-30.384-11.839Q-29.919-11.839-29.628-12.233Q-29.337-12.628-29.337-13.112Q-29.337-13.405-29.472-13.673Q-29.606-13.940-29.847-14.106Q-30.087-14.272-30.384-14.272Q-30.688-14.272-30.927-14.106Q-31.165-13.940-31.300-13.673Q-31.435-13.405-31.435-13.112Q-31.435-12.632-31.142-12.235Q-30.849-11.839-30.384-11.839M-27.060-12.424L-27.060-14.206L-27.810-14.206Q-28.009-14.229-28.060-14.448L-28.060-14.534Q-28.009-14.745-27.810-14.768L-27.060-14.768L-27.060-15.518Q-27.009-15.725-26.810-15.753L-26.665-15.753Q-26.470-15.725-26.419-15.518L-26.419-14.768L-25.060-14.768Q-24.868-14.749-24.810-14.534L-24.810-14.448Q-24.864-14.229-25.060-14.206L-26.419-14.206L-26.419-12.456Q-26.419-11.839-25.845-11.839Q-25.595-11.839-25.431-12.024Q-25.267-12.210-25.267-12.456Q-25.267-12.549-25.194-12.620Q-25.122-12.690-25.021-12.702L-24.876-12.702Q-24.677-12.678-24.626-12.471L-24.626-12.424Q-24.626-12.100-24.810-11.837Q-24.993-11.573-25.286-11.426Q-25.579-11.280-25.899-11.280Q-26.411-11.280-26.735-11.590Q-27.060-11.901-27.060-12.424\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(178.174 1.667)\">\u003Cpath d=\"M-21.015-10.190Q-21.015-10.389-20.878-10.538Q-20.741-10.686-20.542-10.686Q-20.405-10.686-20.310-10.600Q-20.214-10.514-20.214-10.382Q-20.214-10.260-20.280-10.151Q-20.347-10.042-20.456-9.983Q-20.362-9.964-20.187-9.954Q-20.011-9.944-19.792-9.944Q-19.417-9.944-19.097-10.204Q-18.776-10.464-18.687-10.831L-18.464-11.725Q-18.913-11.319-19.390-11.319Q-19.749-11.319-20.019-11.495Q-20.288-11.671-20.431-11.964Q-20.573-12.257-20.573-12.616Q-20.573-13.132-20.296-13.655Q-20.019-14.178-19.550-14.512Q-19.081-14.846-18.558-14.846Q-18.401-14.846-18.255-14.794Q-18.108-14.741-17.987-14.637Q-17.866-14.534-17.792-14.409Q-17.765-14.534-17.659-14.614Q-17.554-14.694-17.425-14.694Q-17.312-14.694-17.231-14.624Q-17.151-14.553-17.151-14.440Q-17.151-14.417-17.153-14.401Q-17.155-14.385-17.159-14.366L-18.062-10.776Q-18.151-10.424-18.425-10.180Q-18.698-9.936-19.069-9.811Q-19.440-9.686-19.800-9.686Q-20.288-9.686-20.651-9.774Q-21.015-9.862-21.015-10.190M-19.374-11.577Q-19.077-11.577-18.802-11.776Q-18.526-11.975-18.327-12.264L-17.901-13.952Q-17.964-14.217-18.140-14.405Q-18.315-14.592-18.573-14.592Q-18.913-14.592-19.183-14.319Q-19.452-14.046-19.608-13.671Q-19.687-13.471-19.757-13.225Q-19.827-12.979-19.874-12.715Q-19.921-12.452-19.921-12.288Q-19.921-12.104-19.862-11.942Q-19.804-11.780-19.679-11.678Q-19.554-11.577-19.374-11.577\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(178.174 1.667)\">\u003Cpath d=\"M-16.011-9.913Q-16.011-9.936-15.980-9.983Q-15.687-10.245-15.521-10.612Q-15.355-10.979-15.355-11.366L-15.355-11.424Q-15.483-11.319-15.651-11.319Q-15.843-11.319-15.980-11.452Q-16.116-11.585-16.116-11.784Q-16.116-11.975-15.980-12.108Q-15.843-12.241-15.651-12.241Q-15.351-12.241-15.226-11.971Q-15.101-11.702-15.101-11.366Q-15.101-10.917-15.282-10.503Q-15.464-10.089-15.804-9.792Q-15.827-9.768-15.866-9.768Q-15.913-9.768-15.962-9.813Q-16.011-9.858-16.011-9.913\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(178.174 1.667)\">\u003Cpath d=\"M-12.120-12.272Q-12.120-12.444-12.077-12.655Q-12.034-12.866-11.973-13.053Q-11.912-13.241-11.815-13.493Q-11.717-13.745-11.643-13.936Q-11.553-14.159-11.553-14.342Q-11.553-14.592-11.721-14.592Q-12.037-14.592-12.246-14.286Q-12.455-13.979-12.561-13.592Q-12.573-13.518-12.643-13.518L-12.745-13.518Q-12.780-13.518-12.807-13.553Q-12.834-13.589-12.834-13.616L-12.834-13.647Q-12.709-14.108-12.412-14.477Q-12.116-14.846-11.705-14.846Q-11.510-14.846-11.336-14.762Q-11.162-14.678-11.061-14.526Q-10.959-14.374-10.959-14.167Q-10.959-14.014-11.018-13.878Q-11.112-13.635-11.237-13.307Q-11.362-12.979-11.434-12.700Q-11.506-12.421-11.506-12.167Q-11.506-11.858-11.352-11.676Q-11.198-11.495-10.897-11.495Q-10.502-11.495-10.120-11.967Q-9.991-12.135-9.844-12.436Q-9.698-12.737-9.602-13.036Q-9.506-13.335-9.506-13.542Q-9.506-13.768-9.580-13.887Q-9.655-14.007-9.791-14.161Q-9.928-14.315-9.928-14.417Q-9.928-14.589-9.787-14.721Q-9.647-14.854-9.491-14.854Q-9.276-14.854-9.182-14.663Q-9.088-14.471-9.088-14.241Q-9.088-13.737-9.317-13.016Q-9.545-12.296-9.965-11.768Q-10.385-11.241-10.912-11.241Q-11.166-11.241-11.387-11.299Q-11.608-11.358-11.770-11.483Q-11.932-11.608-12.026-11.809Q-12.120-12.010-12.120-12.272\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath d=\"M224.57.097h56.905v-22.833h-56.906Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(272.772 6.306)\">\u003Cpath d=\"M-36.009-21.057L-36.009-25.147L-36.431-25.147Q-36.638-25.171-36.681-25.385L-36.681-25.475Q-36.638-25.682-36.431-25.706L-35.614-25.706Q-35.419-25.682-35.368-25.475L-35.368-23.964Q-35.157-24.131-34.894-24.219Q-34.630-24.307-34.360-24.307Q-34.021-24.307-33.724-24.163Q-33.427-24.018-33.212-23.766Q-32.997-23.514-32.882-23.202Q-32.767-22.889-32.767-22.546Q-32.767-22.081-32.993-21.671Q-33.220-21.260-33.606-21.020Q-33.993-20.780-34.462-20.780Q-34.981-20.780-35.368-21.147L-35.368-21.057Q-35.419-20.839-35.614-20.819L-35.759-20.819Q-35.950-20.842-36.009-21.057M-34.505-21.339Q-34.196-21.339-33.946-21.508Q-33.696-21.678-33.552-21.960Q-33.407-22.241-33.407-22.546Q-33.407-22.835-33.532-23.114Q-33.657-23.393-33.890-23.571Q-34.122-23.749-34.423-23.749Q-34.743-23.749-35.005-23.563Q-35.267-23.378-35.368-23.077L-35.368-22.225Q-35.278-21.858-35.058-21.598Q-34.837-21.339-34.505-21.339M-32.099-21.932Q-32.099-22.378-31.685-22.635Q-31.271-22.893-30.730-22.993Q-30.188-23.092-29.681-23.100Q-29.681-23.315-29.815-23.467Q-29.950-23.620-30.157-23.696Q-30.364-23.772-30.575-23.772Q-30.919-23.772-31.079-23.749L-31.079-23.690Q-31.079-23.522-31.198-23.407Q-31.317-23.292-31.481-23.292Q-31.657-23.292-31.772-23.415Q-31.888-23.538-31.888-23.706Q-31.888-24.112-31.507-24.221Q-31.126-24.331-30.567-24.331Q-30.298-24.331-30.030-24.253Q-29.763-24.174-29.538-24.024Q-29.313-23.874-29.177-23.653Q-29.040-23.432-29.040-23.155L-29.040-21.436Q-29.040-21.378-28.513-21.378Q-28.317-21.358-28.267-21.147L-28.267-21.057Q-28.317-20.842-28.513-20.819L-28.657-20.819Q-29.001-20.819-29.230-20.866Q-29.458-20.913-29.603-21.100Q-30.063-20.780-30.771-20.780Q-31.106-20.780-31.411-20.921Q-31.716-21.061-31.907-21.323Q-32.099-21.585-32.099-21.932M-31.458-21.924Q-31.458-21.651-31.216-21.495Q-30.974-21.339-30.688-21.339Q-30.470-21.339-30.237-21.397Q-30.005-21.456-29.843-21.594Q-29.681-21.733-29.681-21.956L-29.681-22.546Q-29.962-22.546-30.378-22.489Q-30.794-22.432-31.126-22.294Q-31.458-22.155-31.458-21.924M-27.700-22.546Q-27.700-23.026-27.456-23.440Q-27.212-23.854-26.796-24.092Q-26.380-24.331-25.899-24.331Q-25.345-24.331-24.966-24.221Q-24.587-24.112-24.587-23.706Q-24.587-23.538-24.700-23.415Q-24.813-23.292-24.985-23.292Q-25.157-23.292-25.276-23.407Q-25.396-23.522-25.396-23.690L-25.396-23.749Q-25.556-23.772-25.892-23.772Q-26.220-23.772-26.487-23.602Q-26.755-23.432-26.907-23.149Q-27.060-22.866-27.060-22.546Q-27.060-22.225-26.888-21.944Q-26.716-21.663-26.431-21.501Q-26.146-21.339-25.817-21.339Q-25.505-21.339-25.378-21.442Q-25.251-21.546-25.134-21.735Q-25.017-21.924-24.899-21.940L-24.731-21.940Q-24.626-21.928-24.562-21.860Q-24.497-21.792-24.497-21.690Q-24.497-21.643-24.517-21.604Q-24.626-21.311-24.829-21.131Q-25.032-20.952-25.308-20.866Q-25.583-20.780-25.899-20.780Q-26.384-20.780-26.800-21.018Q-27.216-21.256-27.458-21.659Q-27.700-22.061-27.700-22.546M-23.868-21.057L-23.868-21.147Q-23.817-21.354-23.622-21.378L-23.157-21.378L-23.157-25.147L-23.622-25.147Q-23.817-25.171-23.868-25.385L-23.868-25.475Q-23.817-25.682-23.622-25.706L-22.876-25.706Q-22.681-25.686-22.630-25.475L-22.630-22.651L-21.485-23.706L-21.782-23.706Q-21.989-23.729-22.028-23.948L-22.028-24.034Q-21.989-24.245-21.782-24.268L-20.333-24.268Q-20.130-24.245-20.083-24.034L-20.083-23.948Q-20.126-23.729-20.333-23.706L-20.700-23.706L-21.646-22.842L-20.493-21.378L-20.157-21.378Q-19.950-21.354-19.907-21.147L-19.907-21.057Q-19.950-20.842-20.157-20.819L-21.325-20.819Q-21.521-20.842-21.571-21.057L-21.571-21.147Q-21.521-21.354-21.325-21.378L-21.165-21.378L-22.028-22.483L-22.630-21.932L-22.630-21.378L-22.165-21.378Q-21.974-21.354-21.915-21.147L-21.915-21.057Q-21.974-20.842-22.165-20.819L-23.622-20.819Q-23.817-20.842-23.868-21.057M-18.759-21.042L-19.224-23.706L-19.384-23.706Q-19.591-23.729-19.630-23.948L-19.630-24.034Q-19.591-24.245-19.384-24.268L-18.216-24.268Q-18.005-24.245-17.966-24.034L-17.966-23.948Q-18.005-23.729-18.216-23.706L-18.688-23.706Q-18.575-23.061-18.497-22.616Q-18.419-22.171-18.368-21.852Q-18.317-21.534-18.317-21.460Q-18.310-21.631-18.024-22.628Q-17.985-22.741-17.888-22.815Q-17.790-22.889-17.669-22.889L-17.591-22.889Q-17.470-22.889-17.372-22.815Q-17.274-22.741-17.239-22.628Q-17.130-22.245-17.048-21.921Q-16.966-21.596-16.966-21.460Q-16.954-21.749-16.606-23.706L-17.079-23.706Q-17.286-23.729-17.325-23.948L-17.325-24.034Q-17.286-24.245-17.079-24.268L-15.903-24.268Q-15.712-24.245-15.653-24.034L-15.653-23.948Q-15.708-23.729-15.903-23.706L-16.071-23.706L-16.536-21.042Q-16.560-20.924-16.647-20.852Q-16.735-20.780-16.856-20.780L-16.997-20.780Q-17.118-20.780-17.214-20.852Q-17.310-20.924-17.353-21.042Q-17.399-21.221-17.472-21.477Q-17.544-21.733-17.587-21.942Q-17.630-22.151-17.630-22.253Q-17.638-21.971-17.919-21.042Q-17.946-20.932-18.044-20.856Q-18.142-20.780-18.263-20.780L-18.438-20.780Q-18.560-20.780-18.647-20.852Q-18.735-20.924-18.759-21.042M-15.114-21.932Q-15.114-22.378-14.700-22.635Q-14.286-22.893-13.745-22.993Q-13.204-23.092-12.696-23.100Q-12.696-23.315-12.831-23.467Q-12.966-23.620-13.173-23.696Q-13.380-23.772-13.591-23.772Q-13.935-23.772-14.095-23.749L-14.095-23.690Q-14.095-23.522-14.214-23.407Q-14.333-23.292-14.497-23.292Q-14.673-23.292-14.788-23.415Q-14.903-23.538-14.903-23.706Q-14.903-24.112-14.522-24.221Q-14.142-24.331-13.583-24.331Q-13.313-24.331-13.046-24.253Q-12.778-24.174-12.554-24.024Q-12.329-23.874-12.192-23.653Q-12.056-23.432-12.056-23.155L-12.056-21.436Q-12.056-21.378-11.528-21.378Q-11.333-21.358-11.282-21.147L-11.282-21.057Q-11.333-20.842-11.528-20.819L-11.673-20.819Q-12.017-20.819-12.245-20.866Q-12.474-20.913-12.618-21.100Q-13.079-20.780-13.786-20.780Q-14.122-20.780-14.427-20.921Q-14.731-21.061-14.923-21.323Q-15.114-21.585-15.114-21.932M-14.474-21.924Q-14.474-21.651-14.231-21.495Q-13.989-21.339-13.704-21.339Q-13.485-21.339-13.253-21.397Q-13.021-21.456-12.858-21.594Q-12.696-21.733-12.696-21.956L-12.696-22.546Q-12.978-22.546-13.394-22.489Q-13.810-22.432-14.142-22.294Q-14.474-22.155-14.474-21.924M-11.052-21.057L-11.052-21.147Q-10.993-21.354-10.802-21.378L-10.091-21.378L-10.091-23.706L-10.802-23.706Q-10.997-23.729-11.052-23.948L-11.052-24.034Q-10.993-24.245-10.802-24.268L-9.700-24.268Q-9.501-24.249-9.450-24.034L-9.450-23.706Q-9.188-23.991-8.833-24.149Q-8.478-24.307-8.091-24.307Q-7.798-24.307-7.563-24.173Q-7.329-24.038-7.329-23.772Q-7.329-23.604-7.438-23.487Q-7.548-23.370-7.716-23.370Q-7.868-23.370-7.983-23.481Q-8.099-23.592-8.099-23.749Q-8.474-23.749-8.788-23.548Q-9.103-23.346-9.276-23.012Q-9.450-22.678-9.450-22.299L-9.450-21.378L-8.505-21.378Q-8.298-21.354-8.259-21.147L-8.259-21.057Q-8.298-20.842-8.505-20.819L-10.802-20.819Q-10.993-20.842-11.052-21.057M-5.165-20.780Q-5.630-20.780-5.995-21.030Q-6.360-21.280-6.565-21.684Q-6.771-22.089-6.771-22.546Q-6.771-22.889-6.646-23.210Q-6.521-23.530-6.288-23.780Q-6.056-24.030-5.751-24.169Q-5.446-24.307-5.091-24.307Q-4.579-24.307-4.173-23.987L-4.173-25.147L-4.595-25.147Q-4.806-25.171-4.845-25.385L-4.845-25.475Q-4.806-25.682-4.595-25.706L-3.782-25.706Q-3.583-25.682-3.532-25.475L-3.532-21.378L-3.106-21.378Q-2.899-21.354-2.860-21.147L-2.860-21.057Q-2.899-20.842-3.106-20.819L-3.923-20.819Q-4.122-20.842-4.173-21.057L-4.173-21.186Q-4.368-20.995-4.630-20.887Q-4.892-20.780-5.165-20.780M-5.126-21.339Q-4.767-21.339-4.515-21.608Q-4.263-21.878-4.173-22.253L-4.173-23.085Q-4.231-23.272-4.358-23.423Q-4.485-23.573-4.663-23.661Q-4.841-23.749-5.036-23.749Q-5.345-23.749-5.595-23.579Q-5.845-23.409-5.989-23.124Q-6.134-22.839-6.134-22.538Q-6.134-22.089-5.849-21.714Q-5.563-21.339-5.126-21.339\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(272.772 6.306)\">\u003Cpath d=\"M-29.970-12.432Q-29.970-12.878-29.556-13.135Q-29.142-13.393-28.601-13.493Q-28.060-13.592-27.552-13.600Q-27.552-13.815-27.687-13.967Q-27.821-14.120-28.028-14.196Q-28.235-14.272-28.446-14.272Q-28.790-14.272-28.950-14.249L-28.950-14.190Q-28.950-14.022-29.069-13.907Q-29.189-13.792-29.353-13.792Q-29.528-13.792-29.644-13.915Q-29.759-14.038-29.759-14.206Q-29.759-14.612-29.378-14.721Q-28.997-14.831-28.439-14.831Q-28.169-14.831-27.901-14.753Q-27.634-14.674-27.409-14.524Q-27.185-14.374-27.048-14.153Q-26.911-13.932-26.911-13.655L-26.911-11.936Q-26.911-11.878-26.384-11.878Q-26.189-11.858-26.138-11.647L-26.138-11.557Q-26.189-11.342-26.384-11.319L-26.528-11.319Q-26.872-11.319-27.101-11.366Q-27.329-11.413-27.474-11.600Q-27.935-11.280-28.642-11.280Q-28.978-11.280-29.282-11.421Q-29.587-11.561-29.778-11.823Q-29.970-12.085-29.970-12.432M-29.329-12.424Q-29.329-12.151-29.087-11.995Q-28.845-11.839-28.560-11.839Q-28.341-11.839-28.108-11.897Q-27.876-11.956-27.714-12.094Q-27.552-12.233-27.552-12.456L-27.552-13.046Q-27.833-13.046-28.249-12.989Q-28.665-12.932-28.997-12.794Q-29.329-12.655-29.329-12.424M-25.892-10.671Q-25.892-10.971-25.743-11.233Q-25.595-11.495-25.345-11.655Q-25.528-11.909-25.528-12.229Q-25.528-12.514-25.376-12.776Q-25.626-13.100-25.626-13.518Q-25.626-13.878-25.435-14.174Q-25.243-14.471-24.925-14.639Q-24.606-14.807-24.243-14.807Q-24.036-14.807-23.833-14.747Q-23.630-14.686-23.466-14.585Q-23.083-14.846-22.610-14.846Q-22.372-14.846-22.190-14.715Q-22.009-14.585-22.009-14.358Q-22.009-14.210-22.110-14.104Q-22.212-13.999-22.368-13.999Q-22.501-13.999-22.597-14.077Q-22.692-14.155-22.724-14.280Q-22.868-14.272-23.067-14.182Q-22.864-13.870-22.864-13.518Q-22.864-13.237-22.976-13.005Q-23.087-12.772-23.282-12.594Q-23.478-12.417-23.730-12.319Q-23.981-12.221-24.243-12.221Q-24.630-12.221-24.962-12.409Q-24.974-12.409-24.983-12.337Q-24.993-12.264-24.993-12.229Q-24.993-12.108-24.927-12.001Q-24.860-11.893-24.739-11.854Q-24.724-11.858-24.710-11.860Q-24.696-11.862-24.673-11.862Q-24.657-11.862-24.626-11.854Q-24.595-11.846-24.587-11.846L-23.993-11.846Q-23.212-11.846-22.671-11.604Q-22.130-11.362-22.130-10.671Q-22.130-10.370-22.310-10.143Q-22.489-9.917-22.786-9.772Q-23.083-9.628-23.403-9.561Q-23.724-9.495-24.009-9.495Q-24.399-9.495-24.841-9.618Q-25.282-9.741-25.587-10.008Q-25.892-10.276-25.892-10.671M-25.353-10.678Q-25.353-10.460-25.112-10.319Q-24.872-10.178-24.548-10.112Q-24.224-10.046-24.009-10.046Q-23.794-10.046-23.470-10.112Q-23.146-10.178-22.905-10.319Q-22.665-10.460-22.665-10.678Q-22.665-10.967-22.890-11.106Q-23.114-11.245-23.396-11.278Q-23.677-11.311-24.024-11.311L-24.642-11.311Q-24.825-11.311-24.987-11.231Q-25.149-11.151-25.251-11.005Q-25.353-10.858-25.353-10.678M-24.243-12.776Q-23.942-12.776-23.724-12.995Q-23.505-13.214-23.505-13.518Q-23.505-13.674-23.560-13.805Q-23.614-13.936-23.720-14.042Q-23.825-14.147-23.956-14.202Q-24.087-14.257-24.243-14.257Q-24.548-14.257-24.767-14.038Q-24.985-13.819-24.985-13.518Q-24.985-13.221-24.763-12.999Q-24.540-12.776-24.243-12.776M-21.478-12.432Q-21.478-12.878-21.064-13.135Q-20.649-13.393-20.108-13.493Q-19.567-13.592-19.060-13.600Q-19.060-13.815-19.194-13.967Q-19.329-14.120-19.536-14.196Q-19.743-14.272-19.954-14.272Q-20.298-14.272-20.458-14.249L-20.458-14.190Q-20.458-14.022-20.577-13.907Q-20.696-13.792-20.860-13.792Q-21.036-13.792-21.151-13.915Q-21.267-14.038-21.267-14.206Q-21.267-14.612-20.886-14.721Q-20.505-14.831-19.946-14.831Q-19.677-14.831-19.409-14.753Q-19.142-14.674-18.917-14.524Q-18.692-14.374-18.556-14.153Q-18.419-13.932-18.419-13.655L-18.419-11.936Q-18.419-11.878-17.892-11.878Q-17.696-11.858-17.646-11.647L-17.646-11.557Q-17.696-11.342-17.892-11.319L-18.036-11.319Q-18.380-11.319-18.608-11.366Q-18.837-11.413-18.981-11.600Q-19.442-11.280-20.149-11.280Q-20.485-11.280-20.790-11.421Q-21.095-11.561-21.286-11.823Q-21.478-12.085-21.478-12.432M-20.837-12.424Q-20.837-12.151-20.595-11.995Q-20.353-11.839-20.067-11.839Q-19.849-11.839-19.616-11.897Q-19.384-11.956-19.222-12.094Q-19.060-12.233-19.060-12.456L-19.060-13.046Q-19.341-13.046-19.757-12.989Q-20.173-12.932-20.505-12.794Q-20.837-12.655-20.837-12.424M-17.021-11.557L-17.021-11.647Q-16.970-11.854-16.774-11.878L-15.735-11.878L-15.735-14.206L-16.708-14.206Q-16.907-14.229-16.958-14.448L-16.958-14.534Q-16.907-14.745-16.708-14.768L-15.341-14.768Q-15.146-14.749-15.095-14.534L-15.095-11.878L-14.181-11.878Q-13.985-11.854-13.935-11.647L-13.935-11.557Q-13.985-11.342-14.181-11.319L-16.774-11.319Q-16.970-11.342-17.021-11.557M-15.989-15.745L-15.989-15.799Q-15.989-15.971-15.853-16.092Q-15.716-16.214-15.540-16.214Q-15.368-16.214-15.231-16.092Q-15.095-15.971-15.095-15.799L-15.095-15.745Q-15.095-15.569-15.231-15.448Q-15.368-15.327-15.540-15.327Q-15.716-15.327-15.853-15.448Q-15.989-15.569-15.989-15.745M-13.321-11.557L-13.321-11.647Q-13.278-11.854-13.071-11.878L-12.649-11.878L-12.649-14.206L-13.071-14.206Q-13.278-14.229-13.321-14.448L-13.321-14.534Q-13.274-14.745-13.071-14.768L-12.255-14.768Q-12.060-14.745-12.009-14.534L-12.009-14.448L-12.017-14.424Q-11.790-14.604-11.517-14.706Q-11.243-14.807-10.950-14.807Q-10.603-14.807-10.364-14.667Q-10.126-14.526-10.011-14.268Q-9.896-14.010-9.896-13.655L-9.896-11.878L-9.470-11.878Q-9.263-11.854-9.224-11.647L-9.224-11.557Q-9.263-11.342-9.470-11.319L-10.864-11.319Q-11.060-11.342-11.110-11.557L-11.110-11.647Q-11.060-11.858-10.864-11.878L-10.536-11.878L-10.536-13.624Q-10.536-13.932-10.626-14.090Q-10.716-14.249-11.009-14.249Q-11.278-14.249-11.507-14.118Q-11.735-13.987-11.872-13.758Q-12.009-13.530-12.009-13.264L-12.009-11.878L-11.583-11.878Q-11.376-11.854-11.337-11.647L-11.337-11.557Q-11.376-11.342-11.583-11.319L-13.071-11.319Q-13.278-11.342-13.321-11.557\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-neutral)\" style=\"stroke-width:.8\">\u003Cpath d=\"M319.263.062h48.37V-22.7h-48.37Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-family=\"cmmi8\" font-size=\"8\">\u003Cg transform=\"translate(374.187 2.733)\">\u003Cpath d=\"M-34.110-11.319L-36.271-11.319Q-36.306-11.319-36.337-11.360Q-36.368-11.401-36.368-11.448L-36.345-11.549Q-36.337-11.596-36.247-11.616Q-35.806-11.616-35.647-11.655Q-35.489-11.694-35.446-11.921L-34.368-16.241Q-34.345-16.311-34.345-16.374Q-34.345-16.436-34.407-16.456Q-34.552-16.487-34.974-16.487Q-35.079-16.514-35.079-16.616L-35.048-16.717Q-35.040-16.764-34.958-16.784L-32.798-16.784Q-32.759-16.784-32.731-16.747Q-32.704-16.710-32.704-16.663L-32.728-16.557Q-32.747-16.503-32.821-16.487Q-33.263-16.487-33.423-16.448Q-33.571-16.413-33.630-16.182L-34.095-14.311L-31.552-14.311L-31.071-16.241Q-31.048-16.311-31.048-16.374Q-31.048-16.436-31.110-16.456Q-31.255-16.487-31.681-16.487Q-31.782-16.514-31.782-16.616L-31.751-16.717Q-31.743-16.764-31.661-16.784L-29.505-16.784Q-29.462-16.784-29.435-16.747Q-29.407-16.710-29.407-16.663L-29.431-16.557Q-29.450-16.503-29.528-16.487Q-29.966-16.487-30.126-16.448Q-30.271-16.413-30.337-16.182L-31.415-11.862Q-31.431-11.768-31.431-11.725Q-31.431-11.686-31.417-11.673Q-31.403-11.659-31.368-11.647Q-31.224-11.616-30.798-11.616Q-30.704-11.589-30.704-11.495L-30.728-11.389Q-30.735-11.339-30.813-11.319L-32.974-11.319Q-33.017-11.319-33.044-11.360Q-33.071-11.401-33.071-11.448L-33.048-11.549Q-33.040-11.596-32.950-11.616Q-32.513-11.616-32.353-11.655Q-32.200-11.690-32.149-11.921L-31.622-14.014L-34.165-14.014L-34.712-11.862Q-34.728-11.768-34.728-11.725Q-34.728-11.667-34.661-11.647Q-34.521-11.616-34.095-11.616Q-33.997-11.589-33.997-11.495L-34.024-11.389Q-34.032-11.339-34.110-11.319\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(374.187 2.733)\">\u003Cpath d=\"M-28.215-12.272Q-28.215-12.444-28.172-12.655Q-28.129-12.866-28.068-13.053Q-28.007-13.241-27.910-13.493Q-27.812-13.745-27.738-13.936Q-27.648-14.159-27.648-14.342Q-27.648-14.592-27.816-14.592Q-28.132-14.592-28.341-14.286Q-28.550-13.979-28.656-13.592Q-28.668-13.518-28.738-13.518L-28.840-13.518Q-28.875-13.518-28.902-13.553Q-28.929-13.589-28.929-13.616L-28.929-13.647Q-28.804-14.108-28.507-14.477Q-28.211-14.846-27.800-14.846Q-27.605-14.846-27.431-14.762Q-27.257-14.678-27.156-14.526Q-27.054-14.374-27.054-14.167Q-27.054-14.014-27.113-13.878Q-27.207-13.635-27.332-13.307Q-27.457-12.979-27.529-12.700Q-27.601-12.421-27.601-12.167Q-27.601-11.858-27.447-11.676Q-27.293-11.495-26.992-11.495Q-26.597-11.495-26.215-11.967Q-26.086-12.135-25.939-12.436Q-25.793-12.737-25.697-13.036Q-25.601-13.335-25.601-13.542Q-25.601-13.768-25.675-13.887Q-25.750-14.007-25.886-14.161Q-26.023-14.315-26.023-14.417Q-26.023-14.589-25.882-14.721Q-25.742-14.854-25.586-14.854Q-25.371-14.854-25.277-14.663Q-25.183-14.471-25.183-14.241Q-25.183-13.737-25.412-13.016Q-25.640-12.296-26.060-11.768Q-26.480-11.241-27.007-11.241Q-27.261-11.241-27.482-11.299Q-27.703-11.358-27.865-11.483Q-28.027-11.608-28.121-11.809Q-28.215-12.010-28.215-12.272\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-7.898-11.32h34.389\"\u002F>\u003Cpath stroke=\"none\" d=\"m29.09-11.32-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M86.796-11.32h34.389\"\u002F>\u003Cpath stroke=\"none\" d=\"m123.785-11.32-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"currentColor\">\u003Cg transform=\"translate(139.965 -5.094)\">\u003Cpath d=\"M-36.457-10.331Q-36.457-10.509-36.336-10.641Q-36.214-10.772-36.037-10.772Q-35.914-10.772-35.832-10.699Q-35.750-10.625-35.750-10.506Q-35.750-10.283-35.955-10.150Q-35.756-10.116-35.350-10.116Q-35.131-10.116-34.919-10.217Q-34.707-10.318-34.555-10.497Q-34.403-10.676-34.352-10.892L-34.153-11.674Q-34.563-11.319-35.001-11.319Q-35.240-11.319-35.440-11.406Q-35.640-11.493-35.785-11.654Q-35.931-11.815-36.008-12.027Q-36.084-12.238-36.084-12.467Q-36.084-12.929-35.832-13.382Q-35.579-13.835-35.157-14.120Q-34.734-14.405-34.273-14.405Q-34.061-14.405-33.873-14.305Q-33.685-14.204-33.565-14.023Q-33.535-14.132-33.444-14.202Q-33.354-14.272-33.237-14.272Q-33.135-14.272-33.066-14.211Q-32.998-14.149-32.998-14.050Q-32.998-13.999-33.005-13.978L-33.791-10.844Q-33.870-10.536-34.117-10.319Q-34.365-10.102-34.697-9.996Q-35.028-9.890-35.356-9.890Q-35.797-9.890-36.127-9.967Q-36.457-10.044-36.457-10.331M-34.987-11.545Q-34.714-11.545-34.463-11.726Q-34.211-11.907-34.027-12.167L-33.671-13.602Q-33.730-13.859-33.888-14.021Q-34.047-14.183-34.287-14.183Q-34.591-14.183-34.842-13.939Q-35.093-13.694-35.233-13.363Q-35.333-13.096-35.415-12.751Q-35.497-12.406-35.497-12.180Q-35.497-11.921-35.365-11.733Q-35.233-11.545-34.987-11.545\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M187.18-11.32h34.39\"\u002F>\u003Cpath stroke=\"none\" d=\"m224.17-11.32-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M281.875-11.32h34.388\"\u002F>\u003Cpath stroke=\"none\" d=\"m318.863-11.32-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-neutral)\" style=\"stroke-width:.8\">\u003Cpath d=\"M131.298-49.108h48.37V-71.87h-48.37Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-size=\"8\">\u003Cg transform=\"translate(175.854 -46.955)\">\u003Cpath d=\"M-35.095-11.542L-35.981-14.206L-36.302-14.206Q-36.501-14.229-36.552-14.448L-36.552-14.534Q-36.501-14.745-36.302-14.768L-35.142-14.768Q-34.946-14.749-34.896-14.534L-34.896-14.448Q-34.946-14.229-35.142-14.206L-35.423-14.206L-34.630-11.831L-33.841-14.206L-34.118-14.206Q-34.317-14.229-34.368-14.448L-34.368-14.534Q-34.317-14.745-34.118-14.768L-32.958-14.768Q-32.763-14.745-32.712-14.534L-32.712-14.448Q-32.763-14.229-32.958-14.206L-33.278-14.206L-34.165-11.542Q-34.208-11.428-34.310-11.354Q-34.411-11.280-34.536-11.280L-34.728-11.280Q-34.845-11.280-34.948-11.352Q-35.052-11.424-35.095-11.542M-28.993-12.807L-31.435-12.807Q-31.380-12.530-31.183-12.307Q-30.985-12.085-30.708-11.962Q-30.431-11.839-30.146-11.839Q-29.673-11.839-29.450-12.128Q-29.442-12.139-29.386-12.245Q-29.329-12.350-29.280-12.393Q-29.231-12.436-29.138-12.448L-28.993-12.448Q-28.802-12.428-28.743-12.214L-28.743-12.159Q-28.810-11.858-29.040-11.661Q-29.271-11.464-29.583-11.372Q-29.896-11.280-30.200-11.280Q-30.685-11.280-31.124-11.508Q-31.563-11.737-31.831-12.137Q-32.099-12.538-32.099-13.030L-32.099-13.089Q-32.099-13.557-31.853-13.960Q-31.606-14.362-31.198-14.596Q-30.790-14.831-30.321-14.831Q-29.817-14.831-29.464-14.608Q-29.110-14.385-28.927-13.997Q-28.743-13.608-28.743-13.104L-28.743-13.046Q-28.802-12.831-28.993-12.807M-31.427-13.358L-29.399-13.358Q-29.446-13.768-29.685-14.020Q-29.923-14.272-30.321-14.272Q-30.716-14.272-31.022-14.010Q-31.329-13.749-31.427-13.358M-27.700-13.046Q-27.700-13.526-27.456-13.940Q-27.212-14.354-26.796-14.592Q-26.380-14.831-25.899-14.831Q-25.345-14.831-24.966-14.721Q-24.587-14.612-24.587-14.206Q-24.587-14.038-24.700-13.915Q-24.813-13.792-24.985-13.792Q-25.157-13.792-25.276-13.907Q-25.396-14.022-25.396-14.190L-25.396-14.249Q-25.556-14.272-25.892-14.272Q-26.220-14.272-26.487-14.102Q-26.755-13.932-26.907-13.649Q-27.060-13.366-27.060-13.046Q-27.060-12.725-26.888-12.444Q-26.716-12.163-26.431-12.001Q-26.146-11.839-25.817-11.839Q-25.505-11.839-25.378-11.942Q-25.251-12.046-25.134-12.235Q-25.017-12.424-24.899-12.440L-24.731-12.440Q-24.626-12.428-24.562-12.360Q-24.497-12.292-24.497-12.190Q-24.497-12.143-24.517-12.104Q-24.626-11.811-24.829-11.632Q-25.032-11.452-25.308-11.366Q-25.583-11.280-25.899-11.280Q-26.384-11.280-26.800-11.518Q-27.216-11.757-27.458-12.159Q-27.700-12.561-27.700-13.046M-22.813-12.424L-22.813-14.206L-23.563-14.206Q-23.763-14.229-23.813-14.448L-23.813-14.534Q-23.763-14.745-23.563-14.768L-22.813-14.768L-22.813-15.518Q-22.763-15.725-22.563-15.753L-22.419-15.753Q-22.224-15.725-22.173-15.518L-22.173-14.768L-20.813-14.768Q-20.622-14.749-20.563-14.534L-20.563-14.448Q-20.618-14.229-20.813-14.206L-22.173-14.206L-22.173-12.456Q-22.173-11.839-21.599-11.839Q-21.349-11.839-21.185-12.024Q-21.021-12.210-21.021-12.456Q-21.021-12.549-20.948-12.620Q-20.876-12.690-20.774-12.702L-20.630-12.702Q-20.431-12.678-20.380-12.471L-20.380-12.424Q-20.380-12.100-20.563-11.837Q-20.747-11.573-21.040-11.426Q-21.333-11.280-21.653-11.280Q-22.165-11.280-22.489-11.590Q-22.813-11.901-22.813-12.424M-17.646-11.280Q-18.118-11.280-18.503-11.524Q-18.888-11.768-19.110-12.178Q-19.333-12.589-19.333-13.046Q-19.333-13.389-19.208-13.712Q-19.083-14.034-18.853-14.288Q-18.622-14.542-18.315-14.686Q-18.009-14.831-17.646-14.831Q-17.282-14.831-16.970-14.684Q-16.657-14.538-16.435-14.292Q-16.212-14.046-16.085-13.725Q-15.958-13.405-15.958-13.046Q-15.958-12.589-16.183-12.176Q-16.407-11.764-16.792-11.522Q-17.177-11.280-17.646-11.280M-17.646-11.839Q-17.181-11.839-16.890-12.233Q-16.599-12.628-16.599-13.112Q-16.599-13.405-16.733-13.673Q-16.868-13.940-17.108-14.106Q-17.349-14.272-17.646-14.272Q-17.950-14.272-18.188-14.106Q-18.427-13.940-18.562-13.673Q-18.696-13.405-18.696-13.112Q-18.696-12.632-18.403-12.235Q-18.110-11.839-17.646-11.839M-15.298-11.557L-15.298-11.647Q-15.239-11.854-15.048-11.878L-14.337-11.878L-14.337-14.206L-15.048-14.206Q-15.243-14.229-15.298-14.448L-15.298-14.534Q-15.239-14.745-15.048-14.768L-13.946-14.768Q-13.747-14.749-13.696-14.534L-13.696-14.206Q-13.435-14.491-13.079-14.649Q-12.724-14.807-12.337-14.807Q-12.044-14.807-11.810-14.673Q-11.575-14.538-11.575-14.272Q-11.575-14.104-11.685-13.987Q-11.794-13.870-11.962-13.870Q-12.114-13.870-12.230-13.981Q-12.345-14.092-12.345-14.249Q-12.720-14.249-13.034-14.048Q-13.349-13.846-13.522-13.512Q-13.696-13.178-13.696-12.799L-13.696-11.878L-12.751-11.878Q-12.544-11.854-12.505-11.647L-12.505-11.557Q-12.544-11.342-12.751-11.319L-15.048-11.319Q-15.239-11.342-15.298-11.557\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(175.854 -46.955)\">\u003Cpath d=\"M-7.479-12.272Q-7.479-12.444-7.437-12.655Q-7.394-12.866-7.333-13.053Q-7.272-13.241-7.175-13.493Q-7.077-13.745-7.003-13.936Q-6.913-14.159-6.913-14.342Q-6.913-14.592-7.081-14.592Q-7.397-14.592-7.606-14.286Q-7.815-13.979-7.921-13.592Q-7.933-13.518-8.003-13.518L-8.104-13.518Q-8.140-13.518-8.167-13.553Q-8.194-13.589-8.194-13.616L-8.194-13.647Q-8.069-14.108-7.772-14.477Q-7.476-14.846-7.065-14.846Q-6.870-14.846-6.696-14.762Q-6.522-14.678-6.421-14.526Q-6.319-14.374-6.319-14.167Q-6.319-14.014-6.378-13.878Q-6.472-13.635-6.597-13.307Q-6.722-12.979-6.794-12.700Q-6.866-12.421-6.866-12.167Q-6.866-11.858-6.712-11.676Q-6.558-11.495-6.257-11.495Q-5.862-11.495-5.479-11.967Q-5.351-12.135-5.204-12.436Q-5.058-12.737-4.962-13.036Q-4.866-13.335-4.866-13.542Q-4.866-13.768-4.940-13.887Q-5.015-14.007-5.151-14.161Q-5.288-14.315-5.288-14.417Q-5.288-14.589-5.147-14.721Q-5.007-14.854-4.851-14.854Q-4.636-14.854-4.542-14.663Q-4.448-14.471-4.448-14.241Q-4.448-13.737-4.677-13.016Q-4.905-12.296-5.325-11.768Q-5.745-11.241-6.272-11.241Q-6.526-11.241-6.747-11.299Q-6.968-11.358-7.130-11.483Q-7.292-11.608-7.386-11.809Q-7.479-12.010-7.479-12.272\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M155.483-48.708V-25.7\"\u002F>\u003Cpath stroke=\"none\" d=\"m155.483-23.1 2.08-4.16-2.08 1.56-2.08-1.56\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmtt8\" font-size=\"7\">\u003Cg transform=\"translate(84.113 19.627)\">\u003Cpath d=\"M-21.076-11.285Q-21.490-11.285-21.826-11.498Q-22.163-11.712-22.358-12.071Q-22.553-12.430-22.553-12.830Q-22.553-13.131-22.443-13.413Q-22.334-13.694-22.132-13.917Q-21.931-14.139-21.662-14.265Q-21.394-14.392-21.076-14.392Q-20.758-14.392-20.485-14.264Q-20.211-14.135-20.016-13.920Q-19.822-13.705-19.711-13.424Q-19.599-13.144-19.599-12.830Q-19.599-12.430-19.796-12.069Q-19.993-11.709-20.329-11.497Q-20.666-11.285-21.076-11.285M-21.076-11.774Q-20.669-11.774-20.415-12.119Q-20.160-12.464-20.160-12.888Q-20.160-13.144-20.278-13.378Q-20.396-13.612-20.606-13.758Q-20.816-13.903-21.076-13.903Q-21.343-13.903-21.551-13.758Q-21.760-13.612-21.878-13.378Q-21.995-13.144-21.995-12.888Q-21.995-12.467-21.739-12.121Q-21.483-11.774-21.076-11.774M-19.155-11.527L-19.155-11.606Q-19.118-11.787-18.936-11.808L-18.567-11.808L-18.567-13.845L-18.936-13.845Q-19.118-13.865-19.155-14.057L-19.155-14.132Q-19.114-14.317-18.936-14.337L-18.222-14.337Q-18.051-14.317-18.007-14.132L-18.007-14.057L-18.014-14.036Q-17.815-14.194-17.576-14.282Q-17.337-14.371-17.080-14.371Q-16.776-14.371-16.568-14.248Q-16.359-14.125-16.258-13.900Q-16.158-13.674-16.158-13.363L-16.158-11.808L-15.785-11.808Q-15.604-11.787-15.570-11.606L-15.570-11.527Q-15.604-11.340-15.785-11.319L-17.005-11.319Q-17.176-11.340-17.221-11.527L-17.221-11.606Q-17.176-11.791-17.005-11.808L-16.718-11.808L-16.718-13.336Q-16.718-13.606-16.797-13.744Q-16.875-13.882-17.132-13.882Q-17.368-13.882-17.567-13.768Q-17.767-13.653-17.887-13.454Q-18.007-13.254-18.007-13.021L-18.007-11.808L-17.634-11.808Q-17.453-11.787-17.419-11.606L-17.419-11.527Q-17.453-11.340-17.634-11.319L-18.936-11.319Q-19.118-11.340-19.155-11.527M-12.429-12.621L-14.565-12.621Q-14.517-12.379-14.344-12.184Q-14.172-11.989-13.929-11.881Q-13.686-11.774-13.437-11.774Q-13.023-11.774-12.828-12.027Q-12.822-12.037-12.772-12.129Q-12.723-12.221-12.680-12.259Q-12.637-12.297-12.555-12.307L-12.429-12.307Q-12.261-12.290-12.210-12.102L-12.210-12.054Q-12.268-11.791-12.470-11.618Q-12.671-11.445-12.945-11.365Q-13.218-11.285-13.485-11.285Q-13.909-11.285-14.293-11.485Q-14.678-11.685-14.912-12.035Q-15.146-12.385-15.146-12.816L-15.146-12.867Q-15.146-13.277-14.931-13.630Q-14.715-13.982-14.358-14.187Q-14.001-14.392-13.591-14.392Q-13.150-14.392-12.840-14.197Q-12.531-14.002-12.370-13.662Q-12.210-13.322-12.210-12.881L-12.210-12.830Q-12.261-12.642-12.429-12.621M-14.558-13.103L-12.784-13.103Q-12.825-13.462-13.034-13.683Q-13.242-13.903-13.591-13.903Q-13.936-13.903-14.204-13.674Q-14.473-13.445-14.558-13.103\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(84.113 19.627)\">\u003Cpath d=\"M-8.838-11.527L-8.838-11.606Q-8.793-11.787-8.622-11.808L-7.850-11.808L-7.850-13.845L-8.598-13.845Q-8.773-13.865-8.817-14.057L-8.817-14.132Q-8.773-14.317-8.598-14.337L-7.850-14.337L-7.850-14.734Q-7.850-15.140-7.494-15.386Q-7.139-15.632-6.719-15.632Q-6.445-15.632-6.232-15.527Q-6.018-15.421-6.018-15.174Q-6.018-15.031-6.114-14.928Q-6.209-14.826-6.353-14.826Q-6.483-14.826-6.589-14.918Q-6.695-15.010-6.695-15.140L-6.766-15.140Q-6.900-15.140-7.019-15.087Q-7.139-15.034-7.214-14.934Q-7.289-14.833-7.289-14.706L-7.289-14.337L-6.387-14.337Q-6.216-14.320-6.172-14.132L-6.172-14.057Q-6.216-13.865-6.387-13.845L-7.289-13.845L-7.289-11.808L-6.520-11.808Q-6.349-11.787-6.305-11.606L-6.305-11.527Q-6.349-11.340-6.520-11.319L-8.622-11.319Q-8.793-11.340-8.838-11.527M-3.547-11.285Q-3.960-11.285-4.297-11.498Q-4.634-11.712-4.828-12.071Q-5.023-12.430-5.023-12.830Q-5.023-13.131-4.914-13.413Q-4.805-13.694-4.603-13.917Q-4.401-14.139-4.133-14.265Q-3.865-14.392-3.547-14.392Q-3.229-14.392-2.955-14.264Q-2.682-14.135-2.487-13.920Q-2.292-13.705-2.181-13.424Q-2.070-13.144-2.070-12.830Q-2.070-12.430-2.267-12.069Q-2.463-11.709-2.800-11.497Q-3.137-11.285-3.547-11.285M-3.547-11.774Q-3.140-11.774-2.885-12.119Q-2.631-12.464-2.631-12.888Q-2.631-13.144-2.749-13.378Q-2.867-13.612-3.077-13.758Q-3.287-13.903-3.547-13.903Q-3.813-13.903-4.022-13.758Q-4.230-13.612-4.348-13.378Q-4.466-13.144-4.466-12.888Q-4.466-12.467-4.210-12.121Q-3.953-11.774-3.547-11.774M-1.493-11.527L-1.493-11.606Q-1.441-11.787-1.274-11.808L-0.652-11.808L-0.652-13.845L-1.274-13.845Q-1.445-13.865-1.493-14.057L-1.493-14.132Q-1.441-14.317-1.274-14.337L-0.310-14.337Q-0.136-14.320-0.091-14.132L-0.091-13.845Q0.138-14.094 0.449-14.233Q0.760-14.371 1.098-14.371Q1.355-14.371 1.560-14.253Q1.765-14.135 1.765-13.903Q1.765-13.756 1.669-13.653Q1.573-13.551 1.426-13.551Q1.293-13.551 1.192-13.648Q1.091-13.746 1.091-13.882Q0.763-13.882 0.488-13.706Q0.213-13.530 0.061-13.238Q-0.091-12.946-0.091-12.614L-0.091-11.808L0.736-11.808Q0.917-11.787 0.951-11.606L0.951-11.527Q0.917-11.340 0.736-11.319L-1.274-11.319Q-1.441-11.340-1.493-11.527M2.910-11.514L2.503-13.845L2.363-13.845Q2.182-13.865 2.148-14.057L2.148-14.132Q2.182-14.317 2.363-14.337L3.385-14.337Q3.569-14.317 3.604-14.132L3.604-14.057Q3.569-13.865 3.385-13.845L2.971-13.845Q3.070-13.281 3.139-12.891Q3.207-12.502 3.252-12.223Q3.296-11.944 3.296-11.880Q3.303-12.030 3.552-12.902Q3.587-13.001 3.672-13.066Q3.757-13.131 3.863-13.131L3.932-13.131Q4.038-13.131 4.123-13.066Q4.209-13.001 4.239-12.902Q4.335-12.567 4.407-12.283Q4.479-11.999 4.479-11.880Q4.489-12.132 4.793-13.845L4.380-13.845Q4.198-13.865 4.164-14.057L4.164-14.132Q4.198-14.317 4.380-14.337L5.408-14.337Q5.576-14.317 5.627-14.132L5.627-14.057Q5.579-13.865 5.408-13.845L5.261-13.845L4.855-11.514Q4.834-11.411 4.757-11.348Q4.680-11.285 4.574-11.285L4.451-11.285Q4.345-11.285 4.262-11.348Q4.178-11.411 4.140-11.514Q4.099-11.671 4.036-11.895Q3.973-12.119 3.935-12.302Q3.898-12.485 3.898-12.573Q3.891-12.327 3.645-11.514Q3.621-11.418 3.535-11.351Q3.450-11.285 3.344-11.285L3.190-11.285Q3.084-11.285 3.007-11.348Q2.930-11.411 2.910-11.514M6.099-12.293Q6.099-12.683 6.461-12.908Q6.823-13.134 7.297-13.221Q7.770-13.308 8.215-13.315Q8.215-13.503 8.097-13.636Q7.979-13.770 7.798-13.836Q7.616-13.903 7.432-13.903Q7.131-13.903 6.991-13.882L6.991-13.831Q6.991-13.684 6.887-13.583Q6.782-13.483 6.639-13.483Q6.485-13.483 6.384-13.590Q6.283-13.698 6.283-13.845Q6.283-14.200 6.617-14.296Q6.950-14.392 7.439-14.392Q7.674-14.392 7.909-14.323Q8.143-14.255 8.339-14.123Q8.536-13.992 8.655-13.799Q8.775-13.606 8.775-13.363L8.775-11.859Q8.775-11.808 9.236-11.808Q9.407-11.791 9.452-11.606L9.452-11.527Q9.407-11.340 9.236-11.319L9.110-11.319Q8.809-11.319 8.609-11.360Q8.409-11.401 8.283-11.565Q7.880-11.285 7.261-11.285Q6.967-11.285 6.700-11.408Q6.434-11.531 6.266-11.760Q6.099-11.989 6.099-12.293M6.659-12.286Q6.659-12.047 6.871-11.910Q7.083-11.774 7.333-11.774Q7.524-11.774 7.727-11.825Q7.931-11.876 8.073-11.997Q8.215-12.119 8.215-12.314L8.215-12.830Q7.968-12.830 7.604-12.780Q7.240-12.731 6.950-12.609Q6.659-12.488 6.659-12.286M9.653-11.527L9.653-11.606Q9.705-11.787 9.872-11.808L10.494-11.808L10.494-13.845L9.872-13.845Q9.701-13.865 9.653-14.057L9.653-14.132Q9.705-14.317 9.872-14.337L10.836-14.337Q11.010-14.320 11.055-14.132L11.055-13.845Q11.284-14.094 11.595-14.233Q11.906-14.371 12.244-14.371Q12.501-14.371 12.706-14.253Q12.911-14.135 12.911-13.903Q12.911-13.756 12.815-13.653Q12.719-13.551 12.572-13.551Q12.439-13.551 12.338-13.648Q12.237-13.746 12.237-13.882Q11.909-13.882 11.634-13.706Q11.359-13.530 11.207-13.238Q11.055-12.946 11.055-12.614L11.055-11.808L11.882-11.808Q12.063-11.787 12.097-11.606L12.097-11.527Q12.063-11.340 11.882-11.319L9.872-11.319Q9.705-11.340 9.653-11.527M14.804-11.285Q14.398-11.285 14.078-11.504Q13.758-11.722 13.579-12.076Q13.400-12.430 13.400-12.830Q13.400-13.131 13.509-13.411Q13.618-13.691 13.822-13.910Q14.025-14.129 14.292-14.250Q14.558-14.371 14.869-14.371Q15.317-14.371 15.673-14.091L15.673-15.106L15.303-15.106Q15.119-15.127 15.085-15.315L15.085-15.393Q15.119-15.574 15.303-15.595L16.014-15.595Q16.189-15.574 16.233-15.393L16.233-11.808L16.606-11.808Q16.787-11.787 16.821-11.606L16.821-11.527Q16.787-11.340 16.606-11.319L15.891-11.319Q15.717-11.340 15.673-11.527L15.673-11.640Q15.502-11.473 15.273-11.379Q15.044-11.285 14.804-11.285M14.839-11.774Q15.153-11.774 15.373-12.009Q15.594-12.245 15.673-12.573L15.673-13.301Q15.621-13.465 15.510-13.597Q15.399-13.729 15.244-13.806Q15.088-13.882 14.917-13.882Q14.647-13.882 14.428-13.734Q14.210-13.585 14.083-13.336Q13.957-13.086 13.957-12.823Q13.957-12.430 14.206-12.102Q14.456-11.774 14.839-11.774\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(84.113 19.627)\">\u003Cpath d=\"M19.633-9.969L19.633-10.044Q19.670-10.236 19.851-10.256L20.220-10.256L20.220-13.845L19.851-13.845Q19.670-13.865 19.633-14.057L19.633-14.132Q19.674-14.317 19.851-14.337L20.566-14.337Q20.737-14.320 20.781-14.132L20.781-14.070Q20.966-14.217 21.196-14.294Q21.427-14.371 21.663-14.371Q21.960-14.371 22.220-14.245Q22.480-14.118 22.668-13.898Q22.856-13.677 22.956-13.404Q23.057-13.131 23.057-12.830Q23.057-12.423 22.859-12.064Q22.661-11.705 22.322-11.495Q21.984-11.285 21.574-11.285Q21.119-11.285 20.781-11.606L20.781-10.256L21.154-10.256Q21.335-10.236 21.369-10.044L21.369-9.969Q21.335-9.784 21.154-9.764L19.851-9.764Q19.674-9.784 19.633-9.969M21.536-11.774Q21.806-11.774 22.025-11.922Q22.244-12.071 22.370-12.317Q22.497-12.563 22.497-12.830Q22.497-13.083 22.387-13.327Q22.278-13.571 22.075-13.727Q21.871-13.882 21.608-13.882Q21.328-13.882 21.099-13.720Q20.870-13.558 20.781-13.295L20.781-12.549Q20.860-12.228 21.053-12.001Q21.246-11.774 21.536-11.774M23.642-12.293Q23.642-12.683 24.004-12.908Q24.366-13.134 24.840-13.221Q25.313-13.308 25.758-13.315Q25.758-13.503 25.640-13.636Q25.522-13.770 25.341-13.836Q25.159-13.903 24.975-13.903Q24.674-13.903 24.534-13.882L24.534-13.831Q24.534-13.684 24.430-13.583Q24.325-13.483 24.182-13.483Q24.028-13.483 23.927-13.590Q23.826-13.698 23.826-13.845Q23.826-14.200 24.160-14.296Q24.493-14.392 24.982-14.392Q25.217-14.392 25.452-14.323Q25.686-14.255 25.882-14.123Q26.079-13.992 26.198-13.799Q26.318-13.606 26.318-13.363L26.318-11.859Q26.318-11.808 26.779-11.808Q26.950-11.791 26.995-11.606L26.995-11.527Q26.950-11.340 26.779-11.319L26.653-11.319Q26.352-11.319 26.152-11.360Q25.952-11.401 25.826-11.565Q25.423-11.285 24.804-11.285Q24.510-11.285 24.243-11.408Q23.977-11.531 23.809-11.760Q23.642-11.989 23.642-12.293M24.202-12.286Q24.202-12.047 24.414-11.910Q24.626-11.774 24.876-11.774Q25.067-11.774 25.270-11.825Q25.474-11.876 25.616-11.997Q25.758-12.119 25.758-12.314L25.758-12.830Q25.511-12.830 25.147-12.780Q24.783-12.731 24.493-12.609Q24.202-12.488 24.202-12.286M27.514-11.493L27.514-12.293Q27.538-12.474 27.723-12.495L27.870-12.495Q28.013-12.474 28.065-12.334Q28.242-11.774 28.878-11.774Q29.059-11.774 29.259-11.804Q29.459-11.835 29.606-11.936Q29.753-12.037 29.753-12.215Q29.753-12.399 29.558-12.498Q29.363-12.597 29.124-12.635L28.512-12.734Q27.514-12.919 27.514-13.551Q27.514-13.804 27.641-13.970Q27.767-14.135 27.977-14.229Q28.188-14.323 28.412-14.358Q28.635-14.392 28.878-14.392Q29.097-14.392 29.266-14.366Q29.435-14.340 29.579-14.272Q29.647-14.375 29.760-14.392L29.828-14.392Q29.914-14.382 29.968-14.327Q30.023-14.272 30.033-14.190L30.033-13.571Q30.023-13.489 29.968-13.431Q29.914-13.373 29.828-13.363L29.681-13.363Q29.599-13.373 29.541-13.431Q29.483-13.489 29.473-13.571Q29.473-13.903 28.864-13.903Q28.560-13.903 28.282-13.831Q28.003-13.759 28.003-13.544Q28.003-13.312 28.591-13.216L29.206-13.110Q29.630-13.038 29.936-12.821Q30.242-12.604 30.242-12.215Q30.242-11.873 30.035-11.661Q29.828-11.449 29.522-11.367Q29.217-11.285 28.878-11.285Q28.372-11.285 28.024-11.507Q27.962-11.398 27.919-11.348Q27.877-11.298 27.784-11.285L27.723-11.285Q27.535-11.305 27.514-11.493M31.230-11.493L31.230-12.293Q31.254-12.474 31.438-12.495L31.585-12.495Q31.729-12.474 31.780-12.334Q31.958-11.774 32.593-11.774Q32.775-11.774 32.975-11.804Q33.175-11.835 33.321-11.936Q33.468-12.037 33.468-12.215Q33.468-12.399 33.274-12.498Q33.079-12.597 32.840-12.635L32.228-12.734Q31.230-12.919 31.230-13.551Q31.230-13.804 31.356-13.970Q31.483-14.135 31.693-14.229Q31.903-14.323 32.127-14.358Q32.351-14.392 32.593-14.392Q32.812-14.392 32.981-14.366Q33.151-14.340 33.294-14.272Q33.363-14.375 33.475-14.392L33.544-14.392Q33.629-14.382 33.684-14.327Q33.738-14.272 33.749-14.190L33.749-13.571Q33.738-13.489 33.684-13.431Q33.629-13.373 33.544-13.363L33.397-13.363Q33.315-13.373 33.257-13.431Q33.198-13.489 33.188-13.571Q33.188-13.903 32.580-13.903Q32.276-13.903 31.997-13.831Q31.718-13.759 31.718-13.544Q31.718-13.312 32.306-13.216L32.922-13.110Q33.345-13.038 33.651-12.821Q33.957-12.604 33.957-12.215Q33.957-11.873 33.750-11.661Q33.544-11.449 33.238-11.367Q32.932-11.285 32.593-11.285Q32.088-11.285 31.739-11.507Q31.677-11.398 31.635-11.348Q31.592-11.298 31.500-11.285L31.438-11.285Q31.250-11.305 31.230-11.493M35.936-10.345Q35.837-10.345 35.758-10.423Q35.680-10.502 35.680-10.598Q35.680-10.752 35.847-10.817Q36.039-10.861 36.189-10.999Q36.340-11.138 36.398-11.333Q36.377-11.333 36.350-11.319L36.295-11.319Q36.093-11.319 35.948-11.461Q35.803-11.603 35.803-11.808Q35.803-12.013 35.946-12.153Q36.090-12.293 36.295-12.293Q36.480-12.293 36.623-12.187Q36.767-12.081 36.845-11.912Q36.924-11.743 36.924-11.558Q36.924-11.141 36.663-10.801Q36.401-10.461 35.994-10.352Q35.974-10.345 35.936-10.345\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(84.113 19.627)\">\u003Cpath d=\"M41.871-12.286L41.871-13.845L41.215-13.845Q41.041-13.865 40.996-14.057L40.996-14.132Q41.041-14.317 41.215-14.337L41.871-14.337L41.871-14.993Q41.916-15.174 42.090-15.198L42.217-15.198Q42.387-15.174 42.432-14.993L42.432-14.337L43.621-14.337Q43.789-14.320 43.840-14.132L43.840-14.057Q43.792-13.865 43.621-13.845L42.432-13.845L42.432-12.314Q42.432-11.774 42.934-11.774Q43.153-11.774 43.297-11.936Q43.440-12.098 43.440-12.314Q43.440-12.396 43.503-12.457Q43.567-12.519 43.655-12.529L43.782-12.529Q43.956-12.508 44.001-12.327L44.001-12.286Q44.001-12.003 43.840-11.772Q43.679-11.541 43.423-11.413Q43.167-11.285 42.886-11.285Q42.439-11.285 42.155-11.557Q41.871-11.828 41.871-12.286M45.419-11.514L45.012-13.845L44.872-13.845Q44.691-13.865 44.657-14.057L44.657-14.132Q44.691-14.317 44.872-14.337L45.894-14.337Q46.079-14.317 46.113-14.132L46.113-14.057Q46.079-13.865 45.894-13.845L45.481-13.845Q45.580-13.281 45.648-12.891Q45.717-12.502 45.761-12.223Q45.805-11.944 45.805-11.880Q45.812-12.030 46.062-12.902Q46.096-13.001 46.181-13.066Q46.267-13.131 46.373-13.131L46.441-13.131Q46.547-13.131 46.633-13.066Q46.718-13.001 46.749-12.902Q46.844-12.567 46.916-12.283Q46.988-11.999 46.988-11.880Q46.998-12.132 47.302-13.845L46.889-13.845Q46.708-13.865 46.674-14.057L46.674-14.132Q46.708-14.317 46.889-14.337L47.918-14.337Q48.085-14.317 48.136-14.132L48.136-14.057Q48.089-13.865 47.918-13.845L47.771-13.845L47.364-11.514Q47.343-11.411 47.267-11.348Q47.190-11.285 47.084-11.285L46.961-11.285Q46.855-11.285 46.771-11.348Q46.687-11.411 46.650-11.514Q46.609-11.671 46.545-11.895Q46.482-12.119 46.445-12.302Q46.407-12.485 46.407-12.573Q46.400-12.327 46.154-11.514Q46.130-11.418 46.045-11.351Q45.959-11.285 45.853-11.285L45.699-11.285Q45.593-11.285 45.517-11.348Q45.440-11.411 45.419-11.514M50.109-11.285Q49.695-11.285 49.358-11.498Q49.022-11.712 48.827-12.071Q48.632-12.430 48.632-12.830Q48.632-13.131 48.741-13.413Q48.851-13.694 49.052-13.917Q49.254-14.139 49.522-14.265Q49.791-14.392 50.109-14.392Q50.426-14.392 50.700-14.264Q50.973-14.135 51.168-13.920Q51.363-13.705 51.474-13.424Q51.585-13.144 51.585-12.830Q51.585-12.430 51.389-12.069Q51.192-11.709 50.855-11.497Q50.519-11.285 50.109-11.285M50.109-11.774Q50.515-11.774 50.770-12.119Q51.025-12.464 51.025-12.888Q51.025-13.144 50.907-13.378Q50.789-13.612 50.579-13.758Q50.368-13.903 50.109-13.903Q49.842-13.903 49.634-13.758Q49.425-13.612 49.307-13.378Q49.189-13.144 49.189-12.888Q49.189-12.467 49.446-12.121Q49.702-11.774 50.109-11.774\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(84.113 19.627)\">\u003Cpath d=\"M55.284-11.527L55.284-15.106L54.915-15.106Q54.734-15.127 54.697-15.315L54.697-15.393Q54.734-15.574 54.915-15.595L55.630-15.595Q55.801-15.574 55.845-15.393L55.845-14.070Q56.030-14.217 56.260-14.294Q56.491-14.371 56.727-14.371Q57.024-14.371 57.284-14.245Q57.544-14.118 57.732-13.898Q57.920-13.677 58.020-13.404Q58.121-13.131 58.121-12.830Q58.121-12.423 57.923-12.064Q57.725-11.705 57.386-11.495Q57.048-11.285 56.638-11.285Q56.183-11.285 55.845-11.606L55.845-11.527Q55.801-11.336 55.630-11.319L55.503-11.319Q55.336-11.340 55.284-11.527M56.600-11.774Q56.870-11.774 57.089-11.922Q57.308-12.071 57.434-12.317Q57.561-12.563 57.561-12.830Q57.561-13.083 57.451-13.327Q57.342-13.571 57.139-13.727Q56.935-13.882 56.672-13.882Q56.392-13.882 56.163-13.720Q55.934-13.558 55.845-13.295L55.845-12.549Q55.924-12.228 56.117-12.001Q56.310-11.774 56.600-11.774M58.706-12.293Q58.706-12.683 59.068-12.908Q59.430-13.134 59.904-13.221Q60.377-13.308 60.822-13.315Q60.822-13.503 60.704-13.636Q60.586-13.770 60.405-13.836Q60.223-13.903 60.039-13.903Q59.738-13.903 59.598-13.882L59.598-13.831Q59.598-13.684 59.494-13.583Q59.389-13.483 59.246-13.483Q59.092-13.483 58.991-13.590Q58.890-13.698 58.890-13.845Q58.890-14.200 59.224-14.296Q59.557-14.392 60.046-14.392Q60.281-14.392 60.516-14.323Q60.750-14.255 60.946-14.123Q61.143-13.992 61.262-13.799Q61.382-13.606 61.382-13.363L61.382-11.859Q61.382-11.808 61.843-11.808Q62.014-11.791 62.059-11.606L62.059-11.527Q62.014-11.340 61.843-11.319L61.717-11.319Q61.416-11.319 61.216-11.360Q61.016-11.401 60.890-11.565Q60.487-11.285 59.868-11.285Q59.574-11.285 59.307-11.408Q59.041-11.531 58.873-11.760Q58.706-11.989 58.706-12.293M59.266-12.286Q59.266-12.047 59.478-11.910Q59.690-11.774 59.940-11.774Q60.131-11.774 60.334-11.825Q60.538-11.876 60.680-11.997Q60.822-12.119 60.822-12.314L60.822-12.830Q60.575-12.830 60.211-12.780Q59.847-12.731 59.557-12.609Q59.266-12.488 59.266-12.286M62.554-12.830Q62.554-13.250 62.768-13.612Q62.982-13.975 63.346-14.183Q63.710-14.392 64.130-14.392Q64.615-14.392 64.947-14.296Q65.279-14.200 65.279-13.845Q65.279-13.698 65.179-13.590Q65.080-13.483 64.930-13.483Q64.780-13.483 64.675-13.583Q64.571-13.684 64.571-13.831L64.571-13.882Q64.431-13.903 64.137-13.903Q63.850-13.903 63.616-13.754Q63.382-13.606 63.248-13.358Q63.115-13.110 63.115-12.830Q63.115-12.549 63.265-12.303Q63.416-12.057 63.665-11.915Q63.915-11.774 64.202-11.774Q64.475-11.774 64.586-11.864Q64.697-11.955 64.800-12.121Q64.903-12.286 65.005-12.300L65.152-12.300Q65.244-12.290 65.301-12.230Q65.357-12.170 65.357-12.081Q65.357-12.040 65.340-12.006Q65.244-11.750 65.067-11.592Q64.889-11.435 64.648-11.360Q64.407-11.285 64.130-11.285Q63.706-11.285 63.342-11.493Q62.978-11.702 62.766-12.054Q62.554-12.406 62.554-12.830M65.907-11.527L65.907-11.606Q65.952-11.787 66.123-11.808L66.530-11.808L66.530-15.106L66.123-15.106Q65.952-15.127 65.907-15.315L65.907-15.393Q65.952-15.574 66.123-15.595L66.776-15.595Q66.947-15.578 66.991-15.393L66.991-12.922L67.992-13.845L67.733-13.845Q67.552-13.865 67.517-14.057L67.517-14.132Q67.552-14.317 67.733-14.337L69.001-14.337Q69.178-14.317 69.219-14.132L69.219-14.057Q69.182-13.865 69.001-13.845L68.679-13.845L67.852-13.090L68.861-11.808L69.155-11.808Q69.336-11.787 69.373-11.606L69.373-11.527Q69.336-11.340 69.155-11.319L68.133-11.319Q67.962-11.340 67.917-11.527L67.917-11.606Q67.962-11.787 68.133-11.808L68.273-11.808L67.517-12.775L66.991-12.293L66.991-11.808L67.398-11.808Q67.565-11.787 67.616-11.606L67.616-11.527Q67.565-11.340 67.398-11.319L66.123-11.319Q65.952-11.340 65.907-11.527M70.378-11.514L69.971-13.845L69.831-13.845Q69.650-13.865 69.616-14.057L69.616-14.132Q69.650-14.317 69.831-14.337L70.853-14.337Q71.038-14.317 71.072-14.132L71.072-14.057Q71.038-13.865 70.853-13.845L70.440-13.845Q70.539-13.281 70.607-12.891Q70.676-12.502 70.720-12.223Q70.764-11.944 70.764-11.880Q70.771-12.030 71.021-12.902Q71.055-13.001 71.140-13.066Q71.226-13.131 71.332-13.131L71.400-13.131Q71.506-13.131 71.592-13.066Q71.677-13.001 71.708-12.902Q71.803-12.567 71.875-12.283Q71.947-11.999 71.947-11.880Q71.957-12.132 72.261-13.845L71.848-13.845Q71.667-13.865 71.633-14.057L71.633-14.132Q71.667-14.317 71.848-14.337L72.877-14.337Q73.044-14.317 73.095-14.132L73.095-14.057Q73.048-13.865 72.877-13.845L72.730-13.845L72.323-11.514Q72.302-11.411 72.226-11.348Q72.149-11.285 72.043-11.285L71.920-11.285Q71.814-11.285 71.730-11.348Q71.646-11.411 71.609-11.514Q71.568-11.671 71.504-11.895Q71.441-12.119 71.404-12.302Q71.366-12.485 71.366-12.573Q71.359-12.327 71.113-11.514Q71.089-11.418 71.004-11.351Q70.918-11.285 70.812-11.285L70.658-11.285Q70.552-11.285 70.476-11.348Q70.399-11.411 70.378-11.514M73.567-12.293Q73.567-12.683 73.929-12.908Q74.292-13.134 74.765-13.221Q75.239-13.308 75.683-13.315Q75.683-13.503 75.565-13.636Q75.447-13.770 75.266-13.836Q75.085-13.903 74.900-13.903Q74.599-13.903 74.459-13.882L74.459-13.831Q74.459-13.684 74.355-13.583Q74.251-13.483 74.107-13.483Q73.953-13.483 73.853-13.590Q73.752-13.698 73.752-13.845Q73.752-14.200 74.085-14.296Q74.418-14.392 74.907-14.392Q75.143-14.392 75.377-14.323Q75.611-14.255 75.808-14.123Q76.004-13.992 76.124-13.799Q76.243-13.606 76.243-13.363L76.243-11.859Q76.243-11.808 76.705-11.808Q76.876-11.791 76.920-11.606L76.920-11.527Q76.876-11.340 76.705-11.319L76.578-11.319Q76.278-11.319 76.078-11.360Q75.878-11.401 75.751-11.565Q75.348-11.285 74.729-11.285Q74.435-11.285 74.169-11.408Q73.902-11.531 73.735-11.760Q73.567-11.989 73.567-12.293M74.128-12.286Q74.128-12.047 74.340-11.910Q74.552-11.774 74.801-11.774Q74.992-11.774 75.196-11.825Q75.399-11.876 75.541-11.997Q75.683-12.119 75.683-12.314L75.683-12.830Q75.437-12.830 75.073-12.780Q74.709-12.731 74.418-12.609Q74.128-12.488 74.128-12.286M77.122-11.527L77.122-11.606Q77.173-11.787 77.341-11.808L77.963-11.808L77.963-13.845L77.341-13.845Q77.170-13.865 77.122-14.057L77.122-14.132Q77.173-14.317 77.341-14.337L78.304-14.337Q78.479-14.320 78.523-14.132L78.523-13.845Q78.752-14.094 79.063-14.233Q79.374-14.371 79.713-14.371Q79.969-14.371 80.174-14.253Q80.379-14.135 80.379-13.903Q80.379-13.756 80.283-13.653Q80.188-13.551 80.041-13.551Q79.907-13.551 79.807-13.648Q79.706-13.746 79.706-13.882Q79.378-13.882 79.103-13.706Q78.827-13.530 78.675-13.238Q78.523-12.946 78.523-12.614L78.523-11.808L79.350-11.808Q79.531-11.787 79.566-11.606L79.566-11.527Q79.531-11.340 79.350-11.319L77.341-11.319Q77.173-11.340 77.122-11.527M82.273-11.285Q81.866-11.285 81.546-11.504Q81.227-11.722 81.047-12.076Q80.868-12.430 80.868-12.830Q80.868-13.131 80.977-13.411Q81.087-13.691 81.290-13.910Q81.493-14.129 81.760-14.250Q82.027-14.371 82.338-14.371Q82.785-14.371 83.141-14.091L83.141-15.106L82.772-15.106Q82.587-15.127 82.553-15.315L82.553-15.393Q82.587-15.574 82.772-15.595L83.483-15.595Q83.657-15.574 83.701-15.393L83.701-11.808L84.074-11.808Q84.255-11.787 84.289-11.606L84.289-11.527Q84.255-11.340 84.074-11.319L83.360-11.319Q83.185-11.340 83.141-11.527L83.141-11.640Q82.970-11.473 82.741-11.379Q82.512-11.285 82.273-11.285M82.307-11.774Q82.621-11.774 82.842-12.009Q83.062-12.245 83.141-12.573L83.141-13.301Q83.090-13.465 82.979-13.597Q82.867-13.729 82.712-13.806Q82.556-13.882 82.385-13.882Q82.115-13.882 81.897-13.734Q81.678-13.585 81.552-13.336Q81.425-13.086 81.425-12.823Q81.425-12.430 81.675-12.102Q81.924-11.774 82.307-11.774\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(84.113 19.627)\">\u003Cpath d=\"M87.105-9.969L87.105-10.044Q87.142-10.236 87.323-10.256L87.692-10.256L87.692-13.845L87.323-13.845Q87.142-13.865 87.105-14.057L87.105-14.132Q87.146-14.317 87.323-14.337L88.038-14.337Q88.209-14.320 88.253-14.132L88.253-14.070Q88.438-14.217 88.668-14.294Q88.899-14.371 89.135-14.371Q89.432-14.371 89.692-14.245Q89.952-14.118 90.140-13.898Q90.328-13.677 90.428-13.404Q90.529-13.131 90.529-12.830Q90.529-12.423 90.331-12.064Q90.133-11.705 89.794-11.495Q89.456-11.285 89.046-11.285Q88.591-11.285 88.253-11.606L88.253-10.256L88.626-10.256Q88.807-10.236 88.841-10.044L88.841-9.969Q88.807-9.784 88.626-9.764L87.323-9.764Q87.146-9.784 87.105-9.969M89.008-11.774Q89.278-11.774 89.497-11.922Q89.716-12.071 89.842-12.317Q89.969-12.563 89.969-12.830Q89.969-13.083 89.859-13.327Q89.750-13.571 89.547-13.727Q89.343-13.882 89.080-13.882Q88.800-13.882 88.571-13.720Q88.342-13.558 88.253-13.295L88.253-12.549Q88.332-12.228 88.525-12.001Q88.718-11.774 89.008-11.774M91.114-12.293Q91.114-12.683 91.476-12.908Q91.838-13.134 92.312-13.221Q92.785-13.308 93.230-13.315Q93.230-13.503 93.112-13.636Q92.994-13.770 92.813-13.836Q92.631-13.903 92.447-13.903Q92.146-13.903 92.006-13.882L92.006-13.831Q92.006-13.684 91.902-13.583Q91.797-13.483 91.654-13.483Q91.500-13.483 91.399-13.590Q91.298-13.698 91.298-13.845Q91.298-14.200 91.632-14.296Q91.965-14.392 92.454-14.392Q92.689-14.392 92.924-14.323Q93.158-14.255 93.354-14.123Q93.551-13.992 93.670-13.799Q93.790-13.606 93.790-13.363L93.790-11.859Q93.790-11.808 94.251-11.808Q94.422-11.791 94.467-11.606L94.467-11.527Q94.422-11.340 94.251-11.319L94.125-11.319Q93.824-11.319 93.624-11.360Q93.424-11.401 93.298-11.565Q92.895-11.285 92.276-11.285Q91.982-11.285 91.715-11.408Q91.449-11.531 91.281-11.760Q91.114-11.989 91.114-12.293M91.674-12.286Q91.674-12.047 91.886-11.910Q92.098-11.774 92.348-11.774Q92.539-11.774 92.742-11.825Q92.946-11.876 93.088-11.997Q93.230-12.119 93.230-12.314L93.230-12.830Q92.983-12.830 92.619-12.780Q92.255-12.731 91.965-12.609Q91.674-12.488 91.674-12.286M94.986-11.493L94.986-12.293Q95.010-12.474 95.195-12.495L95.342-12.495Q95.485-12.474 95.537-12.334Q95.714-11.774 96.350-11.774Q96.531-11.774 96.731-11.804Q96.931-11.835 97.078-11.936Q97.225-12.037 97.225-12.215Q97.225-12.399 97.030-12.498Q96.835-12.597 96.596-12.635L95.984-12.734Q94.986-12.919 94.986-13.551Q94.986-13.804 95.113-13.970Q95.239-14.135 95.449-14.229Q95.660-14.323 95.884-14.358Q96.107-14.392 96.350-14.392Q96.569-14.392 96.738-14.366Q96.907-14.340 97.051-14.272Q97.119-14.375 97.232-14.392L97.300-14.392Q97.386-14.382 97.440-14.327Q97.495-14.272 97.505-14.190L97.505-13.571Q97.495-13.489 97.440-13.431Q97.386-13.373 97.300-13.363L97.153-13.363Q97.071-13.373 97.013-13.431Q96.955-13.489 96.945-13.571Q96.945-13.903 96.336-13.903Q96.032-13.903 95.754-13.831Q95.475-13.759 95.475-13.544Q95.475-13.312 96.063-13.216L96.678-13.110Q97.102-13.038 97.408-12.821Q97.714-12.604 97.714-12.215Q97.714-11.873 97.507-11.661Q97.300-11.449 96.994-11.367Q96.689-11.285 96.350-11.285Q95.844-11.285 95.496-11.507Q95.434-11.398 95.391-11.348Q95.349-11.298 95.256-11.285L95.195-11.285Q95.007-11.305 94.986-11.493M98.702-11.493L98.702-12.293Q98.726-12.474 98.910-12.495L99.057-12.495Q99.201-12.474 99.252-12.334Q99.430-11.774 100.065-11.774Q100.247-11.774 100.447-11.804Q100.647-11.835 100.793-11.936Q100.940-12.037 100.940-12.215Q100.940-12.399 100.746-12.498Q100.551-12.597 100.312-12.635L99.700-12.734Q98.702-12.919 98.702-13.551Q98.702-13.804 98.828-13.970Q98.955-14.135 99.165-14.229Q99.375-14.323 99.599-14.358Q99.823-14.392 100.065-14.392Q100.284-14.392 100.453-14.366Q100.623-14.340 100.766-14.272Q100.835-14.375 100.947-14.392L101.016-14.392Q101.101-14.382 101.156-14.327Q101.210-14.272 101.221-14.190L101.221-13.571Q101.210-13.489 101.156-13.431Q101.101-13.373 101.016-13.363L100.869-13.363Q100.787-13.373 100.729-13.431Q100.670-13.489 100.660-13.571Q100.660-13.903 100.052-13.903Q99.748-13.903 99.469-13.831Q99.190-13.759 99.190-13.544Q99.190-13.312 99.778-13.216L100.394-13.110Q100.817-13.038 101.123-12.821Q101.429-12.604 101.429-12.215Q101.429-11.873 101.222-11.661Q101.016-11.449 100.710-11.367Q100.404-11.285 100.065-11.285Q99.560-11.285 99.211-11.507Q99.149-11.398 99.107-11.348Q99.064-11.298 98.972-11.285L98.910-11.285Q98.722-11.305 98.702-11.493M104.977-12.621L102.841-12.621Q102.889-12.379 103.061-12.184Q103.234-11.989 103.477-11.881Q103.719-11.774 103.969-11.774Q104.382-11.774 104.577-12.027Q104.584-12.037 104.634-12.129Q104.683-12.221 104.726-12.259Q104.769-12.297 104.851-12.307L104.977-12.307Q105.145-12.290 105.196-12.102L105.196-12.054Q105.138-11.791 104.936-11.618Q104.734-11.445 104.461-11.365Q104.188-11.285 103.921-11.285Q103.497-11.285 103.113-11.485Q102.728-11.685 102.494-12.035Q102.260-12.385 102.260-12.816L102.260-12.867Q102.260-13.277 102.475-13.630Q102.690-13.982 103.048-14.187Q103.405-14.392 103.815-14.392Q104.256-14.392 104.565-14.197Q104.875-14.002 105.035-13.662Q105.196-13.322 105.196-12.881L105.196-12.830Q105.145-12.642 104.977-12.621M102.848-13.103L104.622-13.103Q104.581-13.462 104.372-13.683Q104.164-13.903 103.815-13.903Q103.470-13.903 103.201-13.674Q102.933-13.445 102.848-13.103M106.132-11.493L106.132-12.293Q106.156-12.474 106.341-12.495L106.488-12.495Q106.631-12.474 106.683-12.334Q106.860-11.774 107.496-11.774Q107.677-11.774 107.877-11.804Q108.077-11.835 108.224-11.936Q108.371-12.037 108.371-12.215Q108.371-12.399 108.176-12.498Q107.981-12.597 107.742-12.635L107.130-12.734Q106.132-12.919 106.132-13.551Q106.132-13.804 106.259-13.970Q106.385-14.135 106.595-14.229Q106.806-14.323 107.030-14.358Q107.253-14.392 107.496-14.392Q107.715-14.392 107.884-14.366Q108.053-14.340 108.197-14.272Q108.265-14.375 108.378-14.392L108.446-14.392Q108.532-14.382 108.586-14.327Q108.641-14.272 108.651-14.190L108.651-13.571Q108.641-13.489 108.586-13.431Q108.532-13.373 108.446-13.363L108.299-13.363Q108.217-13.373 108.159-13.431Q108.101-13.489 108.091-13.571Q108.091-13.903 107.482-13.903Q107.178-13.903 106.900-13.831Q106.621-13.759 106.621-13.544Q106.621-13.312 107.209-13.216L107.824-13.110Q108.248-13.038 108.554-12.821Q108.860-12.604 108.860-12.215Q108.860-11.873 108.653-11.661Q108.446-11.449 108.140-11.367Q107.835-11.285 107.496-11.285Q106.990-11.285 106.642-11.507Q106.580-11.398 106.537-11.348Q106.495-11.298 106.402-11.285L106.341-11.285Q106.153-11.305 106.132-11.493M110.839-10.345Q110.740-10.345 110.661-10.423Q110.583-10.502 110.583-10.598Q110.583-10.724 110.706-10.799Q111.078-10.946 111.191-11.319Q110.993-11.319 110.849-11.466Q110.706-11.613 110.706-11.808Q110.706-12.013 110.849-12.153Q110.993-12.293 111.198-12.293Q111.464-12.293 111.589-12.071Q111.714-11.849 111.714-11.558Q111.714-11.175 111.500-10.852Q111.287-10.529 110.931-10.365Q110.890-10.345 110.839-10.345M110.706-13.852Q110.706-14.047 110.851-14.192Q110.996-14.337 111.198-14.337Q111.328-14.337 111.439-14.269Q111.550-14.200 111.615-14.091Q111.680-13.982 111.680-13.852Q111.680-13.653 111.534-13.508Q111.389-13.363 111.198-13.363Q111-13.363 110.853-13.510Q110.706-13.657 110.706-13.852\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(84.113 19.627)\">\u003Cpath d=\"M115.858-11.527L115.858-11.606Q115.903-11.787 116.074-11.808L116.313-11.808L116.313-15.106L116.074-15.106Q115.903-15.127 115.858-15.315L115.858-15.393Q115.903-15.574 116.074-15.595L117.109-15.595Q117.284-15.574 117.328-15.393L117.328-15.315Q117.284-15.127 117.109-15.106L116.874-15.106L116.874-13.790L118.302-13.790L118.302-15.106L118.063-15.106Q117.896-15.127 117.844-15.315L117.844-15.393Q117.896-15.574 118.063-15.595L119.099-15.595Q119.266-15.574 119.317-15.393L119.317-15.315Q119.266-15.127 119.099-15.106L118.859-15.106L118.859-11.808L119.099-11.808Q119.266-11.787 119.317-11.606L119.317-11.527Q119.266-11.340 119.099-11.319L118.063-11.319Q117.896-11.340 117.844-11.527L117.844-11.606Q117.896-11.787 118.063-11.808L118.302-11.808L118.302-13.301L116.874-13.301L116.874-11.808L117.109-11.808Q117.284-11.787 117.328-11.606L117.328-11.527Q117.284-11.340 117.109-11.319L116.074-11.319Q115.903-11.340 115.858-11.527\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(84.113 19.627)\">\u003Cpath d=\"M122.170-11.527L122.170-11.606Q122.207-11.787 122.388-11.808L122.757-11.808L122.757-13.845L122.388-13.845Q122.207-13.865 122.170-14.057L122.170-14.132Q122.211-14.317 122.388-14.337L123.103-14.337Q123.274-14.317 123.318-14.132L123.318-14.057L123.311-14.036Q123.509-14.194 123.749-14.282Q123.988-14.371 124.244-14.371Q124.548-14.371 124.757-14.248Q124.965-14.125 125.066-13.900Q125.167-13.674 125.167-13.363L125.167-11.808L125.540-11.808Q125.721-11.787 125.755-11.606L125.755-11.527Q125.721-11.340 125.540-11.319L124.319-11.319Q124.149-11.340 124.104-11.527L124.104-11.606Q124.149-11.791 124.319-11.808L124.607-11.808L124.607-13.336Q124.607-13.606 124.528-13.744Q124.449-13.882 124.193-13.882Q123.957-13.882 123.757-13.768Q123.557-13.653 123.438-13.454Q123.318-13.254 123.318-13.021L123.318-11.808L123.691-11.808Q123.872-11.787 123.906-11.606L123.906-11.527Q123.872-11.340 123.691-11.319L122.388-11.319Q122.207-11.340 122.170-11.527M128.896-12.621L126.760-12.621Q126.808-12.379 126.980-12.184Q127.153-11.989 127.396-11.881Q127.638-11.774 127.888-11.774Q128.301-11.774 128.496-12.027Q128.503-12.037 128.553-12.129Q128.602-12.221 128.645-12.259Q128.688-12.297 128.770-12.307L128.896-12.307Q129.064-12.290 129.115-12.102L129.115-12.054Q129.057-11.791 128.855-11.618Q128.653-11.445 128.380-11.365Q128.107-11.285 127.840-11.285Q127.416-11.285 127.032-11.485Q126.647-11.685 126.413-12.035Q126.179-12.385 126.179-12.816L126.179-12.867Q126.179-13.277 126.394-13.630Q126.609-13.982 126.967-14.187Q127.324-14.392 127.734-14.392Q128.175-14.392 128.484-14.197Q128.794-14.002 128.954-13.662Q129.115-13.322 129.115-12.881L129.115-12.830Q129.064-12.642 128.896-12.621M126.767-13.103L128.541-13.103Q128.500-13.462 128.291-13.683Q128.083-13.903 127.734-13.903Q127.389-13.903 127.120-13.674Q126.852-13.445 126.767-13.103M130.988-11.514L130.212-13.845L129.932-13.845Q129.757-13.865 129.713-14.057L129.713-14.132Q129.757-14.317 129.932-14.337L130.947-14.337Q131.118-14.320 131.162-14.132L131.162-14.057Q131.118-13.865 130.947-13.845L130.701-13.845L131.395-11.767L132.085-13.845L131.842-13.845Q131.668-13.865 131.624-14.057L131.624-14.132Q131.668-14.317 131.842-14.337L132.858-14.337Q133.028-14.317 133.073-14.132L133.073-14.057Q133.028-13.865 132.858-13.845L132.577-13.845L131.801-11.514Q131.764-11.415 131.675-11.350Q131.586-11.285 131.477-11.285L131.309-11.285Q131.207-11.285 131.116-11.348Q131.025-11.411 130.988-11.514M136.327-12.621L134.191-12.621Q134.238-12.379 134.411-12.184Q134.584-11.989 134.826-11.881Q135.069-11.774 135.318-11.774Q135.732-11.774 135.927-12.027Q135.934-12.037 135.983-12.129Q136.033-12.221 136.076-12.259Q136.118-12.297 136.200-12.307L136.327-12.307Q136.494-12.290 136.546-12.102L136.546-12.054Q136.487-11.791 136.286-11.618Q136.084-11.445 135.811-11.365Q135.537-11.285 135.271-11.285Q134.847-11.285 134.462-11.485Q134.078-11.685 133.844-12.035Q133.609-12.385 133.609-12.816L133.609-12.867Q133.609-13.277 133.825-13.630Q134.040-13.982 134.397-14.187Q134.754-14.392 135.165-14.392Q135.606-14.392 135.915-14.197Q136.224-14.002 136.385-13.662Q136.546-13.322 136.546-12.881L136.546-12.830Q136.494-12.642 136.327-12.621M134.197-13.103L135.971-13.103Q135.930-13.462 135.722-13.683Q135.513-13.903 135.165-13.903Q134.819-13.903 134.551-13.674Q134.283-13.445 134.197-13.103M137.164-11.527L137.164-11.606Q137.215-11.787 137.383-11.808L138.005-11.808L138.005-13.845L137.383-13.845Q137.212-13.865 137.164-14.057L137.164-14.132Q137.215-14.317 137.383-14.337L138.347-14.337Q138.521-14.320 138.566-14.132L138.566-13.845Q138.795-14.094 139.106-14.233Q139.417-14.371 139.755-14.371Q140.011-14.371 140.216-14.253Q140.421-14.135 140.421-13.903Q140.421-13.756 140.326-13.653Q140.230-13.551 140.083-13.551Q139.950-13.551 139.849-13.648Q139.748-13.746 139.748-13.882Q139.420-13.882 139.145-13.706Q138.870-13.530 138.718-13.238Q138.566-12.946 138.566-12.614L138.566-11.808L139.393-11.808Q139.574-11.787 139.608-11.606L139.608-11.527Q139.574-11.340 139.393-11.319L137.383-11.319Q137.215-11.340 137.164-11.527\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(84.113 19.627)\">\u003Cpath d=\"M143.872-11.493L143.872-12.293Q143.896-12.474 144.080-12.495L144.227-12.495Q144.371-12.474 144.422-12.334Q144.600-11.774 145.235-11.774Q145.417-11.774 145.617-11.804Q145.817-11.835 145.963-11.936Q146.110-12.037 146.110-12.215Q146.110-12.399 145.916-12.498Q145.721-12.597 145.482-12.635L144.870-12.734Q143.872-12.919 143.872-13.551Q143.872-13.804 143.998-13.970Q144.125-14.135 144.335-14.229Q144.545-14.323 144.769-14.358Q144.993-14.392 145.235-14.392Q145.454-14.392 145.623-14.366Q145.793-14.340 145.936-14.272Q146.005-14.375 146.117-14.392L146.186-14.392Q146.271-14.382 146.326-14.327Q146.380-14.272 146.391-14.190L146.391-13.571Q146.380-13.489 146.326-13.431Q146.271-13.373 146.186-13.363L146.039-13.363Q145.957-13.373 145.899-13.431Q145.840-13.489 145.830-13.571Q145.830-13.903 145.222-13.903Q144.918-13.903 144.639-13.831Q144.360-13.759 144.360-13.544Q144.360-13.312 144.948-13.216L145.564-13.110Q145.987-13.038 146.293-12.821Q146.599-12.604 146.599-12.215Q146.599-11.873 146.392-11.661Q146.186-11.449 145.880-11.367Q145.574-11.285 145.235-11.285Q144.730-11.285 144.381-11.507Q144.319-11.398 144.277-11.348Q144.234-11.298 144.142-11.285L144.080-11.285Q143.892-11.305 143.872-11.493M148.124-12.286L148.124-13.845L147.467-13.845Q147.293-13.865 147.249-14.057L147.249-14.132Q147.293-14.317 147.467-14.337L148.124-14.337L148.124-14.993Q148.168-15.174 148.342-15.198L148.469-15.198Q148.640-15.174 148.684-14.993L148.684-14.337L149.874-14.337Q150.041-14.320 150.092-14.132L150.092-14.057Q150.045-13.865 149.874-13.845L148.684-13.845L148.684-12.314Q148.684-11.774 149.187-11.774Q149.405-11.774 149.549-11.936Q149.692-12.098 149.692-12.314Q149.692-12.396 149.756-12.457Q149.819-12.519 149.908-12.529L150.034-12.529Q150.209-12.508 150.253-12.327L150.253-12.286Q150.253-12.003 150.092-11.772Q149.932-11.541 149.675-11.413Q149.419-11.285 149.139-11.285Q148.691-11.285 148.407-11.557Q148.124-11.828 148.124-12.286M152.646-11.285Q152.232-11.285 151.895-11.498Q151.559-11.712 151.364-12.071Q151.169-12.430 151.169-12.830Q151.169-13.131 151.278-13.413Q151.388-13.694 151.589-13.917Q151.791-14.139 152.059-14.265Q152.328-14.392 152.646-14.392Q152.963-14.392 153.237-14.264Q153.510-14.135 153.705-13.920Q153.900-13.705 154.011-13.424Q154.122-13.144 154.122-12.830Q154.122-12.430 153.926-12.069Q153.729-11.709 153.392-11.497Q153.056-11.285 152.646-11.285M152.646-11.774Q153.052-11.774 153.307-12.119Q153.562-12.464 153.562-12.888Q153.562-13.144 153.444-13.378Q153.326-13.612 153.116-13.758Q152.905-13.903 152.646-13.903Q152.379-13.903 152.171-13.758Q151.962-13.612 151.844-13.378Q151.726-13.144 151.726-12.888Q151.726-12.467 151.983-12.121Q152.239-11.774 152.646-11.774M154.700-11.527L154.700-11.606Q154.751-11.787 154.919-11.808L155.541-11.808L155.541-13.845L154.919-13.845Q154.748-13.865 154.700-14.057L154.700-14.132Q154.751-14.317 154.919-14.337L155.882-14.337Q156.057-14.320 156.101-14.132L156.101-13.845Q156.330-14.094 156.641-14.233Q156.952-14.371 157.291-14.371Q157.547-14.371 157.752-14.253Q157.957-14.135 157.957-13.903Q157.957-13.756 157.861-13.653Q157.766-13.551 157.619-13.551Q157.485-13.551 157.385-13.648Q157.284-13.746 157.284-13.882Q156.956-13.882 156.681-13.706Q156.405-13.530 156.253-13.238Q156.101-12.946 156.101-12.614L156.101-11.808L156.928-11.808Q157.109-11.787 157.144-11.606L157.144-11.527Q157.109-11.340 156.928-11.319L154.919-11.319Q154.751-11.340 154.700-11.527M161.293-12.621L159.157-12.621Q159.205-12.379 159.377-12.184Q159.550-11.989 159.793-11.881Q160.035-11.774 160.285-11.774Q160.698-11.774 160.893-12.027Q160.900-12.037 160.950-12.129Q160.999-12.221 161.042-12.259Q161.085-12.297 161.167-12.307L161.293-12.307Q161.461-12.290 161.512-12.102L161.512-12.054Q161.454-11.791 161.252-11.618Q161.050-11.445 160.777-11.365Q160.504-11.285 160.237-11.285Q159.813-11.285 159.429-11.485Q159.044-11.685 158.810-12.035Q158.576-12.385 158.576-12.816L158.576-12.867Q158.576-13.277 158.791-13.630Q159.006-13.982 159.364-14.187Q159.721-14.392 160.131-14.392Q160.572-14.392 160.881-14.197Q161.191-14.002 161.351-13.662Q161.512-13.322 161.512-12.881L161.512-12.830Q161.461-12.642 161.293-12.621M159.164-13.103L160.938-13.103Q160.897-13.462 160.688-13.683Q160.480-13.903 160.131-13.903Q159.786-13.903 159.517-13.674Q159.249-13.445 159.164-13.103M163.566-11.285Q163.159-11.285 162.840-11.504Q162.520-11.722 162.341-12.076Q162.161-12.430 162.161-12.830Q162.161-13.131 162.271-13.411Q162.380-13.691 162.583-13.910Q162.787-14.129 163.053-14.250Q163.320-14.371 163.631-14.371Q164.079-14.371 164.434-14.091L164.434-15.106L164.065-15.106Q163.880-15.127 163.846-15.315L163.846-15.393Q163.880-15.574 164.065-15.595L164.776-15.595Q164.950-15.574 164.995-15.393L164.995-11.808L165.367-11.808Q165.548-11.787 165.583-11.606L165.583-11.527Q165.548-11.340 165.367-11.319L164.653-11.319Q164.479-11.340 164.434-11.527L164.434-11.640Q164.263-11.473 164.034-11.379Q163.805-11.285 163.566-11.285M163.600-11.774Q163.915-11.774 164.135-12.009Q164.356-12.245 164.434-12.573L164.434-13.301Q164.383-13.465 164.272-13.597Q164.161-13.729 164.005-13.806Q163.850-13.882 163.679-13.882Q163.409-13.882 163.190-13.734Q162.971-13.585 162.845-13.336Q162.718-13.086 162.718-12.823Q162.718-12.430 162.968-12.102Q163.217-11.774 163.600-11.774\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">Pearlmutter's trick computes \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0813em;\">H\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> as the gradient of the scalar \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0435em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8491em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">⊤\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>: forward pass, backward pass for \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">g\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>, contract with \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>, backward again. The matrix \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0813em;\">H\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> is never formed; storage stays \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">O\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:460.832px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 345.624 209.412\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-65.203 117.87V-55.937\"\u002F>\u003Cpath stroke=\"none\" d=\"m-65.203-58.537-2.08 4.16 2.08-1.56 2.08 1.56\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(3.733 -181.918)\">\u003Cpath d=\"M-64.644 116.143Q-64.644 115.663-64.400 115.249Q-64.156 114.835-63.740 114.597Q-63.324 114.358-62.844 114.358Q-62.289 114.358-61.910 114.468Q-61.531 114.577-61.531 114.983Q-61.531 115.151-61.644 115.274Q-61.758 115.397-61.930 115.397Q-62.101 115.397-62.221 115.282Q-62.340 115.167-62.340 114.999L-62.340 114.940Q-62.500 114.917-62.836 114.917Q-63.164 114.917-63.432 115.087Q-63.699 115.257-63.851 115.540Q-64.004 115.823-64.004 116.143Q-64.004 116.464-63.832 116.745Q-63.660 117.026-63.375 117.188Q-63.090 117.350-62.762 117.350Q-62.449 117.350-62.322 117.247Q-62.195 117.143-62.078 116.954Q-61.961 116.765-61.844 116.749L-61.676 116.749Q-61.570 116.761-61.506 116.829Q-61.441 116.897-61.441 116.999Q-61.441 117.046-61.461 117.085Q-61.570 117.378-61.773 117.558Q-61.976 117.737-62.252 117.823Q-62.527 117.909-62.844 117.909Q-63.328 117.909-63.744 117.671Q-64.160 117.433-64.402 117.030Q-64.644 116.628-64.644 116.143M-60.215 117.015L-60.215 114.983L-60.637 114.983Q-60.844 114.960-60.887 114.741L-60.887 114.655Q-60.840 114.444-60.637 114.421L-59.820 114.421Q-59.625 114.444-59.574 114.655L-59.574 116.983Q-59.574 117.218-59.404 117.284Q-59.234 117.350-58.949 117.350Q-58.742 117.350-58.547 117.274Q-58.351 117.198-58.226 117.048Q-58.101 116.897-58.101 116.686L-58.101 114.983L-58.523 114.983Q-58.734 114.960-58.773 114.741L-58.773 114.655Q-58.734 114.444-58.523 114.421L-57.711 114.421Q-57.512 114.444-57.461 114.655L-57.461 117.311L-57.035 117.311Q-56.828 117.335-56.789 117.542L-56.789 117.632Q-56.828 117.847-57.035 117.870L-57.851 117.870Q-58.051 117.847-58.101 117.647Q-58.504 117.909-59.012 117.909Q-59.246 117.909-59.461 117.868Q-59.676 117.827-59.842 117.725Q-60.008 117.624-60.111 117.446Q-60.215 117.268-60.215 117.015M-56.488 117.632L-56.488 117.542Q-56.430 117.335-56.238 117.311L-55.527 117.311L-55.527 114.983L-56.238 114.983Q-56.433 114.960-56.488 114.741L-56.488 114.655Q-56.430 114.444-56.238 114.421L-55.137 114.421Q-54.937 114.440-54.887 114.655L-54.887 114.983Q-54.625 114.698-54.269 114.540Q-53.914 114.382-53.527 114.382Q-53.234 114.382-53 114.516Q-52.766 114.651-52.766 114.917Q-52.766 115.085-52.875 115.202Q-52.984 115.319-53.152 115.319Q-53.305 115.319-53.420 115.208Q-53.535 115.097-53.535 114.940Q-53.910 114.940-54.224 115.141Q-54.539 115.343-54.713 115.677Q-54.887 116.011-54.887 116.390L-54.887 117.311L-53.941 117.311Q-53.734 117.335-53.695 117.542L-53.695 117.632Q-53.734 117.847-53.941 117.870L-56.238 117.870Q-56.430 117.847-56.488 117.632M-50.808 117.647L-51.695 114.983L-52.016 114.983Q-52.215 114.960-52.266 114.741L-52.266 114.655Q-52.215 114.444-52.016 114.421L-50.855 114.421Q-50.660 114.440-50.609 114.655L-50.609 114.741Q-50.660 114.960-50.855 114.983L-51.137 114.983L-50.344 117.358L-49.555 114.983L-49.832 114.983Q-50.031 114.960-50.082 114.741L-50.082 114.655Q-50.031 114.444-49.832 114.421L-48.672 114.421Q-48.476 114.444-48.426 114.655L-48.426 114.741Q-48.476 114.960-48.672 114.983L-48.992 114.983L-49.879 117.647Q-49.922 117.761-50.023 117.835Q-50.125 117.909-50.250 117.909L-50.441 117.909Q-50.558 117.909-50.662 117.837Q-50.766 117.765-50.808 117.647M-47.812 116.757Q-47.812 116.311-47.398 116.054Q-46.984 115.796-46.443 115.696Q-45.902 115.597-45.394 115.589Q-45.394 115.374-45.529 115.222Q-45.664 115.069-45.871 114.993Q-46.078 114.917-46.289 114.917Q-46.633 114.917-46.793 114.940L-46.793 114.999Q-46.793 115.167-46.912 115.282Q-47.031 115.397-47.195 115.397Q-47.371 115.397-47.486 115.274Q-47.601 115.151-47.601 114.983Q-47.601 114.577-47.221 114.468Q-46.840 114.358-46.281 114.358Q-46.012 114.358-45.744 114.436Q-45.476 114.515-45.252 114.665Q-45.027 114.815-44.891 115.036Q-44.754 115.257-44.754 115.534L-44.754 117.253Q-44.754 117.311-44.226 117.311Q-44.031 117.331-43.980 117.542L-43.980 117.632Q-44.031 117.847-44.226 117.870L-44.371 117.870Q-44.715 117.870-44.943 117.823Q-45.172 117.776-45.316 117.589Q-45.777 117.909-46.484 117.909Q-46.820 117.909-47.125 117.768Q-47.430 117.628-47.621 117.366Q-47.812 117.104-47.812 116.757M-47.172 116.765Q-47.172 117.038-46.930 117.194Q-46.687 117.350-46.402 117.350Q-46.183 117.350-45.951 117.292Q-45.719 117.233-45.557 117.095Q-45.394 116.956-45.394 116.733L-45.394 116.143Q-45.676 116.143-46.092 116.200Q-46.508 116.257-46.840 116.395Q-47.172 116.534-47.172 116.765M-42.773 116.765L-42.773 114.983L-43.523 114.983Q-43.723 114.960-43.773 114.741L-43.773 114.655Q-43.723 114.444-43.523 114.421L-42.773 114.421L-42.773 113.671Q-42.723 113.464-42.523 113.436L-42.379 113.436Q-42.183 113.464-42.133 113.671L-42.133 114.421L-40.773 114.421Q-40.582 114.440-40.523 114.655L-40.523 114.741Q-40.578 114.960-40.773 114.983L-42.133 114.983L-42.133 116.733Q-42.133 117.350-41.558 117.350Q-41.308 117.350-41.144 117.165Q-40.980 116.979-40.980 116.733Q-40.980 116.640-40.908 116.569Q-40.836 116.499-40.734 116.487L-40.590 116.487Q-40.391 116.511-40.340 116.718L-40.340 116.765Q-40.340 117.089-40.523 117.352Q-40.707 117.616-41 117.763Q-41.293 117.909-41.613 117.909Q-42.125 117.909-42.449 117.599Q-42.773 117.288-42.773 116.765M-38.984 117.015L-38.984 114.983L-39.406 114.983Q-39.613 114.960-39.656 114.741L-39.656 114.655Q-39.609 114.444-39.406 114.421L-38.590 114.421Q-38.394 114.444-38.344 114.655L-38.344 116.983Q-38.344 117.218-38.174 117.284Q-38.004 117.350-37.719 117.350Q-37.512 117.350-37.316 117.274Q-37.121 117.198-36.996 117.048Q-36.871 116.897-36.871 116.686L-36.871 114.983L-37.293 114.983Q-37.504 114.960-37.543 114.741L-37.543 114.655Q-37.504 114.444-37.293 114.421L-36.480 114.421Q-36.281 114.444-36.230 114.655L-36.230 117.311L-35.805 117.311Q-35.598 117.335-35.558 117.542L-35.558 117.632Q-35.598 117.847-35.805 117.870L-36.621 117.870Q-36.820 117.847-36.871 117.647Q-37.273 117.909-37.781 117.909Q-38.016 117.909-38.230 117.868Q-38.445 117.827-38.611 117.725Q-38.777 117.624-38.881 117.446Q-38.984 117.268-38.984 117.015M-35.258 117.632L-35.258 117.542Q-35.199 117.335-35.008 117.311L-34.297 117.311L-34.297 114.983L-35.008 114.983Q-35.203 114.960-35.258 114.741L-35.258 114.655Q-35.199 114.444-35.008 114.421L-33.906 114.421Q-33.707 114.440-33.656 114.655L-33.656 114.983Q-33.394 114.698-33.039 114.540Q-32.683 114.382-32.297 114.382Q-32.004 114.382-31.769 114.516Q-31.535 114.651-31.535 114.917Q-31.535 115.085-31.644 115.202Q-31.754 115.319-31.922 115.319Q-32.074 115.319-32.189 115.208Q-32.305 115.097-32.305 114.940Q-32.680 114.940-32.994 115.141Q-33.308 115.343-33.482 115.677Q-33.656 116.011-33.656 116.390L-33.656 117.311L-32.711 117.311Q-32.504 117.335-32.465 117.542L-32.465 117.632Q-32.504 117.847-32.711 117.870L-35.008 117.870Q-35.199 117.847-35.258 117.632M-27.723 116.382L-30.164 116.382Q-30.109 116.659-29.912 116.882Q-29.715 117.104-29.437 117.227Q-29.160 117.350-28.875 117.350Q-28.402 117.350-28.180 117.061Q-28.172 117.050-28.115 116.944Q-28.058 116.839-28.010 116.796Q-27.961 116.753-27.867 116.741L-27.723 116.741Q-27.531 116.761-27.473 116.975L-27.473 117.030Q-27.539 117.331-27.769 117.528Q-28 117.725-28.312 117.817Q-28.625 117.909-28.930 117.909Q-29.414 117.909-29.853 117.681Q-30.293 117.452-30.560 117.052Q-30.828 116.651-30.828 116.159L-30.828 116.100Q-30.828 115.632-30.582 115.229Q-30.336 114.827-29.928 114.593Q-29.519 114.358-29.051 114.358Q-28.547 114.358-28.193 114.581Q-27.840 114.804-27.656 115.192Q-27.473 115.581-27.473 116.085L-27.473 116.143Q-27.531 116.358-27.723 116.382M-30.156 115.831L-28.129 115.831Q-28.176 115.421-28.414 115.169Q-28.652 114.917-29.051 114.917Q-29.445 114.917-29.752 115.179Q-30.058 115.440-30.156 115.831\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(3.733 -181.918)\">\u003Cpath d=\"M-22.144 116.143Q-22.144 115.663-21.900 115.249Q-21.656 114.835-21.240 114.597Q-20.824 114.358-20.344 114.358Q-19.789 114.358-19.410 114.468Q-19.031 114.577-19.031 114.983Q-19.031 115.151-19.144 115.274Q-19.258 115.397-19.430 115.397Q-19.601 115.397-19.721 115.282Q-19.840 115.167-19.840 114.999L-19.840 114.940Q-20 114.917-20.336 114.917Q-20.664 114.917-20.932 115.087Q-21.199 115.257-21.351 115.540Q-21.504 115.823-21.504 116.143Q-21.504 116.464-21.332 116.745Q-21.160 117.026-20.875 117.188Q-20.590 117.350-20.262 117.350Q-19.949 117.350-19.822 117.247Q-19.695 117.143-19.578 116.954Q-19.461 116.765-19.344 116.749L-19.176 116.749Q-19.070 116.761-19.006 116.829Q-18.941 116.897-18.941 116.999Q-18.941 117.046-18.961 117.085Q-19.070 117.378-19.273 117.558Q-19.476 117.737-19.752 117.823Q-20.027 117.909-20.344 117.909Q-20.828 117.909-21.244 117.671Q-21.660 117.433-21.902 117.030Q-22.144 116.628-22.144 116.143M-18.051 116.757Q-18.051 116.311-17.637 116.054Q-17.223 115.796-16.682 115.696Q-16.140 115.597-15.633 115.589Q-15.633 115.374-15.767 115.222Q-15.902 115.069-16.109 114.993Q-16.316 114.917-16.527 114.917Q-16.871 114.917-17.031 114.940L-17.031 114.999Q-17.031 115.167-17.150 115.282Q-17.269 115.397-17.433 115.397Q-17.609 115.397-17.724 115.274Q-17.840 115.151-17.840 114.983Q-17.840 114.577-17.459 114.468Q-17.078 114.358-16.519 114.358Q-16.250 114.358-15.982 114.436Q-15.715 114.515-15.490 114.665Q-15.265 114.815-15.129 115.036Q-14.992 115.257-14.992 115.534L-14.992 117.253Q-14.992 117.311-14.465 117.311Q-14.269 117.331-14.219 117.542L-14.219 117.632Q-14.269 117.847-14.465 117.870L-14.609 117.870Q-14.953 117.870-15.182 117.823Q-15.410 117.776-15.555 117.589Q-16.015 117.909-16.723 117.909Q-17.058 117.909-17.363 117.768Q-17.668 117.628-17.859 117.366Q-18.051 117.104-18.051 116.757M-17.410 116.765Q-17.410 117.038-17.168 117.194Q-16.926 117.350-16.640 117.350Q-16.422 117.350-16.189 117.292Q-15.957 117.233-15.795 117.095Q-15.633 116.956-15.633 116.733L-15.633 116.143Q-15.914 116.143-16.330 116.200Q-16.746 116.257-17.078 116.395Q-17.410 116.534-17.410 116.765M-14.140 119.413L-14.140 119.327Q-14.098 119.108-13.890 119.085L-13.469 119.085L-13.469 114.983L-13.890 114.983Q-14.098 114.960-14.140 114.741L-14.140 114.655Q-14.094 114.444-13.890 114.421L-13.074 114.421Q-12.879 114.440-12.828 114.655L-12.828 114.725Q-12.617 114.558-12.353 114.470Q-12.090 114.382-11.820 114.382Q-11.480 114.382-11.183 114.526Q-10.887 114.671-10.672 114.923Q-10.457 115.175-10.342 115.487Q-10.226 115.800-10.226 116.143Q-10.226 116.608-10.453 117.018Q-10.680 117.429-11.066 117.669Q-11.453 117.909-11.922 117.909Q-12.441 117.909-12.828 117.542L-12.828 119.085L-12.402 119.085Q-12.195 119.108-12.156 119.327L-12.156 119.413Q-12.195 119.624-12.402 119.647L-13.890 119.647Q-14.094 119.624-14.140 119.413M-11.965 117.350Q-11.656 117.350-11.406 117.181Q-11.156 117.011-11.012 116.729Q-10.867 116.448-10.867 116.143Q-10.867 115.854-10.992 115.575Q-11.117 115.296-11.349 115.118Q-11.582 114.940-11.883 114.940Q-12.203 114.940-12.465 115.126Q-12.726 115.311-12.828 115.612L-12.828 116.464Q-12.738 116.831-12.517 117.091Q-12.297 117.350-11.965 117.350M-8.765 116.765L-8.765 114.983L-9.515 114.983Q-9.715 114.960-9.765 114.741L-9.765 114.655Q-9.715 114.444-9.515 114.421L-8.765 114.421L-8.765 113.671Q-8.715 113.464-8.515 113.436L-8.371 113.436Q-8.176 113.464-8.125 113.671L-8.125 114.421L-6.765 114.421Q-6.574 114.440-6.515 114.655L-6.515 114.741Q-6.570 114.960-6.765 114.983L-8.125 114.983L-8.125 116.733Q-8.125 117.350-7.551 117.350Q-7.301 117.350-7.137 117.165Q-6.973 116.979-6.973 116.733Q-6.973 116.640-6.900 116.569Q-6.828 116.499-6.726 116.487L-6.582 116.487Q-6.383 116.511-6.332 116.718L-6.332 116.765Q-6.332 117.089-6.515 117.352Q-6.699 117.616-6.992 117.763Q-7.285 117.909-7.605 117.909Q-8.117 117.909-8.441 117.599Q-8.765 117.288-8.765 116.765M-4.976 117.015L-4.976 114.983L-5.398 114.983Q-5.605 114.960-5.648 114.741L-5.648 114.655Q-5.601 114.444-5.398 114.421L-4.582 114.421Q-4.387 114.444-4.336 114.655L-4.336 116.983Q-4.336 117.218-4.166 117.284Q-3.996 117.350-3.711 117.350Q-3.504 117.350-3.308 117.274Q-3.113 117.198-2.988 117.048Q-2.863 116.897-2.863 116.686L-2.863 114.983L-3.285 114.983Q-3.496 114.960-3.535 114.741L-3.535 114.655Q-3.496 114.444-3.285 114.421L-2.473 114.421Q-2.273 114.444-2.223 114.655L-2.223 117.311L-1.797 117.311Q-1.590 117.335-1.551 117.542L-1.551 117.632Q-1.590 117.847-1.797 117.870L-2.613 117.870Q-2.812 117.847-2.863 117.647Q-3.265 117.909-3.773 117.909Q-4.008 117.909-4.223 117.868Q-4.437 117.827-4.603 117.725Q-4.769 117.624-4.873 117.446Q-4.976 117.268-4.976 117.015M-1.250 117.632L-1.250 117.542Q-1.191 117.335-1 117.311L-0.289 117.311L-0.289 114.983L-1 114.983Q-1.195 114.960-1.250 114.741L-1.250 114.655Q-1.191 114.444-1 114.421L0.102 114.421Q0.301 114.440 0.352 114.655L0.352 114.983Q0.613 114.698 0.969 114.540Q1.324 114.382 1.711 114.382Q2.004 114.382 2.238 114.516Q2.473 114.651 2.473 114.917Q2.473 115.085 2.363 115.202Q2.254 115.319 2.086 115.319Q1.934 115.319 1.818 115.208Q1.703 115.097 1.703 114.940Q1.328 114.940 1.014 115.141Q0.699 115.343 0.526 115.677Q0.352 116.011 0.352 116.390L0.352 117.311L1.297 117.311Q1.504 117.335 1.543 117.542L1.543 117.632Q1.504 117.847 1.297 117.870L-1 117.870Q-1.191 117.847-1.250 117.632M6.285 116.382L3.844 116.382Q3.899 116.659 4.096 116.882Q4.293 117.104 4.570 117.227Q4.848 117.350 5.133 117.350Q5.606 117.350 5.828 117.061Q5.836 117.050 5.893 116.944Q5.949 116.839 5.998 116.796Q6.047 116.753 6.141 116.741L6.285 116.741Q6.477 116.761 6.535 116.975L6.535 117.030Q6.469 117.331 6.238 117.528Q6.008 117.725 5.695 117.817Q5.383 117.909 5.078 117.909Q4.594 117.909 4.154 117.681Q3.715 117.452 3.447 117.052Q3.180 116.651 3.180 116.159L3.180 116.100Q3.180 115.632 3.426 115.229Q3.672 114.827 4.080 114.593Q4.488 114.358 4.957 114.358Q5.461 114.358 5.815 114.581Q6.168 114.804 6.352 115.192Q6.535 115.581 6.535 116.085L6.535 116.143Q6.477 116.358 6.285 116.382M3.852 115.831L5.879 115.831Q5.832 115.421 5.594 115.169Q5.356 114.917 4.957 114.917Q4.563 114.917 4.256 115.179Q3.949 115.440 3.852 115.831M8.883 117.909Q8.418 117.909 8.053 117.659Q7.688 117.409 7.483 117.005Q7.277 116.600 7.277 116.143Q7.277 115.800 7.402 115.479Q7.527 115.159 7.760 114.909Q7.992 114.659 8.297 114.520Q8.602 114.382 8.957 114.382Q9.469 114.382 9.875 114.702L9.875 113.542L9.453 113.542Q9.242 113.518 9.203 113.304L9.203 113.214Q9.242 113.007 9.453 112.983L10.266 112.983Q10.465 113.007 10.516 113.214L10.516 117.311L10.942 117.311Q11.149 117.335 11.188 117.542L11.188 117.632Q11.149 117.847 10.942 117.870L10.125 117.870Q9.926 117.847 9.875 117.632L9.875 117.503Q9.680 117.694 9.418 117.802Q9.156 117.909 8.883 117.909M8.922 117.350Q9.281 117.350 9.533 117.081Q9.785 116.811 9.875 116.436L9.875 115.604Q9.817 115.417 9.690 115.266Q9.563 115.116 9.385 115.028Q9.207 114.940 9.012 114.940Q8.703 114.940 8.453 115.110Q8.203 115.280 8.059 115.565Q7.914 115.850 7.914 116.151Q7.914 116.600 8.199 116.975Q8.485 117.350 8.922 117.350\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-65.203 117.87h298.999\"\u002F>\u003Cpath stroke=\"none\" d=\"m236.396 117.87-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmtt8\" font-size=\"8\">\u003Cg transform=\"translate(273.974 8.162)\">\u003Cpath d=\"M-64.644 116.143Q-64.644 115.663-64.400 115.249Q-64.156 114.835-63.740 114.597Q-63.324 114.358-62.844 114.358Q-62.289 114.358-61.910 114.468Q-61.531 114.577-61.531 114.983Q-61.531 115.151-61.644 115.274Q-61.758 115.397-61.930 115.397Q-62.101 115.397-62.221 115.282Q-62.340 115.167-62.340 114.999L-62.340 114.940Q-62.500 114.917-62.836 114.917Q-63.164 114.917-63.432 115.087Q-63.699 115.257-63.851 115.540Q-64.004 115.823-64.004 116.143Q-64.004 116.464-63.832 116.745Q-63.660 117.026-63.375 117.188Q-63.090 117.350-62.762 117.350Q-62.449 117.350-62.322 117.247Q-62.195 117.143-62.078 116.954Q-61.961 116.765-61.844 116.749L-61.676 116.749Q-61.570 116.761-61.506 116.829Q-61.441 116.897-61.441 116.999Q-61.441 117.046-61.461 117.085Q-61.570 117.378-61.773 117.558Q-61.976 117.737-62.252 117.823Q-62.527 117.909-62.844 117.909Q-63.328 117.909-63.744 117.671Q-64.160 117.433-64.402 117.030Q-64.644 116.628-64.644 116.143M-58.836 117.909Q-59.308 117.909-59.693 117.665Q-60.078 117.421-60.301 117.011Q-60.523 116.600-60.523 116.143Q-60.523 115.800-60.398 115.477Q-60.273 115.155-60.043 114.901Q-59.812 114.647-59.506 114.503Q-59.199 114.358-58.836 114.358Q-58.473 114.358-58.160 114.505Q-57.848 114.651-57.625 114.897Q-57.402 115.143-57.275 115.464Q-57.148 115.784-57.148 116.143Q-57.148 116.600-57.373 117.013Q-57.598 117.425-57.982 117.667Q-58.367 117.909-58.836 117.909M-58.836 117.350Q-58.371 117.350-58.080 116.956Q-57.789 116.561-57.789 116.077Q-57.789 115.784-57.924 115.516Q-58.058 115.249-58.299 115.083Q-58.539 114.917-58.836 114.917Q-59.141 114.917-59.379 115.083Q-59.617 115.249-59.752 115.516Q-59.887 115.784-59.887 116.077Q-59.887 116.558-59.594 116.954Q-59.301 117.350-58.836 117.350M-56.125 117.671L-56.125 116.757Q-56.098 116.550-55.887 116.526L-55.719 116.526Q-55.555 116.550-55.496 116.710Q-55.293 117.350-54.566 117.350Q-54.359 117.350-54.131 117.315Q-53.902 117.280-53.734 117.165Q-53.566 117.050-53.566 116.847Q-53.566 116.636-53.789 116.522Q-54.012 116.409-54.285 116.366L-54.984 116.253Q-56.125 116.042-56.125 115.319Q-56.125 115.030-55.980 114.841Q-55.836 114.651-55.596 114.544Q-55.355 114.436-55.099 114.397Q-54.844 114.358-54.566 114.358Q-54.316 114.358-54.123 114.388Q-53.930 114.417-53.766 114.495Q-53.687 114.378-53.558 114.358L-53.480 114.358Q-53.383 114.370-53.320 114.433Q-53.258 114.495-53.246 114.589L-53.246 115.296Q-53.258 115.390-53.320 115.456Q-53.383 115.522-53.480 115.534L-53.648 115.534Q-53.742 115.522-53.808 115.456Q-53.875 115.390-53.887 115.296Q-53.887 114.917-54.582 114.917Q-54.930 114.917-55.248 114.999Q-55.566 115.081-55.566 115.327Q-55.566 115.593-54.894 115.702L-54.191 115.823Q-53.707 115.905-53.357 116.153Q-53.008 116.401-53.008 116.847Q-53.008 117.237-53.244 117.479Q-53.480 117.722-53.830 117.815Q-54.180 117.909-54.566 117.909Q-55.144 117.909-55.543 117.655Q-55.613 117.780-55.662 117.837Q-55.711 117.893-55.816 117.909L-55.887 117.909Q-56.101 117.886-56.125 117.671M-51.266 116.765L-51.266 114.983L-52.016 114.983Q-52.215 114.960-52.266 114.741L-52.266 114.655Q-52.215 114.444-52.016 114.421L-51.266 114.421L-51.266 113.671Q-51.215 113.464-51.016 113.436L-50.871 113.436Q-50.676 113.464-50.625 113.671L-50.625 114.421L-49.266 114.421Q-49.074 114.440-49.016 114.655L-49.016 114.741Q-49.070 114.960-49.266 114.983L-50.625 114.983L-50.625 116.733Q-50.625 117.350-50.051 117.350Q-49.801 117.350-49.637 117.165Q-49.473 116.979-49.473 116.733Q-49.473 116.640-49.400 116.569Q-49.328 116.499-49.226 116.487L-49.082 116.487Q-48.883 116.511-48.832 116.718L-48.832 116.765Q-48.832 117.089-49.016 117.352Q-49.199 117.616-49.492 117.763Q-49.785 117.909-50.105 117.909Q-50.617 117.909-50.941 117.599Q-51.266 117.288-51.266 116.765\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(273.974 8.162)\">\u003Cpath d=\"M-43.883 119.413L-43.883 119.327Q-43.840 119.108-43.633 119.085L-43.211 119.085L-43.211 114.983L-43.633 114.983Q-43.840 114.960-43.883 114.741L-43.883 114.655Q-43.836 114.444-43.633 114.421L-42.816 114.421Q-42.621 114.440-42.570 114.655L-42.570 114.725Q-42.359 114.558-42.096 114.470Q-41.832 114.382-41.562 114.382Q-41.223 114.382-40.926 114.526Q-40.629 114.671-40.414 114.923Q-40.199 115.175-40.084 115.487Q-39.969 115.800-39.969 116.143Q-39.969 116.608-40.195 117.018Q-40.422 117.429-40.808 117.669Q-41.195 117.909-41.664 117.909Q-42.183 117.909-42.570 117.542L-42.570 119.085L-42.144 119.085Q-41.937 119.108-41.898 119.327L-41.898 119.413Q-41.937 119.624-42.144 119.647L-43.633 119.647Q-43.836 119.624-43.883 119.413M-41.707 117.350Q-41.398 117.350-41.148 117.181Q-40.898 117.011-40.754 116.729Q-40.609 116.448-40.609 116.143Q-40.609 115.854-40.734 115.575Q-40.859 115.296-41.092 115.118Q-41.324 114.940-41.625 114.940Q-41.945 114.940-42.207 115.126Q-42.469 115.311-42.570 115.612L-42.570 116.464Q-42.480 116.831-42.260 117.091Q-42.039 117.350-41.707 117.350M-36.195 116.382L-38.637 116.382Q-38.582 116.659-38.385 116.882Q-38.187 117.104-37.910 117.227Q-37.633 117.350-37.348 117.350Q-36.875 117.350-36.652 117.061Q-36.644 117.050-36.588 116.944Q-36.531 116.839-36.482 116.796Q-36.433 116.753-36.340 116.741L-36.195 116.741Q-36.004 116.761-35.945 116.975L-35.945 117.030Q-36.012 117.331-36.242 117.528Q-36.473 117.725-36.785 117.817Q-37.098 117.909-37.402 117.909Q-37.887 117.909-38.326 117.681Q-38.766 117.452-39.033 117.052Q-39.301 116.651-39.301 116.159L-39.301 116.100Q-39.301 115.632-39.055 115.229Q-38.808 114.827-38.400 114.593Q-37.992 114.358-37.523 114.358Q-37.019 114.358-36.666 114.581Q-36.312 114.804-36.129 115.192Q-35.945 115.581-35.945 116.085L-35.945 116.143Q-36.004 116.358-36.195 116.382M-38.629 115.831L-36.601 115.831Q-36.648 115.421-36.887 115.169Q-37.125 114.917-37.523 114.917Q-37.918 114.917-38.224 115.179Q-38.531 115.440-38.629 115.831M-35.238 117.632L-35.238 117.542Q-35.180 117.335-34.988 117.311L-34.277 117.311L-34.277 114.983L-34.988 114.983Q-35.183 114.960-35.238 114.741L-35.238 114.655Q-35.180 114.444-34.988 114.421L-33.887 114.421Q-33.687 114.440-33.637 114.655L-33.637 114.983Q-33.375 114.698-33.019 114.540Q-32.664 114.382-32.277 114.382Q-31.984 114.382-31.750 114.516Q-31.516 114.651-31.516 114.917Q-31.516 115.085-31.625 115.202Q-31.734 115.319-31.902 115.319Q-32.055 115.319-32.170 115.208Q-32.285 115.097-32.285 114.940Q-32.660 114.940-32.974 115.141Q-33.289 115.343-33.463 115.677Q-33.637 116.011-33.637 116.390L-33.637 117.311L-32.691 117.311Q-32.484 117.335-32.445 117.542L-32.445 117.632Q-32.484 117.847-32.691 117.870L-34.988 117.870Q-35.180 117.847-35.238 117.632\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(273.974 8.162)\">\u003Cpath d=\"M-26.367 117.671L-26.367 116.757Q-26.340 116.550-26.129 116.526L-25.961 116.526Q-25.797 116.550-25.738 116.710Q-25.535 117.350-24.808 117.350Q-24.601 117.350-24.373 117.315Q-24.144 117.280-23.976 117.165Q-23.808 117.050-23.808 116.847Q-23.808 116.636-24.031 116.522Q-24.254 116.409-24.527 116.366L-25.226 116.253Q-26.367 116.042-26.367 115.319Q-26.367 115.030-26.223 114.841Q-26.078 114.651-25.838 114.544Q-25.598 114.436-25.342 114.397Q-25.086 114.358-24.808 114.358Q-24.558 114.358-24.365 114.388Q-24.172 114.417-24.008 114.495Q-23.930 114.378-23.801 114.358L-23.723 114.358Q-23.625 114.370-23.562 114.433Q-23.500 114.495-23.488 114.589L-23.488 115.296Q-23.500 115.390-23.562 115.456Q-23.625 115.522-23.723 115.534L-23.890 115.534Q-23.984 115.522-24.051 115.456Q-24.117 115.390-24.129 115.296Q-24.129 114.917-24.824 114.917Q-25.172 114.917-25.490 114.999Q-25.808 115.081-25.808 115.327Q-25.808 115.593-25.137 115.702L-24.433 115.823Q-23.949 115.905-23.599 116.153Q-23.250 116.401-23.250 116.847Q-23.250 117.237-23.486 117.479Q-23.723 117.722-24.072 117.815Q-24.422 117.909-24.808 117.909Q-25.387 117.909-25.785 117.655Q-25.855 117.780-25.904 117.837Q-25.953 117.893-26.058 117.909L-26.129 117.909Q-26.344 117.886-26.367 117.671M-21.508 116.765L-21.508 114.983L-22.258 114.983Q-22.457 114.960-22.508 114.741L-22.508 114.655Q-22.457 114.444-22.258 114.421L-21.508 114.421L-21.508 113.671Q-21.457 113.464-21.258 113.436L-21.113 113.436Q-20.918 113.464-20.867 113.671L-20.867 114.421L-19.508 114.421Q-19.316 114.440-19.258 114.655L-19.258 114.741Q-19.312 114.960-19.508 114.983L-20.867 114.983L-20.867 116.733Q-20.867 117.350-20.293 117.350Q-20.043 117.350-19.879 117.165Q-19.715 116.979-19.715 116.733Q-19.715 116.640-19.642 116.569Q-19.570 116.499-19.469 116.487L-19.324 116.487Q-19.125 116.511-19.074 116.718L-19.074 116.765Q-19.074 117.089-19.258 117.352Q-19.441 117.616-19.734 117.763Q-20.027 117.909-20.348 117.909Q-20.859 117.909-21.183 117.599Q-21.508 117.288-21.508 116.765M-14.949 116.382L-17.390 116.382Q-17.336 116.659-17.139 116.882Q-16.941 117.104-16.664 117.227Q-16.387 117.350-16.101 117.350Q-15.629 117.350-15.406 117.061Q-15.398 117.050-15.342 116.944Q-15.285 116.839-15.236 116.796Q-15.187 116.753-15.094 116.741L-14.949 116.741Q-14.758 116.761-14.699 116.975L-14.699 117.030Q-14.765 117.331-14.996 117.528Q-15.226 117.725-15.539 117.817Q-15.851 117.909-16.156 117.909Q-16.640 117.909-17.080 117.681Q-17.519 117.452-17.787 117.052Q-18.055 116.651-18.055 116.159L-18.055 116.100Q-18.055 115.632-17.808 115.229Q-17.562 114.827-17.154 114.593Q-16.746 114.358-16.277 114.358Q-15.773 114.358-15.420 114.581Q-15.066 114.804-14.883 115.192Q-14.699 115.581-14.699 116.085L-14.699 116.143Q-14.758 116.358-14.949 116.382M-17.383 115.831L-15.355 115.831Q-15.402 115.421-15.640 115.169Q-15.879 114.917-16.277 114.917Q-16.672 114.917-16.978 115.179Q-17.285 115.440-17.383 115.831M-14.144 119.413L-14.144 119.327Q-14.101 119.108-13.894 119.085L-13.473 119.085L-13.473 114.983L-13.894 114.983Q-14.101 114.960-14.144 114.741L-14.144 114.655Q-14.098 114.444-13.894 114.421L-13.078 114.421Q-12.883 114.440-12.832 114.655L-12.832 114.725Q-12.621 114.558-12.357 114.470Q-12.094 114.382-11.824 114.382Q-11.484 114.382-11.187 114.526Q-10.890 114.671-10.676 114.923Q-10.461 115.175-10.346 115.487Q-10.230 115.800-10.230 116.143Q-10.230 116.608-10.457 117.018Q-10.683 117.429-11.070 117.669Q-11.457 117.909-11.926 117.909Q-12.445 117.909-12.832 117.542L-12.832 119.085L-12.406 119.085Q-12.199 119.108-12.160 119.327L-12.160 119.413Q-12.199 119.624-12.406 119.647L-13.894 119.647Q-14.098 119.624-14.144 119.413M-11.969 117.350Q-11.660 117.350-11.410 117.181Q-11.160 117.011-11.015 116.729Q-10.871 116.448-10.871 116.143Q-10.871 115.854-10.996 115.575Q-11.121 115.296-11.353 115.118Q-11.586 114.940-11.887 114.940Q-12.207 114.940-12.469 115.126Q-12.730 115.311-12.832 115.612L-12.832 116.464Q-12.742 116.831-12.521 117.091Q-12.301 117.350-11.969 117.350\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath d=\"M-53.422 105.777h113.81V81.592h-113.81Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"7\">\u003Cg transform=\"translate(38.936 -22.824)\">\u003Cpath d=\"M-64.995 118.437Q-64.995 118.174-64.865 117.945Q-64.735 117.716-64.516 117.576Q-64.677 117.354-64.677 117.074Q-64.677 116.824-64.543 116.595Q-64.762 116.311-64.762 115.946Q-64.762 115.631-64.595 115.371Q-64.427 115.112-64.149 114.965Q-63.870 114.818-63.552 114.818Q-63.371 114.818-63.193 114.871Q-63.016 114.924-62.872 115.013Q-62.537 114.784-62.123 114.784Q-61.915 114.784-61.756 114.898Q-61.597 115.013-61.597 115.211Q-61.597 115.341-61.686 115.433Q-61.775 115.525-61.911 115.525Q-62.028 115.525-62.111 115.457Q-62.195 115.389-62.223 115.279Q-62.349 115.286-62.523 115.365Q-62.346 115.638-62.346 115.946Q-62.346 116.192-62.443 116.395Q-62.540 116.599-62.711 116.754Q-62.882 116.910-63.103 116.995Q-63.323 117.080-63.552 117.080Q-63.891 117.080-64.181 116.916Q-64.191 116.916-64.200 116.980Q-64.208 117.043-64.208 117.074Q-64.208 117.180-64.150 117.274Q-64.092 117.368-63.986 117.402Q-63.973 117.398-63.961 117.397Q-63.949 117.395-63.928 117.395Q-63.914 117.395-63.887 117.402Q-63.860 117.409-63.853 117.409L-63.333 117.409Q-62.650 117.409-62.176 117.620Q-61.703 117.832-61.703 118.437Q-61.703 118.701-61.860 118.899Q-62.017 119.097-62.277 119.224Q-62.537 119.350-62.817 119.408Q-63.098 119.466-63.347 119.466Q-63.689 119.466-64.075 119.359Q-64.461 119.251-64.728 119.017Q-64.995 118.783-64.995 118.437M-64.523 118.431Q-64.523 118.622-64.313 118.745Q-64.102 118.868-63.819 118.926Q-63.535 118.984-63.347 118.984Q-63.159 118.984-62.875 118.926Q-62.592 118.868-62.381 118.745Q-62.171 118.622-62.171 118.431Q-62.171 118.178-62.368 118.056Q-62.564 117.935-62.810 117.906Q-63.057 117.877-63.361 117.877L-63.901 117.877Q-64.061 117.877-64.203 117.947Q-64.345 118.017-64.434 118.145Q-64.523 118.273-64.523 118.431M-63.552 116.595Q-63.289 116.595-63.098 116.404Q-62.906 116.212-62.906 115.946Q-62.906 115.809-62.954 115.694Q-63.002 115.580-63.094 115.488Q-63.186 115.395-63.301 115.348Q-63.415 115.300-63.552 115.300Q-63.819 115.300-64.010 115.491Q-64.202 115.683-64.202 115.946Q-64.202 116.205-64.007 116.400Q-63.812 116.595-63.552 116.595M-61.293 117.662L-61.293 117.583Q-61.242 117.402-61.074 117.381L-60.452 117.381L-60.452 115.344L-61.074 115.344Q-61.245 115.324-61.293 115.132L-61.293 115.057Q-61.242 114.872-61.074 114.852L-60.110 114.852Q-59.936 114.869-59.891 115.057L-59.891 115.344Q-59.662 115.095-59.351 114.956Q-59.040 114.818-58.702 114.818Q-58.446 114.818-58.241 114.936Q-58.036 115.054-58.036 115.286Q-58.036 115.433-58.131 115.536Q-58.227 115.638-58.374 115.638Q-58.507 115.638-58.608 115.541Q-58.709 115.443-58.709 115.307Q-59.037 115.307-59.312 115.483Q-59.587 115.659-59.739 115.951Q-59.891 116.243-59.891 116.575L-59.891 117.381L-59.064 117.381Q-58.883 117.402-58.849 117.583L-58.849 117.662Q-58.883 117.849-59.064 117.870L-61.074 117.870Q-61.242 117.849-61.293 117.662M-57.417 116.896Q-57.417 116.506-57.055 116.281Q-56.692 116.055-56.219 115.968Q-55.745 115.881-55.301 115.874Q-55.301 115.686-55.419 115.553Q-55.537 115.419-55.718 115.353Q-55.899 115.286-56.084 115.286Q-56.385 115.286-56.525 115.307L-56.525 115.358Q-56.525 115.505-56.629 115.606Q-56.733 115.706-56.877 115.706Q-57.031 115.706-57.131 115.599Q-57.232 115.491-57.232 115.344Q-57.232 114.989-56.899 114.893Q-56.566 114.797-56.077 114.797Q-55.841 114.797-55.607 114.866Q-55.373 114.934-55.176 115.066Q-54.980 115.197-54.860 115.390Q-54.741 115.583-54.741 115.826L-54.741 117.330Q-54.741 117.381-54.279 117.381Q-54.108 117.398-54.064 117.583L-54.064 117.662Q-54.108 117.849-54.279 117.870L-54.406 117.870Q-54.706 117.870-54.906 117.829Q-55.106 117.788-55.233 117.624Q-55.636 117.904-56.255 117.904Q-56.549 117.904-56.815 117.781Q-57.082 117.658-57.249 117.429Q-57.417 117.200-57.417 116.896M-56.856 116.903Q-56.856 117.142-56.644 117.279Q-56.432 117.415-56.183 117.415Q-55.992 117.415-55.788 117.364Q-55.585 117.313-55.443 117.192Q-55.301 117.070-55.301 116.875L-55.301 116.359Q-55.547 116.359-55.911 116.409Q-56.275 116.458-56.566 116.580Q-56.856 116.701-56.856 116.903M-52.427 117.904Q-52.833 117.904-53.153 117.685Q-53.473 117.467-53.652 117.113Q-53.831 116.759-53.831 116.359Q-53.831 116.058-53.722 115.778Q-53.613 115.498-53.409 115.279Q-53.206 115.060-52.939 114.939Q-52.673 114.818-52.362 114.818Q-51.914 114.818-51.558 115.098L-51.558 114.083L-51.928 114.083Q-52.112 114.062-52.146 113.874L-52.146 113.796Q-52.112 113.615-51.928 113.594L-51.217 113.594Q-51.042 113.615-50.998 113.796L-50.998 117.381L-50.625 117.381Q-50.444 117.402-50.410 117.583L-50.410 117.662Q-50.444 117.849-50.625 117.870L-51.340 117.870Q-51.514 117.849-51.558 117.662L-51.558 117.549Q-51.729 117.716-51.958 117.810Q-52.187 117.904-52.427 117.904M-52.392 117.415Q-52.078 117.415-51.858 117.180Q-51.637 116.944-51.558 116.616L-51.558 115.888Q-51.610 115.724-51.721 115.592Q-51.832 115.460-51.987 115.383Q-52.143 115.307-52.314 115.307Q-52.584 115.307-52.803 115.455Q-53.021 115.604-53.148 115.853Q-53.274 116.103-53.274 116.366Q-53.274 116.759-53.025 117.087Q-52.775 117.415-52.392 117.415M-49.802 117.662L-49.802 117.583Q-49.757 117.402-49.586 117.381L-48.677 117.381L-48.677 115.344L-49.528 115.344Q-49.703 115.324-49.747 115.132L-49.747 115.057Q-49.703 114.872-49.528 114.852L-48.332 114.852Q-48.161 114.869-48.117 115.057L-48.117 117.381L-47.317 117.381Q-47.146 117.402-47.101 117.583L-47.101 117.662Q-47.146 117.849-47.317 117.870L-49.586 117.870Q-49.757 117.849-49.802 117.662M-48.899 113.997L-48.899 113.950Q-48.899 113.799-48.780 113.693Q-48.660 113.587-48.506 113.587Q-48.356 113.587-48.236 113.693Q-48.117 113.799-48.117 113.950L-48.117 113.997Q-48.117 114.151-48.236 114.257Q-48.356 114.363-48.506 114.363Q-48.660 114.363-48.780 114.257Q-48.899 114.151-48.899 113.997M-43.554 116.568L-45.690 116.568Q-45.642 116.810-45.469 117.005Q-45.297 117.200-45.054 117.308Q-44.811 117.415-44.562 117.415Q-44.148 117.415-43.953 117.162Q-43.947 117.152-43.897 117.060Q-43.848 116.968-43.805 116.930Q-43.762 116.892-43.680 116.882L-43.554 116.882Q-43.386 116.899-43.335 117.087L-43.335 117.135Q-43.393 117.398-43.595 117.571Q-43.796 117.744-44.070 117.824Q-44.343 117.904-44.610 117.904Q-45.034 117.904-45.418 117.704Q-45.803 117.504-46.037 117.154Q-46.271 116.804-46.271 116.373L-46.271 116.322Q-46.271 115.912-46.056 115.559Q-45.840 115.207-45.483 115.002Q-45.126 114.797-44.716 114.797Q-44.275 114.797-43.965 114.992Q-43.656 115.187-43.495 115.527Q-43.335 115.867-43.335 116.308L-43.335 116.359Q-43.386 116.547-43.554 116.568M-45.683 116.086L-43.909 116.086Q-43.950 115.727-44.159 115.506Q-44.367 115.286-44.716 115.286Q-45.061 115.286-45.329 115.515Q-45.598 115.744-45.683 116.086M-42.849 117.662L-42.849 117.583Q-42.812 117.402-42.631 117.381L-42.262 117.381L-42.262 115.344L-42.631 115.344Q-42.812 115.324-42.849 115.132L-42.849 115.057Q-42.808 114.872-42.631 114.852L-41.916 114.852Q-41.745 114.872-41.701 115.057L-41.701 115.132L-41.708 115.153Q-41.510 114.995-41.270 114.907Q-41.031 114.818-40.775 114.818Q-40.471 114.818-40.262 114.941Q-40.054 115.064-39.953 115.289Q-39.852 115.515-39.852 115.826L-39.852 117.381L-39.479 117.381Q-39.298 117.402-39.264 117.583L-39.264 117.662Q-39.298 117.849-39.479 117.870L-40.700 117.870Q-40.870 117.849-40.915 117.662L-40.915 117.583Q-40.870 117.398-40.700 117.381L-40.412 117.381L-40.412 115.853Q-40.412 115.583-40.491 115.445Q-40.570 115.307-40.826 115.307Q-41.062 115.307-41.262 115.421Q-41.462 115.536-41.581 115.735Q-41.701 115.935-41.701 116.168L-41.701 117.381L-41.328 117.381Q-41.147 117.402-41.113 117.583L-41.113 117.662Q-41.147 117.849-41.328 117.870L-42.631 117.870Q-42.812 117.849-42.849 117.662M-38.146 116.903L-38.146 115.344L-38.803 115.344Q-38.977 115.324-39.021 115.132L-39.021 115.057Q-38.977 114.872-38.803 114.852L-38.146 114.852L-38.146 114.196Q-38.102 114.015-37.928 113.991L-37.801 113.991Q-37.630 114.015-37.586 114.196L-37.586 114.852L-36.396 114.852Q-36.229 114.869-36.178 115.057L-36.178 115.132Q-36.225 115.324-36.396 115.344L-37.586 115.344L-37.586 116.875Q-37.586 117.415-37.083 117.415Q-36.865 117.415-36.721 117.253Q-36.578 117.091-36.578 116.875Q-36.578 116.793-36.514 116.732Q-36.451 116.670-36.362 116.660L-36.236 116.660Q-36.061 116.681-36.017 116.862L-36.017 116.903Q-36.017 117.186-36.178 117.417Q-36.338 117.648-36.595 117.776Q-36.851 117.904-37.131 117.904Q-37.579 117.904-37.863 117.632Q-38.146 117.361-38.146 116.903\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(38.936 -22.824)\">\u003Cpath d=\"M-30.104 117.904Q-30.510 117.904-30.830 117.685Q-31.150 117.467-31.329 117.113Q-31.508 116.759-31.508 116.359Q-31.508 116.058-31.399 115.778Q-31.290 115.498-31.086 115.279Q-30.883 115.060-30.616 114.939Q-30.350 114.818-30.039 114.818Q-29.591 114.818-29.235 115.098L-29.235 114.083L-29.605 114.083Q-29.789 114.062-29.823 113.874L-29.823 113.796Q-29.789 113.615-29.605 113.594L-28.894 113.594Q-28.719 113.615-28.675 113.796L-28.675 117.381L-28.302 117.381Q-28.121 117.402-28.087 117.583L-28.087 117.662Q-28.121 117.849-28.302 117.870L-29.017 117.870Q-29.191 117.849-29.235 117.662L-29.235 117.549Q-29.406 117.716-29.635 117.810Q-29.864 117.904-30.104 117.904M-30.069 117.415Q-29.755 117.415-29.535 117.180Q-29.314 116.944-29.235 116.616L-29.235 115.888Q-29.287 115.724-29.398 115.592Q-29.509 115.460-29.664 115.383Q-29.820 115.307-29.991 115.307Q-30.261 115.307-30.480 115.455Q-30.698 115.604-30.825 115.853Q-30.951 116.103-30.951 116.366Q-30.951 116.759-30.702 117.087Q-30.452 117.415-30.069 117.415M-24.946 116.568L-27.082 116.568Q-27.034 116.810-26.862 117.005Q-26.689 117.200-26.446 117.308Q-26.204 117.415-25.954 117.415Q-25.541 117.415-25.346 117.162Q-25.339 117.152-25.289 117.060Q-25.240 116.968-25.197 116.930Q-25.154 116.892-25.072 116.882L-24.946 116.882Q-24.778 116.899-24.727 117.087L-24.727 117.135Q-24.785 117.398-24.987 117.571Q-25.189 117.744-25.462 117.824Q-25.735 117.904-26.002 117.904Q-26.426 117.904-26.810 117.704Q-27.195 117.504-27.429 117.154Q-27.663 116.804-27.663 116.373L-27.663 116.322Q-27.663 115.912-27.448 115.559Q-27.233 115.207-26.875 115.002Q-26.518 114.797-26.108 114.797Q-25.667 114.797-25.358 114.992Q-25.048 115.187-24.888 115.527Q-24.727 115.867-24.727 116.308L-24.727 116.359Q-24.778 116.547-24.946 116.568M-27.075 116.086L-25.301 116.086Q-25.342 115.727-25.551 115.506Q-25.759 115.286-26.108 115.286Q-26.453 115.286-26.722 115.515Q-26.990 115.744-27.075 116.086M-23.791 117.696L-23.791 116.896Q-23.767 116.715-23.582 116.694L-23.435 116.694Q-23.292 116.715-23.240 116.855Q-23.063 117.415-22.427 117.415Q-22.246 117.415-22.046 117.385Q-21.846 117.354-21.699 117.253Q-21.552 117.152-21.552 116.974Q-21.552 116.790-21.747 116.691Q-21.942 116.592-22.181 116.554L-22.793 116.455Q-23.791 116.270-23.791 115.638Q-23.791 115.385-23.664 115.219Q-23.538 115.054-23.328 114.960Q-23.117 114.866-22.893 114.831Q-22.670 114.797-22.427 114.797Q-22.208 114.797-22.039 114.823Q-21.870 114.849-21.726 114.917Q-21.658 114.814-21.545 114.797L-21.477 114.797Q-21.391 114.808-21.337 114.862Q-21.282 114.917-21.272 114.999L-21.272 115.618Q-21.282 115.700-21.337 115.758Q-21.391 115.816-21.477 115.826L-21.624 115.826Q-21.706 115.816-21.764 115.758Q-21.822 115.700-21.832 115.618Q-21.832 115.286-22.441 115.286Q-22.745 115.286-23.023 115.358Q-23.302 115.430-23.302 115.645Q-23.302 115.877-22.714 115.973L-22.099 116.079Q-21.675 116.151-21.369 116.368Q-21.063 116.585-21.063 116.974Q-21.063 117.316-21.270 117.528Q-21.477 117.740-21.783 117.822Q-22.088 117.904-22.427 117.904Q-22.933 117.904-23.281 117.682Q-23.343 117.791-23.386 117.841Q-23.428 117.891-23.521 117.904L-23.582 117.904Q-23.770 117.884-23.791 117.696M-20.099 116.359Q-20.099 115.939-19.886 115.577Q-19.672 115.214-19.308 115.006Q-18.944 114.797-18.524 114.797Q-18.038 114.797-17.707 114.893Q-17.375 114.989-17.375 115.344Q-17.375 115.491-17.474 115.599Q-17.573 115.706-17.724 115.706Q-17.874 115.706-17.978 115.606Q-18.083 115.505-18.083 115.358L-18.083 115.307Q-18.223 115.286-18.517 115.286Q-18.804 115.286-19.038 115.435Q-19.272 115.583-19.405 115.831Q-19.539 116.079-19.539 116.359Q-19.539 116.640-19.388 116.886Q-19.238 117.132-18.988 117.274Q-18.739 117.415-18.452 117.415Q-18.178 117.415-18.067 117.325Q-17.956 117.234-17.854 117.068Q-17.751 116.903-17.649 116.889L-17.502 116.889Q-17.409 116.899-17.353 116.959Q-17.297 117.019-17.297 117.108Q-17.297 117.149-17.314 117.183Q-17.409 117.439-17.587 117.597Q-17.765 117.754-18.006 117.829Q-18.247 117.904-18.524 117.904Q-18.947 117.904-19.311 117.696Q-19.675 117.487-19.887 117.135Q-20.099 116.783-20.099 116.359M-13.800 116.568L-15.936 116.568Q-15.888 116.810-15.716 117.005Q-15.543 117.200-15.300 117.308Q-15.058 117.415-14.808 117.415Q-14.395 117.415-14.200 117.162Q-14.193 117.152-14.143 117.060Q-14.094 116.968-14.051 116.930Q-14.008 116.892-13.926 116.882L-13.800 116.882Q-13.632 116.899-13.581 117.087L-13.581 117.135Q-13.639 117.398-13.841 117.571Q-14.043 117.744-14.316 117.824Q-14.589 117.904-14.856 117.904Q-15.280 117.904-15.664 117.704Q-16.049 117.504-16.283 117.154Q-16.517 116.804-16.517 116.373L-16.517 116.322Q-16.517 115.912-16.302 115.559Q-16.087 115.207-15.729 115.002Q-15.372 114.797-14.962 114.797Q-14.521 114.797-14.212 114.992Q-13.902 115.187-13.742 115.527Q-13.581 115.867-13.581 116.308L-13.581 116.359Q-13.632 116.547-13.800 116.568M-15.929 116.086L-14.155 116.086Q-14.196 115.727-14.405 115.506Q-14.613 115.286-14.962 115.286Q-15.307 115.286-15.576 115.515Q-15.844 115.744-15.929 116.086M-13.096 117.662L-13.096 117.583Q-13.058 117.402-12.877 117.381L-12.508 117.381L-12.508 115.344L-12.877 115.344Q-13.058 115.324-13.096 115.132L-13.096 115.057Q-13.055 114.872-12.877 114.852L-12.163 114.852Q-11.992 114.872-11.947 115.057L-11.947 115.132L-11.954 115.153Q-11.756 114.995-11.517 114.907Q-11.277 114.818-11.021 114.818Q-10.717 114.818-10.508 114.941Q-10.300 115.064-10.199 115.289Q-10.098 115.515-10.098 115.826L-10.098 117.381L-9.726 117.381Q-9.545 117.402-9.510 117.583L-9.510 117.662Q-9.545 117.849-9.726 117.870L-10.946 117.870Q-11.117 117.849-11.161 117.662L-11.161 117.583Q-11.117 117.398-10.946 117.381L-10.659 117.381L-10.659 115.853Q-10.659 115.583-10.737 115.445Q-10.816 115.307-11.072 115.307Q-11.308 115.307-11.508 115.421Q-11.708 115.536-11.828 115.735Q-11.947 115.935-11.947 116.168L-11.947 117.381L-11.575 117.381Q-11.394 117.402-11.359 117.583L-11.359 117.662Q-11.394 117.849-11.575 117.870L-12.877 117.870Q-13.058 117.849-13.096 117.662M-8.393 116.903L-8.393 115.344L-9.049 115.344Q-9.223 115.324-9.268 115.132L-9.268 115.057Q-9.223 114.872-9.049 114.852L-8.393 114.852L-8.393 114.196Q-8.348 114.015-8.174 113.991L-8.047 113.991Q-7.877 114.015-7.832 114.196L-7.832 114.852L-6.643 114.852Q-6.475 114.869-6.424 115.057L-6.424 115.132Q-6.472 115.324-6.643 115.344L-7.832 115.344L-7.832 116.875Q-7.832 117.415-7.330 117.415Q-7.111 117.415-6.967 117.253Q-6.824 117.091-6.824 116.875Q-6.824 116.793-6.761 116.732Q-6.697 116.670-6.609 116.660L-6.482 116.660Q-6.308 116.681-6.263 116.862L-6.263 116.903Q-6.263 117.186-6.424 117.417Q-6.585 117.648-6.841 117.776Q-7.097 117.904-7.378 117.904Q-7.825 117.904-8.109 117.632Q-8.393 117.361-8.393 116.903\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath d=\"M-10.743 71.634h113.81V47.449h-113.81Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"7\">\u003Cg transform=\"translate(83.475 -56.675)\">\u003Cpath d=\"M-65.001 117.662L-65.001 117.583Q-64.950 117.398-64.783 117.381L-64.588 117.381L-63.754 113.721Q-63.733 113.618-63.650 113.553Q-63.566 113.488-63.467 113.488L-63.227 113.488Q-63.128 113.488-63.045 113.555Q-62.961 113.621-62.940 113.721L-62.110 117.381L-61.911 117.381Q-61.730 117.402-61.696 117.583L-61.696 117.662Q-61.741 117.849-61.911 117.870L-62.858 117.870Q-63.029 117.849-63.074 117.662L-63.074 117.583Q-63.029 117.402-62.858 117.381L-62.684 117.381Q-62.797 116.886-62.817 116.814L-63.880 116.814Q-63.918 116.974-64.014 117.381L-63.839 117.381Q-63.665 117.402-63.620 117.583L-63.620 117.662Q-63.665 117.849-63.839 117.870L-64.783 117.870Q-64.964 117.849-65.001 117.662M-63.774 116.322L-62.920 116.322Q-63.340 114.455-63.340 114.155L-63.347 114.155Q-63.347 114.315-63.470 114.913Q-63.593 115.512-63.774 116.322M-59.857 117.904Q-60.264 117.904-60.584 117.685Q-60.903 117.467-61.083 117.113Q-61.262 116.759-61.262 116.359Q-61.262 116.058-61.153 115.778Q-61.043 115.498-60.840 115.279Q-60.637 115.060-60.370 114.939Q-60.103 114.818-59.792 114.818Q-59.345 114.818-58.989 115.098L-58.989 114.083L-59.358 114.083Q-59.543 114.062-59.577 113.874L-59.577 113.796Q-59.543 113.615-59.358 113.594L-58.647 113.594Q-58.473 113.615-58.429 113.796L-58.429 117.381L-58.056 117.381Q-57.875 117.402-57.841 117.583L-57.841 117.662Q-57.875 117.849-58.056 117.870L-58.770 117.870Q-58.945 117.849-58.989 117.662L-58.989 117.549Q-59.160 117.716-59.389 117.810Q-59.618 117.904-59.857 117.904M-59.823 117.415Q-59.509 117.415-59.288 117.180Q-59.068 116.944-58.989 116.616L-58.989 115.888Q-59.040 115.724-59.151 115.592Q-59.263 115.460-59.418 115.383Q-59.574 115.307-59.745 115.307Q-60.015 115.307-60.233 115.455Q-60.452 115.604-60.578 115.853Q-60.705 116.103-60.705 116.366Q-60.705 116.759-60.455 117.087Q-60.206 117.415-59.823 117.415M-57.417 116.896Q-57.417 116.506-57.055 116.281Q-56.692 116.055-56.219 115.968Q-55.745 115.881-55.301 115.874Q-55.301 115.686-55.419 115.553Q-55.537 115.419-55.718 115.353Q-55.899 115.286-56.084 115.286Q-56.385 115.286-56.525 115.307L-56.525 115.358Q-56.525 115.505-56.629 115.606Q-56.733 115.706-56.877 115.706Q-57.031 115.706-57.131 115.599Q-57.232 115.491-57.232 115.344Q-57.232 114.989-56.899 114.893Q-56.566 114.797-56.077 114.797Q-55.841 114.797-55.607 114.866Q-55.373 114.934-55.176 115.066Q-54.980 115.197-54.860 115.390Q-54.741 115.583-54.741 115.826L-54.741 117.330Q-54.741 117.381-54.279 117.381Q-54.108 117.398-54.064 117.583L-54.064 117.662Q-54.108 117.849-54.279 117.870L-54.406 117.870Q-54.706 117.870-54.906 117.829Q-55.106 117.788-55.233 117.624Q-55.636 117.904-56.255 117.904Q-56.549 117.904-56.815 117.781Q-57.082 117.658-57.249 117.429Q-57.417 117.200-57.417 116.896M-56.856 116.903Q-56.856 117.142-56.644 117.279Q-56.432 117.415-56.183 117.415Q-55.992 117.415-55.788 117.364Q-55.585 117.313-55.443 117.192Q-55.301 117.070-55.301 116.875L-55.301 116.359Q-55.547 116.359-55.911 116.409Q-56.275 116.458-56.566 116.580Q-56.856 116.701-56.856 116.903M-54.105 117.662L-54.105 117.583Q-54.060 117.398-53.890 117.381L-53.715 117.381L-53.715 115.344L-53.890 115.344Q-54.071 115.324-54.105 115.132L-54.105 115.057Q-54.060 114.869-53.890 114.852L-53.469 114.852Q-53.302 114.872-53.250 115.057Q-52.970 114.818-52.608 114.818Q-52.440 114.818-52.293 114.913Q-52.146 115.009-52.081 115.160Q-51.928 114.992-51.733 114.905Q-51.538 114.818-51.326 114.818Q-50.960 114.818-50.825 115.117Q-50.690 115.416-50.690 115.826L-50.690 117.381L-50.516 117.381Q-50.331 117.402-50.297 117.583L-50.297 117.662Q-50.342 117.849-50.516 117.870L-51.230 117.870Q-51.401 117.849-51.446 117.662L-51.446 117.583Q-51.401 117.402-51.230 117.381L-51.152 117.381L-51.152 115.853Q-51.152 115.307-51.370 115.307Q-51.661 115.307-51.817 115.578Q-51.972 115.850-51.972 116.168L-51.972 117.381L-51.794 117.381Q-51.613 117.402-51.579 117.583L-51.579 117.662Q-51.623 117.849-51.794 117.870L-52.509 117.870Q-52.683 117.849-52.727 117.662L-52.727 117.583Q-52.683 117.402-52.509 117.381L-52.433 117.381L-52.433 115.853Q-52.433 115.307-52.649 115.307Q-52.939 115.307-53.095 115.580Q-53.250 115.853-53.250 116.168L-53.250 117.381L-53.076 117.381Q-52.895 117.402-52.861 117.583L-52.861 117.662Q-52.905 117.853-53.076 117.870L-53.890 117.870Q-54.071 117.849-54.105 117.662\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(83.475 -56.675)\">\u003Cpath d=\"M-43.885 118.431Q-44.350 118.161-44.700 117.733Q-45.050 117.306-45.235 116.787Q-45.420 116.267-45.420 115.727Q-45.420 115.187-45.237 114.672Q-45.054 114.158-44.707 113.738Q-44.360 113.317-43.892 113.034Q-43.830 113.013-43.803 113.013L-43.724 113.013Q-43.639 113.023-43.581 113.081Q-43.523 113.140-43.523 113.228Q-43.523 113.355-43.611 113.410Q-44.005 113.662-44.290 114.026Q-44.575 114.391-44.717 114.821Q-44.859 115.252-44.859 115.727Q-44.859 116.082-44.775 116.421Q-44.692 116.759-44.536 117.053Q-44.380 117.347-44.153 117.595Q-43.926 117.843-43.622 118.044Q-43.523 118.113-43.523 118.232Q-43.523 118.318-43.581 118.379Q-43.639 118.441-43.724 118.451L-43.803 118.451Q-43.854 118.451-43.885 118.431M-41.263 117.904Q-41.670 117.904-41.990 117.685Q-42.309 117.467-42.489 117.113Q-42.668 116.759-42.668 116.359Q-42.668 116.058-42.559 115.778Q-42.449 115.498-42.246 115.279Q-42.043 115.060-41.776 114.939Q-41.509 114.818-41.198 114.818Q-40.751 114.818-40.395 115.098L-40.395 114.083L-40.764 114.083Q-40.949 114.062-40.983 113.874L-40.983 113.796Q-40.949 113.615-40.764 113.594L-40.053 113.594Q-39.879 113.615-39.835 113.796L-39.835 117.381L-39.462 117.381Q-39.281 117.402-39.247 117.583L-39.247 117.662Q-39.281 117.849-39.462 117.870L-40.176 117.870Q-40.351 117.849-40.395 117.662L-40.395 117.549Q-40.566 117.716-40.795 117.810Q-41.024 117.904-41.263 117.904M-41.229 117.415Q-40.915 117.415-40.694 117.180Q-40.474 116.944-40.395 116.616L-40.395 115.888Q-40.446 115.724-40.557 115.592Q-40.669 115.460-40.824 115.383Q-40.980 115.307-41.151 115.307Q-41.421 115.307-41.639 115.455Q-41.858 115.604-41.984 115.853Q-42.111 116.103-42.111 116.366Q-42.111 116.759-41.861 117.087Q-41.612 117.415-41.229 117.415M-38.638 117.662L-38.638 117.583Q-38.594 117.402-38.423 117.381L-37.514 117.381L-37.514 115.344L-38.365 115.344Q-38.539 115.324-38.584 115.132L-38.584 115.057Q-38.539 114.872-38.365 114.852L-37.169 114.852Q-36.998 114.869-36.953 115.057L-36.953 117.381L-36.153 117.381Q-35.983 117.402-35.938 117.583L-35.938 117.662Q-35.983 117.849-36.153 117.870L-38.423 117.870Q-38.594 117.849-38.638 117.662M-37.736 113.997L-37.736 113.950Q-37.736 113.799-37.616 113.693Q-37.497 113.587-37.343 113.587Q-37.192 113.587-37.073 113.693Q-36.953 113.799-36.953 113.950L-36.953 113.997Q-36.953 114.151-37.073 114.257Q-37.192 114.363-37.343 114.363Q-37.497 114.363-37.616 114.257Q-37.736 114.151-37.736 113.997M-35.108 116.896Q-35.108 116.506-34.745 116.281Q-34.383 116.055-33.910 115.968Q-33.436 115.881-32.992 115.874Q-32.992 115.686-33.110 115.553Q-33.228 115.419-33.409 115.353Q-33.590 115.286-33.775 115.286Q-34.075 115.286-34.215 115.307L-34.215 115.358Q-34.215 115.505-34.320 115.606Q-34.424 115.706-34.567 115.706Q-34.721 115.706-34.822 115.599Q-34.923 115.491-34.923 115.344Q-34.923 114.989-34.590 114.893Q-34.256 114.797-33.768 114.797Q-33.532 114.797-33.298 114.866Q-33.064 114.934-32.867 115.066Q-32.671 115.197-32.551 115.390Q-32.431 115.583-32.431 115.826L-32.431 117.330Q-32.431 117.381-31.970 117.381Q-31.799 117.398-31.755 117.583L-31.755 117.662Q-31.799 117.849-31.970 117.870L-32.096 117.870Q-32.397 117.870-32.597 117.829Q-32.797 117.788-32.923 117.624Q-33.327 117.904-33.945 117.904Q-34.239 117.904-34.506 117.781Q-34.773 117.658-34.940 117.429Q-35.108 117.200-35.108 116.896M-34.547 116.903Q-34.547 117.142-34.335 117.279Q-34.123 117.415-33.874 117.415Q-33.682 117.415-33.479 117.364Q-33.276 117.313-33.134 117.192Q-32.992 117.070-32.992 116.875L-32.992 116.359Q-33.238 116.359-33.602 116.409Q-33.966 116.458-34.256 116.580Q-34.547 116.701-34.547 116.903M-31.539 118.437Q-31.539 118.174-31.409 117.945Q-31.279 117.716-31.061 117.576Q-31.221 117.354-31.221 117.074Q-31.221 116.824-31.088 116.595Q-31.307 116.311-31.307 115.946Q-31.307 115.631-31.139 115.371Q-30.972 115.112-30.693 114.965Q-30.415 114.818-30.097 114.818Q-29.916 114.818-29.738 114.871Q-29.560 114.924-29.417 115.013Q-29.082 114.784-28.668 114.784Q-28.460 114.784-28.301 114.898Q-28.142 115.013-28.142 115.211Q-28.142 115.341-28.231 115.433Q-28.319 115.525-28.456 115.525Q-28.572 115.525-28.656 115.457Q-28.740 115.389-28.767 115.279Q-28.894 115.286-29.068 115.365Q-28.890 115.638-28.890 115.946Q-28.890 116.192-28.988 116.395Q-29.085 116.599-29.256 116.754Q-29.427 116.910-29.647 116.995Q-29.868 117.080-30.097 117.080Q-30.435 117.080-30.726 116.916Q-30.736 116.916-30.744 116.980Q-30.753 117.043-30.753 117.074Q-30.753 117.180-30.695 117.274Q-30.637 117.368-30.531 117.402Q-30.517 117.398-30.505 117.397Q-30.493 117.395-30.473 117.395Q-30.459 117.395-30.432 117.402Q-30.404 117.409-30.398 117.409L-29.878 117.409Q-29.194 117.409-28.721 117.620Q-28.248 117.832-28.248 118.437Q-28.248 118.701-28.405 118.899Q-28.562 119.097-28.822 119.224Q-29.082 119.350-29.362 119.408Q-29.642 119.466-29.892 119.466Q-30.234 119.466-30.620 119.359Q-31.006 119.251-31.273 119.017Q-31.539 118.783-31.539 118.437M-31.067 118.431Q-31.067 118.622-30.857 118.745Q-30.647 118.868-30.363 118.926Q-30.080 118.984-29.892 118.984Q-29.704 118.984-29.420 118.926Q-29.136 118.868-28.926 118.745Q-28.716 118.622-28.716 118.431Q-28.716 118.178-28.912 118.056Q-29.109 117.935-29.355 117.906Q-29.601 117.877-29.905 117.877L-30.445 117.877Q-30.606 117.877-30.748 117.947Q-30.890 118.017-30.979 118.145Q-31.067 118.273-31.067 118.431M-30.097 116.595Q-29.834 116.595-29.642 116.404Q-29.451 116.212-29.451 115.946Q-29.451 115.809-29.499 115.694Q-29.547 115.580-29.639 115.488Q-29.731 115.395-29.846 115.348Q-29.960 115.300-30.097 115.300Q-30.363 115.300-30.555 115.491Q-30.746 115.683-30.746 115.946Q-30.746 116.205-30.551 116.400Q-30.357 116.595-30.097 116.595M-26.176 117.904Q-26.590 117.904-26.927 117.691Q-27.263 117.477-27.458 117.118Q-27.653 116.759-27.653 116.359Q-27.653 116.058-27.544 115.776Q-27.434 115.495-27.233 115.272Q-27.031 115.050-26.763 114.924Q-26.494 114.797-26.176 114.797Q-25.859 114.797-25.585 114.925Q-25.312 115.054-25.117 115.269Q-24.922 115.484-24.811 115.765Q-24.700 116.045-24.700 116.359Q-24.700 116.759-24.896 117.120Q-25.093 117.480-25.430 117.692Q-25.766 117.904-26.176 117.904M-26.176 117.415Q-25.770 117.415-25.515 117.070Q-25.260 116.725-25.260 116.301Q-25.260 116.045-25.378 115.811Q-25.496 115.577-25.706 115.431Q-25.917 115.286-26.176 115.286Q-26.443 115.286-26.651 115.431Q-26.860 115.577-26.978 115.811Q-27.096 116.045-27.096 116.301Q-27.096 116.722-26.839 117.068Q-26.583 117.415-26.176 117.415M-24.255 117.662L-24.255 117.583Q-24.218 117.402-24.037 117.381L-23.668 117.381L-23.668 115.344L-24.037 115.344Q-24.218 115.324-24.255 115.132L-24.255 115.057Q-24.214 114.872-24.037 114.852L-23.322 114.852Q-23.151 114.872-23.107 115.057L-23.107 115.132L-23.114 115.153Q-22.916 114.995-22.676 114.907Q-22.437 114.818-22.181 114.818Q-21.877 114.818-21.668 114.941Q-21.460 115.064-21.359 115.289Q-21.258 115.515-21.258 115.826L-21.258 117.381L-20.885 117.381Q-20.704 117.402-20.670 117.583L-20.670 117.662Q-20.704 117.849-20.885 117.870L-22.106 117.870Q-22.276 117.849-22.321 117.662L-22.321 117.583Q-22.276 117.398-22.106 117.381L-21.818 117.381L-21.818 115.853Q-21.818 115.583-21.897 115.445Q-21.976 115.307-22.232 115.307Q-22.468 115.307-22.668 115.421Q-22.868 115.536-22.987 115.735Q-23.107 115.935-23.107 116.168L-23.107 117.381L-22.734 117.381Q-22.553 117.402-22.519 117.583L-22.519 117.662Q-22.553 117.849-22.734 117.870L-24.037 117.870Q-24.218 117.849-24.255 117.662M-20.246 116.896Q-20.246 116.506-19.884 116.281Q-19.522 116.055-19.048 115.968Q-18.575 115.881-18.130 115.874Q-18.130 115.686-18.248 115.553Q-18.366 115.419-18.547 115.353Q-18.729 115.286-18.913 115.286Q-19.214 115.286-19.354 115.307L-19.354 115.358Q-19.354 115.505-19.458 115.606Q-19.563 115.706-19.706 115.706Q-19.860 115.706-19.961 115.599Q-20.062 115.491-20.062 115.344Q-20.062 114.989-19.728 114.893Q-19.395 114.797-18.906 114.797Q-18.671 114.797-18.436 114.866Q-18.202 114.934-18.006 115.066Q-17.809 115.197-17.690 115.390Q-17.570 115.583-17.570 115.826L-17.570 117.330Q-17.570 117.381-17.109 117.381Q-16.938 117.398-16.893 117.583L-16.893 117.662Q-16.938 117.849-17.109 117.870L-17.235 117.870Q-17.536 117.870-17.736 117.829Q-17.936 117.788-18.062 117.624Q-18.465 117.904-19.084 117.904Q-19.378 117.904-19.645 117.781Q-19.911 117.658-20.079 117.429Q-20.246 117.200-20.246 116.896M-19.686 116.903Q-19.686 117.142-19.474 117.279Q-19.262 117.415-19.012 117.415Q-18.821 117.415-18.618 117.364Q-18.414 117.313-18.272 117.192Q-18.130 117.070-18.130 116.875L-18.130 116.359Q-18.377 116.359-18.741 116.409Q-19.105 116.458-19.395 116.580Q-19.686 116.701-19.686 116.903M-16.493 117.662L-16.493 117.583Q-16.449 117.402-16.278 117.381L-15.311 117.381L-15.311 114.083L-16.278 114.083Q-16.449 114.062-16.493 113.874L-16.493 113.796Q-16.449 113.615-16.278 113.594L-14.969 113.594Q-14.801 113.615-14.750 113.796L-14.750 117.381L-13.786 117.381Q-13.612 117.402-13.567 117.583L-13.567 117.662Q-13.612 117.849-13.786 117.870L-16.278 117.870Q-16.449 117.849-16.493 117.662M-12.262 118.451L-12.337 118.451Q-12.429 118.441-12.489 118.378Q-12.549 118.314-12.549 118.232Q-12.549 118.109-12.457 118.051Q-12.163 117.863-11.927 117.610Q-11.691 117.357-11.530 117.053Q-11.370 116.749-11.291 116.417Q-11.213 116.086-11.213 115.727Q-11.213 115.255-11.356 114.825Q-11.500 114.394-11.778 114.032Q-12.057 113.669-12.443 113.416Q-12.549 113.345-12.549 113.228Q-12.549 113.143-12.489 113.083Q-12.429 113.023-12.337 113.013L-12.262 113.013Q-12.214 113.013-12.183 113.034Q-11.845 113.228-11.547 113.533Q-11.250 113.837-11.057 114.182Q-10.864 114.527-10.758 114.917Q-10.652 115.307-10.652 115.727Q-10.652 116.151-10.758 116.546Q-10.864 116.940-11.060 117.287Q-11.257 117.634-11.541 117.926Q-11.824 118.219-12.176 118.431Q-12.238 118.451-12.262 118.451\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath d=\"M31.936 37.49h113.81V13.307H31.937Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"7\">\u003Cg transform=\"translate(129.872 -90.333)\">\u003Cpath d=\"M-65.042 117.662L-65.042 117.583Q-65.008 117.402-64.824 117.381L-64.608 117.381L-64.608 114.083L-64.824 114.083Q-65.008 114.062-65.042 113.874L-65.042 113.796Q-65.008 113.615-64.824 113.594L-63.921 113.594Q-63.750 113.615-63.706 113.796L-63.706 113.874Q-63.750 114.062-63.921 114.083L-64.147 114.083L-64.147 115.792L-62.766 114.083Q-62.974 114.083-63.019 113.874L-63.019 113.796Q-62.974 113.615-62.804 113.594L-61.983 113.594Q-61.816 113.615-61.765 113.796L-61.765 113.874Q-61.816 114.062-61.983 114.083L-62.171 114.083L-63.221 115.378L-62.076 117.381L-61.911 117.381Q-61.741 117.402-61.696 117.583L-61.696 117.662Q-61.741 117.849-61.911 117.870L-62.640 117.870Q-62.824 117.849-62.858 117.662L-62.858 117.583Q-62.824 117.402-62.640 117.381L-62.605 117.381L-63.532 115.765L-64.147 116.527L-64.147 117.381L-63.921 117.381Q-63.750 117.402-63.706 117.583L-63.706 117.662Q-63.750 117.849-63.921 117.870L-64.824 117.870Q-65.008 117.849-65.042 117.662M-58.429 116L-60.828 116Q-60.937 115.990-61.013 115.915Q-61.088 115.840-61.088 115.727Q-61.088 115.614-61.013 115.542Q-60.937 115.471-60.828 115.460L-58.429 115.460Q-58.319 115.471-58.247 115.542Q-58.176 115.614-58.176 115.727Q-58.176 115.840-58.247 115.915Q-58.319 115.990-58.429 116M-57.591 117.662L-57.591 117.583Q-57.557 117.402-57.372 117.381L-57.078 117.381L-57.078 114.083L-57.372 114.083Q-57.557 114.062-57.591 113.874L-57.591 113.796Q-57.557 113.615-57.372 113.594L-54.525 113.594Q-54.351 113.615-54.307 113.796L-54.307 114.531Q-54.348 114.719-54.525 114.743L-54.652 114.743Q-54.823 114.719-54.867 114.531L-54.867 114.083L-56.518 114.083L-56.518 115.484L-55.636 115.484L-55.636 115.293Q-55.592 115.112-55.421 115.091L-55.294 115.091Q-55.123 115.108-55.079 115.293L-55.079 116.154Q-55.123 116.346-55.294 116.366L-55.421 116.366Q-55.592 116.349-55.636 116.154L-55.636 115.973L-56.518 115.973L-56.518 117.381L-56.091 117.381Q-55.920 117.402-55.875 117.583L-55.875 117.662Q-55.920 117.849-56.091 117.870L-57.372 117.870Q-57.557 117.849-57.591 117.662M-53.855 117.662L-53.855 117.583Q-53.804 117.398-53.637 117.381L-53.442 117.381L-52.608 113.721Q-52.587 113.618-52.504 113.553Q-52.420 113.488-52.321 113.488L-52.081 113.488Q-51.982 113.488-51.899 113.555Q-51.815 113.621-51.794 113.721L-50.964 117.381L-50.766 117.381Q-50.584 117.402-50.550 117.583L-50.550 117.662Q-50.595 117.849-50.766 117.870L-51.712 117.870Q-51.883 117.849-51.928 117.662L-51.928 117.583Q-51.883 117.402-51.712 117.381L-51.538 117.381Q-51.651 116.886-51.671 116.814L-52.734 116.814Q-52.772 116.974-52.868 117.381L-52.693 117.381Q-52.519 117.402-52.474 117.583L-52.474 117.662Q-52.519 117.849-52.693 117.870L-53.637 117.870Q-53.818 117.849-53.855 117.662M-52.628 116.322L-51.774 116.322Q-52.194 114.455-52.194 114.155L-52.201 114.155Q-52.201 114.315-52.324 114.913Q-52.447 115.512-52.628 116.322M-48.359 117.938Q-48.752 117.938-49.072 117.742Q-49.391 117.545-49.614 117.210Q-49.836 116.875-49.945 116.487Q-50.055 116.099-50.055 115.727Q-50.055 115.354-49.945 114.970Q-49.836 114.585-49.614 114.250Q-49.391 113.915-49.077 113.719Q-48.763 113.522-48.359 113.522Q-48.117 113.522-47.891 113.603Q-47.665 113.683-47.477 113.830L-47.361 113.621Q-47.337 113.577-47.289 113.550Q-47.242 113.522-47.190 113.522L-47.122 113.522Q-47.040 113.533-46.985 113.584Q-46.931 113.635-46.920 113.727L-46.920 114.890Q-46.931 114.978-46.985 115.033Q-47.040 115.088-47.122 115.098L-47.269 115.098Q-47.358 115.088-47.412 115.030Q-47.467 114.972-47.477 114.890Q-47.512 114.657-47.619 114.461Q-47.727 114.264-47.905 114.139Q-48.082 114.015-48.311 114.015Q-48.602 114.015-48.822 114.175Q-49.043 114.336-49.193 114.599Q-49.344 114.862-49.419 115.161Q-49.494 115.460-49.494 115.727Q-49.494 115.994-49.421 116.293Q-49.347 116.592-49.197 116.857Q-49.046 117.121-48.821 117.286Q-48.595 117.450-48.305 117.450Q-47.970 117.450-47.724 117.210Q-47.477 116.971-47.477 116.633Q-47.453 116.448-47.269 116.428L-47.122 116.428Q-46.920 116.462-46.920 116.660Q-46.920 116.937-47.040 117.174Q-47.160 117.412-47.365 117.585Q-47.570 117.757-47.824 117.848Q-48.079 117.938-48.359 117.938\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(129.872 -90.333)\">\u003Cpath d=\"M-40.167 118.431Q-40.632 118.161-40.982 117.733Q-41.332 117.306-41.517 116.787Q-41.702 116.267-41.702 115.727Q-41.702 115.187-41.519 114.672Q-41.336 114.158-40.989 113.738Q-40.642 113.317-40.174 113.034Q-40.112 113.013-40.085 113.013L-40.006 113.013Q-39.921 113.023-39.863 113.081Q-39.805 113.140-39.805 113.228Q-39.805 113.355-39.893 113.410Q-40.287 113.662-40.572 114.026Q-40.857 114.391-40.999 114.821Q-41.141 115.252-41.141 115.727Q-41.141 116.082-41.057 116.421Q-40.974 116.759-40.818 117.053Q-40.662 117.347-40.435 117.595Q-40.208 117.843-39.904 118.044Q-39.805 118.113-39.805 118.232Q-39.805 118.318-39.863 118.379Q-39.921 118.441-40.006 118.451L-40.085 118.451Q-40.136 118.451-40.167 118.431M-38.526 117.662L-38.526 114.083L-38.895 114.083Q-39.077 114.062-39.114 113.874L-39.114 113.796Q-39.077 113.615-38.895 113.594L-38.181 113.594Q-38.010 113.615-37.966 113.796L-37.966 115.119Q-37.781 114.972-37.550 114.895Q-37.320 114.818-37.084 114.818Q-36.787 114.818-36.527 114.944Q-36.267 115.071-36.079 115.291Q-35.891 115.512-35.790 115.785Q-35.689 116.058-35.689 116.359Q-35.689 116.766-35.888 117.125Q-36.086 117.484-36.424 117.694Q-36.763 117.904-37.173 117.904Q-37.627 117.904-37.966 117.583L-37.966 117.662Q-38.010 117.853-38.181 117.870L-38.308 117.870Q-38.475 117.849-38.526 117.662M-37.210 117.415Q-36.940 117.415-36.722 117.267Q-36.503 117.118-36.376 116.872Q-36.250 116.626-36.250 116.359Q-36.250 116.106-36.359 115.862Q-36.469 115.618-36.672 115.462Q-36.875 115.307-37.139 115.307Q-37.419 115.307-37.648 115.469Q-37.877 115.631-37.966 115.894L-37.966 116.640Q-37.887 116.961-37.694 117.188Q-37.501 117.415-37.210 117.415M-35.067 117.662L-35.067 117.583Q-35.023 117.402-34.852 117.381L-33.885 117.381L-33.885 114.083L-34.852 114.083Q-35.023 114.062-35.067 113.874L-35.067 113.796Q-35.023 113.615-34.852 113.594L-33.543 113.594Q-33.375 113.615-33.324 113.796L-33.324 117.381L-32.360 117.381Q-32.186 117.402-32.141 117.583L-32.141 117.662Q-32.186 117.849-32.360 117.870L-34.852 117.870Q-35.023 117.849-35.067 117.662M-29.889 117.904Q-30.303 117.904-30.639 117.691Q-30.976 117.477-31.171 117.118Q-31.366 116.759-31.366 116.359Q-31.366 116.058-31.256 115.776Q-31.147 115.495-30.945 115.272Q-30.744 115.050-30.475 114.924Q-30.207 114.797-29.889 114.797Q-29.571 114.797-29.298 114.925Q-29.024 115.054-28.829 115.269Q-28.635 115.484-28.524 115.765Q-28.412 116.045-28.412 116.359Q-28.412 116.759-28.609 117.120Q-28.806 117.480-29.142 117.692Q-29.479 117.904-29.889 117.904M-29.889 117.415Q-29.482 117.415-29.228 117.070Q-28.973 116.725-28.973 116.301Q-28.973 116.045-29.091 115.811Q-29.209 115.577-29.419 115.431Q-29.629 115.286-29.889 115.286Q-30.156 115.286-30.364 115.431Q-30.573 115.577-30.691 115.811Q-30.808 116.045-30.808 116.301Q-30.808 116.722-30.552 117.068Q-30.296 117.415-29.889 117.415M-27.541 116.359Q-27.541 115.939-27.327 115.577Q-27.114 115.214-26.750 115.006Q-26.386 114.797-25.965 114.797Q-25.480 114.797-25.148 114.893Q-24.817 114.989-24.817 115.344Q-24.817 115.491-24.916 115.599Q-25.015 115.706-25.165 115.706Q-25.316 115.706-25.420 115.606Q-25.524 115.505-25.524 115.358L-25.524 115.307Q-25.664 115.286-25.958 115.286Q-26.245 115.286-26.480 115.435Q-26.714 115.583-26.847 115.831Q-26.980 116.079-26.980 116.359Q-26.980 116.640-26.830 116.886Q-26.680 117.132-26.430 117.274Q-26.181 117.415-25.893 117.415Q-25.620 117.415-25.509 117.325Q-25.398 117.234-25.295 117.068Q-25.193 116.903-25.090 116.889L-24.943 116.889Q-24.851 116.899-24.795 116.959Q-24.738 117.019-24.738 117.108Q-24.738 117.149-24.755 117.183Q-24.851 117.439-25.029 117.597Q-25.206 117.754-25.447 117.829Q-25.688 117.904-25.965 117.904Q-26.389 117.904-26.753 117.696Q-27.117 117.487-27.329 117.135Q-27.541 116.783-27.541 116.359M-24.188 117.662L-24.188 117.583Q-24.143 117.402-23.973 117.381L-23.566 117.381L-23.566 114.083L-23.973 114.083Q-24.143 114.062-24.188 113.874L-24.188 113.796Q-24.143 113.615-23.973 113.594L-23.320 113.594Q-23.149 113.611-23.104 113.796L-23.104 116.267L-22.103 115.344L-22.363 115.344Q-22.544 115.324-22.578 115.132L-22.578 115.057Q-22.544 114.872-22.363 114.852L-21.095 114.852Q-20.917 114.872-20.876 115.057L-20.876 115.132Q-20.913 115.324-21.095 115.344L-21.416 115.344L-22.243 116.099L-21.235 117.381L-20.941 117.381Q-20.760 117.402-20.722 117.583L-20.722 117.662Q-20.760 117.849-20.941 117.870L-21.963 117.870Q-22.134 117.849-22.178 117.662L-22.178 117.583Q-22.134 117.402-21.963 117.381L-21.823 117.381L-22.578 116.414L-23.104 116.896L-23.104 117.381L-22.698 117.381Q-22.530 117.402-22.479 117.583L-22.479 117.662Q-22.530 117.849-22.698 117.870L-23.973 117.870Q-24.143 117.849-24.188 117.662M-19.690 118.451L-19.765 118.451Q-19.857 118.441-19.917 118.378Q-19.977 118.314-19.977 118.232Q-19.977 118.109-19.885 118.051Q-19.591 117.863-19.355 117.610Q-19.119 117.357-18.958 117.053Q-18.798 116.749-18.719 116.417Q-18.641 116.086-18.641 115.727Q-18.641 115.255-18.784 114.825Q-18.928 114.394-19.206 114.032Q-19.485 113.669-19.871 113.416Q-19.977 113.345-19.977 113.228Q-19.977 113.143-19.917 113.083Q-19.857 113.023-19.765 113.013L-19.690 113.013Q-19.642 113.013-19.611 113.034Q-19.273 113.228-18.975 113.533Q-18.678 113.837-18.485 114.182Q-18.292 114.527-18.186 114.917Q-18.080 115.307-18.080 115.727Q-18.080 116.151-18.186 116.546Q-18.292 116.940-18.488 117.287Q-18.685 117.634-18.969 117.926Q-19.252 118.219-19.604 118.431Q-19.666 118.451-19.690 118.451\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath d=\"M74.615 3.347h113.81v-24.184H74.616Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"7\">\u003Cg transform=\"translate(165.114 -124.476)\">\u003Cpath d=\"M-65.008 117.662L-65.008 117.583Q-64.957 117.402-64.789 117.381L-64.468 117.381L-64.468 114.083L-64.789 114.083Q-64.957 114.062-65.008 113.874L-65.008 113.796Q-64.957 113.615-64.789 113.594L-63.453 113.594Q-63.282 113.615-63.238 113.796L-63.238 113.874Q-63.282 114.062-63.453 114.083L-63.908 114.083L-63.908 117.381L-62.305 117.381L-62.305 116.821Q-62.260 116.640-62.089 116.616L-61.963 116.616Q-61.788 116.640-61.744 116.821L-61.744 117.662Q-61.788 117.849-61.963 117.870L-64.789 117.870Q-64.957 117.849-65.008 117.662M-58.429 116L-60.828 116Q-60.937 115.990-61.013 115.915Q-61.088 115.840-61.088 115.727Q-61.088 115.614-61.013 115.542Q-60.937 115.471-60.828 115.460L-58.429 115.460Q-58.319 115.471-58.247 115.542Q-58.176 115.614-58.176 115.727Q-58.176 115.840-58.247 115.915Q-58.319 115.990-58.429 116M-57.625 117.662L-57.625 117.583Q-57.591 117.402-57.410 117.381L-57.157 117.381L-57.157 114.083L-57.410 114.083Q-57.591 114.062-57.625 113.874L-57.625 113.796Q-57.591 113.615-57.410 113.594L-55.694 113.594Q-55.458 113.594-55.248 113.678Q-55.038 113.762-54.870 113.910Q-54.703 114.059-54.607 114.259Q-54.512 114.459-54.512 114.691Q-54.512 115.013-54.722 115.266Q-54.932 115.518-55.253 115.631Q-55.007 115.686-54.804 115.838Q-54.600 115.990-54.483 116.214Q-54.365 116.438-54.365 116.687Q-54.365 117.002-54.524 117.274Q-54.682 117.545-54.958 117.708Q-55.233 117.870-55.547 117.870L-57.410 117.870Q-57.591 117.849-57.625 117.662M-56.597 115.888L-56.597 117.381L-55.694 117.381Q-55.496 117.381-55.318 117.291Q-55.141 117.200-55.031 117.043Q-54.922 116.886-54.922 116.687Q-54.922 116.486-55.011 116.303Q-55.099 116.120-55.258 116.004Q-55.417 115.888-55.616 115.888L-56.597 115.888M-56.597 114.083L-56.597 115.399L-55.841 115.399Q-55.540 115.399-55.305 115.194Q-55.069 114.989-55.069 114.691Q-55.069 114.527-55.154 114.387Q-55.240 114.247-55.385 114.165Q-55.530 114.083-55.694 114.083L-56.597 114.083M-53.876 117.662L-53.876 117.583Q-53.842 117.402-53.657 117.381L-53.363 117.381L-53.363 114.083L-53.657 114.083Q-53.842 114.062-53.876 113.874L-53.876 113.796Q-53.842 113.615-53.657 113.594L-50.810 113.594Q-50.636 113.615-50.591 113.796L-50.591 114.531Q-50.632 114.719-50.810 114.743L-50.936 114.743Q-51.107 114.719-51.152 114.531L-51.152 114.083L-52.803 114.083L-52.803 115.484L-51.921 115.484L-51.921 115.293Q-51.876 115.112-51.705 115.091L-51.579 115.091Q-51.408 115.108-51.364 115.293L-51.364 116.154Q-51.408 116.346-51.579 116.366L-51.705 116.366Q-51.876 116.349-51.921 116.154L-51.921 115.973L-52.803 115.973L-52.803 117.381L-52.375 117.381Q-52.204 117.402-52.160 117.583L-52.160 117.662Q-52.204 117.849-52.375 117.870L-53.657 117.870Q-53.842 117.849-53.876 117.662M-48.499 117.938Q-49.005 117.938-49.364 117.593Q-49.723 117.248-49.899 116.734Q-50.075 116.219-50.075 115.727Q-50.075 115.245-49.899 114.729Q-49.723 114.213-49.364 113.868Q-49.005 113.522-48.499 113.522Q-48.062 113.522-47.710 113.809Q-47.669 113.734-47.631 113.671Q-47.594 113.608-47.547 113.570Q-47.501 113.533-47.436 113.522L-47.368 113.522Q-47.283 113.533-47.228 113.587Q-47.173 113.642-47.163 113.727L-47.163 114.890Q-47.173 114.972-47.228 115.030Q-47.283 115.088-47.368 115.098L-47.515 115.098Q-47.686 115.074-47.717 114.917Q-47.765 114.551-47.949 114.283Q-48.134 114.015-48.458 114.015Q-48.804 114.015-49.045 114.298Q-49.286 114.582-49.400 114.985Q-49.515 115.389-49.515 115.727Q-49.515 116.079-49.402 116.481Q-49.289 116.882-49.053 117.166Q-48.817 117.450-48.472 117.450Q-48.137 117.450-47.958 117.140Q-47.778 116.831-47.737 116.441L-48.103 116.441Q-48.287 116.421-48.318 116.233L-48.318 116.154Q-48.284 115.973-48.103 115.953L-47.050 115.953Q-46.869 115.973-46.835 116.154L-46.835 116.233Q-46.866 116.421-47.050 116.441L-47.163 116.441L-47.163 117.675Q-47.207 117.853-47.382 117.884L-47.508 117.884Q-47.679 117.856-47.724 117.675L-47.724 117.631Q-48.048 117.938-48.499 117.938M-46.254 117.730L-46.254 116.653Q-46.230 116.469-46.045 116.448L-45.898 116.448Q-45.816 116.458-45.762 116.510Q-45.707 116.561-45.697 116.653Q-45.697 117.084-45.377 117.267Q-45.057 117.450-44.589 117.450Q-44.384 117.450-44.206 117.356Q-44.029 117.262-43.921 117.092Q-43.813 116.923-43.813 116.722Q-43.813 116.462-43.984 116.267Q-44.155 116.072-44.408 116.014L-45.290 115.812Q-45.553 115.751-45.774 115.589Q-45.994 115.426-46.124 115.192Q-46.254 114.958-46.254 114.684Q-46.254 114.353-46.071 114.086Q-45.888 113.820-45.589 113.671Q-45.290 113.522-44.962 113.522Q-44.370 113.522-43.995 113.809L-43.933 113.649Q-43.868 113.543-43.748 113.522L-43.680 113.522Q-43.595 113.533-43.540 113.587Q-43.485 113.642-43.475 113.727L-43.475 114.804Q-43.495 114.989-43.680 115.013L-43.827 115.013Q-43.998 114.992-44.029 114.831Q-44.083 114.435-44.318 114.225Q-44.552 114.015-44.955 114.015Q-45.136 114.015-45.317 114.091Q-45.498 114.168-45.615 114.315Q-45.731 114.462-45.731 114.650Q-45.731 114.872-45.562 115.045Q-45.392 115.218-45.157 115.272L-44.275 115.471Q-43.988 115.539-43.766 115.717Q-43.543 115.894-43.415 116.152Q-43.287 116.411-43.287 116.687Q-43.287 117.036-43.465 117.321Q-43.642 117.607-43.943 117.773Q-44.244 117.938-44.582 117.938Q-45.283 117.938-45.738 117.662L-45.799 117.815Q-45.861 117.918-45.984 117.938L-46.045 117.938Q-46.233 117.918-46.254 117.730\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(165.114 -124.476)\">\u003Cpath d=\"M-36.448 118.431Q-36.913 118.161-37.263 117.733Q-37.613 117.306-37.798 116.787Q-37.983 116.267-37.983 115.727Q-37.983 115.187-37.800 114.672Q-37.617 114.158-37.270 113.738Q-36.923 113.317-36.455 113.034Q-36.393 113.013-36.366 113.013L-36.287 113.013Q-36.202 113.023-36.144 113.081Q-36.086 113.140-36.086 113.228Q-36.086 113.355-36.174 113.410Q-36.568 113.662-36.853 114.026Q-37.138 114.391-37.280 114.821Q-37.422 115.252-37.422 115.727Q-37.422 116.082-37.338 116.421Q-37.255 116.759-37.099 117.053Q-36.943 117.347-36.716 117.595Q-36.489 117.843-36.185 118.044Q-36.086 118.113-36.086 118.232Q-36.086 118.318-36.144 118.379Q-36.202 118.441-36.287 118.451L-36.366 118.451Q-36.417 118.451-36.448 118.431M-35.064 117.662L-35.064 117.583Q-35.019 117.402-34.848 117.381L-33.881 117.381L-33.881 114.083L-34.848 114.083Q-35.019 114.062-35.064 113.874L-35.064 113.796Q-35.019 113.615-34.848 113.594L-33.539 113.594Q-33.372 113.615-33.320 113.796L-33.320 117.381L-32.357 117.381Q-32.182 117.402-32.138 117.583L-32.138 117.662Q-32.182 117.849-32.357 117.870L-34.848 117.870Q-35.019 117.849-35.064 117.662M-29.885 117.904Q-30.299 117.904-30.636 117.691Q-30.972 117.477-31.167 117.118Q-31.362 116.759-31.362 116.359Q-31.362 116.058-31.253 115.776Q-31.143 115.495-30.942 115.272Q-30.740 115.050-30.472 114.924Q-30.203 114.797-29.885 114.797Q-29.568 114.797-29.294 114.925Q-29.021 115.054-28.826 115.269Q-28.631 115.484-28.520 115.765Q-28.409 116.045-28.409 116.359Q-28.409 116.759-28.605 117.120Q-28.802 117.480-29.139 117.692Q-29.475 117.904-29.885 117.904M-29.885 117.415Q-29.479 117.415-29.224 117.070Q-28.969 116.725-28.969 116.301Q-28.969 116.045-29.087 115.811Q-29.205 115.577-29.415 115.431Q-29.626 115.286-29.885 115.286Q-30.152 115.286-30.360 115.431Q-30.569 115.577-30.687 115.811Q-30.805 116.045-30.805 116.301Q-30.805 116.722-30.548 117.068Q-30.292 117.415-29.885 117.415M-27.144 117.675L-27.551 115.344L-27.691 115.344Q-27.872 115.324-27.906 115.132L-27.906 115.057Q-27.872 114.872-27.691 114.852L-26.669 114.852Q-26.484 114.872-26.450 115.057L-26.450 115.132Q-26.484 115.324-26.669 115.344L-27.083 115.344Q-26.984 115.908-26.915 116.298Q-26.847 116.687-26.802 116.966Q-26.758 117.245-26.758 117.309Q-26.751 117.159-26.502 116.287Q-26.467 116.188-26.382 116.123Q-26.297 116.058-26.191 116.058L-26.122 116.058Q-26.016 116.058-25.931 116.123Q-25.845 116.188-25.815 116.287Q-25.719 116.622-25.647 116.906Q-25.575 117.190-25.575 117.309Q-25.565 117.057-25.261 115.344L-25.674 115.344Q-25.856 115.324-25.890 115.132L-25.890 115.057Q-25.856 114.872-25.674 114.852L-24.646 114.852Q-24.478 114.872-24.427 115.057L-24.427 115.132Q-24.475 115.324-24.646 115.344L-24.793 115.344L-25.199 117.675Q-25.220 117.778-25.297 117.841Q-25.374 117.904-25.480 117.904L-25.603 117.904Q-25.709 117.904-25.792 117.841Q-25.876 117.778-25.914 117.675Q-25.955 117.518-26.018 117.294Q-26.081 117.070-26.119 116.887Q-26.156 116.704-26.156 116.616Q-26.163 116.862-26.409 117.675Q-26.433 117.771-26.519 117.838Q-26.604 117.904-26.710 117.904L-26.864 117.904Q-26.970 117.904-27.047 117.841Q-27.124 117.778-27.144 117.675M-21.252 116L-23.651 116Q-23.760 115.990-23.836 115.915Q-23.911 115.840-23.911 115.727Q-23.911 115.614-23.836 115.542Q-23.760 115.471-23.651 115.460L-21.252 115.460Q-21.142 115.471-21.070 115.542Q-20.999 115.614-20.999 115.727Q-20.999 115.840-21.070 115.915Q-21.142 115.990-21.252 116M-20.401 117.662L-20.401 117.583Q-20.349 117.402-20.182 117.381L-19.560 117.381L-19.560 115.344L-20.182 115.344Q-20.353 115.324-20.401 115.132L-20.401 115.057Q-20.349 114.872-20.182 114.852L-19.218 114.852Q-19.044 114.869-18.999 115.057L-18.999 115.344Q-18.770 115.095-18.459 114.956Q-18.148 114.818-17.810 114.818Q-17.553 114.818-17.348 114.936Q-17.143 115.054-17.143 115.286Q-17.143 115.433-17.239 115.536Q-17.335 115.638-17.482 115.638Q-17.615 115.638-17.716 115.541Q-17.817 115.443-17.817 115.307Q-18.145 115.307-18.420 115.483Q-18.695 115.659-18.847 115.951Q-18.999 116.243-18.999 116.575L-18.999 117.381L-18.172 117.381Q-17.991 117.402-17.957 117.583L-17.957 117.662Q-17.991 117.849-18.172 117.870L-20.182 117.870Q-20.349 117.849-20.401 117.662M-16.525 116.896Q-16.525 116.506-16.162 116.281Q-15.800 116.055-15.327 115.968Q-14.853 115.881-14.409 115.874Q-14.409 115.686-14.527 115.553Q-14.645 115.419-14.826 115.353Q-15.007 115.286-15.192 115.286Q-15.492 115.286-15.632 115.307L-15.632 115.358Q-15.632 115.505-15.737 115.606Q-15.841 115.706-15.984 115.706Q-16.138 115.706-16.239 115.599Q-16.340 115.491-16.340 115.344Q-16.340 114.989-16.007 114.893Q-15.673 114.797-15.185 114.797Q-14.949 114.797-14.715 114.866Q-14.481 114.934-14.284 115.066Q-14.088 115.197-13.968 115.390Q-13.848 115.583-13.848 115.826L-13.848 117.330Q-13.848 117.381-13.387 117.381Q-13.216 117.398-13.172 117.583L-13.172 117.662Q-13.216 117.849-13.387 117.870L-13.513 117.870Q-13.814 117.870-14.014 117.829Q-14.214 117.788-14.340 117.624Q-14.744 117.904-15.362 117.904Q-15.656 117.904-15.923 117.781Q-16.190 117.658-16.357 117.429Q-16.525 117.200-16.525 116.896M-15.964 116.903Q-15.964 117.142-15.752 117.279Q-15.540 117.415-15.291 117.415Q-15.099 117.415-14.896 117.364Q-14.693 117.313-14.551 117.192Q-14.409 117.070-14.409 116.875L-14.409 116.359Q-14.655 116.359-15.019 116.409Q-15.383 116.458-15.673 116.580Q-15.964 116.701-15.964 116.903M-13.103 117.662L-13.103 117.583Q-13.066 117.402-12.884 117.381L-12.515 117.381L-12.515 115.344L-12.884 115.344Q-13.066 115.324-13.103 115.132L-13.103 115.057Q-13.062 114.872-12.884 114.852L-12.170 114.852Q-11.999 114.872-11.955 115.057L-11.955 115.132L-11.962 115.153Q-11.763 114.995-11.524 114.907Q-11.285 114.818-11.028 114.818Q-10.724 114.818-10.516 114.941Q-10.307 115.064-10.206 115.289Q-10.106 115.515-10.106 115.826L-10.106 117.381L-9.733 117.381Q-9.552 117.402-9.518 117.583L-9.518 117.662Q-9.552 117.849-9.733 117.870L-10.953 117.870Q-11.124 117.849-11.169 117.662L-11.169 117.583Q-11.124 117.398-10.953 117.381L-10.666 117.381L-10.666 115.853Q-10.666 115.583-10.745 115.445Q-10.823 115.307-11.080 115.307Q-11.316 115.307-11.516 115.421Q-11.715 115.536-11.835 115.735Q-11.955 115.935-11.955 116.168L-11.955 117.381L-11.582 117.381Q-11.401 117.402-11.367 117.583L-11.367 117.662Q-11.401 117.849-11.582 117.870L-12.884 117.870Q-13.066 117.849-13.103 117.662M-9.323 117.662L-9.323 117.583Q-9.278 117.402-9.108 117.381L-8.701 117.381L-8.701 114.083L-9.108 114.083Q-9.278 114.062-9.323 113.874L-9.323 113.796Q-9.278 113.615-9.108 113.594L-8.455 113.594Q-8.284 113.611-8.239 113.796L-8.239 116.267L-7.238 115.344L-7.498 115.344Q-7.679 115.324-7.713 115.132L-7.713 115.057Q-7.679 114.872-7.498 114.852L-6.230 114.852Q-6.052 114.872-6.011 115.057L-6.011 115.132Q-6.048 115.324-6.230 115.344L-6.551 115.344L-7.378 116.099L-6.370 117.381L-6.076 117.381Q-5.895 117.402-5.857 117.583L-5.857 117.662Q-5.895 117.849-6.076 117.870L-7.098 117.870Q-7.269 117.849-7.313 117.662L-7.313 117.583Q-7.269 117.402-7.098 117.381L-6.958 117.381L-7.713 116.414L-8.239 116.896L-8.239 117.381L-7.833 117.381Q-7.665 117.402-7.614 117.583L-7.614 117.662Q-7.665 117.849-7.833 117.870L-9.108 117.870Q-9.278 117.849-9.323 117.662M-4.825 118.451L-4.900 118.451Q-4.992 118.441-5.052 118.378Q-5.112 118.314-5.112 118.232Q-5.112 118.109-5.020 118.051Q-4.726 117.863-4.490 117.610Q-4.254 117.357-4.093 117.053Q-3.933 116.749-3.854 116.417Q-3.776 116.086-3.776 115.727Q-3.776 115.255-3.919 114.825Q-4.063 114.394-4.341 114.032Q-4.620 113.669-5.006 113.416Q-5.112 113.345-5.112 113.228Q-5.112 113.143-5.052 113.083Q-4.992 113.023-4.900 113.013L-4.825 113.013Q-4.777 113.013-4.746 113.034Q-4.408 113.228-4.110 113.533Q-3.813 113.837-3.620 114.182Q-3.427 114.527-3.321 114.917Q-3.215 115.307-3.215 115.727Q-3.215 116.151-3.321 116.546Q-3.427 116.940-3.623 117.287Q-3.820 117.634-4.104 117.926Q-4.387 118.219-4.739 118.431Q-4.801 118.451-4.825 118.451\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-good)\" stroke=\"var(--tk-good)\" style=\"stroke-width:.8\">\u003Cpath fill=\"var(--tk-soft-good)\" stroke=\"var(--tk-good)\" d=\"M117.294-30.796h113.811V-54.98H117.294Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-family=\"cmtt8\" font-size=\"7\">\u003Cg transform=\"translate(218.95 -158.619)\">\u003Cpath d=\"M-64.923 117.662L-64.923 117.583Q-64.878 117.402-64.707 117.381L-63.935 117.381L-63.935 115.344L-64.683 115.344Q-64.858 115.324-64.902 115.132L-64.902 115.057Q-64.858 114.872-64.683 114.852L-63.935 114.852L-63.935 114.455Q-63.935 114.049-63.579 113.803Q-63.224 113.557-62.804 113.557Q-62.530 113.557-62.317 113.662Q-62.103 113.768-62.103 114.015Q-62.103 114.158-62.199 114.261Q-62.294 114.363-62.438 114.363Q-62.568 114.363-62.674 114.271Q-62.780 114.179-62.780 114.049L-62.851 114.049Q-62.985 114.049-63.104 114.102Q-63.224 114.155-63.299 114.255Q-63.374 114.356-63.374 114.483L-63.374 114.852L-62.472 114.852Q-62.301 114.869-62.257 115.057L-62.257 115.132Q-62.301 115.324-62.472 115.344L-63.374 115.344L-63.374 117.381L-62.605 117.381Q-62.434 117.402-62.390 117.583L-62.390 117.662Q-62.434 117.849-62.605 117.870L-64.707 117.870Q-64.878 117.849-64.923 117.662M-60.838 117.121L-60.838 115.344L-61.207 115.344Q-61.389 115.324-61.426 115.132L-61.426 115.057Q-61.385 114.872-61.207 114.852L-60.493 114.852Q-60.322 114.872-60.278 115.057L-60.278 117.094Q-60.278 117.299-60.129 117.357Q-59.980 117.415-59.731 117.415Q-59.550 117.415-59.379 117.349Q-59.208 117.282-59.099 117.151Q-58.989 117.019-58.989 116.834L-58.989 115.344L-59.358 115.344Q-59.543 115.324-59.577 115.132L-59.577 115.057Q-59.543 114.872-59.358 114.852L-58.647 114.852Q-58.473 114.872-58.429 115.057L-58.429 117.381L-58.056 117.381Q-57.875 117.402-57.841 117.583L-57.841 117.662Q-57.875 117.849-58.056 117.870L-58.770 117.870Q-58.945 117.849-58.989 117.675Q-59.341 117.904-59.786 117.904Q-59.991 117.904-60.179 117.868Q-60.367 117.832-60.512 117.744Q-60.657 117.655-60.748 117.499Q-60.838 117.344-60.838 117.121M-57.379 117.662L-57.379 117.583Q-57.335 117.402-57.164 117.381L-56.197 117.381L-56.197 114.083L-57.164 114.083Q-57.335 114.062-57.379 113.874L-57.379 113.796Q-57.335 113.615-57.164 113.594L-55.855 113.594Q-55.687 113.615-55.636 113.796L-55.636 117.381L-54.672 117.381Q-54.498 117.402-54.453 117.583L-54.453 117.662Q-54.498 117.849-54.672 117.870L-57.164 117.870Q-57.335 117.849-57.379 117.662M-53.664 117.662L-53.664 117.583Q-53.620 117.402-53.449 117.381L-52.481 117.381L-52.481 114.083L-53.449 114.083Q-53.620 114.062-53.664 113.874L-53.664 113.796Q-53.620 113.615-53.449 113.594L-52.140 113.594Q-51.972 113.615-51.921 113.796L-51.921 117.381L-50.957 117.381Q-50.783 117.402-50.738 117.583L-50.738 117.662Q-50.783 117.849-50.957 117.870L-53.449 117.870Q-53.620 117.849-53.664 117.662\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(218.95 -158.619)\">\u003Cpath d=\"M-46.435 117.662L-46.435 117.583Q-46.390 117.402-46.216 117.381L-45.980 117.381L-45.980 114.083L-46.216 114.083Q-46.390 114.062-46.435 113.874L-46.435 113.796Q-46.390 113.615-46.216 113.594L-45.532 113.594Q-45.420 113.594-45.327 113.659Q-45.235 113.724-45.201 113.823Q-44.958 114.514-44.818 114.907Q-44.678 115.300-44.456 115.929Q-44.234 116.558-44.112 116.916Q-43.991 117.275-43.991 117.309L-43.991 114.083L-44.230 114.083Q-44.401 114.062-44.445 113.874L-44.445 113.796Q-44.401 113.615-44.230 113.594L-43.290 113.594Q-43.119 113.615-43.075 113.796L-43.075 113.874Q-43.119 114.062-43.290 114.083L-43.529 114.083L-43.529 117.662Q-43.574 117.849-43.745 117.870L-43.977 117.870Q-44.087 117.870-44.177 117.805Q-44.268 117.740-44.305 117.638Q-44.565 116.896-44.891 115.978Q-45.218 115.060-45.367 114.633Q-45.515 114.206-45.519 114.155L-45.519 117.381L-45.279 117.381Q-45.112 117.402-45.061 117.583L-45.061 117.662Q-45.112 117.849-45.279 117.870L-46.216 117.870Q-46.390 117.849-46.435 117.662M-39.821 116.568L-41.957 116.568Q-41.909 116.810-41.737 117.005Q-41.564 117.200-41.321 117.308Q-41.079 117.415-40.829 117.415Q-40.416 117.415-40.221 117.162Q-40.214 117.152-40.164 117.060Q-40.115 116.968-40.072 116.930Q-40.029 116.892-39.947 116.882L-39.821 116.882Q-39.653 116.899-39.602 117.087L-39.602 117.135Q-39.660 117.398-39.862 117.571Q-40.064 117.744-40.337 117.824Q-40.610 117.904-40.877 117.904Q-41.301 117.904-41.685 117.704Q-42.070 117.504-42.304 117.154Q-42.538 116.804-42.538 116.373L-42.538 116.322Q-42.538 115.912-42.323 115.559Q-42.108 115.207-41.750 115.002Q-41.393 114.797-40.983 114.797Q-40.542 114.797-40.233 114.992Q-39.923 115.187-39.763 115.527Q-39.602 115.867-39.602 116.308L-39.602 116.359Q-39.653 116.547-39.821 116.568M-41.950 116.086L-40.176 116.086Q-40.217 115.727-40.426 115.506Q-40.634 115.286-40.983 115.286Q-41.328 115.286-41.597 115.515Q-41.865 115.744-41.950 116.086M-38.297 117.675L-38.703 115.344L-38.843 115.344Q-39.025 115.324-39.059 115.132L-39.059 115.057Q-39.025 114.872-38.843 114.852L-37.821 114.852Q-37.637 114.872-37.603 115.057L-37.603 115.132Q-37.637 115.324-37.821 115.344L-38.235 115.344Q-38.136 115.908-38.067 116.298Q-37.999 116.687-37.955 116.966Q-37.910 117.245-37.910 117.309Q-37.903 117.159-37.654 116.287Q-37.620 116.188-37.534 116.123Q-37.449 116.058-37.343 116.058L-37.275 116.058Q-37.169 116.058-37.083 116.123Q-36.998 116.188-36.967 116.287Q-36.871 116.622-36.799 116.906Q-36.728 117.190-36.728 117.309Q-36.717 117.057-36.413 115.344L-36.827 115.344Q-37.008 115.324-37.042 115.132L-37.042 115.057Q-37.008 114.872-36.827 114.852L-35.798 114.852Q-35.630 114.872-35.579 115.057L-35.579 115.132Q-35.627 115.324-35.798 115.344L-35.945 115.344L-36.352 117.675Q-36.372 117.778-36.449 117.841Q-36.526 117.904-36.632 117.904L-36.755 117.904Q-36.861 117.904-36.945 117.841Q-37.028 117.778-37.066 117.675Q-37.107 117.518-37.170 117.294Q-37.234 117.070-37.271 116.887Q-37.309 116.704-37.309 116.616Q-37.316 116.862-37.562 117.675Q-37.586 117.771-37.671 117.838Q-37.756 117.904-37.862 117.904L-38.016 117.904Q-38.122 117.904-38.199 117.841Q-38.276 117.778-38.297 117.675M-34.414 116.903L-34.414 115.344L-35.070 115.344Q-35.244 115.324-35.289 115.132L-35.289 115.057Q-35.244 114.872-35.070 114.852L-34.414 114.852L-34.414 114.196Q-34.369 114.015-34.195 113.991L-34.068 113.991Q-33.898 114.015-33.853 114.196L-33.853 114.852L-32.664 114.852Q-32.496 114.869-32.445 115.057L-32.445 115.132Q-32.493 115.324-32.664 115.344L-33.853 115.344L-33.853 116.875Q-33.853 117.415-33.351 117.415Q-33.132 117.415-32.988 117.253Q-32.845 117.091-32.845 116.875Q-32.845 116.793-32.782 116.732Q-32.718 116.670-32.630 116.660L-32.503 116.660Q-32.329 116.681-32.284 116.862L-32.284 116.903Q-32.284 117.186-32.445 117.417Q-32.606 117.648-32.862 117.776Q-33.118 117.904-33.399 117.904Q-33.846 117.904-34.130 117.632Q-34.414 117.361-34.414 116.903M-29.892 117.904Q-30.305 117.904-30.642 117.691Q-30.979 117.477-31.173 117.118Q-31.368 116.759-31.368 116.359Q-31.368 116.058-31.259 115.776Q-31.150 115.495-30.948 115.272Q-30.746 115.050-30.478 114.924Q-30.210 114.797-29.892 114.797Q-29.574 114.797-29.300 114.925Q-29.027 115.054-28.832 115.269Q-28.637 115.484-28.526 115.765Q-28.415 116.045-28.415 116.359Q-28.415 116.759-28.612 117.120Q-28.808 117.480-29.145 117.692Q-29.482 117.904-29.892 117.904M-29.892 117.415Q-29.485 117.415-29.230 117.070Q-28.976 116.725-28.976 116.301Q-28.976 116.045-29.094 115.811Q-29.212 115.577-29.422 115.431Q-29.632 115.286-29.892 115.286Q-30.158 115.286-30.367 115.431Q-30.575 115.577-30.693 115.811Q-30.811 116.045-30.811 116.301Q-30.811 116.722-30.555 117.068Q-30.298 117.415-29.892 117.415M-27.971 117.662L-27.971 117.583Q-27.933 117.402-27.752 117.381L-27.383 117.381L-27.383 115.344L-27.752 115.344Q-27.933 115.324-27.971 115.132L-27.971 115.057Q-27.930 114.872-27.752 114.852L-27.038 114.852Q-26.867 114.872-26.822 115.057L-26.822 115.132L-26.829 115.153Q-26.631 114.995-26.392 114.907Q-26.152 114.818-25.896 114.818Q-25.592 114.818-25.383 114.941Q-25.175 115.064-25.074 115.289Q-24.973 115.515-24.973 115.826L-24.973 117.381L-24.601 117.381Q-24.420 117.402-24.385 117.583L-24.385 117.662Q-24.420 117.849-24.601 117.870L-25.821 117.870Q-25.992 117.849-26.036 117.662L-26.036 117.583Q-25.992 117.398-25.821 117.381L-25.534 117.381L-25.534 115.853Q-25.534 115.583-25.612 115.445Q-25.691 115.307-25.947 115.307Q-26.183 115.307-26.383 115.421Q-26.583 115.536-26.703 115.735Q-26.822 115.935-26.822 116.168L-26.822 117.381L-26.450 117.381Q-26.269 117.402-26.234 117.583L-26.234 117.662Q-26.269 117.849-26.450 117.870L-27.752 117.870Q-27.933 117.849-27.971 117.662\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M24.423 81.45v-7.074\"\u002F>\u003Cpath stroke=\"none\" d=\"m24.423 71.776-2.08 4.16 2.08-1.56 2.08 1.56\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M67.102 47.307v-7.074\"\u002F>\u003Cpath stroke=\"none\" d=\"m67.102 37.633-2.08 4.16 2.08-1.56 2.08 1.56\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M109.78 13.164V6.089\"\u002F>\u003Cpath stroke=\"none\" d=\"m109.78 3.49-2.08 4.16 2.08-1.56 2.08 1.56\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M152.46-20.98v-7.074\"\u002F>\u003Cpath stroke=\"none\" d=\"m152.46-30.654-2.08 4.16 2.08-1.56 2.08 1.56\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"none\" font-size=\"7\">\u003Cg transform=\"translate(129.525 -22.435)\">\u003Cpath d=\"M-62.501 116.093Q-62.501 115.498-62.219 114.922Q-61.937 114.346-61.460 113.905Q-60.984 113.464-60.394 113.206Q-59.804 112.948-59.210 112.948Q-58.786 112.948-58.412 113.088Q-58.037 113.228-57.767 113.483Q-57.497 113.738-57.345 114.098Q-57.193 114.459-57.193 114.890Q-57.193 115.501-57.473 116.065Q-57.754 116.629-58.225 117.068Q-58.697 117.508-59.290 117.759Q-59.883 118.010-60.478 118.010Q-60.898 118.010-61.269 117.873Q-61.640 117.737-61.915 117.485Q-62.190 117.234-62.346 116.879Q-62.501 116.523-62.501 116.093M-60.413 117.750Q-59.883 117.750-59.417 117.465Q-58.950 117.180-58.615 116.720Q-58.280 116.260-58.094 115.708Q-57.907 115.156-57.907 114.643Q-57.907 114.226-58.072 113.898Q-58.236 113.570-58.547 113.386Q-58.858 113.201-59.275 113.201Q-60.006 113.201-60.570 113.657Q-61.134 114.114-61.443 114.828Q-61.753 115.542-61.753 116.246Q-61.753 116.660-61.599 117.004Q-61.445 117.347-61.143 117.549Q-60.840 117.750-60.413 117.750\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(129.525 -22.435)\">\u003Cpath d=\"M-54.248 119.620Q-54.798 119.220-55.169 118.665Q-55.540 118.109-55.721 117.463Q-55.902 116.817-55.902 116.120Q-55.902 115.607-55.802 115.112Q-55.701 114.616-55.496 114.165Q-55.291 113.714-54.978 113.322Q-54.665 112.931-54.248 112.627Q-54.238 112.623-54.231 112.622Q-54.224 112.620-54.214 112.620L-54.146 112.620Q-54.111 112.620-54.089 112.644Q-54.067 112.668-54.067 112.705Q-54.067 112.750-54.094 112.767Q-54.443 113.068-54.696 113.452Q-54.949 113.837-55.101 114.278Q-55.253 114.719-55.325 115.175Q-55.397 115.631-55.397 116.120Q-55.397 117.121-55.087 118.008Q-54.778 118.895-54.094 119.480Q-54.067 119.497-54.067 119.541Q-54.067 119.579-54.089 119.603Q-54.111 119.627-54.146 119.627L-54.214 119.627Q-54.221 119.623-54.229 119.622Q-54.238 119.620-54.248 119.620\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(129.525 -22.435)\">\u003Cpath d=\"M-52.077 117.938Q-52.401 117.938-52.646 117.781Q-52.890 117.624-53.022 117.359Q-53.153 117.094-53.153 116.769Q-53.153 116.301-52.900 115.838Q-52.648 115.375-52.224 115.079Q-51.800 114.784-51.328 114.784Q-51.113 114.784-50.927 114.888Q-50.740 114.992-50.621 115.177L-50.241 113.649Q-50.207 113.546-50.207 113.461Q-50.207 113.369-50.641 113.369Q-50.727 113.341-50.727 113.256L-50.696 113.146Q-50.669 113.099-50.621 113.088L-49.640 113.013Q-49.602 113.013-49.568 113.040Q-49.534 113.068-49.534 113.116L-50.542 117.142Q-50.569 117.299-50.569 117.388Q-50.569 117.515-50.518 117.615Q-50.467 117.716-50.347 117.716Q-50.125 117.716-50.012 117.463Q-49.900 117.210-49.814 116.841Q-49.790 116.780-49.739 116.780L-49.626 116.780Q-49.592 116.780-49.570 116.809Q-49.547 116.838-49.547 116.862Q-49.547 116.875-49.554 116.889Q-49.817 117.938-50.361 117.938Q-50.610 117.938-50.819 117.815Q-51.027 117.692-51.096 117.463Q-51.571 117.938-52.077 117.938M-52.063 117.716Q-51.790 117.716-51.538 117.532Q-51.287 117.347-51.096 117.080L-50.727 115.600Q-50.764 115.440-50.845 115.303Q-50.925 115.166-51.051 115.086Q-51.178 115.006-51.342 115.006Q-51.550 115.006-51.737 115.130Q-51.923 115.255-52.061 115.447Q-52.200 115.638-52.285 115.840Q-52.398 116.134-52.482 116.479Q-52.566 116.824-52.566 117.074Q-52.566 117.330-52.437 117.523Q-52.309 117.716-52.063 117.716\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(129.525 -22.435)\">\u003Cpath d=\"M-48.768 119.627L-48.837 119.627Q-48.871 119.627-48.893 119.601Q-48.915 119.576-48.915 119.541Q-48.915 119.497-48.884 119.480Q-48.529 119.176-48.279 118.786Q-48.030 118.396-47.878 117.964Q-47.726 117.532-47.656 117.063Q-47.586 116.595-47.586 116.120Q-47.586 115.641-47.656 115.175Q-47.726 114.708-47.880 114.273Q-48.033 113.837-48.285 113.449Q-48.536 113.061-48.884 112.767Q-48.915 112.750-48.915 112.705Q-48.915 112.671-48.893 112.646Q-48.871 112.620-48.837 112.620L-48.768 112.620Q-48.758 112.620-48.749 112.622Q-48.741 112.623-48.731 112.627Q-48.187 113.027-47.815 113.580Q-47.442 114.134-47.261 114.780Q-47.080 115.426-47.080 116.120Q-47.080 116.821-47.261 117.468Q-47.442 118.116-47.816 118.670Q-48.191 119.224-48.731 119.620Q-48.741 119.620-48.749 119.622Q-48.758 119.623-48.768 119.627\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-size=\"7\">\u003Cg transform=\"translate(172.204 -56.578)\">\u003Cpath d=\"M-62.501 116.093Q-62.501 115.498-62.219 114.922Q-61.937 114.346-61.460 113.905Q-60.984 113.464-60.394 113.206Q-59.804 112.948-59.210 112.948Q-58.786 112.948-58.412 113.088Q-58.037 113.228-57.767 113.483Q-57.497 113.738-57.345 114.098Q-57.193 114.459-57.193 114.890Q-57.193 115.501-57.473 116.065Q-57.754 116.629-58.225 117.068Q-58.697 117.508-59.290 117.759Q-59.883 118.010-60.478 118.010Q-60.898 118.010-61.269 117.873Q-61.640 117.737-61.915 117.485Q-62.190 117.234-62.346 116.879Q-62.501 116.523-62.501 116.093M-60.413 117.750Q-59.883 117.750-59.417 117.465Q-58.950 117.180-58.615 116.720Q-58.280 116.260-58.094 115.708Q-57.907 115.156-57.907 114.643Q-57.907 114.226-58.072 113.898Q-58.236 113.570-58.547 113.386Q-58.858 113.201-59.275 113.201Q-60.006 113.201-60.570 113.657Q-61.134 114.114-61.443 114.828Q-61.753 115.542-61.753 116.246Q-61.753 116.660-61.599 117.004Q-61.445 117.347-61.143 117.549Q-60.840 117.750-60.413 117.750\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(172.204 -56.578)\">\u003Cpath d=\"M-54.248 119.620Q-54.798 119.220-55.169 118.665Q-55.540 118.109-55.721 117.463Q-55.902 116.817-55.902 116.120Q-55.902 115.607-55.802 115.112Q-55.701 114.616-55.496 114.165Q-55.291 113.714-54.978 113.322Q-54.665 112.931-54.248 112.627Q-54.238 112.623-54.231 112.622Q-54.224 112.620-54.214 112.620L-54.146 112.620Q-54.111 112.620-54.089 112.644Q-54.067 112.668-54.067 112.705Q-54.067 112.750-54.094 112.767Q-54.443 113.068-54.696 113.452Q-54.949 113.837-55.101 114.278Q-55.253 114.719-55.325 115.175Q-55.397 115.631-55.397 116.120Q-55.397 117.121-55.087 118.008Q-54.778 118.895-54.094 119.480Q-54.067 119.497-54.067 119.541Q-54.067 119.579-54.089 119.603Q-54.111 119.627-54.146 119.627L-54.214 119.627Q-54.221 119.623-54.229 119.622Q-54.238 119.620-54.248 119.620\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(172.204 -56.578)\">\u003Cpath d=\"M-52.077 117.938Q-52.401 117.938-52.646 117.781Q-52.890 117.624-53.022 117.359Q-53.153 117.094-53.153 116.769Q-53.153 116.301-52.900 115.838Q-52.648 115.375-52.224 115.079Q-51.800 114.784-51.328 114.784Q-51.113 114.784-50.927 114.888Q-50.740 114.992-50.621 115.177L-50.241 113.649Q-50.207 113.546-50.207 113.461Q-50.207 113.369-50.641 113.369Q-50.727 113.341-50.727 113.256L-50.696 113.146Q-50.669 113.099-50.621 113.088L-49.640 113.013Q-49.602 113.013-49.568 113.040Q-49.534 113.068-49.534 113.116L-50.542 117.142Q-50.569 117.299-50.569 117.388Q-50.569 117.515-50.518 117.615Q-50.467 117.716-50.347 117.716Q-50.125 117.716-50.012 117.463Q-49.900 117.210-49.814 116.841Q-49.790 116.780-49.739 116.780L-49.626 116.780Q-49.592 116.780-49.570 116.809Q-49.547 116.838-49.547 116.862Q-49.547 116.875-49.554 116.889Q-49.817 117.938-50.361 117.938Q-50.610 117.938-50.819 117.815Q-51.027 117.692-51.096 117.463Q-51.571 117.938-52.077 117.938M-52.063 117.716Q-51.790 117.716-51.538 117.532Q-51.287 117.347-51.096 117.080L-50.727 115.600Q-50.764 115.440-50.845 115.303Q-50.925 115.166-51.051 115.086Q-51.178 115.006-51.342 115.006Q-51.550 115.006-51.737 115.130Q-51.923 115.255-52.061 115.447Q-52.200 115.638-52.285 115.840Q-52.398 116.134-52.482 116.479Q-52.566 116.824-52.566 117.074Q-52.566 117.330-52.437 117.523Q-52.309 117.716-52.063 117.716\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(172.204 -56.578)\">\u003Cpath d=\"M-48.768 119.627L-48.837 119.627Q-48.871 119.627-48.893 119.601Q-48.915 119.576-48.915 119.541Q-48.915 119.497-48.884 119.480Q-48.529 119.176-48.279 118.786Q-48.030 118.396-47.878 117.964Q-47.726 117.532-47.656 117.063Q-47.586 116.595-47.586 116.120Q-47.586 115.641-47.656 115.175Q-47.726 114.708-47.880 114.273Q-48.033 113.837-48.285 113.449Q-48.536 113.061-48.884 112.767Q-48.915 112.750-48.915 112.705Q-48.915 112.671-48.893 112.646Q-48.871 112.620-48.837 112.620L-48.768 112.620Q-48.758 112.620-48.749 112.622Q-48.741 112.623-48.731 112.627Q-48.187 113.027-47.815 113.580Q-47.442 114.134-47.261 114.780Q-47.080 115.426-47.080 116.120Q-47.080 116.821-47.261 117.468Q-47.442 118.116-47.816 118.670Q-48.191 119.224-48.731 119.620Q-48.741 119.620-48.749 119.622Q-48.758 119.623-48.768 119.627\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-size=\"7\">\u003Cg transform=\"translate(214.883 -90.721)\">\u003Cpath d=\"M-62.501 116.093Q-62.501 115.498-62.219 114.922Q-61.937 114.346-61.460 113.905Q-60.984 113.464-60.394 113.206Q-59.804 112.948-59.210 112.948Q-58.786 112.948-58.412 113.088Q-58.037 113.228-57.767 113.483Q-57.497 113.738-57.345 114.098Q-57.193 114.459-57.193 114.890Q-57.193 115.501-57.473 116.065Q-57.754 116.629-58.225 117.068Q-58.697 117.508-59.290 117.759Q-59.883 118.010-60.478 118.010Q-60.898 118.010-61.269 117.873Q-61.640 117.737-61.915 117.485Q-62.190 117.234-62.346 116.879Q-62.501 116.523-62.501 116.093M-60.413 117.750Q-59.883 117.750-59.417 117.465Q-58.950 117.180-58.615 116.720Q-58.280 116.260-58.094 115.708Q-57.907 115.156-57.907 114.643Q-57.907 114.226-58.072 113.898Q-58.236 113.570-58.547 113.386Q-58.858 113.201-59.275 113.201Q-60.006 113.201-60.570 113.657Q-61.134 114.114-61.443 114.828Q-61.753 115.542-61.753 116.246Q-61.753 116.660-61.599 117.004Q-61.445 117.347-61.143 117.549Q-60.840 117.750-60.413 117.750\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(214.883 -90.721)\">\u003Cpath d=\"M-54.248 119.620Q-54.798 119.220-55.169 118.665Q-55.540 118.109-55.721 117.463Q-55.902 116.817-55.902 116.120Q-55.902 115.607-55.802 115.112Q-55.701 114.616-55.496 114.165Q-55.291 113.714-54.978 113.322Q-54.665 112.931-54.248 112.627Q-54.238 112.623-54.231 112.622Q-54.224 112.620-54.214 112.620L-54.146 112.620Q-54.111 112.620-54.089 112.644Q-54.067 112.668-54.067 112.705Q-54.067 112.750-54.094 112.767Q-54.443 113.068-54.696 113.452Q-54.949 113.837-55.101 114.278Q-55.253 114.719-55.325 115.175Q-55.397 115.631-55.397 116.120Q-55.397 117.121-55.087 118.008Q-54.778 118.895-54.094 119.480Q-54.067 119.497-54.067 119.541Q-54.067 119.579-54.089 119.603Q-54.111 119.627-54.146 119.627L-54.214 119.627Q-54.221 119.623-54.229 119.622Q-54.238 119.620-54.248 119.620\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(214.883 -90.721)\">\u003Cpath d=\"M-52.077 117.938Q-52.401 117.938-52.646 117.781Q-52.890 117.624-53.022 117.359Q-53.153 117.094-53.153 116.769Q-53.153 116.301-52.900 115.838Q-52.648 115.375-52.224 115.079Q-51.800 114.784-51.328 114.784Q-51.113 114.784-50.927 114.888Q-50.740 114.992-50.621 115.177L-50.241 113.649Q-50.207 113.546-50.207 113.461Q-50.207 113.369-50.641 113.369Q-50.727 113.341-50.727 113.256L-50.696 113.146Q-50.669 113.099-50.621 113.088L-49.640 113.013Q-49.602 113.013-49.568 113.040Q-49.534 113.068-49.534 113.116L-50.542 117.142Q-50.569 117.299-50.569 117.388Q-50.569 117.515-50.518 117.615Q-50.467 117.716-50.347 117.716Q-50.125 117.716-50.012 117.463Q-49.900 117.210-49.814 116.841Q-49.790 116.780-49.739 116.780L-49.626 116.780Q-49.592 116.780-49.570 116.809Q-49.547 116.838-49.547 116.862Q-49.547 116.875-49.554 116.889Q-49.817 117.938-50.361 117.938Q-50.610 117.938-50.819 117.815Q-51.027 117.692-51.096 117.463Q-51.571 117.938-52.077 117.938M-52.063 117.716Q-51.790 117.716-51.538 117.532Q-51.287 117.347-51.096 117.080L-50.727 115.600Q-50.764 115.440-50.845 115.303Q-50.925 115.166-51.051 115.086Q-51.178 115.006-51.342 115.006Q-51.550 115.006-51.737 115.130Q-51.923 115.255-52.061 115.447Q-52.200 115.638-52.285 115.840Q-52.398 116.134-52.482 116.479Q-52.566 116.824-52.566 117.074Q-52.566 117.330-52.437 117.523Q-52.309 117.716-52.063 117.716\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(214.883 -90.721)\">\u003Cpath d=\"M-48.768 119.627L-48.837 119.627Q-48.871 119.627-48.893 119.601Q-48.915 119.576-48.915 119.541Q-48.915 119.497-48.884 119.480Q-48.529 119.176-48.279 118.786Q-48.030 118.396-47.878 117.964Q-47.726 117.532-47.656 117.063Q-47.586 116.595-47.586 116.120Q-47.586 115.641-47.656 115.175Q-47.726 114.708-47.880 114.273Q-48.033 113.837-48.285 113.449Q-48.536 113.061-48.884 112.767Q-48.915 112.750-48.915 112.705Q-48.915 112.671-48.893 112.646Q-48.871 112.620-48.837 112.620L-48.768 112.620Q-48.758 112.620-48.749 112.622Q-48.741 112.623-48.731 112.627Q-48.187 113.027-47.815 113.580Q-47.442 114.134-47.261 114.780Q-47.080 115.426-47.080 116.120Q-47.080 116.821-47.261 117.468Q-47.442 118.116-47.816 118.670Q-48.191 119.224-48.731 119.620Q-48.741 119.620-48.749 119.622Q-48.758 119.623-48.768 119.627\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-size=\"7\">\u003Cg transform=\"translate(257.562 -124.865)\">\u003Cpath d=\"M-62.501 116.093Q-62.501 115.498-62.219 114.922Q-61.937 114.346-61.460 113.905Q-60.984 113.464-60.394 113.206Q-59.804 112.948-59.210 112.948Q-58.786 112.948-58.412 113.088Q-58.037 113.228-57.767 113.483Q-57.497 113.738-57.345 114.098Q-57.193 114.459-57.193 114.890Q-57.193 115.501-57.473 116.065Q-57.754 116.629-58.225 117.068Q-58.697 117.508-59.290 117.759Q-59.883 118.010-60.478 118.010Q-60.898 118.010-61.269 117.873Q-61.640 117.737-61.915 117.485Q-62.190 117.234-62.346 116.879Q-62.501 116.523-62.501 116.093M-60.413 117.750Q-59.883 117.750-59.417 117.465Q-58.950 117.180-58.615 116.720Q-58.280 116.260-58.094 115.708Q-57.907 115.156-57.907 114.643Q-57.907 114.226-58.072 113.898Q-58.236 113.570-58.547 113.386Q-58.858 113.201-59.275 113.201Q-60.006 113.201-60.570 113.657Q-61.134 114.114-61.443 114.828Q-61.753 115.542-61.753 116.246Q-61.753 116.660-61.599 117.004Q-61.445 117.347-61.143 117.549Q-60.840 117.750-60.413 117.750\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(257.562 -124.865)\">\u003Cpath d=\"M-54.248 119.620Q-54.798 119.220-55.169 118.665Q-55.540 118.109-55.721 117.463Q-55.902 116.817-55.902 116.120Q-55.902 115.607-55.802 115.112Q-55.701 114.616-55.496 114.165Q-55.291 113.714-54.978 113.322Q-54.665 112.931-54.248 112.627Q-54.238 112.623-54.231 112.622Q-54.224 112.620-54.214 112.620L-54.146 112.620Q-54.111 112.620-54.089 112.644Q-54.067 112.668-54.067 112.705Q-54.067 112.750-54.094 112.767Q-54.443 113.068-54.696 113.452Q-54.949 113.837-55.101 114.278Q-55.253 114.719-55.325 115.175Q-55.397 115.631-55.397 116.120Q-55.397 117.121-55.087 118.008Q-54.778 118.895-54.094 119.480Q-54.067 119.497-54.067 119.541Q-54.067 119.579-54.089 119.603Q-54.111 119.627-54.146 119.627L-54.214 119.627Q-54.221 119.623-54.229 119.622Q-54.238 119.620-54.248 119.620\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(257.562 -124.865)\">\u003Cpath d=\"M-52.784 117.716Q-52.784 117.668-52.777 117.644L-52.265 115.580Q-52.231 115.453-52.231 115.337Q-52.231 115.197-52.284 115.101Q-52.337 115.006-52.460 115.006Q-52.682 115.006-52.783 115.233Q-52.883 115.460-52.993 115.881Q-53.003 115.946-53.065 115.946L-53.174 115.946Q-53.205 115.946-53.229 115.915Q-53.253 115.884-53.253 115.860L-53.253 115.833Q-53.140 115.399-52.959 115.091Q-52.777 114.784-52.446 114.784Q-52.261 114.784-52.082 114.859Q-51.902 114.934-51.790 115.074Q-51.677 115.214-51.677 115.406Q-51.530 115.221-51.349 115.081Q-51.168 114.941-50.954 114.862Q-50.740 114.784-50.508 114.784Q-50.142 114.784-49.864 114.958Q-49.585 115.132-49.585 115.477Q-49.356 115.163-49.042 114.973Q-48.727 114.784-48.358 114.784Q-47.948 114.784-47.692 114.980Q-47.435 115.177-47.435 115.573Q-47.435 115.857-47.563 116.255Q-47.692 116.653-47.890 117.142Q-47.968 117.344-47.968 117.484Q-47.968 117.716-47.797 117.716Q-47.524 117.716-47.329 117.436Q-47.134 117.156-47.056 116.841Q-47.032 116.780-46.981 116.780L-46.868 116.780Q-46.834 116.780-46.813 116.805Q-46.793 116.831-46.793 116.862Q-46.793 116.875-46.799 116.889Q-46.861 117.139-47.003 117.381Q-47.145 117.624-47.355 117.781Q-47.565 117.938-47.815 117.938Q-48.095 117.938-48.296 117.776Q-48.498 117.614-48.498 117.344Q-48.498 117.227-48.450 117.094Q-48.245 116.575-48.114 116.170Q-47.982 115.765-47.982 115.477Q-47.982 115.266-48.076 115.136Q-48.170 115.006-48.372 115.006Q-49.127 115.006-49.653 116.021L-50.067 117.682Q-50.091 117.788-50.185 117.863Q-50.279 117.938-50.395 117.938Q-50.494 117.938-50.564 117.877Q-50.634 117.815-50.634 117.716Q-50.634 117.668-50.628 117.644L-50.187 115.881Q-50.129 115.597-50.129 115.477Q-50.129 115.269-50.224 115.137Q-50.320 115.006-50.522 115.006Q-51.277 115.006-51.803 116.021L-52.217 117.682Q-52.241 117.788-52.335 117.863Q-52.429 117.938-52.545 117.938Q-52.644 117.938-52.714 117.877Q-52.784 117.815-52.784 117.716M-44.984 117.938Q-45.309 117.938-45.554 117.781Q-45.798 117.624-45.930 117.359Q-46.061 117.094-46.061 116.769Q-46.061 116.301-45.808 115.838Q-45.555 115.375-45.131 115.079Q-44.708 114.784-44.236 114.784Q-44.021 114.784-43.834 114.888Q-43.648 114.992-43.528 115.177L-43.149 113.649Q-43.115 113.546-43.115 113.461Q-43.115 113.369-43.549 113.369Q-43.634 113.341-43.634 113.256L-43.604 113.146Q-43.576 113.099-43.528 113.088L-42.547 113.013Q-42.510 113.013-42.476 113.040Q-42.442 113.068-42.442 113.116L-43.450 117.142Q-43.477 117.299-43.477 117.388Q-43.477 117.515-43.426 117.615Q-43.375 117.716-43.255 117.716Q-43.033 117.716-42.920 117.463Q-42.807 117.210-42.722 116.841Q-42.698 116.780-42.647 116.780L-42.534 116.780Q-42.500 116.780-42.477 116.809Q-42.455 116.838-42.455 116.862Q-42.455 116.875-42.462 116.889Q-42.725 117.938-43.269 117.938Q-43.518 117.938-43.727 117.815Q-43.935 117.692-44.004 117.463Q-44.479 117.938-44.984 117.938M-44.971 117.716Q-44.697 117.716-44.446 117.532Q-44.195 117.347-44.004 117.080L-43.634 115.600Q-43.672 115.440-43.752 115.303Q-43.833 115.166-43.959 115.086Q-44.086 115.006-44.250 115.006Q-44.458 115.006-44.644 115.130Q-44.831 115.255-44.969 115.447Q-45.108 115.638-45.193 115.840Q-45.306 116.134-45.390 116.479Q-45.473 116.824-45.473 117.074Q-45.473 117.330-45.345 117.523Q-45.217 117.716-44.971 117.716\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(257.562 -124.865)\">\u003Cpath d=\"M-41.672 119.627L-41.741 119.627Q-41.775 119.627-41.797 119.601Q-41.819 119.576-41.819 119.541Q-41.819 119.497-41.788 119.480Q-41.433 119.176-41.183 118.786Q-40.934 118.396-40.782 117.964Q-40.630 117.532-40.560 117.063Q-40.490 116.595-40.490 116.120Q-40.490 115.641-40.560 115.175Q-40.630 114.708-40.784 114.273Q-40.937 113.837-41.189 113.449Q-41.440 113.061-41.788 112.767Q-41.819 112.750-41.819 112.705Q-41.819 112.671-41.797 112.646Q-41.775 112.620-41.741 112.620L-41.672 112.620Q-41.662 112.620-41.653 112.622Q-41.645 112.623-41.635 112.627Q-41.091 113.027-40.719 113.580Q-40.346 114.134-40.165 114.780Q-39.984 115.426-39.984 116.120Q-39.984 116.821-40.165 117.468Q-40.346 118.116-40.720 118.670Q-41.095 119.224-41.635 119.620Q-41.645 119.620-41.653 119.622Q-41.662 119.623-41.672 119.627\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\">\u003Cg transform=\"translate(300.241 -158.513)\">\u003Cpath d=\"M-62.501 116.093Q-62.501 115.498-62.219 114.922Q-61.937 114.346-61.460 113.905Q-60.984 113.464-60.394 113.206Q-59.804 112.948-59.210 112.948Q-58.786 112.948-58.412 113.088Q-58.037 113.228-57.767 113.483Q-57.497 113.738-57.345 114.098Q-57.193 114.459-57.193 114.890Q-57.193 115.501-57.473 116.065Q-57.754 116.629-58.225 117.068Q-58.697 117.508-59.290 117.759Q-59.883 118.010-60.478 118.010Q-60.898 118.010-61.269 117.873Q-61.640 117.737-61.915 117.485Q-62.190 117.234-62.346 116.879Q-62.501 116.523-62.501 116.093M-60.413 117.750Q-59.883 117.750-59.417 117.465Q-58.950 117.180-58.615 116.720Q-58.280 116.260-58.094 115.708Q-57.907 115.156-57.907 114.643Q-57.907 114.226-58.072 113.898Q-58.236 113.570-58.547 113.386Q-58.858 113.201-59.275 113.201Q-60.006 113.201-60.570 113.657Q-61.134 114.114-61.443 114.828Q-61.753 115.542-61.753 116.246Q-61.753 116.660-61.599 117.004Q-61.445 117.347-61.143 117.549Q-60.840 117.750-60.413 117.750\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(300.241 -158.513)\">\u003Cpath d=\"M-54.248 119.620Q-54.798 119.220-55.169 118.665Q-55.540 118.109-55.721 117.463Q-55.902 116.817-55.902 116.120Q-55.902 115.607-55.802 115.112Q-55.701 114.616-55.496 114.165Q-55.291 113.714-54.978 113.322Q-54.665 112.931-54.248 112.627Q-54.238 112.623-54.231 112.622Q-54.224 112.620-54.214 112.620L-54.146 112.620Q-54.111 112.620-54.089 112.644Q-54.067 112.668-54.067 112.705Q-54.067 112.750-54.094 112.767Q-54.443 113.068-54.696 113.452Q-54.949 113.837-55.101 114.278Q-55.253 114.719-55.325 115.175Q-55.397 115.631-55.397 116.120Q-55.397 117.121-55.087 118.008Q-54.778 118.895-54.094 119.480Q-54.067 119.497-54.067 119.541Q-54.067 119.579-54.089 119.603Q-54.111 119.627-54.146 119.627L-54.214 119.627Q-54.221 119.623-54.229 119.622Q-54.238 119.620-54.248 119.620\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(300.241 -158.513)\">\u003Cpath d=\"M-52.077 117.938Q-52.401 117.938-52.646 117.781Q-52.890 117.624-53.022 117.359Q-53.153 117.094-53.153 116.769Q-53.153 116.301-52.900 115.838Q-52.648 115.375-52.224 115.079Q-51.800 114.784-51.328 114.784Q-51.113 114.784-50.927 114.888Q-50.740 114.992-50.621 115.177L-50.241 113.649Q-50.207 113.546-50.207 113.461Q-50.207 113.369-50.641 113.369Q-50.727 113.341-50.727 113.256L-50.696 113.146Q-50.669 113.099-50.621 113.088L-49.640 113.013Q-49.602 113.013-49.568 113.040Q-49.534 113.068-49.534 113.116L-50.542 117.142Q-50.569 117.299-50.569 117.388Q-50.569 117.515-50.518 117.615Q-50.467 117.716-50.347 117.716Q-50.125 117.716-50.012 117.463Q-49.900 117.210-49.814 116.841Q-49.790 116.780-49.739 116.780L-49.626 116.780Q-49.592 116.780-49.570 116.809Q-49.547 116.838-49.547 116.862Q-49.547 116.875-49.554 116.889Q-49.817 117.938-50.361 117.938Q-50.610 117.938-50.819 117.815Q-51.027 117.692-51.096 117.463Q-51.571 117.938-52.077 117.938M-52.063 117.716Q-51.790 117.716-51.538 117.532Q-51.287 117.347-51.096 117.080L-50.727 115.600Q-50.764 115.440-50.845 115.303Q-50.925 115.166-51.051 115.086Q-51.178 115.006-51.342 115.006Q-51.550 115.006-51.737 115.130Q-51.923 115.255-52.061 115.447Q-52.200 115.638-52.285 115.840Q-52.398 116.134-52.482 116.479Q-52.566 116.824-52.566 117.074Q-52.566 117.330-52.437 117.523Q-52.309 117.716-52.063 117.716\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(300.241 -158.513)\">\u003Cpath d=\"M-48.618 114.513Q-48.364 114.737-47.729 114.737Q-47.507 114.737-47.342 114.638Q-47.177 114.540-47.093 114.364Q-47.009 114.188-47.009 113.973Q-47.009 113.751-47.096 113.579Q-47.182 113.407-47.348 113.307Q-47.514 113.206-47.734 113.206L-48.159 113.206Q-48.213 113.194-48.225 113.143L-48.225 113.082Q-48.213 113.038-48.159 113.021L-47.800 112.997Q-47.614 112.997-47.458 112.885Q-47.302 112.774-47.215 112.595Q-47.129 112.415-47.129 112.237Q-47.129 112.079-47.209 111.963Q-47.290 111.847-47.424 111.789Q-47.558 111.732-47.724 111.732Q-48.239 111.732-48.474 111.961Q-48.366 111.983-48.300 112.063Q-48.235 112.142-48.235 112.252Q-48.235 112.376-48.320 112.462Q-48.405 112.547-48.535 112.547Q-48.657 112.547-48.742 112.462Q-48.828 112.376-48.828 112.252Q-48.828 111.996-48.656 111.831Q-48.484 111.666-48.230 111.594Q-47.976 111.522-47.724 111.522Q-47.492 111.522-47.223 111.598Q-46.953 111.673-46.766 111.834Q-46.579 111.996-46.579 112.237Q-46.579 112.555-46.798 112.778Q-47.016 113.001-47.343 113.102Q-47.173 113.136-47.010 113.208Q-46.848 113.280-46.709 113.391Q-46.570 113.502-46.487 113.648Q-46.404 113.795-46.404 113.973Q-46.404 114.210-46.522 114.395Q-46.640 114.581-46.843 114.709Q-47.046 114.837-47.278 114.900Q-47.509 114.962-47.724 114.962Q-48.015 114.962-48.305 114.897Q-48.596 114.832-48.800 114.663Q-49.004 114.493-49.004 114.203Q-49.004 114.068-48.913 113.978Q-48.823 113.888-48.684 113.888Q-48.596 113.888-48.524 113.929Q-48.452 113.971-48.410 114.043Q-48.369 114.115-48.369 114.203Q-48.369 114.320-48.437 114.404Q-48.506 114.488-48.618 114.513\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.150\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(300.241 -158.513)\">\u003Cpath d=\"M-44.865 119.627L-44.934 119.627Q-44.968 119.627-44.990 119.601Q-45.012 119.576-45.012 119.541Q-45.012 119.497-44.981 119.480Q-44.626 119.176-44.376 118.786Q-44.127 118.396-43.975 117.964Q-43.823 117.532-43.753 117.063Q-43.683 116.595-43.683 116.120Q-43.683 115.641-43.753 115.175Q-43.823 114.708-43.977 114.273Q-44.130 113.837-44.382 113.449Q-44.633 113.061-44.981 112.767Q-45.012 112.750-45.012 112.705Q-45.012 112.671-44.990 112.646Q-44.968 112.620-44.934 112.620L-44.865 112.620Q-44.855 112.620-44.846 112.622Q-44.838 112.623-44.828 112.627Q-44.284 113.027-43.912 113.580Q-43.539 114.134-43.358 114.780Q-43.177 115.426-43.177 116.120Q-43.177 116.821-43.358 117.468Q-43.539 118.116-43.913 118.670Q-44.288 119.224-44.828 119.620Q-44.838 119.620-44.846 119.622Q-44.855 119.623-44.865 119.627\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">The cost\u002Faccuracy ladder. Climbing captures more of the true curvature \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0813em;\">H\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> (diagonal \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.3669em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">→\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> block \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.3669em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">→\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> low-rank \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.3669em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">→\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> full) at rising per-step cost (\u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">O\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">→\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">O\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">→\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">O\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">→\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.0641em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">O\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">d\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8141em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">3\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>). Deep learning lives on the lowest rungs.\u003C\u002Ffigcaption>",1785117765665]