[{"data":1,"prerenderedAt":17269},["ShallowReactive",2],{"lesson:\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td":3,"course-wordcounts":11489,"ref-card-index":12401,"nav:reinforcement-learning":17083,"tikz:d95d098bccb57803ca45d490014e3c9937e7ad7b469c31a24c67feee8589de20":17264,"tikz:cbb69ce3577e35b807b8c382534d64bd127f1a6aecd1d437c8b9dbb453f7d5ee":17265,"tikz:8e8d5b2a8003797447945f15fcd4b2d1950c19a213823669158862176d2b2ac7":17266,"tikz:7e758feba09c8380d82572e09d8640b050875d1c5cdb7b231c694626aa8e2f0f":17267,"tikz:f8f108bf91266823c1fc5715f78cc455441761e193e2d20146aa9f2a2bfb0178":17268},{"id":4,"title":5,"blurb":6,"body":7,"brief":11459,"category":11460,"description":11461,"draft":11462,"extension":11463,"meta":11464,"module":11467,"navigation":11468,"path":11469,"practice":11470,"rawbody":11471,"readingTime":11472,"seo":11477,"sources":11478,"status":11483,"stem":11484,"summary":11485,"topics":11486,"__hash__":11488},"course\u002F07.reinforcement-learning\u002F03.approximation\u002F06.bellman-error-and-gradient-td.md","Value-Function Geometry and Gradient-TD Methods","",{"type":8,"value":9,"toc":11443},"minimark",[10,20,25,455,663,667,707,934,958,1467,1584,1589,1651,2082,2563,2778,2954,3172,3606,3931,4102,4106,4195,4314,4317,4925,5010,5336,5357,5754,5940,6362,6829,7180,7183,7507,7511,7606,8249,8308,8312,8328,8334,8338,8664,9818,9821,9844,9847,9851,9860,9982,9985,9995,9999,10002,10154,10211,10214,10272,10276,10308],[11,12,13,14,19],"p",{},"This builds on\n",[15,16,18],"a",{"href":17},"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad","off-policy methods and the deadly triad",",\nwhich showed that function approximation, bootstrapping, and off-policy training\ntogether can send the weights to infinity — on a seven-state problem with linearly\nindependent features, no less. That lesson named the failure but did not explain\nit. This lesson does, by picturing value functions as points in a vector space,\nand from that picture builds the two families of methods that make off-policy\nlearning stable.",[21,22,24],"h2",{"id":23},"linear-value-function-geometry","Linear value-function geometry",[11,26,27,28,32,33,43,44,234,235,253,254,431,432,454],{},"To understand ",[29,30,31],"em",{},"why"," the triad diverges and how to fix it, it helps to picture value\nfunctions as vectors.",[34,35,36],"sup",{},[15,37,42],{"href":38,"ariaDescribedBy":39,"dataFootnoteRef":6,"id":41},"#user-content-fn-sb-geom",[40],"footnote-label","user-content-fnref-sb-geom","1"," Enumerate the states ",[45,46,49],"span",{"className":47},[48],"katex",[45,50,54,83],{"className":51,"ariaHidden":53},[52],"katex-html","true",[45,55,58,63,70,75,80],{"className":56},[57],"base",[45,59],{"className":60,"style":62},[61],"strut","height:0.6833em;",[45,64,69],{"className":65,"style":68},[66,67],"mord","mathcal","margin-right:0.075em;","S",[45,71],{"className":72,"style":74},[73],"mspace","margin-right:0.2778em;",[45,76,79],{"className":77},[78],"mrel","=",[45,81],{"className":82,"style":74},[73],[45,84,86,90,95,153,158,162,167,170,173,176,229],{"className":85},[57],[45,87],{"className":88,"style":89},[61],"height:1.1052em;vertical-align:-0.3552em;",[45,91,94],{"className":92},[93],"mopen","{",[45,96,98,103],{"className":97},[66],[45,99,102],{"className":100},[66,101],"mathnormal","s",[45,104,107],{"className":105},[106],"msupsub",[45,108,112,144],{"className":109},[110,111],"vlist-t","vlist-t2",[45,113,116,139],{"className":114},[115],"vlist-r",[45,117,121],{"className":118,"style":120},[119],"vlist","height:0.3011em;",[45,122,124,129],{"style":123},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[45,125],{"className":126,"style":128},[127],"pstrut","height:2.7em;",[45,130,136],{"className":131},[132,133,134,135],"sizing","reset-size6","size3","mtight",[45,137,42],{"className":138},[66,135],[45,140,143],{"className":141},[142],"vlist-s","​",[45,145,147],{"className":146},[115],[45,148,151],{"className":149,"style":150},[119],"height:0.15em;",[45,152],{},[45,154,157],{"className":155},[156],"mpunct",",",[45,159],{"className":160,"style":161},[73],"margin-right:0.1667em;",[45,163,166],{"className":164},[165],"minner","…",[45,168],{"className":169,"style":161},[73],[45,171,157],{"className":172},[156],[45,174],{"className":175,"style":161},[73],[45,177,179,182],{"className":178},[66],[45,180,102],{"className":181},[66,101],[45,183,185],{"className":184},[106],[45,186,188,220],{"className":187},[110,111],[45,189,191,217],{"className":190},[115],[45,192,195],{"className":193,"style":194},[119],"height:0.3448em;",[45,196,198,201],{"style":197},"top:-2.5198em;margin-left:0em;margin-right:0.05em;",[45,199],{"className":200,"style":128},[127],[45,202,204],{"className":203},[132,133,134,135],[45,205,207,211,214],{"className":206},[66,135],[45,208,210],{"className":209},[66,135],"∣",[45,212,69],{"className":213,"style":68},[66,67,135],[45,215,210],{"className":216},[66,135],[45,218,143],{"className":219},[142],[45,221,223],{"className":222},[115],[45,224,227],{"className":225,"style":226},[119],"height:0.3552em;",[45,228],{},[45,230,233],{"className":231},[232],"mclose","}","; then any value function ",[45,236,238],{"className":237},[48],[45,239,241],{"className":240,"ariaHidden":53},[52],[45,242,244,248],{"className":243},[57],[45,245],{"className":246,"style":247},[61],"height:0.4306em;",[45,249,252],{"className":250,"style":251},[66,101],"margin-right:0.0359em;","v"," is the vector ",[45,255,257],{"className":256},[48],[45,258,260],{"className":259,"ariaHidden":53},[52],[45,261,263,267,271,274,278,318,322,325,328,331,334,337,340,343,346,395,398],{"className":262},[57],[45,264],{"className":265,"style":266},[61],"height:1.2043em;vertical-align:-0.3552em;",[45,268,270],{"className":269},[93],"[",[45,272,252],{"className":273,"style":251},[66,101],[45,275,277],{"className":276},[93],"(",[45,279,281,284],{"className":280},[66],[45,282,102],{"className":283},[66,101],[45,285,287],{"className":286},[106],[45,288,290,310],{"className":289},[110,111],[45,291,293,307],{"className":292},[115],[45,294,296],{"className":295,"style":120},[119],[45,297,298,301],{"style":123},[45,299],{"className":300,"style":128},[127],[45,302,304],{"className":303},[132,133,134,135],[45,305,42],{"className":306},[66,135],[45,308,143],{"className":309},[142],[45,311,313],{"className":312},[115],[45,314,316],{"className":315,"style":150},[119],[45,317],{},[45,319,321],{"className":320},[232],")",[45,323,157],{"className":324},[156],[45,326],{"className":327,"style":161},[73],[45,329,166],{"className":330},[165],[45,332],{"className":333,"style":161},[73],[45,335,157],{"className":336},[156],[45,338],{"className":339,"style":161},[73],[45,341,252],{"className":342,"style":251},[66,101],[45,344,277],{"className":345},[93],[45,347,349,352],{"className":348},[66],[45,350,102],{"className":351},[66,101],[45,353,355],{"className":354},[106],[45,356,358,387],{"className":357},[110,111],[45,359,361,384],{"className":360},[115],[45,362,364],{"className":363,"style":194},[119],[45,365,366,369],{"style":197},[45,367],{"className":368,"style":128},[127],[45,370,372],{"className":371},[132,133,134,135],[45,373,375,378,381],{"className":374},[66,135],[45,376,210],{"className":377},[66,135],[45,379,69],{"className":380,"style":68},[66,67,135],[45,382,210],{"className":383},[66,135],[45,385,143],{"className":386},[142],[45,388,390],{"className":389},[115],[45,391,393],{"className":392,"style":226},[119],[45,394],{},[45,396,321],{"className":397},[232],[45,399,401,405],{"className":400},[232],[45,402,404],{"className":403},[232],"]",[45,406,408],{"className":407},[106],[45,409,411],{"className":410},[110],[45,412,414],{"className":413},[115],[45,415,418],{"className":416,"style":417},[119],"height:0.8491em;",[45,419,421,424],{"style":420},"top:-3.063em;margin-right:0.05em;",[45,422],{"className":423,"style":128},[127],[45,425,427],{"className":426},[132,133,134,135],[45,428,430],{"className":429},[66,135],"⊤"," in an ",[45,433,435],{"className":434},[48],[45,436,438],{"className":437,"ariaHidden":53},[52],[45,439,441,445,448,451],{"className":440},[57],[45,442],{"className":443,"style":444},[61],"height:1em;vertical-align:-0.25em;",[45,446,210],{"className":447},[66],[45,449,69],{"className":450,"style":68},[66,67],[45,452,210],{"className":453},[66],"-dimensional space. The whole space\nholds every conceivable value function; most correspond to no policy at all.",[11,456,457,458,500,501,517,518,521,522,603,604,658,659,662],{},"A linear approximator with ",[45,459,461],{"className":460},[48],[45,462,464,485],{"className":463,"ariaHidden":53},[52],[45,465,467,471,475,478,482],{"className":466},[57],[45,468],{"className":469,"style":470},[61],"height:0.7335em;vertical-align:-0.0391em;",[45,472,474],{"className":473},[66,101],"d",[45,476],{"className":477,"style":74},[73],[45,479,481],{"className":480},[78],"\u003C",[45,483],{"className":484,"style":74},[73],[45,486,488,491,494,497],{"className":487},[57],[45,489],{"className":490,"style":444},[61],[45,492,210],{"className":493},[66],[45,495,69],{"className":496,"style":68},[66,67],[45,498,210],{"className":499},[66]," parameters can only reach a\n",[45,502,504],{"className":503},[48],[45,505,507],{"className":506,"ariaHidden":53},[52],[45,508,510,514],{"className":509},[57],[45,511],{"className":512,"style":513},[61],"height:0.6944em;",[45,515,474],{"className":516},[66,101],"-dimensional ",[29,519,520],{},"subspace"," of that space — the plane of all ",[45,523,525],{"className":524},[48],[45,526,528,592],{"className":527,"ariaHidden":53},[52],[45,529,531,535,583,586,589],{"className":530},[57],[45,532],{"className":533,"style":534},[61],"height:0.5806em;vertical-align:-0.15em;",[45,536,538,541],{"className":537},[66],[45,539,252],{"className":540,"style":251},[66,101],[45,542,544],{"className":543},[106],[45,545,547,575],{"className":546},[110,111],[45,548,550,572],{"className":549},[115],[45,551,554],{"className":552,"style":553},[119],"height:0.1611em;",[45,555,557,560],{"style":556},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[45,558],{"className":559,"style":128},[127],[45,561,563],{"className":562},[132,133,134,135],[45,564,566],{"className":565},[66,135],[45,567,571],{"className":568,"style":570},[66,569,135],"mathbf","margin-right:0.016em;","w",[45,573,143],{"className":574},[142],[45,576,578],{"className":577},[115],[45,579,581],{"className":580,"style":150},[119],[45,582],{},[45,584],{"className":585,"style":74},[73],[45,587,79],{"className":588},[78],[45,590],{"className":591,"style":74},[73],[45,593,595,599],{"className":594},[57],[45,596],{"className":597,"style":598},[61],"height:0.6861em;",[45,600,602],{"className":601,"style":570},[66,569],"Xw"," representable by some weight vector. The true value ",[45,605,607],{"className":606},[48],[45,608,610],{"className":609,"ariaHidden":53},[52],[45,611,613,616],{"className":612},[57],[45,614],{"className":615,"style":534},[61],[45,617,619,622],{"className":618},[66],[45,620,252],{"className":621,"style":251},[66,101],[45,623,625],{"className":624},[106],[45,626,628,650],{"className":627},[110,111],[45,629,631,647],{"className":630},[115],[45,632,635],{"className":633,"style":634},[119],"height:0.1514em;",[45,636,637,640],{"style":556},[45,638],{"className":639,"style":128},[127],[45,641,643],{"className":642},[132,133,134,135],[45,644,646],{"className":645,"style":251},[66,101,135],"π",[45,648,143],{"className":649},[142],[45,651,653],{"className":652},[115],[45,654,656],{"className":655,"style":150},[119],[45,657],{}," of a\npolicy generally lies ",[29,660,661],{},"outside"," this subspace, too complex to represent exactly.",[664,665],"tikz-figure",{"hash":666},"d95d098bccb57803ca45d490014e3c9937e7ad7b469c31a24c67feee8589de20",[11,668,669,670,687,688],{},"To measure how far apart two value functions are, Euclidean distance is wrong: some\nstates matter more because they are visited more often. Weight each state by the\ndistribution ",[45,671,673],{"className":672},[48],[45,674,676],{"className":675,"ariaHidden":53},[52],[45,677,679,683],{"className":678},[57],[45,680],{"className":681,"style":682},[61],"height:0.625em;vertical-align:-0.1944em;",[45,684,686],{"className":685},[66,101],"μ"," (typically the on-policy distribution) and use the ",[689,690,691,706],"strong",{},[45,692,694],{"className":693},[48],[45,695,697],{"className":696,"ariaHidden":53},[52],[45,698,700,703],{"className":699},[57],[45,701],{"className":702,"style":682},[61],[45,704,686],{"className":705},[66,101],"-norm",[45,708,711],{"className":709},[710],"katex-display",[45,712,714],{"className":713},[48],[45,715,717,803],{"className":716,"ariaHidden":53},[52],[45,718,720,724,728,731,787,790,793,797,800],{"className":719},[57],[45,721],{"className":722,"style":723},[61],"height:1.2472em;vertical-align:-0.3831em;",[45,725,727],{"className":726},[93],"∥",[45,729,252],{"className":730,"style":251},[66,101],[45,732,734,737],{"className":733},[232],[45,735,727],{"className":736},[232],[45,738,740],{"className":739},[106],[45,741,743,778],{"className":742},[110,111],[45,744,746,775],{"className":745},[115],[45,747,750,762],{"className":748,"style":749},[119],"height:0.8641em;",[45,751,753,756],{"style":752},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[45,754],{"className":755,"style":128},[127],[45,757,759],{"className":758},[132,133,134,135],[45,760,686],{"className":761},[66,101,135],[45,763,765,768],{"style":764},"top:-3.113em;margin-right:0.05em;",[45,766],{"className":767,"style":128},[127],[45,769,771],{"className":770},[132,133,134,135],[45,772,774],{"className":773},[66,135],"2",[45,776,143],{"className":777},[142],[45,779,781],{"className":780},[115],[45,782,785],{"className":783,"style":784},[119],"height:0.3831em;",[45,786],{},[45,788],{"className":789,"style":74},[73],[45,791],{"className":792,"style":74},[73],[45,794,796],{"className":795},[78],"≐",[45,798],{"className":799,"style":74},[73],[45,801],{"className":802,"style":74},[73],[45,804,806,810,874,877,880,883,886,889,892,895,898,901,930],{"className":805},[57],[45,807],{"className":808,"style":809},[61],"height:2.3717em;vertical-align:-1.3217em;",[45,811,815],{"className":812},[813,814],"mop","op-limits",[45,816,818,865],{"className":817},[110,111],[45,819,821,862],{"className":820},[115],[45,822,825,848],{"className":823,"style":824},[119],"height:1.05em;",[45,826,828,832],{"style":827},"top:-1.8557em;margin-left:0em;",[45,829],{"className":830,"style":831},[127],"height:3.05em;",[45,833,835],{"className":834},[132,133,134,135],[45,836,838,841,845],{"className":837},[66,135],[45,839,102],{"className":840},[66,101,135],[45,842,844],{"className":843},[78,135],"∈",[45,846,69],{"className":847,"style":68},[66,67,135],[45,849,851,854],{"style":850},"top:-3.05em;",[45,852],{"className":853,"style":831},[127],[45,855,856],{},[45,857,861],{"className":858},[813,859,860],"op-symbol","large-op","∑",[45,863,143],{"className":864},[142],[45,866,868],{"className":867},[115],[45,869,872],{"className":870,"style":871},[119],"height:1.3217em;",[45,873],{},[45,875],{"className":876,"style":161},[73],[45,878,686],{"className":879},[66,101],[45,881,277],{"className":882},[93],[45,884,102],{"className":885},[66,101],[45,887,321],{"className":888},[232],[45,890],{"className":891,"style":161},[73],[45,893,252],{"className":894,"style":251},[66,101],[45,896,277],{"className":897},[93],[45,899,102],{"className":900},[66,101],[45,902,904,907],{"className":903},[232],[45,905,321],{"className":906},[232],[45,908,910],{"className":909},[106],[45,911,913],{"className":912},[110],[45,914,916],{"className":915},[115],[45,917,919],{"className":918,"style":749},[119],[45,920,921,924],{"style":764},[45,922],{"className":923,"style":128},[127],[45,925,927],{"className":926},[132,133,134,135],[45,928,774],{"className":929},[66,135],[45,931,933],{"className":932},[66],".",[11,935,936,937,940,941,957],{},"The ",[689,938,939],{},"projection operator"," ",[45,942,944],{"className":943},[48],[45,945,947],{"className":946,"ariaHidden":53},[52],[45,948,950,953],{"className":949},[57],[45,951],{"className":952,"style":62},[61],[45,954,956],{"className":955},[66],"Π"," takes any value function to the representable one\nclosest in this norm:",[45,959,961],{"className":960},[710],[45,962,964],{"className":963},[48],[45,965,967,994,1077,1206,1334],{"className":966,"ariaHidden":53},[52],[45,968,970,973,976,979,982,985,988,991],{"className":969},[57],[45,971],{"className":972,"style":62},[61],[45,974,956],{"className":975},[66],[45,977,252],{"className":978,"style":251},[66,101],[45,980],{"className":981,"style":74},[73],[45,983],{"className":984,"style":74},[73],[45,986,796],{"className":987},[78],[45,989],{"className":990,"style":74},[73],[45,992],{"className":993,"style":74},[73],[45,995,997,1001,1044,1048,1056,1059,1062,1065,1068,1071,1074],{"className":996},[57],[45,998],{"className":999,"style":1000},[61],"height:0.8444em;vertical-align:-0.15em;",[45,1002,1004,1007],{"className":1003},[66],[45,1005,252],{"className":1006,"style":251},[66,101],[45,1008,1010],{"className":1009},[106],[45,1011,1013,1036],{"className":1012},[110,111],[45,1014,1016,1033],{"className":1015},[115],[45,1017,1019],{"className":1018,"style":553},[119],[45,1020,1021,1024],{"style":556},[45,1022],{"className":1023,"style":128},[127],[45,1025,1027],{"className":1026},[132,133,134,135],[45,1028,1030],{"className":1029},[66,135],[45,1031,571],{"className":1032,"style":570},[66,569,135],[45,1034,143],{"className":1035},[142],[45,1037,1039],{"className":1038},[115],[45,1040,1042],{"className":1041,"style":150},[119],[45,1043],{},[45,1045],{"className":1046,"style":1047},[73],"margin-right:1em;",[45,1049,1052],{"className":1050},[66,1051],"text",[45,1053,1055],{"className":1054},[66],"where",[45,1057],{"className":1058,"style":1047},[73],[45,1060,571],{"className":1061,"style":570},[66,569],[45,1063],{"className":1064,"style":74},[73],[45,1066],{"className":1067,"style":74},[73],[45,1069,79],{"className":1070},[78],[45,1072],{"className":1073,"style":74},[73],[45,1075],{"className":1076,"style":74},[73],[45,1078,1080,1084,1188,1191,1194,1198,1203],{"className":1079},[57],[45,1081],{"className":1082,"style":1083},[61],"height:1.7852em;vertical-align:-1.0352em;",[45,1085,1087],{"className":1086},[813,814],[45,1088,1090,1179],{"className":1089},[110,111],[45,1091,1093,1176],{"className":1092},[115],[45,1094,1097,1152],{"className":1095,"style":1096},[119],"height:0.6679em;",[45,1098,1100,1104],{"style":1099},"top:-2.0921em;margin-left:0em;",[45,1101],{"className":1102,"style":1103},[127],"height:3em;",[45,1105,1107],{"className":1106},[132,133,134,135],[45,1108,1110,1113,1116],{"className":1109},[66,135],[45,1111,571],{"className":1112,"style":570},[66,569,135],[45,1114,844],{"className":1115},[78,135],[45,1117,1119,1124],{"className":1118},[66,135],[45,1120,1123],{"className":1121},[66,1122,135],"mathbb","R",[45,1125,1127],{"className":1126},[106],[45,1128,1130],{"className":1129},[110],[45,1131,1133],{"className":1132},[115],[45,1134,1137],{"className":1135,"style":1136},[119],"height:0.782em;",[45,1138,1140,1144],{"style":1139},"top:-2.786em;margin-right:0.0714em;",[45,1141],{"className":1142,"style":1143},[127],"height:2.5em;",[45,1145,1149],{"className":1146},[132,1147,1148,135],"reset-size3","size1",[45,1150,474],{"className":1151},[66,101,135],[45,1153,1155,1158],{"style":1154},"top:-3em;",[45,1156],{"className":1157,"style":1103},[127],[45,1159,1160],{},[45,1161,1163,1169,1172],{"className":1162},[813],[45,1164,1168],{"className":1165,"style":1167},[66,1166],"mathrm","margin-right:0.0139em;","arg",[45,1170],{"className":1171,"style":161},[73],[45,1173,1175],{"className":1174},[66,1166],"min",[45,1177,143],{"className":1178},[142],[45,1180,1182],{"className":1181},[115],[45,1183,1186],{"className":1184,"style":1185},[119],"height:1.0352em;",[45,1187],{},[45,1189,727],{"className":1190},[93],[45,1192,252],{"className":1193,"style":251},[66,101],[45,1195],{"className":1196,"style":1197},[73],"margin-right:0.2222em;",[45,1199,1202],{"className":1200},[1201],"mbin","−",[45,1204],{"className":1205,"style":1197},[73],[45,1207,1209,1212,1255,1306,1309,1313,1316,1319,1322,1325,1328,1331],{"className":1208},[57],[45,1210],{"className":1211,"style":723},[61],[45,1213,1215,1218],{"className":1214},[66],[45,1216,252],{"className":1217,"style":251},[66,101],[45,1219,1221],{"className":1220},[106],[45,1222,1224,1247],{"className":1223},[110,111],[45,1225,1227,1244],{"className":1226},[115],[45,1228,1230],{"className":1229,"style":553},[119],[45,1231,1232,1235],{"style":556},[45,1233],{"className":1234,"style":128},[127],[45,1236,1238],{"className":1237},[132,133,134,135],[45,1239,1241],{"className":1240},[66,135],[45,1242,571],{"className":1243,"style":570},[66,569,135],[45,1245,143],{"className":1246},[142],[45,1248,1250],{"className":1249},[115],[45,1251,1253],{"className":1252,"style":150},[119],[45,1254],{},[45,1256,1258,1261],{"className":1257},[232],[45,1259,727],{"className":1260},[232],[45,1262,1264],{"className":1263},[106],[45,1265,1267,1298],{"className":1266},[110,111],[45,1268,1270,1295],{"className":1269},[115],[45,1271,1273,1284],{"className":1272,"style":749},[119],[45,1274,1275,1278],{"style":752},[45,1276],{"className":1277,"style":128},[127],[45,1279,1281],{"className":1280},[132,133,134,135],[45,1282,686],{"className":1283},[66,101,135],[45,1285,1286,1289],{"style":764},[45,1287],{"className":1288,"style":128},[127],[45,1290,1292],{"className":1291},[132,133,134,135],[45,1293,774],{"className":1294},[66,135],[45,1296,143],{"className":1297},[142],[45,1299,1301],{"className":1300},[115],[45,1302,1304],{"className":1303,"style":784},[119],[45,1305],{},[45,1307,157],{"className":1308},[156],[45,1310],{"className":1311,"style":1312},[73],"margin-right:2em;",[45,1314],{"className":1315,"style":161},[73],[45,1317,956],{"className":1318},[66],[45,1320],{"className":1321,"style":74},[73],[45,1323],{"className":1324,"style":74},[73],[45,1326,796],{"className":1327},[78],[45,1329],{"className":1330,"style":74},[73],[45,1332],{"className":1333,"style":74},[73],[45,1335,1337,1341,1345,1352,1382,1386,1389,1392,1432,1461,1464],{"className":1336},[57],[45,1338],{"className":1339,"style":1340},[61],"height:1.404em;vertical-align:-0.35em;",[45,1342,1344],{"className":1343},[66,569],"X",[45,1346,1348],{"className":1347},[93],[45,1349,277],{"className":1350},[1351,1148],"delimsizing",[45,1353,1355,1358],{"className":1354},[66],[45,1356,1344],{"className":1357},[66,569],[45,1359,1361],{"className":1360},[106],[45,1362,1364],{"className":1363},[110],[45,1365,1367],{"className":1366},[115],[45,1368,1371],{"className":1369,"style":1370},[119],"height:0.8991em;",[45,1372,1373,1376],{"style":764},[45,1374],{"className":1375,"style":128},[127],[45,1377,1379],{"className":1378},[132,133,134,135],[45,1380,430],{"className":1381},[66,135],[45,1383,1385],{"className":1384},[66,569],"D",[45,1387],{"className":1388,"style":161},[73],[45,1390,1344],{"className":1391},[66,569],[45,1393,1395,1401],{"className":1394},[232],[45,1396,1398],{"className":1397},[232],[45,1399,321],{"className":1400},[1351,1148],[45,1402,1404],{"className":1403},[106],[45,1405,1407],{"className":1406},[110],[45,1408,1410],{"className":1409},[115],[45,1411,1414],{"className":1412,"style":1413},[119],"height:1.054em;",[45,1415,1417,1420],{"style":1416},"top:-3.3029em;margin-right:0.05em;",[45,1418],{"className":1419,"style":128},[127],[45,1421,1423],{"className":1422},[132,133,134,135],[45,1424,1426,1429],{"className":1425},[66,135],[45,1427,1202],{"className":1428},[66,135],[45,1430,42],{"className":1431},[66,135],[45,1433,1435,1438],{"className":1434},[66],[45,1436,1344],{"className":1437},[66,569],[45,1439,1441],{"className":1440},[106],[45,1442,1444],{"className":1443},[110],[45,1445,1447],{"className":1446},[115],[45,1448,1450],{"className":1449,"style":1370},[119],[45,1451,1452,1455],{"style":764},[45,1453],{"className":1454,"style":128},[127],[45,1456,1458],{"className":1457},[132,133,134,135],[45,1459,430],{"className":1460},[66,135],[45,1462,1385],{"className":1463},[66,569],[45,1465,157],{"className":1466},[156],[11,1468,1469,1470,1485,1486,1501,1502,1526,1527,1583],{},"where ",[45,1471,1473],{"className":1472},[48],[45,1474,1476],{"className":1475,"ariaHidden":53},[52],[45,1477,1479,1482],{"className":1478},[57],[45,1480],{"className":1481,"style":598},[61],[45,1483,1344],{"className":1484},[66,569]," stacks the feature vectors as rows and ",[45,1487,1489],{"className":1488},[48],[45,1490,1492],{"className":1491,"ariaHidden":53},[52],[45,1493,1495,1498],{"className":1494},[57],[45,1496],{"className":1497,"style":598},[61],[45,1499,1385],{"className":1500},[66,569]," is the diagonal\nmatrix of the weights ",[45,1503,1505],{"className":1504},[48],[45,1506,1508],{"className":1507,"ariaHidden":53},[52],[45,1509,1511,1514,1517,1520,1523],{"className":1510},[57],[45,1512],{"className":1513,"style":444},[61],[45,1515,686],{"className":1516},[66,101],[45,1518,277],{"className":1519},[93],[45,1521,102],{"className":1522},[66,101],[45,1524,321],{"className":1525},[232],". The projection of the true value, ",[45,1528,1530],{"className":1529},[48],[45,1531,1533],{"className":1532,"ariaHidden":53},[52],[45,1534,1536,1540,1543],{"className":1535},[57],[45,1537],{"className":1538,"style":1539},[61],"height:0.8333em;vertical-align:-0.15em;",[45,1541,956],{"className":1542},[66],[45,1544,1546,1549],{"className":1545},[66],[45,1547,252],{"className":1548,"style":251},[66,101],[45,1550,1552],{"className":1551},[106],[45,1553,1555,1575],{"className":1554},[110,111],[45,1556,1558,1572],{"className":1557},[115],[45,1559,1561],{"className":1560,"style":634},[119],[45,1562,1563,1566],{"style":556},[45,1564],{"className":1565,"style":128},[127],[45,1567,1569],{"className":1568},[132,133,134,135],[45,1570,646],{"className":1571,"style":251},[66,101,135],[45,1573,143],{"className":1574},[142],[45,1576,1578],{"className":1577},[115],[45,1579,1581],{"className":1580,"style":150},[119],[45,1582],{},", is the\nbest representable approximation in the value-error sense, and the point Monte Carlo\nmethods converge to.",[1585,1586,1588],"h3",{"id":1587},"bellman-error-versus-projected-bellman-error","Bellman error versus projected Bellman error",[11,1590,1591,1592,940,1595,1650],{},"TD methods aim somewhere else. The ",[689,1593,1594],{},"Bellman operator",[45,1596,1598],{"className":1597},[48],[45,1599,1601],{"className":1600,"ariaHidden":53},[52],[45,1602,1604,1607],{"className":1603},[57],[45,1605],{"className":1606,"style":1539},[61],[45,1608,1610,1615],{"className":1609},[66],[45,1611,1614],{"className":1612,"style":1613},[66,101],"margin-right:0.0502em;","B",[45,1616,1618],{"className":1617},[106],[45,1619,1621,1642],{"className":1620},[110,111],[45,1622,1624,1639],{"className":1623},[115],[45,1625,1627],{"className":1626,"style":634},[119],[45,1628,1630,1633],{"style":1629},"top:-2.55em;margin-left:-0.0502em;margin-right:0.05em;",[45,1631],{"className":1632,"style":128},[127],[45,1634,1636],{"className":1635},[132,133,134,135],[45,1637,646],{"className":1638,"style":251},[66,101,135],[45,1640,143],{"className":1641},[142],[45,1643,1645],{"className":1644},[115],[45,1646,1648],{"className":1647,"style":150},[119],[45,1649],{}," applies one sweep of\nthe Bellman equation:",[45,1652,1654],{"className":1653},[710],[45,1655,1657],{"className":1656},[48],[45,1658,1660,1739,1813,1977,2018],{"className":1659,"ariaHidden":53},[52],[45,1661,1663,1666,1669,1709,1712,1715,1718,1721,1724,1727,1730,1733,1736],{"className":1662},[57],[45,1664],{"className":1665,"style":444},[61],[45,1667,277],{"className":1668},[93],[45,1670,1672,1675],{"className":1671},[66],[45,1673,1614],{"className":1674,"style":1613},[66,101],[45,1676,1678],{"className":1677},[106],[45,1679,1681,1701],{"className":1680},[110,111],[45,1682,1684,1698],{"className":1683},[115],[45,1685,1687],{"className":1686,"style":634},[119],[45,1688,1689,1692],{"style":1629},[45,1690],{"className":1691,"style":128},[127],[45,1693,1695],{"className":1694},[132,133,134,135],[45,1696,646],{"className":1697,"style":251},[66,101,135],[45,1699,143],{"className":1700},[142],[45,1702,1704],{"className":1703},[115],[45,1705,1707],{"className":1706,"style":150},[119],[45,1708],{},[45,1710,252],{"className":1711,"style":251},[66,101],[45,1713,321],{"className":1714},[232],[45,1716,277],{"className":1717},[93],[45,1719,102],{"className":1720},[66,101],[45,1722,321],{"className":1723},[232],[45,1725],{"className":1726,"style":74},[73],[45,1728],{"className":1729,"style":74},[73],[45,1731,796],{"className":1732},[78],[45,1734],{"className":1735,"style":74},[73],[45,1737],{"className":1738,"style":74},[73],[45,1740,1742,1746,1792,1795,1798,1801,1804,1807,1810],{"className":1741},[57],[45,1743],{"className":1744,"style":1745},[61],"height:2.3em;vertical-align:-1.25em;",[45,1747,1749],{"className":1748},[813,814],[45,1750,1752,1783],{"className":1751},[110,111],[45,1753,1755,1780],{"className":1754},[115],[45,1756,1758,1770],{"className":1757,"style":824},[119],[45,1759,1761,1764],{"style":1760},"top:-1.9em;margin-left:0em;",[45,1762],{"className":1763,"style":831},[127],[45,1765,1767],{"className":1766},[132,133,134,135],[45,1768,15],{"className":1769},[66,101,135],[45,1771,1772,1775],{"style":850},[45,1773],{"className":1774,"style":831},[127],[45,1776,1777],{},[45,1778,861],{"className":1779},[813,859,860],[45,1781,143],{"className":1782},[142],[45,1784,1786],{"className":1785},[115],[45,1787,1790],{"className":1788,"style":1789},[119],"height:1.25em;",[45,1791],{},[45,1793],{"className":1794,"style":161},[73],[45,1796,646],{"className":1797,"style":251},[66,101],[45,1799,277],{"className":1800},[93],[45,1802,15],{"className":1803},[66,101],[45,1805],{"className":1806,"style":74},[73],[45,1808,210],{"className":1809},[78],[45,1811],{"className":1812,"style":74},[73],[45,1814,1816,1820,1823,1826,1829,1917,1920,1923,1926,1959,1962,1965,1968,1971,1974],{"className":1815},[57],[45,1817],{"className":1818,"style":1819},[61],"height:2.4801em;vertical-align:-1.4301em;",[45,1821,102],{"className":1822},[66,101],[45,1824,321],{"className":1825},[232],[45,1827],{"className":1828,"style":161},[73],[45,1830,1832],{"className":1831},[813,814],[45,1833,1835,1908],{"className":1834},[110,111],[45,1836,1838,1905],{"className":1837},[115],[45,1839,1841,1895],{"className":1840,"style":824},[119],[45,1842,1844,1847],{"style":1843},"top:-1.856em;margin-left:0em;",[45,1845],{"className":1846,"style":831},[127],[45,1848,1850],{"className":1849},[132,133,134,135],[45,1851,1853,1887,1890],{"className":1852},[66,135],[45,1854,1856,1859],{"className":1855},[66,135],[45,1857,102],{"className":1858},[66,101,135],[45,1860,1862],{"className":1861},[106],[45,1863,1865],{"className":1864},[110],[45,1866,1868],{"className":1867},[115],[45,1869,1872],{"className":1870,"style":1871},[119],"height:0.6828em;",[45,1873,1874,1877],{"style":1139},[45,1875],{"className":1876,"style":1143},[127],[45,1878,1880],{"className":1879},[132,1147,1148,135],[45,1881,1883],{"className":1882},[66,135],[45,1884,1886],{"className":1885},[66,135],"′",[45,1888,157],{"className":1889},[156,135],[45,1891,1894],{"className":1892,"style":1893},[66,101,135],"margin-right:0.0278em;","r",[45,1896,1897,1900],{"style":850},[45,1898],{"className":1899,"style":831},[127],[45,1901,1902],{},[45,1903,861],{"className":1904},[813,859,860],[45,1906,143],{"className":1907},[142],[45,1909,1911],{"className":1910},[115],[45,1912,1915],{"className":1913,"style":1914},[119],"height:1.4301em;",[45,1916],{},[45,1918],{"className":1919,"style":161},[73],[45,1921,11],{"className":1922},[66,101],[45,1924,277],{"className":1925},[93],[45,1927,1929,1932],{"className":1928},[66],[45,1930,102],{"className":1931},[66,101],[45,1933,1935],{"className":1934},[106],[45,1936,1938],{"className":1937},[110],[45,1939,1941],{"className":1940},[115],[45,1942,1945],{"className":1943,"style":1944},[119],"height:0.8019em;",[45,1946,1947,1950],{"style":764},[45,1948],{"className":1949,"style":128},[127],[45,1951,1953],{"className":1952},[132,133,134,135],[45,1954,1956],{"className":1955},[66,135],[45,1957,1886],{"className":1958},[66,135],[45,1960,157],{"className":1961},[156],[45,1963],{"className":1964,"style":161},[73],[45,1966,1894],{"className":1967,"style":1893},[66,101],[45,1969],{"className":1970,"style":74},[73],[45,1972,210],{"className":1973},[78],[45,1975],{"className":1976,"style":74},[73],[45,1978,1980,1984,1987,1990,1993,1996,1999,2005,2008,2011,2015],{"className":1979},[57],[45,1981],{"className":1982,"style":1983},[61],"height:1.2em;vertical-align:-0.35em;",[45,1985,102],{"className":1986},[66,101],[45,1988,157],{"className":1989},[156],[45,1991],{"className":1992,"style":161},[73],[45,1994,15],{"className":1995},[66,101],[45,1997,321],{"className":1998},[232],[45,2000,2002],{"className":2001},[93],[45,2003,270],{"className":2004},[1351,1148],[45,2006,1894],{"className":2007,"style":1893},[66,101],[45,2009],{"className":2010,"style":1197},[73],[45,2012,2014],{"className":2013},[1201],"+",[45,2016],{"className":2017,"style":1197},[73],[45,2019,2021,2024,2029,2032,2035,2038,2070,2073,2079],{"className":2020},[57],[45,2022],{"className":2023,"style":1983},[61],[45,2025,2028],{"className":2026,"style":2027},[66,101],"margin-right:0.0556em;","γ",[45,2030],{"className":2031,"style":161},[73],[45,2033,252],{"className":2034,"style":251},[66,101],[45,2036,277],{"className":2037},[93],[45,2039,2041,2044],{"className":2040},[66],[45,2042,102],{"className":2043},[66,101],[45,2045,2047],{"className":2046},[106],[45,2048,2050],{"className":2049},[110],[45,2051,2053],{"className":2052},[115],[45,2054,2056],{"className":2055,"style":1944},[119],[45,2057,2058,2061],{"style":764},[45,2059],{"className":2060,"style":128},[127],[45,2062,2064],{"className":2063},[132,133,134,135],[45,2065,2067],{"className":2066},[66,135],[45,2068,1886],{"className":2069},[66,135],[45,2071,321],{"className":2072},[232],[45,2074,2076],{"className":2075},[232],[45,2077,404],{"className":2078},[1351,1148],[45,2080,933],{"className":2081},[66],[11,2083,2084,2085,2232,2233,2288,2289,2538,2539,2542,2543,2558,2559,2562],{},"Its unique fixed point is the true value: ",[45,2086,2088],{"className":2087},[48],[45,2089,2091,2146],{"className":2090,"ariaHidden":53},[52],[45,2092,2094,2097,2137,2140,2143],{"className":2093},[57],[45,2095],{"className":2096,"style":534},[61],[45,2098,2100,2103],{"className":2099},[66],[45,2101,252],{"className":2102,"style":251},[66,101],[45,2104,2106],{"className":2105},[106],[45,2107,2109,2129],{"className":2108},[110,111],[45,2110,2112,2126],{"className":2111},[115],[45,2113,2115],{"className":2114,"style":634},[119],[45,2116,2117,2120],{"style":556},[45,2118],{"className":2119,"style":128},[127],[45,2121,2123],{"className":2122},[132,133,134,135],[45,2124,646],{"className":2125,"style":251},[66,101,135],[45,2127,143],{"className":2128},[142],[45,2130,2132],{"className":2131},[115],[45,2133,2135],{"className":2134,"style":150},[119],[45,2136],{},[45,2138],{"className":2139,"style":74},[73],[45,2141,79],{"className":2142},[78],[45,2144],{"className":2145,"style":74},[73],[45,2147,2149,2152,2192],{"className":2148},[57],[45,2150],{"className":2151,"style":1539},[61],[45,2153,2155,2158],{"className":2154},[66],[45,2156,1614],{"className":2157,"style":1613},[66,101],[45,2159,2161],{"className":2160},[106],[45,2162,2164,2184],{"className":2163},[110,111],[45,2165,2167,2181],{"className":2166},[115],[45,2168,2170],{"className":2169,"style":634},[119],[45,2171,2172,2175],{"style":1629},[45,2173],{"className":2174,"style":128},[127],[45,2176,2178],{"className":2177},[132,133,134,135],[45,2179,646],{"className":2180,"style":251},[66,101,135],[45,2182,143],{"className":2183},[142],[45,2185,2187],{"className":2186},[115],[45,2188,2190],{"className":2189,"style":150},[119],[45,2191],{},[45,2193,2195,2198],{"className":2194},[66],[45,2196,252],{"className":2197,"style":251},[66,101],[45,2199,2201],{"className":2200},[106],[45,2202,2204,2224],{"className":2203},[110,111],[45,2205,2207,2221],{"className":2206},[115],[45,2208,2210],{"className":2209,"style":634},[119],[45,2211,2212,2215],{"style":556},[45,2213],{"className":2214,"style":128},[127],[45,2216,2218],{"className":2217},[132,133,134,135],[45,2219,646],{"className":2220,"style":251},[66,101,135],[45,2222,143],{"className":2223},[142],[45,2225,2227],{"className":2226},[115],[45,2228,2230],{"className":2229,"style":150},[119],[45,2231],{},", which is just the\nBellman equation rewritten. For any representable ",[45,2234,2236],{"className":2235},[48],[45,2237,2239],{"className":2238,"ariaHidden":53},[52],[45,2240,2242,2245],{"className":2241},[57],[45,2243],{"className":2244,"style":534},[61],[45,2246,2248,2251],{"className":2247},[66],[45,2249,252],{"className":2250,"style":251},[66,101],[45,2252,2254],{"className":2253},[106],[45,2255,2257,2280],{"className":2256},[110,111],[45,2258,2260,2277],{"className":2259},[115],[45,2261,2263],{"className":2262,"style":553},[119],[45,2264,2265,2268],{"style":556},[45,2266],{"className":2267,"style":128},[127],[45,2269,2271],{"className":2270},[132,133,134,135],[45,2272,2274],{"className":2273},[66,135],[45,2275,571],{"className":2276,"style":570},[66,569,135],[45,2278,143],{"className":2279},[142],[45,2281,2283],{"className":2282},[115],[45,2284,2286],{"className":2285,"style":150},[119],[45,2287],{},", the difference\n",[45,2290,2292],{"className":2291},[48],[45,2293,2295,2391,2489],{"className":2294,"ariaHidden":53},[52],[45,2296,2298,2302,2382,2385,2388],{"className":2297},[57],[45,2299],{"className":2300,"style":2301},[61],"height:0.9812em;vertical-align:-0.15em;",[45,2303,2305,2344],{"className":2304},[66],[45,2306,2309],{"className":2307},[66,2308],"accent",[45,2310,2312],{"className":2311},[110],[45,2313,2315],{"className":2314},[115],[45,2316,2319,2329],{"className":2317,"style":2318},[119],"height:0.8312em;",[45,2320,2321,2324],{"style":1154},[45,2322],{"className":2323,"style":1103},[127],[45,2325,2328],{"className":2326,"style":2327},[66,101],"margin-right:0.0379em;","δ",[45,2330,2332,2335],{"style":2331},"top:-3.2634em;",[45,2333],{"className":2334,"style":1103},[127],[45,2336,2340],{"className":2337,"style":2339},[2338],"accent-body","left:-0.1944em;",[45,2341,2343],{"className":2342},[66],"ˉ",[45,2345,2347],{"className":2346},[106],[45,2348,2350,2374],{"className":2349},[110,111],[45,2351,2353,2371],{"className":2352},[115],[45,2354,2356],{"className":2355,"style":553},[119],[45,2357,2359,2362],{"style":2358},"top:-2.55em;margin-left:-0.0379em;margin-right:0.05em;",[45,2360],{"className":2361,"style":128},[127],[45,2363,2365],{"className":2364},[132,133,134,135],[45,2366,2368],{"className":2367},[66,135],[45,2369,571],{"className":2370,"style":570},[66,569,135],[45,2372,143],{"className":2373},[142],[45,2375,2377],{"className":2376},[115],[45,2378,2380],{"className":2379,"style":150},[119],[45,2381],{},[45,2383],{"className":2384,"style":74},[73],[45,2386,79],{"className":2387},[78],[45,2389],{"className":2390,"style":74},[73],[45,2392,2394,2397,2437,2480,2483,2486],{"className":2393},[57],[45,2395],{"className":2396,"style":1539},[61],[45,2398,2400,2403],{"className":2399},[66],[45,2401,1614],{"className":2402,"style":1613},[66,101],[45,2404,2406],{"className":2405},[106],[45,2407,2409,2429],{"className":2408},[110,111],[45,2410,2412,2426],{"className":2411},[115],[45,2413,2415],{"className":2414,"style":634},[119],[45,2416,2417,2420],{"style":1629},[45,2418],{"className":2419,"style":128},[127],[45,2421,2423],{"className":2422},[132,133,134,135],[45,2424,646],{"className":2425,"style":251},[66,101,135],[45,2427,143],{"className":2428},[142],[45,2430,2432],{"className":2431},[115],[45,2433,2435],{"className":2434,"style":150},[119],[45,2436],{},[45,2438,2440,2443],{"className":2439},[66],[45,2441,252],{"className":2442,"style":251},[66,101],[45,2444,2446],{"className":2445},[106],[45,2447,2449,2472],{"className":2448},[110,111],[45,2450,2452,2469],{"className":2451},[115],[45,2453,2455],{"className":2454,"style":553},[119],[45,2456,2457,2460],{"style":556},[45,2458],{"className":2459,"style":128},[127],[45,2461,2463],{"className":2462},[132,133,134,135],[45,2464,2466],{"className":2465},[66,135],[45,2467,571],{"className":2468,"style":570},[66,569,135],[45,2470,143],{"className":2471},[142],[45,2473,2475],{"className":2474},[115],[45,2476,2478],{"className":2477,"style":150},[119],[45,2479],{},[45,2481],{"className":2482,"style":1197},[73],[45,2484,1202],{"className":2485},[1201],[45,2487],{"className":2488,"style":1197},[73],[45,2490,2492,2495],{"className":2491},[57],[45,2493],{"className":2494,"style":534},[61],[45,2496,2498,2501],{"className":2497},[66],[45,2499,252],{"className":2500,"style":251},[66,101],[45,2502,2504],{"className":2503},[106],[45,2505,2507,2530],{"className":2506},[110,111],[45,2508,2510,2527],{"className":2509},[115],[45,2511,2513],{"className":2512,"style":553},[119],[45,2514,2515,2518],{"style":556},[45,2516],{"className":2517,"style":128},[127],[45,2519,2521],{"className":2520},[132,133,134,135],[45,2522,2524],{"className":2523},[66,135],[45,2525,571],{"className":2526,"style":570},[66,569,135],[45,2528,143],{"className":2529},[142],[45,2531,2533],{"className":2532},[115],[45,2534,2536],{"className":2535,"style":150},[119],[45,2537],{}," is the ",[689,2540,2541],{},"Bellman error\nvector"," — the expectation of the TD error at each state — and its squared ",[45,2544,2546],{"className":2545},[48],[45,2547,2549],{"className":2548,"ariaHidden":53},[52],[45,2550,2552,2555],{"className":2551},[57],[45,2553],{"className":2554,"style":682},[61],[45,2556,686],{"className":2557},[66,101],"-norm is\nthe ",[689,2560,2561],{},"mean square Bellman error",":",[45,2564,2566],{"className":2565},[710],[45,2567,2569],{"className":2568},[48],[45,2570,2572,2644],{"className":2571,"ariaHidden":53},[52],[45,2573,2575,2579,2620,2623,2626,2629,2632,2635,2638,2641],{"className":2574},[57],[45,2576],{"className":2577,"style":2578},[61],"height:1.1333em;vertical-align:-0.25em;",[45,2580,2583],{"className":2581},[66,2582],"overline",[45,2584,2586],{"className":2585},[110],[45,2587,2589],{"className":2588},[115],[45,2590,2593,2609],{"className":2591,"style":2592},[119],"height:0.8833em;",[45,2594,2595,2598],{"style":1154},[45,2596],{"className":2597,"style":1103},[127],[45,2599,2601,2604],{"className":2600},[66],[45,2602,1614],{"className":2603,"style":1613},[66,101],[45,2605,2608],{"className":2606,"style":2607},[66,101],"margin-right:0.0576em;","E",[45,2610,2612,2615],{"style":2611},"top:-3.8033em;",[45,2613],{"className":2614,"style":1103},[127],[45,2616],{"className":2617,"style":2619},[2618],"overline-line","border-bottom-width:0.04em;",[45,2621,277],{"className":2622},[93],[45,2624,571],{"className":2625,"style":570},[66,569],[45,2627,321],{"className":2628},[232],[45,2630],{"className":2631,"style":74},[73],[45,2633],{"className":2634,"style":74},[73],[45,2636,79],{"className":2637},[78],[45,2639],{"className":2640,"style":74},[73],[45,2642],{"className":2643,"style":74},[73],[45,2645,2647,2650,2653,2724,2775],{"className":2646},[57],[45,2648],{"className":2649,"style":723},[61],[45,2651,727],{"className":2652},[93],[45,2654,2656,2687],{"className":2655},[66],[45,2657,2659],{"className":2658},[66,2308],[45,2660,2662],{"className":2661},[110],[45,2663,2665],{"className":2664},[115],[45,2666,2668,2676],{"className":2667,"style":2318},[119],[45,2669,2670,2673],{"style":1154},[45,2671],{"className":2672,"style":1103},[127],[45,2674,2328],{"className":2675,"style":2327},[66,101],[45,2677,2678,2681],{"style":2331},[45,2679],{"className":2680,"style":1103},[127],[45,2682,2684],{"className":2683,"style":2339},[2338],[45,2685,2343],{"className":2686},[66],[45,2688,2690],{"className":2689},[106],[45,2691,2693,2716],{"className":2692},[110,111],[45,2694,2696,2713],{"className":2695},[115],[45,2697,2699],{"className":2698,"style":553},[119],[45,2700,2701,2704],{"style":2358},[45,2702],{"className":2703,"style":128},[127],[45,2705,2707],{"className":2706},[132,133,134,135],[45,2708,2710],{"className":2709},[66,135],[45,2711,571],{"className":2712,"style":570},[66,569,135],[45,2714,143],{"className":2715},[142],[45,2717,2719],{"className":2718},[115],[45,2720,2722],{"className":2721,"style":150},[119],[45,2723],{},[45,2725,2727,2730],{"className":2726},[232],[45,2728,727],{"className":2729},[232],[45,2731,2733],{"className":2732},[106],[45,2734,2736,2767],{"className":2735},[110,111],[45,2737,2739,2764],{"className":2738},[115],[45,2740,2742,2753],{"className":2741,"style":749},[119],[45,2743,2744,2747],{"style":752},[45,2745],{"className":2746,"style":128},[127],[45,2748,2750],{"className":2749},[132,133,134,135],[45,2751,686],{"className":2752},[66,101,135],[45,2754,2755,2758],{"style":764},[45,2756],{"className":2757,"style":128},[127],[45,2759,2761],{"className":2760},[132,133,134,135],[45,2762,774],{"className":2763},[66,135],[45,2765,143],{"className":2766},[142],[45,2768,2770],{"className":2769},[115],[45,2771,2773],{"className":2772,"style":784},[119],[45,2774],{},[45,2776,933],{"className":2777},[66],[11,2779,2780,2781,2833,2834,2836,2837,2889,2890,2942,2943,2946,2947,2950,2951,2562],{},"Here is the geometric crux. Applying ",[45,2782,2784],{"className":2783},[48],[45,2785,2787],{"className":2786,"ariaHidden":53},[52],[45,2788,2790,2793],{"className":2789},[57],[45,2791],{"className":2792,"style":1539},[61],[45,2794,2796,2799],{"className":2795},[66],[45,2797,1614],{"className":2798,"style":1613},[66,101],[45,2800,2802],{"className":2801},[106],[45,2803,2805,2825],{"className":2804},[110,111],[45,2806,2808,2822],{"className":2807},[115],[45,2809,2811],{"className":2810,"style":634},[119],[45,2812,2813,2816],{"style":1629},[45,2814],{"className":2815,"style":128},[127],[45,2817,2819],{"className":2818},[132,133,134,135],[45,2820,646],{"className":2821,"style":251},[66,101,135],[45,2823,143],{"className":2824},[142],[45,2826,2828],{"className":2827},[115],[45,2829,2831],{"className":2830,"style":150},[119],[45,2832],{}," to a representable value function generally\nproduces one ",[29,2835,661],{}," the subspace (the gray arrow leaving the plane in the figure). In\npure dynamic programming, without approximation, you follow ",[45,2838,2840],{"className":2839},[48],[45,2841,2843],{"className":2842,"ariaHidden":53},[52],[45,2844,2846,2849],{"className":2845},[57],[45,2847],{"className":2848,"style":1539},[61],[45,2850,2852,2855],{"className":2851},[66],[45,2853,1614],{"className":2854,"style":1613},[66,101],[45,2856,2858],{"className":2857},[106],[45,2859,2861,2881],{"className":2860},[110,111],[45,2862,2864,2878],{"className":2863},[115],[45,2865,2867],{"className":2866,"style":634},[119],[45,2868,2869,2872],{"style":1629},[45,2870],{"className":2871,"style":128},[127],[45,2873,2875],{"className":2874},[132,133,134,135],[45,2876,646],{"className":2877,"style":251},[66,101,135],[45,2879,143],{"className":2880},[142],[45,2882,2884],{"className":2883},[115],[45,2885,2887],{"className":2886,"style":150},[119],[45,2888],{}," repeatedly and\nconverge to ",[45,2891,2893],{"className":2892},[48],[45,2894,2896],{"className":2895,"ariaHidden":53},[52],[45,2897,2899,2902],{"className":2898},[57],[45,2900],{"className":2901,"style":534},[61],[45,2903,2905,2908],{"className":2904},[66],[45,2906,252],{"className":2907,"style":251},[66,101],[45,2909,2911],{"className":2910},[106],[45,2912,2914,2934],{"className":2913},[110,111],[45,2915,2917,2931],{"className":2916},[115],[45,2918,2920],{"className":2919,"style":634},[119],[45,2921,2922,2925],{"style":556},[45,2923],{"className":2924,"style":128},[127],[45,2926,2928],{"className":2927},[132,133,134,135],[45,2929,646],{"className":2930,"style":251},[66,101,135],[45,2932,143],{"className":2933},[142],[45,2935,2937],{"className":2936},[115],[45,2938,2940],{"className":2939,"style":150},[119],[45,2941],{},". With approximation you cannot: after one Bellman step you must\n",[29,2944,2945],{},"project"," back into the subspace to have a representable function, then step out again,\nthen project back. The quantity TD methods drive to zero is the residual ",[29,2948,2949],{},"inside"," the\nsubspace — the ",[689,2952,2953],{},"projected Bellman error",[45,2955,2957],{"className":2956},[710],[45,2958,2960],{"className":2959},[48],[45,2961,2963,3032],{"className":2962,"ariaHidden":53},[52],[45,2964,2966,2969,3008,3011,3014,3017,3020,3023,3026,3029],{"className":2965},[57],[45,2967],{"className":2968,"style":2578},[61],[45,2970,2972],{"className":2971},[66,2582],[45,2973,2975],{"className":2974},[110],[45,2976,2978],{"className":2977},[115],[45,2979,2981,3000],{"className":2980,"style":2592},[119],[45,2982,2983,2986],{"style":1154},[45,2984],{"className":2985,"style":1103},[127],[45,2987,2989,2994,2997],{"className":2988},[66],[45,2990,2993],{"className":2991,"style":2992},[66,101],"margin-right:0.1389em;","P",[45,2995,1614],{"className":2996,"style":1613},[66,101],[45,2998,2608],{"className":2999,"style":2607},[66,101],[45,3001,3002,3005],{"style":2611},[45,3003],{"className":3004,"style":1103},[127],[45,3006],{"className":3007,"style":2619},[2618],[45,3009,277],{"className":3010},[93],[45,3012,571],{"className":3013,"style":570},[66,569],[45,3015,321],{"className":3016},[232],[45,3018],{"className":3019,"style":74},[73],[45,3021],{"className":3022,"style":74},[73],[45,3024,79],{"className":3025},[78],[45,3027],{"className":3028,"style":74},[73],[45,3030],{"className":3031,"style":74},[73],[45,3033,3035,3038,3041,3044,3047,3118,3169],{"className":3034},[57],[45,3036],{"className":3037,"style":723},[61],[45,3039,727],{"className":3040},[93],[45,3042,956],{"className":3043},[66],[45,3045],{"className":3046,"style":161},[73],[45,3048,3050,3081],{"className":3049},[66],[45,3051,3053],{"className":3052},[66,2308],[45,3054,3056],{"className":3055},[110],[45,3057,3059],{"className":3058},[115],[45,3060,3062,3070],{"className":3061,"style":2318},[119],[45,3063,3064,3067],{"style":1154},[45,3065],{"className":3066,"style":1103},[127],[45,3068,2328],{"className":3069,"style":2327},[66,101],[45,3071,3072,3075],{"style":2331},[45,3073],{"className":3074,"style":1103},[127],[45,3076,3078],{"className":3077,"style":2339},[2338],[45,3079,2343],{"className":3080},[66],[45,3082,3084],{"className":3083},[106],[45,3085,3087,3110],{"className":3086},[110,111],[45,3088,3090,3107],{"className":3089},[115],[45,3091,3093],{"className":3092,"style":553},[119],[45,3094,3095,3098],{"style":2358},[45,3096],{"className":3097,"style":128},[127],[45,3099,3101],{"className":3100},[132,133,134,135],[45,3102,3104],{"className":3103},[66,135],[45,3105,571],{"className":3106,"style":570},[66,569,135],[45,3108,143],{"className":3109},[142],[45,3111,3113],{"className":3112},[115],[45,3114,3116],{"className":3115,"style":150},[119],[45,3117],{},[45,3119,3121,3124],{"className":3120},[232],[45,3122,727],{"className":3123},[232],[45,3125,3127],{"className":3126},[106],[45,3128,3130,3161],{"className":3129},[110,111],[45,3131,3133,3158],{"className":3132},[115],[45,3134,3136,3147],{"className":3135,"style":749},[119],[45,3137,3138,3141],{"style":752},[45,3139],{"className":3140,"style":128},[127],[45,3142,3144],{"className":3143},[132,133,134,135],[45,3145,686],{"className":3146},[66,101,135],[45,3148,3149,3152],{"style":764},[45,3150],{"className":3151,"style":128},[127],[45,3153,3155],{"className":3154},[132,133,134,135],[45,3156,774],{"className":3157},[66,135],[45,3159,143],{"className":3160},[142],[45,3162,3164],{"className":3163},[115],[45,3165,3167],{"className":3166,"style":784},[119],[45,3168],{},[45,3170,933],{"className":3171},[66],[3173,3174,3176],"callout",{"type":3175},"definition",[11,3177,3178,3231,3232,3247,3248,3251,3252,3397,3398,3460,3461,3510,3511,3557,3558,3605],{},[689,3179,3180,3181,3230],{},"Definition (Projected Bellman error, ",[45,3182,3184],{"className":3183},[48],[45,3185,3187],{"className":3186,"ariaHidden":53},[52],[45,3188,3190,3193],{"className":3189},[57],[45,3191],{"className":3192,"style":2592},[61],[45,3194,3196],{"className":3195},[66,2582],[45,3197,3199],{"className":3198},[110],[45,3200,3202],{"className":3201},[115],[45,3203,3205,3222],{"className":3204,"style":2592},[119],[45,3206,3207,3210],{"style":1154},[45,3208],{"className":3209,"style":1103},[127],[45,3211,3213,3216,3219],{"className":3212},[66],[45,3214,2993],{"className":3215,"style":2992},[66,101],[45,3217,1614],{"className":3218,"style":1613},[66,101],[45,3220,2608],{"className":3221,"style":2607},[66,101],[45,3223,3224,3227],{"style":2611},[45,3225],{"className":3226,"style":1103},[127],[45,3228],{"className":3229,"style":2619},[2618],")."," The squared ",[45,3233,3235],{"className":3234},[48],[45,3236,3238],{"className":3237,"ariaHidden":53},[52],[45,3239,3241,3244],{"className":3240},[57],[45,3242],{"className":3243,"style":682},[61],[45,3245,686],{"className":3246},[66,101],"-norm of\nthe Bellman error vector ",[29,3249,3250],{},"after projecting it back into the representable subspace",",\n",[45,3253,3255],{"className":3254},[48],[45,3256,3258],{"className":3257,"ariaHidden":53},[52],[45,3259,3261,3265,3268,3271,3274,3345],{"className":3260},[57],[45,3262],{"className":3263,"style":3264},[61],"height:1.2143em;vertical-align:-0.3831em;",[45,3266,727],{"className":3267},[93],[45,3269,956],{"className":3270},[66],[45,3272],{"className":3273,"style":161},[73],[45,3275,3277,3308],{"className":3276},[66],[45,3278,3280],{"className":3279},[66,2308],[45,3281,3283],{"className":3282},[110],[45,3284,3286],{"className":3285},[115],[45,3287,3289,3297],{"className":3288,"style":2318},[119],[45,3290,3291,3294],{"style":1154},[45,3292],{"className":3293,"style":1103},[127],[45,3295,2328],{"className":3296,"style":2327},[66,101],[45,3298,3299,3302],{"style":2331},[45,3300],{"className":3301,"style":1103},[127],[45,3303,3305],{"className":3304,"style":2339},[2338],[45,3306,2343],{"className":3307},[66],[45,3309,3311],{"className":3310},[106],[45,3312,3314,3337],{"className":3313},[110,111],[45,3315,3317,3334],{"className":3316},[115],[45,3318,3320],{"className":3319,"style":553},[119],[45,3321,3322,3325],{"style":2358},[45,3323],{"className":3324,"style":128},[127],[45,3326,3328],{"className":3327},[132,133,134,135],[45,3329,3331],{"className":3330},[66,135],[45,3332,571],{"className":3333,"style":570},[66,569,135],[45,3335,143],{"className":3336},[142],[45,3338,3340],{"className":3339},[115],[45,3341,3343],{"className":3342,"style":150},[119],[45,3344],{},[45,3346,3348,3351],{"className":3347},[232],[45,3349,727],{"className":3350},[232],[45,3352,3354],{"className":3353},[106],[45,3355,3357,3389],{"className":3356},[110,111],[45,3358,3360,3386],{"className":3359},[115],[45,3361,3364,3375],{"className":3362,"style":3363},[119],"height:0.8141em;",[45,3365,3366,3369],{"style":752},[45,3367],{"className":3368,"style":128},[127],[45,3370,3372],{"className":3371},[132,133,134,135],[45,3373,686],{"className":3374},[66,101,135],[45,3376,3377,3380],{"style":420},[45,3378],{"className":3379,"style":128},[127],[45,3381,3383],{"className":3382},[132,133,134,135],[45,3384,774],{"className":3385},[66,135],[45,3387,143],{"className":3388},[142],[45,3390,3392],{"className":3391},[115],[45,3393,3395],{"className":3394,"style":784},[119],[45,3396],{},". With linear approximation there is\nalways a weight vector ",[45,3399,3401],{"className":3400},[48],[45,3402,3404],{"className":3403,"ariaHidden":53},[52],[45,3405,3407,3411],{"className":3406},[57],[45,3408],{"className":3409,"style":3410},[61],"height:0.5944em;vertical-align:-0.15em;",[45,3412,3414,3417],{"className":3413},[66],[45,3415,571],{"className":3416,"style":570},[66,569],[45,3418,3420],{"className":3419},[106],[45,3421,3423,3452],{"className":3422},[110,111],[45,3424,3426,3449],{"className":3425},[115],[45,3427,3430],{"className":3428,"style":3429},[119],"height:0.3283em;",[45,3431,3433,3436],{"style":3432},"top:-2.55em;margin-left:-0.016em;margin-right:0.05em;",[45,3434],{"className":3435,"style":128},[127],[45,3437,3439],{"className":3438},[132,133,134,135],[45,3440,3442],{"className":3441},[66,135],[45,3443,3445],{"className":3444},[66,1051,135],[45,3446,3448],{"className":3447},[66,135],"TD",[45,3450,143],{"className":3451},[142],[45,3453,3455],{"className":3454},[115],[45,3456,3458],{"className":3457,"style":150},[119],[45,3459],{}," making it zero — the TD fixed point.\nMinimizing ",[45,3462,3464],{"className":3463},[48],[45,3465,3467],{"className":3466,"ariaHidden":53},[52],[45,3468,3470,3473],{"className":3469},[57],[45,3471],{"className":3472,"style":2592},[61],[45,3474,3476],{"className":3475},[66,2582],[45,3477,3479],{"className":3478},[110],[45,3480,3482],{"className":3481},[115],[45,3483,3485,3502],{"className":3484,"style":2592},[119],[45,3486,3487,3490],{"style":1154},[45,3488],{"className":3489,"style":1103},[127],[45,3491,3493,3496,3499],{"className":3492},[66],[45,3494,2993],{"className":3495,"style":2992},[66,101],[45,3497,1614],{"className":3498,"style":1613},[66,101],[45,3500,2608],{"className":3501,"style":2607},[66,101],[45,3503,3504,3507],{"style":2611},[45,3505],{"className":3506,"style":1103},[127],[45,3508],{"className":3509,"style":2619},[2618]," (rather than ",[45,3512,3514],{"className":3513},[48],[45,3515,3517],{"className":3516,"ariaHidden":53},[52],[45,3518,3520,3523],{"className":3519},[57],[45,3521],{"className":3522,"style":2592},[61],[45,3524,3526],{"className":3525},[66,2582],[45,3527,3529],{"className":3528},[110],[45,3530,3532],{"className":3531},[115],[45,3533,3535,3549],{"className":3534,"style":2592},[119],[45,3536,3537,3540],{"style":1154},[45,3538],{"className":3539,"style":1103},[127],[45,3541,3543,3546],{"className":3542},[66],[45,3544,1614],{"className":3545,"style":1613},[66,101],[45,3547,2608],{"className":3548,"style":2607},[66,101],[45,3550,3551,3554],{"style":2611},[45,3552],{"className":3553,"style":1103},[127],[45,3555],{"className":3556,"style":2619},[2618]," or ",[45,3559,3561],{"className":3560},[48],[45,3562,3564],{"className":3563,"ariaHidden":53},[52],[45,3565,3567,3570],{"className":3566},[57],[45,3568],{"className":3569,"style":2592},[61],[45,3571,3573],{"className":3572},[66,2582],[45,3574,3576],{"className":3575},[110],[45,3577,3579],{"className":3578},[115],[45,3580,3582,3597],{"className":3581,"style":2592},[119],[45,3583,3584,3587],{"style":1154},[45,3585],{"className":3586,"style":1103},[127],[45,3588,3590,3594],{"className":3589},[66],[45,3591,3593],{"className":3592,"style":1197},[66,101],"V",[45,3595,2608],{"className":3596,"style":2607},[66,101],[45,3598,3599,3602],{"style":2611},[45,3600],{"className":3601,"style":1103},[127],[45,3603],{"className":3604,"style":2619},[2618],") is the\ntarget of stable off-policy methods.",[11,3607,3608,3609,3612,3613,3659,3660,3715,3716,3762,3763,3812,3813,3871,3872,3930],{},"The three objectives generally have ",[29,3610,3611],{},"different"," minimizers: ",[45,3614,3616],{"className":3615},[48],[45,3617,3619],{"className":3618,"ariaHidden":53},[52],[45,3620,3622,3625],{"className":3621},[57],[45,3623],{"className":3624,"style":2592},[61],[45,3626,3628],{"className":3627},[66,2582],[45,3629,3631],{"className":3630},[110],[45,3632,3634],{"className":3633},[115],[45,3635,3637,3651],{"className":3636,"style":2592},[119],[45,3638,3639,3642],{"style":1154},[45,3640],{"className":3641,"style":1103},[127],[45,3643,3645,3648],{"className":3644},[66],[45,3646,3593],{"className":3647,"style":1197},[66,101],[45,3649,2608],{"className":3650,"style":2607},[66,101],[45,3652,3653,3656],{"style":2611},[45,3654],{"className":3655,"style":1103},[127],[45,3657],{"className":3658,"style":2619},[2618]," is minimized\nat ",[45,3661,3663],{"className":3662},[48],[45,3664,3666],{"className":3665,"ariaHidden":53},[52],[45,3667,3669,3672,3675],{"className":3668},[57],[45,3670],{"className":3671,"style":1539},[61],[45,3673,956],{"className":3674},[66],[45,3676,3678,3681],{"className":3677},[66],[45,3679,252],{"className":3680,"style":251},[66,101],[45,3682,3684],{"className":3683},[106],[45,3685,3687,3707],{"className":3686},[110,111],[45,3688,3690,3704],{"className":3689},[115],[45,3691,3693],{"className":3692,"style":634},[119],[45,3694,3695,3698],{"style":556},[45,3696],{"className":3697,"style":128},[127],[45,3699,3701],{"className":3700},[132,133,134,135],[45,3702,646],{"className":3703,"style":251},[66,101,135],[45,3705,143],{"className":3706},[142],[45,3708,3710],{"className":3709},[115],[45,3711,3713],{"className":3712,"style":150},[119],[45,3714],{},", ",[45,3717,3719],{"className":3718},[48],[45,3720,3722],{"className":3721,"ariaHidden":53},[52],[45,3723,3725,3728],{"className":3724},[57],[45,3726],{"className":3727,"style":2592},[61],[45,3729,3731],{"className":3730},[66,2582],[45,3732,3734],{"className":3733},[110],[45,3735,3737],{"className":3736},[115],[45,3738,3740,3754],{"className":3739,"style":2592},[119],[45,3741,3742,3745],{"style":1154},[45,3743],{"className":3744,"style":1103},[127],[45,3746,3748,3751],{"className":3747},[66],[45,3749,1614],{"className":3750,"style":1613},[66,101],[45,3752,2608],{"className":3753,"style":2607},[66,101],[45,3755,3756,3759],{"style":2611},[45,3757],{"className":3758,"style":1103},[127],[45,3760],{"className":3761,"style":2619},[2618]," at some other point, and ",[45,3764,3766],{"className":3765},[48],[45,3767,3769],{"className":3768,"ariaHidden":53},[52],[45,3770,3772,3775],{"className":3771},[57],[45,3773],{"className":3774,"style":2592},[61],[45,3776,3778],{"className":3777},[66,2582],[45,3779,3781],{"className":3780},[110],[45,3782,3784],{"className":3783},[115],[45,3785,3787,3804],{"className":3786,"style":2592},[119],[45,3788,3789,3792],{"style":1154},[45,3790],{"className":3791,"style":1103},[127],[45,3793,3795,3798,3801],{"className":3794},[66],[45,3796,2993],{"className":3797,"style":2992},[66,101],[45,3799,1614],{"className":3800,"style":1613},[66,101],[45,3802,2608],{"className":3803,"style":2607},[66,101],[45,3805,3806,3809],{"style":2611},[45,3807],{"className":3808,"style":1103},[127],[45,3810],{"className":3811,"style":2619},[2618]," at the TD fixed\npoint ",[45,3814,3816],{"className":3815},[48],[45,3817,3819],{"className":3818,"ariaHidden":53},[52],[45,3820,3822,3825],{"className":3821},[57],[45,3823],{"className":3824,"style":3410},[61],[45,3826,3828,3831],{"className":3827},[66],[45,3829,571],{"className":3830,"style":570},[66,569],[45,3832,3834],{"className":3833},[106],[45,3835,3837,3863],{"className":3836},[110,111],[45,3838,3840,3860],{"className":3839},[115],[45,3841,3843],{"className":3842,"style":3429},[119],[45,3844,3845,3848],{"style":3432},[45,3846],{"className":3847,"style":128},[127],[45,3849,3851],{"className":3850},[132,133,134,135],[45,3852,3854],{"className":3853},[66,135],[45,3855,3857],{"className":3856},[66,1051,135],[45,3858,3448],{"className":3859},[66,135],[45,3861,143],{"className":3862},[142],[45,3864,3866],{"className":3865},[115],[45,3867,3869],{"className":3868,"style":150},[119],[45,3870],{},". Semi-gradient TD, when it converges, converges to\n",[45,3873,3875],{"className":3874},[48],[45,3876,3878],{"className":3877,"ariaHidden":53},[52],[45,3879,3881,3884],{"className":3880},[57],[45,3882],{"className":3883,"style":3410},[61],[45,3885,3887,3890],{"className":3886},[66],[45,3888,571],{"className":3889,"style":570},[66,569],[45,3891,3893],{"className":3892},[106],[45,3894,3896,3922],{"className":3895},[110,111],[45,3897,3899,3919],{"className":3898},[115],[45,3900,3902],{"className":3901,"style":3429},[119],[45,3903,3904,3907],{"style":3432},[45,3905],{"className":3906,"style":128},[127],[45,3908,3910],{"className":3909},[132,133,134,135],[45,3911,3913],{"className":3912},[66,135],[45,3914,3916],{"className":3915},[66,1051,135],[45,3917,3448],{"className":3918},[66,135],[45,3920,143],{"className":3921},[142],[45,3923,3925],{"className":3924},[115],[45,3926,3928],{"className":3927,"style":150},[119],[45,3929],{}," — but under off-policy training it need not converge at all.",[11,3932,3933,3934,3937,3938,3941,3942,3988,3989,3992,3993,4039,4040,4047,4048,4097,4098,4101],{},"A tempting fix is to do true gradient descent on the Bellman error itself (the\n",[689,3935,3936],{},"residual-gradient"," algorithm), which is a genuine SGD method and converges. But two\nresults rule it out as a general approach. First, on the A-split example it converges to the\n",[29,3939,3940],{},"wrong"," values, minimizing a temporal-smoothing surrogate rather than accuracy. Second,\nand more fundamentally, the ",[45,3943,3945],{"className":3944},[48],[45,3946,3948],{"className":3947,"ariaHidden":53},[52],[45,3949,3951,3954],{"className":3950},[57],[45,3952],{"className":3953,"style":2592},[61],[45,3955,3957],{"className":3956},[66,2582],[45,3958,3960],{"className":3959},[110],[45,3961,3963],{"className":3962},[115],[45,3964,3966,3980],{"className":3965,"style":2592},[119],[45,3967,3968,3971],{"style":1154},[45,3969],{"className":3970,"style":1103},[127],[45,3972,3974,3977],{"className":3973},[66],[45,3975,1614],{"className":3976,"style":1613},[66,101],[45,3978,2608],{"className":3979,"style":2607},[66,101],[45,3981,3982,3985],{"style":2611},[45,3983],{"className":3984,"style":1103},[127],[45,3986],{"className":3987,"style":2619},[2618]," is ",[689,3990,3991],{},"not learnable",": two Markov reward\nprocesses can produce identical data yet have different ",[45,3994,3996],{"className":3995},[48],[45,3997,3999],{"className":3998,"ariaHidden":53},[52],[45,4000,4002,4005],{"className":4001},[57],[45,4003],{"className":4004,"style":2592},[61],[45,4006,4008],{"className":4007},[66,2582],[45,4009,4011],{"className":4010},[110],[45,4012,4014],{"className":4013},[115],[45,4015,4017,4031],{"className":4016,"style":2592},[119],[45,4018,4019,4022],{"style":1154},[45,4020],{"className":4021,"style":1103},[127],[45,4023,4025,4028],{"className":4024},[66],[45,4026,1614],{"className":4027,"style":1613},[66,101],[45,4029,2608],{"className":4030,"style":2607},[66,101],[45,4032,4033,4036],{"style":2611},[45,4034],{"className":4035,"style":1103},[127],[45,4037],{"className":4038,"style":2619},[2618],"-minimizing weight\nvectors, so no algorithm working from feature vectors, actions, and rewards alone can\nminimize it.",[34,4041,4042],{},[15,4043,774],{"href":4044,"ariaDescribedBy":4045,"dataFootnoteRef":6,"id":4046},"#user-content-fn-sb-be",[40],"user-content-fnref-sb-be"," That leaves the ",[45,4049,4051],{"className":4050},[48],[45,4052,4054],{"className":4053,"ariaHidden":53},[52],[45,4055,4057,4060],{"className":4056},[57],[45,4058],{"className":4059,"style":2592},[61],[45,4061,4063],{"className":4062},[66,2582],[45,4064,4066],{"className":4065},[110],[45,4067,4069],{"className":4068},[115],[45,4070,4072,4089],{"className":4071,"style":2592},[119],[45,4073,4074,4077],{"style":1154},[45,4075],{"className":4076,"style":1103},[127],[45,4078,4080,4083,4086],{"className":4079},[66],[45,4081,2993],{"className":4082,"style":2992},[66,101],[45,4084,1614],{"className":4085,"style":1613},[66,101],[45,4087,2608],{"className":4088,"style":2607},[66,101],[45,4090,4091,4094],{"style":2611},[45,4092],{"className":4093,"style":1103},[127],[45,4095],{"className":4096,"style":2619},[2618],", which ",[29,4099,4100],{},"is"," learnable, as the\nobjective worth pursuing.",[21,4103,4105],{"id":4104},"gradient-td-methods","Gradient-TD methods",[11,4107,4108,4110,4111,933,4160,4168,4169,4194],{},[689,4109,4105],{}," perform true stochastic gradient descent on the\n",[45,4112,4114],{"className":4113},[48],[45,4115,4117],{"className":4116,"ariaHidden":53},[52],[45,4118,4120,4123],{"className":4119},[57],[45,4121],{"className":4122,"style":2592},[61],[45,4124,4126],{"className":4125},[66,2582],[45,4127,4129],{"className":4128},[110],[45,4130,4132],{"className":4131},[115],[45,4133,4135,4152],{"className":4134,"style":2592},[119],[45,4136,4137,4140],{"style":1154},[45,4138],{"className":4139,"style":1103},[127],[45,4141,4143,4146,4149],{"className":4142},[66],[45,4144,2993],{"className":4145,"style":2992},[66,101],[45,4147,1614],{"className":4148,"style":1613},[66,101],[45,4150,2608],{"className":4151,"style":2607},[66,101],[45,4153,4154,4157],{"style":2611},[45,4155],{"className":4156,"style":1103},[127],[45,4158],{"className":4159,"style":2619},[2618],[34,4161,4162],{},[15,4163,4167],{"href":4164,"ariaDescribedBy":4165,"dataFootnoteRef":6,"id":4166},"#user-content-fn-sb-gtd",[40],"user-content-fnref-sb-gtd","3"," Because they follow an actual gradient, they inherit SGD's\nrobust convergence — stable even under off-policy training and nonlinear approximation —\nat ",[45,4170,4172],{"className":4171},[48],[45,4173,4175],{"className":4174,"ariaHidden":53},[52],[45,4176,4178,4181,4185,4188,4191],{"className":4177},[57],[45,4179],{"className":4180,"style":444},[61],[45,4182,4184],{"className":4183,"style":1893},[66,101],"O",[45,4186,277],{"className":4187},[93],[45,4189,474],{"className":4190},[66,101],[45,4192,321],{"className":4193},[232]," cost per step, only about double that of semi-gradient TD.",[11,4196,4197,4198,4247,4248,4313],{},"The derivation expands the ",[45,4199,4201],{"className":4200},[48],[45,4202,4204],{"className":4203,"ariaHidden":53},[52],[45,4205,4207,4210],{"className":4206},[57],[45,4208],{"className":4209,"style":2592},[61],[45,4211,4213],{"className":4212},[66,2582],[45,4214,4216],{"className":4215},[110],[45,4217,4219],{"className":4218},[115],[45,4220,4222,4239],{"className":4221,"style":2592},[119],[45,4223,4224,4227],{"style":1154},[45,4225],{"className":4226,"style":1103},[127],[45,4228,4230,4233,4236],{"className":4229},[66],[45,4231,2993],{"className":4232,"style":2992},[66,101],[45,4234,1614],{"className":4235,"style":1613},[66,101],[45,4237,2608],{"className":4238,"style":2607},[66,101],[45,4240,4241,4244],{"style":2611},[45,4242],{"className":4243,"style":1103},[127],[45,4245],{"className":4246,"style":2619},[2618]," in matrix form and differentiates. The\nkey point in the algebra: the gradient factors into three pieces, and\ntwo of them share a random quantity (",[45,4249,4251],{"className":4250},[48],[45,4252,4254],{"className":4253,"ariaHidden":53},[52],[45,4255,4257,4261],{"className":4256},[57],[45,4258],{"className":4259,"style":4260},[61],"height:0.6528em;vertical-align:-0.2083em;",[45,4262,4264,4268],{"className":4263},[66],[45,4265,4267],{"className":4266},[66,569],"x",[45,4269,4271],{"className":4270},[106],[45,4272,4274,4304],{"className":4273},[110,111],[45,4275,4277,4301],{"className":4276},[115],[45,4278,4280],{"className":4279,"style":120},[119],[45,4281,4282,4285],{"style":123},[45,4283],{"className":4284,"style":128},[127],[45,4286,4288],{"className":4287},[132,133,134,135],[45,4289,4291,4295,4298],{"className":4290},[66,135],[45,4292,4294],{"className":4293},[66,101,135],"t",[45,4296,2014],{"className":4297},[1201,135],[45,4299,42],{"className":4300},[66,135],[45,4302,143],{"className":4303},[142],[45,4305,4307],{"className":4306},[115],[45,4308,4311],{"className":4309,"style":4310},[119],"height:0.2083em;",[45,4312],{},"), so sampling each\nindependently and multiplying would double-count the noise. Gradient-TD works\naround that one obstacle.",[11,4315,4316],{},"The gradient turns out to be a product of three expectations,",[45,4318,4320],{"className":4319},[710],[45,4321,4323],{"className":4322},[48],[45,4324,4326,4397,4527],{"className":4325,"ariaHidden":53},[52],[45,4327,4329,4332,4336,4373,4376,4379,4382,4385,4388,4391,4394],{"className":4328},[57],[45,4330],{"className":4331,"style":2578},[61],[45,4333,4335],{"className":4334},[66],"∇",[45,4337,4339],{"className":4338},[66,2582],[45,4340,4342],{"className":4341},[110],[45,4343,4345],{"className":4344},[115],[45,4346,4348,4365],{"className":4347,"style":2592},[119],[45,4349,4350,4353],{"style":1154},[45,4351],{"className":4352,"style":1103},[127],[45,4354,4356,4359,4362],{"className":4355},[66],[45,4357,2993],{"className":4358,"style":2992},[66,101],[45,4360,1614],{"className":4361,"style":1613},[66,101],[45,4363,2608],{"className":4364,"style":2607},[66,101],[45,4366,4367,4370],{"style":2611},[45,4368],{"className":4369,"style":1103},[127],[45,4371],{"className":4372,"style":2619},[2618],[45,4374,277],{"className":4375},[93],[45,4377,571],{"className":4378,"style":570},[66,569],[45,4380,321],{"className":4381},[232],[45,4383],{"className":4384,"style":74},[73],[45,4386],{"className":4387,"style":74},[73],[45,4389,79],{"className":4390},[78],[45,4392],{"className":4393,"style":74},[73],[45,4395],{"className":4396,"style":74},[73],[45,4398,4400,4403,4406,4409,4412,4418,4460,4463,4466,4469,4518,4521,4524],{"className":4399},[57],[45,4401],{"className":4402,"style":1983},[61],[45,4404,774],{"className":4405},[66],[45,4407],{"className":4408,"style":161},[73],[45,4410,2608],{"className":4411},[66,1122],[45,4413,4415],{"className":4414},[93],[45,4416,270],{"className":4417},[1351,1148],[45,4419,4421,4425],{"className":4420},[66],[45,4422,4424],{"className":4423},[66,101],"ρ",[45,4426,4428],{"className":4427},[106],[45,4429,4431,4452],{"className":4430},[110,111],[45,4432,4434,4449],{"className":4433},[115],[45,4435,4438],{"className":4436,"style":4437},[119],"height:0.2806em;",[45,4439,4440,4443],{"style":123},[45,4441],{"className":4442,"style":128},[127],[45,4444,4446],{"className":4445},[132,133,134,135],[45,4447,4294],{"className":4448},[66,101,135],[45,4450,143],{"className":4451},[142],[45,4453,4455],{"className":4454},[115],[45,4456,4458],{"className":4457,"style":150},[119],[45,4459],{},[45,4461,277],{"className":4462},[93],[45,4464,2028],{"className":4465,"style":2027},[66,101],[45,4467],{"className":4468,"style":161},[73],[45,4470,4472,4475],{"className":4471},[66],[45,4473,4267],{"className":4474},[66,569],[45,4476,4478],{"className":4477},[106],[45,4479,4481,4510],{"className":4480},[110,111],[45,4482,4484,4507],{"className":4483},[115],[45,4485,4487],{"className":4486,"style":120},[119],[45,4488,4489,4492],{"style":123},[45,4490],{"className":4491,"style":128},[127],[45,4493,4495],{"className":4494},[132,133,134,135],[45,4496,4498,4501,4504],{"className":4497},[66,135],[45,4499,4294],{"className":4500},[66,101,135],[45,4502,2014],{"className":4503},[1201,135],[45,4505,42],{"className":4506},[66,135],[45,4508,143],{"className":4509},[142],[45,4511,4513],{"className":4512},[115],[45,4514,4516],{"className":4515,"style":4310},[119],[45,4517],{},[45,4519],{"className":4520,"style":1197},[73],[45,4522,1202],{"className":4523},[1201],[45,4525],{"className":4526,"style":1197},[73],[45,4528,4530,4533,4573,4576,4579,4631,4637,4640,4643,4649,4689,4740,4778,4781,4784,4790,4830,4833,4873,4876,4916,4922],{"className":4529},[57],[45,4531],{"className":4532,"style":1340},[61],[45,4534,4536,4539],{"className":4535},[66],[45,4537,4267],{"className":4538},[66,569],[45,4540,4542],{"className":4541},[106],[45,4543,4545,4565],{"className":4544},[110,111],[45,4546,4548,4562],{"className":4547},[115],[45,4549,4551],{"className":4550,"style":4437},[119],[45,4552,4553,4556],{"style":123},[45,4554],{"className":4555,"style":128},[127],[45,4557,4559],{"className":4558},[132,133,134,135],[45,4560,4294],{"className":4561},[66,101,135],[45,4563,143],{"className":4564},[142],[45,4566,4568],{"className":4567},[115],[45,4569,4571],{"className":4570,"style":150},[119],[45,4572],{},[45,4574,321],{"className":4575},[232],[45,4577],{"className":4578,"style":161},[73],[45,4580,4582,4585],{"className":4581},[66],[45,4583,4267],{"className":4584},[66,569],[45,4586,4588],{"className":4587},[106],[45,4589,4591,4622],{"className":4590},[110,111],[45,4592,4594,4619],{"className":4593},[115],[45,4595,4597,4608],{"className":4596,"style":1370},[119],[45,4598,4599,4602],{"style":752},[45,4600],{"className":4601,"style":128},[127],[45,4603,4605],{"className":4604},[132,133,134,135],[45,4606,4294],{"className":4607},[66,101,135],[45,4609,4610,4613],{"style":764},[45,4611],{"className":4612,"style":128},[127],[45,4614,4616],{"className":4615},[132,133,134,135],[45,4617,430],{"className":4618},[66,135],[45,4620,143],{"className":4621},[142],[45,4623,4625],{"className":4624},[115],[45,4626,4629],{"className":4627,"style":4628},[119],"height:0.247em;",[45,4630],{},[45,4632,4634],{"className":4633},[232],[45,4635,404],{"className":4636},[1351,1148],[45,4638],{"className":4639,"style":74},[73],[45,4641,2608],{"className":4642},[66,1122],[45,4644,4646],{"className":4645},[93],[45,4647,270],{"className":4648},[1351,1148],[45,4650,4652,4655],{"className":4651},[66],[45,4653,4267],{"className":4654},[66,569],[45,4656,4658],{"className":4657},[106],[45,4659,4661,4681],{"className":4660},[110,111],[45,4662,4664,4678],{"className":4663},[115],[45,4665,4667],{"className":4666,"style":4437},[119],[45,4668,4669,4672],{"style":123},[45,4670],{"className":4671,"style":128},[127],[45,4673,4675],{"className":4674},[132,133,134,135],[45,4676,4294],{"className":4677},[66,101,135],[45,4679,143],{"className":4680},[142],[45,4682,4684],{"className":4683},[115],[45,4685,4687],{"className":4686,"style":150},[119],[45,4688],{},[45,4690,4692,4695],{"className":4691},[66],[45,4693,4267],{"className":4694},[66,569],[45,4696,4698],{"className":4697},[106],[45,4699,4701,4732],{"className":4700},[110,111],[45,4702,4704,4729],{"className":4703},[115],[45,4705,4707,4718],{"className":4706,"style":1370},[119],[45,4708,4709,4712],{"style":752},[45,4710],{"className":4711,"style":128},[127],[45,4713,4715],{"className":4714},[132,133,134,135],[45,4716,4294],{"className":4717},[66,101,135],[45,4719,4720,4723],{"style":764},[45,4721],{"className":4722,"style":128},[127],[45,4724,4726],{"className":4725},[132,133,134,135],[45,4727,430],{"className":4728},[66,135],[45,4730,143],{"className":4731},[142],[45,4733,4735],{"className":4734},[115],[45,4736,4738],{"className":4737,"style":4628},[119],[45,4739],{},[45,4741,4743,4749],{"className":4742},[232],[45,4744,4746],{"className":4745},[232],[45,4747,404],{"className":4748},[1351,1148],[45,4750,4752],{"className":4751},[106],[45,4753,4755],{"className":4754},[110],[45,4756,4758],{"className":4757},[115],[45,4759,4761],{"className":4760,"style":1413},[119],[45,4762,4763,4766],{"style":1416},[45,4764],{"className":4765,"style":128},[127],[45,4767,4769],{"className":4768},[132,133,134,135],[45,4770,4772,4775],{"className":4771},[66,135],[45,4773,1202],{"className":4774},[66,135],[45,4776,42],{"className":4777},[66,135],[45,4779],{"className":4780,"style":74},[73],[45,4782,2608],{"className":4783},[66,1122],[45,4785,4787],{"className":4786},[93],[45,4788,270],{"className":4789},[1351,1148],[45,4791,4793,4796],{"className":4792},[66],[45,4794,4424],{"className":4795},[66,101],[45,4797,4799],{"className":4798},[106],[45,4800,4802,4822],{"className":4801},[110,111],[45,4803,4805,4819],{"className":4804},[115],[45,4806,4808],{"className":4807,"style":4437},[119],[45,4809,4810,4813],{"style":123},[45,4811],{"className":4812,"style":128},[127],[45,4814,4816],{"className":4815},[132,133,134,135],[45,4817,4294],{"className":4818},[66,101,135],[45,4820,143],{"className":4821},[142],[45,4823,4825],{"className":4824},[115],[45,4826,4828],{"className":4827,"style":150},[119],[45,4829],{},[45,4831],{"className":4832,"style":161},[73],[45,4834,4836,4839],{"className":4835},[66],[45,4837,2328],{"className":4838,"style":2327},[66,101],[45,4840,4842],{"className":4841},[106],[45,4843,4845,4865],{"className":4844},[110,111],[45,4846,4848,4862],{"className":4847},[115],[45,4849,4851],{"className":4850,"style":4437},[119],[45,4852,4853,4856],{"style":2358},[45,4854],{"className":4855,"style":128},[127],[45,4857,4859],{"className":4858},[132,133,134,135],[45,4860,4294],{"className":4861},[66,101,135],[45,4863,143],{"className":4864},[142],[45,4866,4868],{"className":4867},[115],[45,4869,4871],{"className":4870,"style":150},[119],[45,4872],{},[45,4874],{"className":4875,"style":161},[73],[45,4877,4879,4882],{"className":4878},[66],[45,4880,4267],{"className":4881},[66,569],[45,4883,4885],{"className":4884},[106],[45,4886,4888,4908],{"className":4887},[110,111],[45,4889,4891,4905],{"className":4890},[115],[45,4892,4894],{"className":4893,"style":4437},[119],[45,4895,4896,4899],{"style":123},[45,4897],{"className":4898,"style":128},[127],[45,4900,4902],{"className":4901},[132,133,134,135],[45,4903,4294],{"className":4904},[66,101,135],[45,4906,143],{"className":4907},[142],[45,4909,4911],{"className":4910},[115],[45,4912,4914],{"className":4913,"style":150},[119],[45,4915],{},[45,4917,4919],{"className":4918},[232],[45,4920,404],{"className":4921},[1351,1148],[45,4923,157],{"className":4924},[156],[11,4926,4927,4928,4989,4990,4993,4994,157],{},"and the trouble is that the first and last factors both depend on the next feature\n",[45,4929,4931],{"className":4930},[48],[45,4932,4934],{"className":4933,"ariaHidden":53},[52],[45,4935,4937,4940],{"className":4936},[57],[45,4938],{"className":4939,"style":4260},[61],[45,4941,4943,4946],{"className":4942},[66],[45,4944,4267],{"className":4945},[66,569],[45,4947,4949],{"className":4948},[106],[45,4950,4952,4981],{"className":4951},[110,111],[45,4953,4955,4978],{"className":4954},[115],[45,4956,4958],{"className":4957,"style":120},[119],[45,4959,4960,4963],{"style":123},[45,4961],{"className":4962,"style":128},[127],[45,4964,4966],{"className":4965},[132,133,134,135],[45,4967,4969,4972,4975],{"className":4968},[66,135],[45,4970,4294],{"className":4971},[66,101,135],[45,4973,2014],{"className":4974},[1201,135],[45,4976,42],{"className":4977},[66,135],[45,4979,143],{"className":4980},[142],[45,4982,4984],{"className":4983},[115],[45,4985,4987],{"className":4986,"style":4310},[119],[45,4988],{},", so sampling them independently and multiplying gives a ",[29,4991,4992],{},"biased","\nestimate — the same double-sampling problem that afflicts the residual-gradient method.\nThe Gradient-TD solution is to estimate and store the product of the last two factors as a\nsecond parameter vector ",[45,4995,4997],{"className":4996},[48],[45,4998,5000],{"className":4999,"ariaHidden":53},[52],[45,5001,5003,5007],{"className":5002},[57],[45,5004],{"className":5005,"style":5006},[61],"height:0.4444em;",[45,5008,252],{"className":5009,"style":570},[66,569],[45,5011,5013],{"className":5012},[710],[45,5014,5016],{"className":5015},[48],[45,5017,5019,5045],{"className":5018,"ariaHidden":53},[52],[45,5020,5022,5026,5029,5032,5035,5039,5042],{"className":5021},[57],[45,5023],{"className":5024,"style":5025},[61],"height:0.4831em;",[45,5027,252],{"className":5028,"style":570},[66,569],[45,5030],{"className":5031,"style":74},[73],[45,5033],{"className":5034,"style":74},[73],[45,5036,5038],{"className":5037},[78],"≈",[45,5040],{"className":5041,"style":74},[73],[45,5043],{"className":5044,"style":74},[73],[45,5046,5048,5051,5054,5060,5100,5151,5189,5192,5195,5201,5241,5244,5284,5287,5327,5333],{"className":5047},[57],[45,5049],{"className":5050,"style":1340},[61],[45,5052,2608],{"className":5053},[66,1122],[45,5055,5057],{"className":5056},[93],[45,5058,270],{"className":5059},[1351,1148],[45,5061,5063,5066],{"className":5062},[66],[45,5064,4267],{"className":5065},[66,569],[45,5067,5069],{"className":5068},[106],[45,5070,5072,5092],{"className":5071},[110,111],[45,5073,5075,5089],{"className":5074},[115],[45,5076,5078],{"className":5077,"style":4437},[119],[45,5079,5080,5083],{"style":123},[45,5081],{"className":5082,"style":128},[127],[45,5084,5086],{"className":5085},[132,133,134,135],[45,5087,4294],{"className":5088},[66,101,135],[45,5090,143],{"className":5091},[142],[45,5093,5095],{"className":5094},[115],[45,5096,5098],{"className":5097,"style":150},[119],[45,5099],{},[45,5101,5103,5106],{"className":5102},[66],[45,5104,4267],{"className":5105},[66,569],[45,5107,5109],{"className":5108},[106],[45,5110,5112,5143],{"className":5111},[110,111],[45,5113,5115,5140],{"className":5114},[115],[45,5116,5118,5129],{"className":5117,"style":1370},[119],[45,5119,5120,5123],{"style":752},[45,5121],{"className":5122,"style":128},[127],[45,5124,5126],{"className":5125},[132,133,134,135],[45,5127,4294],{"className":5128},[66,101,135],[45,5130,5131,5134],{"style":764},[45,5132],{"className":5133,"style":128},[127],[45,5135,5137],{"className":5136},[132,133,134,135],[45,5138,430],{"className":5139},[66,135],[45,5141,143],{"className":5142},[142],[45,5144,5146],{"className":5145},[115],[45,5147,5149],{"className":5148,"style":4628},[119],[45,5150],{},[45,5152,5154,5160],{"className":5153},[232],[45,5155,5157],{"className":5156},[232],[45,5158,404],{"className":5159},[1351,1148],[45,5161,5163],{"className":5162},[106],[45,5164,5166],{"className":5165},[110],[45,5167,5169],{"className":5168},[115],[45,5170,5172],{"className":5171,"style":1413},[119],[45,5173,5174,5177],{"style":1416},[45,5175],{"className":5176,"style":128},[127],[45,5178,5180],{"className":5179},[132,133,134,135],[45,5181,5183,5186],{"className":5182},[66,135],[45,5184,1202],{"className":5185},[66,135],[45,5187,42],{"className":5188},[66,135],[45,5190],{"className":5191,"style":161},[73],[45,5193,2608],{"className":5194},[66,1122],[45,5196,5198],{"className":5197},[93],[45,5199,270],{"className":5200},[1351,1148],[45,5202,5204,5207],{"className":5203},[66],[45,5205,4424],{"className":5206},[66,101],[45,5208,5210],{"className":5209},[106],[45,5211,5213,5233],{"className":5212},[110,111],[45,5214,5216,5230],{"className":5215},[115],[45,5217,5219],{"className":5218,"style":4437},[119],[45,5220,5221,5224],{"style":123},[45,5222],{"className":5223,"style":128},[127],[45,5225,5227],{"className":5226},[132,133,134,135],[45,5228,4294],{"className":5229},[66,101,135],[45,5231,143],{"className":5232},[142],[45,5234,5236],{"className":5235},[115],[45,5237,5239],{"className":5238,"style":150},[119],[45,5240],{},[45,5242],{"className":5243,"style":161},[73],[45,5245,5247,5250],{"className":5246},[66],[45,5248,2328],{"className":5249,"style":2327},[66,101],[45,5251,5253],{"className":5252},[106],[45,5254,5256,5276],{"className":5255},[110,111],[45,5257,5259,5273],{"className":5258},[115],[45,5260,5262],{"className":5261,"style":4437},[119],[45,5263,5264,5267],{"style":2358},[45,5265],{"className":5266,"style":128},[127],[45,5268,5270],{"className":5269},[132,133,134,135],[45,5271,4294],{"className":5272},[66,101,135],[45,5274,143],{"className":5275},[142],[45,5277,5279],{"className":5278},[115],[45,5280,5282],{"className":5281,"style":150},[119],[45,5283],{},[45,5285],{"className":5286,"style":161},[73],[45,5288,5290,5293],{"className":5289},[66],[45,5291,4267],{"className":5292},[66,569],[45,5294,5296],{"className":5295},[106],[45,5297,5299,5319],{"className":5298},[110,111],[45,5300,5302,5316],{"className":5301},[115],[45,5303,5305],{"className":5304,"style":4437},[119],[45,5306,5307,5310],{"style":123},[45,5308],{"className":5309,"style":128},[127],[45,5311,5313],{"className":5312},[132,133,134,135],[45,5314,4294],{"className":5315},[66,101,135],[45,5317,143],{"className":5318},[142],[45,5320,5322],{"className":5321},[115],[45,5323,5325],{"className":5324,"style":150},[119],[45,5326],{},[45,5328,5330],{"className":5329},[232],[45,5331,404],{"className":5332},[1351,1148],[45,5334,157],{"className":5335},[156],[11,5337,5338,5339,2562],{},"which is the solution to a least-squares problem and is learned by an\nimportance-weighted LMS rule on a second, faster step size ",[45,5340,5342],{"className":5341},[48],[45,5343,5345],{"className":5344,"ariaHidden":53},[52],[45,5346,5348,5352],{"className":5347},[57],[45,5349],{"className":5350,"style":5351},[61],"height:0.8889em;vertical-align:-0.1944em;",[45,5353,5356],{"className":5354,"style":5355},[66,101],"margin-right:0.0528em;","β",[45,5358,5360],{"className":5359},[710],[45,5361,5363],{"className":5362},[48],[45,5364,5366,5437,5493,5603],{"className":5365,"ariaHidden":53},[52],[45,5367,5369,5373,5422,5425,5428,5431,5434],{"className":5368},[57],[45,5370],{"className":5371,"style":5372},[61],"height:0.8813em;vertical-align:-0.2083em;",[45,5374,5376,5379],{"className":5375},[66],[45,5377,252],{"className":5378,"style":570},[66,569],[45,5380,5382],{"className":5381},[106],[45,5383,5385,5414],{"className":5384},[110,111],[45,5386,5388,5411],{"className":5387},[115],[45,5389,5391],{"className":5390,"style":120},[119],[45,5392,5393,5396],{"style":3432},[45,5394],{"className":5395,"style":128},[127],[45,5397,5399],{"className":5398},[132,133,134,135],[45,5400,5402,5405,5408],{"className":5401},[66,135],[45,5403,4294],{"className":5404},[66,101,135],[45,5406,2014],{"className":5407},[1201,135],[45,5409,42],{"className":5410},[66,135],[45,5412,143],{"className":5413},[142],[45,5415,5417],{"className":5416},[115],[45,5418,5420],{"className":5419,"style":4310},[119],[45,5421],{},[45,5423],{"className":5424,"style":74},[73],[45,5426],{"className":5427,"style":74},[73],[45,5429,796],{"className":5430},[78],[45,5432],{"className":5433,"style":74},[73],[45,5435],{"className":5436,"style":74},[73],[45,5438,5440,5444,5484,5487,5490],{"className":5439},[57],[45,5441],{"className":5442,"style":5443},[61],"height:0.7333em;vertical-align:-0.15em;",[45,5445,5447,5450],{"className":5446},[66],[45,5448,252],{"className":5449,"style":570},[66,569],[45,5451,5453],{"className":5452},[106],[45,5454,5456,5476],{"className":5455},[110,111],[45,5457,5459,5473],{"className":5458},[115],[45,5460,5462],{"className":5461,"style":4437},[119],[45,5463,5464,5467],{"style":3432},[45,5465],{"className":5466,"style":128},[127],[45,5468,5470],{"className":5469},[132,133,134,135],[45,5471,4294],{"className":5472},[66,101,135],[45,5474,143],{"className":5475},[142],[45,5477,5479],{"className":5478},[115],[45,5480,5482],{"className":5481,"style":150},[119],[45,5483],{},[45,5485],{"className":5486,"style":1197},[73],[45,5488,2014],{"className":5489},[1201],[45,5491],{"className":5492,"style":1197},[73],[45,5494,5496,5499,5502,5505,5545,5548,5554,5594,5597,5600],{"className":5495},[57],[45,5497],{"className":5498,"style":1983},[61],[45,5500,5356],{"className":5501,"style":5355},[66,101],[45,5503],{"className":5504,"style":161},[73],[45,5506,5508,5511],{"className":5507},[66],[45,5509,4424],{"className":5510},[66,101],[45,5512,5514],{"className":5513},[106],[45,5515,5517,5537],{"className":5516},[110,111],[45,5518,5520,5534],{"className":5519},[115],[45,5521,5523],{"className":5522,"style":4437},[119],[45,5524,5525,5528],{"style":123},[45,5526],{"className":5527,"style":128},[127],[45,5529,5531],{"className":5530},[132,133,134,135],[45,5532,4294],{"className":5533},[66,101,135],[45,5535,143],{"className":5536},[142],[45,5538,5540],{"className":5539},[115],[45,5541,5543],{"className":5542,"style":150},[119],[45,5544],{},[45,5546],{"className":5547,"style":161},[73],[45,5549,5551],{"className":5550},[93],[45,5552,277],{"className":5553},[1351,1148],[45,5555,5557,5560],{"className":5556},[66],[45,5558,2328],{"className":5559,"style":2327},[66,101],[45,5561,5563],{"className":5562},[106],[45,5564,5566,5586],{"className":5565},[110,111],[45,5567,5569,5583],{"className":5568},[115],[45,5570,5572],{"className":5571,"style":4437},[119],[45,5573,5574,5577],{"style":2358},[45,5575],{"className":5576,"style":128},[127],[45,5578,5580],{"className":5579},[132,133,134,135],[45,5581,4294],{"className":5582},[66,101,135],[45,5584,143],{"className":5585},[142],[45,5587,5589],{"className":5588},[115],[45,5590,5592],{"className":5591,"style":150},[119],[45,5593],{},[45,5595],{"className":5596,"style":1197},[73],[45,5598,1202],{"className":5599},[1201],[45,5601],{"className":5602,"style":1197},[73],[45,5604,5606,5610,5662,5702,5708,5711,5751],{"className":5605},[57],[45,5607],{"className":5608,"style":5609},[61],"height:1.2491em;vertical-align:-0.35em;",[45,5611,5613,5616],{"className":5612},[66],[45,5614,252],{"className":5615,"style":570},[66,569],[45,5617,5619],{"className":5618},[106],[45,5620,5622,5654],{"className":5621},[110,111],[45,5623,5625,5651],{"className":5624},[115],[45,5626,5628,5640],{"className":5627,"style":1370},[119],[45,5629,5631,5634],{"style":5630},"top:-2.453em;margin-left:-0.016em;margin-right:0.05em;",[45,5632],{"className":5633,"style":128},[127],[45,5635,5637],{"className":5636},[132,133,134,135],[45,5638,4294],{"className":5639},[66,101,135],[45,5641,5642,5645],{"style":764},[45,5643],{"className":5644,"style":128},[127],[45,5646,5648],{"className":5647},[132,133,134,135],[45,5649,430],{"className":5650},[66,135],[45,5652,143],{"className":5653},[142],[45,5655,5657],{"className":5656},[115],[45,5658,5660],{"className":5659,"style":4628},[119],[45,5661],{},[45,5663,5665,5668],{"className":5664},[66],[45,5666,4267],{"className":5667},[66,569],[45,5669,5671],{"className":5670},[106],[45,5672,5674,5694],{"className":5673},[110,111],[45,5675,5677,5691],{"className":5676},[115],[45,5678,5680],{"className":5679,"style":4437},[119],[45,5681,5682,5685],{"style":123},[45,5683],{"className":5684,"style":128},[127],[45,5686,5688],{"className":5687},[132,133,134,135],[45,5689,4294],{"className":5690},[66,101,135],[45,5692,143],{"className":5693},[142],[45,5695,5697],{"className":5696},[115],[45,5698,5700],{"className":5699,"style":150},[119],[45,5701],{},[45,5703,5705],{"className":5704},[232],[45,5706,321],{"className":5707},[1351,1148],[45,5709],{"className":5710,"style":161},[73],[45,5712,5714,5717],{"className":5713},[66],[45,5715,4267],{"className":5716},[66,569],[45,5718,5720],{"className":5719},[106],[45,5721,5723,5743],{"className":5722},[110,111],[45,5724,5726,5740],{"className":5725},[115],[45,5727,5729],{"className":5728,"style":4437},[119],[45,5730,5731,5734],{"style":123},[45,5732],{"className":5733,"style":128},[127],[45,5735,5737],{"className":5736},[132,133,134,135],[45,5738,4294],{"className":5739},[66,101,135],[45,5741,143],{"className":5742},[142],[45,5744,5746],{"className":5745},[115],[45,5747,5749],{"className":5748,"style":150},[119],[45,5750],{},[45,5752,933],{"className":5753},[66],[11,5755,5756,5757,5809,5810,5834,5835,5939],{},"Given ",[45,5758,5760],{"className":5759},[48],[45,5761,5763],{"className":5762,"ariaHidden":53},[52],[45,5764,5766,5769],{"className":5765},[57],[45,5767],{"className":5768,"style":3410},[61],[45,5770,5772,5775],{"className":5771},[66],[45,5773,252],{"className":5774,"style":570},[66,569],[45,5776,5778],{"className":5777},[106],[45,5779,5781,5801],{"className":5780},[110,111],[45,5782,5784,5798],{"className":5783},[115],[45,5785,5787],{"className":5786,"style":4437},[119],[45,5788,5789,5792],{"style":3432},[45,5790],{"className":5791,"style":128},[127],[45,5793,5795],{"className":5794},[132,133,134,135],[45,5796,4294],{"className":5797},[66,101,135],[45,5799,143],{"className":5800},[142],[45,5802,5804],{"className":5803},[115],[45,5805,5807],{"className":5806,"style":150},[119],[45,5808],{},", the main weight vector is updated by a rule that is ",[45,5811,5813],{"className":5812},[48],[45,5814,5816],{"className":5815,"ariaHidden":53},[52],[45,5817,5819,5822,5825,5828,5831],{"className":5818},[57],[45,5820],{"className":5821,"style":444},[61],[45,5823,4184],{"className":5824,"style":1893},[66,101],[45,5826,277],{"className":5827},[93],[45,5829,474],{"className":5830},[66,101],[45,5832,321],{"className":5833},[232]," if the\ninner product ",[45,5836,5838],{"className":5837},[48],[45,5839,5841],{"className":5840,"ariaHidden":53},[52],[45,5842,5844,5848,5899],{"className":5843},[57],[45,5845],{"className":5846,"style":5847},[61],"height:1.0961em;vertical-align:-0.247em;",[45,5849,5851,5854],{"className":5850},[66],[45,5852,4267],{"className":5853},[66,569],[45,5855,5857],{"className":5856},[106],[45,5858,5860,5891],{"className":5859},[110,111],[45,5861,5863,5888],{"className":5862},[115],[45,5864,5866,5877],{"className":5865,"style":417},[119],[45,5867,5868,5871],{"style":752},[45,5869],{"className":5870,"style":128},[127],[45,5872,5874],{"className":5873},[132,133,134,135],[45,5875,4294],{"className":5876},[66,101,135],[45,5878,5879,5882],{"style":420},[45,5880],{"className":5881,"style":128},[127],[45,5883,5885],{"className":5884},[132,133,134,135],[45,5886,430],{"className":5887},[66,135],[45,5889,143],{"className":5890},[142],[45,5892,5894],{"className":5893},[115],[45,5895,5897],{"className":5896,"style":4628},[119],[45,5898],{},[45,5900,5902,5905],{"className":5901},[66],[45,5903,252],{"className":5904,"style":570},[66,569],[45,5906,5908],{"className":5907},[106],[45,5909,5911,5931],{"className":5910},[110,111],[45,5912,5914,5928],{"className":5913},[115],[45,5915,5917],{"className":5916,"style":4437},[119],[45,5918,5919,5922],{"style":3432},[45,5920],{"className":5921,"style":128},[127],[45,5923,5925],{"className":5924},[132,133,134,135],[45,5926,4294],{"className":5927},[66,101,135],[45,5929,143],{"className":5930},[142],[45,5932,5934],{"className":5933},[115],[45,5935,5937],{"className":5936,"style":150},[119],[45,5938],{}," is formed first. Two members of the family:",[45,5941,5943],{"className":5942},[710],[45,5944,5946],{"className":5945},[48],[45,5947,5949,6030,6085,6194],{"className":5948,"ariaHidden":53},[52],[45,5950,5952,5956,5963,5966,6015,6018,6021,6024,6027],{"className":5951},[57],[45,5953],{"className":5954,"style":5955},[61],"height:0.8917em;vertical-align:-0.2083em;",[45,5957,5959],{"className":5958},[66,1051],[45,5960,5962],{"className":5961},[66],"GTD2:",[45,5964],{"className":5965,"style":1047},[73],[45,5967,5969,5972],{"className":5968},[66],[45,5970,571],{"className":5971,"style":570},[66,569],[45,5973,5975],{"className":5974},[106],[45,5976,5978,6007],{"className":5977},[110,111],[45,5979,5981,6004],{"className":5980},[115],[45,5982,5984],{"className":5983,"style":120},[119],[45,5985,5986,5989],{"style":3432},[45,5987],{"className":5988,"style":128},[127],[45,5990,5992],{"className":5991},[132,133,134,135],[45,5993,5995,5998,6001],{"className":5994},[66,135],[45,5996,4294],{"className":5997},[66,101,135],[45,5999,2014],{"className":6000},[1201,135],[45,6002,42],{"className":6003},[66,135],[45,6005,143],{"className":6006},[142],[45,6008,6010],{"className":6009},[115],[45,6011,6013],{"className":6012,"style":4310},[119],[45,6014],{},[45,6016],{"className":6017,"style":74},[73],[45,6019],{"className":6020,"style":74},[73],[45,6022,79],{"className":6023},[78],[45,6025],{"className":6026,"style":74},[73],[45,6028],{"className":6029,"style":74},[73],[45,6031,6033,6036,6076,6079,6082],{"className":6032},[57],[45,6034],{"className":6035,"style":5443},[61],[45,6037,6039,6042],{"className":6038},[66],[45,6040,571],{"className":6041,"style":570},[66,569],[45,6043,6045],{"className":6044},[106],[45,6046,6048,6068],{"className":6047},[110,111],[45,6049,6051,6065],{"className":6050},[115],[45,6052,6054],{"className":6053,"style":4437},[119],[45,6055,6056,6059],{"style":3432},[45,6057],{"className":6058,"style":128},[127],[45,6060,6062],{"className":6061},[132,133,134,135],[45,6063,4294],{"className":6064},[66,101,135],[45,6066,143],{"className":6067},[142],[45,6069,6071],{"className":6070},[115],[45,6072,6074],{"className":6073,"style":150},[119],[45,6075],{},[45,6077],{"className":6078,"style":1197},[73],[45,6080,2014],{"className":6081},[1201],[45,6083],{"className":6084,"style":1197},[73],[45,6086,6088,6091,6096,6099,6139,6142,6145,6185,6188,6191],{"className":6087},[57],[45,6089],{"className":6090,"style":444},[61],[45,6092,6095],{"className":6093,"style":6094},[66,101],"margin-right:0.0037em;","α",[45,6097],{"className":6098,"style":161},[73],[45,6100,6102,6105],{"className":6101},[66],[45,6103,4424],{"className":6104},[66,101],[45,6106,6108],{"className":6107},[106],[45,6109,6111,6131],{"className":6110},[110,111],[45,6112,6114,6128],{"className":6113},[115],[45,6115,6117],{"className":6116,"style":4437},[119],[45,6118,6119,6122],{"style":123},[45,6120],{"className":6121,"style":128},[127],[45,6123,6125],{"className":6124},[132,133,134,135],[45,6126,4294],{"className":6127},[66,101,135],[45,6129,143],{"className":6130},[142],[45,6132,6134],{"className":6133},[115],[45,6135,6137],{"className":6136,"style":150},[119],[45,6138],{},[45,6140],{"className":6141,"style":161},[73],[45,6143,277],{"className":6144},[93],[45,6146,6148,6151],{"className":6147},[66],[45,6149,4267],{"className":6150},[66,569],[45,6152,6154],{"className":6153},[106],[45,6155,6157,6177],{"className":6156},[110,111],[45,6158,6160,6174],{"className":6159},[115],[45,6161,6163],{"className":6162,"style":4437},[119],[45,6164,6165,6168],{"style":123},[45,6166],{"className":6167,"style":128},[127],[45,6169,6171],{"className":6170},[132,133,134,135],[45,6172,4294],{"className":6173},[66,101,135],[45,6175,143],{"className":6176},[142],[45,6178,6180],{"className":6179},[115],[45,6181,6183],{"className":6182,"style":150},[119],[45,6184],{},[45,6186],{"className":6187,"style":1197},[73],[45,6189,1202],{"className":6190},[1201],[45,6192],{"className":6193,"style":1197},[73],[45,6195,6197,6201,6204,6207,6256,6259,6262,6265,6316,6356,6359],{"className":6196},[57],[45,6198],{"className":6199,"style":6200},[61],"height:1.1491em;vertical-align:-0.25em;",[45,6202,2028],{"className":6203,"style":2027},[66,101],[45,6205],{"className":6206,"style":161},[73],[45,6208,6210,6213],{"className":6209},[66],[45,6211,4267],{"className":6212},[66,569],[45,6214,6216],{"className":6215},[106],[45,6217,6219,6248],{"className":6218},[110,111],[45,6220,6222,6245],{"className":6221},[115],[45,6223,6225],{"className":6224,"style":120},[119],[45,6226,6227,6230],{"style":123},[45,6228],{"className":6229,"style":128},[127],[45,6231,6233],{"className":6232},[132,133,134,135],[45,6234,6236,6239,6242],{"className":6235},[66,135],[45,6237,4294],{"className":6238},[66,101,135],[45,6240,2014],{"className":6241},[1201,135],[45,6243,42],{"className":6244},[66,135],[45,6246,143],{"className":6247},[142],[45,6249,6251],{"className":6250},[115],[45,6252,6254],{"className":6253,"style":4310},[119],[45,6255],{},[45,6257,321],{"className":6258},[232],[45,6260],{"className":6261,"style":161},[73],[45,6263,277],{"className":6264},[93],[45,6266,6268,6271],{"className":6267},[66],[45,6269,4267],{"className":6270},[66,569],[45,6272,6274],{"className":6273},[106],[45,6275,6277,6308],{"className":6276},[110,111],[45,6278,6280,6305],{"className":6279},[115],[45,6281,6283,6294],{"className":6282,"style":1370},[119],[45,6284,6285,6288],{"style":752},[45,6286],{"className":6287,"style":128},[127],[45,6289,6291],{"className":6290},[132,133,134,135],[45,6292,4294],{"className":6293},[66,101,135],[45,6295,6296,6299],{"style":764},[45,6297],{"className":6298,"style":128},[127],[45,6300,6302],{"className":6301},[132,133,134,135],[45,6303,430],{"className":6304},[66,135],[45,6306,143],{"className":6307},[142],[45,6309,6311],{"className":6310},[115],[45,6312,6314],{"className":6313,"style":4628},[119],[45,6315],{},[45,6317,6319,6322],{"className":6318},[66],[45,6320,252],{"className":6321,"style":570},[66,569],[45,6323,6325],{"className":6324},[106],[45,6326,6328,6348],{"className":6327},[110,111],[45,6329,6331,6345],{"className":6330},[115],[45,6332,6334],{"className":6333,"style":4437},[119],[45,6335,6336,6339],{"style":3432},[45,6337],{"className":6338,"style":128},[127],[45,6340,6342],{"className":6341},[132,133,134,135],[45,6343,4294],{"className":6344},[66,101,135],[45,6346,143],{"className":6347},[142],[45,6349,6351],{"className":6350},[115],[45,6352,6354],{"className":6353,"style":150},[119],[45,6355],{},[45,6357,321],{"className":6358},[232],[45,6360,157],{"className":6361},[156],[45,6363,6365],{"className":6364},[710],[45,6366,6368],{"className":6367},[48],[45,6369,6371,6451,6506,6659],{"className":6370,"ariaHidden":53},[52],[45,6372,6374,6377,6384,6387,6436,6439,6442,6445,6448],{"className":6373},[57],[45,6375],{"className":6376,"style":444},[61],[45,6378,6380],{"className":6379},[66,1051],[45,6381,6383],{"className":6382},[66],"TDC \u002F GTD(0):",[45,6385],{"className":6386,"style":1047},[73],[45,6388,6390,6393],{"className":6389},[66],[45,6391,571],{"className":6392,"style":570},[66,569],[45,6394,6396],{"className":6395},[106],[45,6397,6399,6428],{"className":6398},[110,111],[45,6400,6402,6425],{"className":6401},[115],[45,6403,6405],{"className":6404,"style":120},[119],[45,6406,6407,6410],{"style":3432},[45,6408],{"className":6409,"style":128},[127],[45,6411,6413],{"className":6412},[132,133,134,135],[45,6414,6416,6419,6422],{"className":6415},[66,135],[45,6417,4294],{"className":6418},[66,101,135],[45,6420,2014],{"className":6421},[1201,135],[45,6423,42],{"className":6424},[66,135],[45,6426,143],{"className":6427},[142],[45,6429,6431],{"className":6430},[115],[45,6432,6434],{"className":6433,"style":4310},[119],[45,6435],{},[45,6437],{"className":6438,"style":74},[73],[45,6440],{"className":6441,"style":74},[73],[45,6443,79],{"className":6444},[78],[45,6446],{"className":6447,"style":74},[73],[45,6449],{"className":6450,"style":74},[73],[45,6452,6454,6457,6497,6500,6503],{"className":6453},[57],[45,6455],{"className":6456,"style":5443},[61],[45,6458,6460,6463],{"className":6459},[66],[45,6461,571],{"className":6462,"style":570},[66,569],[45,6464,6466],{"className":6465},[106],[45,6467,6469,6489],{"className":6468},[110,111],[45,6470,6472,6486],{"className":6471},[115],[45,6473,6475],{"className":6474,"style":4437},[119],[45,6476,6477,6480],{"style":3432},[45,6478],{"className":6479,"style":128},[127],[45,6481,6483],{"className":6482},[132,133,134,135],[45,6484,4294],{"className":6485},[66,101,135],[45,6487,143],{"className":6488},[142],[45,6490,6492],{"className":6491},[115],[45,6493,6495],{"className":6494,"style":150},[119],[45,6496],{},[45,6498],{"className":6499,"style":1197},[73],[45,6501,2014],{"className":6502},[1201],[45,6504],{"className":6505,"style":1197},[73],[45,6507,6509,6512,6515,6518,6558,6561,6567,6607,6610,6650,6653,6656],{"className":6508},[57],[45,6510],{"className":6511,"style":1983},[61],[45,6513,6095],{"className":6514,"style":6094},[66,101],[45,6516],{"className":6517,"style":161},[73],[45,6519,6521,6524],{"className":6520},[66],[45,6522,4424],{"className":6523},[66,101],[45,6525,6527],{"className":6526},[106],[45,6528,6530,6550],{"className":6529},[110,111],[45,6531,6533,6547],{"className":6532},[115],[45,6534,6536],{"className":6535,"style":4437},[119],[45,6537,6538,6541],{"style":123},[45,6539],{"className":6540,"style":128},[127],[45,6542,6544],{"className":6543},[132,133,134,135],[45,6545,4294],{"className":6546},[66,101,135],[45,6548,143],{"className":6549},[142],[45,6551,6553],{"className":6552},[115],[45,6554,6556],{"className":6555,"style":150},[119],[45,6557],{},[45,6559],{"className":6560,"style":161},[73],[45,6562,6564],{"className":6563},[93],[45,6565,277],{"className":6566},[1351,1148],[45,6568,6570,6573],{"className":6569},[66],[45,6571,2328],{"className":6572,"style":2327},[66,101],[45,6574,6576],{"className":6575},[106],[45,6577,6579,6599],{"className":6578},[110,111],[45,6580,6582,6596],{"className":6581},[115],[45,6583,6585],{"className":6584,"style":4437},[119],[45,6586,6587,6590],{"style":2358},[45,6588],{"className":6589,"style":128},[127],[45,6591,6593],{"className":6592},[132,133,134,135],[45,6594,4294],{"className":6595},[66,101,135],[45,6597,143],{"className":6598},[142],[45,6600,6602],{"className":6601},[115],[45,6603,6605],{"className":6604,"style":150},[119],[45,6606],{},[45,6608],{"className":6609,"style":161},[73],[45,6611,6613,6616],{"className":6612},[66],[45,6614,4267],{"className":6615},[66,569],[45,6617,6619],{"className":6618},[106],[45,6620,6622,6642],{"className":6621},[110,111],[45,6623,6625,6639],{"className":6624},[115],[45,6626,6628],{"className":6627,"style":4437},[119],[45,6629,6630,6633],{"style":123},[45,6631],{"className":6632,"style":128},[127],[45,6634,6636],{"className":6635},[132,133,134,135],[45,6637,4294],{"className":6638},[66,101,135],[45,6640,143],{"className":6641},[142],[45,6643,6645],{"className":6644},[115],[45,6646,6648],{"className":6647,"style":150},[119],[45,6649],{},[45,6651],{"className":6652,"style":1197},[73],[45,6654,1202],{"className":6655},[1201],[45,6657],{"className":6658,"style":1197},[73],[45,6660,6662,6665,6668,6671,6720,6723,6726,6777,6817,6820,6826],{"className":6661},[57],[45,6663],{"className":6664,"style":5609},[61],[45,6666,2028],{"className":6667,"style":2027},[66,101],[45,6669],{"className":6670,"style":161},[73],[45,6672,6674,6677],{"className":6673},[66],[45,6675,4267],{"className":6676},[66,569],[45,6678,6680],{"className":6679},[106],[45,6681,6683,6712],{"className":6682},[110,111],[45,6684,6686,6709],{"className":6685},[115],[45,6687,6689],{"className":6688,"style":120},[119],[45,6690,6691,6694],{"style":123},[45,6692],{"className":6693,"style":128},[127],[45,6695,6697],{"className":6696},[132,133,134,135],[45,6698,6700,6703,6706],{"className":6699},[66,135],[45,6701,4294],{"className":6702},[66,101,135],[45,6704,2014],{"className":6705},[1201,135],[45,6707,42],{"className":6708},[66,135],[45,6710,143],{"className":6711},[142],[45,6713,6715],{"className":6714},[115],[45,6716,6718],{"className":6717,"style":4310},[119],[45,6719],{},[45,6721],{"className":6722,"style":161},[73],[45,6724,277],{"className":6725},[93],[45,6727,6729,6732],{"className":6728},[66],[45,6730,4267],{"className":6731},[66,569],[45,6733,6735],{"className":6734},[106],[45,6736,6738,6769],{"className":6737},[110,111],[45,6739,6741,6766],{"className":6740},[115],[45,6742,6744,6755],{"className":6743,"style":1370},[119],[45,6745,6746,6749],{"style":752},[45,6747],{"className":6748,"style":128},[127],[45,6750,6752],{"className":6751},[132,133,134,135],[45,6753,4294],{"className":6754},[66,101,135],[45,6756,6757,6760],{"style":764},[45,6758],{"className":6759,"style":128},[127],[45,6761,6763],{"className":6762},[132,133,134,135],[45,6764,430],{"className":6765},[66,135],[45,6767,143],{"className":6768},[142],[45,6770,6772],{"className":6771},[115],[45,6773,6775],{"className":6774,"style":4628},[119],[45,6776],{},[45,6778,6780,6783],{"className":6779},[66],[45,6781,252],{"className":6782,"style":570},[66,569],[45,6784,6786],{"className":6785},[106],[45,6787,6789,6809],{"className":6788},[110,111],[45,6790,6792,6806],{"className":6791},[115],[45,6793,6795],{"className":6794,"style":4437},[119],[45,6796,6797,6800],{"style":3432},[45,6798],{"className":6799,"style":128},[127],[45,6801,6803],{"className":6802},[132,133,134,135],[45,6804,4294],{"className":6805},[66,101,135],[45,6807,143],{"className":6808},[142],[45,6810,6812],{"className":6811},[115],[45,6813,6815],{"className":6814,"style":150},[119],[45,6816],{},[45,6818,321],{"className":6819},[232],[45,6821,6823],{"className":6822},[232],[45,6824,321],{"className":6825},[1351,1148],[45,6827,933],{"className":6828},[66],[11,6830,6831,6834,6835,6970,6971,7179],{},[689,6832,6833],{},"TDC"," (TD with gradient correction) is the semi-gradient TD update\n",[45,6836,6838],{"className":6837},[48],[45,6839,6841],{"className":6840,"ariaHidden":53},[52],[45,6842,6844,6847,6850,6890,6930],{"className":6843},[57],[45,6845],{"className":6846,"style":5351},[61],[45,6848,6095],{"className":6849,"style":6094},[66,101],[45,6851,6853,6856],{"className":6852},[66],[45,6854,4424],{"className":6855},[66,101],[45,6857,6859],{"className":6858},[106],[45,6860,6862,6882],{"className":6861},[110,111],[45,6863,6865,6879],{"className":6864},[115],[45,6866,6868],{"className":6867,"style":4437},[119],[45,6869,6870,6873],{"style":123},[45,6871],{"className":6872,"style":128},[127],[45,6874,6876],{"className":6875},[132,133,134,135],[45,6877,4294],{"className":6878},[66,101,135],[45,6880,143],{"className":6881},[142],[45,6883,6885],{"className":6884},[115],[45,6886,6888],{"className":6887,"style":150},[119],[45,6889],{},[45,6891,6893,6896],{"className":6892},[66],[45,6894,2328],{"className":6895,"style":2327},[66,101],[45,6897,6899],{"className":6898},[106],[45,6900,6902,6922],{"className":6901},[110,111],[45,6903,6905,6919],{"className":6904},[115],[45,6906,6908],{"className":6907,"style":4437},[119],[45,6909,6910,6913],{"style":2358},[45,6911],{"className":6912,"style":128},[127],[45,6914,6916],{"className":6915},[132,133,134,135],[45,6917,4294],{"className":6918},[66,101,135],[45,6920,143],{"className":6921},[142],[45,6923,6925],{"className":6924},[115],[45,6926,6928],{"className":6927,"style":150},[119],[45,6929],{},[45,6931,6933,6936],{"className":6932},[66],[45,6934,4267],{"className":6935},[66,569],[45,6937,6939],{"className":6938},[106],[45,6940,6942,6962],{"className":6941},[110,111],[45,6943,6945,6959],{"className":6944},[115],[45,6946,6948],{"className":6947,"style":4437},[119],[45,6949,6950,6953],{"style":123},[45,6951],{"className":6952,"style":128},[127],[45,6954,6956],{"className":6955},[132,133,134,135],[45,6957,4294],{"className":6958},[66,101,135],[45,6960,143],{"className":6961},[142],[45,6963,6965],{"className":6964},[115],[45,6966,6968],{"className":6967,"style":150},[119],[45,6969],{}," plus a correction term ",[45,6972,6974],{"className":6973},[48],[45,6975,6977],{"className":6976,"ariaHidden":53},[52],[45,6978,6980,6984,6987,6990,7030,7033,7082,7085,7136,7176],{"className":6979},[57],[45,6981],{"className":6982,"style":6983},[61],"height:1.0991em;vertical-align:-0.25em;",[45,6985,1202],{"className":6986},[66],[45,6988,6095],{"className":6989,"style":6094},[66,101],[45,6991,6993,6996],{"className":6992},[66],[45,6994,4424],{"className":6995},[66,101],[45,6997,6999],{"className":6998},[106],[45,7000,7002,7022],{"className":7001},[110,111],[45,7003,7005,7019],{"className":7004},[115],[45,7006,7008],{"className":7007,"style":4437},[119],[45,7009,7010,7013],{"style":123},[45,7011],{"className":7012,"style":128},[127],[45,7014,7016],{"className":7015},[132,133,134,135],[45,7017,4294],{"className":7018},[66,101,135],[45,7020,143],{"className":7021},[142],[45,7023,7025],{"className":7024},[115],[45,7026,7028],{"className":7027,"style":150},[119],[45,7029],{},[45,7031,2028],{"className":7032,"style":2027},[66,101],[45,7034,7036,7039],{"className":7035},[66],[45,7037,4267],{"className":7038},[66,569],[45,7040,7042],{"className":7041},[106],[45,7043,7045,7074],{"className":7044},[110,111],[45,7046,7048,7071],{"className":7047},[115],[45,7049,7051],{"className":7050,"style":120},[119],[45,7052,7053,7056],{"style":123},[45,7054],{"className":7055,"style":128},[127],[45,7057,7059],{"className":7058},[132,133,134,135],[45,7060,7062,7065,7068],{"className":7061},[66,135],[45,7063,4294],{"className":7064},[66,101,135],[45,7066,2014],{"className":7067},[1201,135],[45,7069,42],{"className":7070},[66,135],[45,7072,143],{"className":7073},[142],[45,7075,7077],{"className":7076},[115],[45,7078,7080],{"className":7079,"style":4310},[119],[45,7081],{},[45,7083,277],{"className":7084},[93],[45,7086,7088,7091],{"className":7087},[66],[45,7089,4267],{"className":7090},[66,569],[45,7092,7094],{"className":7093},[106],[45,7095,7097,7128],{"className":7096},[110,111],[45,7098,7100,7125],{"className":7099},[115],[45,7101,7103,7114],{"className":7102,"style":417},[119],[45,7104,7105,7108],{"style":752},[45,7106],{"className":7107,"style":128},[127],[45,7109,7111],{"className":7110},[132,133,134,135],[45,7112,4294],{"className":7113},[66,101,135],[45,7115,7116,7119],{"style":420},[45,7117],{"className":7118,"style":128},[127],[45,7120,7122],{"className":7121},[132,133,134,135],[45,7123,430],{"className":7124},[66,135],[45,7126,143],{"className":7127},[142],[45,7129,7131],{"className":7130},[115],[45,7132,7134],{"className":7133,"style":4628},[119],[45,7135],{},[45,7137,7139,7142],{"className":7138},[66],[45,7140,252],{"className":7141,"style":570},[66,569],[45,7143,7145],{"className":7144},[106],[45,7146,7148,7168],{"className":7147},[110,111],[45,7149,7151,7165],{"className":7150},[115],[45,7152,7154],{"className":7153,"style":4437},[119],[45,7155,7156,7159],{"style":3432},[45,7157],{"className":7158,"style":128},[127],[45,7160,7162],{"className":7161},[132,133,134,135],[45,7163,4294],{"className":7164},[66,101,135],[45,7166,143],{"className":7167},[142],[45,7169,7171],{"className":7170},[115],[45,7172,7174],{"className":7173,"style":150},[119],[45,7175],{},[45,7177,321],{"className":7178},[232]," that supplies the missing half of the\ngradient. That single correction is the difference between divergence and convergence.",[664,7181],{"hash":7182},"cbb69ce3577e35b807b8c382534d64bd127f1a6aecd1d437c8b9dbb453f7d5ee",[11,7184,7185,7186,7235,7236,7239,7240,7256,7257,7306,7307,7353,7354,7357,7358,7373,7374,7389,7390,7405,7406,7421,7422,7425,7426,7461,7462,7502,7503,933],{},"On Baird's counterexample TDC does exactly what the theory promises: the\n",[45,7187,7189],{"className":7188},[48],[45,7190,7192],{"className":7191,"ariaHidden":53},[52],[45,7193,7195,7198],{"className":7194},[57],[45,7196],{"className":7197,"style":2592},[61],[45,7199,7201],{"className":7200},[66,2582],[45,7202,7204],{"className":7203},[110],[45,7205,7207],{"className":7206},[115],[45,7208,7210,7227],{"className":7209,"style":2592},[119],[45,7211,7212,7215],{"style":1154},[45,7213],{"className":7214,"style":1103},[127],[45,7216,7218,7221,7224],{"className":7217},[66],[45,7219,2993],{"className":7220,"style":2992},[66,101],[45,7222,1614],{"className":7223,"style":1613},[66,101],[45,7225,2608],{"className":7226,"style":2607},[66,101],[45,7228,7229,7232],{"style":2611},[45,7230],{"className":7231,"style":1103},[127],[45,7233],{"className":7234,"style":2619},[2618]," decays to zero and the weights stop growing. Notice the components do\n",[29,7237,7238],{},"not"," all reach zero, so the value error stays near ",[45,7241,7243],{"className":7242},[48],[45,7244,7246],{"className":7245,"ariaHidden":53},[52],[45,7247,7249,7253],{"className":7248},[57],[45,7250],{"className":7251,"style":7252},[61],"height:0.6444em;",[45,7254,774],{"className":7255},[66]," — the ",[45,7258,7260],{"className":7259},[48],[45,7261,7263],{"className":7262,"ariaHidden":53},[52],[45,7264,7266,7269],{"className":7265},[57],[45,7267],{"className":7268,"style":2592},[61],[45,7270,7272],{"className":7271},[66,2582],[45,7273,7275],{"className":7274},[110],[45,7276,7278],{"className":7277},[115],[45,7279,7281,7298],{"className":7280,"style":2592},[119],[45,7282,7283,7286],{"style":1154},[45,7284],{"className":7285,"style":1103},[127],[45,7287,7289,7292,7295],{"className":7288},[66],[45,7290,2993],{"className":7291,"style":2992},[66,101],[45,7293,1614],{"className":7294,"style":1613},[66,101],[45,7296,2608],{"className":7297,"style":2607},[66,101],[45,7299,7300,7303],{"style":2611},[45,7301],{"className":7302,"style":1103},[127],[45,7304],{"className":7305,"style":2619},[2618]," fixed point\nis not the best approximation in the ",[45,7308,7310],{"className":7309},[48],[45,7311,7313],{"className":7312,"ariaHidden":53},[52],[45,7314,7316,7319],{"className":7315},[57],[45,7317],{"className":7318,"style":2592},[61],[45,7320,7322],{"className":7321},[66,2582],[45,7323,7325],{"className":7324},[110],[45,7326,7328],{"className":7327},[115],[45,7329,7331,7345],{"className":7330,"style":2592},[119],[45,7332,7333,7336],{"style":1154},[45,7334],{"className":7335,"style":1103},[127],[45,7337,7339,7342],{"className":7338},[66],[45,7340,3593],{"className":7341,"style":1197},[66,101],[45,7343,2608],{"className":7344,"style":2607},[66,101],[45,7346,7347,7350],{"style":2611},[45,7348],{"className":7349,"style":1103},[127],[45,7351],{"className":7352,"style":2619},[2618]," sense — but the divergence is gone.\nGTD2 and TDC are ",[689,7355,7356],{},"two-time-scale"," methods: the correction estimate ",[45,7359,7361],{"className":7360},[48],[45,7362,7364],{"className":7363,"ariaHidden":53},[52],[45,7365,7367,7370],{"className":7366},[57],[45,7368],{"className":7369,"style":5006},[61],[45,7371,252],{"className":7372,"style":570},[66,569]," (fast,\nstep size ",[45,7375,7377],{"className":7376},[48],[45,7378,7380],{"className":7379,"ariaHidden":53},[52],[45,7381,7383,7386],{"className":7382},[57],[45,7384],{"className":7385,"style":5351},[61],[45,7387,5356],{"className":7388,"style":5355},[66,101],") must track ahead of the primary weights ",[45,7391,7393],{"className":7392},[48],[45,7394,7396],{"className":7395,"ariaHidden":53},[52],[45,7397,7399,7402],{"className":7398},[57],[45,7400],{"className":7401,"style":5006},[61],[45,7403,571],{"className":7404,"style":570},[66,569]," (slow, step size\n",[45,7407,7409],{"className":7408},[48],[45,7410,7412],{"className":7411,"ariaHidden":53},[52],[45,7413,7415,7418],{"className":7414},[57],[45,7416],{"className":7417,"style":247},[61],[45,7419,6095],{"className":7420,"style":6094},[66,101],"), which the convergence proofs formalize as a ",[29,7423,7424],{},"cascade"," requiring ",[45,7427,7429],{"className":7428},[48],[45,7430,7432,7451],{"className":7431,"ariaHidden":53},[52],[45,7433,7435,7438,7441,7444,7448],{"className":7434},[57],[45,7436],{"className":7437,"style":5351},[61],[45,7439,5356],{"className":7440,"style":5355},[66,101],[45,7442],{"className":7443,"style":74},[73],[45,7445,7447],{"className":7446},[78],"→",[45,7449],{"className":7450,"style":74},[73],[45,7452,7454,7457],{"className":7453},[57],[45,7455],{"className":7456,"style":7252},[61],[45,7458,7460],{"className":7459},[66],"0","\nand ",[45,7463,7465],{"className":7464},[48],[45,7466,7468,7493],{"className":7467,"ariaHidden":53},[52],[45,7469,7471,7474,7477,7481,7484,7487,7490],{"className":7470},[57],[45,7472],{"className":7473,"style":444},[61],[45,7475,6095],{"className":7476,"style":6094},[66,101],[45,7478,7480],{"className":7479},[66],"\u002F",[45,7482,5356],{"className":7483,"style":5355},[66,101],[45,7485],{"className":7486,"style":74},[73],[45,7488,7447],{"className":7489},[78],[45,7491],{"className":7492,"style":74},[73],[45,7494,7496,7499],{"className":7495},[57],[45,7497],{"className":7498,"style":7252},[61],[45,7500,7460],{"className":7501},[66],". Gradient-TD is currently the best-understood and most widely\nused family of stable off-policy methods, with extensions to action values, control, and\n",[15,7504,7506],{"href":7505},"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces","eligibility traces",[21,7508,7510],{"id":7509},"a-note-on-emphatic-td","A note on emphatic-TD",[11,7512,7513,7514,7517,7518,7521,7522,7530,7531,7546,7547,940,7550,7605],{},"Gradient-TD attacks the second part of the problem — the wrong update distribution — by\nbuilding a true gradient method. ",[689,7515,7516],{},"Emphatic-TD"," attacks it differently: keep the simple,\ncomputationally cheap semi-gradient update, but ",[29,7519,7520],{},"reweight the states"," so the update\ndistribution recovers the on-policy properties that make linear semi-gradient TD\nstable.",[34,7523,7524],{},[15,7525,7529],{"href":7526,"ariaDescribedBy":7527,"dataFootnoteRef":6,"id":7528},"#user-content-fn-sb-etd",[40],"user-content-fnref-sb-etd","4"," Linear semi-gradient TD is stable on-policy precisely because the state\ndistribution matches the transitions under ",[45,7532,7534],{"className":7533},[48],[45,7535,7537],{"className":7536,"ariaHidden":53},[52],[45,7538,7540,7543],{"className":7539},[57],[45,7541],{"className":7542,"style":247},[61],[45,7544,646],{"className":7545,"style":251},[66,101],"; emphatic-TD manufactures that match by\nscaling each update with an ",[689,7548,7549],{},"emphasis",[45,7551,7553],{"className":7552},[48],[45,7554,7556],{"className":7555,"ariaHidden":53},[52],[45,7557,7559,7562],{"className":7558},[57],[45,7560],{"className":7561,"style":1539},[61],[45,7563,7565,7570],{"className":7564},[66],[45,7566,7569],{"className":7567,"style":7568},[66,101],"margin-right:0.109em;","M",[45,7571,7573],{"className":7572},[106],[45,7574,7576,7597],{"className":7575},[110,111],[45,7577,7579,7594],{"className":7578},[115],[45,7580,7582],{"className":7581,"style":4437},[119],[45,7583,7585,7588],{"style":7584},"top:-2.55em;margin-left:-0.109em;margin-right:0.05em;",[45,7586],{"className":7587,"style":128},[127],[45,7589,7591],{"className":7590},[132,133,134,135],[45,7592,4294],{"className":7593},[66,101,135],[45,7595,143],{"className":7596},[142],[45,7598,7600],{"className":7599},[115],[45,7601,7603],{"className":7602,"style":150},[119],[45,7604],{}," that accumulates responsibility for the\ncurrent state through a recursive trace,",[45,7607,7609],{"className":7608},[710],[45,7610,7612],{"className":7611},[48],[45,7613,7615,7685,7740,8074,8196],{"className":7614,"ariaHidden":53},[52],[45,7616,7618,7621,7670,7673,7676,7679,7682],{"className":7617},[57],[45,7619],{"className":7620,"style":4260},[61],[45,7622,7624,7627],{"className":7623},[66],[45,7625,571],{"className":7626,"style":570},[66,569],[45,7628,7630],{"className":7629},[106],[45,7631,7633,7662],{"className":7632},[110,111],[45,7634,7636,7659],{"className":7635},[115],[45,7637,7639],{"className":7638,"style":120},[119],[45,7640,7641,7644],{"style":3432},[45,7642],{"className":7643,"style":128},[127],[45,7645,7647],{"className":7646},[132,133,134,135],[45,7648,7650,7653,7656],{"className":7649},[66,135],[45,7651,4294],{"className":7652},[66,101,135],[45,7654,2014],{"className":7655},[1201,135],[45,7657,42],{"className":7658},[66,135],[45,7660,143],{"className":7661},[142],[45,7663,7665],{"className":7664},[115],[45,7666,7668],{"className":7667,"style":4310},[119],[45,7669],{},[45,7671],{"className":7672,"style":74},[73],[45,7674],{"className":7675,"style":74},[73],[45,7677,79],{"className":7678},[78],[45,7680],{"className":7681,"style":74},[73],[45,7683],{"className":7684,"style":74},[73],[45,7686,7688,7691,7731,7734,7737],{"className":7687},[57],[45,7689],{"className":7690,"style":5443},[61],[45,7692,7694,7697],{"className":7693},[66],[45,7695,571],{"className":7696,"style":570},[66,569],[45,7698,7700],{"className":7699},[106],[45,7701,7703,7723],{"className":7702},[110,111],[45,7704,7706,7720],{"className":7705},[115],[45,7707,7709],{"className":7708,"style":4437},[119],[45,7710,7711,7714],{"style":3432},[45,7712],{"className":7713,"style":128},[127],[45,7715,7717],{"className":7716},[132,133,134,135],[45,7718,4294],{"className":7719},[66,101,135],[45,7721,143],{"className":7722},[142],[45,7724,7726],{"className":7725},[115],[45,7727,7729],{"className":7728,"style":150},[119],[45,7730],{},[45,7732],{"className":7733,"style":1197},[73],[45,7735,2014],{"className":7736},[1201],[45,7738],{"className":7739,"style":1197},[73],[45,7741,7743,7746,7749,7752,7792,7795,7835,7838,7878,7881,7884,7917,7920,7961,7964,7967,8007,8010,8013,8016,8019,8059,8062,8065,8068,8071],{"className":7742},[57],[45,7744],{"className":7745,"style":444},[61],[45,7747,6095],{"className":7748,"style":6094},[66,101],[45,7750],{"className":7751,"style":161},[73],[45,7753,7755,7758],{"className":7754},[66],[45,7756,7569],{"className":7757,"style":7568},[66,101],[45,7759,7761],{"className":7760},[106],[45,7762,7764,7784],{"className":7763},[110,111],[45,7765,7767,7781],{"className":7766},[115],[45,7768,7770],{"className":7769,"style":4437},[119],[45,7771,7772,7775],{"style":7584},[45,7773],{"className":7774,"style":128},[127],[45,7776,7778],{"className":7777},[132,133,134,135],[45,7779,4294],{"className":7780},[66,101,135],[45,7782,143],{"className":7783},[142],[45,7785,7787],{"className":7786},[115],[45,7788,7790],{"className":7789,"style":150},[119],[45,7791],{},[45,7793],{"className":7794,"style":161},[73],[45,7796,7798,7801],{"className":7797},[66],[45,7799,4424],{"className":7800},[66,101],[45,7802,7804],{"className":7803},[106],[45,7805,7807,7827],{"className":7806},[110,111],[45,7808,7810,7824],{"className":7809},[115],[45,7811,7813],{"className":7812,"style":4437},[119],[45,7814,7815,7818],{"style":123},[45,7816],{"className":7817,"style":128},[127],[45,7819,7821],{"className":7820},[132,133,134,135],[45,7822,4294],{"className":7823},[66,101,135],[45,7825,143],{"className":7826},[142],[45,7828,7830],{"className":7829},[115],[45,7831,7833],{"className":7832,"style":150},[119],[45,7834],{},[45,7836],{"className":7837,"style":161},[73],[45,7839,7841,7844],{"className":7840},[66],[45,7842,2328],{"className":7843,"style":2327},[66,101],[45,7845,7847],{"className":7846},[106],[45,7848,7850,7870],{"className":7849},[110,111],[45,7851,7853,7867],{"className":7852},[115],[45,7854,7856],{"className":7855,"style":4437},[119],[45,7857,7858,7861],{"style":2358},[45,7859],{"className":7860,"style":128},[127],[45,7862,7864],{"className":7863},[132,133,134,135],[45,7865,4294],{"className":7866},[66,101,135],[45,7868,143],{"className":7869},[142],[45,7871,7873],{"className":7872},[115],[45,7874,7876],{"className":7875,"style":150},[119],[45,7877],{},[45,7879],{"className":7880,"style":161},[73],[45,7882,4335],{"className":7883},[66],[45,7885,7887],{"className":7886},[66,2308],[45,7888,7890],{"className":7889},[110],[45,7891,7893],{"className":7892},[115],[45,7894,7896,7904],{"className":7895,"style":513},[119],[45,7897,7898,7901],{"style":1154},[45,7899],{"className":7900,"style":1103},[127],[45,7902,252],{"className":7903,"style":251},[66,101],[45,7905,7906,7909],{"style":1154},[45,7907],{"className":7908,"style":1103},[127],[45,7910,7913],{"className":7911,"style":7912},[2338],"left:-0.2222em;",[45,7914,7916],{"className":7915},[66],"^",[45,7918,277],{"className":7919},[93],[45,7921,7923,7926],{"className":7922},[66],[45,7924,69],{"className":7925,"style":2607},[66,101],[45,7927,7929],{"className":7928},[106],[45,7930,7932,7953],{"className":7931},[110,111],[45,7933,7935,7950],{"className":7934},[115],[45,7936,7938],{"className":7937,"style":4437},[119],[45,7939,7941,7944],{"style":7940},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[45,7942],{"className":7943,"style":128},[127],[45,7945,7947],{"className":7946},[132,133,134,135],[45,7948,4294],{"className":7949},[66,101,135],[45,7951,143],{"className":7952},[142],[45,7954,7956],{"className":7955},[115],[45,7957,7959],{"className":7958,"style":150},[119],[45,7960],{},[45,7962,157],{"className":7963},[156],[45,7965],{"className":7966,"style":161},[73],[45,7968,7970,7973],{"className":7969},[66],[45,7971,571],{"className":7972,"style":570},[66,569],[45,7974,7976],{"className":7975},[106],[45,7977,7979,7999],{"className":7978},[110,111],[45,7980,7982,7996],{"className":7981},[115],[45,7983,7985],{"className":7984,"style":4437},[119],[45,7986,7987,7990],{"style":3432},[45,7988],{"className":7989,"style":128},[127],[45,7991,7993],{"className":7992},[132,133,134,135],[45,7994,4294],{"className":7995},[66,101,135],[45,7997,143],{"className":7998},[142],[45,8000,8002],{"className":8001},[115],[45,8003,8005],{"className":8004,"style":150},[119],[45,8006],{},[45,8008,321],{"className":8009},[232],[45,8011,157],{"className":8012},[156],[45,8014],{"className":8015,"style":1312},[73],[45,8017],{"className":8018,"style":161},[73],[45,8020,8022,8025],{"className":8021},[66],[45,8023,7569],{"className":8024,"style":7568},[66,101],[45,8026,8028],{"className":8027},[106],[45,8029,8031,8051],{"className":8030},[110,111],[45,8032,8034,8048],{"className":8033},[115],[45,8035,8037],{"className":8036,"style":4437},[119],[45,8038,8039,8042],{"style":7584},[45,8040],{"className":8041,"style":128},[127],[45,8043,8045],{"className":8044},[132,133,134,135],[45,8046,4294],{"className":8047},[66,101,135],[45,8049,143],{"className":8050},[142],[45,8052,8054],{"className":8053},[115],[45,8055,8057],{"className":8056,"style":150},[119],[45,8058],{},[45,8060],{"className":8061,"style":74},[73],[45,8063],{"className":8064,"style":74},[73],[45,8066,79],{"className":8067},[78],[45,8069],{"className":8070,"style":74},[73],[45,8072],{"className":8073,"style":74},[73],[45,8075,8077,8080,8083,8086,8135,8138,8187,8190,8193],{"className":8076},[57],[45,8078],{"className":8079,"style":5955},[61],[45,8081,2028],{"className":8082,"style":2027},[66,101],[45,8084],{"className":8085,"style":161},[73],[45,8087,8089,8092],{"className":8088},[66],[45,8090,4424],{"className":8091},[66,101],[45,8093,8095],{"className":8094},[106],[45,8096,8098,8127],{"className":8097},[110,111],[45,8099,8101,8124],{"className":8100},[115],[45,8102,8104],{"className":8103,"style":120},[119],[45,8105,8106,8109],{"style":123},[45,8107],{"className":8108,"style":128},[127],[45,8110,8112],{"className":8111},[132,133,134,135],[45,8113,8115,8118,8121],{"className":8114},[66,135],[45,8116,4294],{"className":8117},[66,101,135],[45,8119,1202],{"className":8120},[1201,135],[45,8122,42],{"className":8123},[66,135],[45,8125,143],{"className":8126},[142],[45,8128,8130],{"className":8129},[115],[45,8131,8133],{"className":8132,"style":4310},[119],[45,8134],{},[45,8136],{"className":8137,"style":161},[73],[45,8139,8141,8144],{"className":8140},[66],[45,8142,7569],{"className":8143,"style":7568},[66,101],[45,8145,8147],{"className":8146},[106],[45,8148,8150,8179],{"className":8149},[110,111],[45,8151,8153,8176],{"className":8152},[115],[45,8154,8156],{"className":8155,"style":120},[119],[45,8157,8158,8161],{"style":7584},[45,8159],{"className":8160,"style":128},[127],[45,8162,8164],{"className":8163},[132,133,134,135],[45,8165,8167,8170,8173],{"className":8166},[66,135],[45,8168,4294],{"className":8169},[66,101,135],[45,8171,1202],{"className":8172},[1201,135],[45,8174,42],{"className":8175},[66,135],[45,8177,143],{"className":8178},[142],[45,8180,8182],{"className":8181},[115],[45,8183,8185],{"className":8184,"style":4310},[119],[45,8186],{},[45,8188],{"className":8189,"style":1197},[73],[45,8191,2014],{"className":8192},[1201],[45,8194],{"className":8195,"style":1197},[73],[45,8197,8199,8203,8246],{"className":8198},[57],[45,8200],{"className":8201,"style":8202},[61],"height:0.8778em;vertical-align:-0.1944em;",[45,8204,8206,8211],{"className":8205},[66],[45,8207,8210],{"className":8208,"style":8209},[66,101],"margin-right:0.0785em;","I",[45,8212,8214],{"className":8213},[106],[45,8215,8217,8238],{"className":8216},[110,111],[45,8218,8220,8235],{"className":8219},[115],[45,8221,8223],{"className":8222,"style":4437},[119],[45,8224,8226,8229],{"style":8225},"top:-2.55em;margin-left:-0.0785em;margin-right:0.05em;",[45,8227],{"className":8228,"style":128},[127],[45,8230,8232],{"className":8231},[132,133,134,135],[45,8233,4294],{"className":8234},[66,101,135],[45,8236,143],{"className":8237},[142],[45,8239,8241],{"className":8240},[115],[45,8242,8244],{"className":8243,"style":150},[119],[45,8245],{},[45,8247,157],{"className":8248},[156],[11,8250,8251,8252,940,8255,8307],{},"with the ",[689,8253,8254],{},"interest",[45,8256,8258],{"className":8257},[48],[45,8259,8261],{"className":8260,"ariaHidden":53},[52],[45,8262,8264,8267],{"className":8263},[57],[45,8265],{"className":8266,"style":1539},[61],[45,8268,8270,8273],{"className":8269},[66],[45,8271,8210],{"className":8272,"style":8209},[66,101],[45,8274,8276],{"className":8275},[106],[45,8277,8279,8299],{"className":8278},[110,111],[45,8280,8282,8296],{"className":8281},[115],[45,8283,8285],{"className":8284,"style":4437},[119],[45,8286,8287,8290],{"style":8225},[45,8288],{"className":8289,"style":128},[127],[45,8291,8293],{"className":8292},[132,133,134,135],[45,8294,4294],{"className":8295},[66,101,135],[45,8297,143],{"className":8298},[142],[45,8300,8302],{"className":8301},[115],[45,8303,8305],{"className":8304,"style":150},[119],[45,8306],{}," specifying how much we care about accurately valuing each\nstate. In expectation, emphatic-TD converges to zero value error on Baird's\ncounterexample; in practice its variance is so high that consistent results are hard to\nobtain, which is why the next section turns to variance-reduction techniques for all\nof these methods.",[21,8309,8311],{"id":8310},"reducing-variance","Reducing variance",[11,8313,8314,8315,8318,8319,8327],{},"Stability is only half the problem. Gradient-TD and emphatic-TD keep the weights\nfrom diverging, but off-policy learning is ",[29,8316,8317],{},"inherently"," higher-variance than\non-policy learning, and that variance is what makes these methods slow and their\nresults erratic in practice.",[34,8320,8321],{},[15,8322,8326],{"href":8323,"ariaDescribedBy":8324,"dataFootnoteRef":6,"id":8325},"#user-content-fn-sb-variance",[40],"user-content-fnref-sb-variance","5"," The intuition is plain: if the data you\nreceive is only loosely related to the policy you are trying to evaluate, you\nshould expect to learn less about it. In the extreme you learn nothing — you\ncannot learn to drive by cooking dinner. Only when the target and behavior\npolicies visit similar states and take similar actions is real progress possible.",[11,8329,8330,8331,933],{},"Yet that overlap is what off-policy learning is for: any policy has many\nneighbors, similar-but-not-identical policies with considerable overlap, and the\nwhole reason to learn off-policy is to generalize across that vast family of\nrelated policies from one stream of experience. So the variance cannot simply be\navoided; it has to be ",[29,8332,8333],{},"managed",[1585,8335,8337],{"id":8336},"where-the-variance-comes-from","Where the variance comes from",[11,8339,8340,8341,8621,8622,8637,8638,8653,8654,8657,8658],{},"The specific source is the importance-sampling ratio. As we saw, off-policy\ntargets are corrected by products of per-step ratios ",[45,8342,8344],{"className":8343},[48],[45,8345,8347,8402,8464,8572],{"className":8346,"ariaHidden":53},[52],[45,8348,8350,8353,8393,8396,8399],{"className":8349},[57],[45,8351],{"className":8352,"style":682},[61],[45,8354,8356,8359],{"className":8355},[66],[45,8357,4424],{"className":8358},[66,101],[45,8360,8362],{"className":8361},[106],[45,8363,8365,8385],{"className":8364},[110,111],[45,8366,8368,8382],{"className":8367},[115],[45,8369,8371],{"className":8370,"style":4437},[119],[45,8372,8373,8376],{"style":123},[45,8374],{"className":8375,"style":128},[127],[45,8377,8379],{"className":8378},[132,133,134,135],[45,8380,4294],{"className":8381},[66,101,135],[45,8383,143],{"className":8384},[142],[45,8386,8388],{"className":8387},[115],[45,8389,8391],{"className":8390,"style":150},[119],[45,8392],{},[45,8394],{"className":8395,"style":74},[73],[45,8397,79],{"className":8398},[78],[45,8400],{"className":8401,"style":74},[73],[45,8403,8405,8408,8411,8414,8455,8458,8461],{"className":8404},[57],[45,8406],{"className":8407,"style":444},[61],[45,8409,646],{"className":8410,"style":251},[66,101],[45,8412,277],{"className":8413},[93],[45,8415,8417,8421],{"className":8416},[66],[45,8418,8420],{"className":8419},[66,101],"A",[45,8422,8424],{"className":8423},[106],[45,8425,8427,8447],{"className":8426},[110,111],[45,8428,8430,8444],{"className":8429},[115],[45,8431,8433],{"className":8432,"style":4437},[119],[45,8434,8435,8438],{"style":123},[45,8436],{"className":8437,"style":128},[127],[45,8439,8441],{"className":8440},[132,133,134,135],[45,8442,4294],{"className":8443},[66,101,135],[45,8445,143],{"className":8446},[142],[45,8448,8450],{"className":8449},[115],[45,8451,8453],{"className":8452,"style":150},[119],[45,8454],{},[45,8456],{"className":8457,"style":74},[73],[45,8459,210],{"className":8460},[78],[45,8462],{"className":8463,"style":74},[73],[45,8465,8467,8470,8510,8513,8516,8520,8523,8563,8566,8569],{"className":8466},[57],[45,8468],{"className":8469,"style":444},[61],[45,8471,8473,8476],{"className":8472},[66],[45,8474,69],{"className":8475,"style":2607},[66,101],[45,8477,8479],{"className":8478},[106],[45,8480,8482,8502],{"className":8481},[110,111],[45,8483,8485,8499],{"className":8484},[115],[45,8486,8488],{"className":8487,"style":4437},[119],[45,8489,8490,8493],{"style":7940},[45,8491],{"className":8492,"style":128},[127],[45,8494,8496],{"className":8495},[132,133,134,135],[45,8497,4294],{"className":8498},[66,101,135],[45,8500,143],{"className":8501},[142],[45,8503,8505],{"className":8504},[115],[45,8506,8508],{"className":8507,"style":150},[119],[45,8509],{},[45,8511,321],{"className":8512},[232],[45,8514,7480],{"className":8515},[66],[45,8517,8519],{"className":8518},[66,101],"b",[45,8521,277],{"className":8522},[93],[45,8524,8526,8529],{"className":8525},[66],[45,8527,8420],{"className":8528},[66,101],[45,8530,8532],{"className":8531},[106],[45,8533,8535,8555],{"className":8534},[110,111],[45,8536,8538,8552],{"className":8537},[115],[45,8539,8541],{"className":8540,"style":4437},[119],[45,8542,8543,8546],{"style":123},[45,8544],{"className":8545,"style":128},[127],[45,8547,8549],{"className":8548},[132,133,134,135],[45,8550,4294],{"className":8551},[66,101,135],[45,8553,143],{"className":8554},[142],[45,8556,8558],{"className":8557},[115],[45,8559,8561],{"className":8560,"style":150},[119],[45,8562],{},[45,8564],{"className":8565,"style":74},[73],[45,8567,210],{"className":8568},[78],[45,8570],{"className":8571,"style":74},[73],[45,8573,8575,8578,8618],{"className":8574},[57],[45,8576],{"className":8577,"style":444},[61],[45,8579,8581,8584],{"className":8580},[66],[45,8582,69],{"className":8583,"style":2607},[66,101],[45,8585,8587],{"className":8586},[106],[45,8588,8590,8610],{"className":8589},[110,111],[45,8591,8593,8607],{"className":8592},[115],[45,8594,8596],{"className":8595,"style":4437},[119],[45,8597,8598,8601],{"style":7940},[45,8599],{"className":8600,"style":128},[127],[45,8602,8604],{"className":8603},[132,133,134,135],[45,8605,4294],{"className":8606},[66,101,135],[45,8608,143],{"className":8609},[142],[45,8611,8613],{"className":8612},[115],[45,8614,8616],{"className":8615,"style":150},[119],[45,8617],{},[45,8619,321],{"className":8620},[232],". Each ratio is ",[45,8623,8625],{"className":8624},[48],[45,8626,8628],{"className":8627,"ariaHidden":53},[52],[45,8629,8631,8634],{"className":8630},[57],[45,8632],{"className":8633,"style":7252},[61],[45,8635,42],{"className":8636},[66]," in expectation, and successive ratios are\nuncorrelated, so their products are ",[45,8639,8641],{"className":8640},[48],[45,8642,8644],{"className":8643,"ariaHidden":53},[52],[45,8645,8647,8650],{"className":8646},[57],[45,8648],{"className":8649,"style":7252},[61],[45,8651,42],{"className":8652},[66]," in expectation too — but their ",[29,8655,8656],{},"values","\nswing from zero up to very large numbers, and a product of such terms has very\nhigh variance.",[34,8659,8660],{},[15,8661,8326],{"href":8323,"ariaDescribedBy":8662,"dataFootnoteRef":6,"id":8663},[40],"user-content-fnref-sb-variance-2",[11,8665,8666,8667,8728,8729,8827,8828,8926,8927,8980,8981,8827,9033,9086,9087,9139,9140,9155,9156,9245,9246,9281,9282,9351,9352,9367,9368,9371,9372,9424,9425,9492,9493,9561,9562,9614,9615,9630,9631,9683,9684,9785,9786,9801,9802,9817],{},"A worked case shows how fast the product blows up. Let the behavior policy be\nuniform over two actions, ",[45,8668,8670],{"className":8669},[48],[45,8671,8673,8697,8718],{"className":8672,"ariaHidden":53},[52],[45,8674,8676,8679,8682,8685,8688,8691,8694],{"className":8675},[57],[45,8677],{"className":8678,"style":444},[61],[45,8680,8519],{"className":8681},[66,101],[45,8683,277],{"className":8684},[93],[45,8686,15],{"className":8687},[66,101],[45,8689],{"className":8690,"style":74},[73],[45,8692,210],{"className":8693},[78],[45,8695],{"className":8696,"style":74},[73],[45,8698,8700,8703,8706,8709,8712,8715],{"className":8699},[57],[45,8701],{"className":8702,"style":444},[61],[45,8704,102],{"className":8705},[66,101],[45,8707,321],{"className":8708},[232],[45,8710],{"className":8711,"style":74},[73],[45,8713,79],{"className":8714},[78],[45,8716],{"className":8717,"style":74},[73],[45,8719,8721,8724],{"className":8720},[57],[45,8722],{"className":8723,"style":7252},[61],[45,8725,8727],{"className":8726},[66],"0.5",", while the target policy is nearly\ndeterministic, ",[45,8730,8732],{"className":8731},[48],[45,8733,8735,8796,8817],{"className":8734,"ariaHidden":53},[52],[45,8736,8738,8741,8744,8747,8787,8790,8793],{"className":8737},[57],[45,8739],{"className":8740,"style":444},[61],[45,8742,646],{"className":8743,"style":251},[66,101],[45,8745,277],{"className":8746},[93],[45,8748,8750,8753],{"className":8749},[66],[45,8751,15],{"className":8752},[66,101],[45,8754,8756],{"className":8755},[106],[45,8757,8759,8779],{"className":8758},[110,111],[45,8760,8762,8776],{"className":8761},[115],[45,8763,8765],{"className":8764,"style":120},[119],[45,8766,8767,8770],{"style":123},[45,8768],{"className":8769,"style":128},[127],[45,8771,8773],{"className":8772},[132,133,134,135],[45,8774,42],{"className":8775},[66,135],[45,8777,143],{"className":8778},[142],[45,8780,8782],{"className":8781},[115],[45,8783,8785],{"className":8784,"style":150},[119],[45,8786],{},[45,8788],{"className":8789,"style":74},[73],[45,8791,210],{"className":8792},[78],[45,8794],{"className":8795,"style":74},[73],[45,8797,8799,8802,8805,8808,8811,8814],{"className":8798},[57],[45,8800],{"className":8801,"style":444},[61],[45,8803,102],{"className":8804},[66,101],[45,8806,321],{"className":8807},[232],[45,8809],{"className":8810,"style":74},[73],[45,8812,79],{"className":8813},[78],[45,8815],{"className":8816,"style":74},[73],[45,8818,8820,8823],{"className":8819},[57],[45,8821],{"className":8822,"style":7252},[61],[45,8824,8826],{"className":8825},[66],"0.9"," and ",[45,8829,8831],{"className":8830},[48],[45,8832,8834,8895,8916],{"className":8833,"ariaHidden":53},[52],[45,8835,8837,8840,8843,8846,8886,8889,8892],{"className":8836},[57],[45,8838],{"className":8839,"style":444},[61],[45,8841,646],{"className":8842,"style":251},[66,101],[45,8844,277],{"className":8845},[93],[45,8847,8849,8852],{"className":8848},[66],[45,8850,15],{"className":8851},[66,101],[45,8853,8855],{"className":8854},[106],[45,8856,8858,8878],{"className":8857},[110,111],[45,8859,8861,8875],{"className":8860},[115],[45,8862,8864],{"className":8863,"style":120},[119],[45,8865,8866,8869],{"style":123},[45,8867],{"className":8868,"style":128},[127],[45,8870,8872],{"className":8871},[132,133,134,135],[45,8873,774],{"className":8874},[66,135],[45,8876,143],{"className":8877},[142],[45,8879,8881],{"className":8880},[115],[45,8882,8884],{"className":8883,"style":150},[119],[45,8885],{},[45,8887],{"className":8888,"style":74},[73],[45,8890,210],{"className":8891},[78],[45,8893],{"className":8894,"style":74},[73],[45,8896,8898,8901,8904,8907,8910,8913],{"className":8897},[57],[45,8899],{"className":8900,"style":444},[61],[45,8902,102],{"className":8903},[66,101],[45,8905,321],{"className":8906},[232],[45,8908],{"className":8909,"style":74},[73],[45,8911,79],{"className":8912},[78],[45,8914],{"className":8915,"style":74},[73],[45,8917,8919,8922],{"className":8918},[57],[45,8920],{"className":8921,"style":7252},[61],[45,8923,8925],{"className":8924},[66],"0.1",". The per-step\nratio is then ",[45,8928,8930],{"className":8929},[48],[45,8931,8933,8951,8970],{"className":8932,"ariaHidden":53},[52],[45,8934,8936,8939,8942,8945,8948],{"className":8935},[57],[45,8937],{"className":8938,"style":682},[61],[45,8940,4424],{"className":8941},[66,101],[45,8943],{"className":8944,"style":74},[73],[45,8946,79],{"className":8947},[78],[45,8949],{"className":8950,"style":74},[73],[45,8952,8954,8957,8961,8964,8967],{"className":8953},[57],[45,8955],{"className":8956,"style":444},[61],[45,8958,8960],{"className":8959},[66],"0.9\u002F0.5",[45,8962],{"className":8963,"style":74},[73],[45,8965,79],{"className":8966},[78],[45,8968],{"className":8969,"style":74},[73],[45,8971,8973,8976],{"className":8972},[57],[45,8974],{"className":8975,"style":7252},[61],[45,8977,8979],{"className":8978},[66],"1.8"," when the behavior policy happens to sample\n",[45,8982,8984],{"className":8983},[48],[45,8985,8987],{"className":8986,"ariaHidden":53},[52],[45,8988,8990,8993],{"className":8989},[57],[45,8991],{"className":8992,"style":534},[61],[45,8994,8996,8999],{"className":8995},[66],[45,8997,15],{"className":8998},[66,101],[45,9000,9002],{"className":9001},[106],[45,9003,9005,9025],{"className":9004},[110,111],[45,9006,9008,9022],{"className":9007},[115],[45,9009,9011],{"className":9010,"style":120},[119],[45,9012,9013,9016],{"style":123},[45,9014],{"className":9015,"style":128},[127],[45,9017,9019],{"className":9018},[132,133,134,135],[45,9020,42],{"className":9021},[66,135],[45,9023,143],{"className":9024},[142],[45,9026,9028],{"className":9027},[115],[45,9029,9031],{"className":9030,"style":150},[119],[45,9032],{},[45,9034,9036],{"className":9035},[48],[45,9037,9039,9057,9076],{"className":9038,"ariaHidden":53},[52],[45,9040,9042,9045,9048,9051,9054],{"className":9041},[57],[45,9043],{"className":9044,"style":682},[61],[45,9046,4424],{"className":9047},[66,101],[45,9049],{"className":9050,"style":74},[73],[45,9052,79],{"className":9053},[78],[45,9055],{"className":9056,"style":74},[73],[45,9058,9060,9063,9067,9070,9073],{"className":9059},[57],[45,9061],{"className":9062,"style":444},[61],[45,9064,9066],{"className":9065},[66],"0.1\u002F0.5",[45,9068],{"className":9069,"style":74},[73],[45,9071,79],{"className":9072},[78],[45,9074],{"className":9075,"style":74},[73],[45,9077,9079,9082],{"className":9078},[57],[45,9080],{"className":9081,"style":7252},[61],[45,9083,9085],{"className":9084},[66],"0.2"," when it samples ",[45,9088,9090],{"className":9089},[48],[45,9091,9093],{"className":9092,"ariaHidden":53},[52],[45,9094,9096,9099],{"className":9095},[57],[45,9097],{"className":9098,"style":534},[61],[45,9100,9102,9105],{"className":9101},[66],[45,9103,15],{"className":9104},[66,101],[45,9106,9108],{"className":9107},[106],[45,9109,9111,9131],{"className":9110},[110,111],[45,9112,9114,9128],{"className":9113},[115],[45,9115,9117],{"className":9116,"style":120},[119],[45,9118,9119,9122],{"style":123},[45,9120],{"className":9121,"style":128},[127],[45,9123,9125],{"className":9124},[132,133,134,135],[45,9126,774],{"className":9127},[66,135],[45,9129,143],{"className":9130},[142],[45,9132,9134],{"className":9133},[115],[45,9135,9137],{"className":9136,"style":150},[119],[45,9138],{},"; each occurs with\nprobability ",[45,9141,9143],{"className":9142},[48],[45,9144,9146],{"className":9145,"ariaHidden":53},[52],[45,9147,9149,9152],{"className":9148},[57],[45,9150],{"className":9151,"style":7252},[61],[45,9153,8727],{"className":9154},[66],", and the mean is ",[45,9157,9159],{"className":9158},[48],[45,9160,9162,9181,9200,9218,9236],{"className":9161,"ariaHidden":53},[52],[45,9163,9165,9168,9171,9174,9178],{"className":9164},[57],[45,9166],{"className":9167,"style":7252},[61],[45,9169,8727],{"className":9170},[66],[45,9172],{"className":9173,"style":1197},[73],[45,9175,9177],{"className":9176},[1201],"⋅",[45,9179],{"className":9180,"style":1197},[73],[45,9182,9184,9188,9191,9194,9197],{"className":9183},[57],[45,9185],{"className":9186,"style":9187},[61],"height:0.7278em;vertical-align:-0.0833em;",[45,9189,8979],{"className":9190},[66],[45,9192],{"className":9193,"style":1197},[73],[45,9195,2014],{"className":9196},[1201],[45,9198],{"className":9199,"style":1197},[73],[45,9201,9203,9206,9209,9212,9215],{"className":9202},[57],[45,9204],{"className":9205,"style":7252},[61],[45,9207,8727],{"className":9208},[66],[45,9210],{"className":9211,"style":1197},[73],[45,9213,9177],{"className":9214},[1201],[45,9216],{"className":9217,"style":1197},[73],[45,9219,9221,9224,9227,9230,9233],{"className":9220},[57],[45,9222],{"className":9223,"style":7252},[61],[45,9225,9085],{"className":9226},[66],[45,9228],{"className":9229,"style":74},[73],[45,9231,79],{"className":9232},[78],[45,9234],{"className":9235,"style":74},[73],[45,9237,9239,9242],{"className":9238},[57],[45,9240],{"className":9241,"style":7252},[61],[45,9243,42],{"className":9244},[66],", as it must\nbe. Over a horizon of ",[45,9247,9249],{"className":9248},[48],[45,9250,9252,9271],{"className":9251,"ariaHidden":53},[52],[45,9253,9255,9258,9262,9265,9268],{"className":9254},[57],[45,9256],{"className":9257,"style":247},[61],[45,9259,9261],{"className":9260},[66,101],"n",[45,9263],{"className":9264,"style":74},[73],[45,9266,79],{"className":9267},[78],[45,9269],{"className":9270,"style":74},[73],[45,9272,9274,9277],{"className":9273},[57],[45,9275],{"className":9276,"style":7252},[61],[45,9278,9280],{"className":9279},[66],"10"," steps the cumulative ratio ",[45,9283,9285],{"className":9284},[48],[45,9286,9288],{"className":9287,"ariaHidden":53},[52],[45,9289,9291,9295],{"className":9290},[57],[45,9292],{"className":9293,"style":9294},[61],"height:0.6389em;vertical-align:-0.2083em;",[45,9296,9298,9301],{"className":9297},[66],[45,9299,4424],{"className":9300},[66,101],[45,9302,9304],{"className":9303},[106],[45,9305,9307,9343],{"className":9306},[110,111],[45,9308,9310,9340],{"className":9309},[115],[45,9311,9313],{"className":9312,"style":120},[119],[45,9314,9315,9318],{"style":123},[45,9316],{"className":9317,"style":128},[127],[45,9319,9321],{"className":9320},[132,133,134,135],[45,9322,9324,9327,9330,9333,9336],{"className":9323},[66,135],[45,9325,4294],{"className":9326},[66,101,135],[45,9328,2562],{"className":9329},[78,135],[45,9331,4294],{"className":9332},[66,101,135],[45,9334,2014],{"className":9335},[1201,135],[45,9337,9339],{"className":9338},[66,135],"9",[45,9341,143],{"className":9342},[142],[45,9344,9346],{"className":9345},[115],[45,9347,9349],{"className":9348,"style":4310},[119],[45,9350],{}," is a\nproduct of ten such draws. Its mean stays exactly ",[45,9353,9355],{"className":9354},[48],[45,9356,9358],{"className":9357,"ariaHidden":53},[52],[45,9359,9361,9364],{"className":9360},[57],[45,9362],{"className":9363,"style":7252},[61],[45,9365,42],{"className":9366},[66],", but its ",[29,9369,9370],{},"spread"," is enormous:\nan all-",[45,9373,9375],{"className":9374},[48],[45,9376,9378],{"className":9377,"ariaHidden":53},[52],[45,9379,9381,9384],{"className":9380},[57],[45,9382],{"className":9383,"style":534},[61],[45,9385,9387,9390],{"className":9386},[66],[45,9388,15],{"className":9389},[66,101],[45,9391,9393],{"className":9392},[106],[45,9394,9396,9416],{"className":9395},[110,111],[45,9397,9399,9413],{"className":9398},[115],[45,9400,9402],{"className":9401,"style":120},[119],[45,9403,9404,9407],{"style":123},[45,9405],{"className":9406,"style":128},[127],[45,9408,9410],{"className":9409},[132,133,134,135],[45,9411,42],{"className":9412},[66,135],[45,9414,143],{"className":9415},[142],[45,9417,9419],{"className":9418},[115],[45,9420,9422],{"className":9421,"style":150},[119],[45,9423],{}," trajectory (probability ",[45,9426,9428],{"className":9427},[48],[45,9429,9431,9482],{"className":9430,"ariaHidden":53},[52],[45,9432,9434,9437,9441,9473,9476,9479],{"className":9433},[57],[45,9435],{"className":9436,"style":3363},[61],[45,9438,9440],{"className":9439},[66],"0.",[45,9442,9444,9447],{"className":9443},[66],[45,9445,8326],{"className":9446},[66],[45,9448,9450],{"className":9449},[106],[45,9451,9453],{"className":9452},[110],[45,9454,9456],{"className":9455},[115],[45,9457,9459],{"className":9458,"style":3363},[119],[45,9460,9461,9464],{"style":420},[45,9462],{"className":9463,"style":128},[127],[45,9465,9467],{"className":9466},[132,133,134,135],[45,9468,9470],{"className":9469},[66,135],[45,9471,9280],{"className":9472},[66,135],[45,9474],{"className":9475,"style":74},[73],[45,9477,5038],{"className":9478},[78],[45,9480],{"className":9481,"style":74},[73],[45,9483,9485,9488],{"className":9484},[57],[45,9486],{"className":9487,"style":7252},[61],[45,9489,9491],{"className":9490},[66],"0.001",") carries\n",[45,9494,9496],{"className":9495},[48],[45,9497,9499,9551],{"className":9498,"ariaHidden":53},[52],[45,9500,9502,9505,9509,9542,9545,9548],{"className":9501},[57],[45,9503],{"className":9504,"style":3363},[61],[45,9506,9508],{"className":9507},[66],"1.",[45,9510,9512,9516],{"className":9511},[66],[45,9513,9515],{"className":9514},[66],"8",[45,9517,9519],{"className":9518},[106],[45,9520,9522],{"className":9521},[110],[45,9523,9525],{"className":9524},[115],[45,9526,9528],{"className":9527,"style":3363},[119],[45,9529,9530,9533],{"style":420},[45,9531],{"className":9532,"style":128},[127],[45,9534,9536],{"className":9535},[132,133,134,135],[45,9537,9539],{"className":9538},[66,135],[45,9540,9280],{"className":9541},[66,135],[45,9543],{"className":9544,"style":74},[73],[45,9546,5038],{"className":9547},[78],[45,9549],{"className":9550,"style":74},[73],[45,9552,9554,9557],{"className":9553},[57],[45,9555],{"className":9556,"style":7252},[61],[45,9558,9560],{"className":9559},[66],"357",", while a trajectory with even one ",[45,9563,9565],{"className":9564},[48],[45,9566,9568],{"className":9567,"ariaHidden":53},[52],[45,9569,9571,9574],{"className":9570},[57],[45,9572],{"className":9573,"style":534},[61],[45,9575,9577,9580],{"className":9576},[66],[45,9578,15],{"className":9579},[66,101],[45,9581,9583],{"className":9582},[106],[45,9584,9586,9606],{"className":9585},[110,111],[45,9587,9589,9603],{"className":9588},[115],[45,9590,9592],{"className":9591,"style":120},[119],[45,9593,9594,9597],{"style":123},[45,9595],{"className":9596,"style":128},[127],[45,9598,9600],{"className":9599},[132,133,134,135],[45,9601,774],{"className":9602},[66,135],[45,9604,143],{"className":9605},[142],[45,9607,9609],{"className":9608},[115],[45,9610,9612],{"className":9611,"style":150},[119],[45,9613],{}," is scaled down by\nthe ",[45,9616,9618],{"className":9617},[48],[45,9619,9621],{"className":9620,"ariaHidden":53},[52],[45,9622,9624,9627],{"className":9623},[57],[45,9625],{"className":9626,"style":7252},[61],[45,9628,9085],{"className":9629},[66]," factor, and a mostly-",[45,9632,9634],{"className":9633},[48],[45,9635,9637],{"className":9636,"ariaHidden":53},[52],[45,9638,9640,9643],{"className":9639},[57],[45,9641],{"className":9642,"style":534},[61],[45,9644,9646,9649],{"className":9645},[66],[45,9647,15],{"className":9648},[66,101],[45,9650,9652],{"className":9651},[106],[45,9653,9655,9675],{"className":9654},[110,111],[45,9656,9658,9672],{"className":9657},[115],[45,9659,9661],{"className":9660,"style":120},[119],[45,9662,9663,9666],{"style":123},[45,9664],{"className":9665,"style":128},[127],[45,9667,9669],{"className":9668},[132,133,134,135],[45,9670,774],{"className":9671},[66,135],[45,9673,143],{"className":9674},[142],[45,9676,9678],{"className":9677},[115],[45,9679,9681],{"className":9680,"style":150},[119],[45,9682],{}," trajectory carries a ratio near ",[45,9685,9687],{"className":9686},[48],[45,9688,9690,9740],{"className":9689,"ariaHidden":53},[52],[45,9691,9693,9696,9699,9731,9734,9737],{"className":9692},[57],[45,9694],{"className":9695,"style":3363},[61],[45,9697,9440],{"className":9698},[66],[45,9700,9702,9705],{"className":9701},[66],[45,9703,774],{"className":9704},[66],[45,9706,9708],{"className":9707},[106],[45,9709,9711],{"className":9710},[110],[45,9712,9714],{"className":9713},[115],[45,9715,9717],{"className":9716,"style":3363},[119],[45,9718,9719,9722],{"style":420},[45,9720],{"className":9721,"style":128},[127],[45,9723,9725],{"className":9724},[132,133,134,135],[45,9726,9728],{"className":9727},[66,135],[45,9729,9280],{"className":9730},[66,135],[45,9732],{"className":9733,"style":74},[73],[45,9735,5038],{"className":9736},[78],[45,9738],{"className":9739,"style":74},[73],[45,9741,9743,9746,9749],{"className":9742},[57],[45,9744],{"className":9745,"style":3363},[61],[45,9747,42],{"className":9748},[66],[45,9750,9752,9755],{"className":9751},[66],[45,9753,7460],{"className":9754},[66],[45,9756,9758],{"className":9757},[106],[45,9759,9761],{"className":9760},[110],[45,9762,9764],{"className":9763},[115],[45,9765,9767],{"className":9766,"style":3363},[119],[45,9768,9769,9772],{"style":420},[45,9770],{"className":9771,"style":128},[127],[45,9773,9775],{"className":9774},[132,133,134,135],[45,9776,9778,9781],{"className":9777},[66,135],[45,9779,1202],{"className":9780},[66,135],[45,9782,9784],{"className":9783},[66,135],"7",". The variance of the product grows geometrically in ",[45,9787,9789],{"className":9788},[48],[45,9790,9792],{"className":9791,"ariaHidden":53},[52],[45,9793,9795,9798],{"className":9794},[57],[45,9796],{"className":9797,"style":247},[61],[45,9799,9261],{"className":9800},[66,101],": a rare\ntrajectory contributes a step roughly ",[45,9803,9805],{"className":9804},[48],[45,9806,9808],{"className":9807,"ariaHidden":53},[52],[45,9809,9811,9814],{"className":9810},[57],[45,9812],{"className":9813,"style":7252},[61],[45,9815,9560],{"className":9816},[66]," times the nominal size, and that single\nstep dominates the average of thousands of ordinary ones.",[664,9819],{"hash":9820},"8e8d5b2a8003797447945f15fcd4b2d1950c19a213823669158862176d2b2ac7",[11,9822,9823,9824,9839,9840,9843],{},"The damage is done because these ratios multiply the step size in the SGD update:\na large ",[45,9825,9827],{"className":9826},[48],[45,9828,9830],{"className":9829,"ariaHidden":53},[52],[45,9831,9833,9836],{"className":9832},[57],[45,9834],{"className":9835,"style":682},[61],[45,9837,4424],{"className":9838},[66,101]," means a large step, which stochastic gradient\ndescent cannot tolerate. SGD relies on averaging many small, noisy steps to estimate\nthe true gradient; an occasional enormous step can move the parameter vector into\na region of the space with a completely different gradient, and the averaging\nargument breaks. The obvious defense — shrinking the step size until no single step\nis dangerous — makes the ",[29,9841,9842],{},"expected"," step tiny and learning slow.",[664,9845],{"hash":9846},"7e758feba09c8380d82572e09d8640b050875d1c5cdb7b231c694626aa8e2f0f",[1585,9848,9850],{"id":9849},"five-ideas-for-taming-it","Five ideas for taming it",[11,9852,9853,9854],{},"No single technique suffices, but several complementary strategies each reduce\nthe variance.",[34,9855,9856],{},[15,9857,8326],{"href":8323,"ariaDescribedBy":9858,"dataFootnoteRef":6,"id":9859},[40],"user-content-fnref-sb-variance-3",[9861,9862,9863,9900,9915,9921,9936],"ul",{},[9864,9865,9866,9869,9870,9874,9875,9899],"li",{},[689,9867,9868],{},"Weighted importance sampling."," Chapter 5 already showed that\n",[15,9871,9873],{"href":9872},"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods","weighted importance sampling","\nis far better behaved than the ordinary kind — its updates have much lower\nvariance because the largest weight any sample can carry is bounded. The catch is\nthat adapting weighted importance sampling to function approximation is genuinely\nhard, and can probably only be done approximately, at ",[45,9876,9878],{"className":9877},[48],[45,9879,9881],{"className":9880,"ariaHidden":53},[52],[45,9882,9884,9887,9890,9893,9896],{"className":9883},[57],[45,9885],{"className":9886,"style":444},[61],[45,9888,4184],{"className":9889,"style":1893},[66,101],[45,9891,277],{"className":9892},[93],[45,9894,474],{"className":9895},[66,101],[45,9897,321],{"className":9898},[232]," cost.",[9864,9901,9902,9905,9906,9910,9911,9914],{},[689,9903,9904],{},"Avoiding ratios entirely."," The\n",[15,9907,9909],{"href":9908},"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping","tree-backup","\nalgorithm performs off-policy learning ",[29,9912,9913],{},"without"," any importance-sampling ratios\nat all, by backing up the expected value under the target policy rather than\ncorrecting sampled actions. Extending that idea to the function-approximation\ncase yields stable and more efficient methods, since where there are no ratios\nthere is no ratio-variance.",[9864,9916,9917,9920],{},[689,9918,9919],{},"Momentum and averaging."," Because the trouble is occasional large, uncorrelated\nsteps, optimizer machinery that smooths the update sequence helps directly:\nmomentum, Polyak–Ruppert averaging (averaging the iterates rather than trusting\nthe latest one), and their extensions all damp the effect of a single outlier\nstep.",[9864,9922,9923,9926,9927,9930,9931,9935],{},[689,9924,9925],{},"Adaptive, per-component step sizes."," Rather than one global step size\nlimited by the worst-case ratio, methods that set a ",[29,9928,9929],{},"separate",",\nadaptively-tuned step size for each component of the parameter vector — and\n",[9932,9933,9934],"q",{},"importance-weight-aware"," updates designed with these products in mind — let the\nlearning rate respond to the local variance instead of being throttled globally.",[9864,9937,9938,9941,9942,9945,9946,9949,9950,9965,9966,9981],{},[689,9939,9940],{},"Constraining the target toward the behavior."," A complementary strategy defines\nthe target policy partly ",[29,9943,9944],{},"by reference to"," the behavior policy, so that it can\nnever differ enough to create huge ratios in the first place. ",[9932,9947,9948],{},"Recognizer","-style\nconstructions cap how far ",[45,9951,9953],{"className":9952},[48],[45,9954,9956],{"className":9955,"ariaHidden":53},[52],[45,9957,9959,9962],{"className":9958},[57],[45,9960],{"className":9961,"style":247},[61],[45,9963,646],{"className":9964,"style":251},[66,101]," may stray from ",[45,9967,9969],{"className":9968},[48],[45,9970,9972],{"className":9971,"ariaHidden":53},[52],[45,9973,9975,9978],{"className":9974},[57],[45,9976],{"className":9977,"style":513},[61],[45,9979,8519],{"className":9980},[66,101],", trading a little generality\nfor a lot of stability.",[664,9983],{"hash":9984},"f8f108bf91266823c1fc5715f78cc455441761e193e2d20146aa9f2a2bfb0178",[11,9986,9987,9988,9991,9992,9994],{},"None of these is decisive on its own, and the chapter is candid that off-policy\nvariance ",[9932,9989,9990],{},"will probably always remain a challenge."," Which of these methods combine\neffectively with the stable Gradient-TD and emphatic-TD updates is still an open\nquestion — the whole area is young. But the framing is what matters: stability and\nvariance are ",[29,9993,9929],{}," problems. Gradient-TD and emphatic-TD solve the first;\nthese techniques address the second; and a usable off-policy method under function\napproximation generally needs both.",[21,9996,9998],{"id":9997},"the-triad-in-deep-reinforcement-learning","The triad in deep reinforcement learning",[11,10000,10001],{},"Sutton and Barto present the triad as a theoretical hazard demonstrated on small\nlinear problems. The obvious question — does it actually break the deep networks\nthat now dominate practice? — was answered empirically after the second edition, and\nthe answer is more nuanced than the theory alone suggests.",[11,10003,10004,10005,10008,10009,10017,10018,10051,10052,10086,10087,10149,10150,10153],{},"Van Hasselt, Doron, Strub, Hessel, Sonnerat, and Modayil (2018, ",[9932,10006,10007],{},"Deep Reinforcement Learning and the Deadly Triad",") ran a controlled study on the 57-game Atari suite,\nvarying the ingredients of the triad in a deep Q-network and measuring how often the\naction-value estimates blew up.",[34,10010,10011],{},[15,10012,10016],{"href":10013,"ariaDescribedBy":10014,"dataFootnoteRef":6,"id":10015},"#user-content-fn-vh-triad",[40],"user-content-fnref-vh-triad","6"," With rewards clipped to ",[45,10019,10021],{"className":10020},[48],[45,10022,10024],{"className":10023,"ariaHidden":53},[52],[45,10025,10027,10030,10033,10036,10039,10042,10045,10048],{"className":10026},[57],[45,10028],{"className":10029,"style":444},[61],[45,10031,270],{"className":10032},[93],[45,10034,1202],{"className":10035},[66],[45,10037,42],{"className":10038},[66],[45,10040,157],{"className":10041},[156],[45,10043],{"className":10044,"style":161},[73],[45,10046,42],{"className":10047},[66],[45,10049,404],{"className":10050},[232]," and\n",[45,10053,10055],{"className":10054},[48],[45,10056,10058,10076],{"className":10057,"ariaHidden":53},[52],[45,10059,10061,10064,10067,10070,10073],{"className":10060},[57],[45,10062],{"className":10063,"style":682},[61],[45,10065,2028],{"className":10066,"style":2027},[66,101],[45,10068],{"className":10069,"style":74},[73],[45,10071,79],{"className":10072},[78],[45,10074],{"className":10075,"style":74},[73],[45,10077,10079,10082],{"className":10078},[57],[45,10080],{"className":10081,"style":7252},[61],[45,10083,10085],{"className":10084},[66],"0.99",", the largest possible true action value is about ",[45,10088,10090],{"className":10089},[48],[45,10091,10093,10118,10139],{"className":10092,"ariaHidden":53},[52],[45,10094,10096,10099,10103,10106,10109,10112,10115],{"className":10095},[57],[45,10097],{"className":10098,"style":444},[61],[45,10100,10102],{"className":10101},[66],"1\u002F",[45,10104,277],{"className":10105},[93],[45,10107,42],{"className":10108},[66],[45,10110],{"className":10111,"style":1197},[73],[45,10113,1202],{"className":10114},[1201],[45,10116],{"className":10117,"style":1197},[73],[45,10119,10121,10124,10127,10130,10133,10136],{"className":10120},[57],[45,10122],{"className":10123,"style":444},[61],[45,10125,2028],{"className":10126,"style":2027},[66,101],[45,10128,321],{"className":10129},[232],[45,10131],{"className":10132,"style":74},[73],[45,10134,79],{"className":10135},[78],[45,10137],{"className":10138,"style":74},[73],[45,10140,10142,10145],{"className":10141},[57],[45,10143],{"className":10144,"style":7252},[61],[45,10146,10148],{"className":10147},[66],"100",";\nthey flagged any estimate that exceeded roughly that bound as ",[689,10151,10152],{},"soft divergence",".\nTheir findings sharpen Sutton and Barto's picture on four points.",[9861,10155,10156,10162,10187,10198],{},[9864,10157,10158,10161],{},[689,10159,10160],{},"Unbounded divergence is rare in practice."," Soft divergence — estimates growing\nwell past the reachable range — did occur, but the values essentially never ran off\nto infinity (floating-point NaN). The deep networks in current use apparently sit in\na regime where the triad distorts the estimates without making them diverge, which is\nwhy DQN works at all despite carrying all three ingredients.",[9864,10163,10164,10170,10171,10186],{},[689,10165,10166,10167,933],{},"Longer multi-step returns diverge ",[29,10168,10169],{},"less"," Increasing the bootstrapping horizon\n(using ",[45,10172,10174],{"className":10173},[48],[45,10175,10177],{"className":10176,"ariaHidden":53},[52],[45,10178,10180,10183],{"className":10179},[57],[45,10181],{"className":10182,"style":247},[61],[45,10184,9261],{"className":10185},[66,101],"-step returns instead of one-step targets) reduced both the frequency of\nsoft divergence and the number of unstable runs, and improved control performance.\nMore of the target comes from real returns and less from the bootstrapped estimate,\nweakening the leg of the triad that drives instability.",[9864,10188,10189,10192,10193,10197],{},[689,10190,10191],{},"Correcting overestimation helps."," Target networks and\n",[15,10194,10196],{"href":10195},"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks","double Q-learning"," — which was\ndesigned to cut the maximization bias in the Q-learning target — were the most stable\nvariants. Reducing overestimation and decoupling the bootstrap target from the\nweights being updated both damp the runaway feedback.",[9864,10199,10200,10203,10204,10206,10207,10210],{},[689,10201,10202],{},"Bigger networks are not automatically safer."," The intuition that a\nhigher-capacity approximator, closer to tabular, should be more stable did ",[29,10205,7238],{}," hold:\nlarger networks showed ",[29,10208,10209],{},"more"," soft divergence, even though they also reached the best\nfinal performance. Capacity and stability trade off rather than align.",[11,10212,10213],{},"These results reconcile the theory with practice. The triad's danger is real — van\nHasselt's own linear controls still diverge to infinity — but the specific\narchitecture of deep Q-learning (target networks, experience replay, clipped rewards,\nmulti-step returns) keeps the deep case in the soft-divergence regime where learning,\nthough sometimes unstable, does not explode.",[11,10215,10216,10217,940,10220,10223,10224,10240,10241,10244,10245,10252,10253,940,10256,10259,10260,10263,10264,10271],{},"On the theory side, two lines extend Sutton and Barto's stable methods. Sutton, Mahmood, and\nWhite (2016, ",[9932,10218,10219],{},"An Emphatic Approach to the Problem of Off-policy Temporal-Difference Learning,",[29,10221,10222],{},"JMLR",") gave the emphatic-TD method of §11.8 its formal footing: they proved\nthat emphasizing linear TD(",[45,10225,10227],{"className":10226},[48],[45,10228,10230],{"className":10229,"ariaHidden":53},[52],[45,10231,10233,10236],{"className":10232},[57],[45,10234],{"className":10235,"style":513},[61],[45,10237,10239],{"className":10238},[66,101],"λ",")'s updates by the emphasis weighting makes the\nexpected update ",[689,10242,10243],{},"stable"," under off-policy training, with a single learned weight\nvector and a single step size — simpler than the two-time-scale Gradient-TD family,\nat the cost of the higher variance the lesson already flagged.",[34,10246,10247],{},[15,10248,9784],{"href":10249,"ariaDescribedBy":10250,"dataFootnoteRef":6,"id":10251},"#user-content-fn-smw-emphatic",[40],"user-content-fnref-smw-emphatic"," More\nrecently, Zhang, Yao, and Whiteson (2021, ",[9932,10254,10255],{},"Breaking the Deadly Triad with a Target Network,",[29,10257,10258],{},"ICML",") proved that a ",[689,10261,10262],{},"target network"," — the same trick DQN uses\nheuristically — combined with a projection or ratio clipping yields a genuinely\nconvergent off-policy prediction and control algorithm under linear approximation,\nturning DQN's empirical stabilizer into a method with a convergence guarantee.",[34,10265,10266],{},[15,10267,9515],{"href":10268,"ariaDescribedBy":10269,"dataFootnoteRef":6,"id":10270},"#user-content-fn-zyw-target",[40],"user-content-fnref-zyw-target","\nTogether these results close much of the gap between Sutton and Barto's cautionary\ncounterexamples and the systems that run in production: the triad is still dangerous,\nbut there are now several principled ways to avoid its divergence.",[21,10273,10275],{"id":10274},"where-this-leaves-us","Where this leaves us",[11,10277,10278,10279,10303,10304,10307],{},"Off-policy learning with approximation is the point where the clean guarantees of\ntabular RL give out. The first part of the challenge — correcting update targets with\nimportance sampling — is routine, if variance-prone. The second part — the instability of\nbootstrapping under an off-policy update distribution — is the deadly triad, and it is\nreal enough to send a linearly-independent seven-state problem to infinity. Two responses\nrestore stability without abandoning any leg of the triad: Gradient-TD methods, which do\ntrue SGD on the projected Bellman error at ",[45,10280,10282],{"className":10281},[48],[45,10283,10285],{"className":10284,"ariaHidden":53},[52],[45,10286,10288,10291,10294,10297,10300],{"className":10287},[57],[45,10289],{"className":10290,"style":444},[61],[45,10292,4184],{"className":10293,"style":1893},[66,101],[45,10295,277],{"className":10296},[93],[45,10298,474],{"className":10299},[66,101],[45,10301,321],{"className":10302},[232]," cost, and emphatic-TD methods, which\nreweight updates back toward the on-policy distribution. Both remain relatively young and\nunsettled. The most consequential practical answer arrives later:\n",[15,10305,10306],{"href":10195},"deep Q-networks"," keep function\napproximation, bootstrapping, and off-policy replay all at once, and stabilize the triad\nempirically with a target network and a replay buffer rather than by giving anything up.",[10309,10310,10313,10318],"section",{"className":10311,"dataFootnotes":6},[10312],"footnotes",[21,10314,10317],{"className":10315,"id":40},[10316],"sr-only","Footnotes",[10319,10320,10321,10598,10850,10977,11265,11323,11393,11426],"ol",{},[9864,10322,10324,10327,10328,10343,10344,10359,10360,10375,10376,10428,10429,10481,10482,10531,10532,10590,10591],{"id":10323},"user-content-fn-sb-geom",[689,10325,10326],{},"Sutton & Barto",", §11.4 — Linear Value-function Geometry, and Figure 11.3: value functions as vectors, the ",[45,10329,10331],{"className":10330},[48],[45,10332,10334],{"className":10333,"ariaHidden":53},[52],[45,10335,10337,10340],{"className":10336},[57],[45,10338],{"className":10339,"style":513},[61],[45,10341,474],{"className":10342},[66,101],"-dimensional representable subspace, the ",[45,10345,10347],{"className":10346},[48],[45,10348,10350],{"className":10349,"ariaHidden":53},[52],[45,10351,10353,10356],{"className":10352},[57],[45,10354],{"className":10355,"style":682},[61],[45,10357,686],{"className":10358},[66,101],"-norm (11.11), the projection operator ",[45,10361,10363],{"className":10362},[48],[45,10364,10366],{"className":10365,"ariaHidden":53},[52],[45,10367,10369,10372],{"className":10368},[57],[45,10370],{"className":10371,"style":62},[61],[45,10373,956],{"className":10374},[66]," (11.12) and its matrix form (11.14)–(11.16), the Bellman operator ",[45,10377,10379],{"className":10378},[48],[45,10380,10382],{"className":10381,"ariaHidden":53},[52],[45,10383,10385,10388],{"className":10384},[57],[45,10386],{"className":10387,"style":1539},[61],[45,10389,10391,10394],{"className":10390},[66],[45,10392,1614],{"className":10393,"style":1613},[66,101],[45,10395,10397],{"className":10396},[106],[45,10398,10400,10420],{"className":10399},[110,111],[45,10401,10403,10417],{"className":10402},[115],[45,10404,10406],{"className":10405,"style":634},[119],[45,10407,10408,10411],{"style":1629},[45,10409],{"className":10410,"style":128},[127],[45,10412,10414],{"className":10413},[132,133,134,135],[45,10415,646],{"className":10416,"style":251},[66,101,135],[45,10418,143],{"className":10419},[142],[45,10421,10423],{"className":10422},[115],[45,10424,10426],{"className":10425,"style":150},[119],[45,10427],{}," (11.20) with fixed point ",[45,10430,10432],{"className":10431},[48],[45,10433,10435],{"className":10434,"ariaHidden":53},[52],[45,10436,10438,10441],{"className":10437},[57],[45,10439],{"className":10440,"style":534},[61],[45,10442,10444,10447],{"className":10443},[66],[45,10445,252],{"className":10446,"style":251},[66,101],[45,10448,10450],{"className":10449},[106],[45,10451,10453,10473],{"className":10452},[110,111],[45,10454,10456,10470],{"className":10455},[115],[45,10457,10459],{"className":10458,"style":634},[119],[45,10460,10461,10464],{"style":556},[45,10462],{"className":10463,"style":128},[127],[45,10465,10467],{"className":10466},[132,133,134,135],[45,10468,646],{"className":10469,"style":251},[66,101,135],[45,10471,143],{"className":10472},[142],[45,10474,10476],{"className":10475},[115],[45,10477,10479],{"className":10478,"style":150},[119],[45,10480],{}," (11.21), the Bellman error vector (11.17)–(11.19), and the projected Bellman error ",[45,10483,10485],{"className":10484},[48],[45,10486,10488],{"className":10487,"ariaHidden":53},[52],[45,10489,10491,10494],{"className":10490},[57],[45,10492],{"className":10493,"style":2592},[61],[45,10495,10497],{"className":10496},[66,2582],[45,10498,10500],{"className":10499},[110],[45,10501,10503],{"className":10502},[115],[45,10504,10506,10523],{"className":10505,"style":2592},[119],[45,10507,10508,10511],{"style":1154},[45,10509],{"className":10510,"style":1103},[127],[45,10512,10514,10517,10520],{"className":10513},[66],[45,10515,2993],{"className":10516,"style":2992},[66,101],[45,10518,1614],{"className":10519,"style":1613},[66,101],[45,10521,2608],{"className":10522,"style":2607},[66,101],[45,10524,10525,10528],{"style":2611},[45,10526],{"className":10527,"style":1103},[127],[45,10529],{"className":10530,"style":2619},[2618]," (11.22) with its TD fixed point ",[45,10533,10535],{"className":10534},[48],[45,10536,10538],{"className":10537,"ariaHidden":53},[52],[45,10539,10541,10544],{"className":10540},[57],[45,10542],{"className":10543,"style":3410},[61],[45,10545,10547,10550],{"className":10546},[66],[45,10548,571],{"className":10549,"style":570},[66,569],[45,10551,10553],{"className":10552},[106],[45,10554,10556,10582],{"className":10555},[110,111],[45,10557,10559,10579],{"className":10558},[115],[45,10560,10562],{"className":10561,"style":3429},[119],[45,10563,10564,10567],{"style":3432},[45,10565],{"className":10566,"style":128},[127],[45,10568,10570],{"className":10569},[132,133,134,135],[45,10571,10573],{"className":10572},[66,135],[45,10574,10576],{"className":10575},[66,1051,135],[45,10577,3448],{"className":10578},[66,135],[45,10580,143],{"className":10581},[142],[45,10583,10585],{"className":10584},[115],[45,10586,10588],{"className":10587,"style":150},[119],[45,10589],{},". ",[15,10592,10597],{"href":10593,"ariaLabel":10594,"className":10595,"dataFootnoteBackref":6},"#user-content-fnref-sb-geom","Back to reference 1",[10596],"data-footnote-backref","↩",[9864,10599,10601,10603,10604,10650,10651,10697,10698,10744,10745,8827,10794,10844,10845],{"id":10600},"user-content-fn-sb-be",[689,10602,10326],{},", §11.5 — Gradient Descent in the Bellman Error (the naive residual-gradient algorithm (11.23), the A-split and A-presplit examples showing ",[45,10605,10607],{"className":10606},[48],[45,10608,10610],{"className":10609,"ariaHidden":53},[52],[45,10611,10613,10616],{"className":10612},[57],[45,10614],{"className":10615,"style":2592},[61],[45,10617,10619],{"className":10618},[66,2582],[45,10620,10622],{"className":10621},[110],[45,10623,10625],{"className":10624},[115],[45,10626,10628,10642],{"className":10627,"style":2592},[119],[45,10629,10630,10633],{"style":1154},[45,10631],{"className":10632,"style":1103},[127],[45,10634,10636,10639],{"className":10635},[66],[45,10637,1614],{"className":10638,"style":1613},[66,101],[45,10640,2608],{"className":10641,"style":2607},[66,101],[45,10643,10644,10647],{"style":2611},[45,10645],{"className":10646,"style":1103},[127],[45,10648],{"className":10649,"style":2619},[2618]," minimization finds the wrong values), and §11.6 — The Bellman Error is Not Learnable: two MRPs generating identical data with different ",[45,10652,10654],{"className":10653},[48],[45,10655,10657],{"className":10656,"ariaHidden":53},[52],[45,10658,10660,10663],{"className":10659},[57],[45,10661],{"className":10662,"style":2592},[61],[45,10664,10666],{"className":10665},[66,2582],[45,10667,10669],{"className":10668},[110],[45,10670,10672],{"className":10671},[115],[45,10673,10675,10689],{"className":10674,"style":2592},[119],[45,10676,10677,10680],{"style":1154},[45,10678],{"className":10679,"style":1103},[127],[45,10681,10683,10686],{"className":10682},[66],[45,10684,1614],{"className":10685,"style":1613},[66,101],[45,10687,2608],{"className":10688,"style":2607},[66,101],[45,10690,10691,10694],{"style":2611},[45,10692],{"className":10693,"style":1103},[127],[45,10695],{"className":10696,"style":2619},[2618],"-minimizers, so ",[45,10699,10701],{"className":10700},[48],[45,10702,10704],{"className":10703,"ariaHidden":53},[52],[45,10705,10707,10710],{"className":10706},[57],[45,10708],{"className":10709,"style":2592},[61],[45,10711,10713],{"className":10712},[66,2582],[45,10714,10716],{"className":10715},[110],[45,10717,10719],{"className":10718},[115],[45,10720,10722,10736],{"className":10721,"style":2592},[119],[45,10723,10724,10727],{"style":1154},[45,10725],{"className":10726,"style":1103},[127],[45,10728,10730,10733],{"className":10729},[66],[45,10731,1614],{"className":10732,"style":1613},[66,101],[45,10734,2608],{"className":10735,"style":2607},[66,101],[45,10737,10738,10741],{"style":2611},[45,10739],{"className":10740,"style":1103},[127],[45,10742],{"className":10743,"style":2619},[2618]," cannot be estimated from feature vectors, actions, and rewards alone, while ",[45,10746,10748],{"className":10747},[48],[45,10749,10751],{"className":10750,"ariaHidden":53},[52],[45,10752,10754,10757],{"className":10753},[57],[45,10755],{"className":10756,"style":2592},[61],[45,10758,10760],{"className":10759},[66,2582],[45,10761,10763],{"className":10762},[110],[45,10764,10766],{"className":10765},[115],[45,10767,10769,10786],{"className":10768,"style":2592},[119],[45,10770,10771,10774],{"style":1154},[45,10772],{"className":10773,"style":1103},[127],[45,10775,10777,10780,10783],{"className":10776},[66],[45,10778,2993],{"className":10779,"style":2992},[66,101],[45,10781,1614],{"className":10782,"style":1613},[66,101],[45,10784,2608],{"className":10785,"style":2607},[66,101],[45,10787,10788,10791],{"style":2611},[45,10789],{"className":10790,"style":1103},[127],[45,10792],{"className":10793,"style":2619},[2618],[45,10795,10797],{"className":10796},[48],[45,10798,10800],{"className":10799,"ariaHidden":53},[52],[45,10801,10803,10806],{"className":10802},[57],[45,10804],{"className":10805,"style":2592},[61],[45,10807,10809],{"className":10808},[66,2582],[45,10810,10812],{"className":10811},[110],[45,10813,10815],{"className":10814},[115],[45,10816,10818,10836],{"className":10817,"style":2592},[119],[45,10819,10820,10823],{"style":1154},[45,10821],{"className":10822,"style":1103},[127],[45,10824,10826,10830,10833],{"className":10825},[66],[45,10827,10829],{"className":10828,"style":2992},[66,101],"T",[45,10831,1385],{"className":10832,"style":1893},[66,101],[45,10834,2608],{"className":10835,"style":2607},[66,101],[45,10837,10838,10841],{"style":2611},[45,10839],{"className":10840,"style":1103},[127],[45,10842],{"className":10843,"style":2619},[2618]," are learnable (Figure 11.4). ",[15,10846,10597],{"href":10847,"ariaLabel":10848,"className":10849,"dataFootnoteBackref":6},"#user-content-fnref-sb-be","Back to reference 2",[10596],[9864,10851,10853,10855,10856,10905,10906,10921,10922,10971,10972],{"id":10852},"user-content-fn-sb-gtd",[689,10854,10326],{},", §11.7 — Gradient-TD Methods: SGD on the ",[45,10857,10859],{"className":10858},[48],[45,10860,10862],{"className":10861,"ariaHidden":53},[52],[45,10863,10865,10868],{"className":10864},[57],[45,10866],{"className":10867,"style":2592},[61],[45,10869,10871],{"className":10870},[66,2582],[45,10872,10874],{"className":10873},[110],[45,10875,10877],{"className":10876},[115],[45,10878,10880,10897],{"className":10879,"style":2592},[119],[45,10881,10882,10885],{"style":1154},[45,10883],{"className":10884,"style":1103},[127],[45,10886,10888,10891,10894],{"className":10887},[66],[45,10889,2993],{"className":10890,"style":2992},[66,101],[45,10892,1614],{"className":10893,"style":1613},[66,101],[45,10895,2608],{"className":10896,"style":2607},[66,101],[45,10898,10899,10902],{"style":2611},[45,10900],{"className":10901,"style":1103},[127],[45,10903],{"className":10904,"style":2619},[2618],", the matrix expansion (11.25)–(11.27), the product-of-expectations gradient and its double-sampling obstacle, the stored auxiliary vector ",[45,10907,10909],{"className":10908},[48],[45,10910,10912],{"className":10911,"ariaHidden":53},[52],[45,10913,10915,10918],{"className":10914},[57],[45,10916],{"className":10917,"style":5006},[61],[45,10919,252],{"className":10920,"style":570},[66,569]," (11.28) learned by an importance-weighted LMS rule, the GTD2 and TDC \u002F GTD(0) updates (11.29 and following), Figure 11.5 showing TDC drive the ",[45,10923,10925],{"className":10924},[48],[45,10926,10928],{"className":10927,"ariaHidden":53},[52],[45,10929,10931,10934],{"className":10930},[57],[45,10932],{"className":10933,"style":2592},[61],[45,10935,10937],{"className":10936},[66,2582],[45,10938,10940],{"className":10939},[110],[45,10941,10943],{"className":10942},[115],[45,10944,10946,10963],{"className":10945,"style":2592},[119],[45,10947,10948,10951],{"style":1154},[45,10949],{"className":10950,"style":1103},[127],[45,10952,10954,10957,10960],{"className":10953},[66],[45,10955,2993],{"className":10956,"style":2992},[66,101],[45,10958,1614],{"className":10959,"style":1613},[66,101],[45,10961,2608],{"className":10962,"style":2607},[66,101],[45,10964,10965,10968],{"style":2611},[45,10966],{"className":10967,"style":1103},[127],[45,10969],{"className":10970,"style":2619},[2618]," to zero on Baird's counterexample, and the two-time-scale \u002F cascade convergence argument. ",[15,10973,10597],{"href":10974,"ariaLabel":10975,"className":10976,"dataFootnoteBackref":6},"#user-content-fnref-sb-gtd","Back to reference 3",[10596],[9864,10978,10980,10982,10983,11206,11207,11259,11260],{"id":10979},"user-content-fn-sb-etd",[689,10981,10326],{},", §11.8 — Emphatic-TD Methods: reweighting states to recover the on-policy distribution's stability under semi-gradient TD, the emphasis ",[45,10984,10986],{"className":10985},[48],[45,10987,10989,11044,11160],{"className":10988,"ariaHidden":53},[52],[45,10990,10992,10995,11035,11038,11041],{"className":10991},[57],[45,10993],{"className":10994,"style":1539},[61],[45,10996,10998,11001],{"className":10997},[66],[45,10999,7569],{"className":11000,"style":7568},[66,101],[45,11002,11004],{"className":11003},[106],[45,11005,11007,11027],{"className":11006},[110,111],[45,11008,11010,11024],{"className":11009},[115],[45,11011,11013],{"className":11012,"style":4437},[119],[45,11014,11015,11018],{"style":7584},[45,11016],{"className":11017,"style":128},[127],[45,11019,11021],{"className":11020},[132,133,134,135],[45,11022,4294],{"className":11023},[66,101,135],[45,11025,143],{"className":11026},[142],[45,11028,11030],{"className":11029},[115],[45,11031,11033],{"className":11032,"style":150},[119],[45,11034],{},[45,11036],{"className":11037,"style":74},[73],[45,11039,79],{"className":11040},[78],[45,11042],{"className":11043,"style":74},[73],[45,11045,11047,11050,11053,11102,11151,11154,11157],{"className":11046},[57],[45,11048],{"className":11049,"style":5955},[61],[45,11051,2028],{"className":11052,"style":2027},[66,101],[45,11054,11056,11059],{"className":11055},[66],[45,11057,4424],{"className":11058},[66,101],[45,11060,11062],{"className":11061},[106],[45,11063,11065,11094],{"className":11064},[110,111],[45,11066,11068,11091],{"className":11067},[115],[45,11069,11071],{"className":11070,"style":120},[119],[45,11072,11073,11076],{"style":123},[45,11074],{"className":11075,"style":128},[127],[45,11077,11079],{"className":11078},[132,133,134,135],[45,11080,11082,11085,11088],{"className":11081},[66,135],[45,11083,4294],{"className":11084},[66,101,135],[45,11086,1202],{"className":11087},[1201,135],[45,11089,42],{"className":11090},[66,135],[45,11092,143],{"className":11093},[142],[45,11095,11097],{"className":11096},[115],[45,11098,11100],{"className":11099,"style":4310},[119],[45,11101],{},[45,11103,11105,11108],{"className":11104},[66],[45,11106,7569],{"className":11107,"style":7568},[66,101],[45,11109,11111],{"className":11110},[106],[45,11112,11114,11143],{"className":11113},[110,111],[45,11115,11117,11140],{"className":11116},[115],[45,11118,11120],{"className":11119,"style":120},[119],[45,11121,11122,11125],{"style":7584},[45,11123],{"className":11124,"style":128},[127],[45,11126,11128],{"className":11127},[132,133,134,135],[45,11129,11131,11134,11137],{"className":11130},[66,135],[45,11132,4294],{"className":11133},[66,101,135],[45,11135,1202],{"className":11136},[1201,135],[45,11138,42],{"className":11139},[66,135],[45,11141,143],{"className":11142},[142],[45,11144,11146],{"className":11145},[115],[45,11147,11149],{"className":11148,"style":4310},[119],[45,11150],{},[45,11152],{"className":11153,"style":1197},[73],[45,11155,2014],{"className":11156},[1201],[45,11158],{"className":11159,"style":1197},[73],[45,11161,11163,11166],{"className":11162},[57],[45,11164],{"className":11165,"style":1539},[61],[45,11167,11169,11172],{"className":11168},[66],[45,11170,8210],{"className":11171,"style":8209},[66,101],[45,11173,11175],{"className":11174},[106],[45,11176,11178,11198],{"className":11177},[110,111],[45,11179,11181,11195],{"className":11180},[115],[45,11182,11184],{"className":11183,"style":4437},[119],[45,11185,11186,11189],{"style":8225},[45,11187],{"className":11188,"style":128},[127],[45,11190,11192],{"className":11191},[132,133,134,135],[45,11193,4294],{"className":11194},[66,101,135],[45,11196,143],{"className":11197},[142],[45,11199,11201],{"className":11200},[115],[45,11202,11204],{"className":11203,"style":150},[119],[45,11205],{}," and interest ",[45,11208,11210],{"className":11209},[48],[45,11211,11213],{"className":11212,"ariaHidden":53},[52],[45,11214,11216,11219],{"className":11215},[57],[45,11217],{"className":11218,"style":1539},[61],[45,11220,11222,11225],{"className":11221},[66],[45,11223,8210],{"className":11224,"style":8209},[66,101],[45,11226,11228],{"className":11227},[106],[45,11229,11231,11251],{"className":11230},[110,111],[45,11232,11234,11248],{"className":11233},[115],[45,11235,11237],{"className":11236,"style":4437},[119],[45,11238,11239,11242],{"style":8225},[45,11240],{"className":11241,"style":128},[127],[45,11243,11245],{"className":11244},[132,133,134,135],[45,11246,4294],{"className":11247},[66,101,135],[45,11249,143],{"className":11250},[142],[45,11252,11254],{"className":11253},[115],[45,11255,11257],{"className":11256,"style":150},[119],[45,11258],{},", the one-step emphatic-TD update, Figure 11.6 showing convergence in expectation on Baird's counterexample, and the high-variance caveat that motivates the variance-reduction methods of §11.9. ",[15,11261,10597],{"href":11262,"ariaLabel":11263,"className":11264,"dataFootnoteBackref":6},"#user-content-fnref-sb-etd","Back to reference 4",[10596],[9864,11266,11268,11270,11271,11274,11275,11299,11300,11303,11304,940,11309,940,11316],{"id":11267},"user-content-fn-sb-variance",[689,11269,10326],{},", §11.9 — Reducing Variance: off-policy learning as inherently higher-variance than on-policy (",[9932,11272,11273],{},"you can't learn to drive by cooking dinner","); the raison d'être of generalizing to related-but-not-identical policies; the source of variance in products of policy ratios (mean 1 by (5.13), uncorrelated, but of high variance) multiplying the SGD step size and producing occasional very large steps; and the surveyed remedies — weighted importance sampling with its ",[45,11276,11278],{"className":11277},[48],[45,11279,11281],{"className":11280,"ariaHidden":53},[52],[45,11282,11284,11287,11290,11293,11296],{"className":11283},[57],[45,11285],{"className":11286,"style":444},[61],[45,11288,4184],{"className":11289,"style":1893},[66,101],[45,11291,277],{"className":11292},[93],[45,11294,474],{"className":11295},[66,101],[45,11297,321],{"className":11298},[232],"-approximation difficulty (Mahmood and Sutton, 2015), the tree-backup route to ratio-free off-policy learning (Munos et al., 2016; Mahmood, Yu and Sutton, 2017), momentum and Polyak–Ruppert averaging, adaptive per-component and importance-weight-aware step sizes (Jacobs, 1988; Karampatziakis and Langford, 2010), and constraining the target to the behavior via ",[9932,11301,11302],{},"recognizers"," (Precup et al., 2006). ",[15,11305,10597],{"href":11306,"ariaLabel":11307,"className":11308,"dataFootnoteBackref":6},"#user-content-fnref-sb-variance","Back to reference 5",[10596],[15,11310,10597,11314],{"href":11311,"ariaLabel":11312,"className":11313,"dataFootnoteBackref":6},"#user-content-fnref-sb-variance-2","Back to reference 5-2",[10596],[34,11315,774],{},[15,11317,10597,11321],{"href":11318,"ariaLabel":11319,"className":11320,"dataFootnoteBackref":6},"#user-content-fnref-sb-variance-3","Back to reference 5-3",[10596],[34,11322,4167],{},[9864,11324,11326,3715,11329,11332,11333,11387,11388],{"id":11325},"user-content-fn-vh-triad",[689,11327,11328],{},"H. van Hasselt, Y. Doron, F. Strub, M. Hessel, N. Sonnerat, and J. Modayil",[9932,11330,11331],{},"Deep Reinforcement Learning and the Deadly Triad,"," arXiv:1812.02648 (2018). A controlled study on the 57-game Atari suite measuring how the triad's ingredients affect divergence of a deep Q-network: unbounded divergence is rare (soft divergence, above the reachable value bound ",[45,11334,11336],{"className":11335},[48],[45,11337,11339,11351,11375],{"className":11338,"ariaHidden":53},[52],[45,11340,11342,11345,11348],{"className":11341},[57],[45,11343],{"className":11344,"style":5025},[61],[45,11346,5038],{"className":11347},[78],[45,11349],{"className":11350,"style":74},[73],[45,11352,11354,11357,11360,11363,11366,11369,11372],{"className":11353},[57],[45,11355],{"className":11356,"style":444},[61],[45,11358,10102],{"className":11359},[66],[45,11361,277],{"className":11362},[93],[45,11364,42],{"className":11365},[66],[45,11367],{"className":11368,"style":1197},[73],[45,11370,1202],{"className":11371},[1201],[45,11373],{"className":11374,"style":1197},[73],[45,11376,11378,11381,11384],{"className":11377},[57],[45,11379],{"className":11380,"style":444},[61],[45,11382,2028],{"className":11383,"style":2027},[66,101],[45,11385,321],{"className":11386},[232],", is common but values stay finite); longer multi-step returns reduce instability; target networks and double Q-learning are the most stable; and larger networks show more soft divergence though they perform best. Grounds the claim that DQN's architecture keeps the deep triad in a soft-divergence regime. ",[15,11389,10597],{"href":11390,"ariaLabel":11391,"className":11392,"dataFootnoteBackref":6},"#user-content-fnref-vh-triad","Back to reference 6",[10596],[9864,11394,11396,3715,11399,940,11401,11404,11405,11420,11421],{"id":11395},"user-content-fn-smw-emphatic",[689,11397,11398],{},"R. S. Sutton, A. R. Mahmood, and M. White",[9932,11400,10219],{},[29,11402,11403],{},"Journal of Machine Learning Research"," 17(73):1–29 (2016). Proves that emphasizing linear TD(",[45,11406,11408],{"className":11407},[48],[45,11409,11411],{"className":11410,"ariaHidden":53},[52],[45,11412,11414,11417],{"className":11413},[57],[45,11415],{"className":11416,"style":513},[61],[45,11418,10239],{"className":11419},[66,101],")'s updates makes its expected update stable under off-policy training, with one learned weight vector and one step size — simpler than Gradient-TD; the formal basis for the emphatic-TD method of §11.8. ",[15,11422,10597],{"href":11423,"ariaLabel":11424,"className":11425,"dataFootnoteBackref":6},"#user-content-fnref-smw-emphatic","Back to reference 7",[10596],[9864,11427,11429,3715,11432,940,11434,11437,11438],{"id":11428},"user-content-fn-zyw-target",[689,11430,11431],{},"S. Zhang, H. Yao, and S. Whiteson",[9932,11433,10255],{},[29,11435,11436],{},"Proceedings of the 38th International Conference on Machine Learning"," (ICML 2021). Shows that a target network combined with a projection \u002F ratio truncation gives a provably convergent off-policy prediction and control algorithm under linear function approximation, supplying a convergence guarantee for the target-network trick that DQN uses only heuristically. ",[15,11439,10597],{"href":11440,"ariaLabel":11441,"className":11442,"dataFootnoteBackref":6},"#user-content-fnref-zyw-target","Back to reference 8",[10596],{"title":6,"searchDepth":11444,"depth":11444,"links":11445},2,[11446,11450,11451,11452,11456,11457,11458],{"id":23,"depth":11444,"text":24,"children":11447},[11448],{"id":1587,"depth":11449,"text":1588},3,{"id":4104,"depth":11444,"text":4105},{"id":7509,"depth":11444,"text":7510},{"id":8310,"depth":11444,"text":8311,"children":11453},[11454,11455],{"id":8336,"depth":11449,"text":8337},{"id":9849,"depth":11449,"text":9850},{"id":9997,"depth":11444,"text":9998},{"id":10274,"depth":11444,"text":10275},{"id":40,"depth":11444,"text":10317},[],"computer-science","This builds on\noff-policy methods and the deadly triad,\nwhich showed that function approximation, bootstrapping, and off-policy training\ntogether can send the weights to infinity — on a seven-state problem with linearly\nindependent features, no less. That lesson named the failure but did not explain\nit. This lesson does, by picturing value functions as points in a vector space,\nand from that picture builds the two families of methods that make off-policy\nlearning stable.",false,"md",{"moduleNumber":11449,"lessonNumber":11465,"order":11466},6,306,"Approximate Solution Methods",true,"\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td",[],"---\ntitle: \"Value-Function Geometry and Gradient-TD Methods\"\nmodule: Approximate Solution Methods\nmoduleNumber: 3\nlessonNumber: 6\norder: 306\nsummary: >\n  Why does the deadly triad diverge, and how do you stop it? This lesson develops\n  the geometry that explains the failure: value functions as vectors, the\n  projection operator onto the representable subspace, and the split between the\n  Bellman error, the value error, and the projected Bellman error: the three\n  objectives have different minimizers. The projected Bellman error is the\n  learnable one, and Gradient-TD methods (GTD2, TDC) do true stochastic\n  gradient descent on it, staying stable even off-policy at $O(d)$ cost. Emphatic\n  TD reweights states instead, and a survey of variance-reduction techniques\n  closes the gap between stability and usable learning.\ntopics: [Approximation]\nsources:\n  - book: Sutton & Barto\n    ref: \"Ch. 11 — Off-policy Methods with Approximation; §11.4 Linear Value-function Geometry; §11.5 Gradient Descent in the Bellman Error\"\n  - book: Sutton & Barto\n    ref: \"§11.7 Gradient-TD Methods; §11.8 Emphatic-TD Methods; §11.9 Reducing Variance\"\n---\n\nThis builds on\n[off-policy methods and the deadly triad](\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad),\nwhich showed that function approximation, bootstrapping, and off-policy training\ntogether can send the weights to infinity — on a seven-state problem with linearly\nindependent features, no less. That lesson named the failure but did not explain\nit. This lesson does, by picturing value functions as points in a vector space,\nand from that picture builds the two families of methods that make off-policy\nlearning stable.\n\n## Linear value-function geometry\n\nTo understand _why_ the triad diverges and how to fix it, it helps to picture value\nfunctions as vectors.[^sb-geom] Enumerate the states $\\mathcal{S} = \\{s_1, \\dots,\ns_{|\\mathcal{S}|}\\}$; then any value function $v$ is the vector $[v(s_1), \\dots,\nv(s_{|\\mathcal{S}|})]^\\top$ in an $|\\mathcal{S}|$-dimensional space. The whole space\nholds every conceivable value function; most correspond to no policy at all.\n\nA linear approximator with $d \u003C |\\mathcal{S}|$ parameters can only reach a\n$d$-dimensional _subspace_ of that space — the plane of all $v_{\\mathbf{w}} =\n\\mathbf{X}\\mathbf{w}$ representable by some weight vector. The true value $v_\\pi$ of a\npolicy generally lies _outside_ this subspace, too complex to represent exactly.\n\n$$\n% caption: Linear value-function geometry. The true value $v_\\pi$ lies outside the\n% subspace of representable functions; the projection operator $\\Pi$ drops it to the\n% closest representable point $\\Pi v_\\pi$, and the Bellman operator $B_\\pi$ (gray) takes\n% points out of the subspace before projection maps them back.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  % the representable subspace (a slanted plane)\n  \\draw[black, fill=black!4] (-0.4,0) -- (6.6,0) -- (5.4,1.5) -- (-1.6,1.5) -- cycle;\n  \\node[anchor=north west, font=\\scriptsize, black] at (-1.5,0.02) {subspace: values representable as v(w)};\n  % point v_w in the plane\n  \\fill[acc] (1.4,0.7) circle (2pt);\n  \\node[acc, anchor=north, font=\\scriptsize] at (1.4,0.58) {v(w)};\n  % projection of v_pi into plane\n  \\fill[black] (4.1,0.75) circle (2pt);\n  \\node[black, anchor=north west, font=\\scriptsize] at (4.15,0.72) {proj of v-pi};\n  % true value above\n  \\fill[black] (4.6,3.35) circle (2.4pt);\n  \\node[black, anchor=west, font=\\footnotesize] at (4.72,3.4) {v-pi};\n  % value error line (vertical drop)\n  \\draw[black, thick] (4.6,3.35) -- (4.1,0.78);\n  \\node[black, anchor=south west, font=\\scriptsize] at (4.72,2.0) {value error VE};\n  % Bellman image of v_w outside the plane\n  \\fill[red] (2.55,2.55) circle (2pt);\n  \\node[red, anchor=south west, font=\\scriptsize] at (2.6,2.58) {B-pi v(w)};\n  \\draw[red, thick] (1.4,0.7) -- (2.55,2.55);\n  \\node[red, anchor=south east, font=\\scriptsize] at (1.55,1.85) {Bellman error BE};\n  % projection of Bellman image back into plane\n  \\fill[red] (2.55,0.68) circle (2pt);\n  \\node[red, anchor=north, font=\\scriptsize] at (2.55,0.6) {proj B-pi v(w)};\n  \\draw[red, dashed] (2.55,2.55) -- (2.55,0.68);\n  % PBE arrow inside plane\n  \\draw[->, red, thick] (1.4,0.7) -- (2.5,0.68);\n  \\node[red, anchor=south, font=\\scriptsize] at (1.95,0.72) {PBE};\n\\end{tikzpicture}\n$$\n\nTo measure how far apart two value functions are, Euclidean distance is wrong: some\nstates matter more because they are visited more often. Weight each state by the\ndistribution $\\mu$ (typically the on-policy distribution) and use the **$\\mu$-norm**\n\n$$\n\\lVert v \\rVert_\\mu^2 \\;\\doteq\\; \\sum_{s \\in \\mathcal{S}} \\mu(s)\\, v(s)^2.\n$$\n\nThe **projection operator** $\\Pi$ takes any value function to the representable one\nclosest in this norm:\n\n$$\n\\Pi v \\;\\doteq\\; v_{\\mathbf{w}}\n\\quad\\text{where}\\quad\n\\mathbf{w} \\;=\\; \\argmin_{\\mathbf{w} \\in \\mathbb{R}^d} \\lVert v - v_{\\mathbf{w}} \\rVert_\\mu^2,\n\\qquad\n\\Pi \\;\\doteq\\; \\mathbf{X}\\bigl(\\mathbf{X}^\\top \\mathbf{D}\\, \\mathbf{X}\\bigr)^{-1}\\mathbf{X}^\\top \\mathbf{D},\n$$\n\nwhere $\\mathbf{X}$ stacks the feature vectors as rows and $\\mathbf{D}$ is the diagonal\nmatrix of the weights $\\mu(s)$. The projection of the true value, $\\Pi v_\\pi$, is the\nbest representable approximation in the value-error sense, and the point Monte Carlo\nmethods converge to.\n\n### Bellman error versus projected Bellman error\n\nTD methods aim somewhere else. The **Bellman operator** $B_\\pi$ applies one sweep of\nthe Bellman equation:\n\n$$\n(B_\\pi v)(s) \\;\\doteq\\; \\sum_a \\pi(a \\mid s) \\sum_{s', r} p(s', r \\mid s, a)\\bigl[r + \\gamma\\, v(s')\\bigr].\n$$\n\nIts unique fixed point is the true value: $v_\\pi = B_\\pi v_\\pi$, which is just the\nBellman equation rewritten. For any representable $v_{\\mathbf{w}}$, the difference\n$\\bar\\delta_{\\mathbf{w}} = B_\\pi v_{\\mathbf{w}} - v_{\\mathbf{w}}$ is the **Bellman error\nvector** — the expectation of the TD error at each state — and its squared $\\mu$-norm is\nthe **mean square Bellman error**:\n\n$$\n\\overline{BE}(\\mathbf{w}) \\;=\\; \\lVert \\bar\\delta_{\\mathbf{w}} \\rVert_\\mu^2.\n$$\n\nHere is the geometric crux. Applying $B_\\pi$ to a representable value function generally\nproduces one _outside_ the subspace (the gray arrow leaving the plane in the figure). In\npure dynamic programming, without approximation, you follow $B_\\pi$ repeatedly and\nconverge to $v_\\pi$. With approximation you cannot: after one Bellman step you must\n_project_ back into the subspace to have a representable function, then step out again,\nthen project back. The quantity TD methods drive to zero is the residual _inside_ the\nsubspace — the **projected Bellman error**:\n\n$$\n\\overline{PBE}(\\mathbf{w}) \\;=\\; \\lVert \\Pi\\, \\bar\\delta_{\\mathbf{w}} \\rVert_\\mu^2.\n$$\n\n> **Definition (Projected Bellman error, $\\overline{PBE}$).** The squared $\\mu$-norm of\n> the Bellman error vector _after projecting it back into the representable subspace_,\n> $\\lVert \\Pi\\, \\bar\\delta_{\\mathbf{w}} \\rVert_\\mu^2$. With linear approximation there is\n> always a weight vector $\\mathbf{w}_{\\text{TD}}$ making it zero — the TD fixed point.\n> Minimizing $\\overline{PBE}$ (rather than $\\overline{BE}$ or $\\overline{VE}$) is the\n> target of stable off-policy methods.\n\nThe three objectives generally have _different_ minimizers: $\\overline{VE}$ is minimized\nat $\\Pi v_\\pi$, $\\overline{BE}$ at some other point, and $\\overline{PBE}$ at the TD fixed\npoint $\\mathbf{w}_{\\text{TD}}$. Semi-gradient TD, when it converges, converges to\n$\\mathbf{w}_{\\text{TD}}$ — but under off-policy training it need not converge at all.\n\nA tempting fix is to do true gradient descent on the Bellman error itself (the\n**residual-gradient** algorithm), which is a genuine SGD method and converges. But two\nresults rule it out as a general approach. First, on the A-split example it converges to the\n_wrong_ values, minimizing a temporal-smoothing surrogate rather than accuracy. Second,\nand more fundamentally, the $\\overline{BE}$ is **not learnable**: two Markov reward\nprocesses can produce identical data yet have different $\\overline{BE}$-minimizing weight\nvectors, so no algorithm working from feature vectors, actions, and rewards alone can\nminimize it.[^sb-be] That leaves the $\\overline{PBE}$, which _is_ learnable, as the\nobjective worth pursuing.\n\n## Gradient-TD methods\n\n**Gradient-TD methods** perform true stochastic gradient descent on the\n$\\overline{PBE}$.[^sb-gtd] Because they follow an actual gradient, they inherit SGD's\nrobust convergence — stable even under off-policy training and nonlinear approximation —\nat $O(d)$ cost per step, only about double that of semi-gradient TD.\n\nThe derivation expands the $\\overline{PBE}$ in matrix form and differentiates. The\nkey point in the algebra: the gradient factors into three pieces, and\ntwo of them share a random quantity ($\\mathbf{x}_{t+1}$), so sampling each\nindependently and multiplying would double-count the noise. Gradient-TD works\naround that one obstacle.\n\nThe gradient turns out to be a product of three expectations,\n\n$$\n\\nabla \\overline{PBE}(\\mathbf{w}) \\;=\\; 2\\, \\mathbb{E}\\bigl[\\rho_t (\\gamma\\, \\mathbf{x}_{t+1} - \\mathbf{x}_t)\\, \\mathbf{x}_t^\\top\\bigr]\\; \\mathbb{E}\\bigl[\\mathbf{x}_t \\mathbf{x}_t^\\top\\bigr]^{-1}\\; \\mathbb{E}\\bigl[\\rho_t\\, \\delta_t\\, \\mathbf{x}_t\\bigr],\n$$\n\nand the trouble is that the first and last factors both depend on the next feature\n$\\mathbf{x}_{t+1}$, so sampling them independently and multiplying gives a _biased_\nestimate — the same double-sampling problem that afflicts the residual-gradient method.\nThe Gradient-TD solution is to estimate and store the product of the last two factors as a\nsecond parameter vector $\\mathbf{v}$,\n\n$$\n\\mathbf{v} \\;\\approx\\; \\mathbb{E}\\bigl[\\mathbf{x}_t \\mathbf{x}_t^\\top\\bigr]^{-1}\\, \\mathbb{E}\\bigl[\\rho_t\\, \\delta_t\\, \\mathbf{x}_t\\bigr],\n$$\n\nwhich is the solution to a least-squares problem and is learned by an\nimportance-weighted LMS rule on a second, faster step size $\\beta$:\n\n$$\n\\mathbf{v}_{t+1} \\;\\doteq\\; \\mathbf{v}_t + \\beta\\, \\rho_t\\, \\bigl(\\delta_t - \\mathbf{v}_t^\\top \\mathbf{x}_t\\bigr)\\, \\mathbf{x}_t.\n$$\n\nGiven $\\mathbf{v}_t$, the main weight vector is updated by a rule that is $O(d)$ if the\ninner product $\\mathbf{x}_t^\\top \\mathbf{v}_t$ is formed first. Two members of the family:\n\n$$\n\\text{GTD2:}\\quad \\mathbf{w}_{t+1} \\;=\\; \\mathbf{w}_t + \\alpha\\, \\rho_t\\, (\\mathbf{x}_t - \\gamma\\, \\mathbf{x}_{t+1})\\, (\\mathbf{x}_t^\\top \\mathbf{v}_t),\n$$\n\n$$\n\\text{TDC \u002F GTD(0):}\\quad \\mathbf{w}_{t+1} \\;=\\; \\mathbf{w}_t + \\alpha\\, \\rho_t\\, \\bigl(\\delta_t\\, \\mathbf{x}_t - \\gamma\\, \\mathbf{x}_{t+1}\\, (\\mathbf{x}_t^\\top \\mathbf{v}_t)\\bigr).\n$$\n\n**TDC** (TD with gradient correction) is the semi-gradient TD update\n$\\alpha \\rho_t \\delta_t \\mathbf{x}_t$ plus a correction term $-\\alpha \\rho_t \\gamma\n\\mathbf{x}_{t+1}(\\mathbf{x}_t^\\top \\mathbf{v}_t)$ that supplies the missing half of the\ngradient. That single correction is the difference between divergence and convergence.\n\n$$\n% caption: TDC on Baird's counterexample. The $\\overline{PBE}$ falls to zero and the\n% weight block $w_1$ through $w_6$ settles; the individual components need not reach\n% zero, so the value error $\\overline{VE}$ stays near 2, but the system is stable.\n% Step sizes $\\alpha = 0.005$, $\\beta = 0.05$.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  % axes\n  \\draw[->, black] (0,0) -- (6.4,0) node[anchor=north east, font=\\scriptsize] {sweeps};\n  \\draw[->, black] (0,-0.2) -- (0,4.0) node[anchor=south east, font=\\scriptsize] {value};\n  \\draw[black, dashed] (0,0.3) -- (6.0,0.3);\n  \\node[anchor=east, font=\\scriptsize] at (-0.08,0.3) {0};\n  % PBE decays to zero\n  \\draw[red, very thick] (0,2.7) .. controls (0.9,1.1) and (1.6,0.4) .. (6.0,0.3);\n  \\node[red, anchor=west, font=\\scriptsize] at (6.05,0.3) {PBE};\n  % VE settles near 2\n  \\draw[acc, very thick] (0,3.4) .. controls (0.8,2.2) and (1.6,1.75) .. (6.0,1.7);\n  \\node[acc, anchor=west, font=\\scriptsize] at (6.05,1.7) {VE};\n  % weight block settles near a positive constant\n  \\draw[black, thick] (0,2.35) .. controls (0.9,1.5) and (1.8,1.35) .. (6.0,1.35);\n  \\node[black, anchor=west, font=\\scriptsize] at (6.05,1.35) {w1-w6};\n\\end{tikzpicture}\n$$\n\nOn Baird's counterexample TDC does exactly what the theory promises: the\n$\\overline{PBE}$ decays to zero and the weights stop growing. Notice the components do\n_not_ all reach zero, so the value error stays near $2$ — the $\\overline{PBE}$ fixed point\nis not the best approximation in the $\\overline{VE}$ sense — but the divergence is gone.\nGTD2 and TDC are **two-time-scale** methods: the correction estimate $\\mathbf{v}$ (fast,\nstep size $\\beta$) must track ahead of the primary weights $\\mathbf{w}$ (slow, step size\n$\\alpha$), which the convergence proofs formalize as a _cascade_ requiring $\\beta \\to 0$\nand $\\alpha \u002F \\beta \\to 0$. Gradient-TD is currently the best-understood and most widely\nused family of stable off-policy methods, with extensions to action values, control, and\n[eligibility traces](\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces).\n\n## A note on emphatic-TD\n\nGradient-TD attacks the second part of the problem — the wrong update distribution — by\nbuilding a true gradient method. **Emphatic-TD** attacks it differently: keep the simple,\ncomputationally cheap semi-gradient update, but _reweight the states_ so the update\ndistribution recovers the on-policy properties that make linear semi-gradient TD\nstable.[^sb-etd] Linear semi-gradient TD is stable on-policy precisely because the state\ndistribution matches the transitions under $\\pi$; emphatic-TD manufactures that match by\nscaling each update with an **emphasis** $M_t$ that accumulates responsibility for the\ncurrent state through a recursive trace,\n\n$$\n\\mathbf{w}_{t+1} \\;=\\; \\mathbf{w}_t + \\alpha\\, M_t\\, \\rho_t\\, \\delta_t\\, \\nabla \\hat{v}(S_t, \\mathbf{w}_t),\n\\qquad\nM_t \\;=\\; \\gamma\\, \\rho_{t-1}\\, M_{t-1} + I_t,\n$$\n\nwith the **interest** $I_t$ specifying how much we care about accurately valuing each\nstate. In expectation, emphatic-TD converges to zero value error on Baird's\ncounterexample; in practice its variance is so high that consistent results are hard to\nobtain, which is why the next section turns to variance-reduction techniques for all\nof these methods.\n\n## Reducing variance\n\nStability is only half the problem. Gradient-TD and emphatic-TD keep the weights\nfrom diverging, but off-policy learning is _inherently_ higher-variance than\non-policy learning, and that variance is what makes these methods slow and their\nresults erratic in practice.[^sb-variance] The intuition is plain: if the data you\nreceive is only loosely related to the policy you are trying to evaluate, you\nshould expect to learn less about it. In the extreme you learn nothing — you\ncannot learn to drive by cooking dinner. Only when the target and behavior\npolicies visit similar states and take similar actions is real progress possible.\n\nYet that overlap is what off-policy learning is for: any policy has many\nneighbors, similar-but-not-identical policies with considerable overlap, and the\nwhole reason to learn off-policy is to generalize across that vast family of\nrelated policies from one stream of experience. So the variance cannot simply be\navoided; it has to be _managed_.\n\n### Where the variance comes from\n\nThe specific source is the importance-sampling ratio. As we saw, off-policy\ntargets are corrected by products of per-step ratios $\\rho_t = \\pi(A_t\\mid\nS_t)\u002Fb(A_t\\mid S_t)$. Each ratio is $1$ in expectation, and successive ratios are\nuncorrelated, so their products are $1$ in expectation too — but their _values_\nswing from zero up to very large numbers, and a product of such terms has very\nhigh variance.[^sb-variance]\n\nA worked case shows how fast the product blows up. Let the behavior policy be\nuniform over two actions, $b(a \\mid s) = 0.5$, while the target policy is nearly\ndeterministic, $\\pi(a_1 \\mid s) = 0.9$ and $\\pi(a_2 \\mid s) = 0.1$. The per-step\nratio is then $\\rho = 0.9\u002F0.5 = 1.8$ when the behavior policy happens to sample\n$a_1$ and $\\rho = 0.1\u002F0.5 = 0.2$ when it samples $a_2$; each occurs with\nprobability $0.5$, and the mean is $0.5 \\cdot 1.8 + 0.5 \\cdot 0.2 = 1$, as it must\nbe. Over a horizon of $n = 10$ steps the cumulative ratio $\\rho_{t:t+9}$ is a\nproduct of ten such draws. Its mean stays exactly $1$, but its _spread_ is enormous:\nan all-$a_1$ trajectory (probability $0.5^{10} \\approx 0.001$) carries\n$1.8^{10} \\approx 357$, while a trajectory with even one $a_2$ is scaled down by\nthe $0.2$ factor, and a mostly-$a_2$ trajectory carries a ratio near $0.2^{10}\n\\approx 10^{-7}$. The variance of the product grows geometrically in $n$: a rare\ntrajectory contributes a step roughly $357$ times the nominal size, and that single\nstep dominates the average of thousands of ordinary ones.\n\n$$\n% caption: The distribution of the 10-step importance ratio $\\rho_{t:t+9}$ for\n% $b = (0.5, 0.5)$, $\\pi = (0.9, 0.1)$. Almost all mass sits below 1 (products of\n% mostly-$0.2$ factors), so the mean of exactly 1 is held up by a thin tail reaching\n% $1.8^{10} \\approx 357$. That heavy tail is the variance the SGD update inherits.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  \\draw[->, black] (0,0) -- (6.6,0) node[anchor=north east, font=\\scriptsize] {ratio value (log scale)};\n  \\draw[->, black] (0,0) -- (0,3.6) node[anchor=south east, font=\\scriptsize] {probability};\n  % tall bars near small values\n  \\foreach \\x\u002F\\h in {0.5\u002F3.1, 1.1\u002F2.1, 1.7\u002F1.25, 2.3\u002F0.7, 2.9\u002F0.38} {\n    \\draw[acc, very thick] (\\x,0) -- (\\x,\\h);\n  }\n  % thin tail bars toward the large ratio\n  \\foreach \\x\u002F\\h in {3.6\u002F0.22, 4.3\u002F0.12, 5.0\u002F0.06, 5.7\u002F0.03} {\n    \\draw[red, very thick] (\\x,0) -- (\\x,\\h);\n  }\n  \\node[anchor=north, font=\\scriptsize] at (0.5,-0.08) {tiny};\n  \\draw[black, dashed] (2.05,0) -- (2.05,3.3);\n  \\node[black, anchor=south, font=\\scriptsize] at (2.05,3.3) {mean = 1};\n  \\node[red, anchor=south, font=\\scriptsize] at (5.7,0.1) {357};\n  \\node[red, anchor=west, font=\\scriptsize] at (4.4,1.6) {heavy tail};\n\\end{tikzpicture}\n$$\n\nThe damage is done because these ratios multiply the step size in the SGD update:\na large $\\rho$ means a large step, which stochastic gradient\ndescent cannot tolerate. SGD relies on averaging many small, noisy steps to estimate\nthe true gradient; an occasional enormous step can move the parameter vector into\na region of the space with a completely different gradient, and the averaging\nargument breaks. The obvious defense — shrinking the step size until no single step\nis dangerous — makes the _expected_ step tiny and learning slow.\n\n$$\n% caption: Why importance-sampling variance hurts SGD. Small ratios give safe\n% steps (blue); an occasional large ratio produces a giant step (red) that\n% overshoots into a region with a different gradient, and no amount of averaging\n% recovers a good direction. Shrinking the step size to contain the outlier makes\n% the ordinary steps uselessly small.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  % loss contours (a bowl)\n  \\foreach \\r in {0.7,1.4,2.1} \\draw[black] (0,0) ellipse ({\\r*1.5} and \\r);\n  \\fill[black] (0,0) circle (2pt);\n  \\node[anchor=north, font=\\scriptsize] at (0,-2.3) {minimum};\n  % start point\n  \\coordinate (p0) at (-3.3,1.7);\n  \\fill[acc] (p0) circle (2pt);\n  \\node[acc, anchor=south, font=\\scriptsize] at (-3.3,1.85) {start};\n  % small safe steps\n  \\coordinate (p1) at (-2.5,1.25);\n  \\coordinate (p2) at (-1.8,0.9);\n  \\draw[acc, thick, ->] (p0) -- (p1);\n  \\draw[acc, thick, ->] (p1) -- (p2);\n  \\node[acc, anchor=east, font=\\scriptsize] at (-2.6,0.95) {small ratios: safe};\n  % giant step overshoots\n  \\draw[red, very thick, ->] (p2) -- (3.6,-1.9);\n  \\node[red, anchor=west, font=\\scriptsize] at (1.2,-1.0) {large ratio: overshoot};\n\\end{tikzpicture}\n$$\n\n### Five ideas for taming it\n\nNo single technique suffices, but several complementary strategies each reduce\nthe variance.[^sb-variance]\n\n- **Weighted importance sampling.** Chapter 5 already showed that\n  [weighted importance sampling](\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods)\n  is far better behaved than the ordinary kind — its updates have much lower\n  variance because the largest weight any sample can carry is bounded. The catch is\n  that adapting weighted importance sampling to function approximation is genuinely\n  hard, and can probably only be done approximately, at $O(d)$ cost.\n\n- **Avoiding ratios entirely.** The\n  [tree-backup](\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping)\n  algorithm performs off-policy learning _without_ any importance-sampling ratios\n  at all, by backing up the expected value under the target policy rather than\n  correcting sampled actions. Extending that idea to the function-approximation\n  case yields stable and more efficient methods, since where there are no ratios\n  there is no ratio-variance.\n\n- **Momentum and averaging.** Because the trouble is occasional large, uncorrelated\n  steps, optimizer machinery that smooths the update sequence helps directly:\n  momentum, Polyak–Ruppert averaging (averaging the iterates rather than trusting\n  the latest one), and their extensions all damp the effect of a single outlier\n  step.\n\n- **Adaptive, per-component step sizes.** Rather than one global step size\n  limited by the worst-case ratio, methods that set a _separate_,\n  adaptively-tuned step size for each component of the parameter vector — and\n  \"importance-weight-aware\" updates designed with these products in mind — let the\n  learning rate respond to the local variance instead of being throttled globally.\n\n- **Constraining the target toward the behavior.** A complementary strategy defines\n  the target policy partly _by reference to_ the behavior policy, so that it can\n  never differ enough to create huge ratios in the first place. \"Recognizer\"-style\n  constructions cap how far $\\pi$ may stray from $b$, trading a little generality\n  for a lot of stability.\n\n$$\n% caption: Five levers on off-policy variance. Some cut the variance at its source\n% (weighted IS, tree-backup, constrained targets); others absorb it after the fact\n% (momentum\u002Faveraging, adaptive step sizes). They are complementary and can be\n% combined.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize,\n  src\u002F.style={draw, minimum width=34mm, minimum height=9mm, align=center, font=\\scriptsize},\n  abs\u002F.style={draw, minimum width=34mm, minimum height=9mm, align=center, font=\\scriptsize}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  \\node[font=\\footnotesize, text=acc] at (-3.0,3.0) {cut at the source};\n  \\node[src, draw=acc] (a) at (-3.0,2.15) {weighted IS};\n  \\node[src, draw=acc] (b) at (-3.0,1.05) {tree-backup: no ratios};\n  \\node[src, draw=acc] (c) at (-3.0,-0.05) {constrain target to b};\n  \\node[font=\\footnotesize, text=red] at (3.0,3.0) {absorb after the fact};\n  \\node[abs, draw=red] (d) at (3.0,2.15) {momentum \u002F averaging};\n  \\node[abs, draw=red] (e) at (3.0,1.05) {adaptive step sizes};\n  \\node[align=center, font=\\footnotesize] (v) at (0,-1.5) {lower-variance\\\\of\\\u002Ff-policy update};\n  \\draw[->, acc] (a) -- (v);\n  \\draw[->, acc] (b) -- (v);\n  \\draw[->, acc] (c) -- (v);\n  \\draw[->, red] (d) -- (v);\n  \\draw[->, red] (e) -- (v);\n\\end{tikzpicture}\n$$\n\nNone of these is decisive on its own, and the chapter is candid that off-policy\nvariance \"will probably always remain a challenge.\" Which of these methods combine\neffectively with the stable Gradient-TD and emphatic-TD updates is still an open\nquestion — the whole area is young. But the framing is what matters: stability and\nvariance are _separate_ problems. Gradient-TD and emphatic-TD solve the first;\nthese techniques address the second; and a usable off-policy method under function\napproximation generally needs both.\n\n## The triad in deep reinforcement learning\n\nSutton and Barto present the triad as a theoretical hazard demonstrated on small\nlinear problems. The obvious question — does it actually break the deep networks\nthat now dominate practice? — was answered empirically after the second edition, and\nthe answer is more nuanced than the theory alone suggests.\n\nVan Hasselt, Doron, Strub, Hessel, Sonnerat, and Modayil (2018, \"Deep Reinforcement\nLearning and the Deadly Triad\") ran a controlled study on the 57-game Atari suite,\nvarying the ingredients of the triad in a deep Q-network and measuring how often the\naction-value estimates blew up.[^vh-triad] With rewards clipped to $[-1, 1]$ and\n$\\gamma = 0.99$, the largest possible true action value is about $1\u002F(1-\\gamma) = 100$;\nthey flagged any estimate that exceeded roughly that bound as **soft divergence**.\nTheir findings sharpen Sutton and Barto's picture on four points.\n\n- **Unbounded divergence is rare in practice.** Soft divergence — estimates growing\n  well past the reachable range — did occur, but the values essentially never ran off\n  to infinity (floating-point NaN). The deep networks in current use apparently sit in\n  a regime where the triad distorts the estimates without making them diverge, which is\n  why DQN works at all despite carrying all three ingredients.\n- **Longer multi-step returns diverge _less_.** Increasing the bootstrapping horizon\n  (using $n$-step returns instead of one-step targets) reduced both the frequency of\n  soft divergence and the number of unstable runs, and improved control performance.\n  More of the target comes from real returns and less from the bootstrapped estimate,\n  weakening the leg of the triad that drives instability.\n- **Correcting overestimation helps.** Target networks and\n  [double Q-learning](\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks) — which was\n  designed to cut the maximization bias in the Q-learning target — were the most stable\n  variants. Reducing overestimation and decoupling the bootstrap target from the\n  weights being updated both damp the runaway feedback.\n- **Bigger networks are not automatically safer.** The intuition that a\n  higher-capacity approximator, closer to tabular, should be more stable did _not_ hold:\n  larger networks showed _more_ soft divergence, even though they also reached the best\n  final performance. Capacity and stability trade off rather than align.\n\nThese results reconcile the theory with practice. The triad's danger is real — van\nHasselt's own linear controls still diverge to infinity — but the specific\narchitecture of deep Q-learning (target networks, experience replay, clipped rewards,\nmulti-step returns) keeps the deep case in the soft-divergence regime where learning,\nthough sometimes unstable, does not explode.\n\nOn the theory side, two lines extend Sutton and Barto's stable methods. Sutton, Mahmood, and\nWhite (2016, \"An Emphatic Approach to the Problem of Off-policy Temporal-Difference\nLearning,\" _JMLR_) gave the emphatic-TD method of §11.8 its formal footing: they proved\nthat emphasizing linear TD($\\lambda$)'s updates by the emphasis weighting makes the\nexpected update **stable** under off-policy training, with a single learned weight\nvector and a single step size — simpler than the two-time-scale Gradient-TD family,\nat the cost of the higher variance the lesson already flagged.[^smw-emphatic] More\nrecently, Zhang, Yao, and Whiteson (2021, \"Breaking the Deadly Triad with a Target\nNetwork,\" _ICML_) proved that a **target network** — the same trick DQN uses\nheuristically — combined with a projection or ratio clipping yields a genuinely\nconvergent off-policy prediction and control algorithm under linear approximation,\nturning DQN's empirical stabilizer into a method with a convergence guarantee.[^zyw-target]\nTogether these results close much of the gap between Sutton and Barto's cautionary\ncounterexamples and the systems that run in production: the triad is still dangerous,\nbut there are now several principled ways to avoid its divergence.\n\n## Where this leaves us\n\nOff-policy learning with approximation is the point where the clean guarantees of\ntabular RL give out. The first part of the challenge — correcting update targets with\nimportance sampling — is routine, if variance-prone. The second part — the instability of\nbootstrapping under an off-policy update distribution — is the deadly triad, and it is\nreal enough to send a linearly-independent seven-state problem to infinity. Two responses\nrestore stability without abandoning any leg of the triad: Gradient-TD methods, which do\ntrue SGD on the projected Bellman error at $O(d)$ cost, and emphatic-TD methods, which\nreweight updates back toward the on-policy distribution. Both remain relatively young and\nunsettled. The most consequential practical answer arrives later:\n[deep Q-networks](\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks) keep function\napproximation, bootstrapping, and off-policy replay all at once, and stabilize the triad\nempirically with a target network and a replay buffer rather than by giving anything up.\n\n[^sb-geom]: **Sutton & Barto**, §11.4 — Linear Value-function Geometry, and Figure 11.3: value functions as vectors, the $d$-dimensional representable subspace, the $\\mu$-norm (11.11), the projection operator $\\Pi$ (11.12) and its matrix form (11.14)–(11.16), the Bellman operator $B_\\pi$ (11.20) with fixed point $v_\\pi$ (11.21), the Bellman error vector (11.17)–(11.19), and the projected Bellman error $\\overline{PBE}$ (11.22) with its TD fixed point $\\mathbf{w}_{\\text{TD}}$.\n[^sb-be]: **Sutton & Barto**, §11.5 — Gradient Descent in the Bellman Error (the naive residual-gradient algorithm (11.23), the A-split and A-presplit examples showing $\\overline{BE}$ minimization finds the wrong values), and §11.6 — The Bellman Error is Not Learnable: two MRPs generating identical data with different $\\overline{BE}$-minimizers, so $\\overline{BE}$ cannot be estimated from feature vectors, actions, and rewards alone, while $\\overline{PBE}$ and $\\overline{TDE}$ are learnable (Figure 11.4).\n[^sb-gtd]: **Sutton & Barto**, §11.7 — Gradient-TD Methods: SGD on the $\\overline{PBE}$, the matrix expansion (11.25)–(11.27), the product-of-expectations gradient and its double-sampling obstacle, the stored auxiliary vector $\\mathbf{v}$ (11.28) learned by an importance-weighted LMS rule, the GTD2 and TDC \u002F GTD(0) updates (11.29 and following), Figure 11.5 showing TDC drive the $\\overline{PBE}$ to zero on Baird's counterexample, and the two-time-scale \u002F cascade convergence argument.\n[^sb-variance]: **Sutton & Barto**, §11.9 — Reducing Variance: off-policy learning as inherently higher-variance than on-policy (\"you can't learn to drive by cooking dinner\"); the raison d'être of generalizing to related-but-not-identical policies; the source of variance in products of policy ratios (mean 1 by (5.13), uncorrelated, but of high variance) multiplying the SGD step size and producing occasional very large steps; and the surveyed remedies — weighted importance sampling with its $O(d)$-approximation difficulty (Mahmood and Sutton, 2015), the tree-backup route to ratio-free off-policy learning (Munos et al., 2016; Mahmood, Yu and Sutton, 2017), momentum and Polyak–Ruppert averaging, adaptive per-component and importance-weight-aware step sizes (Jacobs, 1988; Karampatziakis and Langford, 2010), and constraining the target to the behavior via \"recognizers\" (Precup et al., 2006).\n[^sb-etd]: **Sutton & Barto**, §11.8 — Emphatic-TD Methods: reweighting states to recover the on-policy distribution's stability under semi-gradient TD, the emphasis $M_t = \\gamma \\rho_{t-1} M_{t-1} + I_t$ and interest $I_t$, the one-step emphatic-TD update, Figure 11.6 showing convergence in expectation on Baird's counterexample, and the high-variance caveat that motivates the variance-reduction methods of §11.9.\n[^vh-triad]: **H. van Hasselt, Y. Doron, F. Strub, M. Hessel, N. Sonnerat, and J. Modayil**, \"Deep Reinforcement Learning and the Deadly Triad,\" arXiv:1812.02648 (2018). A controlled study on the 57-game Atari suite measuring how the triad's ingredients affect divergence of a deep Q-network: unbounded divergence is rare (soft divergence, above the reachable value bound $\\approx 1\u002F(1-\\gamma)$, is common but values stay finite); longer multi-step returns reduce instability; target networks and double Q-learning are the most stable; and larger networks show more soft divergence though they perform best. Grounds the claim that DQN's architecture keeps the deep triad in a soft-divergence regime.\n[^smw-emphatic]: **R. S. Sutton, A. R. Mahmood, and M. White**, \"An Emphatic Approach to the Problem of Off-policy Temporal-Difference Learning,\" _Journal of Machine Learning Research_ 17(73):1–29 (2016). Proves that emphasizing linear TD($\\lambda$)'s updates makes its expected update stable under off-policy training, with one learned weight vector and one step size — simpler than Gradient-TD; the formal basis for the emphatic-TD method of §11.8.\n[^zyw-target]: **S. Zhang, H. Yao, and S. Whiteson**, \"Breaking the Deadly Triad with a Target Network,\" _Proceedings of the 38th International Conference on Machine Learning_ (ICML 2021). Shows that a target network combined with a projection \u002F ratio truncation gives a provably convergent off-policy prediction and control algorithm under linear function approximation, supplying a convergence guarantee for the target-network trick that DQN uses only heuristically.\n",{"text":11473,"minutes":11474,"time":11475,"words":11476},"16 min read",15.73,943800,3146,{"title":5,"description":11461},[11479,11481],{"book":10326,"ref":11480},"Ch. 11 — Off-policy Methods with Approximation; §11.4 Linear Value-function Geometry; §11.5 Gradient Descent in the Bellman Error",{"book":10326,"ref":11482},"§11.7 Gradient-TD Methods; §11.8 Emphatic-TD Methods; §11.9 Reducing Variance","available","07.reinforcement-learning\u002F03.approximation\u002F06.bellman-error-and-gradient-td","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",[11487],"Approximation","hUdBf7f1wOPdbzqaNaOidecJHSK438so9bMI3HiUJrk",{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":11490,"\u002Falgorithms\u002Ffoundations\u002Fproof-techniques":11491,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":11492,"\u002Falgorithms\u002Ffoundations\u002Fgrowth-rates-and-loop-analysis":11493,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":11494,"\u002Falgorithms\u002Ffoundations\u002Famortized-analysis":11495,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":11496,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":11497,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":11498,"\u002Falgorithms\u002Fdivide-and-conquer\u002Ffast-multiplication":11499,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":11500,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":11501,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":11502,"\u002Falgorithms\u002Fsorting\u002Fexternal-sorting":11503,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":11504,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":11505,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":11506,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":11507,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":11508,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":11509,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":11510,"\u002Falgorithms\u002Fdata-structures\u002Fspatial-data-structures":11511,"\u002Falgorithms\u002Fdata-structures\u002Fskip-lists-and-probabilistic-structures":11512,"\u002Falgorithms\u002Fdata-structures\u002Fb-trees":11513,"\u002Falgorithms\u002Fdata-structures\u002Fdata-stream-algorithms":11514,"\u002Falgorithms\u002Fdata-structures\u002Fstreaming-sketches":11515,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":11516,"\u002Falgorithms\u002Fsequences\u002Fprefix-sums":11517,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":11518,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":11519,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":11520,"\u002Falgorithms\u002Fsequences\u002Fkmp-and-z-function":11521,"\u002Falgorithms\u002Fsequences\u002Ftries":11522,"\u002Falgorithms\u002Fsequences\u002Fsuffix-arrays-and-aho-corasick":11523,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":11524,"\u002Falgorithms\u002Fgraphs\u002Fdepth-first-search":11525,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":11526,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":11527,"\u002Falgorithms\u002Fgraphs\u002Fkruskal-and-prim":11528,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":11529,"\u002Falgorithms\u002Fgraphs\u002Fall-pairs-and-negative-weights":11530,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":11531,"\u002Falgorithms\u002Fgraphs\u002Fmax-flow-min-cut":11532,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":11533,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":11534,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":11535,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":11536,"\u002Falgorithms\u002Fgraphs\u002Fbipartite-matching":11537,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":11538,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":11539,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":11540,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":11541,"\u002Falgorithms\u002Fgreedy\u002Fstable-matching":11542,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":11543,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":11544,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":11545,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":11546,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":11547,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":11548,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":11549,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":11550,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":11551,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":11552,"\u002Falgorithms\u002Fdynamic-programming\u002Fdigit-and-probability-dp":11553,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":11554,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":11555,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":11556,"\u002Falgorithms\u002Fbacktracking\u002Fgraph-backtracking":11557,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":11558,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":11559,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":11560,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":11561,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmatrix-exponentiation":11562,"\u002Falgorithms\u002Fmathematical-algorithms\u002Ffast-fourier-transform":11563,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fgradient-descent":11564,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":11565,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":11566,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":11567,"\u002Falgorithms\u002Fcomputational-geometry\u002Fpolygons-and-proximity":11568,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":11569,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":11570,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":11571,"\u002Falgorithms\u002Fintractability\u002Fapproximation-algorithms":11572,"\u002Falgorithms":11573,"\u002Fcalculus\u002Flimits-and-continuity\u002Ffunctions-and-models":11574,"\u002Fcalculus\u002Flimits-and-continuity\u002Fthe-limit-of-a-function":11575,"\u002Fcalculus\u002Flimits-and-continuity\u002Flimit-laws-and-the-precise-definition":11576,"\u002Fcalculus\u002Flimits-and-continuity\u002Fcontinuity":11577,"\u002Fcalculus\u002Fderivatives\u002Fthe-derivative-and-rates-of-change":11578,"\u002Fcalculus\u002Fderivatives\u002Fdifferentiation-rules-and-the-chain-rule":11579,"\u002Fcalculus\u002Fderivatives\u002Fimplicit-differentiation-and-related-rates":11580,"\u002Fcalculus\u002Fderivatives\u002Flinear-approximations-and-differentials":11581,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fextrema-and-the-mean-value-theorem":11582,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fhow-derivatives-shape-a-graph":11583,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fcurve-sketching-and-optimization":11584,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fnewtons-method-and-antiderivatives":11585,"\u002Fcalculus\u002Fintegrals\u002Farea-and-the-definite-integral":11586,"\u002Fcalculus\u002Fintegrals\u002Fthe-fundamental-theorem-of-calculus":11587,"\u002Fcalculus\u002Fintegrals\u002Fthe-substitution-rule":11588,"\u002Fcalculus\u002Fapplications-of-integration\u002Fareas-and-volumes":11589,"\u002Fcalculus\u002Fapplications-of-integration\u002Fwork-average-value-and-arc-length":11590,"\u002Fcalculus\u002Fapplications-of-integration\u002Fphysics-economics-and-probability":11591,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Finverse-functions-logarithms-and-exponentials":11592,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Fgrowth-decay-inverse-trig-and-hyperbolic-functions":11593,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Flhospitals-rule":11594,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fintegration-by-parts":11595,"\u002Fcalculus\u002Ftechniques-of-integration\u002Ftrigonometric-integrals-and-substitution":11596,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fpartial-fractions-and-integration-strategy":11597,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fapproximate-and-improper-integrals":11598,"\u002Fcalculus\u002Fparametric-and-polar\u002Fparametric-curves-and-their-calculus":11599,"\u002Fcalculus\u002Fparametric-and-polar\u002Fpolar-coordinates":11600,"\u002Fcalculus\u002Fparametric-and-polar\u002Fconic-sections":11601,"\u002Fcalculus\u002Fsequences-and-series\u002Fsequences":11602,"\u002Fcalculus\u002Fsequences-and-series\u002Fseries-and-the-integral-test":11603,"\u002Fcalculus\u002Fsequences-and-series\u002Fthe-convergence-tests":11604,"\u002Fcalculus\u002Fsequences-and-series\u002Fpower-series":11605,"\u002Fcalculus\u002Fsequences-and-series\u002Ftaylor-and-maclaurin-series":11606,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvectors-and-the-dot-product":11607,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fthe-cross-product-lines-and-planes":11588,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fcylinders-and-quadric-surfaces":11608,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvector-functions-and-space-curves":11609,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Farc-length-curvature-and-motion":11610,"\u002Fcalculus\u002Fpartial-derivatives\u002Ffunctions-of-several-variables":11578,"\u002Fcalculus\u002Fpartial-derivatives\u002Fpartial-derivatives":11611,"\u002Fcalculus\u002Fpartial-derivatives\u002Ftangent-planes-and-the-chain-rule":11612,"\u002Fcalculus\u002Fpartial-derivatives\u002Fdirectional-derivatives-and-the-gradient":11613,"\u002Fcalculus\u002Fpartial-derivatives\u002Foptimization-and-lagrange-multipliers":11614,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fdouble-integrals":11615,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Ftriple-integrals-and-coordinate-systems":11616,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fvector-fields-and-line-integrals":11617,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fgreens-theorem-curl-and-divergence":11618,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fsurface-integrals":11619,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fstokes-and-the-divergence-theorem":11620,"\u002Fcalculus":11621,"\u002Fmechanics\u002Ffoundations\u002Fmeasurement-and-dimensions":11622,"\u002Fmechanics\u002Ffoundations\u002Fvector-algebra":11623,"\u002Fmechanics\u002Fkinematics\u002Fone-dimensional-motion":11624,"\u002Fmechanics\u002Fkinematics\u002Fmotion-graphs":11625,"\u002Fmechanics\u002Fkinematics\u002Fprojectile-motion":11626,"\u002Fmechanics\u002Fkinematics\u002Frelative-motion":11627,"\u002Fmechanics\u002Fkinematics\u002Fcircular-motion":11628,"\u002Fmechanics\u002Fdynamics\u002Fnewtons-laws":11629,"\u002Fmechanics\u002Fdynamics\u002Ffree-body-diagrams":11630,"\u002Fmechanics\u002Fdynamics\u002Ffriction-and-curved-motion":11631,"\u002Fmechanics\u002Fdynamics\u002Fnumerical-dynamics":11632,"\u002Fmechanics\u002Fdynamics\u002Fcenter-of-mass-systems":11633,"\u002Fmechanics\u002Fenergy\u002Fwork-and-kinetic-energy":11634,"\u002Fmechanics\u002Fenergy\u002Fpotential-energy":11635,"\u002Fmechanics\u002Fenergy\u002Fmultiparticle-work":11636,"\u002Fmechanics\u002Fenergy\u002Fmass-energy-and-binding":11637,"\u002Fmechanics\u002Fenergy\u002Fphotons-and-quantization":11638,"\u002Fmechanics\u002Fmomentum\u002Fmomentum-and-collisions":11639,"\u002Fmechanics\u002Fmomentum\u002Fcenter-of-mass-collisions":11640,"\u002Fmechanics\u002Fmomentum\u002Frocket-propulsion":11641,"\u002Fmechanics\u002Frotation\u002Frotational-inertia":11642,"\u002Fmechanics\u002Frotation\u002Frotational-dynamics":11643,"\u002Fmechanics\u002Frotation\u002Frolling-motion":11644,"\u002Fmechanics\u002Frotation\u002Fangular-momentum":11645,"\u002Fmechanics\u002Frotation\u002Frolling-resistance":11646,"\u002Fmechanics\u002Frotation\u002Fgyroscopic-precession":11647,"\u002Fmechanics\u002Fgravity-and-matter\u002Fkeplerian-orbits":11648,"\u002Fmechanics\u002Fgravity-and-matter\u002Fgravitational-fields":11649,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstatic-equilibrium":11650,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-statics":11651,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-flow":11652,"\u002Fmechanics\u002Fgravity-and-matter\u002Forbital-motion":11653,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstress-and-elasticity":11654,"\u002Fmechanics\u002Foscillations-waves\u002Fdamped-oscillators":11655,"\u002Fmechanics\u002Foscillations-waves\u002Ftravelling-waves":11656,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-superposition":11657,"\u002Fmechanics\u002Foscillations-waves\u002Fstanding-waves":11658,"\u002Fmechanics\u002Foscillations-waves\u002Fsound-waves":11659,"\u002Fmechanics\u002Foscillations-waves\u002Fdoppler-effect":11660,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-packets":11661,"\u002Fmechanics\u002Foscillations-waves\u002Fbeats-and-coupling":11662,"\u002Fmechanics\u002Foscillations-waves\u002Fsimple-harmonic-motion":11663,"\u002Fmechanics\u002Foscillations-waves\u002Fpendulum-motion":11664,"\u002Fmechanics\u002Foscillations-waves\u002Fdriven-oscillators":11665,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-boundaries":11666,"\u002Fmechanics\u002Fthermodynamics\u002Fkinetic-theory-of-ideal-gases":11667,"\u002Fmechanics\u002Fthermodynamics\u002Ffirst-law-of-thermodynamics":11668,"\u002Fmechanics\u002Fthermodynamics\u002Fentropy-and-the-second-law":11669,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-processes":11670,"\u002Fmechanics\u002Fthermodynamics\u002Fphase-changes":11671,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-machines":11672,"\u002Fmechanics":11673,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcharge-and-conductors":11674,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcoulombs-law":11675,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-and-force":11676,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-maps":11677,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-dipoles":11678,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fcontinuous-charge-fields":11679,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fgauss-law-and-conductors":11680,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpoint-charge-potential":11681,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpotential-gradients-and-equipotentials":11682,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Felectrostatic-energy-and-pressure":11683,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Flaplace-boundary-problems":11684,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fcontinuous-charge-potentials":11685,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitance-fundamentals":11662,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-networks":11686,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-energy-and-force":11687,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fdielectric-polarization-and-breakdown":11688,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fcurrent-and-resistance":11658,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fkirchhoff-network-analysis":11523,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Frc-transients":11689,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-trajectories":11649,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fhall-effect":11690,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-force-on-conductors":11691,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-dipoles":11692,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmass-spectrometry":11693,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmoving-charge-fields":11694,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fbiot-savart-law":11695,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fcircular-current-loops":11696,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Famperes-law":11697,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fgauss-law-for-magnetism":11698,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmagnetic-materials":11623,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-flux":11699,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Ffaradays-law":11700,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Flenzs-law":11701,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmotional-emf":11702,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Feddy-currents":11703,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fself-inductance":11704,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-energy":11705,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Frl-circuits":11706,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-fundamentals":11641,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Freactance":11640,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Frlc-resonance":11707,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-power":11708,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Ftransformers":11709,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdisplacement-current":11710,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-waves":11711,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-momentum":11712,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdipole-radiation":11713,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fpolarization":11714,"\u002Felectricity-and-magnetism\u002Foptics\u002Freflection-and-refraction":11715,"\u002Felectricity-and-magnetism\u002Foptics\u002Fthin-lenses":11667,"\u002Felectricity-and-magnetism\u002Foptics\u002Fspherical-mirrors":11665,"\u002Felectricity-and-magnetism":11716,"\u002Flinear-algebra\u002Flinear-systems\u002Fsystems-and-echelon-forms":11717,"\u002Flinear-algebra\u002Flinear-systems\u002Fvector-and-matrix-equations":11718,"\u002Flinear-algebra\u002Flinear-systems\u002Fsolution-sets-and-applications":11719,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-independence":11720,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-transformations":11721,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-operations":11722,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-inverse-and-invertibility":11723,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fpartitioned-matrices-and-lu":11724,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fsubspaces-dimension-rank":11725,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fapplications-leontief-and-graphics":11575,"\u002Flinear-algebra\u002Fdeterminants\u002Fdeterminants-and-cofactors":11726,"\u002Flinear-algebra\u002Fdeterminants\u002Fproperties-of-determinants":11727,"\u002Flinear-algebra\u002Fdeterminants\u002Fcramer-volume-and-area":11579,"\u002Flinear-algebra\u002Fvector-spaces\u002Fvector-spaces-and-subspaces":11728,"\u002Flinear-algebra\u002Fvector-spaces\u002Fnull-and-column-spaces":11729,"\u002Flinear-algebra\u002Fvector-spaces\u002Fbases-and-independent-sets":11730,"\u002Flinear-algebra\u002Fvector-spaces\u002Fcoordinate-systems":11731,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdimension-and-rank":11732,"\u002Flinear-algebra\u002Fvector-spaces\u002Fchange-of-basis":11733,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdifference-equations-and-markov":11734,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-eigenvalues":11735,"\u002Flinear-algebra\u002Feigenvalues\u002Fthe-characteristic-equation":11736,"\u002Flinear-algebra\u002Feigenvalues\u002Fdiagonalization":11737,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-linear-transformations":11738,"\u002Flinear-algebra\u002Feigenvalues\u002Fcomplex-eigenvalues":11739,"\u002Flinear-algebra\u002Feigenvalues\u002Fdynamical-systems":11740,"\u002Flinear-algebra\u002Feigenvalues\u002Fpower-method":11741,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-length-orthogonality":11742,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Forthogonal-sets-and-projections":11743,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fgram-schmidt-and-qr":11744,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-problems":11745,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-applications":11746,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-spaces":11747,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fdiagonalizing-symmetric-matrices":11614,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fquadratic-forms":11748,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fconstrained-optimization":11749,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsingular-value-decomposition":11750,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsvd-applications-pca-imaging":11751,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-thinking-and-matrix-computation":11752,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Flu-and-cholesky":11753,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fconditioning-and-floating-point":11754,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fstability-and-error-analysis":11755,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fqr-and-numerical-least-squares":11756,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-eigenvalues-and-svd":11757,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-combinations":11758,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-independence-and-barycentric-coordinates":11759,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fconvex-combinations-and-convex-sets":11760,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fhyperplanes-and-polytopes":11761,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fcurves-and-surfaces":11762,"\u002Flinear-algebra":11763,"\u002Ftheory-of-computation":11764,"\u002Fcomputer-architecture\u002Ffoundations\u002Fbits-bytes-and-words":11765,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-representation":11766,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-arithmetic":11767,"\u002Fcomputer-architecture\u002Ffoundations\u002Ffloating-point":11768,"\u002Fcomputer-architecture\u002Ffoundations\u002Fboolean-algebra-and-bit-manipulation":11769,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fthe-machines-view":11770,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fdata-movement":11771,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farithmetic-and-logic":11772,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fcontrol-flow":11773,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fprocedures":11774,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farrays-structs-and-alignment":11775,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fmemory-layout-and-buffer-overflows":11776,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fwhat-an-isa-is":11777,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Finstruction-formats-and-operands":11778,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Faddressing-modes":11779,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fthe-y86-64-instruction-set":11780,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fy86-64-programming":11781,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Ftransistors-gates-and-boolean-functions":11782,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fcombinational-logic-and-hcl":11783,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmultiplexers-decoders-and-the-alu":11784,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmemory-elements-latches-flip-flops-and-clocking":11785,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fregister-files-and-random-access-memory":11786,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-fetch-decode-execute-cycle":11787,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-seq-stages":11788,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fcontrol-logic-and-sequencing":11789,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fassembling-seq":11790,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Ftracing-a-program":11791,"\u002Fcomputer-architecture\u002Fpipelining\u002Fpipelining-principles":11792,"\u002Fcomputer-architecture\u002Fpipelining\u002Ffrom-seq-to-pipe":11793,"\u002Fcomputer-architecture\u002Fpipelining\u002Fdata-hazards-stalling-and-forwarding":11794,"\u002Fcomputer-architecture\u002Fpipelining\u002Fcontrol-hazards-and-branch-prediction":11795,"\u002Fcomputer-architecture\u002Fpipelining\u002Fthe-complete-pipe-processor":11796,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fstorage-technologies-and-the-latency-gap":11797,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Flocality":11798,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-memories-direct-mapped":11799,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fset-associative-and-write-policies":11800,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-performance-and-cache-friendly-code":11801,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Faddress-spaces-and-translation":11802,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fpage-tables-and-page-faults":11803,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fthe-tlb-and-multi-level-page-tables":11804,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Fexceptional-control-flow":11805,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Finterrupts-and-the-kernel":11806,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fprocesses-threads-and-parallelism":11807,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fhardware-multithreading":11808,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fcache-coherence":11809,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmemory-consistency-and-synchronization":11810,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmulticore-organization":11811,"\u002Fcomputer-architecture\u002Fcapstone\u002Fthe-whole-machine":11812,"\u002Fcomputer-architecture\u002Fcapstone\u002Fassembling-a-complete-cpu":11813,"\u002Fcomputer-architecture":11764,"\u002Fdifferential-equations\u002Ffoundations\u002Fmodels-and-direction-fields":11814,"\u002Fdifferential-equations\u002Ffoundations\u002Fclassification-and-terminology":11815,"\u002Fdifferential-equations\u002Ffirst-order\u002Flinear-first-order-integrating-factors":11816,"\u002Fdifferential-equations\u002Ffirst-order\u002Fseparable-and-exact":11579,"\u002Fdifferential-equations\u002Ffirst-order\u002Fmodeling-first-order":11817,"\u002Fdifferential-equations\u002Ffirst-order\u002Fautonomous-and-population-dynamics":11578,"\u002Fdifferential-equations\u002Ffirst-order\u002Fexistence-uniqueness-euler":11585,"\u002Fdifferential-equations\u002Ffirst-order\u002Ffirst-order-difference-equations":11818,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhomogeneous-constant-coefficients":11819,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fcomplex-and-repeated-roots":11619,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fnonhomogeneous-undetermined-coefficients":11820,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fvariation-of-parameters":11821,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fmechanical-electrical-vibrations":11822,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhigher-order-linear":11823,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fpower-series-ordinary-points":11824,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fregular-singular-frobenius":11825,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fbessel-and-special-functions":11826,"\u002Fdifferential-equations\u002Flaplace\u002Flaplace-definition-ivps":11827,"\u002Fdifferential-equations\u002Flaplace\u002Fstep-impulse-convolution":11828,"\u002Fdifferential-equations\u002Fsystems\u002Fmatrices-eigenvalues-review":11829,"\u002Fdifferential-equations\u002Fsystems\u002Fconstant-coefficient-systems-phase-portraits":11830,"\u002Fdifferential-equations\u002Fsystems\u002Frepeated-eigenvalues-fundamental-matrices":11831,"\u002Fdifferential-equations\u002Fnumerical\u002Feuler-and-runge-kutta":11826,"\u002Fdifferential-equations\u002Fnumerical\u002Fmultistep-systems-stability":11832,"\u002Fdifferential-equations\u002Fnonlinear\u002Fphase-plane-autonomous-stability":11833,"\u002Fdifferential-equations\u002Fnonlinear\u002Flocally-linear-and-liapunov":11834,"\u002Fdifferential-equations\u002Fnonlinear\u002Fcompeting-species-predator-prey-limit-cycles":11835,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Ffourier-series":11836,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fheat-wave-laplace-equations":11837,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fsturm-liouville":11838,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fcalculus-of-variations":11839,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fhistorical-notes":11840,"\u002Fdifferential-equations":11841,"\u002Frelativity\u002Ffoundations\u002Fspecial-relativity-postulates":11842,"\u002Frelativity\u002Ffoundations\u002Florentz-transformation-spacetime":11843,"\u002Frelativity\u002Ffoundations\u002Ftime-dilation-length-contraction":11844,"\u002Frelativity\u002Ffoundations\u002Frelativistic-momentum-energy":11845,"\u002Frelativity\u002Ffoundations\u002Fgeneral-relativity":11724,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fminkowski-spacetime-and-the-interval":11846,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Ffour-vectors-and-index-notation":11847,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fthe-lorentz-group-and-rapidity":11848,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fdoppler-aberration-and-appearance":11849,"\u002Frelativity\u002Frelativistic-dynamics\u002Ffour-momentum-force-and-accelerated-motion":11850,"\u002Frelativity\u002Frelativistic-dynamics\u002Fparticle-decays-and-two-body-kinematics":11851,"\u002Frelativity\u002Frelativistic-dynamics\u002Fcollisions-thresholds-and-the-cm-frame":11594,"\u002Frelativity\u002Frelativistic-dynamics\u002Fmandelstam-variables-and-invariants":11852,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ffour-current-and-the-four-potential":11853,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fthe-electromagnetic-field-tensor":11854,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ftransformation-of-electric-and-magnetic-fields":11855,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fcovariant-maxwell-and-the-stress-energy-tensor":11856,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-equivalence-principle-formalized":11857,"\u002Frelativity\u002Fcurved-spacetime\u002Fmanifolds-vectors-and-the-metric":11858,"\u002Frelativity\u002Fcurved-spacetime\u002Fcovariant-derivative-and-christoffel-symbols":11859,"\u002Frelativity\u002Fcurved-spacetime\u002Fgeodesics-and-the-geodesic-equation":11860,"\u002Frelativity\u002Fcurved-spacetime\u002Fcurvature-riemann-and-geodesic-deviation":11861,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-einstein-field-equations":11816,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fthe-schwarzschild-metric":11862,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fgeodesics-and-orbits-in-schwarzschild":11863,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Flight-bending-and-null-geodesics":11864,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fperihelion-precession-of-mercury":11865,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fdeflection-of-light-and-gravitational-lensing":11866,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fgravitational-redshift-and-shapiro-delay":11745,"\u002Frelativity\u002Ftests-of-general-relativity\u002Frelativity-in-technology-gps":11867,"\u002Frelativity\u002Fblack-holes\u002Fhorizons-and-coordinate-singularities":11868,"\u002Frelativity\u002Fblack-holes\u002Frotating-and-charged-black-holes":11755,"\u002Frelativity\u002Fblack-holes\u002Fblack-hole-thermodynamics":11869,"\u002Frelativity\u002Fgravitational-waves\u002Flinearized-gravity-and-wave-solutions":11870,"\u002Frelativity\u002Fgravitational-waves\u002Fgeneration-and-the-quadrupole-formula":11871,"\u002Frelativity\u002Fgravitational-waves\u002Fdetection-ligo-and-the-first-events":11872,"\u002Frelativity\u002Fcosmological-bridge\u002Fthe-cosmological-principle-and-flrw-metric":11873,"\u002Frelativity\u002Fcosmological-bridge\u002Ffriedmann-equations-and-cosmic-dynamics":11874,"\u002Frelativity":11875,"\u002Fphysical-computing":11764,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fblackbody-radiation-and-the-planck-quantum":11876,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-photoelectric-effect-and-the-photon":11855,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fx-rays-and-the-compton-effect":11877,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-old-quantum-theory-bohr-and-sommerfeld":11878,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fde-broglie-waves-and-electron-diffraction":11879,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fwave-packets-and-the-probability-interpretation":11880,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fthe-uncertainty-principle":11881,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-schrodinger-equation-in-one-dimension":11882,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-free-particle-and-wave-packet-dynamics":11883,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fparticle-in-infinite-and-finite-square-wells":11831,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Foperators-expectation-values-and-the-harmonic-oscillator":11755,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-dirac-delta-potential":11884,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fbarrier-penetration-and-quantum-tunneling":11885,"\u002Fquantum-mechanics\u002Fformalism\u002Fhilbert-space-and-dirac-notation":11886,"\u002Fquantum-mechanics\u002Fformalism\u002Fobservables-hermitian-operators-and-eigenvalues":11887,"\u002Fquantum-mechanics\u002Fformalism\u002Fthe-postulates-and-quantum-measurement":11883,"\u002Fquantum-mechanics\u002Fformalism\u002Fposition-momentum-and-continuous-spectra":11888,"\u002Fquantum-mechanics\u002Fformalism\u002Fcommutators-and-the-generalized-uncertainty-principle":11863,"\u002Fquantum-mechanics\u002Fformalism\u002Ftime-evolution-schrodinger-and-heisenberg-pictures":11616,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fladder-operators-and-the-number-states":11889,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fcoherent-and-squeezed-states":11890,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fsymmetries-generators-and-conservation-laws":11596,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fparity-time-reversal-and-discrete-symmetries":11891,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Forbital-angular-momentum-and-spherical-harmonics":11892,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Fthe-angular-momentum-algebra":11893,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Faddition-of-angular-momenta-and-clebsch-gordan":11894,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-schrodinger-equation-in-three-dimensions":11895,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-hydrogen-atom":11896,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-isotropic-oscillator-and-hidden-symmetry":11897,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-half-pauli-matrices-and-stern-gerlach":11898,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-in-a-magnetic-field-precession-and-resonance":11899,"\u002Fquantum-mechanics\u002Fspin\u002Ftwo-level-systems-and-the-bloch-sphere":11855,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fidentical-particles-and-exchange-symmetry":11900,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fthe-pauli-principle-atoms-and-the-periodic-table":11901,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ftime-independent-perturbation-theory":11902,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ffine-structure-and-the-real-hydrogen-atom":11889,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-zeeman-and-stark-effects":11577,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-variational-method":11903,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-wkb-approximation":11904,"\u002Fquantum-mechanics":11905,"\u002Freal-analysis\u002Ffoundations\u002Fsets-logic-functions":11839,"\u002Freal-analysis\u002Ffoundations\u002Fordered-fields-completeness":11906,"\u002Freal-analysis\u002Ffoundations\u002Fabsolute-value-bounds":11907,"\u002Freal-analysis\u002Ffoundations\u002Fintervals-uncountability":11728,"\u002Freal-analysis\u002Fsequences-series\u002Fsequences-limits":11908,"\u002Freal-analysis\u002Fsequences-series\u002Flimit-laws-monotone":11614,"\u002Freal-analysis\u002Fsequences-series\u002Flimsup-bolzano-weierstrass":11909,"\u002Freal-analysis\u002Fsequences-series\u002Fcauchy-completeness":11910,"\u002Freal-analysis\u002Fsequences-series\u002Fseries-convergence":11751,"\u002Freal-analysis\u002Fsequences-series\u002Fabsolute-conditional-rearrangement":11861,"\u002Freal-analysis\u002Fmetric-spaces\u002Fmetric-spaces-norms":11911,"\u002Freal-analysis\u002Fmetric-spaces\u002Fopen-closed-sets":11912,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconvergence-completeness":11913,"\u002Freal-analysis\u002Fmetric-spaces\u002Fcompactness":11914,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconnectedness":11915,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-of-functions":11888,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuous-functions":11916,"\u002Freal-analysis\u002Fcontinuity\u002Fevt-ivt":11721,"\u002Freal-analysis\u002Fcontinuity\u002Funiform-continuity":11917,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuity-metric-spaces":11918,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-infinity-monotone":11575,"\u002Freal-analysis\u002Fdifferentiation\u002Fthe-derivative":11919,"\u002Freal-analysis\u002Fdifferentiation\u002Fmean-value-theorem":11920,"\u002Freal-analysis\u002Fdifferentiation\u002Ftaylors-theorem":11877,"\u002Freal-analysis\u002Fdifferentiation\u002Finverse-function-1d":11604,"\u002Freal-analysis\u002Friemann-integration\u002Fdarboux-integral":11755,"\u002Freal-analysis\u002Friemann-integration\u002Fintegrability-classes":11921,"\u002Freal-analysis\u002Friemann-integration\u002Fproperties-of-the-integral":11922,"\u002Freal-analysis\u002Friemann-integration\u002Ffundamental-theorem":11740,"\u002Freal-analysis\u002Friemann-integration\u002Flog-exp-improper":11865,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpointwise-uniform-convergence":11923,"\u002Freal-analysis\u002Ffunction-sequences\u002Finterchange-of-limits":11924,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpower-series-weierstrass":11925,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpicard-ode":11760,"\u002Freal-analysis\u002Fseveral-variables\u002Fdifferentiability-rn":11926,"\u002Freal-analysis\u002Fseveral-variables\u002Fgradient-chain-rule":11927,"\u002Freal-analysis\u002Fseveral-variables\u002Fhigher-derivatives-taylor-extrema":11928,"\u002Freal-analysis\u002Fseveral-variables\u002Finverse-implicit-theorems":11928,"\u002Freal-analysis\u002Fseveral-variables\u002Fmultiple-integrals":11929,"\u002Freal-analysis":11930,"\u002Fabstract-algebra\u002Ffoundations\u002Fsets-functions-relations":11931,"\u002Fabstract-algebra\u002Ffoundations\u002Fintegers-and-modular-arithmetic":11932,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fgroup-axioms-and-first-examples":11933,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fdihedral-and-symmetric-groups":11934,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fmatrix-and-quaternion-groups":11935,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fhomomorphisms-and-group-actions":11936,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fsubgroups-and-substructures":11937,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcyclic-groups":11938,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fgeneration-and-subgroup-lattices":11939,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcosets-lagrange-and-normal-subgroups":11940,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fisomorphism-theorems":11904,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcomposition-series-and-the-alternating-group":11941,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Factions-and-cayleys-theorem":11933,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fconjugation-and-the-class-equation":11850,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fsylow-theorems":11942,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fautomorphisms-and-simple-groups":11943,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fdirect-products-and-finite-abelian-groups":11944,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fsemidirect-products":11945,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fnilpotent-and-solvable-groups":11946,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fclassifying-small-groups":11947,"\u002Fabstract-algebra\u002Fring-theory\u002Frings-definitions-and-examples":11948,"\u002Fabstract-algebra\u002Fring-theory\u002Fideals-quotients-and-homomorphisms":11949,"\u002Fabstract-algebra\u002Fring-theory\u002Ffractions-and-the-chinese-remainder-theorem":11943,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Feuclidean-domains-pids-ufds":11950,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fpolynomial-rings-over-fields":11919,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fgauss-lemma-and-unique-factorization":11951,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Firreducibility-criteria-and-groebner":11952,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fintroduction-to-modules":11953,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ffree-modules-and-direct-sums":11954,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ftensor-products-and-exact-sequences":11955,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fvector-spaces-and-linear-maps":11956,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fstructure-theorem-over-pids":11957,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Frational-canonical-form":11958,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fjordan-canonical-form":11959,"\u002Fabstract-algebra\u002Ffield-theory\u002Ffield-extensions-and-algebraic-elements":11960,"\u002Fabstract-algebra\u002Ffield-theory\u002Fstraightedge-and-compass-constructions":11593,"\u002Fabstract-algebra\u002Ffield-theory\u002Fsplitting-fields-and-algebraic-closure":11961,"\u002Fabstract-algebra\u002Ffield-theory\u002Fseparable-and-cyclotomic-extensions":11962,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fthe-galois-correspondence":11835,"\u002Fabstract-algebra\u002Fgalois-theory\u002Ffinite-fields":11963,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fcyclotomic-and-abelian-extensions":11964,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fgalois-groups-of-polynomials":11892,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fsolvability-by-radicals-and-the-quintic":11964,"\u002Fabstract-algebra\u002Fcapstone\u002Fcommutative-algebra-and-algebraic-geometry":11965,"\u002Fabstract-algebra\u002Fcapstone\u002Frepresentation-and-character-theory":11966,"\u002Fabstract-algebra":11967,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fatomic-spectra-rutherford":11968,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-model-hydrogen":11969,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fx-ray-spectra-franck-hertz":11970,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-sommerfeld-old-quantum-theory":11971,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fold-quantum-theory-limits-wkb":11972,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fschrodinger-3d-hydrogen":11884,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fhydrogen-wave-functions":11973,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fradial-equation-in-full":11974,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fsymmetry-degeneracy-runge-lenz":11975,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fexpectation-values-virial":11976,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fquantum-defects-alkali-spectra":11977,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Frydberg-atoms":11978,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Frelativistic-kinetic-correction":11979,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fspin-orbit-thomas-precession":11718,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdarwin-term-fine-structure-formula":11847,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdirac-equation-hydrogen":11593,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Flamb-shift-qed":11980,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fhyperfine-structure-21cm":11577,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fnuclear-effects-isotope-shift":11981,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fperiodic-table-atomic-spectra":11982,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fcentral-field-self-consistent":11615,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fidentical-particles-hartree-fock":11909,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhelium-two-electron-atom":11983,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fls-jj-coupling-term-symbols":11984,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhund-rules-ground-terms":11985,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fzeeman-effect":11986,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fpaschen-back-intermediate":11987,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fstark-effect-polarizability":11988,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Ftime-dependent-perturbation-golden-rule":11989,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fdipole-approximation-einstein-coefficients":11990,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fselection-rules-forbidden-transitions":11991,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Flifetimes-and-line-shapes":11992,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Flaser-principles":11993,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fspectroscopy-techniques":11994,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fline-catalog-nist-asd":11995,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Flaser-cooling-doppler":11996,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fsub-doppler-trapping":11940,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fbose-einstein-condensation":11997,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Foptical-clocks-precision":11998,"\u002Fatomic-physics":11999,"\u002Fdatabases":11764,"\u002Fcategory-theory\u002Ffoundations\u002Fwhat-is-a-category":12000,"\u002Fcategory-theory\u002Ffoundations\u002Fexamples-of-categories":12001,"\u002Fcategory-theory\u002Ffoundations\u002Fspecial-morphisms":12002,"\u002Fcategory-theory\u002Ffoundations\u002Ffunctors":11931,"\u002Fcategory-theory\u002Ffoundations\u002Fnatural-transformations":12003,"\u002Fcategory-theory\u002Ffoundations\u002Fsize-and-set-theory":12004,"\u002Fcategory-theory\u002Funiversal-properties\u002Funiversal-properties":12005,"\u002Fcategory-theory\u002Funiversal-properties\u002Fproducts-and-coproducts":12006,"\u002Fcategory-theory\u002Funiversal-properties\u002Fconstructions-on-categories":12007,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Frepresentable-functors":12008,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-lemma":12009,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-consequences":12010,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits":12011,"\u002Fcategory-theory\u002Flimits-colimits\u002Fproducts-equalizers-pullbacks":12012,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcolimits":12013,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcomputing-limits":12014,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits-and-functors":12015,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions":12016,"\u002Fcategory-theory\u002Fadjunctions\u002Funits-and-counits":12017,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions-via-universal-arrows":12018,"\u002Fcategory-theory\u002Fadjunctions\u002Ffree-forgetful-adjunctions":12019,"\u002Fcategory-theory\u002Fadjoints-limits\u002Flimits-via-adjoints":12020,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fpresheaf-limits-colimits":12021,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoints-preserve-limits":12022,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoint-functor-theorem":12013,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fmonads":12023,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-eilenberg-moore":12024,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fkleisli-and-programming":12025,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-for-endofunctors":12026,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Fcartesian-closed-categories":12027,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Flambda-calculus-correspondence":11972,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Ffixed-points-and-recursion":12028,"\u002Fcategory-theory":12029,"\u002Fdeep-learning\u002Fmathematical-background\u002Flinear-algebra-for-deep-learning":12030,"\u002Fdeep-learning\u002Fmathematical-background\u002Fprobability-and-information-theory":12031,"\u002Fdeep-learning\u002Fmathematical-background\u002Fnumerical-computation":12032,"\u002Fdeep-learning\u002Fmathematical-background\u002Fcalculus":12033,"\u002Fdeep-learning\u002Ffoundations\u002Fwhat-is-deep-learning":12034,"\u002Fdeep-learning\u002Ffoundations\u002Fmachine-learning-refresher":12035,"\u002Fdeep-learning\u002Ffoundations\u002Flinear-models-and-the-perceptron":11994,"\u002Fdeep-learning\u002Fneural-networks\u002Fthe-multilayer-perceptron":12036,"\u002Fdeep-learning\u002Fneural-networks\u002Factivation-functions":12037,"\u002Fdeep-learning\u002Fneural-networks\u002Funiversal-approximation":12038,"\u002Fdeep-learning\u002Fneural-networks\u002Fbackpropagation":12039,"\u002Fdeep-learning\u002Fneural-networks\u002Floss-functions-and-output-units":12040,"\u002Fdeep-learning\u002Foptimization\u002Fgradient-descent-and-sgd":12041,"\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods":12042,"\u002Fdeep-learning\u002Foptimization\u002Finitialization":12043,"\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape":12044,"\u002Fdeep-learning\u002Foptimization\u002Fsecond-order-and-approximate-methods":12045,"\u002Fdeep-learning\u002Fregularization\u002Fregularization-overview":12046,"\u002Fdeep-learning\u002Fregularization\u002Fdropout-and-data-augmentation":12047,"\u002Fdeep-learning\u002Fregularization\u002Fearly-stopping-and-parameter-sharing":12048,"\u002Fdeep-learning\u002Fregularization\u002Fnormalization":12049,"\u002Fdeep-learning\u002Farchitectures\u002Fconvolutional-networks":12050,"\u002Fdeep-learning\u002Farchitectures\u002Fcnn-architectures":12051,"\u002Fdeep-learning\u002Farchitectures\u002Frecurrent-networks":12052,"\u002Fdeep-learning\u002Farchitectures\u002Flstm-and-gru":12053,"\u002Fdeep-learning\u002Farchitectures\u002Fattention-and-transformers":12054,"\u002Fdeep-learning\u002Farchitectures\u002Fthe-transformer-architecture":12055,"\u002Fdeep-learning\u002Farchitectures\u002Ftransformers-in-practice":12056,"\u002Fdeep-learning\u002Farchitectures\u002Fgraph-neural-networks":12057,"\u002Fdeep-learning\u002Farchitectures\u002Fstate-space-models":12058,"\u002Fdeep-learning\u002Ftheory\u002Fgeneralization-theory":12059,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-robustness":12060,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-defenses":12061,"\u002Fdeep-learning\u002Ftheory\u002Fbayesian-and-ensemble-methods":12062,"\u002Fdeep-learning\u002Ftheory\u002Fdeep-equilibrium-models":12006,"\u002Fdeep-learning\u002Fgenerative-models\u002Flinear-factor-models":12063,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoencoders":12064,"\u002Fdeep-learning\u002Fgenerative-models\u002Fvariational-autoencoders":12065,"\u002Fdeep-learning\u002Fgenerative-models\u002Fgenerative-adversarial-networks":12066,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoregressive-and-normalizing-flows":12067,"\u002Fdeep-learning\u002Fgenerative-models\u002Fenergy-based-and-boltzmann-machines":12068,"\u002Fdeep-learning\u002Fgenerative-models\u002Fdiffusion-and-score-based-models":12069,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fstructured-probabilistic-models":11558,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fmonte-carlo-and-mcmc":12070,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fapproximate-inference":12071,"\u002Fdeep-learning\u002Fpractical\u002Fpractical-methodology":11794,"\u002Fdeep-learning\u002Fpractical\u002Fhyperparameters-and-debugging":12072,"\u002Fdeep-learning\u002Fpractical\u002Frepresentation-learning":12073,"\u002Fdeep-learning\u002Fpractical\u002Ftransfer-learning":12074,"\u002Fdeep-learning\u002Fpractical\u002Fapplications":12075,"\u002Fdeep-learning\u002Fpractical\u002Fmodel-compression-and-distillation":12076,"\u002Fdeep-learning\u002Fpractical\u002Fmeta-learning-and-few-shot":12077,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models":12078,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment":12079,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fseq2seq-pretraining-and-bart":12080,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ftext-to-text-transfer-and-conditional-generation":12081,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fspeech-and-audio-models":12082,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fself-supervised-speech-and-synthesis":12083,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents":11767,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fagent-memory-retrieval-and-orchestration":12084,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmixture-of-experts":12085,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmultimodal-models":12086,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ffusion-and-vision-language-models":12087,"\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning":11804,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fmodel-free-prediction-and-control":12088,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fdeep-q-networks":12089,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic":12090,"\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback":12091,"\u002Fdeep-learning":11764,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fequilibrium-state-variables-zeroth-law":12092,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Ffirst-law-heat-and-work":11889,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fsecond-law-entropy-and-the-carnot-bound":12093,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fthermodynamic-potentials-and-maxwell-relations":12094,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fstability-response-functions-and-the-third-law":12095,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fclassical-statistics-and-equipartition":12096,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fphase-space-and-liouvilles-theorem":12097,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fensembles-and-the-equal-probability-postulate":12098,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fstatistical-entropy-boltzmann-and-gibbs":12099,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fmicrocanonical-ensemble-and-entropy":12100,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fequilibrium-conditions-temperature-pressure-chemical-potential":11936,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fideal-gas-phase-space-and-the-sackur-tetrode-entropy":12101,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Ftwo-state-systems-paramagnets-and-negative-temperature":12102,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fcanonical-ensemble-and-the-boltzmann-distribution":12103,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fpartition-function-and-the-helmholtz-free-energy":11823,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fenergy-fluctuations-and-ensemble-equivalence":11616,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fthe-einstein-solid-and-harmonic-systems":12104,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fparamagnetism-and-the-schottky-anomaly":12105,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fideal-gas-partition-function-and-the-gibbs-paradox":12106,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fequipartition-and-the-virial-theorem":11750,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fmolecular-gases-rotation-and-vibration":12107,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fgrand-canonical-ensemble-and-the-grand-partition-function":12108,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fchemical-potential-fugacity-and-number-fluctuations":11828,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fensemble-summary-and-the-thermodynamic-web":11755,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fquantum-statistics-bose-einstein-and-fermi-dirac":12109,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fderiving-the-quantum-distributions":11724,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fthe-classical-limit-and-quantum-concentration":11594,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fideal-quantum-gases-general-framework":11610,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-and-the-fermion-gas":12110,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthe-photon-gas-and-plancks-radiation-law":12111,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fblackbody-thermodynamics-and-radiation-pressure":12112,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fphonons-and-the-debye-model":11742,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-derived":12113,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthermodynamics-of-the-bose-gas-and-superfluidity":11604,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fthe-ideal-fermi-gas-at-zero-temperature":12114,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fsommerfeld-expansion-and-electrons-in-metals":12115,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":12116,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fneutron-stars-and-nuclear-matter":12117,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-cluster-expansion-and-virial-coefficients":11938,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-van-der-waals-gas-and-liquid-gas-coexistence":12118,"\u002Fstatistical-mechanics\u002Finteractions\u002Fquantum-gases-with-interactions-and-exchange":12119,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fphases-coexistence-and-classification":12120,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-ising-model-and-exact-solutions":12121,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fmean-field-theory-and-the-weiss-model":11884,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fcritical-exponents-and-landau-theory":12122,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-renormalization-group-idea":11862,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fthermodynamic-fluctuations-and-response":12123,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fbrownian-motion-and-the-langevin-equation":11723,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Flinear-response-and-the-fluctuation-dissipation-theorem":12124,"\u002Fstatistical-mechanics":12125,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fbonding-mechanisms":12126,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fmolecular-orbitals-and-h2-plus":11589,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fhydrogen-molecule-and-exchange":11849,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fvan-der-waals-forces":12127,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Frotational-vibrational-spectra":12128,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Fanharmonicity-and-rovibrational-structure":12129,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Framan-and-electronic-bands":12130,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Flasers-and-masers":12131,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fstructure-of-solids":12132,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fbravais-lattices-and-crystal-systems":11722,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Freciprocal-lattice-and-brillouin-zones":12133,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fdiffraction-and-structure-factors":12134,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonon-dispersion":12135,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonons-quantization-and-dos":12136,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fdebye-einstein-heat-capacity":11865,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fanharmonicity-and-thermal-transport":12137,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ffree-electron-gas-and-conduction":12138,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fsommerfeld-model-and-heat-capacity":12139,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ftransport-and-the-hall-effect":12140,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fscreening-and-plasmons":12141,"\u002Fcondensed-matter\u002Fband-theory\u002Fblochs-theorem-and-energy-bands":11920,"\u002Fcondensed-matter\u002Fband-theory\u002Fnearly-free-electron-model":11864,"\u002Fcondensed-matter\u002Fband-theory\u002Ftight-binding-method":12142,"\u002Fcondensed-matter\u002Fband-theory\u002Ffermi-surfaces-and-semiclassical-dynamics":12143,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fsemiconductor-bands-and-junctions":12144,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fintrinsic-and-extrinsic-semiconductors":12145,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fcarrier-transport-and-recombination":11855,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fthe-pn-junction":12146,"\u002Fcondensed-matter\u002Fsemiconductors\u002Ftransistors-and-optoelectronics":12147,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fdielectrics-and-polarization":12092,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fferroelectrics-and-piezoelectrics":11992,"\u002Fcondensed-matter\u002Fmagnetism\u002Fdiamagnetism-and-paramagnetism":11719,"\u002Fcondensed-matter\u002Fmagnetism\u002Fexchange-and-ferromagnetism":12148,"\u002Fcondensed-matter\u002Fmagnetism\u002Fantiferromagnetism-and-domains":11585,"\u002Fcondensed-matter\u002Fmagnetism\u002Fspin-waves-and-magnons":12149,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fsuperconductivity-phenomenology":12150,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Flondon-theory-and-the-meissner-effect":11730,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fginzburg-landau-theory":12151,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fbcs-theory":11985,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fjosephson-and-high-tc":12152,"\u002Fcondensed-matter\u002Fnanostructures\u002Fquantum-wells-wires-and-dots":11577,"\u002Fcondensed-matter\u002Fnanostructures\u002Finteger-quantum-hall-effect":12153,"\u002Fcondensed-matter\u002Fnanostructures\u002Ffractional-quantum-hall-and-topology":11588,"\u002Fcondensed-matter\u002Fnanostructures\u002Fgraphene-and-dirac-materials":12154,"\u002Fcondensed-matter":11905,"\u002Flogic\u002Ffoundations\u002Flogic-as-a-mathematical-model":12155,"\u002Flogic\u002Fsentential-logic\u002Fformal-languages-and-well-formed-formulas":12156,"\u002Flogic\u002Fsentential-logic\u002Ftruth-assignments-and-tautologies":12157,"\u002Flogic\u002Fsentential-logic\u002Funique-readability-and-parsing":12158,"\u002Flogic\u002Fsentential-logic\u002Finduction-and-recursion":11602,"\u002Flogic\u002Fsentential-logic\u002Fexpressive-completeness-and-normal-forms":12159,"\u002Flogic\u002Fsentential-logic\u002Fboolean-circuits":12160,"\u002Flogic\u002Fsentential-logic\u002Fcompactness-and-effectiveness":11602,"\u002Flogic\u002Ffirst-order-languages\u002Ffirst-order-languages":12161,"\u002Flogic\u002Ffirst-order-languages\u002Fstructures-truth-and-satisfaction":12015,"\u002Flogic\u002Ffirst-order-languages\u002Fdefinability-and-elementary-equivalence":12162,"\u002Flogic\u002Ffirst-order-languages\u002Fterms-substitution-and-parsing":12163,"\u002Flogic\u002Fdeductive-calculus\u002Fa-deductive-calculus":12164,"\u002Flogic\u002Fdeductive-calculus\u002Fdeduction-theorem-and-derived-rules":12162,"\u002Flogic\u002Fdeductive-calculus\u002Fsoundness":12165,"\u002Flogic\u002Fdeductive-calculus\u002Fcompleteness-and-consistency":12166,"\u002Flogic\u002Fmodels-and-theories\u002Fcompactness-and-lowenheim-skolem":12167,"\u002Flogic\u002Fmodels-and-theories\u002Ftheories-elementary-classes-and-categoricity":12168,"\u002Flogic\u002Fmodels-and-theories\u002Finterpretations-between-theories":12169,"\u002Flogic\u002Fmodels-and-theories\u002Fnonstandard-analysis":12170,"\u002Flogic\u002Farithmetic-and-definability\u002Fdefinability-in-arithmetic":12171,"\u002Flogic\u002Farithmetic-and-definability\u002Fnatural-numbers-with-successor":12172,"\u002Flogic\u002Farithmetic-and-definability\u002Fpresburger-and-reducts":12093,"\u002Flogic\u002Farithmetic-and-definability\u002Fa-subtheory-and-representability":12173,"\u002Flogic\u002Fincompleteness\u002Farithmetization-of-syntax":12166,"\u002Flogic\u002Fincompleteness\u002Fincompleteness-and-undecidability":12174,"\u002Flogic\u002Fincompleteness\u002Fsecond-incompleteness-theorem":12175,"\u002Flogic\u002Fcomputability-and-representability\u002Frecursive-functions":11808,"\u002Flogic\u002Fcomputability-and-representability\u002Frepresenting-exponentiation":12176,"\u002Flogic\u002Fsecond-order-logic\u002Fsecond-order-languages":11993,"\u002Flogic\u002Fsecond-order-logic\u002Fskolem-functions-and-many-sorted-logic":12177,"\u002Flogic\u002Fsecond-order-logic\u002Fgeneral-structures":12178,"\u002Flogic":12179,"\u002Freinforcement-learning\u002Ffoundations\u002Fwhat-is-reinforcement-learning":12180,"\u002Freinforcement-learning\u002Ffoundations\u002Fa-brief-history-of-rl":12181,"\u002Freinforcement-learning\u002Ffoundations\u002Fmulti-armed-bandits":11794,"\u002Freinforcement-learning\u002Ffoundations\u002Fbandit-exploration-algorithms":12182,"\u002Freinforcement-learning\u002Ffoundations\u002Fmarkov-decision-processes":12183,"\u002Freinforcement-learning\u002Ffoundations\u002Fvalue-functions-and-optimality":12184,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdynamic-programming":12185,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdp-async-and-gpi":12175,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods":12186,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-off-policy":12187,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftemporal-difference-learning":12188,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftd-control-sarsa-and-q-learning":12086,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping":12189,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-off-policy-methods":12190,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-and-learning":12191,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-focusing-and-decision-time":12192,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdecision-time-planning":12193,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-tree-search":11811,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-prediction":12194,"\u002Freinforcement-learning\u002Fapproximation\u002Ffeature-construction-and-nonlinear":12195,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-control":12196,"\u002Freinforcement-learning\u002Fapproximation\u002Faverage-reward-control":12197,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad":12002,"\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td":11476,"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces":12174,"\u002Freinforcement-learning\u002Fapproximation\u002Ftrue-online-and-sarsa-lambda":12198,"\u002Freinforcement-learning\u002Fapproximation\u002Fpolicy-gradient-methods":12199,"\u002Freinforcement-learning\u002Fapproximation\u002Factor-critic-and-continuous-actions":12200,"\u002Freinforcement-learning\u002Fapproximation\u002Fleast-squares-and-memory-based-methods":11822,"\u002Freinforcement-learning\u002Fapproximation\u002Fmemory-and-kernel-methods":12201,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces":11952,"\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces":12202,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks":12203,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdqn-improvements":11784,"\u002Freinforcement-learning\u002Fdeep-rl\u002Factor-critic-and-ppo":12204,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fppo-and-continuous-control":12205,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fcase-studies":12206,"\u002Freinforcement-learning\u002Fdeep-rl\u002Frl-beyond-games":12207,"\u002Freinforcement-learning\u002Fdeep-rl\u002Ffrontiers":12208,"\u002Freinforcement-learning\u002Fdeep-rl\u002Freward-design-and-open-problems":12062,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow":12209,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow-part-2":12210,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control":12211,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control-part-2":12097,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl":12212,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl-part-2":12213,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration":12214,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration-part-2":11811,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl":11878,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2":12215,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl":12216,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl-part-2":12217,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl":12218,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl-part-2":12219,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl":12220,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl-part-2":12221,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Frlhf-and-language-models":12222,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps":12223,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps-part-2":12224,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl":12225,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl-part-2":12226,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmeta-rl-and-generalization":12227,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fpsychology-of-reinforcement":12228,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Finstrumental-conditioning-and-control":12229,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-and-td-error":12230,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-in-the-brain":12231,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fanimal-learning-and-cognition":12232,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fcognitive-maps-and-planning":12233,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fneuroscience-of-reinforcement":12234,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fseveral-learning-systems":12235,"\u002Freinforcement-learning":11764,"\u002Fartificial-intelligence\u002Ffoundations\u002Fwhat-is-ai":12236,"\u002Fartificial-intelligence\u002Ffoundations\u002Ffoundations-of-ai":12237,"\u002Fartificial-intelligence\u002Ffoundations\u002Fintelligent-agents":12238,"\u002Fartificial-intelligence\u002Ffoundations\u002Fagent-architectures":12239,"\u002Fartificial-intelligence\u002Fsearch\u002Funinformed-search":12240,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-strategies-compared":12241,"\u002Fartificial-intelligence\u002Fsearch\u002Finformed-search":12242,"\u002Fartificial-intelligence\u002Fsearch\u002Fheuristic-functions":12243,"\u002Fartificial-intelligence\u002Fsearch\u002Flocal-search":12244,"\u002Fartificial-intelligence\u002Fsearch\u002Fpopulation-and-continuous-search":12245,"\u002Fartificial-intelligence\u002Fsearch\u002Fadversarial-search":12246,"\u002Fartificial-intelligence\u002Fsearch\u002Fgames-of-chance-and-imperfect-information":12247,"\u002Fartificial-intelligence\u002Fsearch\u002Fconstraint-satisfaction":12248,"\u002Fartificial-intelligence\u002Fsearch\u002Fcsp-search-and-structure":12091,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-under-uncertainty":11946,"\u002Fartificial-intelligence\u002Fsearch\u002Fbelief-state-and-online-search":12249,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-logic":12250,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-inference":12251,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic":12252,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic-in-use":12253,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Finference-and-resolution":12254,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-resolution":12073,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fclassical-planning":12255,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-graphs-and-graphplan":12256,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-in-the-real-world":12257,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-under-uncertainty":12258,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fknowledge-representation":12259,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Freasoning-systems-and-defaults":12260,"\u002Fartificial-intelligence\u002Funcertainty\u002Fprobability-and-bayes":12261,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayes-rule-and-naive-bayes":12262,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayesian-networks":12263,"\u002Fartificial-intelligence\u002Funcertainty\u002Finference-in-bayesian-networks":12264,"\u002Fartificial-intelligence\u002Funcertainty\u002Freasoning-over-time":12265,"\u002Fartificial-intelligence\u002Funcertainty\u002Ftracking-and-data-association":12266,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmaking-decisions":12005,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmarkov-decision-processes":12256,"\u002Fartificial-intelligence\u002Funcertainty\u002Fdecision-networks-and-game-theory":12267,"\u002Fartificial-intelligence\u002Funcertainty\u002Fgame-theory-and-mechanism-design":11545,"\u002Fartificial-intelligence\u002Flearning\u002Flearning-from-examples":12268,"\u002Fartificial-intelligence\u002Flearning\u002Ftheory-and-model-families":12269,"\u002Fartificial-intelligence\u002Flearning\u002Fprobabilistic-learning":12270,"\u002Fartificial-intelligence\u002Flearning\u002Fexpectation-maximization":12271,"\u002Fartificial-intelligence\u002Flearning\u002Freinforcement-learning":12272,"\u002Fartificial-intelligence\u002Flearning\u002Fgeneralization-and-policy-search":12054,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-in-learning":12273,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-based-learning-methods":12274,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fvision-and-perception":12275,"\u002Fartificial-intelligence\u002Ffrontiers\u002Freconstructing-the-3d-world":12276,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobotics":12277,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobot-planning-and-control":12278,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnatural-language-in-ai":12279,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnlp-grammar-translation-and-speech":12280,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fphilosophy-and-future":12281,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fai-ethics-and-future":12282,"\u002Fartificial-intelligence":11764,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-constituents-nuclide-chart":12021,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-size-charge-distributions":12283,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-masses-binding-energy":11581,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fsemi-empirical-mass-formula":11579,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-moments-multipoles":12115,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnuclear-force-shell-overview":12284,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fthe-deuteron":11607,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnucleon-nucleon-scattering":11917,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fmeson-theory-isospin":12285,"\u002Fnuclear-physics\u002Fnuclear-models\u002Ffermi-gas-model":12286,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fliquid-drop-collective-coordinates":12287,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fshell-model-single-particle":11977,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fcollective-model-rotations-vibrations":12288,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-law-modes":12289,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-kinetics-equilibrium":12290,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-decay-gamow-theory":12176,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-fine-structure-hindrance":12291,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fbeta-decay-energetics-neutrino":12292,"\u002Fnuclear-physics\u002Fbeta-decay\u002Ffermi-theory-beta-decay":11591,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fweak-interaction-parity-violation":11820,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fdouble-beta-decay-neutrino-mass":12293,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fgamma-multipole-radiation":11921,"\u002Fnuclear-physics\u002Fgamma-decay\u002Finternal-conversion-isomers":12294,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fangular-correlations-mossbauer":12295,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Freaction-kinematics-cross-sections":11816,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fcompound-nucleus-resonances":11904,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fdirect-reactions-optical-model":12296,"\u002Fnuclear-physics\u002Ffission\u002Ffission-barrier-dynamics":12297,"\u002Fnuclear-physics\u002Ffission\u002Fchain-reactions-reactor-physics":12298,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Ffusion-reactions-confinement":11586,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fstellar-nucleosynthesis":11961,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fbig-bang-nucleosynthesis":11823,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fcharged-particle-stopping-power":12299,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fphoton-neutron-interactions":11871,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fradiation-detectors":11935,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fdosimetry-radiation-biology":12300,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fnuclear-applications-dating-medicine":12301,"\u002Fnuclear-physics":12302,"\u002Fnatural-language-processing\u002Ffoundations\u002Fwhat-is-nlp":12303,"\u002Fnatural-language-processing\u002Ffoundations\u002Fregex-and-text-normalization":12304,"\u002Fnatural-language-processing\u002Ffoundations\u002Fminimum-edit-distance":11942,"\u002Fnatural-language-processing\u002Ffoundations\u002Fn-gram-language-models":12305,"\u002Fnatural-language-processing\u002Ffoundations\u002Fsmoothing-and-backoff":12306,"\u002Fnatural-language-processing\u002Fclassification\u002Fnaive-bayes-and-sentiment":12307,"\u002Fnatural-language-processing\u002Fclassification\u002Fevaluating-classifiers":11790,"\u002Fnatural-language-processing\u002Fclassification\u002Flogistic-regression":12308,"\u002Fnatural-language-processing\u002Fclassification\u002Fsentiment-and-affect-lexicons":12309,"\u002Fnatural-language-processing\u002Fsemantics\u002Fvector-semantics-and-embeddings":12087,"\u002Fnatural-language-processing\u002Fsemantics\u002Fstatic-word-embeddings":12310,"\u002Fnatural-language-processing\u002Fsemantics\u002Fneural-language-models":12255,"\u002Fnatural-language-processing\u002Fsequences\u002Fsequence-labeling":12311,"\u002Fnatural-language-processing\u002Fsequences\u002Fcrfs-and-neural-taggers":12312,"\u002Fnatural-language-processing\u002Fsequences\u002Frnns-and-lstms":12313,"\u002Fnatural-language-processing\u002Ftransformers\u002Ftransformers-and-attention":12314,"\u002Fnatural-language-processing\u002Ftransformers\u002Fthe-transformer-architecture":12315,"\u002Fnatural-language-processing\u002Ftransformers\u002Flarge-language-models":12316,"\u002Fnatural-language-processing\u002Ftransformers\u002Fllm-pretraining-and-scaling":12317,"\u002Fnatural-language-processing\u002Ftransformers\u002Ffine-tuning-and-prompting":11768,"\u002Fnatural-language-processing\u002Ftransformers\u002Fprompting-and-alignment":12318,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-parsing":12319,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcky-scoring-and-evaluation":12261,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdependency-parsing":12320,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fgraph-based-and-neural-dependency-parsing":12321,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fword-senses-and-wsd":12322,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fwsd-in-practice-and-induction":12323,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-roles-and-information-extraction":12324,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Frelations-events-and-templates":12325,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoreference-and-discourse":12326,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoherence-and-discourse-structure":12327,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Flogical-semantics":12186,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcompositional-semantics-and-description-logics":12328,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-parsing":12329,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fneural-semantic-parsing":12330,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Finformation-extraction":12331,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftimes-events-and-templates":12332,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdiscourse-coherence":12333,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fentity-based-and-global-coherence":12334,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-grammars":12335,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftreebanks-and-lexicalized-grammars":12336,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation":12337,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation-decoding-and-evaluation":12338,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering":12339,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering-knowledge-and-llms":12040,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-and-chatbots":12206,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-systems-and-assistants":11796,"\u002Fnatural-language-processing\u002Fapplications\u002Ftext-summarization":12340,"\u002Fnatural-language-processing\u002Fapplications\u002Fabstractive-summarization-and-evaluation":12341,"\u002Fnatural-language-processing\u002Fspeech\u002Fphonetics":12342,"\u002Fnatural-language-processing\u002Fspeech\u002Facoustic-phonetics":12343,"\u002Fnatural-language-processing\u002Fspeech\u002Fautomatic-speech-recognition":12054,"\u002Fnatural-language-processing\u002Fspeech\u002Fasr-evaluation-and-applications":12344,"\u002Fnatural-language-processing":11764,"\u002Fparticle-physics\u002Ffoundations\u002Fhistorical-overview-particle-zoo":12345,"\u002Fparticle-physics\u002Ffoundations\u002Fparticle-physics-basic-concepts":11962,"\u002Fparticle-physics\u002Ffoundations\u002Ffundamental-interactions-force-carriers":12346,"\u002Fparticle-physics\u002Funits-kinematics\u002Fnatural-units-and-scales":12107,"\u002Fparticle-physics\u002Funits-kinematics\u002Ffour-vectors-invariant-mass":12347,"\u002Fparticle-physics\u002Funits-kinematics\u002Fdecay-scattering-kinematics-mandelstam":12348,"\u002Fparticle-physics\u002Funits-kinematics\u002Fcross-sections-golden-rule":12296,"\u002Fparticle-physics\u002Fsymmetries\u002Fconservation-laws-symmetries":12349,"\u002Fparticle-physics\u002Fsymmetries\u002Fdiscrete-symmetries-cpt":12350,"\u002Fparticle-physics\u002Fsymmetries\u002Fparity-violation-weak":11912,"\u002Fparticle-physics\u002Fsymmetries\u002Fsu2-su3-flavor-symmetry":11614,"\u002Fparticle-physics\u002Fquark-model\u002Feightfold-way-su3":12351,"\u002Fparticle-physics\u002Fquark-model\u002Fmeson-spectroscopy":11866,"\u002Fparticle-physics\u002Fquark-model\u002Fbaryon-spectroscopy":12352,"\u002Fparticle-physics\u002Fquark-model\u002Fcolor-confinement-exotics":11899,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fklein-gordon-equation":12353,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fdirac-equation-spinors":12153,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fantiparticles-hole-theory":12354,"\u002Fparticle-physics\u002Fqed\u002Ffeynman-rules-qed":12355,"\u002Fparticle-physics\u002Fqed\u002Fqed-tree-processes":12122,"\u002Fparticle-physics\u002Fqed\u002Frenormalization-running-coupling":12356,"\u002Fparticle-physics\u002Fqed\u002Felectron-g-2":11585,"\u002Fparticle-physics\u002Fweak-interaction\u002Fva-structure-weak":12357,"\u002Fparticle-physics\u002Fweak-interaction\u002Fw-z-bosons-decays":12358,"\u002Fparticle-physics\u002Fweak-interaction\u002Fckm-matrix":12359,"\u002Fparticle-physics\u002Fweak-interaction\u002Fcp-violation-kaons-b-mesons":11903,"\u002Fparticle-physics\u002Fqcd\u002Fcolor-su3-gluons":12119,"\u002Fparticle-physics\u002Fqcd\u002Fasymptotic-freedom-confinement":12360,"\u002Fparticle-physics\u002Fqcd\u002Fdeep-inelastic-scattering-partons":12361,"\u002Fparticle-physics\u002Fqcd\u002Fjets-hadronization":11978,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Felectroweak-su2-u1":12362,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fspontaneous-symmetry-breaking":12023,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-mechanism":12363,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-boson-discovery":12364,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fstandard-model":11912,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-oscillations":11945,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-mass-pmns":12365,"\u002Fparticle-physics\u002Fneutrinos\u002Fdirac-majorana-experiments":12366,"\u002Fparticle-physics\u002Fexperiment\u002Faccelerators-luminosity":12367,"\u002Fparticle-physics\u002Fexperiment\u002Fdetectors-subsystems":12001,"\u002Fparticle-physics\u002Fexperiment\u002Fhow-discoveries-are-made":12368,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fbeyond-standard-model":11742,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fgrand-unified-theories":12369,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fsupersymmetry":11845,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fhierarchy-problem-naturalness":12370,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fdark-matter-candidates":12371,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fmatter-antimatter-open-questions":12169,"\u002Fparticle-physics":12372,"\u002Fastrophysics-cosmology\u002Forientation\u002Fthe-sun-and-stars":11963,"\u002Fastrophysics-cosmology\u002Forientation\u002Fstellar-death-final-states":12117,"\u002Fastrophysics-cosmology\u002Forientation\u002Fgalaxies-and-cosmology":12373,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fmagnitudes-fluxes-and-the-distance-modulus":11902,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fstellar-spectra-and-spectral-classification":12374,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Ftelescopes-and-detectors-across-the-spectrum":11962,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fthe-cosmic-distance-ladder":11874,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fblackbody-radiation-and-specific-intensity":12375,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fradiative-transfer-and-the-transfer-equation":12376,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fspectral-line-formation-and-broadening":12377,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fopacity-and-the-rosseland-mean":12378,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fhydrostatic-equilibrium-and-the-virial-theorem":12379,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equations-of-stellar-structure":12168,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equation-of-state-and-polytropes":12099,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-standard-solar-model":12380,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fthermonuclear-reaction-rates-and-the-gamow-peak":12381,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhydrogen-burning-pp-chains-and-cno":12382,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhelium-burning-and-the-triple-alpha-process":12383,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fadvanced-burning-and-neutron-capture-nucleosynthesis":12295,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fphases-of-the-interstellar-medium":12384,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fmolecular-clouds-and-gravitational-collapse":11858,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fprotostars-and-the-pre-main-sequence":11921,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-main-sequence-and-its-structure":11903,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fpost-main-sequence-low-mass-evolution":12385,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-evolution-of-massive-stars":11987,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fstellar-pulsation-and-the-instability-strip":11603,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":12386,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fcore-collapse-supernovae":12387,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fthermonuclear-supernovae-type-ia":12118,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fneutron-stars-and-pulsars":12388,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fblack-holes-schwarzschild-and-kerr":12389,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fbinary-systems-and-mass-transfer":12390,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Faccreting-compact-objects":12391,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fgravitational-waves-from-inspiraling-binaries":11584,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fmultimessenger-astronomy-and-gamma-ray-bursts":12392,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fthe-milky-way":12393,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-morphology-and-classification":11845,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-rotation-curves-and-dark-matter":12394,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Factive-galactic-nuclei-and-supermassive-black-holes":12395,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-clusters-and-large-scale-structure":12111,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-expanding-universe-and-hubbles-law":11766,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-frw-metric-and-cosmological-redshift":12396,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-friedmann-equations-and-cosmic-dynamics":12397,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fcosmological-models-and-distances":12023,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fdark-energy-and-the-accelerating-universe":12009,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fthe-thermal-history-of-the-universe":12021,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fbig-bang-nucleosynthesis":12118,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Frecombination-and-the-cosmic-microwave-background":12398,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcmb-anisotropies-and-cosmological-parameters":11583,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcosmic-inflation":11775,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fstructure-formation-and-the-growth-of-perturbations":12399,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fdark-matter-dark-energy-and-open-questions":11862,"\u002Fastrophysics-cosmology":11967,"\u002Fcolophon":12400,"\u002F":11764},4250,4808,3626,2682,4109,4786,3878,3875,3751,3415,4067,3153,3000,4042,5461,5808,3961,3749,4327,5067,4246,4655,4154,5436,2640,4003,3601,2158,4331,4189,2273,3252,4633,4964,4172,3131,5524,3160,4031,2309,4207,3226,2648,4842,5340,3307,5701,4977,4039,2615,3472,4460,3848,4075,4400,3382,3010,3602,3737,3740,3707,3922,5191,4043,3804,4542,4214,5062,2850,4361,3443,3627,4044,3766,4140,3860,4006,5199,4334,5234,3651,5509,5680,153,1375,1073,1093,1125,1146,1014,1132,876,1541,1189,1173,984,1402,1301,950,1268,1063,1107,1408,1161,925,1012,866,964,1090,1142,1085,1020,1207,973,980,728,764,1225,1329,796,929,801,878,774,1044,1488,1175,1130,890,814,870,154,4073,5140,4961,5127,4870,5382,5195,4955,5369,4501,5576,3824,4132,4289,4307,4570,3403,5084,5105,5201,5116,5341,5175,5368,5188,5211,5499,5155,4981,5125,5415,5255,5304,5130,5167,5552,5164,5094,5239,5036,5190,5004,5099,5035,5159,5088,5026,4937,5023,5264,5244,133,5114,5078,5043,5312,5170,5342,5139,5151,5049,5212,5013,5068,5079,5102,5121,5081,5029,5379,5854,5110,2139,3798,5055,5364,4984,4935,4895,4972,5289,5112,5156,4987,5031,5025,5149,5302,5042,5002,4979,4922,4960,5279,126,1877,1180,1129,907,958,1112,1300,1053,1250,1181,1241,1234,966,1050,734,1190,484,1082,926,733,761,571,607,798,804,952,977,731,784,645,771,1017,742,1004,1000,1562,1254,1288,1101,1011,1486,1061,856,992,1169,988,137,0,2037,1782,2384,2254,2123,2332,1643,1714,2089,1751,1367,1660,2511,1998,1892,1854,1791,2438,2487,1917,2375,2525,2266,1845,2275,1810,1631,2310,2166,2233,2113,2505,2347,2672,2112,2473,2592,2380,3013,2513,3256,3218,2194,2173,2205,2326,2081,3342,3152,1799,1670,1027,960,1095,1291,986,897,1209,1055,1817,1801,1593,1465,1196,1464,1201,1230,1435,1684,1461,1926,1500,1409,1284,1774,1869,162,1487,1122,1188,1001,1351,982,1005,979,1325,1046,943,1279,824,1008,989,1798,1277,1025,987,1043,1211,1074,981,939,1002,739,1139,1108,1013,1070,978,1458,1317,157,1357,1077,2355,1116,1037,1178,1637,1314,1109,1056,1702,1474,1071,1158,832,993,1404,1024,1068,1339,1106,1264,1248,913,1848,1328,1633,1224,1143,135,1378,959,1028,998,911,1527,1203,1266,1483,1165,990,938,965,1257,1418,1099,942,1352,956,1035,1398,1003,1094,1292,138,1721,1827,1449,1354,1148,1184,1285,1281,1213,1290,1271,1252,1274,1778,1591,1503,1437,1571,1584,1957,1117,1781,1648,1342,1667,1510,1965,1607,1365,1849,1259,1303,1356,1238,2208,1564,173,1671,1286,1227,1638,1529,668,1078,918,709,865,880,940,1534,1015,874,922,841,794,1194,822,1105,1658,1359,1296,1438,1921,1844,1570,1429,1324,1400,140,1787,1558,1654,1492,1747,2224,2002,2009,1323,1349,1785,1573,1722,1829,1353,1548,1552,1583,1624,1585,1245,1364,1514,1343,1397,1355,2211,1481,1770,160,2388,2293,2256,2552,2569,2478,2039,2496,2578,2814,2519,2461,2587,2492,2714,3278,2654,3050,2447,2849,2238,2369,2061,2214,2602,2563,2186,2985,2749,3364,2038,2282,2409,2126,2573,2206,2176,2268,2182,2402,2705,2633,2414,2213,2801,3313,3410,3195,1952,2017,1509,2537,2645,2027,2415,2838,2356,1906,3184,2950,2807,2954,1683,1316,1034,1138,1763,1822,1705,1246,1701,1097,1104,1187,1032,1083,1228,916,1489,1033,1652,997,692,837,1023,888,864,1089,1231,1214,1675,1156,1075,1520,1309,139,1205,1051,735,1123,1072,915,567,768,825,1253,983,1007,762,1058,861,862,971,1208,1149,1145,1029,1084,927,810,838,857,807,936,949,2321,1622,1069,1113,1057,854,1958,1528,1618,2049,1432,1679,1796,1685,1346,1275,1476,1505,1610,2018,1599,1215,1838,1909,132,3902,2215,2240,3266,3208,3073,2454,2969,2451,1875,2728,1884,2371,2516,2842,1690,1904,2346,1386,2607,1966,2668,1665,2885,1606,2577,3074,2869,2403,2433,2082,1939,1587,2460,2747,2032,2642,1619,3123,1993,2090,2339,3829,1737,2622,2340,2322,3828,4409,2305,3411,2510,4527,3030,3569,3043,2457,1946,2277,2044,2909,1693,1945,2093,2399,2115,2898,2742,2242,3895,3378,3376,2769,2223,3062,3262,2651,2949,2768,3128,2423,1977,2087,2866,3388,2830,2210,2489,2884,3945,2099,2713,3402,1692,2931,4195,3989,3206,4391,3004,3704,3494,2902,999,881,901,919,748,869,1018,1045,1049,1333,954,1092,1019,976,1771,1480,1396,953,1026,161,3533,2495,1818,3007,2595,3427,3537,2216,1895,2304,3396,1739,2073,1962,2203,1767,2666,2264,2276,2852,1807,3735,1560,4144,1669,1676,1972,2418,3291,1525,2040,2766,2337,2220,2800,3001,2078,1759,2836,1896,2026,1758,1543,1047,896,946,1060,1384,1482,815,1414,1322,1440,1240,1468,1098,1133,847,1009,1381,1052,1191,1258,1370,1712,1441,1199,957,1079,150,1262,1417,1368,1219,1136,1064,1463,1636,1059,931,1115,1736,1174,1376,1363,1411,1247,1746,1313,1299,1617,1102,1076,1495,1265,1193,1263,80,{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":12402,"\u002Falgorithms\u002Ffoundations\u002Fproof-techniques":12407,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":12411,"\u002Falgorithms\u002Ffoundations\u002Fgrowth-rates-and-loop-analysis":12415,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":12419,"\u002Falgorithms\u002Ffoundations\u002Famortized-analysis":12423,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":12427,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":12432,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":12436,"\u002Falgorithms\u002Fdivide-and-conquer\u002Ffast-multiplication":12440,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":12444,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":12449,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":12453,"\u002Falgorithms\u002Fsorting\u002Fexternal-sorting":12457,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":12461,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":12466,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":12470,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":12474,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":12478,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":12482,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":12486,"\u002Falgorithms\u002Fdata-structures\u002Fspatial-data-structures":12490,"\u002Falgorithms\u002Fdata-structures\u002Fskip-lists-and-probabilistic-structures":12494,"\u002Falgorithms\u002Fdata-structures\u002Fb-trees":12498,"\u002Falgorithms\u002Fdata-structures\u002Fdata-stream-algorithms":12502,"\u002Falgorithms\u002Fdata-structures\u002Fstreaming-sketches":12506,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":12510,"\u002Falgorithms\u002Fsequences\u002Fprefix-sums":12515,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":12519,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":12523,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":12527,"\u002Falgorithms\u002Fsequences\u002Fkmp-and-z-function":12531,"\u002Falgorithms\u002Fsequences\u002Ftries":12535,"\u002Falgorithms\u002Fsequences\u002Fsuffix-arrays-and-aho-corasick":12539,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":12543,"\u002Falgorithms\u002Fgraphs\u002Fdepth-first-search":12548,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":12552,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":12556,"\u002Falgorithms\u002Fgraphs\u002Fkruskal-and-prim":12560,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":12564,"\u002Falgorithms\u002Fgraphs\u002Fall-pairs-and-negative-weights":12568,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":12572,"\u002Falgorithms\u002Fgraphs\u002Fmax-flow-min-cut":12576,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":12580,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":12584,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":12588,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":12592,"\u002Falgorithms\u002Fgraphs\u002Fbipartite-matching":12596,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":12600,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":12605,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":12609,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":12613,"\u002Falgorithms\u002Fgreedy\u002Fstable-matching":12617,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":12621,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":12626,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":12630,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":12634,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":12638,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":12642,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":12646,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":12650,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":12654,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":12658,"\u002Falgorithms\u002Fdynamic-programming\u002Fdigit-and-probability-dp":12662,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":12666,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":12671,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":12675,"\u002Falgorithms\u002Fbacktracking\u002Fgraph-backtracking":12679,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":12683,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":12688,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":12692,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":12696,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmatrix-exponentiation":12700,"\u002Falgorithms\u002Fmathematical-algorithms\u002Ffast-fourier-transform":12704,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fgradient-descent":12708,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":12712,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":12717,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":12721,"\u002Falgorithms\u002Fcomputational-geometry\u002Fpolygons-and-proximity":12725,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":12729,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":12734,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":12738,"\u002Falgorithms\u002Fintractability\u002Fapproximation-algorithms":12742,"\u002Falgorithms":12746,"\u002Fcalculus\u002Flimits-and-continuity\u002Ffunctions-and-models":12749,"\u002Fcalculus\u002Flimits-and-continuity\u002Fthe-limit-of-a-function":12754,"\u002Fcalculus\u002Flimits-and-continuity\u002Flimit-laws-and-the-precise-definition":12758,"\u002Fcalculus\u002Flimits-and-continuity\u002Fcontinuity":12762,"\u002Fcalculus\u002Fderivatives\u002Fthe-derivative-and-rates-of-change":12766,"\u002Fcalculus\u002Fderivatives\u002Fdifferentiation-rules-and-the-chain-rule":12771,"\u002Fcalculus\u002Fderivatives\u002Fimplicit-differentiation-and-related-rates":12775,"\u002Fcalculus\u002Fderivatives\u002Flinear-approximations-and-differentials":12779,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fextrema-and-the-mean-value-theorem":12783,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fhow-derivatives-shape-a-graph":12788,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fcurve-sketching-and-optimization":12792,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fnewtons-method-and-antiderivatives":12796,"\u002Fcalculus\u002Fintegrals\u002Farea-and-the-definite-integral":12800,"\u002Fcalculus\u002Fintegrals\u002Fthe-fundamental-theorem-of-calculus":12805,"\u002Fcalculus\u002Fintegrals\u002Fthe-substitution-rule":12809,"\u002Fcalculus\u002Fapplications-of-integration\u002Fareas-and-volumes":12813,"\u002Fcalculus\u002Fapplications-of-integration\u002Fwork-average-value-and-arc-length":12818,"\u002Fcalculus\u002Fapplications-of-integration\u002Fphysics-economics-and-probability":12822,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Finverse-functions-logarithms-and-exponentials":12826,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Fgrowth-decay-inverse-trig-and-hyperbolic-functions":12831,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Flhospitals-rule":12835,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fintegration-by-parts":12839,"\u002Fcalculus\u002Ftechniques-of-integration\u002Ftrigonometric-integrals-and-substitution":12844,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fpartial-fractions-and-integration-strategy":12848,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fapproximate-and-improper-integrals":12852,"\u002Fcalculus\u002Fparametric-and-polar\u002Fparametric-curves-and-their-calculus":12856,"\u002Fcalculus\u002Fparametric-and-polar\u002Fpolar-coordinates":12861,"\u002Fcalculus\u002Fparametric-and-polar\u002Fconic-sections":12865,"\u002Fcalculus\u002Fsequences-and-series\u002Fsequences":12869,"\u002Fcalculus\u002Fsequences-and-series\u002Fseries-and-the-integral-test":12874,"\u002Fcalculus\u002Fsequences-and-series\u002Fthe-convergence-tests":12878,"\u002Fcalculus\u002Fsequences-and-series\u002Fpower-series":12882,"\u002Fcalculus\u002Fsequences-and-series\u002Ftaylor-and-maclaurin-series":12886,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvectors-and-the-dot-product":12890,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fthe-cross-product-lines-and-planes":12895,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fcylinders-and-quadric-surfaces":12899,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvector-functions-and-space-curves":12903,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Farc-length-curvature-and-motion":12907,"\u002Fcalculus\u002Fpartial-derivatives\u002Ffunctions-of-several-variables":12911,"\u002Fcalculus\u002Fpartial-derivatives\u002Fpartial-derivatives":12916,"\u002Fcalculus\u002Fpartial-derivatives\u002Ftangent-planes-and-the-chain-rule":12919,"\u002Fcalculus\u002Fpartial-derivatives\u002Fdirectional-derivatives-and-the-gradient":12923,"\u002Fcalculus\u002Fpartial-derivatives\u002Foptimization-and-lagrange-multipliers":12927,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fdouble-integrals":12931,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Ftriple-integrals-and-coordinate-systems":12936,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fvector-fields-and-line-integrals":12940,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fgreens-theorem-curl-and-divergence":12944,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fsurface-integrals":12948,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fstokes-and-the-divergence-theorem":12952,"\u002Fcalculus":12956,"\u002Fmechanics\u002Ffoundations\u002Fmeasurement-and-dimensions":12959,"\u002Fmechanics\u002Ffoundations\u002Fvector-algebra":12963,"\u002Fmechanics\u002Fkinematics\u002Fone-dimensional-motion":12967,"\u002Fmechanics\u002Fkinematics\u002Fmotion-graphs":12972,"\u002Fmechanics\u002Fkinematics\u002Fprojectile-motion":12976,"\u002Fmechanics\u002Fkinematics\u002Frelative-motion":12980,"\u002Fmechanics\u002Fkinematics\u002Fcircular-motion":12984,"\u002Fmechanics\u002Fdynamics\u002Fnewtons-laws":12988,"\u002Fmechanics\u002Fdynamics\u002Ffree-body-diagrams":12993,"\u002Fmechanics\u002Fdynamics\u002Ffriction-and-curved-motion":12997,"\u002Fmechanics\u002Fdynamics\u002Fnumerical-dynamics":13001,"\u002Fmechanics\u002Fdynamics\u002Fcenter-of-mass-systems":13005,"\u002Fmechanics\u002Fenergy\u002Fwork-and-kinetic-energy":13009,"\u002Fmechanics\u002Fenergy\u002Fpotential-energy":13014,"\u002Fmechanics\u002Fenergy\u002Fmultiparticle-work":13018,"\u002Fmechanics\u002Fenergy\u002Fmass-energy-and-binding":13022,"\u002Fmechanics\u002Fenergy\u002Fphotons-and-quantization":13026,"\u002Fmechanics\u002Fmomentum\u002Fmomentum-and-collisions":13030,"\u002Fmechanics\u002Fmomentum\u002Fcenter-of-mass-collisions":13035,"\u002Fmechanics\u002Fmomentum\u002Frocket-propulsion":13039,"\u002Fmechanics\u002Frotation\u002Frotational-inertia":13043,"\u002Fmechanics\u002Frotation\u002Frotational-dynamics":13048,"\u002Fmechanics\u002Frotation\u002Frolling-motion":13052,"\u002Fmechanics\u002Frotation\u002Fangular-momentum":13056,"\u002Fmechanics\u002Frotation\u002Frolling-resistance":13060,"\u002Fmechanics\u002Frotation\u002Fgyroscopic-precession":13064,"\u002Fmechanics\u002Fgravity-and-matter\u002Fkeplerian-orbits":13068,"\u002Fmechanics\u002Fgravity-and-matter\u002Fgravitational-fields":13073,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstatic-equilibrium":13077,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-statics":13081,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-flow":13085,"\u002Fmechanics\u002Fgravity-and-matter\u002Forbital-motion":13089,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstress-and-elasticity":13093,"\u002Fmechanics\u002Foscillations-waves\u002Fdamped-oscillators":13097,"\u002Fmechanics\u002Foscillations-waves\u002Ftravelling-waves":13102,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-superposition":13106,"\u002Fmechanics\u002Foscillations-waves\u002Fstanding-waves":13110,"\u002Fmechanics\u002Foscillations-waves\u002Fsound-waves":13114,"\u002Fmechanics\u002Foscillations-waves\u002Fdoppler-effect":13118,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-packets":13122,"\u002Fmechanics\u002Foscillations-waves\u002Fbeats-and-coupling":13126,"\u002Fmechanics\u002Foscillations-waves\u002Fsimple-harmonic-motion":13130,"\u002Fmechanics\u002Foscillations-waves\u002Fpendulum-motion":13134,"\u002Fmechanics\u002Foscillations-waves\u002Fdriven-oscillators":13138,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-boundaries":13142,"\u002Fmechanics\u002Fthermodynamics\u002Fkinetic-theory-of-ideal-gases":13146,"\u002Fmechanics\u002Fthermodynamics\u002Ffirst-law-of-thermodynamics":13151,"\u002Fmechanics\u002Fthermodynamics\u002Fentropy-and-the-second-law":13155,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-processes":13159,"\u002Fmechanics\u002Fthermodynamics\u002Fphase-changes":13163,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-machines":13167,"\u002Fmechanics":13171,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcharge-and-conductors":13174,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcoulombs-law":13179,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-and-force":13183,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-maps":13187,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-dipoles":13191,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fcontinuous-charge-fields":13195,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fgauss-law-and-conductors":13200,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpoint-charge-potential":13204,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpotential-gradients-and-equipotentials":13209,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Felectrostatic-energy-and-pressure":13213,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Flaplace-boundary-problems":13217,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fcontinuous-charge-potentials":13221,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitance-fundamentals":13225,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-networks":13230,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-energy-and-force":13234,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fdielectric-polarization-and-breakdown":13238,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fcurrent-and-resistance":13242,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fkirchhoff-network-analysis":13247,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Frc-transients":13251,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-trajectories":13255,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fhall-effect":13260,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-force-on-conductors":13264,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-dipoles":13268,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmass-spectrometry":13272,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmoving-charge-fields":13276,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fbiot-savart-law":13281,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fcircular-current-loops":13285,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Famperes-law":13289,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fgauss-law-for-magnetism":13293,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmagnetic-materials":13297,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-flux":13301,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Ffaradays-law":13306,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Flenzs-law":13310,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmotional-emf":13314,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Feddy-currents":13318,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fself-inductance":13322,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-energy":13326,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Frl-circuits":13330,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-fundamentals":13334,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Freactance":13339,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Frlc-resonance":13343,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-power":13347,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Ftransformers":13351,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdisplacement-current":13355,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-waves":13360,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-momentum":13364,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdipole-radiation":13368,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fpolarization":13372,"\u002Felectricity-and-magnetism\u002Foptics\u002Freflection-and-refraction":13376,"\u002Felectricity-and-magnetism\u002Foptics\u002Fthin-lenses":13381,"\u002Felectricity-and-magnetism\u002Foptics\u002Fspherical-mirrors":13385,"\u002Felectricity-and-magnetism":13389,"\u002Flinear-algebra\u002Flinear-systems\u002Fsystems-and-echelon-forms":13392,"\u002Flinear-algebra\u002Flinear-systems\u002Fvector-and-matrix-equations":13397,"\u002Flinear-algebra\u002Flinear-systems\u002Fsolution-sets-and-applications":13401,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-independence":13405,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-transformations":13409,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-operations":13413,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-inverse-and-invertibility":13418,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fpartitioned-matrices-and-lu":13422,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fsubspaces-dimension-rank":13426,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fapplications-leontief-and-graphics":13430,"\u002Flinear-algebra\u002Fdeterminants\u002Fdeterminants-and-cofactors":13434,"\u002Flinear-algebra\u002Fdeterminants\u002Fproperties-of-determinants":13439,"\u002Flinear-algebra\u002Fdeterminants\u002Fcramer-volume-and-area":13443,"\u002Flinear-algebra\u002Fvector-spaces\u002Fvector-spaces-and-subspaces":13447,"\u002Flinear-algebra\u002Fvector-spaces\u002Fnull-and-column-spaces":13452,"\u002Flinear-algebra\u002Fvector-spaces\u002Fbases-and-independent-sets":13456,"\u002Flinear-algebra\u002Fvector-spaces\u002Fcoordinate-systems":13460,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdimension-and-rank":13464,"\u002Flinear-algebra\u002Fvector-spaces\u002Fchange-of-basis":13468,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdifference-equations-and-markov":13472,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-eigenvalues":13476,"\u002Flinear-algebra\u002Feigenvalues\u002Fthe-characteristic-equation":13481,"\u002Flinear-algebra\u002Feigenvalues\u002Fdiagonalization":13485,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-linear-transformations":13489,"\u002Flinear-algebra\u002Feigenvalues\u002Fcomplex-eigenvalues":13493,"\u002Flinear-algebra\u002Feigenvalues\u002Fdynamical-systems":13497,"\u002Flinear-algebra\u002Feigenvalues\u002Fpower-method":13501,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-length-orthogonality":13505,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Forthogonal-sets-and-projections":13510,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fgram-schmidt-and-qr":13514,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-problems":13518,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-applications":13522,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-spaces":13526,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fdiagonalizing-symmetric-matrices":13530,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fquadratic-forms":13535,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fconstrained-optimization":13539,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsingular-value-decomposition":13543,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsvd-applications-pca-imaging":13547,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-thinking-and-matrix-computation":13551,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Flu-and-cholesky":13556,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fconditioning-and-floating-point":13560,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fstability-and-error-analysis":13564,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fqr-and-numerical-least-squares":13568,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-eigenvalues-and-svd":13572,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-combinations":13576,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-independence-and-barycentric-coordinates":13581,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fconvex-combinations-and-convex-sets":13585,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fhyperplanes-and-polytopes":13589,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fcurves-and-surfaces":13593,"\u002Flinear-algebra":13597,"\u002Ftheory-of-computation":13600,"\u002Fcomputer-architecture\u002Ffoundations\u002Fbits-bytes-and-words":13603,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-representation":13607,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-arithmetic":13611,"\u002Fcomputer-architecture\u002Ffoundations\u002Ffloating-point":13615,"\u002Fcomputer-architecture\u002Ffoundations\u002Fboolean-algebra-and-bit-manipulation":13619,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fthe-machines-view":13623,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fdata-movement":13628,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farithmetic-and-logic":13632,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fcontrol-flow":13636,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fprocedures":13640,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farrays-structs-and-alignment":13644,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fmemory-layout-and-buffer-overflows":13648,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fwhat-an-isa-is":13652,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Finstruction-formats-and-operands":13657,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Faddressing-modes":13661,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fthe-y86-64-instruction-set":13665,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fy86-64-programming":13669,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Ftransistors-gates-and-boolean-functions":13673,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fcombinational-logic-and-hcl":13678,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmultiplexers-decoders-and-the-alu":13682,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmemory-elements-latches-flip-flops-and-clocking":13686,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fregister-files-and-random-access-memory":13690,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-fetch-decode-execute-cycle":13694,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-seq-stages":13699,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fcontrol-logic-and-sequencing":13703,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fassembling-seq":13707,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Ftracing-a-program":13711,"\u002Fcomputer-architecture\u002Fpipelining\u002Fpipelining-principles":13715,"\u002Fcomputer-architecture\u002Fpipelining\u002Ffrom-seq-to-pipe":13720,"\u002Fcomputer-architecture\u002Fpipelining\u002Fdata-hazards-stalling-and-forwarding":13724,"\u002Fcomputer-architecture\u002Fpipelining\u002Fcontrol-hazards-and-branch-prediction":13728,"\u002Fcomputer-architecture\u002Fpipelining\u002Fthe-complete-pipe-processor":13732,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fstorage-technologies-and-the-latency-gap":13736,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Flocality":13741,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-memories-direct-mapped":13745,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fset-associative-and-write-policies":13749,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-performance-and-cache-friendly-code":13753,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Faddress-spaces-and-translation":13757,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fpage-tables-and-page-faults":13762,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fthe-tlb-and-multi-level-page-tables":13766,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Fexceptional-control-flow":13770,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Finterrupts-and-the-kernel":13775,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fprocesses-threads-and-parallelism":13779,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fhardware-multithreading":13784,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fcache-coherence":13788,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmemory-consistency-and-synchronization":13792,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmulticore-organization":13796,"\u002Fcomputer-architecture\u002Fcapstone\u002Fthe-whole-machine":13800,"\u002Fcomputer-architecture\u002Fcapstone\u002Fassembling-a-complete-cpu":13805,"\u002Fcomputer-architecture":13809,"\u002Fdifferential-equations\u002Ffoundations\u002Fmodels-and-direction-fields":13812,"\u002Fdifferential-equations\u002Ffoundations\u002Fclassification-and-terminology":13816,"\u002Fdifferential-equations\u002Ffirst-order\u002Flinear-first-order-integrating-factors":13820,"\u002Fdifferential-equations\u002Ffirst-order\u002Fseparable-and-exact":13825,"\u002Fdifferential-equations\u002Ffirst-order\u002Fmodeling-first-order":13829,"\u002Fdifferential-equations\u002Ffirst-order\u002Fautonomous-and-population-dynamics":13833,"\u002Fdifferential-equations\u002Ffirst-order\u002Fexistence-uniqueness-euler":13837,"\u002Fdifferential-equations\u002Ffirst-order\u002Ffirst-order-difference-equations":13841,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhomogeneous-constant-coefficients":13845,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fcomplex-and-repeated-roots":13850,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fnonhomogeneous-undetermined-coefficients":13854,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fvariation-of-parameters":13858,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fmechanical-electrical-vibrations":13862,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhigher-order-linear":13866,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fpower-series-ordinary-points":13870,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fregular-singular-frobenius":13875,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fbessel-and-special-functions":13879,"\u002Fdifferential-equations\u002Flaplace\u002Flaplace-definition-ivps":13883,"\u002Fdifferential-equations\u002Flaplace\u002Fstep-impulse-convolution":13888,"\u002Fdifferential-equations\u002Fsystems\u002Fmatrices-eigenvalues-review":13892,"\u002Fdifferential-equations\u002Fsystems\u002Fconstant-coefficient-systems-phase-portraits":13897,"\u002Fdifferential-equations\u002Fsystems\u002Frepeated-eigenvalues-fundamental-matrices":13901,"\u002Fdifferential-equations\u002Fnumerical\u002Feuler-and-runge-kutta":13905,"\u002Fdifferential-equations\u002Fnumerical\u002Fmultistep-systems-stability":13910,"\u002Fdifferential-equations\u002Fnonlinear\u002Fphase-plane-autonomous-stability":13914,"\u002Fdifferential-equations\u002Fnonlinear\u002Flocally-linear-and-liapunov":13919,"\u002Fdifferential-equations\u002Fnonlinear\u002Fcompeting-species-predator-prey-limit-cycles":13923,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Ffourier-series":13927,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fheat-wave-laplace-equations":13932,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fsturm-liouville":13936,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fcalculus-of-variations":13940,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fhistorical-notes":13945,"\u002Fdifferential-equations":13949,"\u002Frelativity\u002Ffoundations\u002Fspecial-relativity-postulates":13952,"\u002Frelativity\u002Ffoundations\u002Florentz-transformation-spacetime":13957,"\u002Frelativity\u002Ffoundations\u002Ftime-dilation-length-contraction":13961,"\u002Frelativity\u002Ffoundations\u002Frelativistic-momentum-energy":13965,"\u002Frelativity\u002Ffoundations\u002Fgeneral-relativity":13969,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fminkowski-spacetime-and-the-interval":13973,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Ffour-vectors-and-index-notation":13978,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fthe-lorentz-group-and-rapidity":13982,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fdoppler-aberration-and-appearance":13986,"\u002Frelativity\u002Frelativistic-dynamics\u002Ffour-momentum-force-and-accelerated-motion":13990,"\u002Frelativity\u002Frelativistic-dynamics\u002Fparticle-decays-and-two-body-kinematics":13995,"\u002Frelativity\u002Frelativistic-dynamics\u002Fcollisions-thresholds-and-the-cm-frame":13999,"\u002Frelativity\u002Frelativistic-dynamics\u002Fmandelstam-variables-and-invariants":14003,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ffour-current-and-the-four-potential":14007,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fthe-electromagnetic-field-tensor":14012,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ftransformation-of-electric-and-magnetic-fields":14016,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fcovariant-maxwell-and-the-stress-energy-tensor":14020,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-equivalence-principle-formalized":14024,"\u002Frelativity\u002Fcurved-spacetime\u002Fmanifolds-vectors-and-the-metric":14029,"\u002Frelativity\u002Fcurved-spacetime\u002Fcovariant-derivative-and-christoffel-symbols":14033,"\u002Frelativity\u002Fcurved-spacetime\u002Fgeodesics-and-the-geodesic-equation":14037,"\u002Frelativity\u002Fcurved-spacetime\u002Fcurvature-riemann-and-geodesic-deviation":14041,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-einstein-field-equations":14045,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fthe-schwarzschild-metric":14049,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fgeodesics-and-orbits-in-schwarzschild":14054,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Flight-bending-and-null-geodesics":14058,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fperihelion-precession-of-mercury":14062,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fdeflection-of-light-and-gravitational-lensing":14067,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fgravitational-redshift-and-shapiro-delay":14071,"\u002Frelativity\u002Ftests-of-general-relativity\u002Frelativity-in-technology-gps":14075,"\u002Frelativity\u002Fblack-holes\u002Fhorizons-and-coordinate-singularities":14079,"\u002Frelativity\u002Fblack-holes\u002Frotating-and-charged-black-holes":14084,"\u002Frelativity\u002Fblack-holes\u002Fblack-hole-thermodynamics":14088,"\u002Frelativity\u002Fgravitational-waves\u002Flinearized-gravity-and-wave-solutions":14092,"\u002Frelativity\u002Fgravitational-waves\u002Fgeneration-and-the-quadrupole-formula":14097,"\u002Frelativity\u002Fgravitational-waves\u002Fdetection-ligo-and-the-first-events":14101,"\u002Frelativity\u002Fcosmological-bridge\u002Fthe-cosmological-principle-and-flrw-metric":14105,"\u002Frelativity\u002Fcosmological-bridge\u002Ffriedmann-equations-and-cosmic-dynamics":14110,"\u002Frelativity":14114,"\u002Fphysical-computing":14117,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fblackbody-radiation-and-the-planck-quantum":14120,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-photoelectric-effect-and-the-photon":14125,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fx-rays-and-the-compton-effect":14129,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-old-quantum-theory-bohr-and-sommerfeld":14133,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fde-broglie-waves-and-electron-diffraction":14137,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fwave-packets-and-the-probability-interpretation":14142,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fthe-uncertainty-principle":14146,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-schrodinger-equation-in-one-dimension":14150,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-free-particle-and-wave-packet-dynamics":14155,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fparticle-in-infinite-and-finite-square-wells":14159,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Foperators-expectation-values-and-the-harmonic-oscillator":14163,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-dirac-delta-potential":14167,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fbarrier-penetration-and-quantum-tunneling":14171,"\u002Fquantum-mechanics\u002Fformalism\u002Fhilbert-space-and-dirac-notation":14175,"\u002Fquantum-mechanics\u002Fformalism\u002Fobservables-hermitian-operators-and-eigenvalues":14180,"\u002Fquantum-mechanics\u002Fformalism\u002Fthe-postulates-and-quantum-measurement":14184,"\u002Fquantum-mechanics\u002Fformalism\u002Fposition-momentum-and-continuous-spectra":14188,"\u002Fquantum-mechanics\u002Fformalism\u002Fcommutators-and-the-generalized-uncertainty-principle":14192,"\u002Fquantum-mechanics\u002Fformalism\u002Ftime-evolution-schrodinger-and-heisenberg-pictures":14196,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fladder-operators-and-the-number-states":14200,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fcoherent-and-squeezed-states":14205,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fsymmetries-generators-and-conservation-laws":14209,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fparity-time-reversal-and-discrete-symmetries":14213,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Forbital-angular-momentum-and-spherical-harmonics":14217,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Fthe-angular-momentum-algebra":14221,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Faddition-of-angular-momenta-and-clebsch-gordan":14225,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-schrodinger-equation-in-three-dimensions":14229,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-hydrogen-atom":14234,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-isotropic-oscillator-and-hidden-symmetry":14238,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-half-pauli-matrices-and-stern-gerlach":14242,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-in-a-magnetic-field-precession-and-resonance":14247,"\u002Fquantum-mechanics\u002Fspin\u002Ftwo-level-systems-and-the-bloch-sphere":14251,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fidentical-particles-and-exchange-symmetry":14255,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fthe-pauli-principle-atoms-and-the-periodic-table":14260,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ftime-independent-perturbation-theory":14264,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ffine-structure-and-the-real-hydrogen-atom":14269,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-zeeman-and-stark-effects":14273,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-variational-method":14277,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-wkb-approximation":14281,"\u002Fquantum-mechanics":14285,"\u002Freal-analysis\u002Ffoundations\u002Fsets-logic-functions":14288,"\u002Freal-analysis\u002Ffoundations\u002Fordered-fields-completeness":14293,"\u002Freal-analysis\u002Ffoundations\u002Fabsolute-value-bounds":14297,"\u002Freal-analysis\u002Ffoundations\u002Fintervals-uncountability":14301,"\u002Freal-analysis\u002Fsequences-series\u002Fsequences-limits":14305,"\u002Freal-analysis\u002Fsequences-series\u002Flimit-laws-monotone":14310,"\u002Freal-analysis\u002Fsequences-series\u002Flimsup-bolzano-weierstrass":14314,"\u002Freal-analysis\u002Fsequences-series\u002Fcauchy-completeness":14318,"\u002Freal-analysis\u002Fsequences-series\u002Fseries-convergence":14322,"\u002Freal-analysis\u002Fsequences-series\u002Fabsolute-conditional-rearrangement":14326,"\u002Freal-analysis\u002Fmetric-spaces\u002Fmetric-spaces-norms":14330,"\u002Freal-analysis\u002Fmetric-spaces\u002Fopen-closed-sets":14335,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconvergence-completeness":14339,"\u002Freal-analysis\u002Fmetric-spaces\u002Fcompactness":14343,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconnectedness":14347,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-of-functions":14351,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuous-functions":14355,"\u002Freal-analysis\u002Fcontinuity\u002Fevt-ivt":14359,"\u002Freal-analysis\u002Fcontinuity\u002Funiform-continuity":14363,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuity-metric-spaces":14367,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-infinity-monotone":14371,"\u002Freal-analysis\u002Fdifferentiation\u002Fthe-derivative":14375,"\u002Freal-analysis\u002Fdifferentiation\u002Fmean-value-theorem":14380,"\u002Freal-analysis\u002Fdifferentiation\u002Ftaylors-theorem":14384,"\u002Freal-analysis\u002Fdifferentiation\u002Finverse-function-1d":14388,"\u002Freal-analysis\u002Friemann-integration\u002Fdarboux-integral":14392,"\u002Freal-analysis\u002Friemann-integration\u002Fintegrability-classes":14397,"\u002Freal-analysis\u002Friemann-integration\u002Fproperties-of-the-integral":14401,"\u002Freal-analysis\u002Friemann-integration\u002Ffundamental-theorem":14405,"\u002Freal-analysis\u002Friemann-integration\u002Flog-exp-improper":14408,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpointwise-uniform-convergence":14412,"\u002Freal-analysis\u002Ffunction-sequences\u002Finterchange-of-limits":14417,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpower-series-weierstrass":14421,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpicard-ode":14425,"\u002Freal-analysis\u002Fseveral-variables\u002Fdifferentiability-rn":14429,"\u002Freal-analysis\u002Fseveral-variables\u002Fgradient-chain-rule":14434,"\u002Freal-analysis\u002Fseveral-variables\u002Fhigher-derivatives-taylor-extrema":14438,"\u002Freal-analysis\u002Fseveral-variables\u002Finverse-implicit-theorems":14442,"\u002Freal-analysis\u002Fseveral-variables\u002Fmultiple-integrals":14446,"\u002Freal-analysis":14450,"\u002Fabstract-algebra\u002Ffoundations\u002Fsets-functions-relations":14453,"\u002Fabstract-algebra\u002Ffoundations\u002Fintegers-and-modular-arithmetic":14457,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fgroup-axioms-and-first-examples":14461,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fdihedral-and-symmetric-groups":14466,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fmatrix-and-quaternion-groups":14470,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fhomomorphisms-and-group-actions":14474,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fsubgroups-and-substructures":14478,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcyclic-groups":14483,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fgeneration-and-subgroup-lattices":14487,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcosets-lagrange-and-normal-subgroups":14491,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fisomorphism-theorems":14495,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcomposition-series-and-the-alternating-group":14499,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Factions-and-cayleys-theorem":14503,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fconjugation-and-the-class-equation":14508,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fsylow-theorems":14512,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fautomorphisms-and-simple-groups":14516,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fdirect-products-and-finite-abelian-groups":14520,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fsemidirect-products":14525,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fnilpotent-and-solvable-groups":14529,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fclassifying-small-groups":14533,"\u002Fabstract-algebra\u002Fring-theory\u002Frings-definitions-and-examples":14537,"\u002Fabstract-algebra\u002Fring-theory\u002Fideals-quotients-and-homomorphisms":14542,"\u002Fabstract-algebra\u002Fring-theory\u002Ffractions-and-the-chinese-remainder-theorem":14546,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Feuclidean-domains-pids-ufds":14550,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fpolynomial-rings-over-fields":14555,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fgauss-lemma-and-unique-factorization":14559,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Firreducibility-criteria-and-groebner":14563,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fintroduction-to-modules":14567,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ffree-modules-and-direct-sums":14572,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ftensor-products-and-exact-sequences":14576,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fvector-spaces-and-linear-maps":14580,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fstructure-theorem-over-pids":14584,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Frational-canonical-form":14589,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fjordan-canonical-form":14593,"\u002Fabstract-algebra\u002Ffield-theory\u002Ffield-extensions-and-algebraic-elements":14597,"\u002Fabstract-algebra\u002Ffield-theory\u002Fstraightedge-and-compass-constructions":14602,"\u002Fabstract-algebra\u002Ffield-theory\u002Fsplitting-fields-and-algebraic-closure":14606,"\u002Fabstract-algebra\u002Ffield-theory\u002Fseparable-and-cyclotomic-extensions":14610,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fthe-galois-correspondence":14614,"\u002Fabstract-algebra\u002Fgalois-theory\u002Ffinite-fields":14619,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fcyclotomic-and-abelian-extensions":14623,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fgalois-groups-of-polynomials":14627,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fsolvability-by-radicals-and-the-quintic":14631,"\u002Fabstract-algebra\u002Fcapstone\u002Fcommutative-algebra-and-algebraic-geometry":14635,"\u002Fabstract-algebra\u002Fcapstone\u002Frepresentation-and-character-theory":14640,"\u002Fabstract-algebra":14644,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fatomic-spectra-rutherford":14647,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-model-hydrogen":14652,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fx-ray-spectra-franck-hertz":14656,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-sommerfeld-old-quantum-theory":14660,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fold-quantum-theory-limits-wkb":14664,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fschrodinger-3d-hydrogen":14668,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fhydrogen-wave-functions":14673,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fradial-equation-in-full":14677,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fsymmetry-degeneracy-runge-lenz":14681,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fexpectation-values-virial":14685,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fquantum-defects-alkali-spectra":14689,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Frydberg-atoms":14693,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Frelativistic-kinetic-correction":14697,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fspin-orbit-thomas-precession":14702,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdarwin-term-fine-structure-formula":14706,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdirac-equation-hydrogen":14710,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Flamb-shift-qed":14714,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fhyperfine-structure-21cm":14719,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fnuclear-effects-isotope-shift":14723,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fperiodic-table-atomic-spectra":14727,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fcentral-field-self-consistent":14732,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fidentical-particles-hartree-fock":14736,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhelium-two-electron-atom":14740,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fls-jj-coupling-term-symbols":14744,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhund-rules-ground-terms":14748,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fzeeman-effect":14752,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fpaschen-back-intermediate":14757,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fstark-effect-polarizability":14761,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Ftime-dependent-perturbation-golden-rule":14765,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fdipole-approximation-einstein-coefficients":14770,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fselection-rules-forbidden-transitions":14774,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Flifetimes-and-line-shapes":14778,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Flaser-principles":14782,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fspectroscopy-techniques":14787,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fline-catalog-nist-asd":14791,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Flaser-cooling-doppler":14795,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fsub-doppler-trapping":14800,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fbose-einstein-condensation":14804,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Foptical-clocks-precision":14808,"\u002Fatomic-physics":14812,"\u002Fdatabases":14815,"\u002Fcategory-theory\u002Ffoundations\u002Fwhat-is-a-category":14818,"\u002Fcategory-theory\u002Ffoundations\u002Fexamples-of-categories":14822,"\u002Fcategory-theory\u002Ffoundations\u002Fspecial-morphisms":14826,"\u002Fcategory-theory\u002Ffoundations\u002Ffunctors":14830,"\u002Fcategory-theory\u002Ffoundations\u002Fnatural-transformations":14834,"\u002Fcategory-theory\u002Ffoundations\u002Fsize-and-set-theory":14838,"\u002Fcategory-theory\u002Funiversal-properties\u002Funiversal-properties":14842,"\u002Fcategory-theory\u002Funiversal-properties\u002Fproducts-and-coproducts":14847,"\u002Fcategory-theory\u002Funiversal-properties\u002Fconstructions-on-categories":14851,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Frepresentable-functors":14855,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-lemma":14860,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-consequences":14864,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits":14868,"\u002Fcategory-theory\u002Flimits-colimits\u002Fproducts-equalizers-pullbacks":14873,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcolimits":14877,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcomputing-limits":14881,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits-and-functors":14885,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions":14889,"\u002Fcategory-theory\u002Fadjunctions\u002Funits-and-counits":14894,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions-via-universal-arrows":14898,"\u002Fcategory-theory\u002Fadjunctions\u002Ffree-forgetful-adjunctions":14902,"\u002Fcategory-theory\u002Fadjoints-limits\u002Flimits-via-adjoints":14906,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fpresheaf-limits-colimits":14911,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoints-preserve-limits":14915,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoint-functor-theorem":14919,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fmonads":14923,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-eilenberg-moore":14928,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fkleisli-and-programming":14932,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-for-endofunctors":14936,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Fcartesian-closed-categories":14940,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Flambda-calculus-correspondence":14945,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Ffixed-points-and-recursion":14949,"\u002Fcategory-theory":14953,"\u002Fdeep-learning\u002Fmathematical-background\u002Flinear-algebra-for-deep-learning":14956,"\u002Fdeep-learning\u002Fmathematical-background\u002Fprobability-and-information-theory":14960,"\u002Fdeep-learning\u002Fmathematical-background\u002Fnumerical-computation":14964,"\u002Fdeep-learning\u002Fmathematical-background\u002Fcalculus":14968,"\u002Fdeep-learning\u002Ffoundations\u002Fwhat-is-deep-learning":14971,"\u002Fdeep-learning\u002Ffoundations\u002Fmachine-learning-refresher":14975,"\u002Fdeep-learning\u002Ffoundations\u002Flinear-models-and-the-perceptron":14979,"\u002Fdeep-learning\u002Fneural-networks\u002Fthe-multilayer-perceptron":14983,"\u002Fdeep-learning\u002Fneural-networks\u002Factivation-functions":14988,"\u002Fdeep-learning\u002Fneural-networks\u002Funiversal-approximation":14992,"\u002Fdeep-learning\u002Fneural-networks\u002Fbackpropagation":14996,"\u002Fdeep-learning\u002Fneural-networks\u002Floss-functions-and-output-units":15000,"\u002Fdeep-learning\u002Foptimization\u002Fgradient-descent-and-sgd":15004,"\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods":15009,"\u002Fdeep-learning\u002Foptimization\u002Finitialization":15013,"\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape":15017,"\u002Fdeep-learning\u002Foptimization\u002Fsecond-order-and-approximate-methods":15021,"\u002Fdeep-learning\u002Fregularization\u002Fregularization-overview":15025,"\u002Fdeep-learning\u002Fregularization\u002Fdropout-and-data-augmentation":15030,"\u002Fdeep-learning\u002Fregularization\u002Fearly-stopping-and-parameter-sharing":15034,"\u002Fdeep-learning\u002Fregularization\u002Fnormalization":15038,"\u002Fdeep-learning\u002Farchitectures\u002Fconvolutional-networks":15042,"\u002Fdeep-learning\u002Farchitectures\u002Fcnn-architectures":15047,"\u002Fdeep-learning\u002Farchitectures\u002Frecurrent-networks":15051,"\u002Fdeep-learning\u002Farchitectures\u002Flstm-and-gru":15055,"\u002Fdeep-learning\u002Farchitectures\u002Fattention-and-transformers":15059,"\u002Fdeep-learning\u002Farchitectures\u002Fthe-transformer-architecture":15063,"\u002Fdeep-learning\u002Farchitectures\u002Ftransformers-in-practice":15067,"\u002Fdeep-learning\u002Farchitectures\u002Fgraph-neural-networks":15071,"\u002Fdeep-learning\u002Farchitectures\u002Fstate-space-models":15075,"\u002Fdeep-learning\u002Ftheory\u002Fgeneralization-theory":15079,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-robustness":15084,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-defenses":15088,"\u002Fdeep-learning\u002Ftheory\u002Fbayesian-and-ensemble-methods":15092,"\u002Fdeep-learning\u002Ftheory\u002Fdeep-equilibrium-models":15096,"\u002Fdeep-learning\u002Fgenerative-models\u002Flinear-factor-models":15100,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoencoders":15105,"\u002Fdeep-learning\u002Fgenerative-models\u002Fvariational-autoencoders":15109,"\u002Fdeep-learning\u002Fgenerative-models\u002Fgenerative-adversarial-networks":15113,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoregressive-and-normalizing-flows":15117,"\u002Fdeep-learning\u002Fgenerative-models\u002Fenergy-based-and-boltzmann-machines":15121,"\u002Fdeep-learning\u002Fgenerative-models\u002Fdiffusion-and-score-based-models":15125,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fstructured-probabilistic-models":15129,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fmonte-carlo-and-mcmc":15134,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fapproximate-inference":15138,"\u002Fdeep-learning\u002Fpractical\u002Fpractical-methodology":15142,"\u002Fdeep-learning\u002Fpractical\u002Fhyperparameters-and-debugging":15147,"\u002Fdeep-learning\u002Fpractical\u002Frepresentation-learning":15151,"\u002Fdeep-learning\u002Fpractical\u002Ftransfer-learning":15155,"\u002Fdeep-learning\u002Fpractical\u002Fapplications":15159,"\u002Fdeep-learning\u002Fpractical\u002Fmodel-compression-and-distillation":15163,"\u002Fdeep-learning\u002Fpractical\u002Fmeta-learning-and-few-shot":15167,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models":15171,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment":15176,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fseq2seq-pretraining-and-bart":15180,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ftext-to-text-transfer-and-conditional-generation":15184,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fspeech-and-audio-models":15188,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fself-supervised-speech-and-synthesis":15192,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents":15196,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fagent-memory-retrieval-and-orchestration":15200,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmixture-of-experts":15204,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmultimodal-models":15208,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ffusion-and-vision-language-models":15212,"\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning":15216,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fmodel-free-prediction-and-control":15221,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fdeep-q-networks":15225,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic":15229,"\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback":15233,"\u002Fdeep-learning":15237,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fequilibrium-state-variables-zeroth-law":15240,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Ffirst-law-heat-and-work":15244,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fsecond-law-entropy-and-the-carnot-bound":15248,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fthermodynamic-potentials-and-maxwell-relations":15252,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fstability-response-functions-and-the-third-law":15256,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fclassical-statistics-and-equipartition":15260,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fphase-space-and-liouvilles-theorem":15265,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fensembles-and-the-equal-probability-postulate":15269,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fstatistical-entropy-boltzmann-and-gibbs":15273,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fmicrocanonical-ensemble-and-entropy":15277,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fequilibrium-conditions-temperature-pressure-chemical-potential":15282,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fideal-gas-phase-space-and-the-sackur-tetrode-entropy":15286,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Ftwo-state-systems-paramagnets-and-negative-temperature":15290,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fcanonical-ensemble-and-the-boltzmann-distribution":15294,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fpartition-function-and-the-helmholtz-free-energy":15299,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fenergy-fluctuations-and-ensemble-equivalence":15303,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fthe-einstein-solid-and-harmonic-systems":15307,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fparamagnetism-and-the-schottky-anomaly":15311,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fideal-gas-partition-function-and-the-gibbs-paradox":15315,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fequipartition-and-the-virial-theorem":15320,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fmolecular-gases-rotation-and-vibration":15324,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fgrand-canonical-ensemble-and-the-grand-partition-function":15328,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fchemical-potential-fugacity-and-number-fluctuations":15333,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fensemble-summary-and-the-thermodynamic-web":15337,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fquantum-statistics-bose-einstein-and-fermi-dirac":15341,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fderiving-the-quantum-distributions":15346,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fthe-classical-limit-and-quantum-concentration":15350,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fideal-quantum-gases-general-framework":15354,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-and-the-fermion-gas":15358,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthe-photon-gas-and-plancks-radiation-law":15363,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fblackbody-thermodynamics-and-radiation-pressure":15367,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fphonons-and-the-debye-model":15371,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-derived":15375,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthermodynamics-of-the-bose-gas-and-superfluidity":15379,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fthe-ideal-fermi-gas-at-zero-temperature":15383,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fsommerfeld-expansion-and-electrons-in-metals":15388,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":15392,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fneutron-stars-and-nuclear-matter":15396,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-cluster-expansion-and-virial-coefficients":15400,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-van-der-waals-gas-and-liquid-gas-coexistence":15405,"\u002Fstatistical-mechanics\u002Finteractions\u002Fquantum-gases-with-interactions-and-exchange":15409,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fphases-coexistence-and-classification":15413,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-ising-model-and-exact-solutions":15418,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fmean-field-theory-and-the-weiss-model":15422,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fcritical-exponents-and-landau-theory":15426,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-renormalization-group-idea":15430,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fthermodynamic-fluctuations-and-response":15434,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fbrownian-motion-and-the-langevin-equation":15439,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Flinear-response-and-the-fluctuation-dissipation-theorem":15443,"\u002Fstatistical-mechanics":15447,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fbonding-mechanisms":15450,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fmolecular-orbitals-and-h2-plus":15455,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fhydrogen-molecule-and-exchange":15459,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fvan-der-waals-forces":15463,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Frotational-vibrational-spectra":15467,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Fanharmonicity-and-rovibrational-structure":15472,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Framan-and-electronic-bands":15476,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Flasers-and-masers":15480,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fstructure-of-solids":15484,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fbravais-lattices-and-crystal-systems":15489,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Freciprocal-lattice-and-brillouin-zones":15493,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fdiffraction-and-structure-factors":15497,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonon-dispersion":15501,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonons-quantization-and-dos":15506,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fdebye-einstein-heat-capacity":15510,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fanharmonicity-and-thermal-transport":15514,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ffree-electron-gas-and-conduction":15518,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fsommerfeld-model-and-heat-capacity":15523,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ftransport-and-the-hall-effect":15527,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fscreening-and-plasmons":15531,"\u002Fcondensed-matter\u002Fband-theory\u002Fblochs-theorem-and-energy-bands":15535,"\u002Fcondensed-matter\u002Fband-theory\u002Fnearly-free-electron-model":15540,"\u002Fcondensed-matter\u002Fband-theory\u002Ftight-binding-method":15544,"\u002Fcondensed-matter\u002Fband-theory\u002Ffermi-surfaces-and-semiclassical-dynamics":15548,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fsemiconductor-bands-and-junctions":15552,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fintrinsic-and-extrinsic-semiconductors":15557,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fcarrier-transport-and-recombination":15561,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fthe-pn-junction":15565,"\u002Fcondensed-matter\u002Fsemiconductors\u002Ftransistors-and-optoelectronics":15569,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fdielectrics-and-polarization":15573,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fferroelectrics-and-piezoelectrics":15578,"\u002Fcondensed-matter\u002Fmagnetism\u002Fdiamagnetism-and-paramagnetism":15582,"\u002Fcondensed-matter\u002Fmagnetism\u002Fexchange-and-ferromagnetism":15587,"\u002Fcondensed-matter\u002Fmagnetism\u002Fantiferromagnetism-and-domains":15591,"\u002Fcondensed-matter\u002Fmagnetism\u002Fspin-waves-and-magnons":15595,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fsuperconductivity-phenomenology":15599,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Flondon-theory-and-the-meissner-effect":15604,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fginzburg-landau-theory":15608,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fbcs-theory":15612,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fjosephson-and-high-tc":15616,"\u002Fcondensed-matter\u002Fnanostructures\u002Fquantum-wells-wires-and-dots":15620,"\u002Fcondensed-matter\u002Fnanostructures\u002Finteger-quantum-hall-effect":15625,"\u002Fcondensed-matter\u002Fnanostructures\u002Ffractional-quantum-hall-and-topology":15629,"\u002Fcondensed-matter\u002Fnanostructures\u002Fgraphene-and-dirac-materials":15633,"\u002Fcondensed-matter":15637,"\u002Flogic\u002Ffoundations\u002Flogic-as-a-mathematical-model":15640,"\u002Flogic\u002Fsentential-logic\u002Fformal-languages-and-well-formed-formulas":15644,"\u002Flogic\u002Fsentential-logic\u002Ftruth-assignments-and-tautologies":15649,"\u002Flogic\u002Fsentential-logic\u002Funique-readability-and-parsing":15653,"\u002Flogic\u002Fsentential-logic\u002Finduction-and-recursion":15657,"\u002Flogic\u002Fsentential-logic\u002Fexpressive-completeness-and-normal-forms":15661,"\u002Flogic\u002Fsentential-logic\u002Fboolean-circuits":15665,"\u002Flogic\u002Fsentential-logic\u002Fcompactness-and-effectiveness":15669,"\u002Flogic\u002Ffirst-order-languages\u002Ffirst-order-languages":15673,"\u002Flogic\u002Ffirst-order-languages\u002Fstructures-truth-and-satisfaction":15678,"\u002Flogic\u002Ffirst-order-languages\u002Fdefinability-and-elementary-equivalence":15682,"\u002Flogic\u002Ffirst-order-languages\u002Fterms-substitution-and-parsing":15686,"\u002Flogic\u002Fdeductive-calculus\u002Fa-deductive-calculus":15690,"\u002Flogic\u002Fdeductive-calculus\u002Fdeduction-theorem-and-derived-rules":15695,"\u002Flogic\u002Fdeductive-calculus\u002Fsoundness":15699,"\u002Flogic\u002Fdeductive-calculus\u002Fcompleteness-and-consistency":15703,"\u002Flogic\u002Fmodels-and-theories\u002Fcompactness-and-lowenheim-skolem":15707,"\u002Flogic\u002Fmodels-and-theories\u002Ftheories-elementary-classes-and-categoricity":15712,"\u002Flogic\u002Fmodels-and-theories\u002Finterpretations-between-theories":15716,"\u002Flogic\u002Fmodels-and-theories\u002Fnonstandard-analysis":15720,"\u002Flogic\u002Farithmetic-and-definability\u002Fdefinability-in-arithmetic":15724,"\u002Flogic\u002Farithmetic-and-definability\u002Fnatural-numbers-with-successor":15729,"\u002Flogic\u002Farithmetic-and-definability\u002Fpresburger-and-reducts":15733,"\u002Flogic\u002Farithmetic-and-definability\u002Fa-subtheory-and-representability":15737,"\u002Flogic\u002Fincompleteness\u002Farithmetization-of-syntax":15741,"\u002Flogic\u002Fincompleteness\u002Fincompleteness-and-undecidability":15746,"\u002Flogic\u002Fincompleteness\u002Fsecond-incompleteness-theorem":15750,"\u002Flogic\u002Fcomputability-and-representability\u002Frecursive-functions":15754,"\u002Flogic\u002Fcomputability-and-representability\u002Frepresenting-exponentiation":15759,"\u002Flogic\u002Fsecond-order-logic\u002Fsecond-order-languages":15763,"\u002Flogic\u002Fsecond-order-logic\u002Fskolem-functions-and-many-sorted-logic":15768,"\u002Flogic\u002Fsecond-order-logic\u002Fgeneral-structures":15772,"\u002Flogic":15776,"\u002Freinforcement-learning\u002Ffoundations\u002Fwhat-is-reinforcement-learning":15779,"\u002Freinforcement-learning\u002Ffoundations\u002Fa-brief-history-of-rl":15783,"\u002Freinforcement-learning\u002Ffoundations\u002Fmulti-armed-bandits":15787,"\u002Freinforcement-learning\u002Ffoundations\u002Fbandit-exploration-algorithms":15791,"\u002Freinforcement-learning\u002Ffoundations\u002Fmarkov-decision-processes":15795,"\u002Freinforcement-learning\u002Ffoundations\u002Fvalue-functions-and-optimality":15799,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdynamic-programming":15803,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdp-async-and-gpi":15807,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods":15811,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-off-policy":15814,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftemporal-difference-learning":15818,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftd-control-sarsa-and-q-learning":15822,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping":15826,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-off-policy-methods":15829,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-and-learning":15833,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-focusing-and-decision-time":15837,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdecision-time-planning":15841,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-tree-search":15845,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-prediction":15849,"\u002Freinforcement-learning\u002Fapproximation\u002Ffeature-construction-and-nonlinear":15853,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-control":15857,"\u002Freinforcement-learning\u002Fapproximation\u002Faverage-reward-control":15861,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad":15865,"\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td":15868,"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces":15869,"\u002Freinforcement-learning\u002Fapproximation\u002Ftrue-online-and-sarsa-lambda":15872,"\u002Freinforcement-learning\u002Fapproximation\u002Fpolicy-gradient-methods":15876,"\u002Freinforcement-learning\u002Fapproximation\u002Factor-critic-and-continuous-actions":15880,"\u002Freinforcement-learning\u002Fapproximation\u002Fleast-squares-and-memory-based-methods":15884,"\u002Freinforcement-learning\u002Fapproximation\u002Fmemory-and-kernel-methods":15888,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces":15892,"\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces":15896,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks":15900,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdqn-improvements":15903,"\u002Freinforcement-learning\u002Fdeep-rl\u002Factor-critic-and-ppo":15907,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fppo-and-continuous-control":15911,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fcase-studies":15915,"\u002Freinforcement-learning\u002Fdeep-rl\u002Frl-beyond-games":15919,"\u002Freinforcement-learning\u002Fdeep-rl\u002Ffrontiers":15923,"\u002Freinforcement-learning\u002Fdeep-rl\u002Freward-design-and-open-problems":15927,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow":15931,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow-part-2":15936,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control":15940,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control-part-2":15944,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl":15948,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl-part-2":15952,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration":15956,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration-part-2":15960,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl":15964,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2":15968,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl":15972,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl-part-2":15976,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl":15980,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl-part-2":15984,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl":15988,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl-part-2":15992,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Frlhf-and-language-models":15996,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps":16000,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps-part-2":16004,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl":16008,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl-part-2":16012,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmeta-rl-and-generalization":16016,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fpsychology-of-reinforcement":16020,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Finstrumental-conditioning-and-control":16025,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-and-td-error":16029,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-in-the-brain":16033,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fanimal-learning-and-cognition":16037,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fcognitive-maps-and-planning":16041,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fneuroscience-of-reinforcement":16045,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fseveral-learning-systems":16049,"\u002Freinforcement-learning":16053,"\u002Fartificial-intelligence\u002Ffoundations\u002Fwhat-is-ai":16055,"\u002Fartificial-intelligence\u002Ffoundations\u002Ffoundations-of-ai":16059,"\u002Fartificial-intelligence\u002Ffoundations\u002Fintelligent-agents":16063,"\u002Fartificial-intelligence\u002Ffoundations\u002Fagent-architectures":16067,"\u002Fartificial-intelligence\u002Fsearch\u002Funinformed-search":16071,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-strategies-compared":16076,"\u002Fartificial-intelligence\u002Fsearch\u002Finformed-search":16080,"\u002Fartificial-intelligence\u002Fsearch\u002Fheuristic-functions":16084,"\u002Fartificial-intelligence\u002Fsearch\u002Flocal-search":16088,"\u002Fartificial-intelligence\u002Fsearch\u002Fpopulation-and-continuous-search":16092,"\u002Fartificial-intelligence\u002Fsearch\u002Fadversarial-search":16096,"\u002Fartificial-intelligence\u002Fsearch\u002Fgames-of-chance-and-imperfect-information":16100,"\u002Fartificial-intelligence\u002Fsearch\u002Fconstraint-satisfaction":16104,"\u002Fartificial-intelligence\u002Fsearch\u002Fcsp-search-and-structure":16108,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-under-uncertainty":16112,"\u002Fartificial-intelligence\u002Fsearch\u002Fbelief-state-and-online-search":16116,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-logic":16120,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-inference":16125,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic":16129,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic-in-use":16133,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Finference-and-resolution":16137,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-resolution":16141,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fclassical-planning":16145,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-graphs-and-graphplan":16149,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-in-the-real-world":16153,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-under-uncertainty":16157,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fknowledge-representation":16161,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Freasoning-systems-and-defaults":16165,"\u002Fartificial-intelligence\u002Funcertainty\u002Fprobability-and-bayes":16169,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayes-rule-and-naive-bayes":16174,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayesian-networks":16178,"\u002Fartificial-intelligence\u002Funcertainty\u002Finference-in-bayesian-networks":16182,"\u002Fartificial-intelligence\u002Funcertainty\u002Freasoning-over-time":16186,"\u002Fartificial-intelligence\u002Funcertainty\u002Ftracking-and-data-association":16190,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmaking-decisions":16194,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmarkov-decision-processes":16198,"\u002Fartificial-intelligence\u002Funcertainty\u002Fdecision-networks-and-game-theory":16201,"\u002Fartificial-intelligence\u002Funcertainty\u002Fgame-theory-and-mechanism-design":16205,"\u002Fartificial-intelligence\u002Flearning\u002Flearning-from-examples":16209,"\u002Fartificial-intelligence\u002Flearning\u002Ftheory-and-model-families":16214,"\u002Fartificial-intelligence\u002Flearning\u002Fprobabilistic-learning":16218,"\u002Fartificial-intelligence\u002Flearning\u002Fexpectation-maximization":16222,"\u002Fartificial-intelligence\u002Flearning\u002Freinforcement-learning":16226,"\u002Fartificial-intelligence\u002Flearning\u002Fgeneralization-and-policy-search":16229,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-in-learning":16233,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-based-learning-methods":16237,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fvision-and-perception":16241,"\u002Fartificial-intelligence\u002Ffrontiers\u002Freconstructing-the-3d-world":16246,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobotics":16250,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobot-planning-and-control":16254,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnatural-language-in-ai":16258,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnlp-grammar-translation-and-speech":16262,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fphilosophy-and-future":16266,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fai-ethics-and-future":16270,"\u002Fartificial-intelligence":16274,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-constituents-nuclide-chart":16277,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-size-charge-distributions":16282,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-masses-binding-energy":16286,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fsemi-empirical-mass-formula":16290,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-moments-multipoles":16294,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnuclear-force-shell-overview":16298,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fthe-deuteron":16303,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnucleon-nucleon-scattering":16307,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fmeson-theory-isospin":16311,"\u002Fnuclear-physics\u002Fnuclear-models\u002Ffermi-gas-model":16315,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fliquid-drop-collective-coordinates":16320,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fshell-model-single-particle":16324,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fcollective-model-rotations-vibrations":16328,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-law-modes":16332,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-kinetics-equilibrium":16337,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-decay-gamow-theory":16341,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-fine-structure-hindrance":16346,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fbeta-decay-energetics-neutrino":16350,"\u002Fnuclear-physics\u002Fbeta-decay\u002Ffermi-theory-beta-decay":16355,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fweak-interaction-parity-violation":16359,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fdouble-beta-decay-neutrino-mass":16363,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fgamma-multipole-radiation":16367,"\u002Fnuclear-physics\u002Fgamma-decay\u002Finternal-conversion-isomers":16372,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fangular-correlations-mossbauer":16376,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Freaction-kinematics-cross-sections":16380,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fcompound-nucleus-resonances":16385,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fdirect-reactions-optical-model":16389,"\u002Fnuclear-physics\u002Ffission\u002Ffission-barrier-dynamics":16393,"\u002Fnuclear-physics\u002Ffission\u002Fchain-reactions-reactor-physics":16398,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Ffusion-reactions-confinement":16402,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fstellar-nucleosynthesis":16407,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fbig-bang-nucleosynthesis":16411,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fcharged-particle-stopping-power":16415,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fphoton-neutron-interactions":16420,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fradiation-detectors":16424,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fdosimetry-radiation-biology":16428,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fnuclear-applications-dating-medicine":16432,"\u002Fnuclear-physics":16436,"\u002Fnatural-language-processing\u002Ffoundations\u002Fwhat-is-nlp":16439,"\u002Fnatural-language-processing\u002Ffoundations\u002Fregex-and-text-normalization":16443,"\u002Fnatural-language-processing\u002Ffoundations\u002Fminimum-edit-distance":16447,"\u002Fnatural-language-processing\u002Ffoundations\u002Fn-gram-language-models":16451,"\u002Fnatural-language-processing\u002Ffoundations\u002Fsmoothing-and-backoff":16455,"\u002Fnatural-language-processing\u002Fclassification\u002Fnaive-bayes-and-sentiment":16459,"\u002Fnatural-language-processing\u002Fclassification\u002Fevaluating-classifiers":16464,"\u002Fnatural-language-processing\u002Fclassification\u002Flogistic-regression":16468,"\u002Fnatural-language-processing\u002Fclassification\u002Fsentiment-and-affect-lexicons":16472,"\u002Fnatural-language-processing\u002Fsemantics\u002Fvector-semantics-and-embeddings":16476,"\u002Fnatural-language-processing\u002Fsemantics\u002Fstatic-word-embeddings":16481,"\u002Fnatural-language-processing\u002Fsemantics\u002Fneural-language-models":16485,"\u002Fnatural-language-processing\u002Fsequences\u002Fsequence-labeling":16489,"\u002Fnatural-language-processing\u002Fsequences\u002Fcrfs-and-neural-taggers":16493,"\u002Fnatural-language-processing\u002Fsequences\u002Frnns-and-lstms":16497,"\u002Fnatural-language-processing\u002Ftransformers\u002Ftransformers-and-attention":16501,"\u002Fnatural-language-processing\u002Ftransformers\u002Fthe-transformer-architecture":16505,"\u002Fnatural-language-processing\u002Ftransformers\u002Flarge-language-models":16508,"\u002Fnatural-language-processing\u002Ftransformers\u002Fllm-pretraining-and-scaling":16511,"\u002Fnatural-language-processing\u002Ftransformers\u002Ffine-tuning-and-prompting":16515,"\u002Fnatural-language-processing\u002Ftransformers\u002Fprompting-and-alignment":16519,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-parsing":16523,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcky-scoring-and-evaluation":16528,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdependency-parsing":16532,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fgraph-based-and-neural-dependency-parsing":16536,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fword-senses-and-wsd":16540,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fwsd-in-practice-and-induction":16544,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-roles-and-information-extraction":16548,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Frelations-events-and-templates":16552,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoreference-and-discourse":16556,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoherence-and-discourse-structure":16560,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Flogical-semantics":16564,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcompositional-semantics-and-description-logics":16568,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-parsing":16572,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fneural-semantic-parsing":16576,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Finformation-extraction":16580,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftimes-events-and-templates":16584,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdiscourse-coherence":16588,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fentity-based-and-global-coherence":16592,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-grammars":16596,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftreebanks-and-lexicalized-grammars":16600,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation":16604,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation-decoding-and-evaluation":16608,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering":16612,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering-knowledge-and-llms":16616,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-and-chatbots":16620,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-systems-and-assistants":16624,"\u002Fnatural-language-processing\u002Fapplications\u002Ftext-summarization":16628,"\u002Fnatural-language-processing\u002Fapplications\u002Fabstractive-summarization-and-evaluation":16632,"\u002Fnatural-language-processing\u002Fspeech\u002Fphonetics":16636,"\u002Fnatural-language-processing\u002Fspeech\u002Facoustic-phonetics":16641,"\u002Fnatural-language-processing\u002Fspeech\u002Fautomatic-speech-recognition":16645,"\u002Fnatural-language-processing\u002Fspeech\u002Fasr-evaluation-and-applications":16649,"\u002Fnatural-language-processing":16653,"\u002Fparticle-physics\u002Ffoundations\u002Fhistorical-overview-particle-zoo":16656,"\u002Fparticle-physics\u002Ffoundations\u002Fparticle-physics-basic-concepts":16660,"\u002Fparticle-physics\u002Ffoundations\u002Ffundamental-interactions-force-carriers":16664,"\u002Fparticle-physics\u002Funits-kinematics\u002Fnatural-units-and-scales":16668,"\u002Fparticle-physics\u002Funits-kinematics\u002Ffour-vectors-invariant-mass":16673,"\u002Fparticle-physics\u002Funits-kinematics\u002Fdecay-scattering-kinematics-mandelstam":16677,"\u002Fparticle-physics\u002Funits-kinematics\u002Fcross-sections-golden-rule":16681,"\u002Fparticle-physics\u002Fsymmetries\u002Fconservation-laws-symmetries":16685,"\u002Fparticle-physics\u002Fsymmetries\u002Fdiscrete-symmetries-cpt":16690,"\u002Fparticle-physics\u002Fsymmetries\u002Fparity-violation-weak":16694,"\u002Fparticle-physics\u002Fsymmetries\u002Fsu2-su3-flavor-symmetry":16698,"\u002Fparticle-physics\u002Fquark-model\u002Feightfold-way-su3":16702,"\u002Fparticle-physics\u002Fquark-model\u002Fmeson-spectroscopy":16707,"\u002Fparticle-physics\u002Fquark-model\u002Fbaryon-spectroscopy":16711,"\u002Fparticle-physics\u002Fquark-model\u002Fcolor-confinement-exotics":16715,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fklein-gordon-equation":16719,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fdirac-equation-spinors":16724,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fantiparticles-hole-theory":16728,"\u002Fparticle-physics\u002Fqed\u002Ffeynman-rules-qed":16732,"\u002Fparticle-physics\u002Fqed\u002Fqed-tree-processes":16737,"\u002Fparticle-physics\u002Fqed\u002Frenormalization-running-coupling":16741,"\u002Fparticle-physics\u002Fqed\u002Felectron-g-2":16745,"\u002Fparticle-physics\u002Fweak-interaction\u002Fva-structure-weak":16749,"\u002Fparticle-physics\u002Fweak-interaction\u002Fw-z-bosons-decays":16754,"\u002Fparticle-physics\u002Fweak-interaction\u002Fckm-matrix":16758,"\u002Fparticle-physics\u002Fweak-interaction\u002Fcp-violation-kaons-b-mesons":16762,"\u002Fparticle-physics\u002Fqcd\u002Fcolor-su3-gluons":16766,"\u002Fparticle-physics\u002Fqcd\u002Fasymptotic-freedom-confinement":16771,"\u002Fparticle-physics\u002Fqcd\u002Fdeep-inelastic-scattering-partons":16775,"\u002Fparticle-physics\u002Fqcd\u002Fjets-hadronization":16779,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Felectroweak-su2-u1":16783,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fspontaneous-symmetry-breaking":16788,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-mechanism":16792,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-boson-discovery":16796,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fstandard-model":16800,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-oscillations":16804,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-mass-pmns":16809,"\u002Fparticle-physics\u002Fneutrinos\u002Fdirac-majorana-experiments":16813,"\u002Fparticle-physics\u002Fexperiment\u002Faccelerators-luminosity":16817,"\u002Fparticle-physics\u002Fexperiment\u002Fdetectors-subsystems":16822,"\u002Fparticle-physics\u002Fexperiment\u002Fhow-discoveries-are-made":16826,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fbeyond-standard-model":16830,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fgrand-unified-theories":16834,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fsupersymmetry":16838,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fhierarchy-problem-naturalness":16842,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fdark-matter-candidates":16846,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fmatter-antimatter-open-questions":16850,"\u002Fparticle-physics":16854,"\u002Fastrophysics-cosmology\u002Forientation\u002Fthe-sun-and-stars":16857,"\u002Fastrophysics-cosmology\u002Forientation\u002Fstellar-death-final-states":16862,"\u002Fastrophysics-cosmology\u002Forientation\u002Fgalaxies-and-cosmology":16866,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fmagnitudes-fluxes-and-the-distance-modulus":16870,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fstellar-spectra-and-spectral-classification":16875,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Ftelescopes-and-detectors-across-the-spectrum":16879,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fthe-cosmic-distance-ladder":16883,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fblackbody-radiation-and-specific-intensity":16887,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fradiative-transfer-and-the-transfer-equation":16892,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fspectral-line-formation-and-broadening":16896,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fopacity-and-the-rosseland-mean":16900,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fhydrostatic-equilibrium-and-the-virial-theorem":16904,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equations-of-stellar-structure":16909,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equation-of-state-and-polytropes":16913,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-standard-solar-model":16917,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fthermonuclear-reaction-rates-and-the-gamow-peak":16921,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhydrogen-burning-pp-chains-and-cno":16926,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhelium-burning-and-the-triple-alpha-process":16930,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fadvanced-burning-and-neutron-capture-nucleosynthesis":16934,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fphases-of-the-interstellar-medium":16938,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fmolecular-clouds-and-gravitational-collapse":16943,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fprotostars-and-the-pre-main-sequence":16947,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-main-sequence-and-its-structure":16951,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fpost-main-sequence-low-mass-evolution":16956,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-evolution-of-massive-stars":16960,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fstellar-pulsation-and-the-instability-strip":16964,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":16968,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fcore-collapse-supernovae":16972,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fthermonuclear-supernovae-type-ia":16976,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fneutron-stars-and-pulsars":16980,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fblack-holes-schwarzschild-and-kerr":16984,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fbinary-systems-and-mass-transfer":16988,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Faccreting-compact-objects":16993,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fgravitational-waves-from-inspiraling-binaries":16997,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fmultimessenger-astronomy-and-gamma-ray-bursts":17001,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fthe-milky-way":17005,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-morphology-and-classification":17010,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-rotation-curves-and-dark-matter":17014,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Factive-galactic-nuclei-and-supermassive-black-holes":17018,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-clusters-and-large-scale-structure":17022,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-expanding-universe-and-hubbles-law":17026,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-frw-metric-and-cosmological-redshift":17031,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-friedmann-equations-and-cosmic-dynamics":17035,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fcosmological-models-and-distances":17038,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fdark-energy-and-the-accelerating-universe":17042,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fthe-thermal-history-of-the-universe":17046,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fbig-bang-nucleosynthesis":17051,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Frecombination-and-the-cosmic-microwave-background":17055,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcmb-anisotropies-and-cosmological-parameters":17059,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcosmic-inflation":17063,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fstructure-formation-and-the-growth-of-perturbations":17067,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fdark-matter-dark-energy-and-open-questions":17071,"\u002Fastrophysics-cosmology":17075,"\u002Fcolophon":17078,"\u002F":17081},{"path":12403,"title":12404,"module":12405,"summary":12406},"\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":12408,"title":12409,"module":12405,"summary":12410},"\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":12412,"title":12413,"module":12405,"summary":12414},"\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":12416,"title":12417,"module":12405,"summary":12418},"\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":12420,"title":12421,"module":12405,"summary":12422},"\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":12424,"title":12425,"module":12405,"summary":12426},"\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":12428,"title":12429,"module":12430,"summary":12431},"\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":12433,"title":12434,"module":12430,"summary":12435},"\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":12437,"title":12438,"module":12430,"summary":12439},"\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":12441,"title":12442,"module":12430,"summary":12443},"\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":12445,"title":12446,"module":12447,"summary":12448},"\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":12450,"title":12451,"module":12447,"summary":12452},"\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":12454,"title":12455,"module":12447,"summary":12456},"\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":12458,"title":12459,"module":12447,"summary":12460},"\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":12462,"title":12463,"module":12464,"summary":12465},"\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":12467,"title":12468,"module":12464,"summary":12469},"\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":12471,"title":12472,"module":12464,"summary":12473},"\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":12475,"title":12476,"module":12464,"summary":12477},"\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":12479,"title":12480,"module":12464,"summary":12481},"\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":12483,"title":12484,"module":12464,"summary":12485},"\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":12487,"title":12488,"module":12464,"summary":12489},"\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":12491,"title":12492,"module":12464,"summary":12493},"\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":12495,"title":12496,"module":12464,"summary":12497},"\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":12499,"title":12500,"module":12464,"summary":12501},"\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":12503,"title":12504,"module":12464,"summary":12505},"\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":12507,"title":12508,"module":12464,"summary":12509},"\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":12511,"title":12512,"module":12513,"summary":12514},"\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":12516,"title":12517,"module":12513,"summary":12518},"\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":12520,"title":12521,"module":12513,"summary":12522},"\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":12524,"title":12525,"module":12513,"summary":12526},"\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":12528,"title":12529,"module":12513,"summary":12530},"\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":12532,"title":12533,"module":12513,"summary":12534},"\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":12536,"title":12537,"module":12513,"summary":12538},"\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":12540,"title":12541,"module":12513,"summary":12542},"\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":12544,"title":12545,"module":12546,"summary":12547},"\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":12549,"title":12550,"module":12546,"summary":12551},"\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":12553,"title":12554,"module":12546,"summary":12555},"\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":12557,"title":12558,"module":12546,"summary":12559},"\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":12561,"title":12562,"module":12546,"summary":12563},"\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":12565,"title":12566,"module":12546,"summary":12567},"\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":12569,"title":12570,"module":12546,"summary":12571},"\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":12573,"title":12574,"module":12546,"summary":12575},"\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":12577,"title":12578,"module":12546,"summary":12579},"\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":12581,"title":12582,"module":12546,"summary":12583},"\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":12585,"title":12586,"module":12546,"summary":12587},"\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":12589,"title":12590,"module":12546,"summary":12591},"\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":12593,"title":12594,"module":12546,"summary":12595},"\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":12597,"title":12598,"module":12546,"summary":12599},"\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":12601,"title":12602,"module":12603,"summary":12604},"\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":12606,"title":12607,"module":12603,"summary":12608},"\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":12610,"title":12611,"module":12603,"summary":12612},"\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":12614,"title":12615,"module":12603,"summary":12616},"\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":12618,"title":12619,"module":12603,"summary":12620},"\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":12622,"title":12623,"module":12624,"summary":12625},"\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":12627,"title":12628,"module":12624,"summary":12629},"\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":12631,"title":12632,"module":12624,"summary":12633},"\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":12635,"title":12636,"module":12624,"summary":12637},"\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":12639,"title":12640,"module":12624,"summary":12641},"\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":12643,"title":12644,"module":12624,"summary":12645},"\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":12647,"title":12648,"module":12624,"summary":12649},"\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":12651,"title":12652,"module":12624,"summary":12653},"\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":12655,"title":12656,"module":12624,"summary":12657},"\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":12659,"title":12660,"module":12624,"summary":12661},"\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":12663,"title":12664,"module":12624,"summary":12665},"\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":12667,"title":12668,"module":12669,"summary":12670},"\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":12672,"title":12673,"module":12669,"summary":12674},"\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":12676,"title":12677,"module":12669,"summary":12678},"\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":12680,"title":12681,"module":12669,"summary":12682},"\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":12684,"title":12685,"module":12686,"summary":12687},"\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":12689,"title":12690,"module":12686,"summary":12691},"\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":12693,"title":12694,"module":12686,"summary":12695},"\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":12697,"title":12698,"module":12686,"summary":12699},"\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":12701,"title":12702,"module":12686,"summary":12703},"\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":12705,"title":12706,"module":12686,"summary":12707},"\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":12709,"title":12710,"module":12686,"summary":12711},"\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":12713,"title":12714,"module":12715,"summary":12716},"\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":12718,"title":12719,"module":12715,"summary":12720},"\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":12722,"title":12723,"module":12715,"summary":12724},"\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":12726,"title":12727,"module":12715,"summary":12728},"\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":12730,"title":12731,"module":12732,"summary":12733},"\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":12735,"title":12736,"module":12732,"summary":12737},"\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":12739,"title":12740,"module":12732,"summary":12741},"\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":12743,"title":12744,"module":12732,"summary":12745},"\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":12747,"title":12748,"module":6,"summary":6},"\u002Falgorithms","Algorithms",{"path":12750,"title":12751,"module":12752,"summary":12753},"\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":12755,"title":12756,"module":12752,"summary":12757},"\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":12759,"title":12760,"module":12752,"summary":12761},"\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":12763,"title":12764,"module":12752,"summary":12765},"\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":12767,"title":12768,"module":12769,"summary":12770},"\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":12772,"title":12773,"module":12769,"summary":12774},"\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":12776,"title":12777,"module":12769,"summary":12778},"\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":12780,"title":12781,"module":12769,"summary":12782},"\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":12784,"title":12785,"module":12786,"summary":12787},"\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":12789,"title":12790,"module":12786,"summary":12791},"\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":12793,"title":12794,"module":12786,"summary":12795},"\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":12797,"title":12798,"module":12786,"summary":12799},"\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":12801,"title":12802,"module":12803,"summary":12804},"\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":12806,"title":12807,"module":12803,"summary":12808},"\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":12810,"title":12811,"module":12803,"summary":12812},"\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":12814,"title":12815,"module":12816,"summary":12817},"\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":12819,"title":12820,"module":12816,"summary":12821},"\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":12823,"title":12824,"module":12816,"summary":12825},"\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":12827,"title":12828,"module":12829,"summary":12830},"\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":12832,"title":12833,"module":12829,"summary":12834},"\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":12836,"title":12837,"module":12829,"summary":12838},"\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":12840,"title":12841,"module":12842,"summary":12843},"\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":12845,"title":12846,"module":12842,"summary":12847},"\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":12849,"title":12850,"module":12842,"summary":12851},"\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":12853,"title":12854,"module":12842,"summary":12855},"\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":12857,"title":12858,"module":12859,"summary":12860},"\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":12862,"title":12863,"module":12859,"summary":12864},"\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":12866,"title":12867,"module":12859,"summary":12868},"\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":12870,"title":12871,"module":12872,"summary":12873},"\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":12875,"title":12876,"module":12872,"summary":12877},"\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":12879,"title":12880,"module":12872,"summary":12881},"\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":12883,"title":12884,"module":12872,"summary":12885},"\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":12887,"title":12888,"module":12872,"summary":12889},"\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":12891,"title":12892,"module":12893,"summary":12894},"\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":12896,"title":12897,"module":12893,"summary":12898},"\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":12900,"title":12901,"module":12893,"summary":12902},"\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":12904,"title":12905,"module":12893,"summary":12906},"\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":12908,"title":12909,"module":12893,"summary":12910},"\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":12912,"title":12913,"module":12914,"summary":12915},"\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":12917,"title":12914,"module":12914,"summary":12918},"\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":12920,"title":12921,"module":12914,"summary":12922},"\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":12924,"title":12925,"module":12914,"summary":12926},"\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":12928,"title":12929,"module":12914,"summary":12930},"\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":12932,"title":12933,"module":12934,"summary":12935},"\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":12937,"title":12938,"module":12934,"summary":12939},"\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":12941,"title":12942,"module":12934,"summary":12943},"\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":12945,"title":12946,"module":12934,"summary":12947},"\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":12949,"title":12950,"module":12934,"summary":12951},"\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":12953,"title":12954,"module":12934,"summary":12955},"\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":12957,"title":12958,"module":6,"summary":6},"\u002Fcalculus","Calculus",{"path":12960,"title":12961,"module":12405,"summary":12962},"\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":12964,"title":12965,"module":12405,"summary":12966},"\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":12968,"title":12969,"module":12970,"summary":12971},"\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":12973,"title":12974,"module":12970,"summary":12975},"\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":12977,"title":12978,"module":12970,"summary":12979},"\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":12981,"title":12982,"module":12970,"summary":12983},"\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":12985,"title":12986,"module":12970,"summary":12987},"\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":12989,"title":12990,"module":12991,"summary":12992},"\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":12994,"title":12995,"module":12991,"summary":12996},"\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":12998,"title":12999,"module":12991,"summary":13000},"\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":13002,"title":13003,"module":12991,"summary":13004},"\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":13006,"title":13007,"module":12991,"summary":13008},"\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":13010,"title":13011,"module":13012,"summary":13013},"\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":13015,"title":13016,"module":13012,"summary":13017},"\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":13019,"title":13020,"module":13012,"summary":13021},"\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":13023,"title":13024,"module":13012,"summary":13025},"\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":13027,"title":13028,"module":13012,"summary":13029},"\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":13031,"title":13032,"module":13033,"summary":13034},"\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":13036,"title":13037,"module":13033,"summary":13038},"\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":13040,"title":13041,"module":13033,"summary":13042},"\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":13044,"title":13045,"module":13046,"summary":13047},"\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":13049,"title":13050,"module":13046,"summary":13051},"\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":13053,"title":13054,"module":13046,"summary":13055},"\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":13057,"title":13058,"module":13046,"summary":13059},"\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":13061,"title":13062,"module":13046,"summary":13063},"\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":13065,"title":13066,"module":13046,"summary":13067},"\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":13069,"title":13070,"module":13071,"summary":13072},"\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":13074,"title":13075,"module":13071,"summary":13076},"\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":13078,"title":13079,"module":13071,"summary":13080},"\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":13082,"title":13083,"module":13071,"summary":13084},"\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":13086,"title":13087,"module":13071,"summary":13088},"\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":13090,"title":13091,"module":13071,"summary":13092},"\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":13094,"title":13095,"module":13071,"summary":13096},"\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":13098,"title":13099,"module":13100,"summary":13101},"\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":13103,"title":13104,"module":13100,"summary":13105},"\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":13107,"title":13108,"module":13100,"summary":13109},"\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":13111,"title":13112,"module":13100,"summary":13113},"\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":13115,"title":13116,"module":13100,"summary":13117},"\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":13119,"title":13120,"module":13100,"summary":13121},"\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":13123,"title":13124,"module":13100,"summary":13125},"\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":13127,"title":13128,"module":13100,"summary":13129},"\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":13131,"title":13132,"module":13100,"summary":13133},"\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":13135,"title":13136,"module":13100,"summary":13137},"\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":13139,"title":13140,"module":13100,"summary":13141},"\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":13143,"title":13144,"module":13100,"summary":13145},"\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":13147,"title":13148,"module":13149,"summary":13150},"\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":13152,"title":13153,"module":13149,"summary":13154},"\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":13156,"title":13157,"module":13149,"summary":13158},"\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":13160,"title":13161,"module":13149,"summary":13162},"\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":13164,"title":13165,"module":13149,"summary":13166},"\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":13168,"title":13169,"module":13149,"summary":13170},"\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":13172,"title":13173,"module":6,"summary":6},"\u002Fmechanics","Mechanics & Dynamics",{"path":13175,"title":13176,"module":13177,"summary":13178},"\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":13180,"title":13181,"module":13177,"summary":13182},"\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":13184,"title":13185,"module":13177,"summary":13186},"\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":13188,"title":13189,"module":13177,"summary":13190},"\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":13192,"title":13193,"module":13177,"summary":13194},"\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":13196,"title":13197,"module":13198,"summary":13199},"\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":13201,"title":13202,"module":13198,"summary":13203},"\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":13205,"title":13206,"module":13207,"summary":13208},"\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":13210,"title":13211,"module":13207,"summary":13212},"\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":13214,"title":13215,"module":13207,"summary":13216},"\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":13218,"title":13219,"module":13207,"summary":13220},"\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":13222,"title":13223,"module":13207,"summary":13224},"\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":13226,"title":13227,"module":13228,"summary":13229},"\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":13231,"title":13232,"module":13228,"summary":13233},"\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":13235,"title":13236,"module":13228,"summary":13237},"\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":13239,"title":13240,"module":13228,"summary":13241},"\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":13243,"title":13244,"module":13245,"summary":13246},"\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":13248,"title":13249,"module":13245,"summary":13250},"\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":13252,"title":13253,"module":13245,"summary":13254},"\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":13256,"title":13257,"module":13258,"summary":13259},"\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":13261,"title":13262,"module":13258,"summary":13263},"\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":13265,"title":13266,"module":13258,"summary":13267},"\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":13269,"title":13270,"module":13258,"summary":13271},"\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":13273,"title":13274,"module":13258,"summary":13275},"\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":13277,"title":13278,"module":13279,"summary":13280},"\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":13282,"title":13283,"module":13279,"summary":13284},"\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":13286,"title":13287,"module":13279,"summary":13288},"\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":13290,"title":13291,"module":13279,"summary":13292},"\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":13294,"title":13295,"module":13279,"summary":13296},"\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":13298,"title":13299,"module":13279,"summary":13300},"\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":13302,"title":13303,"module":13304,"summary":13305},"\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":13307,"title":13308,"module":13304,"summary":13309},"\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":13311,"title":13312,"module":13304,"summary":13313},"\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":13315,"title":13316,"module":13304,"summary":13317},"\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":13319,"title":13320,"module":13304,"summary":13321},"\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":13323,"title":13324,"module":13304,"summary":13325},"\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":13327,"title":13328,"module":13304,"summary":13329},"\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":13331,"title":13332,"module":13304,"summary":13333},"\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":13335,"title":13336,"module":13337,"summary":13338},"\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":13340,"title":13341,"module":13337,"summary":13342},"\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":13344,"title":13345,"module":13337,"summary":13346},"\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":13348,"title":13349,"module":13337,"summary":13350},"\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":13352,"title":13353,"module":13337,"summary":13354},"\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":13356,"title":13357,"module":13358,"summary":13359},"\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":13361,"title":13362,"module":13358,"summary":13363},"\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":13365,"title":13366,"module":13358,"summary":13367},"\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":13369,"title":13370,"module":13358,"summary":13371},"\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":13373,"title":13374,"module":13358,"summary":13375},"\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":13377,"title":13378,"module":13379,"summary":13380},"\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":13382,"title":13383,"module":13379,"summary":13384},"\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":13386,"title":13387,"module":13379,"summary":13388},"\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":13390,"title":13391,"module":6,"summary":6},"\u002Felectricity-and-magnetism","Electricity & Magnetism",{"path":13393,"title":13394,"module":13395,"summary":13396},"\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":13398,"title":13399,"module":13395,"summary":13400},"\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":13402,"title":13403,"module":13395,"summary":13404},"\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":13406,"title":13407,"module":13395,"summary":13408},"\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":13410,"title":13411,"module":13395,"summary":13412},"\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":13414,"title":13415,"module":13416,"summary":13417},"\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":13419,"title":13420,"module":13416,"summary":13421},"\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":13423,"title":13424,"module":13416,"summary":13425},"\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":13427,"title":13428,"module":13416,"summary":13429},"\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":13431,"title":13432,"module":13416,"summary":13433},"\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":13435,"title":13436,"module":13437,"summary":13438},"\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":13440,"title":13441,"module":13437,"summary":13442},"\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":13444,"title":13445,"module":13437,"summary":13446},"\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":13448,"title":13449,"module":13450,"summary":13451},"\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":13453,"title":13454,"module":13450,"summary":13455},"\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":13457,"title":13458,"module":13450,"summary":13459},"\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":13461,"title":13462,"module":13450,"summary":13463},"\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":13465,"title":13466,"module":13450,"summary":13467},"\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":13469,"title":13470,"module":13450,"summary":13471},"\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":13473,"title":13474,"module":13450,"summary":13475},"\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":13477,"title":13478,"module":13479,"summary":13480},"\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":13482,"title":13483,"module":13479,"summary":13484},"\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":13486,"title":13487,"module":13479,"summary":13488},"\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":13490,"title":13491,"module":13479,"summary":13492},"\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":13494,"title":13495,"module":13479,"summary":13496},"\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":13498,"title":13499,"module":13479,"summary":13500},"\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":13502,"title":13503,"module":13479,"summary":13504},"\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":13506,"title":13507,"module":13508,"summary":13509},"\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":13511,"title":13512,"module":13508,"summary":13513},"\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":13515,"title":13516,"module":13508,"summary":13517},"\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":13519,"title":13520,"module":13508,"summary":13521},"\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":13523,"title":13524,"module":13508,"summary":13525},"\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":13527,"title":13528,"module":13508,"summary":13529},"\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":13531,"title":13532,"module":13533,"summary":13534},"\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":13536,"title":13537,"module":13533,"summary":13538},"\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":13540,"title":13541,"module":13533,"summary":13542},"\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":13544,"title":13545,"module":13533,"summary":13546},"\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":13548,"title":13549,"module":13533,"summary":13550},"\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":13552,"title":13553,"module":13554,"summary":13555},"\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":13557,"title":13558,"module":13554,"summary":13559},"\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":13561,"title":13562,"module":13554,"summary":13563},"\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":13565,"title":13566,"module":13554,"summary":13567},"\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":13569,"title":13570,"module":13554,"summary":13571},"\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":13573,"title":13574,"module":13554,"summary":13575},"\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":13577,"title":13578,"module":13579,"summary":13580},"\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":13582,"title":13583,"module":13579,"summary":13584},"\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":13586,"title":13587,"module":13579,"summary":13588},"\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":13590,"title":13591,"module":13579,"summary":13592},"\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":13594,"title":13595,"module":13579,"summary":13596},"\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":13598,"title":13599,"module":6,"summary":6},"\u002Flinear-algebra","Linear Algebra",{"path":13601,"title":13602,"module":6,"summary":6},"\u002Ftheory-of-computation","Theory of Computation",{"path":13604,"title":13605,"module":12405,"summary":13606},"\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":13608,"title":13609,"module":12405,"summary":13610},"\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":13612,"title":13613,"module":12405,"summary":13614},"\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":13616,"title":13617,"module":12405,"summary":13618},"\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":13620,"title":13621,"module":12405,"summary":13622},"\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":13624,"title":13625,"module":13626,"summary":13627},"\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":13629,"title":13630,"module":13626,"summary":13631},"\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":13633,"title":13634,"module":13626,"summary":13635},"\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":13637,"title":13638,"module":13626,"summary":13639},"\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":13641,"title":13642,"module":13626,"summary":13643},"\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":13645,"title":13646,"module":13626,"summary":13647},"\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":13649,"title":13650,"module":13626,"summary":13651},"\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":13653,"title":13654,"module":13655,"summary":13656},"\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":13658,"title":13659,"module":13655,"summary":13660},"\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":13662,"title":13663,"module":13655,"summary":13664},"\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":13666,"title":13667,"module":13655,"summary":13668},"\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":13670,"title":13671,"module":13655,"summary":13672},"\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":13674,"title":13675,"module":13676,"summary":13677},"\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":13679,"title":13680,"module":13676,"summary":13681},"\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":13683,"title":13684,"module":13676,"summary":13685},"\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":13687,"title":13688,"module":13676,"summary":13689},"\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":13691,"title":13692,"module":13676,"summary":13693},"\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":13695,"title":13696,"module":13697,"summary":13698},"\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":13700,"title":13701,"module":13697,"summary":13702},"\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":13704,"title":13705,"module":13697,"summary":13706},"\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":13708,"title":13709,"module":13697,"summary":13710},"\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":13712,"title":13713,"module":13697,"summary":13714},"\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":13716,"title":13717,"module":13718,"summary":13719},"\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":13721,"title":13722,"module":13718,"summary":13723},"\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":13725,"title":13726,"module":13718,"summary":13727},"\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":13729,"title":13730,"module":13718,"summary":13731},"\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":13733,"title":13734,"module":13718,"summary":13735},"\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":13737,"title":13738,"module":13739,"summary":13740},"\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":13742,"title":13743,"module":13739,"summary":13744},"\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":13746,"title":13747,"module":13739,"summary":13748},"\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":13750,"title":13751,"module":13739,"summary":13752},"\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":13754,"title":13755,"module":13739,"summary":13756},"\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":13758,"title":13759,"module":13760,"summary":13761},"\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":13763,"title":13764,"module":13760,"summary":13765},"\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":13767,"title":13768,"module":13760,"summary":13769},"\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":13771,"title":13772,"module":13773,"summary":13774},"\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":13776,"title":13777,"module":13773,"summary":13778},"\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":13780,"title":13781,"module":13782,"summary":13783},"\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":13785,"title":13786,"module":13782,"summary":13787},"\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":13789,"title":13790,"module":13782,"summary":13791},"\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":13793,"title":13794,"module":13782,"summary":13795},"\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":13797,"title":13798,"module":13782,"summary":13799},"\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":13801,"title":13802,"module":13803,"summary":13804},"\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":13806,"title":13807,"module":13803,"summary":13808},"\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":13810,"title":13811,"module":6,"summary":6},"\u002Fcomputer-architecture","Computer Architecture",{"path":13813,"title":13814,"module":12405,"summary":13815},"\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":13817,"title":13818,"module":12405,"summary":13819},"\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":13821,"title":13822,"module":13823,"summary":13824},"\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":13826,"title":13827,"module":13823,"summary":13828},"\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":13830,"title":13831,"module":13823,"summary":13832},"\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":13834,"title":13835,"module":13823,"summary":13836},"\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":13838,"title":13839,"module":13823,"summary":13840},"\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":13842,"title":13843,"module":13823,"summary":13844},"\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":13846,"title":13847,"module":13848,"summary":13849},"\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":13851,"title":13852,"module":13848,"summary":13853},"\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":13855,"title":13856,"module":13848,"summary":13857},"\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":13859,"title":13860,"module":13848,"summary":13861},"\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":13863,"title":13864,"module":13848,"summary":13865},"\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":13867,"title":13868,"module":13848,"summary":13869},"\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":13871,"title":13872,"module":13873,"summary":13874},"\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":13876,"title":13877,"module":13873,"summary":13878},"\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":13880,"title":13881,"module":13873,"summary":13882},"\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":13884,"title":13885,"module":13886,"summary":13887},"\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":13889,"title":13890,"module":13886,"summary":13891},"\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":13893,"title":13894,"module":13895,"summary":13896},"\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":13898,"title":13899,"module":13895,"summary":13900},"\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":13902,"title":13903,"module":13895,"summary":13904},"\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":13906,"title":13907,"module":13908,"summary":13909},"\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":13911,"title":13912,"module":13908,"summary":13913},"\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":13915,"title":13916,"module":13917,"summary":13918},"\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":13920,"title":13921,"module":13917,"summary":13922},"\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":13924,"title":13925,"module":13917,"summary":13926},"\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":13928,"title":13929,"module":13930,"summary":13931},"\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":13933,"title":13934,"module":13930,"summary":13935},"\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":13937,"title":13938,"module":13930,"summary":13939},"\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":13941,"title":13942,"module":13943,"summary":13944},"\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":13946,"title":13947,"module":13943,"summary":13948},"\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":13950,"title":13951,"module":6,"summary":6},"\u002Fdifferential-equations","Differential Equations",{"path":13953,"title":13954,"module":13955,"summary":13956},"\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":13958,"title":13959,"module":13955,"summary":13960},"\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":13962,"title":13963,"module":13955,"summary":13964},"\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":13966,"title":13967,"module":13955,"summary":13968},"\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":13970,"title":13971,"module":13955,"summary":13972},"\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":13974,"title":13975,"module":13976,"summary":13977},"\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":13979,"title":13980,"module":13976,"summary":13981},"\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":13983,"title":13984,"module":13976,"summary":13985},"\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":13987,"title":13988,"module":13976,"summary":13989},"\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":13991,"title":13992,"module":13993,"summary":13994},"\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":13996,"title":13997,"module":13993,"summary":13998},"\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":14000,"title":14001,"module":13993,"summary":14002},"\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":14004,"title":14005,"module":13993,"summary":14006},"\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":14008,"title":14009,"module":14010,"summary":14011},"\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":14013,"title":14014,"module":14010,"summary":14015},"\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":14017,"title":14018,"module":14010,"summary":14019},"\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":14021,"title":14022,"module":14010,"summary":14023},"\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":14025,"title":14026,"module":14027,"summary":14028},"\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":14030,"title":14031,"module":14027,"summary":14032},"\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":14034,"title":14035,"module":14027,"summary":14036},"\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":14038,"title":14039,"module":14027,"summary":14040},"\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":14042,"title":14043,"module":14027,"summary":14044},"\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":14046,"title":14047,"module":14027,"summary":14048},"\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":14050,"title":14051,"module":14052,"summary":14053},"\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":14055,"title":14056,"module":14052,"summary":14057},"\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":14059,"title":14060,"module":14052,"summary":14061},"\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":14063,"title":14064,"module":14065,"summary":14066},"\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":14068,"title":14069,"module":14065,"summary":14070},"\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":14072,"title":14073,"module":14065,"summary":14074},"\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":14076,"title":14077,"module":14065,"summary":14078},"\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":14080,"title":14081,"module":14082,"summary":14083},"\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":14085,"title":14086,"module":14082,"summary":14087},"\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":14089,"title":14090,"module":14082,"summary":14091},"\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":14093,"title":14094,"module":14095,"summary":14096},"\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":14098,"title":14099,"module":14095,"summary":14100},"\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":14102,"title":14103,"module":14095,"summary":14104},"\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":14106,"title":14107,"module":14108,"summary":14109},"\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":14111,"title":14112,"module":14108,"summary":14113},"\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":14115,"title":14116,"module":6,"summary":6},"\u002Frelativity","Relativity",{"path":14118,"title":14119,"module":6,"summary":6},"\u002Fphysical-computing","Physical Computing",{"path":14121,"title":14122,"module":14123,"summary":14124},"\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":14126,"title":14127,"module":14123,"summary":14128},"\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":14130,"title":14131,"module":14123,"summary":14132},"\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":14134,"title":14135,"module":14123,"summary":14136},"\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":14138,"title":14139,"module":14140,"summary":14141},"\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":14143,"title":14144,"module":14140,"summary":14145},"\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":14147,"title":14148,"module":14140,"summary":14149},"\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":14151,"title":14152,"module":14153,"summary":14154},"\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":14156,"title":14157,"module":14153,"summary":14158},"\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":14160,"title":14161,"module":14153,"summary":14162},"\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":14164,"title":14165,"module":14153,"summary":14166},"\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":14168,"title":14169,"module":14153,"summary":14170},"\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":14172,"title":14173,"module":14153,"summary":14174},"\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":14176,"title":14177,"module":14178,"summary":14179},"\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":14181,"title":14182,"module":14178,"summary":14183},"\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":14185,"title":14186,"module":14178,"summary":14187},"\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":14189,"title":14190,"module":14178,"summary":14191},"\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":14193,"title":14194,"module":14178,"summary":14195},"\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":14197,"title":14198,"module":14178,"summary":14199},"\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":14201,"title":14202,"module":14203,"summary":14204},"\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":14206,"title":14207,"module":14203,"summary":14208},"\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":14210,"title":14211,"module":14203,"summary":14212},"\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":14214,"title":14215,"module":14203,"summary":14216},"\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":14218,"title":14219,"module":13058,"summary":14220},"\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":14222,"title":14223,"module":13058,"summary":14224},"\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":14226,"title":14227,"module":13058,"summary":14228},"\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":14230,"title":14231,"module":14232,"summary":14233},"\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":14235,"title":14236,"module":14232,"summary":14237},"\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":14239,"title":14240,"module":14232,"summary":14241},"\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":14243,"title":14244,"module":14245,"summary":14246},"\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":14248,"title":14249,"module":14245,"summary":14250},"\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":14252,"title":14253,"module":14245,"summary":14254},"\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":14256,"title":14257,"module":14258,"summary":14259},"\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":14261,"title":14262,"module":14258,"summary":14263},"\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":14265,"title":14266,"module":14267,"summary":14268},"\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":14270,"title":14271,"module":14267,"summary":14272},"\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":14274,"title":14275,"module":14267,"summary":14276},"\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":14278,"title":14279,"module":14267,"summary":14280},"\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":14282,"title":14283,"module":14267,"summary":14284},"\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":14286,"title":14287,"module":6,"summary":6},"\u002Fquantum-mechanics","Quantum Mechanics",{"path":14289,"title":14290,"module":14291,"summary":14292},"\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":14294,"title":14295,"module":14291,"summary":14296},"\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":14298,"title":14299,"module":14291,"summary":14300},"\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":14302,"title":14303,"module":14291,"summary":14304},"\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":14306,"title":14307,"module":14308,"summary":14309},"\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":14311,"title":14312,"module":14308,"summary":14313},"\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":14315,"title":14316,"module":14308,"summary":14317},"\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":14319,"title":14320,"module":14308,"summary":14321},"\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":14323,"title":14324,"module":14308,"summary":14325},"\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":14327,"title":14328,"module":14308,"summary":14329},"\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":14331,"title":14332,"module":14333,"summary":14334},"\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":14336,"title":14337,"module":14333,"summary":14338},"\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":14340,"title":14341,"module":14333,"summary":14342},"\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":14344,"title":14345,"module":14333,"summary":14346},"\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":14348,"title":14349,"module":14333,"summary":14350},"\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":14352,"title":14353,"module":12752,"summary":14354},"\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":14356,"title":14357,"module":12752,"summary":14358},"\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":14360,"title":14361,"module":12752,"summary":14362},"\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":14364,"title":14365,"module":12752,"summary":14366},"\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":14368,"title":14369,"module":12752,"summary":14370},"\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":14372,"title":14373,"module":12752,"summary":14374},"\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":14376,"title":14377,"module":14378,"summary":14379},"\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":14381,"title":14382,"module":14378,"summary":14383},"\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":14385,"title":14386,"module":14378,"summary":14387},"\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":14389,"title":14390,"module":14378,"summary":14391},"\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":14393,"title":14394,"module":14395,"summary":14396},"\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":14398,"title":14399,"module":14395,"summary":14400},"\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":14402,"title":14403,"module":14395,"summary":14404},"\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":14406,"title":12807,"module":14395,"summary":14407},"\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":14409,"title":14410,"module":14395,"summary":14411},"\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":14413,"title":14414,"module":14415,"summary":14416},"\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":14418,"title":14419,"module":14415,"summary":14420},"\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":14422,"title":14423,"module":14415,"summary":14424},"\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":14426,"title":14427,"module":14415,"summary":14428},"\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":14430,"title":14431,"module":14432,"summary":14433},"\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":14435,"title":14436,"module":14432,"summary":14437},"\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":14439,"title":14440,"module":14432,"summary":14441},"\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":14443,"title":14444,"module":14432,"summary":14445},"\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":14447,"title":14448,"module":14432,"summary":14449},"\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":14451,"title":14452,"module":6,"summary":6},"\u002Freal-analysis","Real Analysis",{"path":14454,"title":14455,"module":12405,"summary":14456},"\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":14458,"title":14459,"module":12405,"summary":14460},"\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":14462,"title":14463,"module":14464,"summary":14465},"\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":14467,"title":14468,"module":14464,"summary":14469},"\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":14471,"title":14472,"module":14464,"summary":14473},"\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":14475,"title":14476,"module":14464,"summary":14477},"\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":14479,"title":14480,"module":14481,"summary":14482},"\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":14484,"title":14485,"module":14481,"summary":14486},"\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":14488,"title":14489,"module":14481,"summary":14490},"\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":14492,"title":14493,"module":14481,"summary":14494},"\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":14496,"title":14497,"module":14481,"summary":14498},"\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":14500,"title":14501,"module":14481,"summary":14502},"\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":14504,"title":14505,"module":14506,"summary":14507},"\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":14509,"title":14510,"module":14506,"summary":14511},"\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":14513,"title":14514,"module":14506,"summary":14515},"\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":14517,"title":14518,"module":14506,"summary":14519},"\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":14521,"title":14522,"module":14523,"summary":14524},"\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":14526,"title":14527,"module":14523,"summary":14528},"\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":14530,"title":14531,"module":14523,"summary":14532},"\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":14534,"title":14535,"module":14523,"summary":14536},"\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":14538,"title":14539,"module":14540,"summary":14541},"\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":14543,"title":14544,"module":14540,"summary":14545},"\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":14547,"title":14548,"module":14540,"summary":14549},"\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":14551,"title":14552,"module":14553,"summary":14554},"\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":14556,"title":14557,"module":14553,"summary":14558},"\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":14560,"title":14561,"module":14553,"summary":14562},"\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":14564,"title":14565,"module":14553,"summary":14566},"\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":14568,"title":14569,"module":14570,"summary":14571},"\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":14573,"title":14574,"module":14570,"summary":14575},"\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":14577,"title":14578,"module":14570,"summary":14579},"\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":14581,"title":14582,"module":14570,"summary":14583},"\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":14585,"title":14586,"module":14587,"summary":14588},"\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":14590,"title":14591,"module":14587,"summary":14592},"\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":14594,"title":14595,"module":14587,"summary":14596},"\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":14598,"title":14599,"module":14600,"summary":14601},"\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":14603,"title":14604,"module":14600,"summary":14605},"\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":14607,"title":14608,"module":14600,"summary":14609},"\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":14611,"title":14612,"module":14600,"summary":14613},"\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":14615,"title":14616,"module":14617,"summary":14618},"\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":14620,"title":14621,"module":14617,"summary":14622},"\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":14624,"title":14625,"module":14617,"summary":14626},"\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":14628,"title":14629,"module":14617,"summary":14630},"\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":14632,"title":14633,"module":14617,"summary":14634},"\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":14636,"title":14637,"module":14638,"summary":14639},"\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":14641,"title":14642,"module":14638,"summary":14643},"\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":14645,"title":14646,"module":6,"summary":6},"\u002Fabstract-algebra","Abstract Algebra",{"path":14648,"title":14649,"module":14650,"summary":14651},"\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":14653,"title":14654,"module":14650,"summary":14655},"\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":14657,"title":14658,"module":14650,"summary":14659},"\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":14661,"title":14662,"module":14650,"summary":14663},"\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":14665,"title":14666,"module":14650,"summary":14667},"\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":14669,"title":14670,"module":14671,"summary":14672},"\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":14674,"title":14675,"module":14671,"summary":14676},"\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":14678,"title":14679,"module":14671,"summary":14680},"\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":14682,"title":14683,"module":14671,"summary":14684},"\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":14686,"title":14687,"module":14671,"summary":14688},"\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":14690,"title":14691,"module":14671,"summary":14692},"\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":14694,"title":14695,"module":14671,"summary":14696},"\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":14698,"title":14699,"module":14700,"summary":14701},"\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":14703,"title":14704,"module":14700,"summary":14705},"\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":14707,"title":14708,"module":14700,"summary":14709},"\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":14711,"title":14712,"module":14700,"summary":14713},"\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":14715,"title":14716,"module":14717,"summary":14718},"\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":14720,"title":14721,"module":14717,"summary":14722},"\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":14724,"title":14725,"module":14717,"summary":14726},"\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":14728,"title":14729,"module":14730,"summary":14731},"\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":14733,"title":14734,"module":14730,"summary":14735},"\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":14737,"title":14738,"module":14730,"summary":14739},"\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":14741,"title":14742,"module":14730,"summary":14743},"\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":14745,"title":14746,"module":14730,"summary":14747},"\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":14749,"title":14750,"module":14730,"summary":14751},"\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":14753,"title":14754,"module":14755,"summary":14756},"\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":14758,"title":14759,"module":14755,"summary":14760},"\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":14762,"title":14763,"module":14755,"summary":14764},"\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":14766,"title":14767,"module":14768,"summary":14769},"\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":14771,"title":14772,"module":14768,"summary":14773},"\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":14775,"title":14776,"module":14768,"summary":14777},"\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":14779,"title":14780,"module":14768,"summary":14781},"\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":14783,"title":14784,"module":14785,"summary":14786},"\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":14788,"title":14789,"module":14785,"summary":14790},"\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":14792,"title":14793,"module":14785,"summary":14794},"\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":14796,"title":14797,"module":14798,"summary":14799},"\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":14801,"title":14802,"module":14798,"summary":14803},"\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":14805,"title":14806,"module":14798,"summary":14807},"\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":14809,"title":14810,"module":14798,"summary":14811},"\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":14813,"title":14814,"module":6,"summary":6},"\u002Fatomic-physics","Atomic Physics",{"path":14816,"title":14817,"module":6,"summary":6},"\u002Fdatabases","Databases",{"path":14819,"title":14820,"module":12405,"summary":14821},"\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":14823,"title":14824,"module":12405,"summary":14825},"\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":14827,"title":14828,"module":12405,"summary":14829},"\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":14831,"title":14832,"module":12405,"summary":14833},"\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":14835,"title":14836,"module":12405,"summary":14837},"\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":14839,"title":14840,"module":12405,"summary":14841},"\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":14843,"title":14844,"module":14845,"summary":14846},"\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":14848,"title":14849,"module":14845,"summary":14850},"\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":14852,"title":14853,"module":14845,"summary":14854},"\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":14856,"title":14857,"module":14858,"summary":14859},"\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":14861,"title":14862,"module":14858,"summary":14863},"\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":14865,"title":14866,"module":14858,"summary":14867},"\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":14869,"title":14870,"module":14871,"summary":14872},"\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":14874,"title":14875,"module":14871,"summary":14876},"\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":14878,"title":14879,"module":14871,"summary":14880},"\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":14882,"title":14883,"module":14871,"summary":14884},"\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":14886,"title":14887,"module":14871,"summary":14888},"\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":14890,"title":14891,"module":14892,"summary":14893},"\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":14895,"title":14896,"module":14892,"summary":14897},"\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":14899,"title":14900,"module":14892,"summary":14901},"\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":14903,"title":14904,"module":14892,"summary":14905},"\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":14907,"title":14908,"module":14909,"summary":14910},"\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":14912,"title":14913,"module":14909,"summary":14914},"\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":14916,"title":14917,"module":14909,"summary":14918},"\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":14920,"title":14921,"module":14909,"summary":14922},"\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":14924,"title":14925,"module":14926,"summary":14927},"\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":14929,"title":14930,"module":14926,"summary":14931},"\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":14933,"title":14934,"module":14926,"summary":14935},"\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":14937,"title":14938,"module":14926,"summary":14939},"\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":14941,"title":14942,"module":14943,"summary":14944},"\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":14946,"title":14947,"module":14943,"summary":14948},"\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":14950,"title":14951,"module":14943,"summary":14952},"\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":14954,"title":14955,"module":6,"summary":6},"\u002Fcategory-theory","Category Theory",{"path":14957,"title":13599,"module":14958,"summary":14959},"\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":14961,"title":14962,"module":14958,"summary":14963},"\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":14965,"title":14966,"module":14958,"summary":14967},"\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":14969,"title":12958,"module":14958,"summary":14970},"\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":14972,"title":14973,"module":12405,"summary":14974},"\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":14976,"title":14977,"module":12405,"summary":14978},"\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":14980,"title":14981,"module":12405,"summary":14982},"\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":14984,"title":14985,"module":14986,"summary":14987},"\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":14989,"title":14990,"module":14986,"summary":14991},"\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":14993,"title":14994,"module":14986,"summary":14995},"\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":14997,"title":14998,"module":14986,"summary":14999},"\u002Fdeep-learning\u002Fneural-networks\u002Fbackpropagation","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":15001,"title":15002,"module":14986,"summary":15003},"\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":15005,"title":15006,"module":15007,"summary":15008},"\u002Fdeep-learning\u002Foptimization\u002Fgradient-descent-and-sgd","Gradient Descent & SGD","Optimization","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":15010,"title":15011,"module":15007,"summary":15012},"\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods","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":15014,"title":15015,"module":15007,"summary":15016},"\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":15018,"title":15019,"module":15007,"summary":15020},"\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape","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":15022,"title":15023,"module":15007,"summary":15024},"\u002Fdeep-learning\u002Foptimization\u002Fsecond-order-and-approximate-methods","Second-Order & 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",{"path":15026,"title":15027,"module":15028,"summary":15029},"\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":15031,"title":15032,"module":15028,"summary":15033},"\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":15035,"title":15036,"module":15028,"summary":15037},"\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":15039,"title":15040,"module":15028,"summary":15041},"\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":15043,"title":15044,"module":15045,"summary":15046},"\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":15048,"title":15049,"module":15045,"summary":15050},"\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":15052,"title":15053,"module":15045,"summary":15054},"\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":15056,"title":15057,"module":15045,"summary":15058},"\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":15060,"title":15061,"module":15045,"summary":15062},"\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":15064,"title":15065,"module":15045,"summary":15066},"\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":15068,"title":15069,"module":15045,"summary":15070},"\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":15072,"title":15073,"module":15045,"summary":15074},"\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":15076,"title":15077,"module":15045,"summary":15078},"\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":15080,"title":15081,"module":15082,"summary":15083},"\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":15085,"title":15086,"module":15082,"summary":15087},"\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":15089,"title":15090,"module":15082,"summary":15091},"\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":15093,"title":15094,"module":15082,"summary":15095},"\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":15097,"title":15098,"module":15082,"summary":15099},"\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":15101,"title":15102,"module":15103,"summary":15104},"\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":15106,"title":15107,"module":15103,"summary":15108},"\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":15110,"title":15111,"module":15103,"summary":15112},"\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":15114,"title":15115,"module":15103,"summary":15116},"\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":15118,"title":15119,"module":15103,"summary":15120},"\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":15122,"title":15123,"module":15103,"summary":15124},"\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":15126,"title":15127,"module":15103,"summary":15128},"\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":15130,"title":15131,"module":15132,"summary":15133},"\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":15135,"title":15136,"module":15132,"summary":15137},"\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":15139,"title":15140,"module":15132,"summary":15141},"\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":15143,"title":15144,"module":15145,"summary":15146},"\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":15148,"title":15149,"module":15145,"summary":15150},"\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":15152,"title":15153,"module":15145,"summary":15154},"\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":15156,"title":15157,"module":15145,"summary":15158},"\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":15160,"title":15161,"module":15145,"summary":15162},"\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":15164,"title":15165,"module":15145,"summary":15166},"\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":15168,"title":15169,"module":15145,"summary":15170},"\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":15172,"title":15173,"module":15174,"summary":15175},"\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":15177,"title":15178,"module":15174,"summary":15179},"\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":15181,"title":15182,"module":15174,"summary":15183},"\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":15185,"title":15186,"module":15174,"summary":15187},"\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":15189,"title":15190,"module":15174,"summary":15191},"\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":15193,"title":15194,"module":15174,"summary":15195},"\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":15197,"title":15198,"module":15174,"summary":15199},"\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":15201,"title":15202,"module":15174,"summary":15203},"\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":15205,"title":15206,"module":15174,"summary":15207},"\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":15209,"title":15210,"module":15174,"summary":15211},"\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":15213,"title":15214,"module":15174,"summary":15215},"\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":15217,"title":15218,"module":15219,"summary":15220},"\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":15222,"title":15223,"module":15219,"summary":15224},"\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":15226,"title":15227,"module":15219,"summary":15228},"\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":15230,"title":15231,"module":15219,"summary":15232},"\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":15234,"title":15235,"module":15219,"summary":15236},"\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":15238,"title":15239,"module":6,"summary":6},"\u002Fdeep-learning","Deep Learning",{"path":15241,"title":15242,"module":13149,"summary":15243},"\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":15245,"title":15246,"module":13149,"summary":15247},"\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":15249,"title":15250,"module":13149,"summary":15251},"\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":15253,"title":15254,"module":13149,"summary":15255},"\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":15257,"title":15258,"module":13149,"summary":15259},"\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":15261,"title":15262,"module":15263,"summary":15264},"\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":15266,"title":15267,"module":15263,"summary":15268},"\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":15270,"title":15271,"module":15263,"summary":15272},"\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":15274,"title":15275,"module":15263,"summary":15276},"\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":15278,"title":15279,"module":15280,"summary":15281},"\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":15283,"title":15284,"module":15280,"summary":15285},"\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":15287,"title":15288,"module":15280,"summary":15289},"\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":15291,"title":15292,"module":15280,"summary":15293},"\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":15295,"title":15296,"module":15297,"summary":15298},"\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":15300,"title":15301,"module":15297,"summary":15302},"\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":15304,"title":15305,"module":15297,"summary":15306},"\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":15308,"title":15309,"module":15297,"summary":15310},"\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":15312,"title":15313,"module":15297,"summary":15314},"\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":15316,"title":15317,"module":15318,"summary":15319},"\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":15321,"title":15322,"module":15318,"summary":15323},"\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":15325,"title":15326,"module":15318,"summary":15327},"\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":15329,"title":15330,"module":15331,"summary":15332},"\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":15334,"title":15335,"module":15331,"summary":15336},"\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":15338,"title":15339,"module":15331,"summary":15340},"\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":15342,"title":15343,"module":15344,"summary":15345},"\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":15347,"title":15348,"module":15344,"summary":15349},"\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":15351,"title":15352,"module":15344,"summary":15353},"\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":15355,"title":15356,"module":15344,"summary":15357},"\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":15359,"title":15360,"module":15361,"summary":15362},"\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":15364,"title":15365,"module":15361,"summary":15366},"\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":15368,"title":15369,"module":15361,"summary":15370},"\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":15372,"title":15373,"module":15361,"summary":15374},"\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":15376,"title":15377,"module":15361,"summary":15378},"\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":15380,"title":15381,"module":15361,"summary":15382},"\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":15384,"title":15385,"module":15386,"summary":15387},"\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":15389,"title":15390,"module":15386,"summary":15391},"\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":15393,"title":15394,"module":15386,"summary":15395},"\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":15397,"title":15398,"module":15386,"summary":15399},"\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":15401,"title":15402,"module":15403,"summary":15404},"\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":15406,"title":15407,"module":15403,"summary":15408},"\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":15410,"title":15411,"module":15403,"summary":15412},"\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":15414,"title":15415,"module":15416,"summary":15417},"\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":15419,"title":15420,"module":15416,"summary":15421},"\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":15423,"title":15424,"module":15416,"summary":15425},"\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":15427,"title":15428,"module":15416,"summary":15429},"\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":15431,"title":15432,"module":15416,"summary":15433},"\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":15435,"title":15436,"module":15437,"summary":15438},"\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":15440,"title":15441,"module":15437,"summary":15442},"\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":15444,"title":15445,"module":15437,"summary":15446},"\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":15448,"title":15449,"module":6,"summary":6},"\u002Fstatistical-mechanics","Statistical Mechanics",{"path":15451,"title":15452,"module":15453,"summary":15454},"\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":15456,"title":15457,"module":15453,"summary":15458},"\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":15460,"title":15461,"module":15453,"summary":15462},"\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":15464,"title":15465,"module":15453,"summary":15466},"\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":15468,"title":15469,"module":15470,"summary":15471},"\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":15473,"title":15474,"module":15470,"summary":15475},"\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":15477,"title":15478,"module":15470,"summary":15479},"\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":15481,"title":15482,"module":15470,"summary":15483},"\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":15485,"title":15486,"module":15487,"summary":15488},"\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":15490,"title":15491,"module":15487,"summary":15492},"\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":15494,"title":15495,"module":15487,"summary":15496},"\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":15498,"title":15499,"module":15487,"summary":15500},"\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":15502,"title":15503,"module":15504,"summary":15505},"\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":15507,"title":15508,"module":15504,"summary":15509},"\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":15511,"title":15512,"module":15504,"summary":15513},"\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":15515,"title":15516,"module":15504,"summary":15517},"\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":15519,"title":15520,"module":15521,"summary":15522},"\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":15524,"title":15525,"module":15521,"summary":15526},"\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":15528,"title":15529,"module":15521,"summary":15530},"\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":15532,"title":15533,"module":15521,"summary":15534},"\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":15536,"title":15537,"module":15538,"summary":15539},"\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":15541,"title":15542,"module":15538,"summary":15543},"\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":15545,"title":15546,"module":15538,"summary":15547},"\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":15549,"title":15550,"module":15538,"summary":15551},"\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":15553,"title":15554,"module":15555,"summary":15556},"\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":15558,"title":15559,"module":15555,"summary":15560},"\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":15562,"title":15563,"module":15555,"summary":15564},"\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":15566,"title":15567,"module":15555,"summary":15568},"\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":15570,"title":15571,"module":15555,"summary":15572},"\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":15574,"title":15575,"module":15576,"summary":15577},"\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":15579,"title":15580,"module":15576,"summary":15581},"\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":15583,"title":15584,"module":15585,"summary":15586},"\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":15588,"title":15589,"module":15585,"summary":15590},"\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":15592,"title":15593,"module":15585,"summary":15594},"\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":15596,"title":15597,"module":15585,"summary":15598},"\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":15600,"title":15601,"module":15602,"summary":15603},"\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":15605,"title":15606,"module":15602,"summary":15607},"\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":15609,"title":15610,"module":15602,"summary":15611},"\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":15613,"title":15614,"module":15602,"summary":15615},"\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":15617,"title":15618,"module":15602,"summary":15619},"\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":15621,"title":15622,"module":15623,"summary":15624},"\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":15626,"title":15627,"module":15623,"summary":15628},"\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":15630,"title":15631,"module":15623,"summary":15632},"\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":15634,"title":15635,"module":15623,"summary":15636},"\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":15638,"title":15639,"module":6,"summary":6},"\u002Fcondensed-matter","Condensed Matter Physics",{"path":15641,"title":15642,"module":12405,"summary":15643},"\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":15645,"title":15646,"module":15647,"summary":15648},"\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":15650,"title":15651,"module":15647,"summary":15652},"\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":15654,"title":15655,"module":15647,"summary":15656},"\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":15658,"title":15659,"module":15647,"summary":15660},"\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":15662,"title":15663,"module":15647,"summary":15664},"\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":15666,"title":15667,"module":15647,"summary":15668},"\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":15670,"title":15671,"module":15647,"summary":15672},"\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":15674,"title":15675,"module":15676,"summary":15677},"\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":15679,"title":15680,"module":15676,"summary":15681},"\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":15683,"title":15684,"module":15676,"summary":15685},"\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":15687,"title":15688,"module":15676,"summary":15689},"\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":15691,"title":15692,"module":15693,"summary":15694},"\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":15696,"title":15697,"module":15693,"summary":15698},"\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":15700,"title":15701,"module":15693,"summary":15702},"\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":15704,"title":15705,"module":15693,"summary":15706},"\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":15708,"title":15709,"module":15710,"summary":15711},"\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":15713,"title":15714,"module":15710,"summary":15715},"\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":15717,"title":15718,"module":15710,"summary":15719},"\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":15721,"title":15722,"module":15710,"summary":15723},"\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":15725,"title":15726,"module":15727,"summary":15728},"\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":15730,"title":15731,"module":15727,"summary":15732},"\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":15734,"title":15735,"module":15727,"summary":15736},"\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":15738,"title":15739,"module":15727,"summary":15740},"\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":15742,"title":15743,"module":15744,"summary":15745},"\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":15747,"title":15748,"module":15744,"summary":15749},"\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":15751,"title":15752,"module":15744,"summary":15753},"\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":15755,"title":15756,"module":15757,"summary":15758},"\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":15760,"title":15761,"module":15757,"summary":15762},"\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":15764,"title":15765,"module":15766,"summary":15767},"\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":15769,"title":15770,"module":15766,"summary":15771},"\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":15773,"title":15774,"module":15766,"summary":15775},"\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":15777,"title":15778,"module":6,"summary":6},"\u002Flogic","Logic",{"path":15780,"title":15781,"module":12405,"summary":15782},"\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":15784,"title":15785,"module":12405,"summary":15786},"\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":15788,"title":15789,"module":12405,"summary":15790},"\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":15792,"title":15793,"module":12405,"summary":15794},"\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":15796,"title":15797,"module":12405,"summary":15798},"\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":15800,"title":15801,"module":12405,"summary":15802},"\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":15804,"title":12624,"module":15805,"summary":15806},"\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":15808,"title":15809,"module":15805,"summary":15810},"\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":9872,"title":15812,"module":15805,"summary":15813},"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":15815,"title":15816,"module":15805,"summary":15817},"\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":15819,"title":15820,"module":15805,"summary":15821},"\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":15823,"title":15824,"module":15805,"summary":15825},"\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":9908,"title":15827,"module":15805,"summary":15828},"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":15830,"title":15831,"module":15805,"summary":15832},"\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":15834,"title":15835,"module":15805,"summary":15836},"\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":15838,"title":15839,"module":15805,"summary":15840},"\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":15842,"title":15843,"module":15805,"summary":15844},"\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":15846,"title":15847,"module":15805,"summary":15848},"\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":15850,"title":15851,"module":11467,"summary":15852},"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-prediction","On-Policy Prediction with Approximation","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":15854,"title":15855,"module":11467,"summary":15856},"\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":15858,"title":15859,"module":11467,"summary":15860},"\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":15862,"title":15863,"module":11467,"summary":15864},"\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":17,"title":15866,"module":11467,"summary":15867},"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":11469,"title":5,"module":11467,"summary":11485},{"path":7505,"title":15870,"module":11467,"summary":15871},"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":15873,"title":15874,"module":11467,"summary":15875},"\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":15877,"title":15878,"module":11467,"summary":15879},"\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":15881,"title":15882,"module":11467,"summary":15883},"\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":15885,"title":15886,"module":11467,"summary":15887},"\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":15889,"title":15890,"module":11467,"summary":15891},"\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":15893,"title":15894,"module":11467,"summary":15895},"\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":15897,"title":15898,"module":11467,"summary":15899},"\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":10195,"title":15227,"module":15901,"summary":15902},"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":15904,"title":15905,"module":15901,"summary":15906},"\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":15908,"title":15909,"module":15901,"summary":15910},"\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":15912,"title":15913,"module":15901,"summary":15914},"\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":15916,"title":15917,"module":15901,"summary":15918},"\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":15920,"title":15921,"module":15901,"summary":15922},"\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":15924,"title":15925,"module":15901,"summary":15926},"\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":15928,"title":15929,"module":15901,"summary":15930},"\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":15932,"title":15933,"module":15934,"summary":15935},"\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":15937,"title":15938,"module":15934,"summary":15939},"\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":15941,"title":15942,"module":15934,"summary":15943},"\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":15945,"title":15946,"module":15934,"summary":15947},"\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":15949,"title":15950,"module":15934,"summary":15951},"\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":15953,"title":15954,"module":15934,"summary":15955},"\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":15957,"title":15958,"module":15934,"summary":15959},"\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":15961,"title":15962,"module":15934,"summary":15963},"\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":15965,"title":15966,"module":15934,"summary":15967},"\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":15969,"title":15970,"module":15934,"summary":15971},"\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":15973,"title":15974,"module":15934,"summary":15975},"\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":15977,"title":15978,"module":15934,"summary":15979},"\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":15981,"title":15982,"module":15934,"summary":15983},"\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":15985,"title":15986,"module":15934,"summary":15987},"\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":15989,"title":15990,"module":15934,"summary":15991},"\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":15993,"title":15994,"module":15934,"summary":15995},"\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":15997,"title":15998,"module":15934,"summary":15999},"\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":16001,"title":16002,"module":15934,"summary":16003},"\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":16005,"title":16006,"module":15934,"summary":16007},"\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":16009,"title":16010,"module":15934,"summary":16011},"\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":16013,"title":16014,"module":15934,"summary":16015},"\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":16017,"title":16018,"module":15934,"summary":16019},"\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":16021,"title":16022,"module":16023,"summary":16024},"\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":16026,"title":16027,"module":16023,"summary":16028},"\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":16030,"title":16031,"module":16023,"summary":16032},"\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":16034,"title":16035,"module":16023,"summary":16036},"\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":16038,"title":16039,"module":16023,"summary":16040},"\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":16042,"title":16043,"module":16023,"summary":16044},"\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":16046,"title":16047,"module":16023,"summary":16048},"\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":16050,"title":16051,"module":16023,"summary":16052},"\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":16054,"title":15219,"module":6,"summary":6},"\u002Freinforcement-learning",{"path":16056,"title":16057,"module":12405,"summary":16058},"\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":16060,"title":16061,"module":12405,"summary":16062},"\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":16064,"title":16065,"module":12405,"summary":16066},"\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":16068,"title":16069,"module":12405,"summary":16070},"\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":16072,"title":16073,"module":16074,"summary":16075},"\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":16077,"title":16078,"module":16074,"summary":16079},"\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":16081,"title":16082,"module":16074,"summary":16083},"\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":16085,"title":16086,"module":16074,"summary":16087},"\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":16089,"title":16090,"module":16074,"summary":16091},"\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":16093,"title":16094,"module":16074,"summary":16095},"\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":16097,"title":16098,"module":16074,"summary":16099},"\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":16101,"title":16102,"module":16074,"summary":16103},"\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":16105,"title":16106,"module":16074,"summary":16107},"\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":16109,"title":16110,"module":16074,"summary":16111},"\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":16113,"title":16114,"module":16074,"summary":16115},"\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":16117,"title":16118,"module":16074,"summary":16119},"\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":16121,"title":16122,"module":16123,"summary":16124},"\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":16126,"title":16127,"module":16123,"summary":16128},"\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":16130,"title":16131,"module":16123,"summary":16132},"\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":16134,"title":16135,"module":16123,"summary":16136},"\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":16138,"title":16139,"module":16123,"summary":16140},"\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":16142,"title":16143,"module":16123,"summary":16144},"\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":16146,"title":16147,"module":16123,"summary":16148},"\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":16150,"title":16151,"module":16123,"summary":16152},"\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":16154,"title":16155,"module":16123,"summary":16156},"\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":16158,"title":16159,"module":16123,"summary":16160},"\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":16162,"title":16163,"module":16123,"summary":16164},"\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":16166,"title":16167,"module":16123,"summary":16168},"\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":16170,"title":16171,"module":16172,"summary":16173},"\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":16175,"title":16176,"module":16172,"summary":16177},"\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":16179,"title":16180,"module":16172,"summary":16181},"\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":16183,"title":16184,"module":16172,"summary":16185},"\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":16187,"title":16188,"module":16172,"summary":16189},"\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":16191,"title":16192,"module":16172,"summary":16193},"\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":16195,"title":16196,"module":16172,"summary":16197},"\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":16199,"title":15797,"module":16172,"summary":16200},"\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":16202,"title":16203,"module":16172,"summary":16204},"\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":16206,"title":16207,"module":16172,"summary":16208},"\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":16210,"title":16211,"module":16212,"summary":16213},"\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":16215,"title":16216,"module":16212,"summary":16217},"\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":16219,"title":16220,"module":16212,"summary":16221},"\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":16223,"title":16224,"module":16212,"summary":16225},"\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":16227,"title":15219,"module":16212,"summary":16228},"\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":16230,"title":16231,"module":16212,"summary":16232},"\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":16234,"title":16235,"module":16212,"summary":16236},"\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":16238,"title":16239,"module":16212,"summary":16240},"\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":16242,"title":16243,"module":16244,"summary":16245},"\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":16247,"title":16248,"module":16244,"summary":16249},"\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":16251,"title":16252,"module":16244,"summary":16253},"\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":16255,"title":16256,"module":16244,"summary":16257},"\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":16259,"title":16260,"module":16244,"summary":16261},"\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":16263,"title":16264,"module":16244,"summary":16265},"\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":16267,"title":16268,"module":16244,"summary":16269},"\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":16271,"title":16272,"module":16244,"summary":16273},"\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":16275,"title":16276,"module":6,"summary":6},"\u002Fartificial-intelligence","Artificial Intelligence",{"path":16278,"title":16279,"module":16280,"summary":16281},"\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":16283,"title":16284,"module":16280,"summary":16285},"\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":16287,"title":16288,"module":16280,"summary":16289},"\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":16291,"title":16292,"module":16280,"summary":16293},"\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":16295,"title":16296,"module":16280,"summary":16297},"\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":16299,"title":16300,"module":16301,"summary":16302},"\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":16304,"title":16305,"module":16301,"summary":16306},"\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":16308,"title":16309,"module":16301,"summary":16310},"\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":16312,"title":16313,"module":16301,"summary":16314},"\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":16316,"title":16317,"module":16318,"summary":16319},"\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":16321,"title":16322,"module":16318,"summary":16323},"\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":16325,"title":16326,"module":16318,"summary":16327},"\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":16329,"title":16330,"module":16318,"summary":16331},"\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":16333,"title":16334,"module":16335,"summary":16336},"\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":16338,"title":16339,"module":16335,"summary":16340},"\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":16342,"title":16343,"module":16344,"summary":16345},"\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":16347,"title":16348,"module":16344,"summary":16349},"\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":16351,"title":16352,"module":16353,"summary":16354},"\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":16356,"title":16357,"module":16353,"summary":16358},"\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":16360,"title":16361,"module":16353,"summary":16362},"\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":16364,"title":16365,"module":16353,"summary":16366},"\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":16368,"title":16369,"module":16370,"summary":16371},"\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":16373,"title":16374,"module":16370,"summary":16375},"\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":16377,"title":16378,"module":16370,"summary":16379},"\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":16381,"title":16382,"module":16383,"summary":16384},"\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":16386,"title":16387,"module":16383,"summary":16388},"\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":16390,"title":16391,"module":16383,"summary":16392},"\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":16394,"title":16395,"module":16396,"summary":16397},"\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":16399,"title":16400,"module":16396,"summary":16401},"\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":16403,"title":16404,"module":16405,"summary":16406},"\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":16408,"title":16409,"module":16405,"summary":16410},"\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":16412,"title":16413,"module":16405,"summary":16414},"\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":16416,"title":16417,"module":16418,"summary":16419},"\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":16421,"title":16422,"module":16418,"summary":16423},"\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":16425,"title":16426,"module":16418,"summary":16427},"\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":16429,"title":16430,"module":16418,"summary":16431},"\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":16433,"title":16434,"module":16418,"summary":16435},"\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":16437,"title":16438,"module":6,"summary":6},"\u002Fnuclear-physics","Nuclear Physics",{"path":16440,"title":16441,"module":12405,"summary":16442},"\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":16444,"title":16445,"module":12405,"summary":16446},"\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":16448,"title":16449,"module":12405,"summary":16450},"\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":16452,"title":16453,"module":12405,"summary":16454},"\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":16456,"title":16457,"module":12405,"summary":16458},"\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":16460,"title":16461,"module":16462,"summary":16463},"\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":16465,"title":16466,"module":16462,"summary":16467},"\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":16469,"title":16470,"module":16462,"summary":16471},"\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":16473,"title":16474,"module":16462,"summary":16475},"\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":16477,"title":16478,"module":16479,"summary":16480},"\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":16482,"title":16483,"module":16479,"summary":16484},"\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":16486,"title":16487,"module":16479,"summary":16488},"\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":16490,"title":16491,"module":12871,"summary":16492},"\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":16494,"title":16495,"module":12871,"summary":16496},"\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":16498,"title":16499,"module":12871,"summary":16500},"\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":16502,"title":16503,"module":13353,"summary":16504},"\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":16506,"title":15065,"module":13353,"summary":16507},"\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":16509,"title":15173,"module":13353,"summary":16510},"\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":16512,"title":16513,"module":13353,"summary":16514},"\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":16516,"title":16517,"module":13353,"summary":16518},"\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":16520,"title":16521,"module":13353,"summary":16522},"\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":16524,"title":16525,"module":16526,"summary":16527},"\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":16529,"title":16530,"module":16526,"summary":16531},"\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":16533,"title":16534,"module":16526,"summary":16535},"\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":16537,"title":16538,"module":16526,"summary":16539},"\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":16541,"title":16542,"module":16526,"summary":16543},"\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":16545,"title":16546,"module":16526,"summary":16547},"\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":16549,"title":16550,"module":16526,"summary":16551},"\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":16553,"title":16554,"module":16526,"summary":16555},"\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":16557,"title":16558,"module":16526,"summary":16559},"\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":16561,"title":16562,"module":16526,"summary":16563},"\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":16565,"title":16566,"module":16526,"summary":16567},"\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":16569,"title":16570,"module":16526,"summary":16571},"\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":16573,"title":16574,"module":16526,"summary":16575},"\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":16577,"title":16578,"module":16526,"summary":16579},"\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":16581,"title":16582,"module":16526,"summary":16583},"\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":16585,"title":16586,"module":16526,"summary":16587},"\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":16589,"title":16590,"module":16526,"summary":16591},"\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":16593,"title":16594,"module":16526,"summary":16595},"\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":16597,"title":16598,"module":16526,"summary":16599},"\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":16601,"title":16602,"module":16526,"summary":16603},"\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":16605,"title":16606,"module":15161,"summary":16607},"\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":16609,"title":16610,"module":15161,"summary":16611},"\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":16613,"title":16614,"module":15161,"summary":16615},"\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":16617,"title":16618,"module":15161,"summary":16619},"\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":16621,"title":16622,"module":15161,"summary":16623},"\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":16625,"title":16626,"module":15161,"summary":16627},"\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":16629,"title":16630,"module":15161,"summary":16631},"\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":16633,"title":16634,"module":15161,"summary":16635},"\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":16637,"title":16638,"module":16639,"summary":16640},"\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":16642,"title":16643,"module":16639,"summary":16644},"\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":16646,"title":16647,"module":16639,"summary":16648},"\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":16650,"title":16651,"module":16639,"summary":16652},"\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":16654,"title":16655,"module":6,"summary":6},"\u002Fnatural-language-processing","Natural Language Processing",{"path":16657,"title":16658,"module":12405,"summary":16659},"\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":16661,"title":16662,"module":12405,"summary":16663},"\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":16665,"title":16666,"module":12405,"summary":16667},"\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":16669,"title":16670,"module":16671,"summary":16672},"\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":16674,"title":16675,"module":16671,"summary":16676},"\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":16678,"title":16679,"module":16671,"summary":16680},"\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":16682,"title":16683,"module":16671,"summary":16684},"\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":16686,"title":16687,"module":16688,"summary":16689},"\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":16691,"title":16692,"module":16688,"summary":16693},"\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":16695,"title":16696,"module":16688,"summary":16697},"\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":16699,"title":16700,"module":16688,"summary":16701},"\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":16703,"title":16704,"module":16705,"summary":16706},"\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":16708,"title":16709,"module":16705,"summary":16710},"\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":16712,"title":16713,"module":16705,"summary":16714},"\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":16716,"title":16717,"module":16705,"summary":16718},"\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":16720,"title":16721,"module":16722,"summary":16723},"\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":16725,"title":16726,"module":16722,"summary":16727},"\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":16729,"title":16730,"module":16722,"summary":16731},"\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":16733,"title":16734,"module":16735,"summary":16736},"\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":16738,"title":16739,"module":16735,"summary":16740},"\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":16742,"title":16743,"module":16735,"summary":16744},"\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":16746,"title":16747,"module":16735,"summary":16748},"\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":16750,"title":16751,"module":16752,"summary":16753},"\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":16755,"title":16756,"module":16752,"summary":16757},"\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":16759,"title":16760,"module":16752,"summary":16761},"\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":16763,"title":16764,"module":16752,"summary":16765},"\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":16767,"title":16768,"module":16769,"summary":16770},"\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":16772,"title":16773,"module":16769,"summary":16774},"\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":16776,"title":16777,"module":16769,"summary":16778},"\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":16780,"title":16781,"module":16769,"summary":16782},"\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":16784,"title":16785,"module":16786,"summary":16787},"\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":16789,"title":16790,"module":16786,"summary":16791},"\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":16793,"title":16794,"module":16786,"summary":16795},"\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":16797,"title":16798,"module":16786,"summary":16799},"\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":16801,"title":16802,"module":16786,"summary":16803},"\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":16805,"title":16806,"module":16807,"summary":16808},"\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":16810,"title":16811,"module":16807,"summary":16812},"\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":16814,"title":16815,"module":16807,"summary":16816},"\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":16818,"title":16819,"module":16820,"summary":16821},"\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":16823,"title":16824,"module":16820,"summary":16825},"\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":16827,"title":16828,"module":16820,"summary":16829},"\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":16831,"title":16832,"module":16832,"summary":16833},"\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":16835,"title":16836,"module":16832,"summary":16837},"\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":16839,"title":16840,"module":16832,"summary":16841},"\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":16843,"title":16844,"module":16832,"summary":16845},"\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":16847,"title":16848,"module":16832,"summary":16849},"\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":16851,"title":16852,"module":16832,"summary":16853},"\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":16855,"title":16856,"module":6,"summary":6},"\u002Fparticle-physics","Particle Physics",{"path":16858,"title":16859,"module":16860,"summary":16861},"\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":16863,"title":16864,"module":16860,"summary":16865},"\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":16867,"title":16868,"module":16860,"summary":16869},"\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":16871,"title":16872,"module":16873,"summary":16874},"\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":16876,"title":16877,"module":16873,"summary":16878},"\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":16880,"title":16881,"module":16873,"summary":16882},"\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":16884,"title":16885,"module":16873,"summary":16886},"\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":16888,"title":16889,"module":16890,"summary":16891},"\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":16893,"title":16894,"module":16890,"summary":16895},"\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":16897,"title":16898,"module":16890,"summary":16899},"\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":16901,"title":16902,"module":16890,"summary":16903},"\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":16905,"title":16906,"module":16907,"summary":16908},"\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":16910,"title":16911,"module":16907,"summary":16912},"\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":16914,"title":16915,"module":16907,"summary":16916},"\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":16918,"title":16919,"module":16907,"summary":16920},"\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":16922,"title":16923,"module":16924,"summary":16925},"\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":16927,"title":16928,"module":16924,"summary":16929},"\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":16931,"title":16932,"module":16924,"summary":16933},"\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":16935,"title":16936,"module":16924,"summary":16937},"\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":16939,"title":16940,"module":16941,"summary":16942},"\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":16944,"title":16945,"module":16941,"summary":16946},"\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":16948,"title":16949,"module":16941,"summary":16950},"\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":16952,"title":16953,"module":16954,"summary":16955},"\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":16957,"title":16958,"module":16954,"summary":16959},"\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":16961,"title":16962,"module":16954,"summary":16963},"\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":16965,"title":16966,"module":16954,"summary":16967},"\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":16969,"title":15394,"module":16970,"summary":16971},"\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":16973,"title":16974,"module":16970,"summary":16975},"\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":16977,"title":16978,"module":16970,"summary":16979},"\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":16981,"title":16982,"module":16970,"summary":16983},"\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":16985,"title":16986,"module":16970,"summary":16987},"\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":16989,"title":16990,"module":16991,"summary":16992},"\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":16994,"title":16995,"module":16991,"summary":16996},"\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":16998,"title":16999,"module":16991,"summary":17000},"\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":17002,"title":17003,"module":16991,"summary":17004},"\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":17006,"title":17007,"module":17008,"summary":17009},"\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":17011,"title":17012,"module":17008,"summary":17013},"\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":17015,"title":17016,"module":17008,"summary":17017},"\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":17019,"title":17020,"module":17008,"summary":17021},"\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":17023,"title":17024,"module":17008,"summary":17025},"\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":17027,"title":17028,"module":17029,"summary":17030},"\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":17032,"title":17033,"module":17029,"summary":17034},"\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":17036,"title":14112,"module":17029,"summary":17037},"\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":17039,"title":17040,"module":17029,"summary":17041},"\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":17043,"title":17044,"module":17029,"summary":17045},"\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":17047,"title":17048,"module":17049,"summary":17050},"\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":17052,"title":17053,"module":17049,"summary":17054},"\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":17056,"title":17057,"module":17049,"summary":17058},"\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":17060,"title":17061,"module":17049,"summary":17062},"\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":17064,"title":17065,"module":17049,"summary":17066},"\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":17068,"title":17069,"module":17049,"summary":17070},"\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":17072,"title":17073,"module":17049,"summary":17074},"\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":17076,"title":17077,"module":6,"summary":6},"\u002Fastrophysics-cosmology","Astrophysics & Cosmology",{"path":17079,"title":17080,"module":6,"summary":6},"\u002Fcolophon","Colophon",{"path":7480,"title":17082,"module":6,"summary":6},"Study Notes",[17084,17102,17136,17169,17189,17244],{"module":12405,"moduleNumber":17085,"slug":17086,"lessons":17087},1,"foundations",[17088,17090,17092,17094,17097,17100],{"title":15781,"path":15780,"lessonNumber":17085,"topics":17089,"summary":15782},[12405],{"title":15785,"path":15784,"lessonNumber":11444,"topics":17091,"summary":15786},[12405],{"title":15789,"path":15788,"lessonNumber":11449,"topics":17093,"summary":15790},[12405],{"title":15793,"path":15792,"lessonNumber":17095,"topics":17096,"summary":15794},4,[12405],{"title":15797,"path":15796,"lessonNumber":17098,"topics":17099,"summary":15798},5,[12405],{"title":15801,"path":15800,"lessonNumber":11465,"topics":17101,"summary":15802},[12405],{"module":15805,"moduleNumber":11444,"slug":17103,"lessons":17104},"tabular-methods",[17105,17108,17110,17112,17114,17116,17118,17121,17124,17127,17130,17133],{"title":12624,"path":15804,"lessonNumber":17085,"topics":17106,"summary":15806},[17107],"Tabular Methods",{"title":15809,"path":15808,"lessonNumber":11444,"topics":17109,"summary":15810},[17107],{"title":15812,"path":9872,"lessonNumber":11449,"topics":17111,"summary":15813},[17107],{"title":15816,"path":15815,"lessonNumber":17095,"topics":17113,"summary":15817},[17107],{"title":15820,"path":15819,"lessonNumber":17098,"topics":17115,"summary":15821},[17107],{"title":15824,"path":15823,"lessonNumber":11465,"topics":17117,"summary":15825},[17107],{"title":15827,"path":9908,"lessonNumber":17119,"topics":17120,"summary":15828},7,[17107],{"title":15831,"path":15830,"lessonNumber":17122,"topics":17123,"summary":15832},8,[17107],{"title":15835,"path":15834,"lessonNumber":17125,"topics":17126,"summary":15836},9,[17107],{"title":15839,"path":15838,"lessonNumber":17128,"topics":17129,"summary":15840},10,[17107],{"title":15843,"path":15842,"lessonNumber":17131,"topics":17132,"summary":15844},11,[17107],{"title":15847,"path":15846,"lessonNumber":17134,"topics":17135,"summary":15848},12,[17107],{"module":11467,"moduleNumber":11449,"slug":17137,"lessons":17138},"approximation",[17139,17141,17143,17145,17147,17149,17151,17153,17155,17157,17159,17161,17163,17166],{"title":15851,"path":15850,"lessonNumber":17085,"topics":17140,"summary":15852},[11487],{"title":15855,"path":15854,"lessonNumber":11444,"topics":17142,"summary":15856},[11487],{"title":15859,"path":15858,"lessonNumber":11449,"topics":17144,"summary":15860},[11487],{"title":15863,"path":15862,"lessonNumber":17095,"topics":17146,"summary":15864},[11487],{"title":15866,"path":17,"lessonNumber":17098,"topics":17148,"summary":15867},[11487],{"title":5,"path":11469,"lessonNumber":11465,"topics":17150,"summary":11485},[11487],{"title":15870,"path":7505,"lessonNumber":17119,"topics":17152,"summary":15871},[11487],{"title":15874,"path":15873,"lessonNumber":17122,"topics":17154,"summary":15875},[11487],{"title":15878,"path":15877,"lessonNumber":17125,"topics":17156,"summary":15879},[11487],{"title":15882,"path":15881,"lessonNumber":17128,"topics":17158,"summary":15883},[11487],{"title":15886,"path":15885,"lessonNumber":17131,"topics":17160,"summary":15887},[11487],{"title":15890,"path":15889,"lessonNumber":17134,"topics":17162,"summary":15891},[11487],{"title":15894,"path":15893,"lessonNumber":17164,"topics":17165,"summary":15895},13,[11487],{"title":15898,"path":15897,"lessonNumber":17167,"topics":17168,"summary":15899},14,[11487],{"module":15901,"moduleNumber":17095,"slug":17170,"lessons":17171},"deep-rl",[17172,17175,17177,17179,17181,17183,17185,17187],{"title":15227,"path":10195,"lessonNumber":17085,"topics":17173,"summary":15902},[17174],"Deep RL",{"title":15905,"path":15904,"lessonNumber":11444,"topics":17176,"summary":15906},[17174],{"title":15909,"path":15908,"lessonNumber":11449,"topics":17178,"summary":15910},[17174],{"title":15913,"path":15912,"lessonNumber":17095,"topics":17180,"summary":15914},[17174],{"title":15917,"path":15916,"lessonNumber":17098,"topics":17182,"summary":15918},[17174],{"title":15921,"path":15920,"lessonNumber":11465,"topics":17184,"summary":15922},[17174],{"title":15925,"path":15924,"lessonNumber":17119,"topics":17186,"summary":15926},[17174],{"title":15929,"path":15928,"lessonNumber":17122,"topics":17188,"summary":15930},[17174],{"module":15934,"moduleNumber":17098,"slug":17190,"lessons":17191},"modern-deep-rl",[17192,17194,17196,17198,17200,17202,17204,17206,17208,17210,17212,17214,17216,17218,17220,17223,17226,17229,17232,17235,17238,17241],{"title":15933,"path":15932,"lessonNumber":17085,"topics":17193,"summary":15935},[17174],{"title":15938,"path":15937,"lessonNumber":11444,"topics":17195,"summary":15939},[17174],{"title":15942,"path":15941,"lessonNumber":11449,"topics":17197,"summary":15943},[17174],{"title":15946,"path":15945,"lessonNumber":17095,"topics":17199,"summary":15947},[17174],{"title":15950,"path":15949,"lessonNumber":17098,"topics":17201,"summary":15951},[17174],{"title":15954,"path":15953,"lessonNumber":11465,"topics":17203,"summary":15955},[17174],{"title":15958,"path":15957,"lessonNumber":17119,"topics":17205,"summary":15959},[17174],{"title":15962,"path":15961,"lessonNumber":17122,"topics":17207,"summary":15963},[17174],{"title":15966,"path":15965,"lessonNumber":17125,"topics":17209,"summary":15967},[17174],{"title":15970,"path":15969,"lessonNumber":17128,"topics":17211,"summary":15971},[17174],{"title":15974,"path":15973,"lessonNumber":17131,"topics":17213,"summary":15975},[17174],{"title":15978,"path":15977,"lessonNumber":17134,"topics":17215,"summary":15979},[17174],{"title":15982,"path":15981,"lessonNumber":17164,"topics":17217,"summary":15983},[17174],{"title":15986,"path":15985,"lessonNumber":17167,"topics":17219,"summary":15987},[17174],{"title":15990,"path":15989,"lessonNumber":17221,"topics":17222,"summary":15991},15,[17174],{"title":15994,"path":15993,"lessonNumber":17224,"topics":17225,"summary":15995},16,[17174],{"title":15998,"path":15997,"lessonNumber":17227,"topics":17228,"summary":15999},17,[17174],{"title":16002,"path":16001,"lessonNumber":17230,"topics":17231,"summary":16003},18,[17174],{"title":16006,"path":16005,"lessonNumber":17233,"topics":17234,"summary":16007},19,[17174],{"title":16010,"path":16009,"lessonNumber":17236,"topics":17237,"summary":16011},20,[17174],{"title":16014,"path":16013,"lessonNumber":17239,"topics":17240,"summary":16015},21,[17174],{"title":16018,"path":16017,"lessonNumber":17242,"topics":17243,"summary":16019},22,[17174],{"module":16023,"moduleNumber":11465,"slug":17245,"lessons":17246},"minds-and-brains",[17247,17250,17252,17254,17256,17258,17260,17262],{"title":16022,"path":16021,"lessonNumber":17085,"topics":17248,"summary":16024},[17249],"Minds and Brains",{"title":16027,"path":16026,"lessonNumber":11444,"topics":17251,"summary":16028},[17249],{"title":16031,"path":16030,"lessonNumber":11449,"topics":17253,"summary":16032},[17249],{"title":16035,"path":16034,"lessonNumber":17095,"topics":17255,"summary":16036},[17249],{"title":16039,"path":16038,"lessonNumber":17098,"topics":17257,"summary":16040},[17249],{"title":16043,"path":16042,"lessonNumber":11465,"topics":17259,"summary":16044},[17249],{"title":16047,"path":16046,"lessonNumber":17119,"topics":17261,"summary":16048},[17249],{"title":16051,"path":16050,"lessonNumber":17122,"topics":17263,"summary":16052},[17249],"\u003Csvg style=\"width:100%;max-width:334.841px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 251.131 122.829\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"var(--tk-soft-neutral)\" d=\"M-31.26 31.262h199.17l-34.144-42.679h-199.17Z\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-39.146 8.214)\">\u003Cpath d=\"M-19.565 31.255L-19.565 30.192Q-19.565 30.168-19.537 30.141Q-19.510 30.114-19.486 30.114L-19.377 30.114Q-19.312 30.114-19.298 30.172Q-19.202 30.606-18.956 30.857Q-18.710 31.108-18.296 31.108Q-17.955 31.108-17.702 30.975Q-17.449 30.842-17.449 30.534Q-17.449 30.377-17.543 30.262Q-17.637 30.148-17.775 30.079Q-17.914 30.011-18.081 29.973L-18.662 29.874Q-19.018 29.806-19.291 29.585Q-19.565 29.365-19.565 29.023Q-19.565 28.774-19.453 28.599Q-19.342 28.425-19.156 28.326Q-18.970 28.227-18.754 28.184Q-18.539 28.141-18.296 28.141Q-17.883 28.141-17.603 28.323L-17.387 28.148Q-17.377 28.145-17.370 28.143Q-17.363 28.141-17.353 28.141L-17.302 28.141Q-17.275 28.141-17.251 28.165Q-17.227 28.189-17.227 28.217L-17.227 29.064Q-17.227 29.085-17.251 29.112Q-17.275 29.139-17.302 29.139L-17.415 29.139Q-17.442 29.139-17.468 29.114Q-17.493 29.088-17.493 29.064Q-17.493 28.828-17.599 28.664Q-17.705 28.500-17.888 28.418Q-18.071 28.336-18.303 28.336Q-18.631 28.336-18.888 28.439Q-19.144 28.541-19.144 28.818Q-19.144 29.013-18.961 29.122Q-18.778 29.232-18.549 29.273L-17.975 29.379Q-17.729 29.427-17.515 29.555Q-17.302 29.683-17.165 29.886Q-17.028 30.090-17.028 30.339Q-17.028 30.852-17.394 31.091Q-17.760 31.330-18.296 31.330Q-18.792 31.330-19.124 31.036L-19.390 31.310Q-19.411 31.330-19.438 31.330L-19.486 31.330Q-19.510 31.330-19.537 31.303Q-19.565 31.276-19.565 31.255M-15.825 30.428L-15.825 28.924Q-15.825 28.654-15.933 28.593Q-16.041 28.531-16.352 28.531L-16.352 28.251L-15.244 28.176L-15.244 30.408L-15.244 30.428Q-15.244 30.708-15.193 30.852Q-15.142 30.995-15 31.052Q-14.858 31.108-14.571 31.108Q-14.318 31.108-14.113 30.968Q-13.908 30.828-13.792 30.602Q-13.675 30.377-13.675 30.127L-13.675 28.924Q-13.675 28.654-13.783 28.593Q-13.891 28.531-14.202 28.531L-14.202 28.251L-13.094 28.176L-13.094 30.589Q-13.094 30.780-13.041 30.862Q-12.988 30.944-12.888 30.963Q-12.787 30.982-12.571 30.982L-12.571 31.262L-13.648 31.330L-13.648 30.766Q-13.757 30.948-13.903 31.071Q-14.048 31.194-14.234 31.262Q-14.421 31.330-14.622 31.330Q-15.825 31.330-15.825 30.428M-11.177 31.262L-11.443 31.262L-11.443 27.154Q-11.443 26.884-11.551 26.822Q-11.659 26.761-11.970 26.761L-11.970 26.480L-10.890 26.405L-10.890 28.575Q-10.681 28.384-10.396 28.280Q-10.110 28.176-9.813 28.176Q-9.495 28.176-9.198 28.297Q-8.900 28.418-8.678 28.634Q-8.456 28.849-8.330 29.134Q-8.203 29.420-8.203 29.751Q-8.203 30.196-8.442 30.560Q-8.682 30.924-9.075 31.127Q-9.468 31.330-9.912 31.330Q-10.107 31.330-10.297 31.274Q-10.486 31.218-10.647 31.113Q-10.808 31.009-10.948 30.848L-11.177 31.262M-10.862 28.917L-10.862 30.534Q-10.726 30.794-10.485 30.951Q-10.244 31.108-9.967 31.108Q-9.673 31.108-9.461 31.001Q-9.249 30.893-9.116 30.701Q-8.983 30.510-8.924 30.271Q-8.866 30.032-8.866 29.751Q-8.866 29.392-8.960 29.088Q-9.054 28.784-9.282 28.591Q-9.509 28.398-9.875 28.398Q-10.175 28.398-10.442 28.534Q-10.709 28.671-10.862 28.917M-7.567 31.255L-7.567 30.192Q-7.567 30.168-7.540 30.141Q-7.513 30.114-7.489 30.114L-7.379 30.114Q-7.315 30.114-7.301 30.172Q-7.205 30.606-6.959 30.857Q-6.713 31.108-6.299 31.108Q-5.958 31.108-5.705 30.975Q-5.452 30.842-5.452 30.534Q-5.452 30.377-5.546 30.262Q-5.640 30.148-5.778 30.079Q-5.917 30.011-6.084 29.973L-6.665 29.874Q-7.021 29.806-7.294 29.585Q-7.567 29.365-7.567 29.023Q-7.567 28.774-7.456 28.599Q-7.345 28.425-7.159 28.326Q-6.973 28.227-6.757 28.184Q-6.542 28.141-6.299 28.141Q-5.886 28.141-5.606 28.323L-5.390 28.148Q-5.380 28.145-5.373 28.143Q-5.366 28.141-5.356 28.141L-5.305 28.141Q-5.277 28.141-5.254 28.165Q-5.230 28.189-5.230 28.217L-5.230 29.064Q-5.230 29.085-5.254 29.112Q-5.277 29.139-5.305 29.139L-5.418 29.139Q-5.445 29.139-5.471 29.114Q-5.496 29.088-5.496 29.064Q-5.496 28.828-5.602 28.664Q-5.708 28.500-5.891 28.418Q-6.074 28.336-6.306 28.336Q-6.634 28.336-6.891 28.439Q-7.147 28.541-7.147 28.818Q-7.147 29.013-6.964 29.122Q-6.781 29.232-6.552 29.273L-5.978 29.379Q-5.732 29.427-5.518 29.555Q-5.305 29.683-5.168 29.886Q-5.031 30.090-5.031 30.339Q-5.031 30.852-5.397 31.091Q-5.763 31.330-6.299 31.330Q-6.795 31.330-7.127 31.036L-7.393 31.310Q-7.414 31.330-7.441 31.330L-7.489 31.330Q-7.513 31.330-7.540 31.303Q-7.567 31.276-7.567 31.255M-2.758 32.619L-4.389 32.619L-4.389 32.339Q-4.160 32.339-4.011 32.304Q-3.862 32.270-3.862 32.130L-3.862 28.784Q-3.862 28.613-3.999 28.572Q-4.136 28.531-4.389 28.531L-4.389 28.251L-3.309 28.176L-3.309 28.582Q-3.087 28.381-2.799 28.278Q-2.512 28.176-2.205 28.176Q-1.777 28.176-1.413 28.389Q-1.049 28.603-0.836 28.967Q-0.622 29.331-0.622 29.751Q-0.622 30.196-0.861 30.560Q-1.101 30.924-1.494 31.127Q-1.887 31.330-2.331 31.330Q-2.598 31.330-2.846 31.230Q-3.093 31.129-3.281 30.948L-3.281 32.130Q-3.281 32.267-3.133 32.303Q-2.984 32.339-2.758 32.339L-2.758 32.619M-3.281 28.931L-3.281 30.541Q-3.148 30.794-2.905 30.951Q-2.663 31.108-2.386 31.108Q-2.058 31.108-1.805 30.907Q-1.552 30.705-1.419 30.387Q-1.285 30.069-1.285 29.751Q-1.285 29.522-1.350 29.293Q-1.415 29.064-1.543 28.866Q-1.671 28.668-1.866 28.548Q-2.061 28.429-2.294 28.429Q-2.587 28.429-2.856 28.558Q-3.124 28.688-3.281 28.931M0.072 30.534Q0.072 30.202 0.296 29.975Q0.519 29.748 0.863 29.620Q1.206 29.491 1.579 29.439Q1.952 29.386 2.256 29.386L2.256 29.133Q2.256 28.928 2.148 28.748Q2.040 28.569 1.859 28.466Q1.678 28.364 1.470 28.364Q1.063 28.364 0.827 28.456Q0.916 28.493 0.962 28.577Q1.008 28.661 1.008 28.763Q1.008 28.859 0.962 28.938Q0.916 29.016 0.836 29.061Q0.755 29.105 0.666 29.105Q0.516 29.105 0.415 29.008Q0.314 28.910 0.314 28.763Q0.314 28.141 1.470 28.141Q1.682 28.141 1.931 28.205Q2.181 28.268 2.382 28.387Q2.584 28.507 2.710 28.692Q2.837 28.876 2.837 29.119L2.837 30.695Q2.837 30.811 2.898 30.907Q2.960 31.002 3.073 31.002Q3.182 31.002 3.247 30.908Q3.312 30.814 3.312 30.695L3.312 30.247L3.579 30.247L3.579 30.695Q3.579 30.965 3.351 31.130Q3.124 31.296 2.844 31.296Q2.635 31.296 2.498 31.142Q2.362 30.989 2.338 30.773Q2.191 31.040 1.909 31.185Q1.627 31.330 1.302 31.330Q1.025 31.330 0.742 31.255Q0.458 31.180 0.265 31.001Q0.072 30.821 0.072 30.534M0.687 30.534Q0.687 30.708 0.788 30.838Q0.889 30.968 1.044 31.038Q1.200 31.108 1.364 31.108Q1.582 31.108 1.791 31.011Q1.999 30.913 2.128 30.732Q2.256 30.551 2.256 30.325L2.256 29.597Q1.931 29.597 1.565 29.688Q1.200 29.779 0.943 29.991Q0.687 30.202 0.687 30.534M3.996 29.751Q3.996 29.423 4.131 29.122Q4.266 28.822 4.501 28.601Q4.737 28.381 5.041 28.261Q5.346 28.141 5.670 28.141Q6.176 28.141 6.525 28.244Q6.873 28.346 6.873 28.722Q6.873 28.869 6.776 28.970Q6.679 29.071 6.532 29.071Q6.378 29.071 6.279 28.972Q6.180 28.873 6.180 28.722Q6.180 28.534 6.320 28.442Q6.118 28.391 5.677 28.391Q5.322 28.391 5.093 28.587Q4.864 28.784 4.763 29.093Q4.662 29.403 4.662 29.751Q4.662 30.100 4.788 30.406Q4.915 30.712 5.170 30.896Q5.424 31.081 5.780 31.081Q6.002 31.081 6.186 30.997Q6.371 30.913 6.506 30.758Q6.641 30.602 6.699 30.394Q6.713 30.339 6.767 30.339L6.880 30.339Q6.911 30.339 6.933 30.363Q6.955 30.387 6.955 30.421L6.955 30.442Q6.870 30.729 6.682 30.927Q6.494 31.125 6.229 31.228Q5.964 31.330 5.670 31.330Q5.240 31.330 4.852 31.124Q4.464 30.917 4.230 30.554Q3.996 30.192 3.996 29.751M7.502 29.727Q7.502 29.406 7.627 29.117Q7.752 28.828 7.977 28.605Q8.203 28.381 8.499 28.261Q8.794 28.141 9.112 28.141Q9.440 28.141 9.702 28.241Q9.963 28.340 10.139 28.522Q10.315 28.705 10.409 28.963Q10.503 29.221 10.503 29.553Q10.503 29.645 10.421 29.666L8.165 29.666L8.165 29.727Q8.165 30.315 8.449 30.698Q8.733 31.081 9.300 31.081Q9.621 31.081 9.890 30.888Q10.158 30.695 10.247 30.380Q10.254 30.339 10.329 30.325L10.421 30.325Q10.503 30.349 10.503 30.421Q10.503 30.428 10.496 30.455Q10.384 30.852 10.013 31.091Q9.642 31.330 9.218 31.330Q8.781 31.330 8.381 31.122Q7.981 30.913 7.742 30.546Q7.502 30.179 7.502 29.727M8.172 29.457L9.987 29.457Q9.987 29.180 9.890 28.928Q9.792 28.675 9.594 28.519Q9.396 28.364 9.112 28.364Q8.835 28.364 8.622 28.522Q8.408 28.681 8.290 28.936Q8.172 29.191 8.172 29.457M11.491 30.842Q11.491 30.674 11.614 30.551Q11.737 30.428 11.912 30.428Q12.079 30.428 12.202 30.551Q12.325 30.674 12.325 30.842Q12.325 31.016 12.202 31.139Q12.079 31.262 11.912 31.262Q11.737 31.262 11.614 31.139Q11.491 31.016 11.491 30.842M11.491 28.658Q11.491 28.490 11.614 28.367Q11.737 28.244 11.912 28.244Q12.079 28.244 12.202 28.367Q12.325 28.490 12.325 28.658Q12.325 28.832 12.202 28.955Q12.079 29.078 11.912 29.078Q11.737 29.078 11.614 28.955Q11.491 28.832 11.491 28.658\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-39.146 8.214)\">\u003Cpath d=\"M18.528 31.235L17.400 28.736Q17.328 28.589 17.198 28.557Q17.068 28.524 16.839 28.524L16.839 28.244L18.353 28.244L18.353 28.524Q18.001 28.524 18.001 28.671Q18.001 28.716 18.012 28.736L18.876 30.654L19.656 28.924Q19.690 28.856 19.690 28.777Q19.690 28.664 19.606 28.594Q19.522 28.524 19.403 28.524L19.403 28.244L20.599 28.244L20.599 28.524Q20.380 28.524 20.209 28.627Q20.039 28.729 19.950 28.924L18.914 31.235Q18.866 31.330 18.760 31.330L18.682 31.330Q18.576 31.330 18.528 31.235\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-39.146 8.214)\">\u003Cpath d=\"M20.768 30.534Q20.768 30.202 20.991 29.975Q21.215 29.748 21.559 29.620Q21.902 29.491 22.275 29.439Q22.647 29.386 22.952 29.386L22.952 29.133Q22.952 28.928 22.844 28.748Q22.736 28.569 22.555 28.466Q22.374 28.364 22.166 28.364Q21.759 28.364 21.523 28.456Q21.612 28.493 21.658 28.577Q21.704 28.661 21.704 28.763Q21.704 28.859 21.658 28.938Q21.612 29.016 21.531 29.061Q21.451 29.105 21.362 29.105Q21.212 29.105 21.111 29.008Q21.010 28.910 21.010 28.763Q21.010 28.141 22.166 28.141Q22.377 28.141 22.627 28.205Q22.876 28.268 23.078 28.387Q23.280 28.507 23.406 28.692Q23.533 28.876 23.533 29.119L23.533 30.695Q23.533 30.811 23.594 30.907Q23.656 31.002 23.769 31.002Q23.878 31.002 23.943 30.908Q24.008 30.814 24.008 30.695L24.008 30.247L24.274 30.247L24.274 30.695Q24.274 30.965 24.047 31.130Q23.820 31.296 23.540 31.296Q23.331 31.296 23.194 31.142Q23.058 30.989 23.034 30.773Q22.887 31.040 22.605 31.185Q22.323 31.330 21.998 31.330Q21.721 31.330 21.437 31.255Q21.154 31.180 20.961 31.001Q20.768 30.821 20.768 30.534M21.383 30.534Q21.383 30.708 21.484 30.838Q21.584 30.968 21.740 31.038Q21.895 31.108 22.060 31.108Q22.278 31.108 22.487 31.011Q22.695 30.913 22.823 30.732Q22.952 30.551 22.952 30.325L22.952 29.597Q22.627 29.597 22.261 29.688Q21.895 29.779 21.639 29.991Q21.383 30.202 21.383 30.534M26.359 31.262L24.756 31.262L24.756 30.982Q24.982 30.982 25.131 30.948Q25.279 30.913 25.279 30.773L25.279 27.154Q25.279 26.884 25.172 26.822Q25.064 26.761 24.756 26.761L24.756 26.480L25.833 26.405L25.833 30.773Q25.833 30.910 25.983 30.946Q26.134 30.982 26.359 30.982L26.359 31.262M27.528 30.428L27.528 28.924Q27.528 28.654 27.421 28.593Q27.313 28.531 27.002 28.531L27.002 28.251L28.109 28.176L28.109 30.408L28.109 30.428Q28.109 30.708 28.161 30.852Q28.212 30.995 28.354 31.052Q28.496 31.108 28.783 31.108Q29.036 31.108 29.241 30.968Q29.446 30.828 29.562 30.602Q29.678 30.377 29.678 30.127L29.678 28.924Q29.678 28.654 29.571 28.593Q29.463 28.531 29.152 28.531L29.152 28.251L30.259 28.176L30.259 30.589Q30.259 30.780 30.312 30.862Q30.365 30.944 30.466 30.963Q30.567 30.982 30.782 30.982L30.782 31.262L29.706 31.330L29.706 30.766Q29.596 30.948 29.451 31.071Q29.306 31.194 29.119 31.262Q28.933 31.330 28.731 31.330Q27.528 31.330 27.528 30.428M31.329 29.727Q31.329 29.406 31.454 29.117Q31.579 28.828 31.804 28.605Q32.030 28.381 32.325 28.261Q32.621 28.141 32.939 28.141Q33.267 28.141 33.529 28.241Q33.790 28.340 33.966 28.522Q34.142 28.705 34.236 28.963Q34.330 29.221 34.330 29.553Q34.330 29.645 34.248 29.666L31.992 29.666L31.992 29.727Q31.992 30.315 32.276 30.698Q32.560 31.081 33.127 31.081Q33.448 31.081 33.717 30.888Q33.985 30.695 34.074 30.380Q34.081 30.339 34.156 30.325L34.248 30.325Q34.330 30.349 34.330 30.421Q34.330 30.428 34.323 30.455Q34.210 30.852 33.840 31.091Q33.469 31.330 33.045 31.330Q32.607 31.330 32.207 31.122Q31.808 30.913 31.568 30.546Q31.329 30.179 31.329 29.727M31.999 29.457L33.814 29.457Q33.814 29.180 33.717 28.928Q33.619 28.675 33.421 28.519Q33.223 28.364 32.939 28.364Q32.662 28.364 32.448 28.522Q32.235 28.681 32.117 28.936Q31.999 29.191 31.999 29.457M34.918 31.255L34.918 30.192Q34.918 30.168 34.945 30.141Q34.973 30.114 34.997 30.114L35.106 30.114Q35.171 30.114 35.185 30.172Q35.280 30.606 35.526 30.857Q35.772 31.108 36.186 31.108Q36.528 31.108 36.781 30.975Q37.034 30.842 37.034 30.534Q37.034 30.377 36.940 30.262Q36.846 30.148 36.707 30.079Q36.569 30.011 36.401 29.973L35.820 29.874Q35.465 29.806 35.191 29.585Q34.918 29.365 34.918 29.023Q34.918 28.774 35.029 28.599Q35.140 28.425 35.326 28.326Q35.513 28.227 35.728 28.184Q35.943 28.141 36.186 28.141Q36.600 28.141 36.880 28.323L37.095 28.148Q37.105 28.145 37.112 28.143Q37.119 28.141 37.129 28.141L37.181 28.141Q37.208 28.141 37.232 28.165Q37.256 28.189 37.256 28.217L37.256 29.064Q37.256 29.085 37.232 29.112Q37.208 29.139 37.181 29.139L37.068 29.139Q37.041 29.139 37.015 29.114Q36.989 29.088 36.989 29.064Q36.989 28.828 36.883 28.664Q36.777 28.500 36.594 28.418Q36.412 28.336 36.179 28.336Q35.851 28.336 35.595 28.439Q35.338 28.541 35.338 28.818Q35.338 29.013 35.521 29.122Q35.704 29.232 35.933 29.273L36.507 29.379Q36.753 29.427 36.967 29.555Q37.181 29.683 37.317 29.886Q37.454 30.090 37.454 30.339Q37.454 30.852 37.088 31.091Q36.723 31.330 36.186 31.330Q35.690 31.330 35.359 31.036L35.092 31.310Q35.072 31.330 35.044 31.330L34.997 31.330Q34.973 31.330 34.945 31.303Q34.918 31.276 34.918 31.255\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-39.146 8.214)\">\u003Cpath d=\"M42.544 31.262L40.808 31.262L40.808 30.982Q41.037 30.982 41.186 30.948Q41.334 30.913 41.334 30.773L41.334 28.924Q41.334 28.654 41.227 28.593Q41.119 28.531 40.808 28.531L40.808 28.251L41.837 28.176L41.837 28.883Q41.967 28.575 42.209 28.376Q42.452 28.176 42.770 28.176Q42.989 28.176 43.160 28.300Q43.331 28.425 43.331 28.637Q43.331 28.774 43.231 28.873Q43.132 28.972 42.999 28.972Q42.862 28.972 42.763 28.873Q42.664 28.774 42.664 28.637Q42.664 28.497 42.763 28.398Q42.473 28.398 42.273 28.594Q42.073 28.791 41.980 29.085Q41.888 29.379 41.888 29.659L41.888 30.773Q41.888 30.982 42.544 30.982L42.544 31.262M43.874 29.727Q43.874 29.406 43.999 29.117Q44.124 28.828 44.349 28.605Q44.575 28.381 44.870 28.261Q45.166 28.141 45.484 28.141Q45.812 28.141 46.074 28.241Q46.335 28.340 46.511 28.522Q46.687 28.705 46.781 28.963Q46.875 29.221 46.875 29.553Q46.875 29.645 46.793 29.666L44.537 29.666L44.537 29.727Q44.537 30.315 44.821 30.698Q45.105 31.081 45.672 31.081Q45.993 31.081 46.261 30.888Q46.530 30.695 46.619 30.380Q46.626 30.339 46.701 30.325L46.793 30.325Q46.875 30.349 46.875 30.421Q46.875 30.428 46.868 30.455Q46.755 30.852 46.385 31.091Q46.014 31.330 45.590 31.330Q45.152 31.330 44.752 31.122Q44.353 30.913 44.113 30.546Q43.874 30.179 43.874 29.727M44.544 29.457L46.359 29.457Q46.359 29.180 46.261 28.928Q46.164 28.675 45.966 28.519Q45.768 28.364 45.484 28.364Q45.207 28.364 44.993 28.522Q44.780 28.681 44.662 28.936Q44.544 29.191 44.544 29.457M49.107 32.619L47.477 32.619L47.477 32.339Q47.706 32.339 47.854 32.304Q48.003 32.270 48.003 32.130L48.003 28.784Q48.003 28.613 47.866 28.572Q47.730 28.531 47.477 28.531L47.477 28.251L48.557 28.176L48.557 28.582Q48.779 28.381 49.066 28.278Q49.353 28.176 49.661 28.176Q50.088 28.176 50.452 28.389Q50.816 28.603 51.030 28.967Q51.243 29.331 51.243 29.751Q51.243 30.196 51.004 30.560Q50.765 30.924 50.372 31.127Q49.979 31.330 49.534 31.330Q49.268 31.330 49.020 31.230Q48.772 31.129 48.584 30.948L48.584 32.130Q48.584 32.267 48.733 32.303Q48.881 32.339 49.107 32.339L49.107 32.619M48.584 28.931L48.584 30.541Q48.717 30.794 48.960 30.951Q49.203 31.108 49.480 31.108Q49.808 31.108 50.061 30.907Q50.313 30.705 50.447 30.387Q50.580 30.069 50.580 29.751Q50.580 29.522 50.515 29.293Q50.450 29.064 50.322 28.866Q50.194 28.668 49.999 28.548Q49.804 28.429 49.572 28.429Q49.278 28.429 49.010 28.558Q48.741 28.688 48.584 28.931M53.629 31.262L51.893 31.262L51.893 30.982Q52.122 30.982 52.270 30.948Q52.419 30.913 52.419 30.773L52.419 28.924Q52.419 28.654 52.311 28.593Q52.204 28.531 51.893 28.531L51.893 28.251L52.921 28.176L52.921 28.883Q53.051 28.575 53.294 28.376Q53.537 28.176 53.855 28.176Q54.073 28.176 54.244 28.300Q54.415 28.425 54.415 28.637Q54.415 28.774 54.316 28.873Q54.217 28.972 54.084 28.972Q53.947 28.972 53.848 28.873Q53.749 28.774 53.749 28.637Q53.749 28.497 53.848 28.398Q53.557 28.398 53.357 28.594Q53.157 28.791 53.065 29.085Q52.973 29.379 52.973 29.659L52.973 30.773Q52.973 30.982 53.629 30.982L53.629 31.262M54.959 29.727Q54.959 29.406 55.083 29.117Q55.208 28.828 55.434 28.605Q55.659 28.381 55.955 28.261Q56.251 28.141 56.568 28.141Q56.897 28.141 57.158 28.241Q57.419 28.340 57.595 28.522Q57.772 28.705 57.865 28.963Q57.959 29.221 57.959 29.553Q57.959 29.645 57.877 29.666L55.622 29.666L55.622 29.727Q55.622 30.315 55.905 30.698Q56.189 31.081 56.756 31.081Q57.078 31.081 57.346 30.888Q57.614 30.695 57.703 30.380Q57.710 30.339 57.785 30.325L57.877 30.325Q57.959 30.349 57.959 30.421Q57.959 30.428 57.953 30.455Q57.840 30.852 57.469 31.091Q57.098 31.330 56.674 31.330Q56.237 31.330 55.837 31.122Q55.437 30.913 55.198 30.546Q54.959 30.179 54.959 29.727M55.628 29.457L57.443 29.457Q57.443 29.180 57.346 28.928Q57.249 28.675 57.050 28.519Q56.852 28.364 56.568 28.364Q56.292 28.364 56.078 28.522Q55.864 28.681 55.746 28.936Q55.628 29.191 55.628 29.457M58.547 31.255L58.547 30.192Q58.547 30.168 58.575 30.141Q58.602 30.114 58.626 30.114L58.735 30.114Q58.800 30.114 58.814 30.172Q58.910 30.606 59.156 30.857Q59.402 31.108 59.815 31.108Q60.157 31.108 60.410 30.975Q60.663 30.842 60.663 30.534Q60.663 30.377 60.569 30.262Q60.475 30.148 60.337 30.079Q60.198 30.011 60.031 29.973L59.450 29.874Q59.094 29.806 58.821 29.585Q58.547 29.365 58.547 29.023Q58.547 28.774 58.658 28.599Q58.770 28.425 58.956 28.326Q59.142 28.227 59.357 28.184Q59.573 28.141 59.815 28.141Q60.229 28.141 60.509 28.323L60.725 28.148Q60.735 28.145 60.742 28.143Q60.749 28.141 60.759 28.141L60.810 28.141Q60.837 28.141 60.861 28.165Q60.885 28.189 60.885 28.217L60.885 29.064Q60.885 29.085 60.861 29.112Q60.837 29.139 60.810 29.139L60.697 29.139Q60.670 29.139 60.644 29.114Q60.619 29.088 60.619 29.064Q60.619 28.828 60.513 28.664Q60.407 28.500 60.224 28.418Q60.041 28.336 59.809 28.336Q59.480 28.336 59.224 28.439Q58.968 28.541 58.968 28.818Q58.968 29.013 59.151 29.122Q59.334 29.232 59.563 29.273L60.137 29.379Q60.383 29.427 60.596 29.555Q60.810 29.683 60.947 29.886Q61.084 30.090 61.084 30.339Q61.084 30.852 60.718 31.091Q60.352 31.330 59.815 31.330Q59.320 31.330 58.988 31.036L58.722 31.310Q58.701 31.330 58.674 31.330L58.626 31.330Q58.602 31.330 58.575 31.303Q58.547 31.276 58.547 31.255M61.671 29.727Q61.671 29.406 61.796 29.117Q61.921 28.828 62.147 28.605Q62.372 28.381 62.668 28.261Q62.963 28.141 63.281 28.141Q63.609 28.141 63.871 28.241Q64.132 28.340 64.308 28.522Q64.484 28.705 64.578 28.963Q64.672 29.221 64.672 29.553Q64.672 29.645 64.590 29.666L62.334 29.666L62.334 29.727Q62.334 30.315 62.618 30.698Q62.902 31.081 63.469 31.081Q63.791 31.081 64.059 30.888Q64.327 30.695 64.416 30.380Q64.423 30.339 64.498 30.325L64.590 30.325Q64.672 30.349 64.672 30.421Q64.672 30.428 64.666 30.455Q64.553 30.852 64.182 31.091Q63.811 31.330 63.387 31.330Q62.950 31.330 62.550 31.122Q62.150 30.913 61.911 30.546Q61.671 30.179 61.671 29.727M62.341 29.457L64.156 29.457Q64.156 29.180 64.059 28.928Q63.961 28.675 63.763 28.519Q63.565 28.364 63.281 28.364Q63.004 28.364 62.791 28.522Q62.577 28.681 62.459 28.936Q62.341 29.191 62.341 29.457M66.942 31.262L65.308 31.262L65.308 30.982Q65.537 30.982 65.686 30.948Q65.834 30.913 65.834 30.773L65.834 28.924Q65.834 28.654 65.727 28.593Q65.619 28.531 65.308 28.531L65.308 28.251L66.368 28.176L66.368 28.825Q66.539 28.517 66.843 28.346Q67.147 28.176 67.492 28.176Q67.998 28.176 68.282 28.399Q68.565 28.623 68.565 29.119L68.565 30.773Q68.565 30.910 68.714 30.946Q68.863 30.982 69.088 30.982L69.088 31.262L67.458 31.262L67.458 30.982Q67.687 30.982 67.836 30.948Q67.984 30.913 67.984 30.773L67.984 29.133Q67.984 28.798 67.865 28.598Q67.745 28.398 67.431 28.398Q67.161 28.398 66.927 28.534Q66.692 28.671 66.554 28.905Q66.416 29.139 66.416 29.413L66.416 30.773Q66.416 30.910 66.566 30.946Q66.716 30.982 66.942 30.982\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-39.146 8.214)\">\u003Cpath d=\"M70.024 30.421L70.024 28.524L69.385 28.524L69.385 28.302Q69.703 28.302 69.920 28.092Q70.137 27.882 70.237 27.572Q70.338 27.263 70.338 26.955L70.605 26.955L70.605 28.244L71.682 28.244L71.682 28.524L70.605 28.524L70.605 30.408Q70.605 30.684 70.709 30.883Q70.813 31.081 71.073 31.081Q71.230 31.081 71.336 30.977Q71.442 30.872 71.492 30.719Q71.541 30.565 71.541 30.408L71.541 29.994L71.808 29.994L71.808 30.421Q71.808 30.647 71.709 30.857Q71.610 31.067 71.425 31.199Q71.241 31.330 71.012 31.330Q70.574 31.330 70.299 31.093Q70.024 30.855 70.024 30.421M72.676 30.534Q72.676 30.202 72.900 29.975Q73.124 29.748 73.467 29.620Q73.811 29.491 74.183 29.439Q74.556 29.386 74.860 29.386L74.860 29.133Q74.860 28.928 74.753 28.748Q74.645 28.569 74.464 28.466Q74.283 28.364 74.074 28.364Q73.667 28.364 73.432 28.456Q73.520 28.493 73.567 28.577Q73.613 28.661 73.613 28.763Q73.613 28.859 73.567 28.938Q73.520 29.016 73.440 29.061Q73.360 29.105 73.271 29.105Q73.121 29.105 73.020 29.008Q72.919 28.910 72.919 28.763Q72.919 28.141 74.074 28.141Q74.286 28.141 74.536 28.205Q74.785 28.268 74.987 28.387Q75.188 28.507 75.315 28.692Q75.441 28.876 75.441 29.119L75.441 30.695Q75.441 30.811 75.503 30.907Q75.564 31.002 75.677 31.002Q75.787 31.002 75.851 30.908Q75.916 30.814 75.916 30.695L75.916 30.247L76.183 30.247L76.183 30.695Q76.183 30.965 75.956 31.130Q75.728 31.296 75.448 31.296Q75.240 31.296 75.103 31.142Q74.966 30.989 74.942 30.773Q74.795 31.040 74.513 31.185Q74.231 31.330 73.907 31.330Q73.630 31.330 73.346 31.255Q73.062 31.180 72.869 31.001Q72.676 30.821 72.676 30.534M73.291 30.534Q73.291 30.708 73.392 30.838Q73.493 30.968 73.649 31.038Q73.804 31.108 73.968 31.108Q74.187 31.108 74.395 31.011Q74.604 30.913 74.732 30.732Q74.860 30.551 74.860 30.325L74.860 29.597Q74.536 29.597 74.170 29.688Q73.804 29.779 73.548 29.991Q73.291 30.202 73.291 30.534M77.407 31.262L77.140 31.262L77.140 27.154Q77.140 26.884 77.032 26.822Q76.925 26.761 76.614 26.761L76.614 26.480L77.694 26.405L77.694 28.575Q77.902 28.384 78.188 28.280Q78.473 28.176 78.770 28.176Q79.088 28.176 79.386 28.297Q79.683 28.418 79.905 28.634Q80.127 28.849 80.254 29.134Q80.380 29.420 80.380 29.751Q80.380 30.196 80.141 30.560Q79.902 30.924 79.509 31.127Q79.116 31.330 78.671 31.330Q78.476 31.330 78.287 31.274Q78.097 31.218 77.936 31.113Q77.776 31.009 77.636 30.848L77.407 31.262M77.721 28.917L77.721 30.534Q77.858 30.794 78.099 30.951Q78.340 31.108 78.617 31.108Q78.911 31.108 79.122 31.001Q79.334 30.893 79.468 30.701Q79.601 30.510 79.659 30.271Q79.717 30.032 79.717 29.751Q79.717 29.392 79.623 29.088Q79.529 28.784 79.302 28.591Q79.075 28.398 78.709 28.398Q78.408 28.398 78.141 28.534Q77.875 28.671 77.721 28.917M82.684 31.262L81.081 31.262L81.081 30.982Q81.307 30.982 81.455 30.948Q81.604 30.913 81.604 30.773L81.604 27.154Q81.604 26.884 81.496 26.822Q81.389 26.761 81.081 26.761L81.081 26.480L82.158 26.405L82.158 30.773Q82.158 30.910 82.308 30.946Q82.458 30.982 82.684 30.982L82.684 31.262M83.238 29.727Q83.238 29.406 83.362 29.117Q83.487 28.828 83.713 28.605Q83.938 28.381 84.234 28.261Q84.530 28.141 84.848 28.141Q85.176 28.141 85.437 28.241Q85.699 28.340 85.875 28.522Q86.051 28.705 86.145 28.963Q86.239 29.221 86.239 29.553Q86.239 29.645 86.157 29.666L83.901 29.666L83.901 29.727Q83.901 30.315 84.184 30.698Q84.468 31.081 85.036 31.081Q85.357 31.081 85.625 30.888Q85.893 30.695 85.982 30.380Q85.989 30.339 86.064 30.325L86.157 30.325Q86.239 30.349 86.239 30.421Q86.239 30.428 86.232 30.455Q86.119 30.852 85.748 31.091Q85.377 31.330 84.954 31.330Q84.516 31.330 84.116 31.122Q83.716 30.913 83.477 30.546Q83.238 30.179 83.238 29.727M83.908 29.457L85.723 29.457Q85.723 29.180 85.625 28.928Q85.528 28.675 85.329 28.519Q85.131 28.364 84.848 28.364Q84.571 28.364 84.357 28.522Q84.143 28.681 84.026 28.936Q83.908 29.191 83.908 29.457\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-39.146 8.214)\">\u003Cpath d=\"M89.598 30.534Q89.598 30.202 89.821 29.975Q90.045 29.748 90.389 29.620Q90.732 29.491 91.105 29.439Q91.477 29.386 91.782 29.386L91.782 29.133Q91.782 28.928 91.674 28.748Q91.566 28.569 91.385 28.466Q91.204 28.364 90.996 28.364Q90.589 28.364 90.353 28.456Q90.442 28.493 90.488 28.577Q90.534 28.661 90.534 28.763Q90.534 28.859 90.488 28.938Q90.442 29.016 90.361 29.061Q90.281 29.105 90.192 29.105Q90.042 29.105 89.941 29.008Q89.840 28.910 89.840 28.763Q89.840 28.141 90.996 28.141Q91.207 28.141 91.457 28.205Q91.706 28.268 91.908 28.387Q92.110 28.507 92.236 28.692Q92.363 28.876 92.363 29.119L92.363 30.695Q92.363 30.811 92.424 30.907Q92.486 31.002 92.599 31.002Q92.708 31.002 92.773 30.908Q92.838 30.814 92.838 30.695L92.838 30.247L93.104 30.247L93.104 30.695Q93.104 30.965 92.877 31.130Q92.650 31.296 92.370 31.296Q92.161 31.296 92.024 31.142Q91.888 30.989 91.864 30.773Q91.717 31.040 91.435 31.185Q91.153 31.330 90.828 31.330Q90.551 31.330 90.267 31.255Q89.984 31.180 89.791 31.001Q89.598 30.821 89.598 30.534M90.213 30.534Q90.213 30.708 90.314 30.838Q90.414 30.968 90.570 31.038Q90.725 31.108 90.890 31.108Q91.108 31.108 91.317 31.011Q91.525 30.913 91.653 30.732Q91.782 30.551 91.782 30.325L91.782 29.597Q91.457 29.597 91.091 29.688Q90.725 29.779 90.469 29.991Q90.213 30.202 90.213 30.534M93.521 31.255L93.521 30.192Q93.521 30.168 93.549 30.141Q93.576 30.114 93.600 30.114L93.709 30.114Q93.774 30.114 93.788 30.172Q93.884 30.606 94.130 30.857Q94.376 31.108 94.789 31.108Q95.131 31.108 95.384 30.975Q95.637 30.842 95.637 30.534Q95.637 30.377 95.543 30.262Q95.449 30.148 95.311 30.079Q95.172 30.011 95.005 29.973L94.424 29.874Q94.068 29.806 93.795 29.585Q93.521 29.365 93.521 29.023Q93.521 28.774 93.632 28.599Q93.744 28.425 93.930 28.326Q94.116 28.227 94.331 28.184Q94.547 28.141 94.789 28.141Q95.203 28.141 95.483 28.323L95.699 28.148Q95.709 28.145 95.716 28.143Q95.723 28.141 95.733 28.141L95.784 28.141Q95.811 28.141 95.835 28.165Q95.859 28.189 95.859 28.217L95.859 29.064Q95.859 29.085 95.835 29.112Q95.811 29.139 95.784 29.139L95.671 29.139Q95.644 29.139 95.618 29.114Q95.593 29.088 95.593 29.064Q95.593 28.828 95.487 28.664Q95.381 28.500 95.198 28.418Q95.015 28.336 94.783 28.336Q94.454 28.336 94.198 28.439Q93.942 28.541 93.942 28.818Q93.942 29.013 94.125 29.122Q94.308 29.232 94.537 29.273L95.111 29.379Q95.357 29.427 95.570 29.555Q95.784 29.683 95.921 29.886Q96.058 30.090 96.058 30.339Q96.058 30.852 95.692 31.091Q95.326 31.330 94.789 31.330Q94.294 31.330 93.962 31.036L93.696 31.310Q93.675 31.330 93.648 31.330L93.600 31.330Q93.576 31.330 93.549 31.303Q93.521 31.276 93.521 31.255\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-39.146 8.214)\">\u003Cpath d=\"M100.977 31.235L99.849 28.736Q99.777 28.589 99.647 28.557Q99.517 28.524 99.288 28.524L99.288 28.244L100.802 28.244L100.802 28.524Q100.450 28.524 100.450 28.671Q100.450 28.716 100.461 28.736L101.325 30.654L102.105 28.924Q102.139 28.856 102.139 28.777Q102.139 28.664 102.055 28.594Q101.971 28.524 101.852 28.524L101.852 28.244L103.048 28.244L103.048 28.524Q102.829 28.524 102.658 28.627Q102.488 28.729 102.399 28.924L101.363 31.235Q101.315 31.330 101.209 31.330L101.131 31.330Q101.025 31.330 100.977 31.235M105.718 33.012Q105.167 32.612 104.796 32.057Q104.426 31.501 104.244 30.855Q104.063 30.209 104.063 29.512Q104.063 28.999 104.164 28.504Q104.265 28.008 104.470 27.557Q104.675 27.106 104.988 26.714Q105.301 26.323 105.718 26.019Q105.728 26.015 105.735 26.014Q105.741 26.012 105.752 26.012L105.820 26.012Q105.854 26.012 105.876 26.036Q105.899 26.060 105.899 26.097Q105.899 26.142 105.871 26.159Q105.523 26.460 105.270 26.844Q105.017 27.229 104.865 27.670Q104.713 28.111 104.641 28.567Q104.569 29.023 104.569 29.512Q104.569 30.513 104.878 31.400Q105.188 32.287 105.871 32.872Q105.899 32.889 105.899 32.933Q105.899 32.971 105.876 32.995Q105.854 33.019 105.820 33.019L105.752 33.019Q105.745 33.015 105.736 33.014Q105.728 33.012 105.718 33.012M108.096 31.235L107.115 28.736Q107.054 28.593 106.936 28.558Q106.818 28.524 106.603 28.524L106.603 28.244L108.083 28.244L108.083 28.524Q107.703 28.524 107.703 28.685Q107.703 28.695 107.717 28.736L108.431 30.568L109.105 28.863Q109.074 28.791 109.074 28.763Q109.074 28.736 109.047 28.736Q108.985 28.589 108.867 28.557Q108.749 28.524 108.537 28.524L108.537 28.244L109.935 28.244L109.935 28.524Q109.559 28.524 109.559 28.685Q109.559 28.716 109.566 28.736L110.322 30.674L111.009 28.924Q111.029 28.873 111.029 28.818Q111.029 28.678 110.916 28.601Q110.803 28.524 110.663 28.524L110.663 28.244L111.884 28.244L111.884 28.524Q111.678 28.524 111.523 28.630Q111.367 28.736 111.296 28.924L110.390 31.235Q110.356 31.330 110.243 31.330L110.175 31.330Q110.065 31.330 110.028 31.235L109.245 29.232L108.459 31.235Q108.425 31.330 108.312 31.330L108.243 31.330Q108.134 31.330 108.096 31.235M112.735 33.019L112.666 33.019Q112.632 33.019 112.610 32.993Q112.588 32.968 112.588 32.933Q112.588 32.889 112.618 32.872Q112.974 32.568 113.223 32.178Q113.473 31.788 113.625 31.356Q113.777 30.924 113.847 30.455Q113.917 29.987 113.917 29.512Q113.917 29.033 113.847 28.567Q113.777 28.100 113.623 27.665Q113.469 27.229 113.218 26.841Q112.967 26.453 112.618 26.159Q112.588 26.142 112.588 26.097Q112.588 26.063 112.610 26.038Q112.632 26.012 112.666 26.012L112.735 26.012Q112.745 26.012 112.753 26.014Q112.762 26.015 112.772 26.019Q113.316 26.419 113.688 26.972Q114.061 27.526 114.242 28.172Q114.423 28.818 114.423 29.512Q114.423 30.213 114.242 30.860Q114.061 31.508 113.687 32.062Q113.312 32.616 112.772 33.012Q112.762 33.012 112.753 33.014Q112.745 33.015 112.735 33.019\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-accent)\" stroke=\"none\" d=\"M21.955 11.346a2 2 0 1 0-4 0 2 2 0 0 0 4 0m-2 0\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cg transform=\"translate(31.754 -7.72)\">\u003Cpath d=\"M-17.975 31.235L-19.103 28.736Q-19.175 28.589-19.305 28.557Q-19.435 28.524-19.664 28.524L-19.664 28.244L-18.150 28.244L-18.150 28.524Q-18.502 28.524-18.502 28.671Q-18.502 28.716-18.491 28.736L-17.627 30.654L-16.847 28.924Q-16.813 28.856-16.813 28.777Q-16.813 28.664-16.897 28.594Q-16.981 28.524-17.100 28.524L-17.100 28.244L-15.904 28.244L-15.904 28.524Q-16.123 28.524-16.294 28.627Q-16.464 28.729-16.553 28.924L-17.589 31.235Q-17.637 31.330-17.743 31.330L-17.821 31.330Q-17.927 31.330-17.975 31.235M-13.234 33.012Q-13.785 32.612-14.156 32.057Q-14.526 31.501-14.708 30.855Q-14.889 30.209-14.889 29.512Q-14.889 28.999-14.788 28.504Q-14.687 28.008-14.482 27.557Q-14.277 27.106-13.964 26.714Q-13.651 26.323-13.234 26.019Q-13.224 26.015-13.217 26.014Q-13.211 26.012-13.200 26.012L-13.132 26.012Q-13.098 26.012-13.076 26.036Q-13.053 26.060-13.053 26.097Q-13.053 26.142-13.081 26.159Q-13.429 26.460-13.682 26.844Q-13.935 27.229-14.087 27.670Q-14.239 28.111-14.311 28.567Q-14.383 29.023-14.383 29.512Q-14.383 30.513-14.074 31.400Q-13.764 32.287-13.081 32.872Q-13.053 32.889-13.053 32.933Q-13.053 32.971-13.076 32.995Q-13.098 33.019-13.132 33.019L-13.200 33.019Q-13.207 33.015-13.216 33.014Q-13.224 33.012-13.234 33.012M-10.856 31.235L-11.837 28.736Q-11.898 28.593-12.016 28.558Q-12.134 28.524-12.349 28.524L-12.349 28.244L-10.869 28.244L-10.869 28.524Q-11.249 28.524-11.249 28.685Q-11.249 28.695-11.235 28.736L-10.521 30.568L-9.847 28.863Q-9.878 28.791-9.878 28.763Q-9.878 28.736-9.905 28.736Q-9.967 28.589-10.085 28.557Q-10.203 28.524-10.415 28.524L-10.415 28.244L-9.017 28.244L-9.017 28.524Q-9.393 28.524-9.393 28.685Q-9.393 28.716-9.386 28.736L-8.630 30.674L-7.943 28.924Q-7.923 28.873-7.923 28.818Q-7.923 28.678-8.036 28.601Q-8.149 28.524-8.289 28.524L-8.289 28.244L-7.068 28.244L-7.068 28.524Q-7.274 28.524-7.429 28.630Q-7.585 28.736-7.656 28.924L-8.562 31.235Q-8.596 31.330-8.709 31.330L-8.777 31.330Q-8.887 31.330-8.924 31.235L-9.707 29.232L-10.493 31.235Q-10.527 31.330-10.640 31.330L-10.709 31.330Q-10.818 31.330-10.856 31.235M-6.217 33.019L-6.286 33.019Q-6.320 33.019-6.342 32.993Q-6.364 32.968-6.364 32.933Q-6.364 32.889-6.334 32.872Q-5.978 32.568-5.729 32.178Q-5.479 31.788-5.327 31.356Q-5.175 30.924-5.105 30.455Q-5.035 29.987-5.035 29.512Q-5.035 29.033-5.105 28.567Q-5.175 28.100-5.329 27.665Q-5.483 27.229-5.734 26.841Q-5.985 26.453-6.334 26.159Q-6.364 26.142-6.364 26.097Q-6.364 26.063-6.342 26.038Q-6.320 26.012-6.286 26.012L-6.217 26.012Q-6.207 26.012-6.199 26.014Q-6.190 26.015-6.180 26.019Q-5.636 26.419-5.264 26.972Q-4.891 27.526-4.710 28.172Q-4.529 28.818-4.529 29.512Q-4.529 30.213-4.710 30.860Q-4.891 31.508-5.265 32.062Q-5.640 32.616-6.180 33.012Q-6.190 33.012-6.199 33.014Q-6.207 33.015-6.217 33.019\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath stroke=\"none\" d=\"M98.778 9.923a2 2 0 1 0-4 0 2 2 0 0 0 4 0m-2 0\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(121.612 -12.092)\">\u003Cpath d=\"M-17.921 32.619L-19.551 32.619L-19.551 32.339Q-19.322 32.339-19.173 32.304Q-19.025 32.270-19.025 32.130L-19.025 28.784Q-19.025 28.613-19.161 28.572Q-19.298 28.531-19.551 28.531L-19.551 28.251L-18.471 28.176L-18.471 28.582Q-18.249 28.381-17.962 28.278Q-17.674 28.176-17.367 28.176Q-16.940 28.176-16.576 28.389Q-16.212 28.603-15.998 28.967Q-15.784 29.331-15.784 29.751Q-15.784 30.196-16.024 30.560Q-16.263 30.924-16.656 31.127Q-17.049 31.330-17.493 31.330Q-17.760 31.330-18.008 31.230Q-18.255 31.129-18.443 30.948L-18.443 32.130Q-18.443 32.267-18.295 32.303Q-18.146 32.339-17.921 32.339L-17.921 32.619M-18.443 28.931L-18.443 30.541Q-18.310 30.794-18.067 30.951Q-17.825 31.108-17.548 31.108Q-17.220 31.108-16.967 30.907Q-16.714 30.705-16.581 30.387Q-16.447 30.069-16.447 29.751Q-16.447 29.522-16.512 29.293Q-16.577 29.064-16.705 28.866Q-16.834 28.668-17.028 28.548Q-17.223 28.429-17.456 28.429Q-17.750 28.429-18.018 28.558Q-18.286 28.688-18.443 28.931M-13.399 31.262L-15.135 31.262L-15.135 30.982Q-14.906 30.982-14.757 30.948Q-14.608 30.913-14.608 30.773L-14.608 28.924Q-14.608 28.654-14.716 28.593Q-14.824 28.531-15.135 28.531L-15.135 28.251L-14.106 28.176L-14.106 28.883Q-13.976 28.575-13.733 28.376Q-13.491 28.176-13.173 28.176Q-12.954 28.176-12.783 28.300Q-12.612 28.425-12.612 28.637Q-12.612 28.774-12.712 28.873Q-12.811 28.972-12.944 28.972Q-13.081 28.972-13.180 28.873Q-13.279 28.774-13.279 28.637Q-13.279 28.497-13.180 28.398Q-13.470 28.398-13.670 28.594Q-13.870 28.791-13.962 29.085Q-14.055 29.379-14.055 29.659L-14.055 30.773Q-14.055 30.982-13.399 30.982L-13.399 31.262M-12.069 29.779Q-12.069 29.437-11.934 29.138Q-11.799 28.839-11.560 28.615Q-11.320 28.391-11.003 28.266Q-10.685 28.141-10.353 28.141Q-9.909 28.141-9.509 28.357Q-9.109 28.572-8.875 28.950Q-8.641 29.327-8.641 29.779Q-8.641 30.120-8.783 30.404Q-8.924 30.688-9.169 30.895Q-9.413 31.101-9.723 31.216Q-10.032 31.330-10.353 31.330Q-10.784 31.330-11.185 31.129Q-11.587 30.927-11.828 30.575Q-12.069 30.223-12.069 29.779M-10.353 31.081Q-9.752 31.081-9.528 30.703Q-9.304 30.325-9.304 29.693Q-9.304 29.081-9.538 28.722Q-9.772 28.364-10.353 28.364Q-11.406 28.364-11.406 29.693Q-11.406 30.325-11.180 30.703Q-10.955 31.081-10.353 31.081\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(121.612 -12.092)\">\u003Cpath d=\"M-8.135 32.110Q-8.135 31.959-8.035 31.862Q-7.934 31.764-7.787 31.764Q-7.698 31.764-7.618 31.809Q-7.537 31.853-7.491 31.932Q-7.445 32.011-7.445 32.110Q-7.445 32.311-7.606 32.410Q-7.465 32.465-7.260 32.465Q-6.990 32.465-6.874 32.192Q-6.758 31.918-6.758 31.604L-6.758 28.924Q-6.758 28.658-6.884 28.594Q-7.011 28.531-7.339 28.531L-7.339 28.251L-6.204 28.176L-6.204 31.624Q-6.204 31.911-6.351 32.156Q-6.498 32.400-6.749 32.545Q-7.001 32.691-7.277 32.691Q-7.606 32.691-7.870 32.549Q-8.135 32.407-8.135 32.110M-7.038 26.955Q-7.038 26.784-6.915 26.665Q-6.792 26.545-6.618 26.545Q-6.450 26.545-6.327 26.665Q-6.204 26.784-6.204 26.955Q-6.204 27.130-6.327 27.253Q-6.450 27.376-6.618 27.376Q-6.792 27.376-6.915 27.253Q-7.038 27.130-7.038 26.955\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(121.612 -12.092)\">\u003Cpath d=\"M-2.474 29.779Q-2.474 29.437-2.339 29.138Q-2.204 28.839-1.964 28.615Q-1.725 28.391-1.407 28.266Q-1.089 28.141-0.758 28.141Q-0.313 28.141 0.086 28.357Q0.486 28.572 0.721 28.950Q0.955 29.327 0.955 29.779Q0.955 30.120 0.813 30.404Q0.671 30.688 0.427 30.895Q0.182 31.101-0.127 31.216Q-0.436 31.330-0.758 31.330Q-1.188 31.330-1.590 31.129Q-1.992 30.927-2.233 30.575Q-2.474 30.223-2.474 29.779M-0.758 31.081Q-0.156 31.081 0.068 30.703Q0.292 30.325 0.292 29.693Q0.292 29.081 0.057 28.722Q-0.177 28.364-0.758 28.364Q-1.810 28.364-1.810 29.693Q-1.810 30.325-1.585 30.703Q-1.359 31.081-0.758 31.081M3.347 31.262L1.614 31.262L1.614 30.982Q1.840 30.982 1.989 30.948Q2.137 30.913 2.137 30.773L2.137 28.524L1.549 28.524L1.549 28.244L2.137 28.244L2.137 27.427Q2.137 27.109 2.315 26.861Q2.493 26.614 2.783 26.473Q3.074 26.333 3.385 26.333Q3.641 26.333 3.845 26.475Q4.048 26.617 4.048 26.860Q4.048 26.996 3.949 27.095Q3.850 27.195 3.713 27.195Q3.576 27.195 3.477 27.095Q3.378 26.996 3.378 26.860Q3.378 26.679 3.518 26.586Q3.440 26.559 3.340 26.559Q3.132 26.559 2.978 26.692Q2.824 26.825 2.744 27.029Q2.664 27.232 2.664 27.441L2.664 28.244L3.552 28.244L3.552 28.524L2.691 28.524L2.691 30.773Q2.691 30.982 3.347 30.982\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(121.612 -12.092)\">\u003Cpath d=\"M8.317 31.235L7.189 28.736Q7.117 28.589 6.987 28.557Q6.857 28.524 6.628 28.524L6.628 28.244L8.142 28.244L8.142 28.524Q7.790 28.524 7.790 28.671Q7.790 28.716 7.801 28.736L8.665 30.654L9.445 28.924Q9.479 28.856 9.479 28.777Q9.479 28.664 9.395 28.594Q9.311 28.524 9.192 28.524L9.192 28.244L10.388 28.244L10.388 28.524Q10.169 28.524 9.998 28.627Q9.828 28.729 9.739 28.924L8.703 31.235Q8.655 31.330 8.549 31.330L8.471 31.330Q8.365 31.330 8.317 31.235M12.811 30.008L10.754 30.008L10.754 29.505L12.811 29.505L12.811 30.008M15.259 32.619L13.628 32.619L13.628 32.339Q13.857 32.339 14.006 32.304Q14.155 32.270 14.155 32.130L14.155 28.784Q14.155 28.613 14.018 28.572Q13.881 28.531 13.628 28.531L13.628 28.251L14.708 28.176L14.708 28.582Q14.931 28.381 15.218 28.278Q15.505 28.176 15.812 28.176Q16.240 28.176 16.604 28.389Q16.968 28.603 17.181 28.967Q17.395 29.331 17.395 29.751Q17.395 30.196 17.156 30.560Q16.916 30.924 16.523 31.127Q16.130 31.330 15.686 31.330Q15.419 31.330 15.172 31.230Q14.924 31.129 14.736 30.948L14.736 32.130Q14.736 32.267 14.884 32.303Q15.033 32.339 15.259 32.339L15.259 32.619M14.736 28.931L14.736 30.541Q14.869 30.794 15.112 30.951Q15.354 31.108 15.631 31.108Q15.959 31.108 16.212 30.907Q16.465 30.705 16.599 30.387Q16.732 30.069 16.732 29.751Q16.732 29.522 16.667 29.293Q16.602 29.064 16.474 28.866Q16.346 28.668 16.151 28.548Q15.956 28.429 15.724 28.429Q15.430 28.429 15.161 28.558Q14.893 28.688 14.736 28.931M19.647 31.262L18.096 31.262L18.096 30.982Q18.321 30.982 18.470 30.948Q18.619 30.913 18.619 30.773L18.619 28.924Q18.619 28.736 18.571 28.652Q18.523 28.569 18.425 28.550Q18.328 28.531 18.116 28.531L18.116 28.251L19.172 28.176L19.172 30.773Q19.172 30.913 19.304 30.948Q19.435 30.982 19.647 30.982L19.647 31.262M18.376 26.955Q18.376 26.784 18.499 26.665Q18.622 26.545 18.793 26.545Q18.960 26.545 19.083 26.665Q19.206 26.784 19.206 26.955Q19.206 27.130 19.083 27.253Q18.960 27.376 18.793 27.376Q18.622 27.376 18.499 27.253Q18.376 27.130 18.376 26.955\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath stroke=\"none\" d=\"M113.404-64.054a2.4 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(137.83 -94.834)\">\u003Cpath d=\"M-17.840 31.231L-19.063 28.375Q-19.145 28.200-19.289 28.155Q-19.434 28.110-19.703 28.110L-19.703 27.813L-17.992 27.813L-17.992 28.110Q-18.414 28.110-18.414 28.293Q-18.414 28.328-18.399 28.375L-17.453 30.567L-16.613 28.590Q-16.574 28.512-16.574 28.422Q-16.574 28.282-16.680 28.196Q-16.785 28.110-16.926 28.110L-16.926 27.813L-15.574 27.813L-15.574 28.110Q-16.098 28.110-16.313 28.590L-17.438 31.231Q-17.500 31.340-17.606 31.340L-17.672 31.340Q-17.785 31.340-17.840 31.231M-13.047 29.813L-15.301 29.813L-15.301 29.262L-13.047 29.262L-13.047 29.813M-10.445 32.813L-12.301 32.813L-12.301 32.520Q-12.031 32.520-11.863 32.475Q-11.695 32.430-11.695 32.254L-11.695 28.430Q-11.695 28.223-11.852 28.170Q-12.008 28.117-12.301 28.117L-12.301 27.821L-11.078 27.735L-11.078 28.200Q-10.848 27.977-10.533 27.856Q-10.219 27.735-9.879 27.735Q-9.406 27.735-9.002 27.981Q-8.598 28.227-8.365 28.643Q-8.133 29.059-8.133 29.535Q-8.133 29.910-8.281 30.239Q-8.430 30.567-8.699 30.819Q-8.969 31.071-9.313 31.205Q-9.656 31.340-10.016 31.340Q-10.305 31.340-10.576 31.219Q-10.848 31.098-11.055 30.887L-11.055 32.254Q-11.055 32.430-10.887 32.475Q-10.719 32.520-10.445 32.520L-10.445 32.813M-11.055 28.598L-11.055 30.438Q-10.902 30.727-10.641 30.907Q-10.379 31.086-10.070 31.086Q-9.785 31.086-9.563 30.948Q-9.340 30.809-9.188 30.578Q-9.035 30.348-8.957 30.076Q-8.879 29.805-8.879 29.535Q-8.879 29.203-9.004 28.846Q-9.129 28.489-9.377 28.252Q-9.625 28.016-9.973 28.016Q-10.297 28.016-10.592 28.172Q-10.887 28.328-11.055 28.598M-5.750 31.262L-7.527 31.262L-7.527 30.965Q-7.254 30.965-7.086 30.918Q-6.918 30.871-6.918 30.703L-6.918 28.567Q-6.918 28.352-6.975 28.256Q-7.031 28.160-7.145 28.139Q-7.258 28.117-7.504 28.117L-7.504 27.821L-6.305 27.735L-6.305 30.703Q-6.305 30.871-6.158 30.918Q-6.012 30.965-5.750 30.965L-5.750 31.262M-7.192 26.340Q-7.192 26.149-7.057 26.018Q-6.922 25.887-6.727 25.887Q-6.606 25.887-6.502 25.950Q-6.399 26.012-6.336 26.116Q-6.274 26.219-6.274 26.340Q-6.274 26.535-6.404 26.670Q-6.535 26.805-6.727 26.805Q-6.926 26.805-7.059 26.672Q-7.192 26.539-7.192 26.340\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M111.004-64.054 96.778 9.069\" style=\"stroke-width:.8\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(137.83 -60.438)\">\u003Cpath d=\"M-17.975 31.235L-19.103 28.736Q-19.175 28.589-19.305 28.557Q-19.435 28.524-19.664 28.524L-19.664 28.244L-18.150 28.244L-18.150 28.524Q-18.502 28.524-18.502 28.671Q-18.502 28.716-18.491 28.736L-17.627 30.654L-16.847 28.924Q-16.813 28.856-16.813 28.777Q-16.813 28.664-16.897 28.594Q-16.981 28.524-17.100 28.524L-17.100 28.244L-15.904 28.244L-15.904 28.524Q-16.123 28.524-16.294 28.627Q-16.464 28.729-16.553 28.924L-17.589 31.235Q-17.637 31.330-17.743 31.330L-17.821 31.330Q-17.927 31.330-17.975 31.235\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(137.83 -60.438)\">\u003Cpath d=\"M-15.735 30.534Q-15.735 30.202-15.512 29.975Q-15.288 29.748-14.944 29.620Q-14.601 29.491-14.228 29.439Q-13.856 29.386-13.551 29.386L-13.551 29.133Q-13.551 28.928-13.659 28.748Q-13.767 28.569-13.948 28.466Q-14.129 28.364-14.337 28.364Q-14.744 28.364-14.980 28.456Q-14.891 28.493-14.845 28.577Q-14.799 28.661-14.799 28.763Q-14.799 28.859-14.845 28.938Q-14.891 29.016-14.972 29.061Q-15.052 29.105-15.141 29.105Q-15.291 29.105-15.392 29.008Q-15.493 28.910-15.493 28.763Q-15.493 28.141-14.337 28.141Q-14.126 28.141-13.876 28.205Q-13.627 28.268-13.425 28.387Q-13.223 28.507-13.097 28.692Q-12.970 28.876-12.970 29.119L-12.970 30.695Q-12.970 30.811-12.909 30.907Q-12.847 31.002-12.734 31.002Q-12.625 31.002-12.560 30.908Q-12.495 30.814-12.495 30.695L-12.495 30.247L-12.229 30.247L-12.229 30.695Q-12.229 30.965-12.456 31.130Q-12.683 31.296-12.963 31.296Q-13.172 31.296-13.309 31.142Q-13.445 30.989-13.469 30.773Q-13.616 31.040-13.898 31.185Q-14.180 31.330-14.505 31.330Q-14.782 31.330-15.066 31.255Q-15.349 31.180-15.542 31.001Q-15.735 30.821-15.735 30.534M-15.120 30.534Q-15.120 30.708-15.019 30.838Q-14.919 30.968-14.763 31.038Q-14.608 31.108-14.443 31.108Q-14.225 31.108-14.016 31.011Q-13.808 30.913-13.680 30.732Q-13.551 30.551-13.551 30.325L-13.551 29.597Q-13.876 29.597-14.242 29.688Q-14.608 29.779-14.864 29.991Q-15.120 30.202-15.120 30.534M-10.144 31.262L-11.747 31.262L-11.747 30.982Q-11.521 30.982-11.372 30.948Q-11.224 30.913-11.224 30.773L-11.224 27.154Q-11.224 26.884-11.331 26.822Q-11.439 26.761-11.747 26.761L-11.747 26.480L-10.670 26.405L-10.670 30.773Q-10.670 30.910-10.520 30.946Q-10.369 30.982-10.144 30.982L-10.144 31.262M-8.975 30.428L-8.975 28.924Q-8.975 28.654-9.082 28.593Q-9.190 28.531-9.501 28.531L-9.501 28.251L-8.394 28.176L-8.394 30.408L-8.394 30.428Q-8.394 30.708-8.342 30.852Q-8.291 30.995-8.149 31.052Q-8.007 31.108-7.720 31.108Q-7.467 31.108-7.262 30.968Q-7.057 30.828-6.941 30.602Q-6.825 30.377-6.825 30.127L-6.825 28.924Q-6.825 28.654-6.932 28.593Q-7.040 28.531-7.351 28.531L-7.351 28.251L-6.244 28.176L-6.244 30.589Q-6.244 30.780-6.191 30.862Q-6.138 30.944-6.037 30.963Q-5.936 30.982-5.721 30.982L-5.721 31.262L-6.797 31.330L-6.797 30.766Q-6.907 30.948-7.052 31.071Q-7.197 31.194-7.384 31.262Q-7.570 31.330-7.772 31.330Q-8.975 31.330-8.975 30.428M-5.174 29.727Q-5.174 29.406-5.049 29.117Q-4.924 28.828-4.699 28.605Q-4.473 28.381-4.178 28.261Q-3.882 28.141-3.564 28.141Q-3.236 28.141-2.974 28.241Q-2.713 28.340-2.537 28.522Q-2.361 28.705-2.267 28.963Q-2.173 29.221-2.173 29.553Q-2.173 29.645-2.255 29.666L-4.511 29.666L-4.511 29.727Q-4.511 30.315-4.227 30.698Q-3.943 31.081-3.376 31.081Q-3.055 31.081-2.786 30.888Q-2.518 30.695-2.429 30.380Q-2.422 30.339-2.347 30.325L-2.255 30.325Q-2.173 30.349-2.173 30.421Q-2.173 30.428-2.180 30.455Q-2.293 30.852-2.663 31.091Q-3.034 31.330-3.458 31.330Q-3.896 31.330-4.296 31.122Q-4.695 30.913-4.935 30.546Q-5.174 30.179-5.174 29.727M-4.504 29.457L-2.689 29.457Q-2.689 29.180-2.786 28.928Q-2.884 28.675-3.082 28.519Q-3.280 28.364-3.564 28.364Q-3.841 28.364-4.055 28.522Q-4.268 28.681-4.386 28.936Q-4.504 29.191-4.504 29.457\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(137.83 -60.438)\">\u003Cpath d=\"M1.082 29.727Q1.082 29.406 1.207 29.117Q1.332 28.828 1.558 28.605Q1.783 28.381 2.079 28.261Q2.374 28.141 2.692 28.141Q3.020 28.141 3.282 28.241Q3.543 28.340 3.719 28.522Q3.895 28.705 3.989 28.963Q4.083 29.221 4.083 29.553Q4.083 29.645 4.001 29.666L1.746 29.666L1.746 29.727Q1.746 30.315 2.029 30.698Q2.313 31.081 2.880 31.081Q3.202 31.081 3.470 30.888Q3.738 30.695 3.827 30.380Q3.834 30.339 3.909 30.325L4.001 30.325Q4.083 30.349 4.083 30.421Q4.083 30.428 4.077 30.455Q3.964 30.852 3.593 31.091Q3.222 31.330 2.798 31.330Q2.361 31.330 1.961 31.122Q1.561 30.913 1.322 30.546Q1.082 30.179 1.082 29.727M1.752 29.457L3.567 29.457Q3.567 29.180 3.470 28.928Q3.372 28.675 3.174 28.519Q2.976 28.364 2.692 28.364Q2.415 28.364 2.202 28.522Q1.988 28.681 1.870 28.936Q1.752 29.191 1.752 29.457M6.421 31.262L4.685 31.262L4.685 30.982Q4.914 30.982 5.063 30.948Q5.211 30.913 5.211 30.773L5.211 28.924Q5.211 28.654 5.104 28.593Q4.996 28.531 4.685 28.531L4.685 28.251L5.714 28.176L5.714 28.883Q5.844 28.575 6.086 28.376Q6.329 28.176 6.647 28.176Q6.866 28.176 7.037 28.300Q7.207 28.425 7.207 28.637Q7.207 28.774 7.108 28.873Q7.009 28.972 6.876 28.972Q6.739 28.972 6.640 28.873Q6.541 28.774 6.541 28.637Q6.541 28.497 6.640 28.398Q6.350 28.398 6.150 28.594Q5.950 28.791 5.857 29.085Q5.765 29.379 5.765 29.659L5.765 30.773Q5.765 30.982 6.421 30.982L6.421 31.262M9.542 31.262L7.806 31.262L7.806 30.982Q8.035 30.982 8.183 30.948Q8.332 30.913 8.332 30.773L8.332 28.924Q8.332 28.654 8.224 28.593Q8.117 28.531 7.806 28.531L7.806 28.251L8.834 28.176L8.834 28.883Q8.964 28.575 9.207 28.376Q9.450 28.176 9.767 28.176Q9.986 28.176 10.157 28.300Q10.328 28.425 10.328 28.637Q10.328 28.774 10.229 28.873Q10.130 28.972 9.997 28.972Q9.860 28.972 9.761 28.873Q9.662 28.774 9.662 28.637Q9.662 28.497 9.761 28.398Q9.470 28.398 9.270 28.594Q9.070 28.791 8.978 29.085Q8.886 29.379 8.886 29.659L8.886 30.773Q8.886 30.982 9.542 30.982L9.542 31.262M10.872 29.779Q10.872 29.437 11.007 29.138Q11.142 28.839 11.381 28.615Q11.620 28.391 11.938 28.266Q12.256 28.141 12.587 28.141Q13.032 28.141 13.432 28.357Q13.831 28.572 14.066 28.950Q14.300 29.327 14.300 29.779Q14.300 30.120 14.158 30.404Q14.016 30.688 13.772 30.895Q13.527 31.101 13.218 31.216Q12.909 31.330 12.587 31.330Q12.157 31.330 11.755 31.129Q11.353 30.927 11.112 30.575Q10.872 30.223 10.872 29.779M12.587 31.081Q13.189 31.081 13.413 30.703Q13.637 30.325 13.637 29.693Q13.637 29.081 13.403 28.722Q13.168 28.364 12.587 28.364Q11.535 28.364 11.535 29.693Q11.535 30.325 11.760 30.703Q11.986 31.081 12.587 31.081M16.644 31.262L14.908 31.262L14.908 30.982Q15.137 30.982 15.286 30.948Q15.434 30.913 15.434 30.773L15.434 28.924Q15.434 28.654 15.327 28.593Q15.219 28.531 14.908 28.531L14.908 28.251L15.937 28.176L15.937 28.883Q16.067 28.575 16.309 28.376Q16.552 28.176 16.870 28.176Q17.089 28.176 17.260 28.300Q17.431 28.425 17.431 28.637Q17.431 28.774 17.331 28.873Q17.232 28.972 17.099 28.972Q16.962 28.972 16.863 28.873Q16.764 28.774 16.764 28.637Q16.764 28.497 16.863 28.398Q16.573 28.398 16.373 28.594Q16.173 28.791 16.080 29.085Q15.988 29.379 15.988 29.659L15.988 30.773Q15.988 30.982 16.644 30.982\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(137.83 -60.438)\">\u003Cpath d=\"M23.171 31.303L21.329 26.972Q21.261 26.829 21.080 26.795Q20.899 26.761 20.635 26.761L20.635 26.480L22.604 26.480L22.604 26.761Q22.050 26.761 22.050 26.935Q22.054 26.945 22.055 26.955Q22.057 26.966 22.057 26.972L23.578 30.548L25.010 27.160Q25.034 27.123 25.034 27.068Q25.034 26.908 24.884 26.834Q24.733 26.761 24.549 26.761L24.549 26.480L26.097 26.480L26.097 26.761Q25.837 26.761 25.627 26.855Q25.417 26.949 25.321 27.160L23.565 31.303Q23.513 31.402 23.418 31.402L23.318 31.402Q23.219 31.402 23.171 31.303M31.101 31.262L26.699 31.262L26.699 30.982Q27.420 30.982 27.420 30.773L27.420 26.972Q27.420 26.761 26.699 26.761L26.699 26.480L30.988 26.480L31.197 28.117L30.934 28.117Q30.876 27.646 30.773 27.381Q30.670 27.116 30.486 26.983Q30.301 26.849 30.030 26.805Q29.758 26.761 29.259 26.761L28.476 26.761Q28.288 26.761 28.199 26.795Q28.110 26.829 28.110 26.972L28.110 28.637L28.685 28.637Q29.074 28.637 29.257 28.586Q29.440 28.534 29.522 28.362Q29.604 28.189 29.604 27.817L29.867 27.817L29.867 29.738L29.604 29.738Q29.604 29.365 29.522 29.192Q29.440 29.020 29.257 28.969Q29.074 28.917 28.685 28.917L28.110 28.917L28.110 30.773Q28.110 30.913 28.199 30.948Q28.288 30.982 28.476 30.982L29.324 30.982Q29.854 30.982 30.163 30.913Q30.472 30.845 30.660 30.678Q30.848 30.510 30.956 30.208Q31.064 29.905 31.149 29.392L31.416 29.392\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-warn)\" stroke=\"none\" d=\"M54.676-41.292a2 2 0 1 0-4 0 2 2 0 0 0 4 0m-2 0\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(77.51 -78.691)\">\u003Cpath d=\"M-16.499 31.262L-19.472 31.262L-19.472 30.982Q-18.751 30.982-18.751 30.773L-18.751 26.972Q-18.751 26.761-19.472 26.761L-19.472 26.480L-16.714 26.480Q-16.447 26.480-16.141 26.555Q-15.836 26.631-15.576 26.778Q-15.316 26.925-15.154 27.154Q-14.991 27.383-14.991 27.677Q-14.991 28.107-15.378 28.389Q-15.764 28.671-16.246 28.763Q-15.938 28.763-15.589 28.928Q-15.241 29.092-15.012 29.370Q-14.783 29.649-14.783 29.967Q-14.783 30.373-15.046 30.667Q-15.309 30.961-15.707 31.112Q-16.106 31.262-16.499 31.262M-18.108 28.890L-18.108 30.773Q-18.108 30.913-18.020 30.948Q-17.931 30.982-17.743 30.982L-16.714 30.982Q-16.427 30.982-16.150 30.855Q-15.873 30.729-15.702 30.495Q-15.531 30.261-15.531 29.967Q-15.531 29.751-15.613 29.555Q-15.695 29.358-15.846 29.208Q-15.996 29.057-16.193 28.974Q-16.389 28.890-16.605 28.890L-18.108 28.890M-18.108 26.972L-18.108 28.664L-16.926 28.664Q-16.639 28.664-16.358 28.545Q-16.078 28.425-15.899 28.198Q-15.719 27.970-15.719 27.677Q-15.719 27.427-15.858 27.213Q-15.996 27-16.225 26.880Q-16.454 26.761-16.714 26.761L-17.743 26.761Q-17.931 26.761-18.020 26.795Q-18.108 26.829-18.108 26.972M-12.096 30.008L-14.154 30.008L-14.154 29.505L-12.096 29.505L-12.096 30.008M-9.649 32.619L-11.279 32.619L-11.279 32.339Q-11.050 32.339-10.902 32.304Q-10.753 32.270-10.753 32.130L-10.753 28.784Q-10.753 28.613-10.890 28.572Q-11.026 28.531-11.279 28.531L-11.279 28.251L-10.199 28.176L-10.199 28.582Q-9.977 28.381-9.690 28.278Q-9.403 28.176-9.095 28.176Q-8.668 28.176-8.304 28.389Q-7.940 28.603-7.726 28.967Q-7.513 29.331-7.513 29.751Q-7.513 30.196-7.752 30.560Q-7.991 30.924-8.384 31.127Q-8.777 31.330-9.222 31.330Q-9.488 31.330-9.736 31.230Q-9.984 31.129-10.172 30.948L-10.172 32.130Q-10.172 32.267-10.023 32.303Q-9.875 32.339-9.649 32.339L-9.649 32.619M-10.172 28.931L-10.172 30.541Q-10.039 30.794-9.796 30.951Q-9.553 31.108-9.276 31.108Q-8.948 31.108-8.695 30.907Q-8.442 30.705-8.309 30.387Q-8.176 30.069-8.176 29.751Q-8.176 29.522-8.241 29.293Q-8.306 29.064-8.434 28.866Q-8.562 28.668-8.757 28.548Q-8.952 28.429-9.184 28.429Q-9.478 28.429-9.746 28.558Q-10.015 28.688-10.172 28.931M-5.260 31.262L-6.812 31.262L-6.812 30.982Q-6.587 30.982-6.438 30.948Q-6.289 30.913-6.289 30.773L-6.289 28.924Q-6.289 28.736-6.337 28.652Q-6.385 28.569-6.482 28.550Q-6.580 28.531-6.792 28.531L-6.792 28.251L-5.735 28.176L-5.735 30.773Q-5.735 30.913-5.604 30.948Q-5.472 30.982-5.260 30.982L-5.260 31.262M-6.532 26.955Q-6.532 26.784-6.409 26.665Q-6.286 26.545-6.115 26.545Q-5.947 26.545-5.824 26.665Q-5.701 26.784-5.701 26.955Q-5.701 27.130-5.824 27.253Q-5.947 27.376-6.115 27.376Q-6.286 27.376-6.409 27.253Q-6.532 27.130-6.532 26.955\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(77.51 -78.691)\">\u003Cpath d=\"M-0.315 31.235L-1.443 28.736Q-1.515 28.589-1.645 28.557Q-1.775 28.524-2.004 28.524L-2.004 28.244L-0.490 28.244L-0.490 28.524Q-0.842 28.524-0.842 28.671Q-0.842 28.716-0.831 28.736L0.033 30.654L0.813 28.924Q0.847 28.856 0.847 28.777Q0.847 28.664 0.763 28.594Q0.679 28.524 0.560 28.524L0.560 28.244L1.756 28.244L1.756 28.524Q1.537 28.524 1.366 28.627Q1.196 28.729 1.107 28.924L0.071 31.235Q0.023 31.330-0.083 31.330L-0.161 31.330Q-0.267 31.330-0.315 31.235M4.426 33.012Q3.875 32.612 3.504 32.057Q3.134 31.501 2.952 30.855Q2.771 30.209 2.771 29.512Q2.771 28.999 2.872 28.504Q2.973 28.008 3.178 27.557Q3.383 27.106 3.696 26.714Q4.009 26.323 4.426 26.019Q4.436 26.015 4.443 26.014Q4.449 26.012 4.460 26.012L4.528 26.012Q4.562 26.012 4.584 26.036Q4.607 26.060 4.607 26.097Q4.607 26.142 4.579 26.159Q4.231 26.460 3.978 26.844Q3.725 27.229 3.573 27.670Q3.421 28.111 3.349 28.567Q3.277 29.023 3.277 29.512Q3.277 30.513 3.586 31.400Q3.896 32.287 4.579 32.872Q4.607 32.889 4.607 32.933Q4.607 32.971 4.584 32.995Q4.562 33.019 4.528 33.019L4.460 33.019Q4.453 33.015 4.444 33.014Q4.436 33.012 4.426 33.012M6.804 31.235L5.823 28.736Q5.762 28.593 5.644 28.558Q5.526 28.524 5.311 28.524L5.311 28.244L6.791 28.244L6.791 28.524Q6.411 28.524 6.411 28.685Q6.411 28.695 6.425 28.736L7.139 30.568L7.813 28.863Q7.782 28.791 7.782 28.763Q7.782 28.736 7.755 28.736Q7.693 28.589 7.575 28.557Q7.457 28.524 7.245 28.524L7.245 28.244L8.643 28.244L8.643 28.524Q8.267 28.524 8.267 28.685Q8.267 28.716 8.274 28.736L9.030 30.674L9.717 28.924Q9.737 28.873 9.737 28.818Q9.737 28.678 9.624 28.601Q9.511 28.524 9.371 28.524L9.371 28.244L10.592 28.244L10.592 28.524Q10.386 28.524 10.231 28.630Q10.075 28.736 10.004 28.924L9.098 31.235Q9.064 31.330 8.951 31.330L8.883 31.330Q8.773 31.330 8.736 31.235L7.953 29.232L7.167 31.235Q7.133 31.330 7.020 31.330L6.951 31.330Q6.842 31.330 6.804 31.235M11.443 33.019L11.374 33.019Q11.340 33.019 11.318 32.993Q11.296 32.968 11.296 32.933Q11.296 32.889 11.326 32.872Q11.682 32.568 11.931 32.178Q12.181 31.788 12.333 31.356Q12.485 30.924 12.555 30.455Q12.625 29.987 12.625 29.512Q12.625 29.033 12.555 28.567Q12.485 28.100 12.331 27.665Q12.177 27.229 11.926 26.841Q11.675 26.453 11.326 26.159Q11.296 26.142 11.296 26.097Q11.296 26.063 11.318 26.038Q11.340 26.012 11.374 26.012L11.443 26.012Q11.453 26.012 11.461 26.014Q11.470 26.015 11.480 26.019Q12.024 26.419 12.396 26.972Q12.769 27.526 12.950 28.172Q13.131 28.818 13.131 29.512Q13.131 30.213 12.950 30.860Q12.769 31.508 12.395 32.062Q12.020 32.616 11.480 33.012Q11.470 33.012 11.461 33.014Q11.453 33.015 11.443 33.019\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"m19.955 11.346 32.72-52.638\" style=\"stroke-width:.8\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-21.348 -56.17)\">\u003Cpath d=\"M-16.499 31.262L-19.472 31.262L-19.472 30.982Q-18.751 30.982-18.751 30.773L-18.751 26.972Q-18.751 26.761-19.472 26.761L-19.472 26.480L-16.714 26.480Q-16.447 26.480-16.141 26.555Q-15.836 26.631-15.576 26.778Q-15.316 26.925-15.154 27.154Q-14.991 27.383-14.991 27.677Q-14.991 28.107-15.378 28.389Q-15.764 28.671-16.246 28.763Q-15.938 28.763-15.589 28.928Q-15.241 29.092-15.012 29.370Q-14.783 29.649-14.783 29.967Q-14.783 30.373-15.046 30.667Q-15.309 30.961-15.707 31.112Q-16.106 31.262-16.499 31.262M-18.108 28.890L-18.108 30.773Q-18.108 30.913-18.020 30.948Q-17.931 30.982-17.743 30.982L-16.714 30.982Q-16.427 30.982-16.150 30.855Q-15.873 30.729-15.702 30.495Q-15.531 30.261-15.531 29.967Q-15.531 29.751-15.613 29.555Q-15.695 29.358-15.846 29.208Q-15.996 29.057-16.193 28.974Q-16.389 28.890-16.605 28.890L-18.108 28.890M-18.108 26.972L-18.108 28.664L-16.926 28.664Q-16.639 28.664-16.358 28.545Q-16.078 28.425-15.899 28.198Q-15.719 27.970-15.719 27.677Q-15.719 27.427-15.858 27.213Q-15.996 27-16.225 26.880Q-16.454 26.761-16.714 26.761L-17.743 26.761Q-17.931 26.761-18.020 26.795Q-18.108 26.829-18.108 26.972M-14.021 29.727Q-14.021 29.406-13.896 29.117Q-13.771 28.828-13.546 28.605Q-13.320 28.381-13.024 28.261Q-12.729 28.141-12.411 28.141Q-12.083 28.141-11.821 28.241Q-11.560 28.340-11.384 28.522Q-11.208 28.705-11.114 28.963Q-11.020 29.221-11.020 29.553Q-11.020 29.645-11.102 29.666L-13.358 29.666L-13.358 29.727Q-13.358 30.315-13.074 30.698Q-12.790 31.081-12.223 31.081Q-11.901 31.081-11.633 30.888Q-11.365 30.695-11.276 30.380Q-11.269 30.339-11.194 30.325L-11.102 30.325Q-11.020 30.349-11.020 30.421Q-11.020 30.428-11.026 30.455Q-11.139 30.852-11.510 31.091Q-11.881 31.330-12.305 31.330Q-12.742 31.330-13.142 31.122Q-13.542 30.913-13.781 30.546Q-14.021 30.179-14.021 29.727M-13.351 29.457L-11.536 29.457Q-11.536 29.180-11.633 28.928Q-11.731 28.675-11.929 28.519Q-12.127 28.364-12.411 28.364Q-12.688 28.364-12.901 28.522Q-13.115 28.681-13.233 28.936Q-13.351 29.191-13.351 29.457M-8.764 31.262L-10.367 31.262L-10.367 30.982Q-10.141 30.982-9.993 30.948Q-9.844 30.913-9.844 30.773L-9.844 27.154Q-9.844 26.884-9.952 26.822Q-10.059 26.761-10.367 26.761L-10.367 26.480L-9.290 26.405L-9.290 30.773Q-9.290 30.910-9.140 30.946Q-8.989 30.982-8.764 30.982L-8.764 31.262M-6.501 31.262L-8.104 31.262L-8.104 30.982Q-7.879 30.982-7.730 30.948Q-7.581 30.913-7.581 30.773L-7.581 27.154Q-7.581 26.884-7.689 26.822Q-7.796 26.761-8.104 26.761L-8.104 26.480L-7.027 26.405L-7.027 30.773Q-7.027 30.910-6.877 30.946Q-6.727 30.982-6.501 30.982L-6.501 31.262M-4.225 31.262L-5.858 31.262L-5.858 30.982Q-5.629 30.982-5.481 30.948Q-5.332 30.913-5.332 30.773L-5.332 28.924Q-5.332 28.654-5.440 28.593Q-5.547 28.531-5.858 28.531L-5.858 28.251L-4.799 28.176L-4.799 28.825Q-4.628 28.517-4.324 28.346Q-4.020 28.176-3.674 28.176Q-3.275 28.176-2.998 28.316Q-2.721 28.456-2.635 28.804Q-2.468 28.511-2.169 28.343Q-1.870 28.176-1.525 28.176Q-1.019 28.176-0.735 28.399Q-0.451 28.623-0.451 29.119L-0.451 30.773Q-0.451 30.910-0.303 30.946Q-0.154 30.982 0.072 30.982L0.072 31.262L-1.559 31.262L-1.559 30.982Q-1.333 30.982-1.183 30.946Q-1.032 30.910-1.032 30.773L-1.032 29.133Q-1.032 28.798-1.152 28.598Q-1.272 28.398-1.586 28.398Q-1.856 28.398-2.090 28.534Q-2.324 28.671-2.463 28.905Q-2.601 29.139-2.601 29.413L-2.601 30.773Q-2.601 30.910-2.452 30.946Q-2.304 30.982-2.078 30.982L-2.078 31.262L-3.709 31.262L-3.709 30.982Q-3.480 30.982-3.331 30.948Q-3.182 30.913-3.182 30.773L-3.182 29.133Q-3.182 28.798-3.302 28.598Q-3.421 28.398-3.736 28.398Q-4.006 28.398-4.240 28.534Q-4.474 28.671-4.613 28.905Q-4.751 29.139-4.751 29.413L-4.751 30.773Q-4.751 30.910-4.601 30.946Q-4.450 30.982-4.225 30.982L-4.225 31.262M0.718 30.534Q0.718 30.202 0.942 29.975Q1.165 29.748 1.509 29.620Q1.852 29.491 2.225 29.439Q2.598 29.386 2.902 29.386L2.902 29.133Q2.902 28.928 2.794 28.748Q2.686 28.569 2.505 28.466Q2.324 28.364 2.116 28.364Q1.709 28.364 1.473 28.456Q1.562 28.493 1.608 28.577Q1.654 28.661 1.654 28.763Q1.654 28.859 1.608 28.938Q1.562 29.016 1.482 29.061Q1.401 29.105 1.312 29.105Q1.162 29.105 1.061 29.008Q0.960 28.910 0.960 28.763Q0.960 28.141 2.116 28.141Q2.328 28.141 2.577 28.205Q2.827 28.268 3.028 28.387Q3.230 28.507 3.356 28.692Q3.483 28.876 3.483 29.119L3.483 30.695Q3.483 30.811 3.544 30.907Q3.606 31.002 3.719 31.002Q3.828 31.002 3.893 30.908Q3.958 30.814 3.958 30.695L3.958 30.247L4.225 30.247L4.225 30.695Q4.225 30.965 3.997 31.130Q3.770 31.296 3.490 31.296Q3.281 31.296 3.144 31.142Q3.008 30.989 2.984 30.773Q2.837 31.040 2.555 31.185Q2.273 31.330 1.948 31.330Q1.671 31.330 1.388 31.255Q1.104 31.180 0.911 31.001Q0.718 30.821 0.718 30.534M1.333 30.534Q1.333 30.708 1.434 30.838Q1.535 30.968 1.690 31.038Q1.846 31.108 2.010 31.108Q2.228 31.108 2.437 31.011Q2.645 30.913 2.774 30.732Q2.902 30.551 2.902 30.325L2.902 29.597Q2.577 29.597 2.211 29.688Q1.846 29.779 1.589 29.991Q1.333 30.202 1.333 30.534M6.323 31.262L4.689 31.262L4.689 30.982Q4.918 30.982 5.067 30.948Q5.216 30.913 5.216 30.773L5.216 28.924Q5.216 28.654 5.108 28.593Q5 28.531 4.689 28.531L4.689 28.251L5.749 28.176L5.749 28.825Q5.920 28.517 6.224 28.346Q6.528 28.176 6.873 28.176Q7.379 28.176 7.663 28.399Q7.947 28.623 7.947 29.119L7.947 30.773Q7.947 30.910 8.095 30.946Q8.244 30.982 8.470 30.982L8.470 31.262L6.839 31.262L6.839 30.982Q7.068 30.982 7.217 30.948Q7.366 30.913 7.366 30.773L7.366 29.133Q7.366 28.798 7.246 28.598Q7.126 28.398 6.812 28.398Q6.542 28.398 6.308 28.534Q6.074 28.671 5.935 28.905Q5.797 29.139 5.797 29.413L5.797 30.773Q5.797 30.910 5.947 30.946Q6.098 30.982 6.323 30.982\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-21.348 -56.17)\">\u003Cpath d=\"M11.735 29.727Q11.735 29.406 11.860 29.117Q11.985 28.828 12.211 28.605Q12.436 28.381 12.732 28.261Q13.027 28.141 13.345 28.141Q13.673 28.141 13.935 28.241Q14.196 28.340 14.372 28.522Q14.548 28.705 14.642 28.963Q14.736 29.221 14.736 29.553Q14.736 29.645 14.654 29.666L12.399 29.666L12.399 29.727Q12.399 30.315 12.682 30.698Q12.966 31.081 13.533 31.081Q13.855 31.081 14.123 30.888Q14.391 30.695 14.480 30.380Q14.487 30.339 14.562 30.325L14.654 30.325Q14.736 30.349 14.736 30.421Q14.736 30.428 14.730 30.455Q14.617 30.852 14.246 31.091Q13.875 31.330 13.451 31.330Q13.014 31.330 12.614 31.122Q12.214 30.913 11.975 30.546Q11.735 30.179 11.735 29.727M12.405 29.457L14.220 29.457Q14.220 29.180 14.123 28.928Q14.025 28.675 13.827 28.519Q13.629 28.364 13.345 28.364Q13.068 28.364 12.855 28.522Q12.641 28.681 12.523 28.936Q12.405 29.191 12.405 29.457M17.074 31.262L15.338 31.262L15.338 30.982Q15.567 30.982 15.716 30.948Q15.864 30.913 15.864 30.773L15.864 28.924Q15.864 28.654 15.757 28.593Q15.649 28.531 15.338 28.531L15.338 28.251L16.367 28.176L16.367 28.883Q16.497 28.575 16.739 28.376Q16.982 28.176 17.300 28.176Q17.519 28.176 17.690 28.300Q17.860 28.425 17.860 28.637Q17.860 28.774 17.761 28.873Q17.662 28.972 17.529 28.972Q17.392 28.972 17.293 28.873Q17.194 28.774 17.194 28.637Q17.194 28.497 17.293 28.398Q17.003 28.398 16.803 28.594Q16.603 28.791 16.510 29.085Q16.418 29.379 16.418 29.659L16.418 30.773Q16.418 30.982 17.074 30.982L17.074 31.262M20.195 31.262L18.459 31.262L18.459 30.982Q18.688 30.982 18.836 30.948Q18.985 30.913 18.985 30.773L18.985 28.924Q18.985 28.654 18.877 28.593Q18.770 28.531 18.459 28.531L18.459 28.251L19.487 28.176L19.487 28.883Q19.617 28.575 19.860 28.376Q20.103 28.176 20.420 28.176Q20.639 28.176 20.810 28.300Q20.981 28.425 20.981 28.637Q20.981 28.774 20.882 28.873Q20.783 28.972 20.649 28.972Q20.513 28.972 20.414 28.873Q20.315 28.774 20.315 28.637Q20.315 28.497 20.414 28.398Q20.123 28.398 19.923 28.594Q19.723 28.791 19.631 29.085Q19.539 29.379 19.539 29.659L19.539 30.773Q19.539 30.982 20.195 30.982L20.195 31.262M21.524 29.779Q21.524 29.437 21.660 29.138Q21.795 28.839 22.034 28.615Q22.273 28.391 22.591 28.266Q22.909 28.141 23.240 28.141Q23.685 28.141 24.085 28.357Q24.484 28.572 24.719 28.950Q24.953 29.327 24.953 29.779Q24.953 30.120 24.811 30.404Q24.669 30.688 24.425 30.895Q24.180 31.101 23.871 31.216Q23.562 31.330 23.240 31.330Q22.810 31.330 22.408 31.129Q22.006 30.927 21.765 30.575Q21.524 30.223 21.524 29.779M23.240 31.081Q23.842 31.081 24.066 30.703Q24.290 30.325 24.290 29.693Q24.290 29.081 24.056 28.722Q23.821 28.364 23.240 28.364Q22.188 28.364 22.188 29.693Q22.188 30.325 22.413 30.703Q22.639 31.081 23.240 31.081M27.297 31.262L25.561 31.262L25.561 30.982Q25.790 30.982 25.939 30.948Q26.087 30.913 26.087 30.773L26.087 28.924Q26.087 28.654 25.980 28.593Q25.872 28.531 25.561 28.531L25.561 28.251L26.590 28.176L26.590 28.883Q26.720 28.575 26.962 28.376Q27.205 28.176 27.523 28.176Q27.742 28.176 27.913 28.300Q28.084 28.425 28.084 28.637Q28.084 28.774 27.984 28.873Q27.885 28.972 27.752 28.972Q27.615 28.972 27.516 28.873Q27.417 28.774 27.417 28.637Q27.417 28.497 27.516 28.398Q27.226 28.398 27.026 28.594Q26.826 28.791 26.733 29.085Q26.641 29.379 26.641 29.659L26.641 30.773Q26.641 30.982 27.297 30.982\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-21.348 -56.17)\">\u003Cpath d=\"M34.453 31.262L31.480 31.262L31.480 30.982Q32.201 30.982 32.201 30.773L32.201 26.972Q32.201 26.761 31.480 26.761L31.480 26.480L34.238 26.480Q34.505 26.480 34.811 26.555Q35.116 26.631 35.376 26.778Q35.636 26.925 35.798 27.154Q35.961 27.383 35.961 27.677Q35.961 28.107 35.574 28.389Q35.188 28.671 34.706 28.763Q35.014 28.763 35.363 28.928Q35.711 29.092 35.940 29.370Q36.169 29.649 36.169 29.967Q36.169 30.373 35.906 30.667Q35.643 30.961 35.245 31.112Q34.846 31.262 34.453 31.262M32.844 28.890L32.844 30.773Q32.844 30.913 32.932 30.948Q33.021 30.982 33.209 30.982L34.238 30.982Q34.525 30.982 34.802 30.855Q35.079 30.729 35.250 30.495Q35.421 30.261 35.421 29.967Q35.421 29.751 35.339 29.555Q35.257 29.358 35.106 29.208Q34.956 29.057 34.759 28.974Q34.563 28.890 34.347 28.890L32.844 28.890M32.844 26.972L32.844 28.664L34.026 28.664Q34.313 28.664 34.594 28.545Q34.874 28.425 35.053 28.198Q35.233 27.970 35.233 27.677Q35.233 27.427 35.094 27.213Q34.956 27 34.727 26.880Q34.498 26.761 34.238 26.761L33.209 26.761Q33.021 26.761 32.932 26.795Q32.844 26.829 32.844 26.972M41.440 31.262L37.037 31.262L37.037 30.982Q37.759 30.982 37.759 30.773L37.759 26.972Q37.759 26.761 37.037 26.761L37.037 26.480L41.327 26.480L41.535 28.117L41.272 28.117Q41.214 27.646 41.112 27.381Q41.009 27.116 40.824 26.983Q40.640 26.849 40.368 26.805Q40.096 26.761 39.597 26.761L38.815 26.761Q38.627 26.761 38.538 26.795Q38.449 26.829 38.449 26.972L38.449 28.637L39.023 28.637Q39.413 28.637 39.596 28.586Q39.779 28.534 39.861 28.362Q39.943 28.189 39.943 27.817L40.206 27.817L40.206 29.738L39.943 29.738Q39.943 29.365 39.861 29.192Q39.779 29.020 39.596 28.969Q39.413 28.917 39.023 28.917L38.449 28.917L38.449 30.773Q38.449 30.913 38.538 30.948Q38.627 30.982 38.815 30.982L39.662 30.982Q40.192 30.982 40.501 30.913Q40.811 30.845 40.999 30.678Q41.187 30.510 41.294 30.208Q41.402 29.905 41.488 29.392L41.754 29.392\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-warn)\" stroke=\"none\" d=\"M54.676 11.915a2 2 0 1 0-4 0 2 2 0 0 0 4 0m-2 0\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(47.079 -8.289)\">\u003Cpath d=\"M-17.921 32.619L-19.551 32.619L-19.551 32.339Q-19.322 32.339-19.173 32.304Q-19.025 32.270-19.025 32.130L-19.025 28.784Q-19.025 28.613-19.161 28.572Q-19.298 28.531-19.551 28.531L-19.551 28.251L-18.471 28.176L-18.471 28.582Q-18.249 28.381-17.962 28.278Q-17.674 28.176-17.367 28.176Q-16.940 28.176-16.576 28.389Q-16.212 28.603-15.998 28.967Q-15.784 29.331-15.784 29.751Q-15.784 30.196-16.024 30.560Q-16.263 30.924-16.656 31.127Q-17.049 31.330-17.493 31.330Q-17.760 31.330-18.008 31.230Q-18.255 31.129-18.443 30.948L-18.443 32.130Q-18.443 32.267-18.295 32.303Q-18.146 32.339-17.921 32.339L-17.921 32.619M-18.443 28.931L-18.443 30.541Q-18.310 30.794-18.067 30.951Q-17.825 31.108-17.548 31.108Q-17.220 31.108-16.967 30.907Q-16.714 30.705-16.581 30.387Q-16.447 30.069-16.447 29.751Q-16.447 29.522-16.512 29.293Q-16.577 29.064-16.705 28.866Q-16.834 28.668-17.028 28.548Q-17.223 28.429-17.456 28.429Q-17.750 28.429-18.018 28.558Q-18.286 28.688-18.443 28.931M-13.399 31.262L-15.135 31.262L-15.135 30.982Q-14.906 30.982-14.757 30.948Q-14.608 30.913-14.608 30.773L-14.608 28.924Q-14.608 28.654-14.716 28.593Q-14.824 28.531-15.135 28.531L-15.135 28.251L-14.106 28.176L-14.106 28.883Q-13.976 28.575-13.733 28.376Q-13.491 28.176-13.173 28.176Q-12.954 28.176-12.783 28.300Q-12.612 28.425-12.612 28.637Q-12.612 28.774-12.712 28.873Q-12.811 28.972-12.944 28.972Q-13.081 28.972-13.180 28.873Q-13.279 28.774-13.279 28.637Q-13.279 28.497-13.180 28.398Q-13.470 28.398-13.670 28.594Q-13.870 28.791-13.962 29.085Q-14.055 29.379-14.055 29.659L-14.055 30.773Q-14.055 30.982-13.399 30.982L-13.399 31.262M-12.069 29.779Q-12.069 29.437-11.934 29.138Q-11.799 28.839-11.560 28.615Q-11.320 28.391-11.003 28.266Q-10.685 28.141-10.353 28.141Q-9.909 28.141-9.509 28.357Q-9.109 28.572-8.875 28.950Q-8.641 29.327-8.641 29.779Q-8.641 30.120-8.783 30.404Q-8.924 30.688-9.169 30.895Q-9.413 31.101-9.723 31.216Q-10.032 31.330-10.353 31.330Q-10.784 31.330-11.185 31.129Q-11.587 30.927-11.828 30.575Q-12.069 30.223-12.069 29.779M-10.353 31.081Q-9.752 31.081-9.528 30.703Q-9.304 30.325-9.304 29.693Q-9.304 29.081-9.538 28.722Q-9.772 28.364-10.353 28.364Q-11.406 28.364-11.406 29.693Q-11.406 30.325-11.180 30.703Q-10.955 31.081-10.353 31.081\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(47.079 -8.289)\">\u003Cpath d=\"M-8.135 32.110Q-8.135 31.959-8.035 31.862Q-7.934 31.764-7.787 31.764Q-7.698 31.764-7.618 31.809Q-7.537 31.853-7.491 31.932Q-7.445 32.011-7.445 32.110Q-7.445 32.311-7.606 32.410Q-7.465 32.465-7.260 32.465Q-6.990 32.465-6.874 32.192Q-6.758 31.918-6.758 31.604L-6.758 28.924Q-6.758 28.658-6.884 28.594Q-7.011 28.531-7.339 28.531L-7.339 28.251L-6.204 28.176L-6.204 31.624Q-6.204 31.911-6.351 32.156Q-6.498 32.400-6.749 32.545Q-7.001 32.691-7.277 32.691Q-7.606 32.691-7.870 32.549Q-8.135 32.407-8.135 32.110M-7.038 26.955Q-7.038 26.784-6.915 26.665Q-6.792 26.545-6.618 26.545Q-6.450 26.545-6.327 26.665Q-6.204 26.784-6.204 26.955Q-6.204 27.130-6.327 27.253Q-6.450 27.376-6.618 27.376Q-6.792 27.376-6.915 27.253Q-7.038 27.130-7.038 26.955\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(47.079 -8.289)\">\u003Cpath d=\"M0.633 31.262L-2.340 31.262L-2.340 30.982Q-1.619 30.982-1.619 30.773L-1.619 26.972Q-1.619 26.761-2.340 26.761L-2.340 26.480L0.418 26.480Q0.685 26.480 0.991 26.555Q1.296 26.631 1.556 26.778Q1.816 26.925 1.978 27.154Q2.141 27.383 2.141 27.677Q2.141 28.107 1.754 28.389Q1.368 28.671 0.886 28.763Q1.194 28.763 1.543 28.928Q1.891 29.092 2.120 29.370Q2.349 29.649 2.349 29.967Q2.349 30.373 2.086 30.667Q1.823 30.961 1.425 31.112Q1.026 31.262 0.633 31.262M-0.976 28.890L-0.976 30.773Q-0.976 30.913-0.888 30.948Q-0.799 30.982-0.611 30.982L0.418 30.982Q0.705 30.982 0.982 30.855Q1.259 30.729 1.430 30.495Q1.601 30.261 1.601 29.967Q1.601 29.751 1.519 29.555Q1.437 29.358 1.286 29.208Q1.136 29.057 0.939 28.974Q0.743 28.890 0.527 28.890L-0.976 28.890M-0.976 26.972L-0.976 28.664L0.206 28.664Q0.493 28.664 0.774 28.545Q1.054 28.425 1.233 28.198Q1.413 27.970 1.413 27.677Q1.413 27.427 1.274 27.213Q1.136 27 0.907 26.880Q0.678 26.761 0.418 26.761L-0.611 26.761Q-0.799 26.761-0.888 26.795Q-0.976 26.829-0.976 26.972M5.036 30.008L2.978 30.008L2.978 29.505L5.036 29.505L5.036 30.008M7.483 32.619L5.853 32.619L5.853 32.339Q6.082 32.339 6.230 32.304Q6.379 32.270 6.379 32.130L6.379 28.784Q6.379 28.613 6.242 28.572Q6.106 28.531 5.853 28.531L5.853 28.251L6.933 28.176L6.933 28.582Q7.155 28.381 7.442 28.278Q7.729 28.176 8.037 28.176Q8.464 28.176 8.828 28.389Q9.192 28.603 9.406 28.967Q9.619 29.331 9.619 29.751Q9.619 30.196 9.380 30.560Q9.141 30.924 8.748 31.127Q8.355 31.330 7.910 31.330Q7.644 31.330 7.396 31.230Q7.148 31.129 6.960 30.948L6.960 32.130Q6.960 32.267 7.109 32.303Q7.257 32.339 7.483 32.339L7.483 32.619M6.960 28.931L6.960 30.541Q7.093 30.794 7.336 30.951Q7.579 31.108 7.856 31.108Q8.184 31.108 8.437 30.907Q8.690 30.705 8.823 30.387Q8.956 30.069 8.956 29.751Q8.956 29.522 8.891 29.293Q8.826 29.064 8.698 28.866Q8.570 28.668 8.375 28.548Q8.180 28.429 7.948 28.429Q7.654 28.429 7.386 28.558Q7.117 28.688 6.960 28.931M11.872 31.262L10.320 31.262L10.320 30.982Q10.545 30.982 10.694 30.948Q10.843 30.913 10.843 30.773L10.843 28.924Q10.843 28.736 10.795 28.652Q10.747 28.569 10.650 28.550Q10.552 28.531 10.340 28.531L10.340 28.251L11.397 28.176L11.397 30.773Q11.397 30.913 11.528 30.948Q11.660 30.982 11.872 30.982L11.872 31.262M10.600 26.955Q10.600 26.784 10.723 26.665Q10.846 26.545 11.017 26.545Q11.185 26.545 11.308 26.665Q11.431 26.784 11.431 26.955Q11.431 27.130 11.308 27.253Q11.185 27.376 11.017 27.376Q10.846 27.376 10.723 27.253Q10.600 27.130 10.600 26.955\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(47.079 -8.289)\">\u003Cpath d=\"M16.817 31.235L15.689 28.736Q15.617 28.589 15.487 28.557Q15.357 28.524 15.128 28.524L15.128 28.244L16.642 28.244L16.642 28.524Q16.290 28.524 16.290 28.671Q16.290 28.716 16.301 28.736L17.165 30.654L17.945 28.924Q17.979 28.856 17.979 28.777Q17.979 28.664 17.895 28.594Q17.811 28.524 17.692 28.524L17.692 28.244L18.888 28.244L18.888 28.524Q18.669 28.524 18.498 28.627Q18.328 28.729 18.239 28.924L17.203 31.235Q17.155 31.330 17.049 31.330L16.971 31.330Q16.865 31.330 16.817 31.235M21.558 33.012Q21.007 32.612 20.636 32.057Q20.266 31.501 20.084 30.855Q19.903 30.209 19.903 29.512Q19.903 28.999 20.004 28.504Q20.105 28.008 20.310 27.557Q20.515 27.106 20.828 26.714Q21.141 26.323 21.558 26.019Q21.568 26.015 21.575 26.014Q21.581 26.012 21.592 26.012L21.660 26.012Q21.694 26.012 21.716 26.036Q21.739 26.060 21.739 26.097Q21.739 26.142 21.711 26.159Q21.363 26.460 21.110 26.844Q20.857 27.229 20.705 27.670Q20.553 28.111 20.481 28.567Q20.409 29.023 20.409 29.512Q20.409 30.513 20.718 31.400Q21.028 32.287 21.711 32.872Q21.739 32.889 21.739 32.933Q21.739 32.971 21.716 32.995Q21.694 33.019 21.660 33.019L21.592 33.019Q21.585 33.015 21.576 33.014Q21.568 33.012 21.558 33.012M23.936 31.235L22.955 28.736Q22.894 28.593 22.776 28.558Q22.658 28.524 22.443 28.524L22.443 28.244L23.923 28.244L23.923 28.524Q23.543 28.524 23.543 28.685Q23.543 28.695 23.557 28.736L24.271 30.568L24.945 28.863Q24.914 28.791 24.914 28.763Q24.914 28.736 24.887 28.736Q24.825 28.589 24.707 28.557Q24.589 28.524 24.377 28.524L24.377 28.244L25.775 28.244L25.775 28.524Q25.399 28.524 25.399 28.685Q25.399 28.716 25.406 28.736L26.162 30.674L26.849 28.924Q26.869 28.873 26.869 28.818Q26.869 28.678 26.756 28.601Q26.643 28.524 26.503 28.524L26.503 28.244L27.724 28.244L27.724 28.524Q27.518 28.524 27.363 28.630Q27.207 28.736 27.136 28.924L26.230 31.235Q26.196 31.330 26.083 31.330L26.015 31.330Q25.905 31.330 25.868 31.235L25.085 29.232L24.299 31.235Q24.265 31.330 24.152 31.330L24.083 31.330Q23.974 31.330 23.936 31.235M28.575 33.019L28.506 33.019Q28.472 33.019 28.450 32.993Q28.428 32.968 28.428 32.933Q28.428 32.889 28.458 32.872Q28.814 32.568 29.063 32.178Q29.313 31.788 29.465 31.356Q29.617 30.924 29.687 30.455Q29.757 29.987 29.757 29.512Q29.757 29.033 29.687 28.567Q29.617 28.100 29.463 27.665Q29.309 27.229 29.058 26.841Q28.807 26.453 28.458 26.159Q28.428 26.142 28.428 26.097Q28.428 26.063 28.450 26.038Q28.472 26.012 28.506 26.012L28.575 26.012Q28.585 26.012 28.593 26.014Q28.602 26.015 28.612 26.019Q29.156 26.419 29.528 26.972Q29.901 27.526 30.082 28.172Q30.263 28.818 30.263 29.512Q30.263 30.213 30.082 30.860Q29.901 31.508 29.527 32.062Q29.152 32.616 28.612 33.012Q28.602 33.012 28.593 33.014Q28.585 33.015 28.575 33.019\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"M52.676-41.292v53.207\" style=\"stroke-dasharray:3.0,3.0\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"m19.955 11.346 28.699.522\"\u002F>\u003Cpath stroke=\"none\" d=\"M51.253 11.915 47.132 9.76l1.522 2.108-1.598 2.05\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg transform=\"translate(47.313 -24.019)\">\u003Cpath d=\"M-17.353 31.262L-19.486 31.262L-19.486 30.982Q-18.765 30.982-18.765 30.773L-18.765 26.972Q-18.765 26.761-19.486 26.761L-19.486 26.480L-16.820 26.480Q-16.410 26.480-15.989 26.634Q-15.569 26.788-15.285 27.092Q-15.002 27.396-15.002 27.810Q-15.002 28.128-15.169 28.374Q-15.337 28.620-15.613 28.786Q-15.890 28.951-16.212 29.035Q-16.533 29.119-16.820 29.119L-18.074 29.119L-18.074 30.773Q-18.074 30.982-17.353 30.982L-17.353 31.262M-18.102 26.972L-18.102 28.869L-17.015 28.869Q-16.406 28.869-16.092 28.632Q-15.777 28.394-15.777 27.810Q-15.777 27.417-15.923 27.183Q-16.068 26.949-16.340 26.855Q-16.611 26.761-17.015 26.761L-17.736 26.761Q-17.924 26.761-18.013 26.795Q-18.102 26.829-18.102 26.972M-11.129 31.262L-14.103 31.262L-14.103 30.982Q-13.381 30.982-13.381 30.773L-13.381 26.972Q-13.381 26.761-14.103 26.761L-14.103 26.480L-11.344 26.480Q-11.078 26.480-10.772 26.555Q-10.466 26.631-10.206 26.778Q-9.946 26.925-9.784 27.154Q-9.622 27.383-9.622 27.677Q-9.622 28.107-10.008 28.389Q-10.394 28.671-10.876 28.763Q-10.568 28.763-10.220 28.928Q-9.871 29.092-9.642 29.370Q-9.413 29.649-9.413 29.967Q-9.413 30.373-9.676 30.667Q-9.940 30.961-10.338 31.112Q-10.736 31.262-11.129 31.262M-12.739 28.890L-12.739 30.773Q-12.739 30.913-12.650 30.948Q-12.561 30.982-12.373 30.982L-11.344 30.982Q-11.057 30.982-10.780 30.855Q-10.504 30.729-10.333 30.495Q-10.162 30.261-10.162 29.967Q-10.162 29.751-10.244 29.555Q-10.326 29.358-10.476 29.208Q-10.627 29.057-10.823 28.974Q-11.020 28.890-11.235 28.890L-12.739 28.890M-12.739 26.972L-12.739 28.664L-11.556 28.664Q-11.269 28.664-10.989 28.545Q-10.709 28.425-10.529 28.198Q-10.350 27.970-10.350 27.677Q-10.350 27.427-10.488 27.213Q-10.627 27-10.856 26.880Q-11.085 26.761-11.344 26.761L-12.373 26.761Q-12.561 26.761-12.650 26.795Q-12.739 26.829-12.739 26.972M-4.143 31.262L-8.545 31.262L-8.545 30.982Q-7.824 30.982-7.824 30.773L-7.824 26.972Q-7.824 26.761-8.545 26.761L-8.545 26.480L-4.255 26.480L-4.047 28.117L-4.310 28.117Q-4.368 27.646-4.471 27.381Q-4.573 27.116-4.758 26.983Q-4.942 26.849-5.214 26.805Q-5.486 26.761-5.985 26.761L-6.768 26.761Q-6.956 26.761-7.045 26.795Q-7.133 26.829-7.133 26.972L-7.133 28.637L-6.559 28.637Q-6.170 28.637-5.987 28.586Q-5.804 28.534-5.722 28.362Q-5.640 28.189-5.640 27.817L-5.377 27.817L-5.377 29.738L-5.640 29.738Q-5.640 29.365-5.722 29.192Q-5.804 29.020-5.987 28.969Q-6.170 28.917-6.559 28.917L-7.133 28.917L-7.133 30.773Q-7.133 30.913-7.045 30.948Q-6.956 30.982-6.768 30.982L-5.920 30.982Q-5.390 30.982-5.081 30.913Q-4.772 30.845-4.584 30.678Q-4.396 30.510-4.288 30.208Q-4.180 29.905-4.095 29.392L-3.828 29.392\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">Linear value-function geometry. The true value \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\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\" style=\"margin-right:0.0359em;\">π\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> lies outside the subspace of representable functions; the projection operator \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\">Π\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> drops it to the closest representable point \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">Π\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\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\" style=\"margin-right:0.0359em;\">π\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>, and the Bellman operator \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8333em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0502em;\">B\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0502em;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\" style=\"margin-right:0.0359em;\">π\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> (gray) takes points out of the subspace before projection maps them back.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:318.152px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 238.614 142.779\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" d=\"M-40.744 53.268h180.097\"\u002F>\u003Cpath stroke=\"none\" d=\"m141.353 53.268-3.2-1.6 1.2 1.6-1.2 1.6\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(155.207 6.547)\">\u003Cpath d=\"M-40.430 53.261L-40.430 52.198Q-40.430 52.174-40.402 52.147Q-40.375 52.120-40.351 52.120L-40.242 52.120Q-40.177 52.120-40.163 52.178Q-40.067 52.612-39.821 52.863Q-39.575 53.114-39.161 53.114Q-38.820 53.114-38.567 52.981Q-38.314 52.848-38.314 52.540Q-38.314 52.383-38.408 52.268Q-38.502 52.154-38.640 52.085Q-38.779 52.017-38.946 51.979L-39.527 51.880Q-39.883 51.812-40.156 51.591Q-40.430 51.371-40.430 51.029Q-40.430 50.780-40.318 50.605Q-40.207 50.431-40.021 50.332Q-39.835 50.233-39.619 50.190Q-39.404 50.147-39.161 50.147Q-38.748 50.147-38.468 50.329L-38.252 50.154Q-38.242 50.151-38.235 50.149Q-38.228 50.147-38.218 50.147L-38.167 50.147Q-38.140 50.147-38.116 50.171Q-38.092 50.195-38.092 50.223L-38.092 51.070Q-38.092 51.091-38.116 51.118Q-38.140 51.145-38.167 51.145L-38.280 51.145Q-38.307 51.145-38.333 51.120Q-38.358 51.094-38.358 51.070Q-38.358 50.834-38.464 50.670Q-38.570 50.506-38.753 50.424Q-38.936 50.342-39.168 50.342Q-39.496 50.342-39.753 50.445Q-40.009 50.547-40.009 50.824Q-40.009 51.019-39.826 51.128Q-39.643 51.238-39.414 51.279L-38.840 51.385Q-38.594 51.433-38.380 51.561Q-38.167 51.689-38.030 51.892Q-37.893 52.096-37.893 52.345Q-37.893 52.858-38.259 53.097Q-38.625 53.336-39.161 53.336Q-39.657 53.336-39.989 53.042L-40.255 53.316Q-40.276 53.336-40.303 53.336L-40.351 53.336Q-40.375 53.336-40.402 53.309Q-40.430 53.282-40.430 53.261M-35.877 53.241L-36.858 50.742Q-36.919 50.599-37.037 50.564Q-37.155 50.530-37.370 50.530L-37.370 50.250L-35.890 50.250L-35.890 50.530Q-36.270 50.530-36.270 50.691Q-36.270 50.701-36.256 50.742L-35.542 52.574L-34.869 50.869Q-34.899 50.797-34.899 50.769Q-34.899 50.742-34.927 50.742Q-34.988 50.595-35.106 50.563Q-35.224 50.530-35.436 50.530L-35.436 50.250L-34.038 50.250L-34.038 50.530Q-34.414 50.530-34.414 50.691Q-34.414 50.722-34.407 50.742L-33.652 52.680L-32.965 50.930Q-32.944 50.879-32.944 50.824Q-32.944 50.684-33.057 50.607Q-33.170 50.530-33.310 50.530L-33.310 50.250L-32.090 50.250L-32.090 50.530Q-32.295 50.530-32.450 50.636Q-32.606 50.742-32.678 50.930L-33.583 53.241Q-33.618 53.336-33.730 53.336L-33.799 53.336Q-33.908 53.336-33.946 53.241L-34.728 51.238L-35.515 53.241Q-35.549 53.336-35.661 53.336L-35.730 53.336Q-35.839 53.336-35.877 53.241\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(155.207 6.547)\">\u003Cpath d=\"M-31.810 51.733Q-31.810 51.412-31.685 51.123Q-31.560 50.834-31.334 50.611Q-31.109 50.387-30.813 50.267Q-30.518 50.147-30.200 50.147Q-29.872 50.147-29.610 50.247Q-29.349 50.346-29.173 50.528Q-28.997 50.711-28.903 50.969Q-28.809 51.227-28.809 51.559Q-28.809 51.651-28.891 51.672L-31.146 51.672L-31.146 51.733Q-31.146 52.321-30.863 52.704Q-30.579 53.087-30.012 53.087Q-29.690 53.087-29.422 52.894Q-29.154 52.701-29.065 52.386Q-29.058 52.345-28.983 52.331L-28.891 52.331Q-28.809 52.355-28.809 52.427Q-28.809 52.434-28.815 52.461Q-28.928 52.858-29.299 53.097Q-29.670 53.336-30.094 53.336Q-30.531 53.336-30.931 53.128Q-31.331 52.919-31.570 52.552Q-31.810 52.185-31.810 51.733M-31.140 51.463L-29.325 51.463Q-29.325 51.186-29.422 50.934Q-29.520 50.681-29.718 50.525Q-29.916 50.370-30.200 50.370Q-30.477 50.370-30.690 50.528Q-30.904 50.687-31.022 50.942Q-31.140 51.197-31.140 51.463M-28.262 51.733Q-28.262 51.412-28.137 51.123Q-28.012 50.834-27.787 50.611Q-27.561 50.387-27.265 50.267Q-26.970 50.147-26.652 50.147Q-26.324 50.147-26.062 50.247Q-25.801 50.346-25.625 50.528Q-25.449 50.711-25.355 50.969Q-25.261 51.227-25.261 51.559Q-25.261 51.651-25.343 51.672L-27.599 51.672L-27.599 51.733Q-27.599 52.321-27.315 52.704Q-27.031 53.087-26.464 53.087Q-26.143 53.087-25.874 52.894Q-25.606 52.701-25.517 52.386Q-25.510 52.345-25.435 52.331L-25.343 52.331Q-25.261 52.355-25.261 52.427Q-25.261 52.434-25.268 52.461Q-25.380 52.858-25.751 53.097Q-26.122 53.336-26.546 53.336Q-26.983 53.336-27.383 53.128Q-27.783 52.919-28.022 52.552Q-28.262 52.185-28.262 51.733M-27.592 51.463L-25.777 51.463Q-25.777 51.186-25.874 50.934Q-25.972 50.681-26.170 50.525Q-26.368 50.370-26.652 50.370Q-26.929 50.370-27.142 50.528Q-27.356 50.687-27.474 50.942Q-27.592 51.197-27.592 51.463M-23.029 54.625L-24.659 54.625L-24.659 54.345Q-24.430 54.345-24.281 54.310Q-24.133 54.276-24.133 54.136L-24.133 50.790Q-24.133 50.619-24.270 50.578Q-24.406 50.537-24.659 50.537L-24.659 50.257L-23.579 50.182L-23.579 50.588Q-23.357 50.387-23.070 50.284Q-22.783 50.182-22.475 50.182Q-22.048 50.182-21.684 50.395Q-21.320 50.609-21.106 50.973Q-20.893 51.337-20.893 51.757Q-20.893 52.202-21.132 52.566Q-21.371 52.930-21.764 53.133Q-22.157 53.336-22.602 53.336Q-22.868 53.336-23.116 53.236Q-23.364 53.135-23.552 52.954L-23.552 54.136Q-23.552 54.273-23.403 54.309Q-23.254 54.345-23.029 54.345L-23.029 54.625M-23.552 50.937L-23.552 52.547Q-23.418 52.800-23.176 52.957Q-22.933 53.114-22.656 53.114Q-22.328 53.114-22.075 52.913Q-21.822 52.711-21.689 52.393Q-21.556 52.075-21.556 51.757Q-21.556 51.528-21.621 51.299Q-21.686 51.070-21.814 50.872Q-21.942 50.674-22.137 50.554Q-22.332 50.435-22.564 50.435Q-22.858 50.435-23.126 50.564Q-23.395 50.694-23.552 50.937M-20.257 53.261L-20.257 52.198Q-20.257 52.174-20.229 52.147Q-20.202 52.120-20.178 52.120L-20.069 52.120Q-20.004 52.120-19.990 52.178Q-19.895 52.612-19.648 52.863Q-19.402 53.114-18.989 53.114Q-18.647 53.114-18.394 52.981Q-18.141 52.848-18.141 52.540Q-18.141 52.383-18.235 52.268Q-18.329 52.154-18.468 52.085Q-18.606 52.017-18.773 51.979L-19.354 51.880Q-19.710 51.812-19.983 51.591Q-20.257 51.371-20.257 51.029Q-20.257 50.780-20.146 50.605Q-20.035 50.431-19.848 50.332Q-19.662 50.233-19.447 50.190Q-19.231 50.147-18.989 50.147Q-18.575 50.147-18.295 50.329L-18.080 50.154Q-18.069 50.151-18.062 50.149Q-18.056 50.147-18.045 50.147L-17.994 50.147Q-17.967 50.147-17.943 50.171Q-17.919 50.195-17.919 50.223L-17.919 51.070Q-17.919 51.091-17.943 51.118Q-17.967 51.145-17.994 51.145L-18.107 51.145Q-18.134 51.145-18.160 51.120Q-18.186 51.094-18.186 51.070Q-18.186 50.834-18.291 50.670Q-18.397 50.506-18.580 50.424Q-18.763 50.342-18.996 50.342Q-19.324 50.342-19.580 50.445Q-19.836 50.547-19.836 50.824Q-19.836 51.019-19.654 51.128Q-19.471 51.238-19.242 51.279L-18.667 51.385Q-18.421 51.433-18.208 51.561Q-17.994 51.689-17.857 51.892Q-17.721 52.096-17.721 52.345Q-17.721 52.858-18.086 53.097Q-18.452 53.336-18.989 53.336Q-19.484 53.336-19.816 53.042L-20.083 53.316Q-20.103 53.336-20.130 53.336L-20.178 53.336Q-20.202 53.336-20.229 53.309Q-20.257 53.282-20.257 53.261\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-40.744 58.958V-58.543\"\u002F>\u003Cpath stroke=\"none\" d=\"m-40.744-60.543-1.6 3.2 1.6-1.2 1.6 1.2\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-21.526 -117.344)\">\u003Cpath d=\"M-38.840 53.241L-39.968 50.742Q-40.040 50.595-40.170 50.563Q-40.300 50.530-40.529 50.530L-40.529 50.250L-39.015 50.250L-39.015 50.530Q-39.367 50.530-39.367 50.677Q-39.367 50.722-39.356 50.742L-38.492 52.660L-37.712 50.930Q-37.678 50.862-37.678 50.783Q-37.678 50.670-37.762 50.600Q-37.846 50.530-37.965 50.530L-37.965 50.250L-36.769 50.250L-36.769 50.530Q-36.988 50.530-37.159 50.633Q-37.329 50.735-37.418 50.930L-38.454 53.241Q-38.502 53.336-38.608 53.336L-38.686 53.336Q-38.792 53.336-38.840 53.241\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-21.526 -117.344)\">\u003Cpath d=\"M-36.600 52.540Q-36.600 52.208-36.377 51.981Q-36.153 51.754-35.809 51.626Q-35.466 51.497-35.093 51.445Q-34.721 51.392-34.416 51.392L-34.416 51.139Q-34.416 50.934-34.524 50.754Q-34.632 50.575-34.813 50.472Q-34.994 50.370-35.202 50.370Q-35.609 50.370-35.845 50.462Q-35.756 50.499-35.710 50.583Q-35.664 50.667-35.664 50.769Q-35.664 50.865-35.710 50.944Q-35.756 51.022-35.837 51.067Q-35.917 51.111-36.006 51.111Q-36.156 51.111-36.257 51.014Q-36.358 50.916-36.358 50.769Q-36.358 50.147-35.202 50.147Q-34.991 50.147-34.741 50.211Q-34.492 50.274-34.290 50.393Q-34.088 50.513-33.962 50.698Q-33.835 50.882-33.835 51.125L-33.835 52.701Q-33.835 52.817-33.774 52.913Q-33.712 53.008-33.599 53.008Q-33.490 53.008-33.425 52.914Q-33.360 52.820-33.360 52.701L-33.360 52.253L-33.094 52.253L-33.094 52.701Q-33.094 52.971-33.321 53.136Q-33.548 53.302-33.828 53.302Q-34.037 53.302-34.174 53.148Q-34.310 52.995-34.334 52.779Q-34.481 53.046-34.763 53.191Q-35.045 53.336-35.370 53.336Q-35.647 53.336-35.931 53.261Q-36.214 53.186-36.407 53.007Q-36.600 52.827-36.600 52.540M-35.985 52.540Q-35.985 52.714-35.884 52.844Q-35.784 52.974-35.628 53.044Q-35.473 53.114-35.308 53.114Q-35.090 53.114-34.881 53.017Q-34.673 52.919-34.545 52.738Q-34.416 52.557-34.416 52.331L-34.416 51.603Q-34.741 51.603-35.107 51.694Q-35.473 51.785-35.729 51.997Q-35.985 52.208-35.985 52.540M-31.009 53.268L-32.612 53.268L-32.612 52.988Q-32.386 52.988-32.237 52.954Q-32.089 52.919-32.089 52.779L-32.089 49.160Q-32.089 48.890-32.196 48.828Q-32.304 48.767-32.612 48.767L-32.612 48.486L-31.535 48.411L-31.535 52.779Q-31.535 52.916-31.385 52.952Q-31.234 52.988-31.009 52.988L-31.009 53.268M-29.840 52.434L-29.840 50.930Q-29.840 50.660-29.947 50.599Q-30.055 50.537-30.366 50.537L-30.366 50.257L-29.259 50.182L-29.259 52.414L-29.259 52.434Q-29.259 52.714-29.207 52.858Q-29.156 53.001-29.014 53.058Q-28.872 53.114-28.585 53.114Q-28.332 53.114-28.127 52.974Q-27.922 52.834-27.806 52.608Q-27.690 52.383-27.690 52.133L-27.690 50.930Q-27.690 50.660-27.797 50.599Q-27.905 50.537-28.216 50.537L-28.216 50.257L-27.109 50.182L-27.109 52.595Q-27.109 52.786-27.056 52.868Q-27.003 52.950-26.902 52.969Q-26.801 52.988-26.586 52.988L-26.586 53.268L-27.662 53.336L-27.662 52.772Q-27.772 52.954-27.917 53.077Q-28.062 53.200-28.249 53.268Q-28.435 53.336-28.637 53.336Q-29.840 53.336-29.840 52.434M-26.039 51.733Q-26.039 51.412-25.914 51.123Q-25.789 50.834-25.564 50.611Q-25.338 50.387-25.043 50.267Q-24.747 50.147-24.429 50.147Q-24.101 50.147-23.839 50.247Q-23.578 50.346-23.402 50.528Q-23.226 50.711-23.132 50.969Q-23.038 51.227-23.038 51.559Q-23.038 51.651-23.120 51.672L-25.376 51.672L-25.376 51.733Q-25.376 52.321-25.092 52.704Q-24.808 53.087-24.241 53.087Q-23.920 53.087-23.651 52.894Q-23.383 52.701-23.294 52.386Q-23.287 52.345-23.212 52.331L-23.120 52.331Q-23.038 52.355-23.038 52.427Q-23.038 52.434-23.045 52.461Q-23.158 52.858-23.528 53.097Q-23.899 53.336-24.323 53.336Q-24.761 53.336-25.160 53.128Q-25.560 52.919-25.800 52.552Q-26.039 52.185-26.039 51.733M-25.369 51.463L-23.554 51.463Q-23.554 51.186-23.651 50.934Q-23.749 50.681-23.947 50.525Q-24.145 50.370-24.429 50.370Q-24.706 50.370-24.920 50.528Q-25.133 50.687-25.251 50.942Q-25.369 51.197-25.369 51.463\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-40.744 44.732h170.716\" style=\"stroke-dasharray:3.0,3.0\"\u002F>\u003Cg transform=\"translate(-9.795 -6.28)\">\u003Cpath d=\"M-38.755 53.408Q-39.390 53.408-39.754 53.063Q-40.119 52.718-40.254 52.193Q-40.389 51.668-40.389 51.043Q-40.389 50.018-40.033 49.319Q-39.678 48.620-38.755 48.620Q-37.828 48.620-37.476 49.319Q-37.124 50.018-37.124 51.043Q-37.124 51.668-37.259 52.193Q-37.394 52.718-37.757 53.063Q-38.119 53.408-38.755 53.408M-38.755 53.183Q-38.317 53.183-38.104 52.808Q-37.890 52.434-37.840 51.967Q-37.791 51.501-37.791 50.923Q-37.791 50.370-37.840 49.942Q-37.890 49.515-38.102 49.180Q-38.314 48.845-38.755 48.845Q-39.097 48.845-39.300 49.052Q-39.503 49.259-39.590 49.571Q-39.678 49.884-39.700 50.200Q-39.722 50.517-39.722 50.923Q-39.722 51.340-39.700 51.682Q-39.678 52.024-39.589 52.372Q-39.500 52.721-39.295 52.952Q-39.090 53.183-38.755 53.183\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"M-40.744-23.554C-15.137 21.97 4.78 41.887 129.972 44.732\" style=\"stroke-width:1.2\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg transform=\"translate(175.672 -6.144)\">\u003Cpath d=\"M-38.218 53.268L-40.351 53.268L-40.351 52.988Q-39.630 52.988-39.630 52.779L-39.630 48.978Q-39.630 48.767-40.351 48.767L-40.351 48.486L-37.685 48.486Q-37.275 48.486-36.854 48.640Q-36.434 48.794-36.150 49.098Q-35.867 49.402-35.867 49.816Q-35.867 50.134-36.034 50.380Q-36.202 50.626-36.478 50.792Q-36.755 50.957-37.077 51.041Q-37.398 51.125-37.685 51.125L-38.939 51.125L-38.939 52.779Q-38.939 52.988-38.218 52.988L-38.218 53.268M-38.967 48.978L-38.967 50.875L-37.880 50.875Q-37.271 50.875-36.957 50.638Q-36.642 50.400-36.642 49.816Q-36.642 49.423-36.788 49.189Q-36.933 48.955-37.205 48.861Q-37.476 48.767-37.880 48.767L-38.601 48.767Q-38.789 48.767-38.878 48.801Q-38.967 48.835-38.967 48.978M-31.994 53.268L-34.968 53.268L-34.968 52.988Q-34.246 52.988-34.246 52.779L-34.246 48.978Q-34.246 48.767-34.968 48.767L-34.968 48.486L-32.209 48.486Q-31.943 48.486-31.637 48.561Q-31.331 48.637-31.071 48.784Q-30.811 48.931-30.649 49.160Q-30.487 49.389-30.487 49.683Q-30.487 50.113-30.873 50.395Q-31.259 50.677-31.741 50.769Q-31.433 50.769-31.085 50.934Q-30.736 51.098-30.507 51.376Q-30.278 51.655-30.278 51.973Q-30.278 52.379-30.541 52.673Q-30.805 52.967-31.203 53.118Q-31.601 53.268-31.994 53.268M-33.604 50.896L-33.604 52.779Q-33.604 52.919-33.515 52.954Q-33.426 52.988-33.238 52.988L-32.209 52.988Q-31.922 52.988-31.645 52.861Q-31.369 52.735-31.198 52.501Q-31.027 52.267-31.027 51.973Q-31.027 51.757-31.109 51.561Q-31.191 51.364-31.341 51.214Q-31.492 51.063-31.688 50.980Q-31.885 50.896-32.100 50.896L-33.604 50.896M-33.604 48.978L-33.604 50.670L-32.421 50.670Q-32.134 50.670-31.854 50.551Q-31.574 50.431-31.394 50.204Q-31.215 49.976-31.215 49.683Q-31.215 49.433-31.353 49.219Q-31.492 49.006-31.721 48.886Q-31.950 48.767-32.209 48.767L-33.238 48.767Q-33.426 48.767-33.515 48.801Q-33.604 48.835-33.604 48.978M-25.008 53.268L-29.410 53.268L-29.410 52.988Q-28.689 52.988-28.689 52.779L-28.689 48.978Q-28.689 48.767-29.410 48.767L-29.410 48.486L-25.120 48.486L-24.912 50.123L-25.175 50.123Q-25.233 49.652-25.336 49.387Q-25.438 49.122-25.623 48.989Q-25.807 48.855-26.079 48.811Q-26.351 48.767-26.850 48.767L-27.633 48.767Q-27.821 48.767-27.910 48.801Q-27.998 48.835-27.998 48.978L-27.998 50.643L-27.424 50.643Q-27.035 50.643-26.852 50.592Q-26.669 50.540-26.587 50.368Q-26.505 50.195-26.505 49.823L-26.242 49.823L-26.242 51.744L-26.505 51.744Q-26.505 51.371-26.587 51.198Q-26.669 51.026-26.852 50.975Q-27.035 50.923-27.424 50.923L-27.998 50.923L-27.998 52.779Q-27.998 52.919-27.910 52.954Q-27.821 52.988-27.633 52.988L-26.785 52.988Q-26.255 52.988-25.946 52.919Q-25.637 52.851-25.449 52.684Q-25.261 52.516-25.153 52.214Q-25.045 51.911-24.960 51.398L-24.693 51.398\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-accent)\" d=\"M-40.744-43.471C-17.982-9.328 4.78 3.476 129.972 4.899\" style=\"stroke-width:1.2\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cg transform=\"translate(175.672 -45.978)\">\u003Cpath d=\"M-37.993 53.309L-39.835 48.978Q-39.903 48.835-40.084 48.801Q-40.265 48.767-40.529 48.767L-40.529 48.486L-38.560 48.486L-38.560 48.767Q-39.114 48.767-39.114 48.941Q-39.110 48.951-39.109 48.961Q-39.107 48.972-39.107 48.978L-37.586 52.554L-36.154 49.166Q-36.130 49.129-36.130 49.074Q-36.130 48.914-36.280 48.840Q-36.431 48.767-36.615 48.767L-36.615 48.486L-35.067 48.486L-35.067 48.767Q-35.327 48.767-35.537 48.861Q-35.747 48.955-35.843 49.166L-37.599 53.309Q-37.651 53.408-37.746 53.408L-37.846 53.408Q-37.945 53.408-37.993 53.309M-30.063 53.268L-34.465 53.268L-34.465 52.988Q-33.744 52.988-33.744 52.779L-33.744 48.978Q-33.744 48.767-34.465 48.767L-34.465 48.486L-30.176 48.486L-29.967 50.123L-30.230 50.123Q-30.288 49.652-30.391 49.387Q-30.494 49.122-30.678 48.989Q-30.863 48.855-31.134 48.811Q-31.406 48.767-31.905 48.767L-32.688 48.767Q-32.876 48.767-32.965 48.801Q-33.054 48.835-33.054 48.978L-33.054 50.643L-32.479 50.643Q-32.090 50.643-31.907 50.592Q-31.724 50.540-31.642 50.368Q-31.560 50.195-31.560 49.823L-31.297 49.823L-31.297 51.744L-31.560 51.744Q-31.560 51.371-31.642 51.198Q-31.724 51.026-31.907 50.975Q-32.090 50.923-32.479 50.923L-33.054 50.923L-33.054 52.779Q-33.054 52.919-32.965 52.954Q-32.876 52.988-32.688 52.988L-31.840 52.988Q-31.310 52.988-31.001 52.919Q-30.692 52.851-30.504 52.684Q-30.316 52.516-30.208 52.214Q-30.100 51.911-30.015 51.398L-29.748 51.398\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-40.744-13.596c25.607 24.185 51.215 28.453 170.716 28.453\" style=\"stroke-width:.8\"\u002F>\u003Cg transform=\"translate(175.672 -36.156)\">\u003Cpath d=\"M-39.042 53.241L-40.023 50.742Q-40.084 50.599-40.202 50.564Q-40.320 50.530-40.536 50.530L-40.536 50.250L-39.056 50.250L-39.056 50.530Q-39.435 50.530-39.435 50.691Q-39.435 50.701-39.421 50.742L-38.707 52.574L-38.034 50.869Q-38.064 50.797-38.064 50.769Q-38.064 50.742-38.092 50.742Q-38.153 50.595-38.271 50.563Q-38.389 50.530-38.601 50.530L-38.601 50.250L-37.203 50.250L-37.203 50.530Q-37.579 50.530-37.579 50.691Q-37.579 50.722-37.572 50.742L-36.817 52.680L-36.130 50.930Q-36.109 50.879-36.109 50.824Q-36.109 50.684-36.222 50.607Q-36.335 50.530-36.475 50.530L-36.475 50.250L-35.255 50.250L-35.255 50.530Q-35.460 50.530-35.615 50.636Q-35.771 50.742-35.843 50.930L-36.748 53.241Q-36.783 53.336-36.895 53.336L-36.964 53.336Q-37.073 53.336-37.111 53.241L-37.893 51.238L-38.680 53.241Q-38.714 53.336-38.827 53.336L-38.895 53.336Q-39.004 53.336-39.042 53.241M-31.714 53.268L-34.243 53.268L-34.243 52.988Q-33.276 52.988-33.276 52.779L-33.276 49.160Q-33.669 49.348-34.291 49.348L-34.291 49.067Q-33.874 49.067-33.510 48.966Q-33.146 48.866-32.890 48.620L-32.763 48.620Q-32.698 48.637-32.681 48.705L-32.681 52.779Q-32.681 52.988-31.714 52.988L-31.714 53.268M-28.860 52.014L-30.917 52.014L-30.917 51.511L-28.860 51.511L-28.860 52.014M-26.669 53.241L-27.650 50.742Q-27.711 50.599-27.829 50.564Q-27.947 50.530-28.162 50.530L-28.162 50.250L-26.682 50.250L-26.682 50.530Q-27.062 50.530-27.062 50.691Q-27.062 50.701-27.048 50.742L-26.334 52.574L-25.661 50.869Q-25.691 50.797-25.691 50.769Q-25.691 50.742-25.719 50.742Q-25.780 50.595-25.898 50.563Q-26.016 50.530-26.228 50.530L-26.228 50.250L-24.830 50.250L-24.830 50.530Q-25.206 50.530-25.206 50.691Q-25.206 50.722-25.199 50.742L-24.444 52.680L-23.757 50.930Q-23.736 50.879-23.736 50.824Q-23.736 50.684-23.849 50.607Q-23.962 50.530-24.102 50.530L-24.102 50.250L-22.882 50.250L-22.882 50.530Q-23.087 50.530-23.242 50.636Q-23.398 50.742-23.470 50.930L-24.375 53.241Q-24.410 53.336-24.522 53.336L-24.591 53.336Q-24.700 53.336-24.738 53.241L-25.520 51.238L-26.306 53.241Q-26.341 53.336-26.453 53.336L-26.522 53.336Q-26.631 53.336-26.669 53.241M-20.677 53.408Q-21.135 53.408-21.453 53.193Q-21.771 52.977-21.952 52.625Q-22.133 52.273-22.210 51.853Q-22.287 51.433-22.287 51.005Q-22.287 50.421-22.034 49.865Q-21.781 49.310-21.311 48.965Q-20.841 48.620-20.243 48.620Q-19.833 48.620-19.549 48.818Q-19.265 49.016-19.265 49.419Q-19.265 49.515-19.312 49.594Q-19.358 49.672-19.438 49.717Q-19.518 49.761-19.607 49.761Q-19.754 49.761-19.855 49.664Q-19.956 49.566-19.956 49.419Q-19.956 49.289-19.865 49.182Q-19.775 49.074-19.641 49.074Q-19.829 48.852-20.243 48.852Q-20.557 48.852-20.831 49.016Q-21.104 49.180-21.272 49.454Q-21.460 49.744-21.525 50.110Q-21.590 50.476-21.590 50.930Q-21.439 50.636-21.174 50.458Q-20.910 50.281-20.595 50.281Q-20.164 50.281-19.816 50.487Q-19.467 50.694-19.267 51.050Q-19.067 51.405-19.067 51.832Q-19.067 52.277-19.284 52.637Q-19.501 52.998-19.874 53.203Q-20.246 53.408-20.677 53.408M-20.677 53.155Q-20.301 53.155-20.098 52.972Q-19.894 52.789-19.831 52.506Q-19.768 52.222-19.768 51.832Q-19.768 51.446-19.823 51.166Q-19.877 50.886-20.072 50.694Q-20.267 50.503-20.636 50.503Q-20.927 50.503-21.139 50.679Q-21.350 50.855-21.458 51.128Q-21.566 51.402-21.566 51.685L-21.566 51.826L-21.566 51.867Q-21.566 52.372-21.354 52.764Q-21.142 53.155-20.677 53.155\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">TDC on Baird's counterexample. The \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord overline\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8833em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1389em;\">P\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0502em;\">B\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">E\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.8033em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> falls to zero and the weight block \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">1\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> through \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0269em;\">w\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0269em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">6\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> settles; the individual components need not reach zero, so the value error \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord overline\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8833em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.2222em;\">V\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">E\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.8033em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"overline-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> stays near 2, but the system is stable. Step sizes \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:0.0037em;\">α\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:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0.005\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>, \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\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:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0.05\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:327.695px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 245.771 136.730\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" d=\"M-20.487 43.248H165.3\"\u002F>\u003Cpath stroke=\"none\" d=\"m167.301 43.248-3.2-1.6 1.2 1.6-1.2 1.6\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(108.677 8.783)\">\u003Cpath d=\"M-18.423 43.248L-20.159 43.248L-20.159 42.968Q-19.930 42.968-19.781 42.934Q-19.633 42.899-19.633 42.759L-19.633 40.910Q-19.633 40.640-19.740 40.579Q-19.848 40.517-20.159 40.517L-20.159 40.237L-19.130 40.162L-19.130 40.869Q-19 40.561-18.758 40.362Q-18.515 40.162-18.197 40.162Q-17.978 40.162-17.807 40.286Q-17.636 40.411-17.636 40.623Q-17.636 40.760-17.736 40.859Q-17.835 40.958-17.968 40.958Q-18.105 40.958-18.204 40.859Q-18.303 40.760-18.303 40.623Q-18.303 40.483-18.204 40.384Q-18.494 40.384-18.694 40.580Q-18.894 40.777-18.987 41.071Q-19.079 41.365-19.079 41.645L-19.079 42.759Q-19.079 42.968-18.423 42.968L-18.423 43.248M-16.994 42.520Q-16.994 42.188-16.770 41.961Q-16.546 41.734-16.203 41.606Q-15.859 41.477-15.487 41.425Q-15.114 41.372-14.810 41.372L-14.810 41.119Q-14.810 40.914-14.917 40.734Q-15.025 40.555-15.206 40.452Q-15.387 40.350-15.596 40.350Q-16.003 40.350-16.238 40.442Q-16.150 40.479-16.103 40.563Q-16.057 40.647-16.057 40.749Q-16.057 40.845-16.103 40.924Q-16.150 41.002-16.230 41.047Q-16.310 41.091-16.399 41.091Q-16.549 41.091-16.650 40.994Q-16.751 40.896-16.751 40.749Q-16.751 40.127-15.596 40.127Q-15.384 40.127-15.134 40.191Q-14.885 40.254-14.683 40.373Q-14.482 40.493-14.355 40.678Q-14.229 40.862-14.229 41.105L-14.229 42.681Q-14.229 42.797-14.167 42.893Q-14.106 42.988-13.993 42.988Q-13.883 42.988-13.819 42.894Q-13.754 42.800-13.754 42.681L-13.754 42.233L-13.487 42.233L-13.487 42.681Q-13.487 42.951-13.714 43.116Q-13.942 43.282-14.222 43.282Q-14.430 43.282-14.567 43.128Q-14.704 42.975-14.728 42.759Q-14.875 43.026-15.157 43.171Q-15.439 43.316-15.763 43.316Q-16.040 43.316-16.324 43.241Q-16.608 43.166-16.801 42.987Q-16.994 42.807-16.994 42.520M-16.379 42.520Q-16.379 42.694-16.278 42.824Q-16.177 42.954-16.021 43.024Q-15.866 43.094-15.702 43.094Q-15.483 43.094-15.275 42.997Q-15.066 42.899-14.938 42.718Q-14.810 42.537-14.810 42.311L-14.810 41.583Q-15.134 41.583-15.500 41.674Q-15.866 41.765-16.122 41.977Q-16.379 42.188-16.379 42.520M-12.544 42.407L-12.544 40.510L-13.183 40.510L-13.183 40.288Q-12.865 40.288-12.648 40.078Q-12.431 39.868-12.330 39.558Q-12.229 39.249-12.229 38.941L-11.963 38.941L-11.963 40.230L-10.886 40.230L-10.886 40.510L-11.963 40.510L-11.963 42.394Q-11.963 42.670-11.858 42.869Q-11.754 43.067-11.494 43.067Q-11.337 43.067-11.231 42.963Q-11.125 42.858-11.076 42.705Q-11.026 42.551-11.026 42.394L-11.026 41.980L-10.759 41.980L-10.759 42.407Q-10.759 42.633-10.859 42.843Q-10.958 43.053-11.142 43.185Q-11.327 43.316-11.556 43.316Q-11.993 43.316-12.268 43.079Q-12.544 42.841-12.544 42.407M-8.333 43.248L-9.884 43.248L-9.884 42.968Q-9.659 42.968-9.510 42.934Q-9.362 42.899-9.362 42.759L-9.362 40.910Q-9.362 40.722-9.409 40.638Q-9.457 40.555-9.555 40.536Q-9.652 40.517-9.864 40.517L-9.864 40.237L-8.808 40.162L-8.808 42.759Q-8.808 42.899-8.676 42.934Q-8.545 42.968-8.333 42.968L-8.333 43.248M-9.604 38.941Q-9.604 38.770-9.481 38.651Q-9.358 38.531-9.187 38.531Q-9.020 38.531-8.897 38.651Q-8.774 38.770-8.774 38.941Q-8.774 39.116-8.897 39.239Q-9.020 39.362-9.187 39.362Q-9.358 39.362-9.481 39.239Q-9.604 39.116-9.604 38.941M-7.728 41.765Q-7.728 41.423-7.593 41.124Q-7.458 40.825-7.218 40.601Q-6.979 40.377-6.661 40.252Q-6.343 40.127-6.012 40.127Q-5.568 40.127-5.168 40.343Q-4.768 40.558-4.534 40.936Q-4.299 41.313-4.299 41.765Q-4.299 42.106-4.441 42.390Q-4.583 42.674-4.828 42.881Q-5.072 43.087-5.381 43.202Q-5.691 43.316-6.012 43.316Q-6.443 43.316-6.844 43.115Q-7.246 42.913-7.487 42.561Q-7.728 42.209-7.728 41.765M-6.012 43.067Q-5.410 43.067-5.186 42.689Q-4.963 42.311-4.963 41.679Q-4.963 41.067-5.197 40.708Q-5.431 40.350-6.012 40.350Q-7.065 40.350-7.065 41.679Q-7.065 42.311-6.839 42.689Q-6.613 43.067-6.012 43.067\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(108.677 8.783)\">\u003Cpath d=\"M0.597 43.221L-0.531 40.722Q-0.603 40.575-0.733 40.543Q-0.863 40.510-1.092 40.510L-1.092 40.230L0.422 40.230L0.422 40.510Q0.070 40.510 0.070 40.657Q0.070 40.702 0.081 40.722L0.945 42.640L1.725 40.910Q1.759 40.842 1.759 40.763Q1.759 40.650 1.675 40.580Q1.591 40.510 1.472 40.510L1.472 40.230L2.668 40.230L2.668 40.510Q2.449 40.510 2.278 40.613Q2.108 40.715 2.019 40.910L0.983 43.221Q0.935 43.316 0.829 43.316L0.751 43.316Q0.645 43.316 0.597 43.221\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(108.677 8.783)\">\u003Cpath d=\"M2.837 42.520Q2.837 42.188 3.060 41.961Q3.284 41.734 3.628 41.606Q3.971 41.477 4.344 41.425Q4.716 41.372 5.021 41.372L5.021 41.119Q5.021 40.914 4.913 40.734Q4.805 40.555 4.624 40.452Q4.443 40.350 4.235 40.350Q3.828 40.350 3.592 40.442Q3.681 40.479 3.727 40.563Q3.773 40.647 3.773 40.749Q3.773 40.845 3.727 40.924Q3.681 41.002 3.600 41.047Q3.520 41.091 3.431 41.091Q3.281 41.091 3.180 40.994Q3.079 40.896 3.079 40.749Q3.079 40.127 4.235 40.127Q4.446 40.127 4.696 40.191Q4.945 40.254 5.147 40.373Q5.349 40.493 5.475 40.678Q5.602 40.862 5.602 41.105L5.602 42.681Q5.602 42.797 5.663 42.893Q5.725 42.988 5.838 42.988Q5.947 42.988 6.012 42.894Q6.077 42.800 6.077 42.681L6.077 42.233L6.343 42.233L6.343 42.681Q6.343 42.951 6.116 43.116Q5.889 43.282 5.609 43.282Q5.400 43.282 5.263 43.128Q5.127 42.975 5.103 42.759Q4.956 43.026 4.674 43.171Q4.392 43.316 4.067 43.316Q3.790 43.316 3.506 43.241Q3.223 43.166 3.030 42.987Q2.837 42.807 2.837 42.520M3.452 42.520Q3.452 42.694 3.553 42.824Q3.653 42.954 3.809 43.024Q3.964 43.094 4.129 43.094Q4.347 43.094 4.556 42.997Q4.764 42.899 4.892 42.718Q5.021 42.537 5.021 42.311L5.021 41.583Q4.696 41.583 4.330 41.674Q3.964 41.765 3.708 41.977Q3.452 42.188 3.452 42.520M8.428 43.248L6.825 43.248L6.825 42.968Q7.051 42.968 7.200 42.934Q7.348 42.899 7.348 42.759L7.348 39.140Q7.348 38.870 7.241 38.808Q7.133 38.747 6.825 38.747L6.825 38.466L7.902 38.391L7.902 42.759Q7.902 42.896 8.052 42.932Q8.203 42.968 8.428 42.968L8.428 43.248M9.597 42.414L9.597 40.910Q9.597 40.640 9.490 40.579Q9.382 40.517 9.071 40.517L9.071 40.237L10.178 40.162L10.178 42.394L10.178 42.414Q10.178 42.694 10.230 42.838Q10.281 42.981 10.423 43.038Q10.565 43.094 10.852 43.094Q11.105 43.094 11.310 42.954Q11.515 42.814 11.631 42.588Q11.747 42.363 11.747 42.113L11.747 40.910Q11.747 40.640 11.640 40.579Q11.532 40.517 11.221 40.517L11.221 40.237L12.328 40.162L12.328 42.575Q12.328 42.766 12.381 42.848Q12.434 42.930 12.535 42.949Q12.636 42.968 12.851 42.968L12.851 43.248L11.775 43.316L11.775 42.752Q11.665 42.934 11.520 43.057Q11.375 43.180 11.188 43.248Q11.002 43.316 10.800 43.316Q9.597 43.316 9.597 42.414M13.398 41.713Q13.398 41.392 13.523 41.103Q13.648 40.814 13.873 40.591Q14.099 40.367 14.394 40.247Q14.690 40.127 15.008 40.127Q15.336 40.127 15.598 40.227Q15.859 40.326 16.035 40.508Q16.211 40.691 16.305 40.949Q16.399 41.207 16.399 41.539Q16.399 41.631 16.317 41.652L14.061 41.652L14.061 41.713Q14.061 42.301 14.345 42.684Q14.629 43.067 15.196 43.067Q15.517 43.067 15.786 42.874Q16.054 42.681 16.143 42.366Q16.150 42.325 16.225 42.311L16.317 42.311Q16.399 42.335 16.399 42.407Q16.399 42.414 16.392 42.441Q16.279 42.838 15.909 43.077Q15.538 43.316 15.114 43.316Q14.676 43.316 14.277 43.108Q13.877 42.899 13.637 42.532Q13.398 42.165 13.398 41.713M14.068 41.443L15.883 41.443Q15.883 41.166 15.786 40.914Q15.688 40.661 15.490 40.505Q15.292 40.350 15.008 40.350Q14.731 40.350 14.517 40.508Q14.304 40.667 14.186 40.922Q14.068 41.177 14.068 41.443\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(108.677 8.783)\">\u003Cpath d=\"M21.825 44.998Q21.275 44.598 20.904 44.043Q20.533 43.487 20.352 42.841Q20.171 42.195 20.171 41.498Q20.171 40.985 20.271 40.490Q20.372 39.994 20.577 39.543Q20.782 39.092 21.095 38.700Q21.408 38.309 21.825 38.005Q21.835 38.001 21.842 38Q21.849 37.998 21.859 37.998L21.927 37.998Q21.962 37.998 21.984 38.022Q22.006 38.046 22.006 38.083Q22.006 38.128 21.979 38.145Q21.630 38.446 21.377 38.830Q21.124 39.215 20.972 39.656Q20.820 40.097 20.748 40.553Q20.676 41.009 20.676 41.498Q20.676 42.499 20.986 43.386Q21.295 44.273 21.979 44.858Q22.006 44.875 22.006 44.919Q22.006 44.957 21.984 44.981Q21.962 45.005 21.927 45.005L21.859 45.005Q21.852 45.001 21.844 45Q21.835 44.998 21.825 44.998M24.484 43.248L22.881 43.248L22.881 42.968Q23.107 42.968 23.255 42.934Q23.404 42.899 23.404 42.759L23.404 39.140Q23.404 38.870 23.296 38.808Q23.189 38.747 22.881 38.747L22.881 38.466L23.958 38.391L23.958 42.759Q23.958 42.896 24.108 42.932Q24.258 42.968 24.484 42.968L24.484 43.248M25.038 41.765Q25.038 41.423 25.173 41.124Q25.308 40.825 25.547 40.601Q25.786 40.377 26.104 40.252Q26.422 40.127 26.754 40.127Q27.198 40.127 27.598 40.343Q27.998 40.558 28.232 40.936Q28.466 41.313 28.466 41.765Q28.466 42.106 28.324 42.390Q28.182 42.674 27.938 42.881Q27.694 43.087 27.384 43.202Q27.075 43.316 26.754 43.316Q26.323 43.316 25.921 43.115Q25.520 42.913 25.279 42.561Q25.038 42.209 25.038 41.765M26.754 43.067Q27.355 43.067 27.579 42.689Q27.803 42.311 27.803 41.679Q27.803 41.067 27.569 40.708Q27.335 40.350 26.754 40.350Q25.701 40.350 25.701 41.679Q25.701 42.311 25.926 42.689Q26.152 43.067 26.754 43.067M29.020 43.781Q29.020 43.535 29.216 43.351Q29.413 43.166 29.669 43.087Q29.532 42.975 29.461 42.814Q29.389 42.653 29.389 42.472Q29.389 42.151 29.601 41.905Q29.266 41.607 29.266 41.197Q29.266 40.736 29.655 40.449Q30.045 40.162 30.524 40.162Q30.995 40.162 31.330 40.408Q31.505 40.254 31.715 40.172Q31.925 40.090 32.154 40.090Q32.318 40.090 32.439 40.197Q32.561 40.305 32.561 40.469Q32.561 40.565 32.489 40.637Q32.417 40.708 32.325 40.708Q32.226 40.708 32.156 40.635Q32.086 40.561 32.086 40.462Q32.086 40.408 32.099 40.377L32.106 40.363Q32.113 40.343 32.121 40.332Q32.130 40.322 32.133 40.315Q31.778 40.315 31.491 40.538Q31.778 40.831 31.778 41.197Q31.778 41.512 31.593 41.744Q31.409 41.977 31.120 42.105Q30.831 42.233 30.524 42.233Q30.322 42.233 30.131 42.183Q29.939 42.134 29.761 42.024Q29.669 42.151 29.669 42.294Q29.669 42.476 29.797 42.611Q29.925 42.746 30.110 42.746L30.742 42.746Q31.190 42.746 31.559 42.817Q31.928 42.889 32.188 43.118Q32.448 43.347 32.448 43.781Q32.448 44.102 32.152 44.304Q31.857 44.506 31.453 44.595Q31.050 44.684 30.735 44.684Q30.418 44.684 30.014 44.595Q29.611 44.506 29.315 44.304Q29.020 44.102 29.020 43.781M29.474 43.781Q29.474 44.010 29.693 44.159Q29.912 44.308 30.204 44.376Q30.496 44.444 30.735 44.444Q30.900 44.444 31.108 44.408Q31.317 44.373 31.523 44.292Q31.730 44.212 31.862 44.084Q31.993 43.956 31.993 43.781Q31.993 43.429 31.612 43.335Q31.231 43.241 30.729 43.241L30.110 43.241Q29.871 43.241 29.673 43.392Q29.474 43.542 29.474 43.781M30.524 41.994Q31.190 41.994 31.190 41.197Q31.190 40.397 30.524 40.397Q29.854 40.397 29.854 41.197Q29.854 41.994 30.524 41.994\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(108.677 8.783)\">\u003Cpath d=\"M35.751 43.241L35.751 42.178Q35.751 42.154 35.779 42.127Q35.806 42.100 35.830 42.100L35.939 42.100Q36.004 42.100 36.018 42.158Q36.114 42.592 36.360 42.843Q36.606 43.094 37.020 43.094Q37.361 43.094 37.614 42.961Q37.867 42.828 37.867 42.520Q37.867 42.363 37.773 42.248Q37.679 42.134 37.541 42.065Q37.402 41.997 37.235 41.959L36.654 41.860Q36.298 41.792 36.025 41.571Q35.751 41.351 35.751 41.009Q35.751 40.760 35.863 40.585Q35.974 40.411 36.160 40.312Q36.346 40.213 36.562 40.170Q36.777 40.127 37.020 40.127Q37.433 40.127 37.713 40.309L37.929 40.134Q37.939 40.131 37.946 40.129Q37.953 40.127 37.963 40.127L38.014 40.127Q38.041 40.127 38.065 40.151Q38.089 40.175 38.089 40.203L38.089 41.050Q38.089 41.071 38.065 41.098Q38.041 41.125 38.014 41.125L37.901 41.125Q37.874 41.125 37.848 41.100Q37.823 41.074 37.823 41.050Q37.823 40.814 37.717 40.650Q37.611 40.486 37.428 40.404Q37.245 40.322 37.013 40.322Q36.685 40.322 36.428 40.425Q36.172 40.527 36.172 40.804Q36.172 40.999 36.355 41.108Q36.538 41.218 36.767 41.259L37.341 41.365Q37.587 41.413 37.801 41.541Q38.014 41.669 38.151 41.872Q38.288 42.076 38.288 42.325Q38.288 42.838 37.922 43.077Q37.556 43.316 37.020 43.316Q36.524 43.316 36.192 43.022L35.926 43.296Q35.905 43.316 35.878 43.316L35.830 43.316Q35.806 43.316 35.779 43.289Q35.751 43.262 35.751 43.241M38.916 41.737Q38.916 41.409 39.052 41.108Q39.187 40.808 39.422 40.587Q39.658 40.367 39.962 40.247Q40.267 40.127 40.591 40.127Q41.097 40.127 41.446 40.230Q41.794 40.332 41.794 40.708Q41.794 40.855 41.697 40.956Q41.600 41.057 41.453 41.057Q41.299 41.057 41.200 40.958Q41.101 40.859 41.101 40.708Q41.101 40.520 41.241 40.428Q41.039 40.377 40.598 40.377Q40.243 40.377 40.014 40.573Q39.785 40.770 39.684 41.079Q39.583 41.389 39.583 41.737Q39.583 42.086 39.709 42.392Q39.836 42.698 40.091 42.882Q40.345 43.067 40.701 43.067Q40.923 43.067 41.107 42.983Q41.292 42.899 41.427 42.744Q41.562 42.588 41.620 42.380Q41.634 42.325 41.688 42.325L41.801 42.325Q41.832 42.325 41.854 42.349Q41.876 42.373 41.876 42.407L41.876 42.428Q41.791 42.715 41.603 42.913Q41.415 43.111 41.150 43.214Q40.885 43.316 40.591 43.316Q40.161 43.316 39.773 43.110Q39.385 42.903 39.151 42.540Q38.916 42.178 38.916 41.737M42.522 42.520Q42.522 42.188 42.746 41.961Q42.970 41.734 43.314 41.606Q43.657 41.477 44.030 41.425Q44.402 41.372 44.707 41.372L44.707 41.119Q44.707 40.914 44.599 40.734Q44.491 40.555 44.310 40.452Q44.129 40.350 43.920 40.350Q43.514 40.350 43.278 40.442Q43.367 40.479 43.413 40.563Q43.459 40.647 43.459 40.749Q43.459 40.845 43.413 40.924Q43.367 41.002 43.286 41.047Q43.206 41.091 43.117 41.091Q42.967 41.091 42.866 40.994Q42.765 40.896 42.765 40.749Q42.765 40.127 43.920 40.127Q44.132 40.127 44.382 40.191Q44.631 40.254 44.833 40.373Q45.035 40.493 45.161 40.678Q45.288 40.862 45.288 41.105L45.288 42.681Q45.288 42.797 45.349 42.893Q45.411 42.988 45.523 42.988Q45.633 42.988 45.698 42.894Q45.763 42.800 45.763 42.681L45.763 42.233L46.029 42.233L46.029 42.681Q46.029 42.951 45.802 43.116Q45.575 43.282 45.294 43.282Q45.086 43.282 44.949 43.128Q44.812 42.975 44.789 42.759Q44.642 43.026 44.360 43.171Q44.078 43.316 43.753 43.316Q43.476 43.316 43.192 43.241Q42.909 43.166 42.716 42.987Q42.522 42.807 42.522 42.520M43.138 42.520Q43.138 42.694 43.239 42.824Q43.339 42.954 43.495 43.024Q43.650 43.094 43.814 43.094Q44.033 43.094 44.242 42.997Q44.450 42.899 44.578 42.718Q44.707 42.537 44.707 42.311L44.707 41.583Q44.382 41.583 44.016 41.674Q43.650 41.765 43.394 41.977Q43.138 42.188 43.138 42.520M48.114 43.248L46.511 43.248L46.511 42.968Q46.737 42.968 46.885 42.934Q47.034 42.899 47.034 42.759L47.034 39.140Q47.034 38.870 46.927 38.808Q46.819 38.747 46.511 38.747L46.511 38.466L47.588 38.391L47.588 42.759Q47.588 42.896 47.738 42.932Q47.889 42.968 48.114 42.968L48.114 43.248M48.668 41.713Q48.668 41.392 48.793 41.103Q48.917 40.814 49.143 40.591Q49.369 40.367 49.664 40.247Q49.960 40.127 50.278 40.127Q50.606 40.127 50.867 40.227Q51.129 40.326 51.305 40.508Q51.481 40.691 51.575 40.949Q51.669 41.207 51.669 41.539Q51.669 41.631 51.587 41.652L49.331 41.652L49.331 41.713Q49.331 42.301 49.615 42.684Q49.898 43.067 50.466 43.067Q50.787 43.067 51.055 42.874Q51.324 42.681 51.413 42.366Q51.419 42.325 51.495 42.311L51.587 42.311Q51.669 42.335 51.669 42.407Q51.669 42.414 51.662 42.441Q51.549 42.838 51.178 43.077Q50.808 43.316 50.384 43.316Q49.946 43.316 49.546 43.108Q49.146 42.899 48.907 42.532Q48.668 42.165 48.668 41.713M49.338 41.443L51.153 41.443Q51.153 41.166 51.055 40.914Q50.958 40.661 50.760 40.505Q50.562 40.350 50.278 40.350Q50.001 40.350 49.787 40.508Q49.574 40.667 49.456 40.922Q49.338 41.177 49.338 41.443M52.578 45.005L52.510 45.005Q52.476 45.005 52.453 44.979Q52.431 44.954 52.431 44.919Q52.431 44.875 52.462 44.858Q52.817 44.554 53.067 44.164Q53.316 43.774 53.468 43.342Q53.621 42.910 53.691 42.441Q53.761 41.973 53.761 41.498Q53.761 41.019 53.691 40.553Q53.621 40.086 53.467 39.651Q53.313 39.215 53.062 38.827Q52.811 38.439 52.462 38.145Q52.431 38.128 52.431 38.083Q52.431 38.049 52.453 38.024Q52.476 37.998 52.510 37.998L52.578 37.998Q52.588 37.998 52.597 38Q52.605 38.001 52.616 38.005Q53.159 38.405 53.532 38.958Q53.904 39.512 54.085 40.158Q54.267 40.804 54.267 41.498Q54.267 42.199 54.085 42.846Q53.904 43.494 53.530 44.048Q53.156 44.602 52.616 44.998Q52.605 44.998 52.597 45Q52.588 45.001 52.578 45.005\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-20.487 43.248v-100.43\"\u002F>\u003Cpath stroke=\"none\" d=\"m-20.487-59.182-1.6 3.2 1.6-1.2 1.6 1.2\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-41.783 -107.324)\">\u003Cpath d=\"M-18.529 44.605L-20.159 44.605L-20.159 44.325Q-19.930 44.325-19.781 44.290Q-19.633 44.256-19.633 44.116L-19.633 40.770Q-19.633 40.599-19.769 40.558Q-19.906 40.517-20.159 40.517L-20.159 40.237L-19.079 40.162L-19.079 40.568Q-18.857 40.367-18.570 40.264Q-18.282 40.162-17.975 40.162Q-17.548 40.162-17.184 40.375Q-16.820 40.589-16.606 40.953Q-16.392 41.317-16.392 41.737Q-16.392 42.182-16.632 42.546Q-16.871 42.910-17.264 43.113Q-17.657 43.316-18.101 43.316Q-18.368 43.316-18.616 43.216Q-18.863 43.115-19.051 42.934L-19.051 44.116Q-19.051 44.253-18.903 44.289Q-18.754 44.325-18.529 44.325L-18.529 44.605M-19.051 40.917L-19.051 42.527Q-18.918 42.780-18.675 42.937Q-18.433 43.094-18.156 43.094Q-17.828 43.094-17.575 42.893Q-17.322 42.691-17.189 42.373Q-17.055 42.055-17.055 41.737Q-17.055 41.508-17.120 41.279Q-17.185 41.050-17.313 40.852Q-17.442 40.654-17.636 40.534Q-17.831 40.415-18.064 40.415Q-18.358 40.415-18.626 40.544Q-18.894 40.674-19.051 40.917M-14.007 43.248L-15.743 43.248L-15.743 42.968Q-15.514 42.968-15.365 42.934Q-15.216 42.899-15.216 42.759L-15.216 40.910Q-15.216 40.640-15.324 40.579Q-15.432 40.517-15.743 40.517L-15.743 40.237L-14.714 40.162L-14.714 40.869Q-14.584 40.561-14.341 40.362Q-14.099 40.162-13.781 40.162Q-13.562 40.162-13.391 40.286Q-13.220 40.411-13.220 40.623Q-13.220 40.760-13.320 40.859Q-13.419 40.958-13.552 40.958Q-13.689 40.958-13.788 40.859Q-13.887 40.760-13.887 40.623Q-13.887 40.483-13.788 40.384Q-14.078 40.384-14.278 40.580Q-14.478 40.777-14.570 41.071Q-14.663 41.365-14.663 41.645L-14.663 42.759Q-14.663 42.968-14.007 42.968L-14.007 43.248M-12.677 41.765Q-12.677 41.423-12.542 41.124Q-12.407 40.825-12.168 40.601Q-11.928 40.377-11.611 40.252Q-11.293 40.127-10.961 40.127Q-10.517 40.127-10.117 40.343Q-9.717 40.558-9.483 40.936Q-9.249 41.313-9.249 41.765Q-9.249 42.106-9.391 42.390Q-9.532 42.674-9.777 42.881Q-10.021 43.087-10.331 43.202Q-10.640 43.316-10.961 43.316Q-11.392 43.316-11.793 43.115Q-12.195 42.913-12.436 42.561Q-12.677 42.209-12.677 41.765M-10.961 43.067Q-10.360 43.067-10.136 42.689Q-9.912 42.311-9.912 41.679Q-9.912 41.067-10.146 40.708Q-10.380 40.350-10.961 40.350Q-12.014 40.350-12.014 41.679Q-12.014 42.311-11.788 42.689Q-11.563 43.067-10.961 43.067M-7.847 43.248L-8.114 43.248L-8.114 39.140Q-8.114 38.870-8.222 38.808Q-8.329 38.747-8.640 38.747L-8.640 38.466L-7.560 38.391L-7.560 40.561Q-7.352 40.370-7.066 40.266Q-6.781 40.162-6.484 40.162Q-6.166 40.162-5.868 40.283Q-5.571 40.404-5.349 40.620Q-5.127 40.835-5 41.120Q-4.874 41.406-4.874 41.737Q-4.874 42.182-5.113 42.546Q-5.352 42.910-5.745 43.113Q-6.138 43.316-6.583 43.316Q-6.778 43.316-6.967 43.260Q-7.157 43.204-7.318 43.099Q-7.478 42.995-7.618 42.834L-7.847 43.248M-7.533 40.903L-7.533 42.520Q-7.396 42.780-7.155 42.937Q-6.914 43.094-6.637 43.094Q-6.343 43.094-6.132 42.987Q-5.920 42.879-5.786 42.687Q-5.653 42.496-5.595 42.257Q-5.537 42.018-5.537 41.737Q-5.537 41.378-5.631 41.074Q-5.725 40.770-5.952 40.577Q-6.179 40.384-6.545 40.384Q-6.846 40.384-7.112 40.520Q-7.379 40.657-7.533 40.903M-4.180 42.520Q-4.180 42.188-3.956 41.961Q-3.732 41.734-3.389 41.606Q-3.045 41.477-2.673 41.425Q-2.300 41.372-1.996 41.372L-1.996 41.119Q-1.996 40.914-2.103 40.734Q-2.211 40.555-2.392 40.452Q-2.573 40.350-2.782 40.350Q-3.189 40.350-3.424 40.442Q-3.336 40.479-3.289 40.563Q-3.243 40.647-3.243 40.749Q-3.243 40.845-3.289 40.924Q-3.336 41.002-3.416 41.047Q-3.496 41.091-3.585 41.091Q-3.736 41.091-3.836 40.994Q-3.937 40.896-3.937 40.749Q-3.937 40.127-2.782 40.127Q-2.570 40.127-2.320 40.191Q-2.071 40.254-1.869 40.373Q-1.668 40.493-1.541 40.678Q-1.415 40.862-1.415 41.105L-1.415 42.681Q-1.415 42.797-1.353 42.893Q-1.292 42.988-1.179 42.988Q-1.070 42.988-1.005 42.894Q-0.940 42.800-0.940 42.681L-0.940 42.233L-0.673 42.233L-0.673 42.681Q-0.673 42.951-0.900 43.116Q-1.128 43.282-1.408 43.282Q-1.616 43.282-1.753 43.128Q-1.890 42.975-1.914 42.759Q-2.061 43.026-2.343 43.171Q-2.625 43.316-2.949 43.316Q-3.226 43.316-3.510 43.241Q-3.794 43.166-3.987 42.987Q-4.180 42.807-4.180 42.520M-3.565 42.520Q-3.565 42.694-3.464 42.824Q-3.363 42.954-3.207 43.024Q-3.052 43.094-2.888 43.094Q-2.669 43.094-2.461 42.997Q-2.252 42.899-2.124 42.718Q-1.996 42.537-1.996 42.311L-1.996 41.583Q-2.320 41.583-2.686 41.674Q-3.052 41.765-3.308 41.977Q-3.565 42.188-3.565 42.520M0.551 43.248L0.284 43.248L0.284 39.140Q0.284 38.870 0.176 38.808Q0.069 38.747-0.242 38.747L-0.242 38.466L0.838 38.391L0.838 40.561Q1.046 40.370 1.332 40.266Q1.617 40.162 1.914 40.162Q2.232 40.162 2.530 40.283Q2.827 40.404 3.049 40.620Q3.271 40.835 3.398 41.120Q3.524 41.406 3.524 41.737Q3.524 42.182 3.285 42.546Q3.046 42.910 2.653 43.113Q2.260 43.316 1.815 43.316Q1.620 43.316 1.431 43.260Q1.241 43.204 1.080 43.099Q0.920 42.995 0.780 42.834L0.551 43.248M0.865 40.903L0.865 42.520Q1.002 42.780 1.243 42.937Q1.484 43.094 1.761 43.094Q2.055 43.094 2.266 42.987Q2.478 42.879 2.612 42.687Q2.745 42.496 2.803 42.257Q2.861 42.018 2.861 41.737Q2.861 41.378 2.767 41.074Q2.673 40.770 2.446 40.577Q2.219 40.384 1.853 40.384Q1.552 40.384 1.285 40.520Q1.019 40.657 0.865 40.903M5.777 43.248L4.225 43.248L4.225 42.968Q4.451 42.968 4.599 42.934Q4.748 42.899 4.748 42.759L4.748 40.910Q4.748 40.722 4.700 40.638Q4.652 40.555 4.555 40.536Q4.457 40.517 4.245 40.517L4.245 40.237L5.302 40.162L5.302 42.759Q5.302 42.899 5.433 42.934Q5.565 42.968 5.777 42.968L5.777 43.248M4.505 38.941Q4.505 38.770 4.628 38.651Q4.751 38.531 4.922 38.531Q5.090 38.531 5.213 38.651Q5.336 38.770 5.336 38.941Q5.336 39.116 5.213 39.239Q5.090 39.362 4.922 39.362Q4.751 39.362 4.628 39.239Q4.505 39.116 4.505 38.941M8.091 43.248L6.488 43.248L6.488 42.968Q6.713 42.968 6.862 42.934Q7.011 42.899 7.011 42.759L7.011 39.140Q7.011 38.870 6.903 38.808Q6.795 38.747 6.488 38.747L6.488 38.466L7.564 38.391L7.564 42.759Q7.564 42.896 7.715 42.932Q7.865 42.968 8.091 42.968L8.091 43.248M10.302 43.248L8.750 43.248L8.750 42.968Q8.976 42.968 9.125 42.934Q9.273 42.899 9.273 42.759L9.273 40.910Q9.273 40.722 9.225 40.638Q9.178 40.555 9.080 40.536Q8.983 40.517 8.771 40.517L8.771 40.237L9.827 40.162L9.827 42.759Q9.827 42.899 9.959 42.934Q10.090 42.968 10.302 42.968L10.302 43.248M9.031 38.941Q9.031 38.770 9.154 38.651Q9.277 38.531 9.448 38.531Q9.615 38.531 9.738 38.651Q9.861 38.770 9.861 38.941Q9.861 39.116 9.738 39.239Q9.615 39.362 9.448 39.362Q9.277 39.362 9.154 39.239Q9.031 39.116 9.031 38.941M11.474 42.407L11.474 40.510L10.835 40.510L10.835 40.288Q11.153 40.288 11.370 40.078Q11.587 39.868 11.688 39.558Q11.789 39.249 11.789 38.941L12.055 38.941L12.055 40.230L13.132 40.230L13.132 40.510L12.055 40.510L12.055 42.394Q12.055 42.670 12.160 42.869Q12.264 43.067 12.524 43.067Q12.681 43.067 12.787 42.963Q12.893 42.858 12.942 42.705Q12.992 42.551 12.992 42.394L12.992 41.980L13.259 41.980L13.259 42.407Q13.259 42.633 13.159 42.843Q13.060 43.053 12.876 43.185Q12.691 43.316 12.462 43.316Q12.025 43.316 11.750 43.079Q11.474 42.841 11.474 42.407\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-41.783 -107.324)\">\u003Cpath d=\"M14.211 44.383Q14.341 44.451 14.478 44.451Q14.649 44.451 14.799 44.362Q14.950 44.273 15.061 44.128Q15.172 43.983 15.250 43.815L15.514 43.248L14.345 40.722Q14.270 40.575 14.140 40.543Q14.010 40.510 13.777 40.510L13.777 40.230L15.298 40.230L15.298 40.510Q14.950 40.510 14.950 40.657Q14.953 40.678 14.955 40.695Q14.957 40.712 14.957 40.722L15.814 42.581L16.587 40.910Q16.621 40.842 16.621 40.763Q16.621 40.650 16.537 40.580Q16.454 40.510 16.341 40.510L16.341 40.230L17.537 40.230L17.537 40.510Q17.318 40.510 17.146 40.614Q16.973 40.719 16.881 40.910L15.544 43.815Q15.374 44.185 15.104 44.431Q14.833 44.677 14.478 44.677Q14.208 44.677 13.989 44.511Q13.770 44.345 13.770 44.082Q13.770 43.945 13.863 43.856Q13.955 43.768 14.095 43.768Q14.232 43.768 14.321 43.856Q14.410 43.945 14.410 44.082Q14.410 44.185 14.357 44.263Q14.304 44.342 14.211 44.383\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-accent)\" d=\"M-6.26 43.248v-88.203M10.811 43.248v-59.75M27.882 43.248V7.682M44.954 43.248V23.331M62.026 43.248V32.436\" style=\"stroke-width:1.2\"\u002F>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"M81.943 43.248v-6.26M101.86 43.248v-3.414M121.777 43.248v-1.707M141.693 43.248v-.853\" style=\"stroke-width:1.2\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(7.33 10.518)\">\u003Cpath d=\"M-19.646 42.407L-19.646 40.510L-20.285 40.510L-20.285 40.288Q-19.967 40.288-19.750 40.078Q-19.533 39.868-19.433 39.558Q-19.332 39.249-19.332 38.941L-19.065 38.941L-19.065 40.230L-17.988 40.230L-17.988 40.510L-19.065 40.510L-19.065 42.394Q-19.065 42.670-18.961 42.869Q-18.857 43.067-18.597 43.067Q-18.440 43.067-18.334 42.963Q-18.228 42.858-18.178 42.705Q-18.129 42.551-18.129 42.394L-18.129 41.980L-17.862 41.980L-17.862 42.407Q-17.862 42.633-17.961 42.843Q-18.060 43.053-18.245 43.185Q-18.429 43.316-18.658 43.316Q-19.096 43.316-19.371 43.079Q-19.646 42.841-19.646 42.407M-15.435 43.248L-16.987 43.248L-16.987 42.968Q-16.761 42.968-16.613 42.934Q-16.464 42.899-16.464 42.759L-16.464 40.910Q-16.464 40.722-16.512 40.638Q-16.560 40.555-16.657 40.536Q-16.755 40.517-16.966 40.517L-16.966 40.237L-15.910 40.162L-15.910 42.759Q-15.910 42.899-15.779 42.934Q-15.647 42.968-15.435 42.968L-15.435 43.248M-16.707 38.941Q-16.707 38.770-16.584 38.651Q-16.461 38.531-16.290 38.531Q-16.122 38.531-15.999 38.651Q-15.876 38.770-15.876 38.941Q-15.876 39.116-15.999 39.239Q-16.122 39.362-16.290 39.362Q-16.461 39.362-16.584 39.239Q-16.707 39.116-16.707 38.941M-13.108 43.248L-14.741 43.248L-14.741 42.968Q-14.512 42.968-14.364 42.934Q-14.215 42.899-14.215 42.759L-14.215 40.910Q-14.215 40.640-14.323 40.579Q-14.430 40.517-14.741 40.517L-14.741 40.237L-13.682 40.162L-13.682 40.811Q-13.511 40.503-13.207 40.332Q-12.903 40.162-12.557 40.162Q-12.051 40.162-11.768 40.385Q-11.484 40.609-11.484 41.105L-11.484 42.759Q-11.484 42.896-11.335 42.932Q-11.187 42.968-10.961 42.968L-10.961 43.248L-12.591 43.248L-12.591 42.968Q-12.362 42.968-12.214 42.934Q-12.065 42.899-12.065 42.759L-12.065 41.119Q-12.065 40.784-12.185 40.584Q-12.304 40.384-12.619 40.384Q-12.889 40.384-13.123 40.520Q-13.357 40.657-13.496 40.891Q-13.634 41.125-13.634 41.399L-13.634 42.759Q-13.634 42.896-13.484 42.932Q-13.333 42.968-13.108 42.968\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(7.33 10.518)\">\u003Cpath d=\"M-10.248 44.383Q-10.118 44.451-9.981 44.451Q-9.810 44.451-9.660 44.362Q-9.509 44.273-9.398 44.128Q-9.287 43.983-9.209 43.815L-8.945 43.248L-10.114 40.722Q-10.189 40.575-10.319 40.543Q-10.449 40.510-10.682 40.510L-10.682 40.230L-9.161 40.230L-9.161 40.510Q-9.509 40.510-9.509 40.657Q-9.506 40.678-9.504 40.695Q-9.502 40.712-9.502 40.722L-8.645 42.581L-7.872 40.910Q-7.838 40.842-7.838 40.763Q-7.838 40.650-7.922 40.580Q-8.005 40.510-8.118 40.510L-8.118 40.230L-6.922 40.230L-6.922 40.510Q-7.141 40.510-7.313 40.614Q-7.486 40.719-7.578 40.910L-8.915 43.815Q-9.085 44.185-9.355 44.431Q-9.626 44.677-9.981 44.677Q-10.251 44.677-10.470 44.511Q-10.689 44.345-10.689 44.082Q-10.689 43.945-10.596 43.856Q-10.504 43.768-10.364 43.768Q-10.227 43.768-10.138 43.856Q-10.049 43.945-10.049 44.082Q-10.049 44.185-10.102 44.263Q-10.155 44.342-10.248 44.383\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M37.841 43.248v-93.894\" style=\"stroke-dasharray:3.0,3.0\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(41.307 -97.427)\">\u003Cpath d=\"M-18.491 43.248L-20.125 43.248L-20.125 42.968Q-19.896 42.968-19.747 42.934Q-19.598 42.899-19.598 42.759L-19.598 40.910Q-19.598 40.640-19.706 40.579Q-19.814 40.517-20.125 40.517L-20.125 40.237L-19.065 40.162L-19.065 40.811Q-18.894 40.503-18.590 40.332Q-18.286 40.162-17.941 40.162Q-17.541 40.162-17.264 40.302Q-16.987 40.442-16.902 40.790Q-16.734 40.497-16.435 40.329Q-16.136 40.162-15.791 40.162Q-15.285 40.162-15.001 40.385Q-14.717 40.609-14.717 41.105L-14.717 42.759Q-14.717 42.896-14.569 42.932Q-14.420 42.968-14.195 42.968L-14.195 43.248L-15.825 43.248L-15.825 42.968Q-15.599 42.968-15.449 42.932Q-15.299 42.896-15.299 42.759L-15.299 41.119Q-15.299 40.784-15.418 40.584Q-15.538 40.384-15.852 40.384Q-16.122 40.384-16.356 40.520Q-16.591 40.657-16.729 40.891Q-16.867 41.125-16.867 41.399L-16.867 42.759Q-16.867 42.896-16.719 42.932Q-16.570 42.968-16.344 42.968L-16.344 43.248L-17.975 43.248L-17.975 42.968Q-17.746 42.968-17.597 42.934Q-17.448 42.899-17.448 42.759L-17.448 41.119Q-17.448 40.784-17.568 40.584Q-17.688 40.384-18.002 40.384Q-18.272 40.384-18.506 40.520Q-18.740 40.657-18.879 40.891Q-19.017 41.125-19.017 41.399L-19.017 42.759Q-19.017 42.896-18.867 42.932Q-18.716 42.968-18.491 42.968L-18.491 43.248M-13.648 41.713Q-13.648 41.392-13.523 41.103Q-13.398 40.814-13.173 40.591Q-12.947 40.367-12.651 40.247Q-12.356 40.127-12.038 40.127Q-11.710 40.127-11.448 40.227Q-11.187 40.326-11.011 40.508Q-10.835 40.691-10.741 40.949Q-10.647 41.207-10.647 41.539Q-10.647 41.631-10.729 41.652L-12.985 41.652L-12.985 41.713Q-12.985 42.301-12.701 42.684Q-12.417 43.067-11.850 43.067Q-11.529 43.067-11.260 42.874Q-10.992 42.681-10.903 42.366Q-10.896 42.325-10.821 42.311L-10.729 42.311Q-10.647 42.335-10.647 42.407Q-10.647 42.414-10.654 42.441Q-10.766 42.838-11.137 43.077Q-11.508 43.316-11.932 43.316Q-12.369 43.316-12.769 43.108Q-13.169 42.899-13.408 42.532Q-13.648 42.165-13.648 41.713M-12.978 41.443L-11.163 41.443Q-11.163 41.166-11.260 40.914Q-11.358 40.661-11.556 40.505Q-11.754 40.350-12.038 40.350Q-12.315 40.350-12.528 40.508Q-12.742 40.667-12.860 40.922Q-12.978 41.177-12.978 41.443M-10.001 42.520Q-10.001 42.188-9.777 41.961Q-9.553 41.734-9.209 41.606Q-8.866 41.477-8.493 41.425Q-8.121 41.372-7.817 41.372L-7.817 41.119Q-7.817 40.914-7.924 40.734Q-8.032 40.555-8.213 40.452Q-8.394 40.350-8.603 40.350Q-9.009 40.350-9.245 40.442Q-9.156 40.479-9.110 40.563Q-9.064 40.647-9.064 40.749Q-9.064 40.845-9.110 40.924Q-9.156 41.002-9.237 41.047Q-9.317 41.091-9.406 41.091Q-9.556 41.091-9.657 40.994Q-9.758 40.896-9.758 40.749Q-9.758 40.127-8.603 40.127Q-8.391 40.127-8.141 40.191Q-7.892 40.254-7.690 40.373Q-7.488 40.493-7.362 40.678Q-7.236 40.862-7.236 41.105L-7.236 42.681Q-7.236 42.797-7.174 42.893Q-7.112 42.988-7 42.988Q-6.890 42.988-6.825 42.894Q-6.760 42.800-6.760 42.681L-6.760 42.233L-6.494 42.233L-6.494 42.681Q-6.494 42.951-6.721 43.116Q-6.948 43.282-7.229 43.282Q-7.437 43.282-7.574 43.128Q-7.711 42.975-7.735 42.759Q-7.882 43.026-8.164 43.171Q-8.445 43.316-8.770 43.316Q-9.047 43.316-9.331 43.241Q-9.614 43.166-9.808 42.987Q-10.001 42.807-10.001 42.520M-9.385 42.520Q-9.385 42.694-9.285 42.824Q-9.184 42.954-9.028 43.024Q-8.873 43.094-8.709 43.094Q-8.490 43.094-8.281 42.997Q-8.073 42.899-7.945 42.718Q-7.817 42.537-7.817 42.311L-7.817 41.583Q-8.141 41.583-8.507 41.674Q-8.873 41.765-9.129 41.977Q-9.385 42.188-9.385 42.520M-4.395 43.248L-6.029 43.248L-6.029 42.968Q-5.800 42.968-5.651 42.934Q-5.503 42.899-5.503 42.759L-5.503 40.910Q-5.503 40.640-5.610 40.579Q-5.718 40.517-6.029 40.517L-6.029 40.237L-4.969 40.162L-4.969 40.811Q-4.799 40.503-4.494 40.332Q-4.190 40.162-3.845 40.162Q-3.339 40.162-3.055 40.385Q-2.772 40.609-2.772 41.105L-2.772 42.759Q-2.772 42.896-2.623 42.932Q-2.474 42.968-2.249 42.968L-2.249 43.248L-3.879 43.248L-3.879 42.968Q-3.650 42.968-3.501 42.934Q-3.353 42.899-3.353 42.759L-3.353 41.119Q-3.353 40.784-3.472 40.584Q-3.592 40.384-3.906 40.384Q-4.176 40.384-4.411 40.520Q-4.645 40.657-4.783 40.891Q-4.922 41.125-4.922 41.399L-4.922 42.759Q-4.922 42.896-4.771 42.932Q-4.621 42.968-4.395 42.968\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(41.307 -97.427)\">\u003Cpath d=\"M6.217 42.441L1.384 42.441Q1.316 42.431 1.270 42.385Q1.224 42.339 1.224 42.267Q1.224 42.202 1.270 42.156Q1.316 42.110 1.384 42.100L6.217 42.100Q6.286 42.110 6.332 42.156Q6.378 42.202 6.378 42.267Q6.378 42.339 6.332 42.385Q6.286 42.431 6.217 42.441M6.217 40.903L1.384 40.903Q1.316 40.893 1.270 40.847Q1.224 40.801 1.224 40.729Q1.224 40.585 1.384 40.561L6.217 40.561Q6.378 40.585 6.378 40.729Q6.378 40.801 6.332 40.847Q6.286 40.893 6.217 40.903\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(41.307 -97.427)\">\u003Cpath d=\"M12.894 43.248L10.364 43.248L10.364 42.968Q11.332 42.968 11.332 42.759L11.332 39.140Q10.939 39.328 10.317 39.328L10.317 39.047Q10.734 39.047 11.098 38.946Q11.462 38.846 11.718 38.600L11.844 38.600Q11.909 38.617 11.926 38.685L11.926 42.759Q11.926 42.968 12.894 42.968\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg transform=\"translate(156.201 -6.378)\">\u003Cpath d=\"M-19.691 42.701Q-19.571 42.858-19.380 42.957Q-19.188 43.057-18.973 43.096Q-18.758 43.135-18.535 43.135Q-18.238 43.135-18.043 42.980Q-17.848 42.824-17.758 42.570Q-17.667 42.315-17.667 42.031Q-17.667 41.737-17.759 41.486Q-17.852 41.235-18.050 41.079Q-18.248 40.924-18.542 40.924L-19.058 40.924Q-19.086 40.924-19.111 40.898Q-19.137 40.873-19.137 40.849L-19.137 40.777Q-19.137 40.746-19.111 40.724Q-19.086 40.702-19.058 40.702L-18.617 40.671Q-18.255 40.671-18.035 40.314Q-17.814 39.956-17.814 39.567Q-17.814 39.239-18.009 39.035Q-18.204 38.832-18.535 38.832Q-18.822 38.832-19.075 38.916Q-19.328 38.999-19.492 39.187Q-19.345 39.187-19.245 39.302Q-19.144 39.416-19.144 39.567Q-19.144 39.717-19.250 39.827Q-19.356 39.936-19.513 39.936Q-19.674 39.936-19.783 39.827Q-19.892 39.717-19.892 39.567Q-19.892 39.242-19.684 39.023Q-19.475 38.805-19.159 38.702Q-18.843 38.600-18.535 38.600Q-18.217 38.600-17.889 38.704Q-17.561 38.808-17.334 39.030Q-17.107 39.252-17.107 39.567Q-17.107 40.001-17.394 40.326Q-17.681 40.650-18.115 40.797Q-17.804 40.862-17.524 41.028Q-17.243 41.194-17.066 41.452Q-16.888 41.710-16.888 42.031Q-16.888 42.441-17.132 42.751Q-17.377 43.060-17.758 43.224Q-18.139 43.388-18.535 43.388Q-18.904 43.388-19.262 43.275Q-19.619 43.163-19.863 42.913Q-20.108 42.664-20.108 42.294Q-20.108 42.123-19.991 42.011Q-19.875 41.898-19.704 41.898Q-19.595 41.898-19.504 41.949Q-19.414 42-19.359 42.093Q-19.304 42.185-19.304 42.294Q-19.304 42.462-19.417 42.581Q-19.530 42.701-19.691 42.701M-15.698 42.486L-15.729 42.486Q-15.592 42.783-15.295 42.959Q-14.998 43.135-14.670 43.135Q-14.307 43.135-14.080 42.957Q-13.853 42.780-13.759 42.491Q-13.665 42.202-13.665 41.840Q-13.665 41.525-13.719 41.240Q-13.774 40.955-13.947 40.749Q-14.119 40.544-14.434 40.544Q-14.707 40.544-14.890 40.611Q-15.073 40.678-15.177 40.767Q-15.281 40.855-15.377 40.965Q-15.473 41.074-15.517 41.084L-15.596 41.084Q-15.668 41.067-15.685 40.996L-15.685 38.678Q-15.685 38.644-15.661 38.622Q-15.637 38.600-15.603 38.600L-15.575 38.600Q-15.288 38.716-15.020 38.770Q-14.752 38.825-14.475 38.825Q-14.198 38.825-13.928 38.770Q-13.658 38.716-13.378 38.600L-13.354 38.600Q-13.320 38.600-13.296 38.623Q-13.272 38.647-13.272 38.678L-13.272 38.747Q-13.272 38.774-13.292 38.794Q-13.566 39.109-13.950 39.285Q-14.335 39.461-14.748 39.461Q-15.087 39.461-15.404 39.375L-15.404 40.657Q-15.008 40.322-14.434 40.322Q-14.030 40.322-13.694 40.532Q-13.357 40.743-13.164 41.095Q-12.971 41.447-12.971 41.847Q-12.971 42.178-13.111 42.464Q-13.251 42.749-13.496 42.959Q-13.740 43.169-14.042 43.279Q-14.345 43.388-14.663 43.388Q-15.022 43.388-15.348 43.224Q-15.674 43.060-15.869 42.768Q-16.064 42.476-16.064 42.113Q-16.064 41.963-15.958 41.857Q-15.852 41.751-15.698 41.751Q-15.545 41.751-15.440 41.855Q-15.336 41.959-15.336 42.113Q-15.336 42.270-15.440 42.378Q-15.545 42.486-15.698 42.486M-11.088 43.040Q-11.088 42.534-10.958 42.026Q-10.828 41.519-10.590 41.057Q-10.353 40.596-10.018 40.175L-9.372 39.362L-10.185 39.362Q-10.770 39.362-11.166 39.370Q-11.563 39.379-11.587 39.399Q-11.689 39.516-11.768 40.042L-12.034 40.042L-11.788 38.518L-11.522 38.518L-11.522 38.538Q-11.522 38.606-11.446 38.649Q-11.371 38.692-11.293 38.699Q-11.101 38.723-10.906 38.729Q-10.712 38.736-10.520 38.738Q-10.329 38.740-10.131 38.740L-8.709 38.740L-8.709 38.928Q-8.719 38.976-8.729 38.986L-9.785 40.309Q-10.004 40.582-10.127 40.895Q-10.250 41.207-10.308 41.556Q-10.366 41.905-10.380 42.236Q-10.394 42.568-10.394 43.040Q-10.394 43.190-10.493 43.289Q-10.592 43.388-10.739 43.388Q-10.889 43.388-10.988 43.289Q-11.088 43.190-11.088 43.040\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(128.725 -43.775)\">\u003Cpath d=\"M-18.491 43.248L-20.125 43.248L-20.125 42.968Q-19.896 42.968-19.747 42.934Q-19.598 42.899-19.598 42.759L-19.598 39.140Q-19.598 38.870-19.706 38.808Q-19.814 38.747-20.125 38.747L-20.125 38.466L-19.045 38.391L-19.045 40.777Q-18.939 40.592-18.761 40.450Q-18.583 40.309-18.375 40.235Q-18.166 40.162-17.941 40.162Q-17.435 40.162-17.151 40.385Q-16.867 40.609-16.867 41.105L-16.867 42.759Q-16.867 42.896-16.719 42.932Q-16.570 42.968-16.344 42.968L-16.344 43.248L-17.975 43.248L-17.975 42.968Q-17.746 42.968-17.597 42.934Q-17.448 42.899-17.448 42.759L-17.448 41.119Q-17.448 40.784-17.568 40.584Q-17.688 40.384-18.002 40.384Q-18.272 40.384-18.506 40.520Q-18.740 40.657-18.879 40.891Q-19.017 41.125-19.017 41.399L-19.017 42.759Q-19.017 42.896-18.867 42.932Q-18.716 42.968-18.491 42.968L-18.491 43.248M-15.798 41.713Q-15.798 41.392-15.673 41.103Q-15.548 40.814-15.322 40.591Q-15.097 40.367-14.801 40.247Q-14.506 40.127-14.188 40.127Q-13.860 40.127-13.598 40.227Q-13.337 40.326-13.161 40.508Q-12.985 40.691-12.891 40.949Q-12.797 41.207-12.797 41.539Q-12.797 41.631-12.879 41.652L-15.134 41.652L-15.134 41.713Q-15.134 42.301-14.851 42.684Q-14.567 43.067-14 43.067Q-13.678 43.067-13.410 42.874Q-13.142 42.681-13.053 42.366Q-13.046 42.325-12.971 42.311L-12.879 42.311Q-12.797 42.335-12.797 42.407Q-12.797 42.414-12.803 42.441Q-12.916 42.838-13.287 43.077Q-13.658 43.316-14.082 43.316Q-14.519 43.316-14.919 43.108Q-15.319 42.899-15.558 42.532Q-15.798 42.165-15.798 41.713M-15.128 41.443L-13.313 41.443Q-13.313 41.166-13.410 40.914Q-13.508 40.661-13.706 40.505Q-13.904 40.350-14.188 40.350Q-14.465 40.350-14.678 40.508Q-14.892 40.667-15.010 40.922Q-15.128 41.177-15.128 41.443M-12.151 42.520Q-12.151 42.188-11.927 41.961Q-11.703 41.734-11.359 41.606Q-11.016 41.477-10.643 41.425Q-10.271 41.372-9.966 41.372L-9.966 41.119Q-9.966 40.914-10.074 40.734Q-10.182 40.555-10.363 40.452Q-10.544 40.350-10.753 40.350Q-11.159 40.350-11.395 40.442Q-11.306 40.479-11.260 40.563Q-11.214 40.647-11.214 40.749Q-11.214 40.845-11.260 40.924Q-11.306 41.002-11.387 41.047Q-11.467 41.091-11.556 41.091Q-11.706 41.091-11.807 40.994Q-11.908 40.896-11.908 40.749Q-11.908 40.127-10.753 40.127Q-10.541 40.127-10.291 40.191Q-10.042 40.254-9.840 40.373Q-9.638 40.493-9.512 40.678Q-9.385 40.862-9.385 41.105L-9.385 42.681Q-9.385 42.797-9.324 42.893Q-9.262 42.988-9.150 42.988Q-9.040 42.988-8.975 42.894Q-8.910 42.800-8.910 42.681L-8.910 42.233L-8.644 42.233L-8.644 42.681Q-8.644 42.951-8.871 43.116Q-9.098 43.282-9.379 43.282Q-9.587 43.282-9.724 43.128Q-9.861 42.975-9.884 42.759Q-10.031 43.026-10.313 43.171Q-10.595 43.316-10.920 43.316Q-11.197 43.316-11.481 43.241Q-11.764 43.166-11.957 42.987Q-12.151 42.807-12.151 42.520M-11.535 42.520Q-11.535 42.694-11.435 42.824Q-11.334 42.954-11.178 43.024Q-11.023 43.094-10.859 43.094Q-10.640 43.094-10.431 42.997Q-10.223 42.899-10.095 42.718Q-9.966 42.537-9.966 42.311L-9.966 41.583Q-10.291 41.583-10.657 41.674Q-11.023 41.765-11.279 41.977Q-11.535 42.188-11.535 42.520\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(128.725 -43.775)\">\u003Cpath d=\"M-6.840 43.221L-7.968 40.722Q-8.040 40.575-8.170 40.543Q-8.300 40.510-8.529 40.510L-8.529 40.230L-7.015 40.230L-7.015 40.510Q-7.367 40.510-7.367 40.657Q-7.367 40.702-7.356 40.722L-6.492 42.640L-5.712 40.910Q-5.678 40.842-5.678 40.763Q-5.678 40.650-5.762 40.580Q-5.846 40.510-5.965 40.510L-5.965 40.230L-4.769 40.230L-4.769 40.510Q-4.988 40.510-5.159 40.613Q-5.329 40.715-5.418 40.910L-6.454 43.221Q-6.502 43.316-6.608 43.316L-6.686 43.316Q-6.792 43.316-6.840 43.221M-3.894 44.383Q-3.764 44.451-3.627 44.451Q-3.456 44.451-3.306 44.362Q-3.156 44.273-3.045 44.128Q-2.933 43.983-2.855 43.815L-2.592 43.248L-3.761 40.722Q-3.836 40.575-3.966 40.543Q-4.096 40.510-4.328 40.510L-4.328 40.230L-2.807 40.230L-2.807 40.510Q-3.156 40.510-3.156 40.657Q-3.152 40.678-3.150 40.695Q-3.149 40.712-3.149 40.722L-2.291 42.581L-1.518 40.910Q-1.484 40.842-1.484 40.763Q-1.484 40.650-1.568 40.580Q-1.652 40.510-1.765 40.510L-1.765 40.230L-0.568 40.230L-0.568 40.510Q-0.787 40.510-0.960 40.614Q-1.132 40.719-1.224 40.910L-2.561 43.815Q-2.732 44.185-3.002 44.431Q-3.272 44.677-3.627 44.677Q-3.897 44.677-4.116 44.511Q-4.335 44.345-4.335 44.082Q-4.335 43.945-4.243 43.856Q-4.150 43.768-4.010 43.768Q-3.873 43.768-3.785 43.856Q-3.696 43.945-3.696 44.082Q-3.696 44.185-3.749 44.263Q-3.802 44.342-3.894 44.383\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(128.725 -43.775)\">\u003Cpath d=\"M3.194 42.407L3.194 40.510L2.555 40.510L2.555 40.288Q2.873 40.288 3.090 40.078Q3.307 39.868 3.407 39.558Q3.508 39.249 3.508 38.941L3.775 38.941L3.775 40.230L4.852 40.230L4.852 40.510L3.775 40.510L3.775 42.394Q3.775 42.670 3.879 42.869Q3.983 43.067 4.243 43.067Q4.400 43.067 4.506 42.963Q4.612 42.858 4.662 42.705Q4.711 42.551 4.711 42.394L4.711 41.980L4.978 41.980L4.978 42.407Q4.978 42.633 4.879 42.843Q4.780 43.053 4.595 43.185Q4.411 43.316 4.182 43.316Q3.744 43.316 3.469 43.079Q3.194 42.841 3.194 42.407M5.846 42.520Q5.846 42.188 6.070 41.961Q6.294 41.734 6.637 41.606Q6.981 41.477 7.353 41.425Q7.726 41.372 8.030 41.372L8.030 41.119Q8.030 40.914 7.923 40.734Q7.815 40.555 7.634 40.452Q7.453 40.350 7.244 40.350Q6.837 40.350 6.602 40.442Q6.690 40.479 6.737 40.563Q6.783 40.647 6.783 40.749Q6.783 40.845 6.737 40.924Q6.690 41.002 6.610 41.047Q6.530 41.091 6.441 41.091Q6.290 41.091 6.190 40.994Q6.089 40.896 6.089 40.749Q6.089 40.127 7.244 40.127Q7.456 40.127 7.706 40.191Q7.955 40.254 8.157 40.373Q8.358 40.493 8.485 40.678Q8.611 40.862 8.611 41.105L8.611 42.681Q8.611 42.797 8.673 42.893Q8.734 42.988 8.847 42.988Q8.957 42.988 9.021 42.894Q9.086 42.800 9.086 42.681L9.086 42.233L9.353 42.233L9.353 42.681Q9.353 42.951 9.126 43.116Q8.898 43.282 8.618 43.282Q8.410 43.282 8.273 43.128Q8.136 42.975 8.112 42.759Q7.965 43.026 7.683 43.171Q7.401 43.316 7.077 43.316Q6.800 43.316 6.516 43.241Q6.232 43.166 6.039 42.987Q5.846 42.807 5.846 42.520M6.461 42.520Q6.461 42.694 6.562 42.824Q6.663 42.954 6.819 43.024Q6.974 43.094 7.138 43.094Q7.357 43.094 7.565 42.997Q7.774 42.899 7.902 42.718Q8.030 42.537 8.030 42.311L8.030 41.583Q7.706 41.583 7.340 41.674Q6.974 41.765 6.718 41.977Q6.461 42.188 6.461 42.520M11.387 43.248L9.835 43.248L9.835 42.968Q10.061 42.968 10.209 42.934Q10.358 42.899 10.358 42.759L10.358 40.910Q10.358 40.722 10.310 40.638Q10.262 40.555 10.165 40.536Q10.067 40.517 9.855 40.517L9.855 40.237L10.912 40.162L10.912 42.759Q10.912 42.899 11.043 42.934Q11.175 42.968 11.387 42.968L11.387 43.248M10.115 38.941Q10.115 38.770 10.238 38.651Q10.361 38.531 10.532 38.531Q10.700 38.531 10.823 38.651Q10.946 38.770 10.946 38.941Q10.946 39.116 10.823 39.239Q10.700 39.362 10.532 39.362Q10.361 39.362 10.238 39.239Q10.115 39.116 10.115 38.941M13.701 43.248L12.098 43.248L12.098 42.968Q12.323 42.968 12.472 42.934Q12.621 42.899 12.621 42.759L12.621 39.140Q12.621 38.870 12.513 38.808Q12.405 38.747 12.098 38.747L12.098 38.466L13.174 38.391L13.174 42.759Q13.174 42.896 13.325 42.932Q13.475 42.968 13.701 42.968\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">The distribution of the 10-step importance ratio \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6389em;vertical-align:-0.2083em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">ρ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\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 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">:\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003Cspan class=\"mbin mtight\">+\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">9\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:0.2083em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> for \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\">b\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=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">0.5\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0.5\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>, \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;\">π\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=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">0.9\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0.1\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>. Almost all mass sits below 1 (products of mostly-\u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0.2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> factors), so the mean of exactly 1 is held up by a thin tail reaching \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\">8\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\">10\u003C\u002Fspan>\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:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">357\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>. That heavy tail is the variance the SGD update inherits.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:359.173px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 269.379 146.825\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" d=\"M104.724-8.46c0-11-13.376-19.918-29.875-19.918S44.974-19.46 44.974-8.46s13.375 19.917 29.875 19.917S104.724 2.54 104.724-8.46Zm-29.875 0\"\u002F>\u003Cpath fill=\"none\" d=\"M134.599-8.46c0-22-26.75-39.834-59.75-39.834s-59.75 17.833-59.75 39.833 26.75 39.834 59.75 39.834 59.75-17.834 59.75-39.834Zm-59.75 0\"\u002F>\u003Cpath fill=\"none\" d=\"M164.475-8.46c0-33-40.127-59.752-89.626-59.752S-14.778-41.46-14.778-8.46 25.35 51.29 74.85 51.29s89.626-26.75 89.626-59.75Zm-89.626 0\"\u002F>\u003Cpath stroke=\"none\" d=\"M76.849-8.46a2 2 0 1 0-4 0 2 2 0 0 0 4 0m-2 0\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-16.427 73.683)\">\u003Cpath d=\"M76.845-8.461L75.211-8.461L75.211-8.741Q75.440-8.741 75.589-8.775Q75.738-8.810 75.738-8.950L75.738-10.799Q75.738-11.069 75.630-11.130Q75.522-11.192 75.211-11.192L75.211-11.472L76.271-11.547L76.271-10.898Q76.442-11.206 76.746-11.377Q77.050-11.547 77.395-11.547Q77.795-11.547 78.072-11.407Q78.349-11.267 78.434-10.919Q78.602-11.212 78.901-11.380Q79.200-11.547 79.545-11.547Q80.051-11.547 80.335-11.324Q80.619-11.100 80.619-10.604L80.619-8.950Q80.619-8.813 80.767-8.777Q80.916-8.741 81.141-8.741L81.141-8.461L79.511-8.461L79.511-8.741Q79.737-8.741 79.887-8.777Q80.037-8.813 80.037-8.950L80.037-10.590Q80.037-10.925 79.918-11.125Q79.798-11.325 79.484-11.325Q79.214-11.325 78.980-11.189Q78.745-11.052 78.607-10.818Q78.469-10.584 78.469-10.310L78.469-8.950Q78.469-8.813 78.617-8.777Q78.766-8.741 78.992-8.741L78.992-8.461L77.361-8.461L77.361-8.741Q77.590-8.741 77.739-8.775Q77.888-8.810 77.888-8.950L77.888-10.590Q77.888-10.925 77.768-11.125Q77.648-11.325 77.334-11.325Q77.064-11.325 76.830-11.189Q76.596-11.052 76.457-10.818Q76.319-10.584 76.319-10.310L76.319-8.950Q76.319-8.813 76.469-8.777Q76.620-8.741 76.845-8.741L76.845-8.461M83.346-8.461L81.794-8.461L81.794-8.741Q82.020-8.741 82.169-8.775Q82.317-8.810 82.317-8.950L82.317-10.799Q82.317-10.987 82.269-11.071Q82.222-11.154 82.124-11.173Q82.027-11.192 81.815-11.192L81.815-11.472L82.871-11.547L82.871-8.950Q82.871-8.810 83.003-8.775Q83.134-8.741 83.346-8.741L83.346-8.461M82.075-12.768Q82.075-12.939 82.198-13.058Q82.321-13.178 82.492-13.178Q82.659-13.178 82.782-13.058Q82.905-12.939 82.905-12.768Q82.905-12.593 82.782-12.470Q82.659-12.347 82.492-12.347Q82.321-12.347 82.198-12.470Q82.075-12.593 82.075-12.768M85.674-8.461L84.040-8.461L84.040-8.741Q84.269-8.741 84.418-8.775Q84.566-8.810 84.566-8.950L84.566-10.799Q84.566-11.069 84.459-11.130Q84.351-11.192 84.040-11.192L84.040-11.472L85.099-11.547L85.099-10.898Q85.270-11.206 85.575-11.377Q85.879-11.547 86.224-11.547Q86.730-11.547 87.014-11.324Q87.297-11.100 87.297-10.604L87.297-8.950Q87.297-8.813 87.446-8.777Q87.595-8.741 87.820-8.741L87.820-8.461L86.190-8.461L86.190-8.741Q86.419-8.741 86.568-8.775Q86.716-8.810 86.716-8.950L86.716-10.590Q86.716-10.925 86.597-11.125Q86.477-11.325 86.162-11.325Q85.892-11.325 85.658-11.189Q85.424-11.052 85.286-10.818Q85.147-10.584 85.147-10.310L85.147-8.950Q85.147-8.813 85.298-8.777Q85.448-8.741 85.674-8.741L85.674-8.461M90.025-8.461L88.473-8.461L88.473-8.741Q88.699-8.741 88.847-8.775Q88.996-8.810 88.996-8.950L88.996-10.799Q88.996-10.987 88.948-11.071Q88.900-11.154 88.803-11.173Q88.705-11.192 88.494-11.192L88.494-11.472L89.550-11.547L89.550-8.950Q89.550-8.810 89.681-8.775Q89.813-8.741 90.025-8.741L90.025-8.461M88.753-12.768Q88.753-12.939 88.876-13.058Q88.999-13.178 89.170-13.178Q89.338-13.178 89.461-13.058Q89.584-12.939 89.584-12.768Q89.584-12.593 89.461-12.470Q89.338-12.347 89.170-12.347Q88.999-12.347 88.876-12.470Q88.753-12.593 88.753-12.768M92.352-8.461L90.719-8.461L90.719-8.741Q90.948-8.741 91.096-8.775Q91.245-8.810 91.245-8.950L91.245-10.799Q91.245-11.069 91.137-11.130Q91.030-11.192 90.719-11.192L90.719-11.472L91.778-11.547L91.778-10.898Q91.949-11.206 92.253-11.377Q92.557-11.547 92.903-11.547Q93.303-11.547 93.579-11.407Q93.856-11.267 93.942-10.919Q94.109-11.212 94.408-11.380Q94.707-11.547 95.053-11.547Q95.558-11.547 95.842-11.324Q96.126-11.100 96.126-10.604L96.126-8.950Q96.126-8.813 96.275-8.777Q96.423-8.741 96.649-8.741L96.649-8.461L95.018-8.461L95.018-8.741Q95.244-8.741 95.394-8.777Q95.545-8.813 95.545-8.950L95.545-10.590Q95.545-10.925 95.425-11.125Q95.306-11.325 94.991-11.325Q94.721-11.325 94.487-11.189Q94.253-11.052 94.114-10.818Q93.976-10.584 93.976-10.310L93.976-8.950Q93.976-8.813 94.125-8.777Q94.273-8.741 94.499-8.741L94.499-8.461L92.869-8.461L92.869-8.741Q93.098-8.741 93.246-8.775Q93.395-8.810 93.395-8.950L93.395-10.590Q93.395-10.925 93.275-11.125Q93.156-11.325 92.841-11.325Q92.571-11.325 92.337-11.189Q92.103-11.052 91.964-10.818Q91.826-10.584 91.826-10.310L91.826-8.950Q91.826-8.813 91.976-8.777Q92.127-8.741 92.352-8.741\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-16.427 73.683)\">\u003Cpath d=\"M97.606-9.295L97.606-10.799Q97.606-11.069 97.498-11.130Q97.390-11.192 97.079-11.192L97.079-11.472L98.187-11.547L98.187-9.315L98.187-9.295Q98.187-9.015 98.238-8.871Q98.289-8.728 98.431-8.671Q98.573-8.615 98.860-8.615Q99.113-8.615 99.318-8.755Q99.523-8.895 99.639-9.121Q99.756-9.346 99.756-9.596L99.756-10.799Q99.756-11.069 99.648-11.130Q99.540-11.192 99.229-11.192L99.229-11.472L100.337-11.547L100.337-9.134Q100.337-8.943 100.390-8.861Q100.443-8.779 100.543-8.760Q100.644-8.741 100.860-8.741L100.860-8.461L99.783-8.393L99.783-8.957Q99.674-8.775 99.528-8.652Q99.383-8.529 99.197-8.461Q99.010-8.393 98.809-8.393Q97.606-8.393 97.606-9.295M103.129-8.461L101.495-8.461L101.495-8.741Q101.724-8.741 101.873-8.775Q102.022-8.810 102.022-8.950L102.022-10.799Q102.022-11.069 101.914-11.130Q101.806-11.192 101.495-11.192L101.495-11.472L102.555-11.547L102.555-10.898Q102.726-11.206 103.030-11.377Q103.334-11.547 103.679-11.547Q104.079-11.547 104.356-11.407Q104.633-11.267 104.718-10.919Q104.886-11.212 105.185-11.380Q105.484-11.547 105.829-11.547Q106.335-11.547 106.619-11.324Q106.903-11.100 106.903-10.604L106.903-8.950Q106.903-8.813 107.051-8.777Q107.200-8.741 107.425-8.741L107.425-8.461L105.795-8.461L105.795-8.741Q106.021-8.741 106.171-8.777Q106.321-8.813 106.321-8.950L106.321-10.590Q106.321-10.925 106.202-11.125Q106.082-11.325 105.768-11.325Q105.498-11.325 105.264-11.189Q105.029-11.052 104.891-10.818Q104.753-10.584 104.753-10.310L104.753-8.950Q104.753-8.813 104.901-8.777Q105.050-8.741 105.276-8.741L105.276-8.461L103.645-8.461L103.645-8.741Q103.874-8.741 104.023-8.775Q104.172-8.810 104.172-8.950L104.172-10.590Q104.172-10.925 104.052-11.125Q103.932-11.325 103.618-11.325Q103.348-11.325 103.114-11.189Q102.880-11.052 102.741-10.818Q102.603-10.584 102.603-10.310L102.603-8.950Q102.603-8.813 102.753-8.777Q102.904-8.741 103.129-8.741\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"var(--tk-accent)\" stroke=\"none\" d=\"M-17.046-56.83a2 2 0 1 0-4 0 2 2 0 0 0 4 0m-2 0\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cg transform=\"translate(-102.159 -56.17)\">\u003Cpath d=\"M75.163-8.468L75.163-9.531Q75.163-9.555 75.191-9.582Q75.218-9.609 75.242-9.609L75.351-9.609Q75.416-9.609 75.430-9.551Q75.526-9.117 75.772-8.866Q76.018-8.615 76.432-8.615Q76.773-8.615 77.026-8.748Q77.279-8.881 77.279-9.189Q77.279-9.346 77.185-9.461Q77.091-9.575 76.953-9.644Q76.814-9.712 76.647-9.750L76.066-9.849Q75.710-9.917 75.437-10.138Q75.163-10.358 75.163-10.700Q75.163-10.949 75.275-11.124Q75.386-11.298 75.572-11.397Q75.758-11.496 75.974-11.539Q76.189-11.582 76.432-11.582Q76.845-11.582 77.125-11.400L77.341-11.575Q77.351-11.578 77.358-11.580Q77.365-11.582 77.375-11.582L77.426-11.582Q77.453-11.582 77.477-11.558Q77.501-11.534 77.501-11.506L77.501-10.659Q77.501-10.638 77.477-10.611Q77.453-10.584 77.426-10.584L77.313-10.584Q77.286-10.584 77.260-10.609Q77.235-10.635 77.235-10.659Q77.235-10.895 77.129-11.059Q77.023-11.223 76.840-11.305Q76.657-11.387 76.425-11.387Q76.097-11.387 75.840-11.284Q75.584-11.182 75.584-10.905Q75.584-10.710 75.767-10.601Q75.950-10.491 76.179-10.450L76.753-10.344Q76.999-10.296 77.213-10.168Q77.426-10.040 77.563-9.837Q77.700-9.633 77.700-9.384Q77.700-8.871 77.334-8.632Q76.968-8.393 76.432-8.393Q75.936-8.393 75.604-8.687L75.338-8.413Q75.317-8.393 75.290-8.393L75.242-8.393Q75.218-8.393 75.191-8.420Q75.163-8.447 75.163-8.468M78.855-9.302L78.855-11.199L78.216-11.199L78.216-11.421Q78.534-11.421 78.751-11.631Q78.968-11.841 79.068-12.151Q79.169-12.460 79.169-12.768L79.436-12.768L79.436-11.479L80.513-11.479L80.513-11.199L79.436-11.199L79.436-9.315Q79.436-9.039 79.540-8.840Q79.644-8.642 79.904-8.642Q80.061-8.642 80.167-8.746Q80.273-8.851 80.323-9.004Q80.372-9.158 80.372-9.315L80.372-9.729L80.639-9.729L80.639-9.302Q80.639-9.076 80.540-8.866Q80.441-8.656 80.256-8.524Q80.072-8.393 79.843-8.393Q79.405-8.393 79.130-8.630Q78.855-8.868 78.855-9.302M81.507-9.189Q81.507-9.521 81.731-9.748Q81.955-9.975 82.298-10.103Q82.642-10.232 83.015-10.284Q83.387-10.337 83.691-10.337L83.691-10.590Q83.691-10.795 83.584-10.975Q83.476-11.154 83.295-11.257Q83.114-11.359 82.905-11.359Q82.498-11.359 82.263-11.267Q82.351-11.230 82.398-11.146Q82.444-11.062 82.444-10.960Q82.444-10.864 82.398-10.785Q82.351-10.707 82.271-10.662Q82.191-10.618 82.102-10.618Q81.952-10.618 81.851-10.715Q81.750-10.813 81.750-10.960Q81.750-11.582 82.905-11.582Q83.117-11.582 83.367-11.518Q83.616-11.455 83.818-11.336Q84.019-11.216 84.146-11.031Q84.272-10.847 84.272-10.604L84.272-9.028Q84.272-8.912 84.334-8.816Q84.395-8.721 84.508-8.721Q84.618-8.721 84.682-8.815Q84.747-8.909 84.747-9.028L84.747-9.476L85.014-9.476L85.014-9.028Q85.014-8.758 84.787-8.593Q84.559-8.427 84.279-8.427Q84.071-8.427 83.934-8.581Q83.797-8.734 83.773-8.950Q83.626-8.683 83.344-8.538Q83.062-8.393 82.738-8.393Q82.461-8.393 82.177-8.468Q81.893-8.543 81.700-8.722Q81.507-8.902 81.507-9.189M82.122-9.189Q82.122-9.015 82.223-8.885Q82.324-8.755 82.480-8.685Q82.635-8.615 82.799-8.615Q83.018-8.615 83.226-8.712Q83.435-8.810 83.563-8.991Q83.691-9.172 83.691-9.398L83.691-10.126Q83.367-10.126 83.001-10.035Q82.635-9.944 82.379-9.732Q82.122-9.521 82.122-9.189M87.181-8.461L85.445-8.461L85.445-8.741Q85.674-8.741 85.822-8.775Q85.971-8.810 85.971-8.950L85.971-10.799Q85.971-11.069 85.863-11.130Q85.756-11.192 85.445-11.192L85.445-11.472L86.474-11.547L86.474-10.840Q86.603-11.148 86.846-11.347Q87.089-11.547 87.407-11.547Q87.625-11.547 87.796-11.423Q87.967-11.298 87.967-11.086Q87.967-10.949 87.868-10.850Q87.769-10.751 87.636-10.751Q87.499-10.751 87.400-10.850Q87.301-10.949 87.301-11.086Q87.301-11.226 87.400-11.325Q87.109-11.325 86.909-11.129Q86.709-10.932 86.617-10.638Q86.525-10.344 86.525-10.064L86.525-8.950Q86.525-8.741 87.181-8.741L87.181-8.461M89.078-9.302L89.078-11.199L88.439-11.199L88.439-11.421Q88.757-11.421 88.974-11.631Q89.191-11.841 89.292-12.151Q89.392-12.460 89.392-12.768L89.659-12.768L89.659-11.479L90.736-11.479L90.736-11.199L89.659-11.199L89.659-9.315Q89.659-9.039 89.763-8.840Q89.868-8.642 90.127-8.642Q90.285-8.642 90.391-8.746Q90.496-8.851 90.546-9.004Q90.596-9.158 90.596-9.315L90.596-9.729L90.862-9.729L90.862-9.302Q90.862-9.076 90.763-8.866Q90.664-8.656 90.479-8.524Q90.295-8.393 90.066-8.393Q89.628-8.393 89.353-8.630Q89.078-8.868 89.078-9.302\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-19.046-56.83 1.451-45.301\"\u002F>\u003Cpath stroke=\"none\" d=\"M3.717-44.027 1.11-47.879l.34 2.578-2.38 1.048\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"m3.717-44.027 17.591 8.796\"\u002F>\u003Cpath stroke=\"none\" d=\"m23.634-34.068-2.79-3.72.464 2.557-2.325 1.163\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-137.119 -24.6)\">\u003Cpath d=\"M75.163-8.468L75.163-9.531Q75.163-9.555 75.191-9.582Q75.218-9.609 75.242-9.609L75.351-9.609Q75.416-9.609 75.430-9.551Q75.526-9.117 75.772-8.866Q76.018-8.615 76.432-8.615Q76.773-8.615 77.026-8.748Q77.279-8.881 77.279-9.189Q77.279-9.346 77.185-9.461Q77.091-9.575 76.953-9.644Q76.814-9.712 76.647-9.750L76.066-9.849Q75.710-9.917 75.437-10.138Q75.163-10.358 75.163-10.700Q75.163-10.949 75.275-11.124Q75.386-11.298 75.572-11.397Q75.758-11.496 75.974-11.539Q76.189-11.582 76.432-11.582Q76.845-11.582 77.125-11.400L77.341-11.575Q77.351-11.578 77.358-11.580Q77.365-11.582 77.375-11.582L77.426-11.582Q77.453-11.582 77.477-11.558Q77.501-11.534 77.501-11.506L77.501-10.659Q77.501-10.638 77.477-10.611Q77.453-10.584 77.426-10.584L77.313-10.584Q77.286-10.584 77.260-10.609Q77.235-10.635 77.235-10.659Q77.235-10.895 77.129-11.059Q77.023-11.223 76.840-11.305Q76.657-11.387 76.425-11.387Q76.097-11.387 75.840-11.284Q75.584-11.182 75.584-10.905Q75.584-10.710 75.767-10.601Q75.950-10.491 76.179-10.450L76.753-10.344Q76.999-10.296 77.213-10.168Q77.426-10.040 77.563-9.837Q77.700-9.633 77.700-9.384Q77.700-8.871 77.334-8.632Q76.968-8.393 76.432-8.393Q75.936-8.393 75.604-8.687L75.338-8.413Q75.317-8.393 75.290-8.393L75.242-8.393Q75.218-8.393 75.191-8.420Q75.163-8.447 75.163-8.468M80.010-8.461L78.376-8.461L78.376-8.741Q78.605-8.741 78.754-8.775Q78.903-8.810 78.903-8.950L78.903-10.799Q78.903-11.069 78.795-11.130Q78.687-11.192 78.376-11.192L78.376-11.472L79.436-11.547L79.436-10.898Q79.607-11.206 79.911-11.377Q80.215-11.547 80.560-11.547Q80.960-11.547 81.237-11.407Q81.514-11.267 81.599-10.919Q81.767-11.212 82.066-11.380Q82.365-11.547 82.710-11.547Q83.216-11.547 83.500-11.324Q83.784-11.100 83.784-10.604L83.784-8.950Q83.784-8.813 83.932-8.777Q84.081-8.741 84.307-8.741L84.307-8.461L82.676-8.461L82.676-8.741Q82.902-8.741 83.052-8.777Q83.203-8.813 83.203-8.950L83.203-10.590Q83.203-10.925 83.083-11.125Q82.963-11.325 82.649-11.325Q82.379-11.325 82.145-11.189Q81.911-11.052 81.772-10.818Q81.634-10.584 81.634-10.310L81.634-8.950Q81.634-8.813 81.782-8.777Q81.931-8.741 82.157-8.741L82.157-8.461L80.526-8.461L80.526-8.741Q80.755-8.741 80.904-8.775Q81.053-8.810 81.053-8.950L81.053-10.590Q81.053-10.925 80.933-11.125Q80.813-11.325 80.499-11.325Q80.229-11.325 79.995-11.189Q79.761-11.052 79.622-10.818Q79.484-10.584 79.484-10.310L79.484-8.950Q79.484-8.813 79.634-8.777Q79.785-8.741 80.010-8.741L80.010-8.461M84.953-9.189Q84.953-9.521 85.176-9.748Q85.400-9.975 85.744-10.103Q86.087-10.232 86.460-10.284Q86.832-10.337 87.137-10.337L87.137-10.590Q87.137-10.795 87.029-10.975Q86.921-11.154 86.740-11.257Q86.559-11.359 86.350-11.359Q85.944-11.359 85.708-11.267Q85.797-11.230 85.843-11.146Q85.889-11.062 85.889-10.960Q85.889-10.864 85.843-10.785Q85.797-10.707 85.716-10.662Q85.636-10.618 85.547-10.618Q85.397-10.618 85.296-10.715Q85.195-10.813 85.195-10.960Q85.195-11.582 86.350-11.582Q86.562-11.582 86.812-11.518Q87.061-11.455 87.263-11.336Q87.465-11.216 87.591-11.031Q87.718-10.847 87.718-10.604L87.718-9.028Q87.718-8.912 87.779-8.816Q87.841-8.721 87.953-8.721Q88.063-8.721 88.128-8.815Q88.193-8.909 88.193-9.028L88.193-9.476L88.459-9.476L88.459-9.028Q88.459-8.758 88.232-8.593Q88.005-8.427 87.724-8.427Q87.516-8.427 87.379-8.581Q87.243-8.734 87.219-8.950Q87.072-8.683 86.790-8.538Q86.508-8.393 86.183-8.393Q85.906-8.393 85.622-8.468Q85.339-8.543 85.146-8.722Q84.953-8.902 84.953-9.189M85.568-9.189Q85.568-9.015 85.669-8.885Q85.769-8.755 85.925-8.685Q86.080-8.615 86.245-8.615Q86.463-8.615 86.672-8.712Q86.880-8.810 87.008-8.991Q87.137-9.172 87.137-9.398L87.137-10.126Q86.812-10.126 86.446-10.035Q86.080-9.944 85.824-9.732Q85.568-9.521 85.568-9.189M90.544-8.461L88.941-8.461L88.941-8.741Q89.167-8.741 89.316-8.775Q89.464-8.810 89.464-8.950L89.464-12.569Q89.464-12.839 89.357-12.901Q89.249-12.962 88.941-12.962L88.941-13.243L90.018-13.318L90.018-8.950Q90.018-8.813 90.168-8.777Q90.319-8.741 90.544-8.741L90.544-8.461M92.807-8.461L91.204-8.461L91.204-8.741Q91.430-8.741 91.578-8.775Q91.727-8.810 91.727-8.950L91.727-12.569Q91.727-12.839 91.619-12.901Q91.512-12.962 91.204-12.962L91.204-13.243L92.281-13.318L92.281-8.950Q92.281-8.813 92.431-8.777Q92.581-8.741 92.807-8.741\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-137.119 -24.6)\">\u003Cpath d=\"M97.858-8.461L96.122-8.461L96.122-8.741Q96.351-8.741 96.500-8.775Q96.648-8.810 96.648-8.950L96.648-10.799Q96.648-11.069 96.541-11.130Q96.433-11.192 96.122-11.192L96.122-11.472L97.151-11.547L97.151-10.840Q97.281-11.148 97.523-11.347Q97.766-11.547 98.084-11.547Q98.303-11.547 98.474-11.423Q98.645-11.298 98.645-11.086Q98.645-10.949 98.545-10.850Q98.446-10.751 98.313-10.751Q98.176-10.751 98.077-10.850Q97.978-10.949 97.978-11.086Q97.978-11.226 98.077-11.325Q97.787-11.325 97.587-11.129Q97.387-10.932 97.294-10.638Q97.202-10.344 97.202-10.064L97.202-8.950Q97.202-8.741 97.858-8.741L97.858-8.461M99.287-9.189Q99.287-9.521 99.511-9.748Q99.735-9.975 100.078-10.103Q100.422-10.232 100.794-10.284Q101.167-10.337 101.471-10.337L101.471-10.590Q101.471-10.795 101.364-10.975Q101.256-11.154 101.075-11.257Q100.894-11.359 100.685-11.359Q100.278-11.359 100.043-11.267Q100.131-11.230 100.178-11.146Q100.224-11.062 100.224-10.960Q100.224-10.864 100.178-10.785Q100.131-10.707 100.051-10.662Q99.971-10.618 99.882-10.618Q99.731-10.618 99.631-10.715Q99.530-10.813 99.530-10.960Q99.530-11.582 100.685-11.582Q100.897-11.582 101.147-11.518Q101.396-11.455 101.598-11.336Q101.799-11.216 101.926-11.031Q102.052-10.847 102.052-10.604L102.052-9.028Q102.052-8.912 102.114-8.816Q102.175-8.721 102.288-8.721Q102.398-8.721 102.462-8.815Q102.527-8.909 102.527-9.028L102.527-9.476L102.794-9.476L102.794-9.028Q102.794-8.758 102.567-8.593Q102.339-8.427 102.059-8.427Q101.851-8.427 101.714-8.581Q101.577-8.734 101.553-8.950Q101.406-8.683 101.124-8.538Q100.842-8.393 100.518-8.393Q100.241-8.393 99.957-8.468Q99.673-8.543 99.480-8.722Q99.287-8.902 99.287-9.189M99.902-9.189Q99.902-9.015 100.003-8.885Q100.104-8.755 100.260-8.685Q100.415-8.615 100.579-8.615Q100.798-8.615 101.006-8.712Q101.215-8.810 101.343-8.991Q101.471-9.172 101.471-9.398L101.471-10.126Q101.147-10.126 100.781-10.035Q100.415-9.944 100.159-9.732Q99.902-9.521 99.902-9.189M103.737-9.302L103.737-11.199L103.098-11.199L103.098-11.421Q103.416-11.421 103.633-11.631Q103.850-11.841 103.951-12.151Q104.052-12.460 104.052-12.768L104.318-12.768L104.318-11.479L105.395-11.479L105.395-11.199L104.318-11.199L104.318-9.315Q104.318-9.039 104.423-8.840Q104.527-8.642 104.787-8.642Q104.944-8.642 105.050-8.746Q105.156-8.851 105.205-9.004Q105.255-9.158 105.255-9.315L105.255-9.729L105.522-9.729L105.522-9.302Q105.522-9.076 105.422-8.866Q105.323-8.656 105.139-8.524Q104.954-8.393 104.725-8.393Q104.288-8.393 104.013-8.630Q103.737-8.868 103.737-9.302M107.948-8.461L106.397-8.461L106.397-8.741Q106.622-8.741 106.771-8.775Q106.919-8.810 106.919-8.950L106.919-10.799Q106.919-10.987 106.872-11.071Q106.824-11.154 106.726-11.173Q106.629-11.192 106.417-11.192L106.417-11.472L107.473-11.547L107.473-8.950Q107.473-8.810 107.605-8.775Q107.736-8.741 107.948-8.741L107.948-8.461M106.677-12.768Q106.677-12.939 106.800-13.058Q106.923-13.178 107.094-13.178Q107.261-13.178 107.384-13.058Q107.507-12.939 107.507-12.768Q107.507-12.593 107.384-12.470Q107.261-12.347 107.094-12.347Q106.923-12.347 106.800-12.470Q106.677-12.593 106.677-12.768M108.553-9.944Q108.553-10.286 108.688-10.585Q108.823-10.884 109.063-11.108Q109.302-11.332 109.620-11.457Q109.938-11.582 110.269-11.582Q110.713-11.582 111.113-11.366Q111.513-11.151 111.747-10.773Q111.981-10.396 111.981-9.944Q111.981-9.603 111.840-9.319Q111.698-9.035 111.453-8.828Q111.209-8.622 110.900-8.507Q110.590-8.393 110.269-8.393Q109.838-8.393 109.437-8.594Q109.035-8.796 108.794-9.148Q108.553-9.500 108.553-9.944M110.269-8.642Q110.871-8.642 111.095-9.020Q111.318-9.398 111.318-10.030Q111.318-10.642 111.084-11.001Q110.850-11.359 110.269-11.359Q109.216-11.359 109.216-10.030Q109.216-9.398 109.442-9.020Q109.668-8.642 110.269-8.642M112.576-8.468L112.576-9.531Q112.576-9.555 112.604-9.582Q112.631-9.609 112.655-9.609L112.764-9.609Q112.829-9.609 112.843-9.551Q112.939-9.117 113.185-8.866Q113.431-8.615 113.844-8.615Q114.186-8.615 114.439-8.748Q114.692-8.881 114.692-9.189Q114.692-9.346 114.598-9.461Q114.504-9.575 114.366-9.644Q114.227-9.712 114.060-9.750L113.479-9.849Q113.123-9.917 112.850-10.138Q112.576-10.358 112.576-10.700Q112.576-10.949 112.687-11.124Q112.798-11.298 112.985-11.397Q113.171-11.496 113.386-11.539Q113.602-11.582 113.844-11.582Q114.258-11.582 114.538-11.400L114.753-11.575Q114.764-11.578 114.771-11.580Q114.777-11.582 114.788-11.582L114.839-11.582Q114.866-11.582 114.890-11.558Q114.914-11.534 114.914-11.506L114.914-10.659Q114.914-10.638 114.890-10.611Q114.866-10.584 114.839-10.584L114.726-10.584Q114.699-10.584 114.673-10.609Q114.648-10.635 114.648-10.659Q114.648-10.895 114.542-11.059Q114.436-11.223 114.253-11.305Q114.070-11.387 113.837-11.387Q113.509-11.387 113.253-11.284Q112.997-11.182 112.997-10.905Q112.997-10.710 113.179-10.601Q113.362-10.491 113.591-10.450L114.166-10.344Q114.412-10.296 114.625-10.168Q114.839-10.040 114.976-9.837Q115.112-9.633 115.112-9.384Q115.112-8.871 114.747-8.632Q114.381-8.393 113.844-8.393Q113.349-8.393 113.017-8.687L112.751-8.413Q112.730-8.393 112.703-8.393L112.655-8.393Q112.631-8.393 112.604-8.420Q112.576-8.447 112.576-8.468M116.141-8.881Q116.141-9.049 116.264-9.172Q116.387-9.295 116.562-9.295Q116.729-9.295 116.852-9.172Q116.975-9.049 116.975-8.881Q116.975-8.707 116.852-8.584Q116.729-8.461 116.562-8.461Q116.387-8.461 116.264-8.584Q116.141-8.707 116.141-8.881M116.141-11.065Q116.141-11.233 116.264-11.356Q116.387-11.479 116.562-11.479Q116.729-11.479 116.852-11.356Q116.975-11.233 116.975-11.065Q116.975-10.891 116.852-10.768Q116.729-10.645 116.562-10.645Q116.387-10.645 116.264-10.768Q116.141-10.891 116.141-11.065\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-137.119 -24.6)\">\u003Cpath d=\"M121.582-8.468L121.582-9.531Q121.582-9.555 121.610-9.582Q121.637-9.609 121.661-9.609L121.770-9.609Q121.835-9.609 121.849-9.551Q121.945-9.117 122.191-8.866Q122.437-8.615 122.851-8.615Q123.192-8.615 123.445-8.748Q123.698-8.881 123.698-9.189Q123.698-9.346 123.604-9.461Q123.510-9.575 123.372-9.644Q123.233-9.712 123.066-9.750L122.485-9.849Q122.129-9.917 121.856-10.138Q121.582-10.358 121.582-10.700Q121.582-10.949 121.694-11.124Q121.805-11.298 121.991-11.397Q122.177-11.496 122.393-11.539Q122.608-11.582 122.851-11.582Q123.264-11.582 123.544-11.400L123.760-11.575Q123.770-11.578 123.777-11.580Q123.784-11.582 123.794-11.582L123.845-11.582Q123.872-11.582 123.896-11.558Q123.920-11.534 123.920-11.506L123.920-10.659Q123.920-10.638 123.896-10.611Q123.872-10.584 123.845-10.584L123.732-10.584Q123.705-10.584 123.679-10.609Q123.654-10.635 123.654-10.659Q123.654-10.895 123.548-11.059Q123.442-11.223 123.259-11.305Q123.076-11.387 122.844-11.387Q122.516-11.387 122.259-11.284Q122.003-11.182 122.003-10.905Q122.003-10.710 122.186-10.601Q122.369-10.491 122.598-10.450L123.172-10.344Q123.418-10.296 123.632-10.168Q123.845-10.040 123.982-9.837Q124.119-9.633 124.119-9.384Q124.119-8.871 123.753-8.632Q123.387-8.393 122.851-8.393Q122.355-8.393 122.023-8.687L121.757-8.413Q121.736-8.393 121.709-8.393L121.661-8.393Q121.637-8.393 121.610-8.420Q121.582-8.447 121.582-8.468M124.806-9.189Q124.806-9.521 125.029-9.748Q125.253-9.975 125.597-10.103Q125.940-10.232 126.313-10.284Q126.685-10.337 126.990-10.337L126.990-10.590Q126.990-10.795 126.882-10.975Q126.774-11.154 126.593-11.257Q126.412-11.359 126.204-11.359Q125.797-11.359 125.561-11.267Q125.650-11.230 125.696-11.146Q125.742-11.062 125.742-10.960Q125.742-10.864 125.696-10.785Q125.650-10.707 125.570-10.662Q125.489-10.618 125.400-10.618Q125.250-10.618 125.149-10.715Q125.048-10.813 125.048-10.960Q125.048-11.582 126.204-11.582Q126.415-11.582 126.665-11.518Q126.914-11.455 127.116-11.336Q127.318-11.216 127.444-11.031Q127.571-10.847 127.571-10.604L127.571-9.028Q127.571-8.912 127.632-8.816Q127.694-8.721 127.807-8.721Q127.916-8.721 127.981-8.815Q128.046-8.909 128.046-9.028L128.046-9.476L128.312-9.476L128.312-9.028Q128.312-8.758 128.085-8.593Q127.858-8.427 127.578-8.427Q127.369-8.427 127.232-8.581Q127.096-8.734 127.072-8.950Q126.925-8.683 126.643-8.538Q126.361-8.393 126.036-8.393Q125.759-8.393 125.476-8.468Q125.192-8.543 124.999-8.722Q124.806-8.902 124.806-9.189M125.421-9.189Q125.421-9.015 125.522-8.885Q125.622-8.755 125.778-8.685Q125.934-8.615 126.098-8.615Q126.316-8.615 126.525-8.712Q126.733-8.810 126.862-8.991Q126.990-9.172 126.990-9.398L126.990-10.126Q126.665-10.126 126.299-10.035Q125.934-9.944 125.677-9.732Q125.421-9.521 125.421-9.189M130.527-8.461L128.794-8.461L128.794-8.741Q129.020-8.741 129.169-8.775Q129.317-8.810 129.317-8.950L129.317-11.199L128.729-11.199L128.729-11.479L129.317-11.479L129.317-12.296Q129.317-12.614 129.495-12.862Q129.673-13.109 129.963-13.250Q130.254-13.390 130.565-13.390Q130.821-13.390 131.025-13.248Q131.228-13.106 131.228-12.863Q131.228-12.727 131.129-12.628Q131.030-12.528 130.893-12.528Q130.756-12.528 130.657-12.628Q130.558-12.727 130.558-12.863Q130.558-13.044 130.698-13.137Q130.620-13.164 130.520-13.164Q130.312-13.164 130.158-13.031Q130.004-12.898 129.924-12.694Q129.844-12.491 129.844-12.282L129.844-11.479L130.732-11.479L130.732-11.199L129.871-11.199L129.871-8.950Q129.871-8.741 130.527-8.741L130.527-8.461M131.166-9.996Q131.166-10.317 131.291-10.606Q131.416-10.895 131.642-11.118Q131.867-11.342 132.163-11.462Q132.458-11.582 132.776-11.582Q133.104-11.582 133.366-11.482Q133.627-11.383 133.803-11.201Q133.979-11.018 134.073-10.760Q134.167-10.502 134.167-10.170Q134.167-10.078 134.085-10.057L131.830-10.057L131.830-9.996Q131.830-9.408 132.113-9.025Q132.397-8.642 132.964-8.642Q133.286-8.642 133.554-8.835Q133.822-9.028 133.911-9.343Q133.918-9.384 133.993-9.398L134.085-9.398Q134.167-9.374 134.167-9.302Q134.167-9.295 134.161-9.268Q134.048-8.871 133.677-8.632Q133.306-8.393 132.882-8.393Q132.445-8.393 132.045-8.601Q131.645-8.810 131.406-9.177Q131.166-9.544 131.166-9.996M131.836-10.266L133.651-10.266Q133.651-10.543 133.554-10.795Q133.456-11.048 133.258-11.204Q133.060-11.359 132.776-11.359Q132.499-11.359 132.286-11.201Q132.072-11.042 131.954-10.787Q131.836-10.532 131.836-10.266\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\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=\"m23.634-34.068 150.804 78.194\"\u002F>\u003Cpath stroke=\"none\" d=\"m177.279 45.6-3.367-4.63.526 3.156-2.883 1.39\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(37.676 30.203)\">\u003Cpath d=\"M76.831-8.461L75.228-8.461L75.228-8.741Q75.454-8.741 75.603-8.775Q75.751-8.810 75.751-8.950L75.751-12.569Q75.751-12.839 75.644-12.901Q75.536-12.962 75.228-12.962L75.228-13.243L76.305-13.318L76.305-8.950Q76.305-8.813 76.455-8.777Q76.606-8.741 76.831-8.741L76.831-8.461M77.484-9.189Q77.484-9.521 77.708-9.748Q77.932-9.975 78.276-10.103Q78.619-10.232 78.992-10.284Q79.364-10.337 79.668-10.337L79.668-10.590Q79.668-10.795 79.561-10.975Q79.453-11.154 79.272-11.257Q79.091-11.359 78.882-11.359Q78.475-11.359 78.240-11.267Q78.328-11.230 78.375-11.146Q78.421-11.062 78.421-10.960Q78.421-10.864 78.375-10.785Q78.328-10.707 78.248-10.662Q78.168-10.618 78.079-10.618Q77.929-10.618 77.828-10.715Q77.727-10.813 77.727-10.960Q77.727-11.582 78.882-11.582Q79.094-11.582 79.344-11.518Q79.593-11.455 79.795-11.336Q79.996-11.216 80.123-11.031Q80.249-10.847 80.249-10.604L80.249-9.028Q80.249-8.912 80.311-8.816Q80.372-8.721 80.485-8.721Q80.595-8.721 80.660-8.815Q80.724-8.909 80.724-9.028L80.724-9.476L80.991-9.476L80.991-9.028Q80.991-8.758 80.764-8.593Q80.537-8.427 80.256-8.427Q80.048-8.427 79.911-8.581Q79.774-8.734 79.750-8.950Q79.603-8.683 79.321-8.538Q79.039-8.393 78.715-8.393Q78.438-8.393 78.154-8.468Q77.870-8.543 77.677-8.722Q77.484-8.902 77.484-9.189M78.099-9.189Q78.099-9.015 78.200-8.885Q78.301-8.755 78.457-8.685Q78.612-8.615 78.776-8.615Q78.995-8.615 79.203-8.712Q79.412-8.810 79.540-8.991Q79.668-9.172 79.668-9.398L79.668-10.126Q79.344-10.126 78.978-10.035Q78.612-9.944 78.356-9.732Q78.099-9.521 78.099-9.189M83.158-8.461L81.422-8.461L81.422-8.741Q81.651-8.741 81.799-8.775Q81.948-8.810 81.948-8.950L81.948-10.799Q81.948-11.069 81.840-11.130Q81.733-11.192 81.422-11.192L81.422-11.472L82.451-11.547L82.451-10.840Q82.580-11.148 82.823-11.347Q83.066-11.547 83.384-11.547Q83.602-11.547 83.773-11.423Q83.944-11.298 83.944-11.086Q83.944-10.949 83.845-10.850Q83.746-10.751 83.613-10.751Q83.476-10.751 83.377-10.850Q83.278-10.949 83.278-11.086Q83.278-11.226 83.377-11.325Q83.086-11.325 82.886-11.129Q82.686-10.932 82.594-10.638Q82.502-10.344 82.502-10.064L82.502-8.950Q82.502-8.741 83.158-8.741L83.158-8.461M84.488-7.928Q84.488-8.174 84.684-8.358Q84.881-8.543 85.137-8.622Q85-8.734 84.929-8.895Q84.857-9.056 84.857-9.237Q84.857-9.558 85.069-9.804Q84.734-10.102 84.734-10.512Q84.734-10.973 85.123-11.260Q85.513-11.547 85.992-11.547Q86.463-11.547 86.798-11.301Q86.973-11.455 87.183-11.537Q87.393-11.619 87.622-11.619Q87.786-11.619 87.907-11.512Q88.029-11.404 88.029-11.240Q88.029-11.144 87.957-11.072Q87.885-11.001 87.793-11.001Q87.694-11.001 87.624-11.074Q87.554-11.148 87.554-11.247Q87.554-11.301 87.567-11.332L87.574-11.346Q87.581-11.366 87.589-11.377Q87.598-11.387 87.601-11.394Q87.246-11.394 86.959-11.171Q87.246-10.878 87.246-10.512Q87.246-10.197 87.061-9.965Q86.877-9.732 86.588-9.604Q86.299-9.476 85.992-9.476Q85.790-9.476 85.599-9.526Q85.407-9.575 85.229-9.685Q85.137-9.558 85.137-9.415Q85.137-9.233 85.265-9.098Q85.393-8.963 85.578-8.963L86.210-8.963Q86.658-8.963 87.027-8.892Q87.396-8.820 87.656-8.591Q87.916-8.362 87.916-7.928Q87.916-7.607 87.620-7.405Q87.325-7.203 86.921-7.114Q86.518-7.025 86.203-7.025Q85.886-7.025 85.482-7.114Q85.079-7.203 84.783-7.405Q84.488-7.607 84.488-7.928M84.942-7.928Q84.942-7.699 85.161-7.550Q85.380-7.401 85.672-7.333Q85.964-7.265 86.203-7.265Q86.368-7.265 86.576-7.301Q86.785-7.336 86.991-7.417Q87.198-7.497 87.330-7.625Q87.461-7.753 87.461-7.928Q87.461-8.280 87.080-8.374Q86.699-8.468 86.197-8.468L85.578-8.468Q85.339-8.468 85.141-8.317Q84.942-8.167 84.942-7.928M85.992-9.715Q86.658-9.715 86.658-10.512Q86.658-11.312 85.992-11.312Q85.322-11.312 85.322-10.512Q85.322-9.715 85.992-9.715M88.470-9.996Q88.470-10.317 88.594-10.606Q88.719-10.895 88.945-11.118Q89.170-11.342 89.466-11.462Q89.762-11.582 90.079-11.582Q90.408-11.582 90.669-11.482Q90.931-11.383 91.107-11.201Q91.283-11.018 91.377-10.760Q91.471-10.502 91.471-10.170Q91.471-10.078 91.389-10.057L89.133-10.057L89.133-9.996Q89.133-9.408 89.416-9.025Q89.700-8.642 90.267-8.642Q90.589-8.642 90.857-8.835Q91.125-9.028 91.214-9.343Q91.221-9.384 91.296-9.398L91.389-9.398Q91.471-9.374 91.471-9.302Q91.471-9.295 91.464-9.268Q91.351-8.871 90.980-8.632Q90.609-8.393 90.185-8.393Q89.748-8.393 89.348-8.601Q88.948-8.810 88.709-9.177Q88.470-9.544 88.470-9.996M89.140-10.266L90.954-10.266Q90.954-10.543 90.857-10.795Q90.760-11.048 90.561-11.204Q90.363-11.359 90.079-11.359Q89.803-11.359 89.589-11.201Q89.375-11.042 89.257-10.787Q89.140-10.532 89.140-10.266\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(37.676 30.203)\">\u003Cpath d=\"M96.524-8.461L94.788-8.461L94.788-8.741Q95.017-8.741 95.166-8.775Q95.314-8.810 95.314-8.950L95.314-10.799Q95.314-11.069 95.207-11.130Q95.099-11.192 94.788-11.192L94.788-11.472L95.817-11.547L95.817-10.840Q95.947-11.148 96.189-11.347Q96.432-11.547 96.750-11.547Q96.969-11.547 97.140-11.423Q97.311-11.298 97.311-11.086Q97.311-10.949 97.211-10.850Q97.112-10.751 96.979-10.751Q96.842-10.751 96.743-10.850Q96.644-10.949 96.644-11.086Q96.644-11.226 96.743-11.325Q96.453-11.325 96.253-11.129Q96.053-10.932 95.960-10.638Q95.868-10.344 95.868-10.064L95.868-8.950Q95.868-8.741 96.524-8.741L96.524-8.461M97.953-9.189Q97.953-9.521 98.177-9.748Q98.401-9.975 98.744-10.103Q99.088-10.232 99.460-10.284Q99.833-10.337 100.137-10.337L100.137-10.590Q100.137-10.795 100.030-10.975Q99.922-11.154 99.741-11.257Q99.560-11.359 99.351-11.359Q98.944-11.359 98.709-11.267Q98.797-11.230 98.844-11.146Q98.890-11.062 98.890-10.960Q98.890-10.864 98.844-10.785Q98.797-10.707 98.717-10.662Q98.637-10.618 98.548-10.618Q98.397-10.618 98.297-10.715Q98.196-10.813 98.196-10.960Q98.196-11.582 99.351-11.582Q99.563-11.582 99.813-11.518Q100.062-11.455 100.264-11.336Q100.465-11.216 100.592-11.031Q100.718-10.847 100.718-10.604L100.718-9.028Q100.718-8.912 100.780-8.816Q100.841-8.721 100.954-8.721Q101.064-8.721 101.128-8.815Q101.193-8.909 101.193-9.028L101.193-9.476L101.460-9.476L101.460-9.028Q101.460-8.758 101.233-8.593Q101.005-8.427 100.725-8.427Q100.517-8.427 100.380-8.581Q100.243-8.734 100.219-8.950Q100.072-8.683 99.790-8.538Q99.508-8.393 99.184-8.393Q98.907-8.393 98.623-8.468Q98.339-8.543 98.146-8.722Q97.953-8.902 97.953-9.189M98.568-9.189Q98.568-9.015 98.669-8.885Q98.770-8.755 98.926-8.685Q99.081-8.615 99.245-8.615Q99.464-8.615 99.672-8.712Q99.881-8.810 100.009-8.991Q100.137-9.172 100.137-9.398L100.137-10.126Q99.813-10.126 99.447-10.035Q99.081-9.944 98.825-9.732Q98.568-9.521 98.568-9.189M102.403-9.302L102.403-11.199L101.764-11.199L101.764-11.421Q102.082-11.421 102.299-11.631Q102.516-11.841 102.617-12.151Q102.718-12.460 102.718-12.768L102.984-12.768L102.984-11.479L104.061-11.479L104.061-11.199L102.984-11.199L102.984-9.315Q102.984-9.039 103.089-8.840Q103.193-8.642 103.453-8.642Q103.610-8.642 103.716-8.746Q103.822-8.851 103.871-9.004Q103.921-9.158 103.921-9.315L103.921-9.729L104.188-9.729L104.188-9.302Q104.188-9.076 104.088-8.866Q103.989-8.656 103.805-8.524Q103.620-8.393 103.391-8.393Q102.954-8.393 102.679-8.630Q102.403-8.868 102.403-9.302M106.614-8.461L105.063-8.461L105.063-8.741Q105.288-8.741 105.437-8.775Q105.585-8.810 105.585-8.950L105.585-10.799Q105.585-10.987 105.538-11.071Q105.490-11.154 105.392-11.173Q105.295-11.192 105.083-11.192L105.083-11.472L106.139-11.547L106.139-8.950Q106.139-8.810 106.271-8.775Q106.402-8.741 106.614-8.741L106.614-8.461M105.343-12.768Q105.343-12.939 105.466-13.058Q105.589-13.178 105.760-13.178Q105.927-13.178 106.050-13.058Q106.173-12.939 106.173-12.768Q106.173-12.593 106.050-12.470Q105.927-12.347 105.760-12.347Q105.589-12.347 105.466-12.470Q105.343-12.593 105.343-12.768M107.219-9.944Q107.219-10.286 107.354-10.585Q107.489-10.884 107.729-11.108Q107.968-11.332 108.286-11.457Q108.604-11.582 108.935-11.582Q109.379-11.582 109.779-11.366Q110.179-11.151 110.413-10.773Q110.647-10.396 110.647-9.944Q110.647-9.603 110.506-9.319Q110.364-9.035 110.119-8.828Q109.875-8.622 109.566-8.507Q109.256-8.393 108.935-8.393Q108.504-8.393 108.103-8.594Q107.701-8.796 107.460-9.148Q107.219-9.500 107.219-9.944M108.935-8.642Q109.537-8.642 109.761-9.020Q109.984-9.398 109.984-10.030Q109.984-10.642 109.750-11.001Q109.516-11.359 108.935-11.359Q107.882-11.359 107.882-10.030Q107.882-9.398 108.108-9.020Q108.334-8.642 108.935-8.642M111.642-8.881Q111.642-9.049 111.765-9.172Q111.888-9.295 112.063-9.295Q112.230-9.295 112.353-9.172Q112.476-9.049 112.476-8.881Q112.476-8.707 112.353-8.584Q112.230-8.461 112.063-8.461Q111.888-8.461 111.765-8.584Q111.642-8.707 111.642-8.881M111.642-11.065Q111.642-11.233 111.765-11.356Q111.888-11.479 112.063-11.479Q112.230-11.479 112.353-11.356Q112.476-11.233 112.476-11.065Q112.476-10.891 112.353-10.768Q112.230-10.645 112.063-10.645Q111.888-10.645 111.765-10.768Q111.642-10.891 111.642-11.065\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(37.676 30.203)\">\u003Cpath d=\"M117.038-9.944Q117.038-10.286 117.173-10.585Q117.308-10.884 117.548-11.108Q117.787-11.332 118.105-11.457Q118.423-11.582 118.754-11.582Q119.199-11.582 119.598-11.366Q119.998-11.151 120.233-10.773Q120.467-10.396 120.467-9.944Q120.467-9.603 120.325-9.319Q120.183-9.035 119.939-8.828Q119.694-8.622 119.385-8.507Q119.076-8.393 118.754-8.393Q118.324-8.393 117.922-8.594Q117.520-8.796 117.279-9.148Q117.038-9.500 117.038-9.944M118.754-8.642Q119.356-8.642 119.580-9.020Q119.804-9.398 119.804-10.030Q119.804-10.642 119.569-11.001Q119.335-11.359 118.754-11.359Q117.702-11.359 117.702-10.030Q117.702-9.398 117.927-9.020Q118.153-8.642 118.754-8.642\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(37.676 30.203)\">\u003Cpath d=\"M122.440-8.488L121.312-10.987Q121.240-11.134 121.110-11.166Q120.980-11.199 120.751-11.199L120.751-11.479L122.265-11.479L122.265-11.199Q121.913-11.199 121.913-11.052Q121.913-11.007 121.924-10.987L122.788-9.069L123.568-10.799Q123.602-10.867 123.602-10.946Q123.602-11.059 123.518-11.129Q123.434-11.199 123.315-11.199L123.315-11.479L124.511-11.479L124.511-11.199Q124.292-11.199 124.121-11.096Q123.951-10.994 123.862-10.799L122.826-8.488Q122.778-8.393 122.672-8.393L122.594-8.393Q122.488-8.393 122.440-8.488\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(37.676 30.203)\">\u003Cpath d=\"M124.795-9.996Q124.795-10.317 124.920-10.606Q125.045-10.895 125.271-11.118Q125.496-11.342 125.792-11.462Q126.087-11.582 126.405-11.582Q126.733-11.582 126.995-11.482Q127.256-11.383 127.432-11.201Q127.608-11.018 127.702-10.760Q127.796-10.502 127.796-10.170Q127.796-10.078 127.714-10.057L125.459-10.057L125.459-9.996Q125.459-9.408 125.742-9.025Q126.026-8.642 126.593-8.642Q126.915-8.642 127.183-8.835Q127.451-9.028 127.540-9.343Q127.547-9.384 127.622-9.398L127.714-9.398Q127.796-9.374 127.796-9.302Q127.796-9.295 127.790-9.268Q127.677-8.871 127.306-8.632Q126.935-8.393 126.511-8.393Q126.074-8.393 125.674-8.601Q125.274-8.810 125.035-9.177Q124.795-9.544 124.795-9.996M125.465-10.266L127.280-10.266Q127.280-10.543 127.183-10.795Q127.085-11.048 126.887-11.204Q126.689-11.359 126.405-11.359Q126.128-11.359 125.915-11.201Q125.701-11.042 125.583-10.787Q125.465-10.532 125.465-10.266M130.134-8.461L128.398-8.461L128.398-8.741Q128.627-8.741 128.776-8.775Q128.924-8.810 128.924-8.950L128.924-10.799Q128.924-11.069 128.817-11.130Q128.709-11.192 128.398-11.192L128.398-11.472L129.427-11.547L129.427-10.840Q129.557-11.148 129.799-11.347Q130.042-11.547 130.360-11.547Q130.579-11.547 130.750-11.423Q130.920-11.298 130.920-11.086Q130.920-10.949 130.821-10.850Q130.722-10.751 130.589-10.751Q130.452-10.751 130.353-10.850Q130.254-10.949 130.254-11.086Q130.254-11.226 130.353-11.325Q130.063-11.325 129.863-11.129Q129.663-10.932 129.570-10.638Q129.478-10.344 129.478-10.064L129.478-8.950Q129.478-8.741 130.134-8.741L130.134-8.461M131.505-8.468L131.505-9.531Q131.505-9.555 131.532-9.582Q131.560-9.609 131.584-9.609L131.693-9.609Q131.758-9.609 131.772-9.551Q131.867-9.117 132.113-8.866Q132.359-8.615 132.773-8.615Q133.115-8.615 133.368-8.748Q133.621-8.881 133.621-9.189Q133.621-9.346 133.527-9.461Q133.433-9.575 133.294-9.644Q133.156-9.712 132.988-9.750L132.407-9.849Q132.052-9.917 131.778-10.138Q131.505-10.358 131.505-10.700Q131.505-10.949 131.616-11.124Q131.727-11.298 131.913-11.397Q132.100-11.496 132.315-11.539Q132.530-11.582 132.773-11.582Q133.187-11.582 133.467-11.400L133.682-11.575Q133.692-11.578 133.699-11.580Q133.706-11.582 133.716-11.582L133.768-11.582Q133.795-11.582 133.819-11.558Q133.843-11.534 133.843-11.506L133.843-10.659Q133.843-10.638 133.819-10.611Q133.795-10.584 133.768-10.584L133.655-10.584Q133.627-10.584 133.602-10.609Q133.576-10.635 133.576-10.659Q133.576-10.895 133.470-11.059Q133.364-11.223 133.181-11.305Q132.999-11.387 132.766-11.387Q132.438-11.387 132.182-11.284Q131.925-11.182 131.925-10.905Q131.925-10.710 132.108-10.601Q132.291-10.491 132.520-10.450L133.094-10.344Q133.340-10.296 133.554-10.168Q133.768-10.040 133.904-9.837Q134.041-9.633 134.041-9.384Q134.041-8.871 133.675-8.632Q133.310-8.393 132.773-8.393Q132.277-8.393 131.946-8.687L131.679-8.413Q131.659-8.393 131.631-8.393L131.584-8.393Q131.560-8.393 131.532-8.420Q131.505-8.447 131.505-8.468M136.352-8.461L134.718-8.461L134.718-8.741Q134.947-8.741 135.095-8.775Q135.244-8.810 135.244-8.950L135.244-12.569Q135.244-12.839 135.137-12.901Q135.029-12.962 134.718-12.962L134.718-13.243L135.798-13.318L135.798-10.932Q135.904-11.117 136.082-11.259Q136.259-11.400 136.468-11.474Q136.676-11.547 136.902-11.547Q137.408-11.547 137.691-11.324Q137.975-11.100 137.975-10.604L137.975-8.950Q137.975-8.813 138.124-8.777Q138.272-8.741 138.498-8.741L138.498-8.461L136.868-8.461L136.868-8.741Q137.097-8.741 137.245-8.775Q137.394-8.810 137.394-8.950L137.394-10.590Q137.394-10.925 137.274-11.125Q137.155-11.325 136.840-11.325Q136.570-11.325 136.336-11.189Q136.102-11.052 135.964-10.818Q135.825-10.584 135.825-10.310L135.825-8.950Q135.825-8.813 135.976-8.777Q136.126-8.741 136.352-8.741L136.352-8.461M139.045-9.944Q139.045-10.286 139.180-10.585Q139.315-10.884 139.554-11.108Q139.793-11.332 140.111-11.457Q140.429-11.582 140.761-11.582Q141.205-11.582 141.605-11.366Q142.005-11.151 142.239-10.773Q142.473-10.396 142.473-9.944Q142.473-9.603 142.331-9.319Q142.189-9.035 141.945-8.828Q141.701-8.622 141.391-8.507Q141.082-8.393 140.761-8.393Q140.330-8.393 139.928-8.594Q139.527-8.796 139.286-9.148Q139.045-9.500 139.045-9.944M140.761-8.642Q141.362-8.642 141.586-9.020Q141.810-9.398 141.810-10.030Q141.810-10.642 141.576-11.001Q141.342-11.359 140.761-11.359Q139.708-11.359 139.708-10.030Q139.708-9.398 139.934-9.020Q140.159-8.642 140.761-8.642\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(37.676 30.203)\">\u003Cpath d=\"M143.262-9.944Q143.262-10.286 143.397-10.585Q143.532-10.884 143.772-11.108Q144.011-11.332 144.329-11.457Q144.647-11.582 144.978-11.582Q145.423-11.582 145.822-11.366Q146.222-11.151 146.457-10.773Q146.691-10.396 146.691-9.944Q146.691-9.603 146.549-9.319Q146.407-9.035 146.163-8.828Q145.918-8.622 145.609-8.507Q145.300-8.393 144.978-8.393Q144.548-8.393 144.146-8.594Q143.744-8.796 143.503-9.148Q143.262-9.500 143.262-9.944M144.978-8.642Q145.580-8.642 145.804-9.020Q146.028-9.398 146.028-10.030Q146.028-10.642 145.793-11.001Q145.559-11.359 144.978-11.359Q143.926-11.359 143.926-10.030Q143.926-9.398 144.151-9.020Q144.377-8.642 144.978-8.642M147.812-9.302L147.812-11.199L147.173-11.199L147.173-11.421Q147.490-11.421 147.708-11.631Q147.925-11.841 148.025-12.151Q148.126-12.460 148.126-12.768L148.393-12.768L148.393-11.479L149.469-11.479L149.469-11.199L148.393-11.199L148.393-9.315Q148.393-9.039 148.497-8.840Q148.601-8.642 148.861-8.642Q149.018-8.642 149.124-8.746Q149.230-8.851 149.280-9.004Q149.329-9.158 149.329-9.315L149.329-9.729L149.596-9.729L149.596-9.302Q149.596-9.076 149.497-8.866Q149.398-8.656 149.213-8.524Q149.029-8.393 148.800-8.393Q148.362-8.393 148.087-8.630Q147.812-8.868 147.812-9.302\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">Why importance-sampling variance hurts SGD. Small ratios give safe steps (blue); an occasional large ratio produces a giant step (red) that overshoots into a region with a different gradient, and no amount of averaging recovers a good direction. Shrinking the step size to contain the outlier makes the ordinary steps uselessly small.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:378.474px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 283.856 151.787\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(-116.789 -82.58)\">\u003Cpath d=\"M71.939 17.472Q71.939 16.976 72.189 16.551Q72.439 16.125 72.859 15.879Q73.279 15.633 73.779 15.633Q74.318 15.633 74.709 15.758Q75.099 15.883 75.099 16.297Q75.099 16.402 75.049 16.494Q74.998 16.586 74.906 16.637Q74.814 16.687 74.705 16.687Q74.599 16.687 74.508 16.637Q74.416 16.586 74.365 16.494Q74.314 16.402 74.314 16.297Q74.314 16.074 74.482 15.969Q74.260 15.910 73.787 15.910Q73.490 15.910 73.275 16.049Q73.060 16.187 72.929 16.418Q72.799 16.648 72.740 16.918Q72.681 17.187 72.681 17.472Q72.681 17.867 72.814 18.217Q72.947 18.566 73.219 18.783Q73.490 19 73.888 19Q74.263 19 74.539 18.783Q74.814 18.566 74.916 18.207Q74.931 18.144 74.994 18.144L75.099 18.144Q75.135 18.144 75.160 18.172Q75.185 18.199 75.185 18.238L75.185 18.262Q75.053 18.742 74.668 19.010Q74.283 19.277 73.779 19.277Q73.416 19.277 73.082 19.140Q72.748 19.004 72.488 18.754Q72.228 18.504 72.084 18.168Q71.939 17.832 71.939 17.472M76.357 18.246L76.357 16.504Q76.357 16.289 76.295 16.193Q76.232 16.097 76.113 16.076Q75.994 16.054 75.748 16.054L75.748 15.758L76.994 15.672L76.994 18.222L76.994 18.246Q76.994 18.558 77.049 18.720Q77.103 18.883 77.254 18.953Q77.404 19.023 77.724 19.023Q78.154 19.023 78.428 18.685Q78.701 18.347 78.701 17.902L78.701 16.504Q78.701 16.289 78.638 16.193Q78.576 16.097 78.457 16.076Q78.338 16.054 78.092 16.054L78.092 15.758L79.338 15.672L79.338 18.457Q79.338 18.668 79.400 18.763Q79.463 18.859 79.582 18.881Q79.701 18.902 79.947 18.902L79.947 19.199L78.724 19.277L78.724 18.656Q78.556 18.945 78.275 19.111Q77.994 19.277 77.674 19.277Q76.357 19.277 76.357 18.246M81.017 18.238L81.017 16.047L80.314 16.047L80.314 15.793Q80.670 15.793 80.912 15.560Q81.154 15.328 81.265 14.980Q81.377 14.633 81.377 14.277L81.658 14.277L81.658 15.750L82.834 15.750L82.834 16.047L81.658 16.047L81.658 18.222Q81.658 18.543 81.777 18.771Q81.896 19 82.178 19Q82.357 19 82.474 18.877Q82.592 18.754 82.644 18.574Q82.697 18.394 82.697 18.222L82.697 17.750L82.978 17.750L82.978 18.238Q82.978 18.492 82.873 18.732Q82.767 18.972 82.570 19.125Q82.373 19.277 82.115 19.277Q81.799 19.277 81.547 19.154Q81.295 19.031 81.156 18.797Q81.017 18.562 81.017 18.238\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-116.789 -82.58)\">\u003Cpath d=\"M86.633 18.367Q86.633 17.883 87.035 17.588Q87.438 17.293 87.988 17.174Q88.539 17.054 89.031 17.054L89.031 16.765Q89.031 16.539 88.916 16.332Q88.801 16.125 88.604 16.006Q88.406 15.887 88.176 15.887Q87.750 15.887 87.465 15.992Q87.535 16.019 87.582 16.074Q87.629 16.129 87.654 16.199Q87.680 16.269 87.680 16.344Q87.680 16.449 87.629 16.541Q87.578 16.633 87.486 16.683Q87.395 16.734 87.289 16.734Q87.184 16.734 87.092 16.683Q87 16.633 86.949 16.541Q86.899 16.449 86.899 16.344Q86.899 15.926 87.287 15.779Q87.676 15.633 88.176 15.633Q88.508 15.633 88.861 15.763Q89.215 15.894 89.443 16.148Q89.672 16.402 89.672 16.750L89.672 18.551Q89.672 18.683 89.744 18.793Q89.817 18.902 89.945 18.902Q90.070 18.902 90.139 18.797Q90.207 18.691 90.207 18.551L90.207 18.039L90.488 18.039L90.488 18.551Q90.488 18.754 90.371 18.912Q90.254 19.070 90.072 19.154Q89.891 19.238 89.688 19.238Q89.457 19.238 89.305 19.066Q89.152 18.894 89.121 18.664Q88.961 18.945 88.652 19.111Q88.344 19.277 87.992 19.277Q87.481 19.277 87.057 19.054Q86.633 18.832 86.633 18.367M87.320 18.367Q87.320 18.652 87.547 18.838Q87.774 19.023 88.067 19.023Q88.313 19.023 88.537 18.906Q88.762 18.789 88.897 18.586Q89.031 18.383 89.031 18.129L89.031 17.297Q88.766 17.297 88.481 17.351Q88.195 17.406 87.924 17.535Q87.652 17.664 87.486 17.871Q87.320 18.078 87.320 18.367M91.406 18.238L91.406 16.047L90.703 16.047L90.703 15.793Q91.059 15.793 91.301 15.560Q91.543 15.328 91.654 14.980Q91.766 14.633 91.766 14.277L92.047 14.277L92.047 15.750L93.223 15.750L93.223 16.047L92.047 16.047L92.047 18.222Q92.047 18.543 92.166 18.771Q92.285 19 92.567 19Q92.746 19 92.863 18.877Q92.981 18.754 93.033 18.574Q93.086 18.394 93.086 18.222L93.086 17.750L93.367 17.750L93.367 18.238Q93.367 18.492 93.262 18.732Q93.156 18.972 92.959 19.125Q92.762 19.277 92.504 19.277Q92.188 19.277 91.936 19.154Q91.684 19.031 91.545 18.797Q91.406 18.562 91.406 18.238\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-116.789 -82.58)\">\u003Cpath d=\"M97.549 18.238L97.549 16.047L96.846 16.047L96.846 15.793Q97.202 15.793 97.444 15.560Q97.686 15.328 97.797 14.980Q97.909 14.633 97.909 14.277L98.190 14.277L98.190 15.750L99.366 15.750L99.366 16.047L98.190 16.047L98.190 18.222Q98.190 18.543 98.309 18.771Q98.428 19 98.709 19Q98.889 19 99.006 18.877Q99.124 18.754 99.176 18.574Q99.229 18.394 99.229 18.222L99.229 17.750L99.510 17.750L99.510 18.238Q99.510 18.492 99.405 18.732Q99.299 18.972 99.102 19.125Q98.905 19.277 98.647 19.277Q98.331 19.277 98.079 19.154Q97.827 19.031 97.688 18.797Q97.549 18.562 97.549 18.238M102.159 19.199L100.303 19.199L100.303 18.902Q100.577 18.902 100.745 18.855Q100.913 18.808 100.913 18.640L100.913 14.480Q100.913 14.265 100.850 14.170Q100.788 14.074 100.668 14.053Q100.549 14.031 100.303 14.031L100.303 13.734L101.526 13.648L101.526 16.351Q101.651 16.140 101.838 15.990Q102.026 15.840 102.252 15.756Q102.479 15.672 102.725 15.672Q103.893 15.672 103.893 16.750L103.893 18.640Q103.893 18.808 104.063 18.855Q104.233 18.902 104.502 18.902L104.502 19.199L102.647 19.199L102.647 18.902Q102.920 18.902 103.088 18.855Q103.256 18.808 103.256 18.640L103.256 16.765Q103.256 16.383 103.135 16.154Q103.014 15.926 102.663 15.926Q102.350 15.926 102.096 16.088Q101.842 16.250 101.696 16.519Q101.549 16.789 101.549 17.086L101.549 18.640Q101.549 18.808 101.719 18.855Q101.889 18.902 102.159 18.902L102.159 19.199M104.948 17.445Q104.948 16.965 105.180 16.549Q105.413 16.133 105.823 15.883Q106.233 15.633 106.709 15.633Q107.440 15.633 107.838 16.074Q108.237 16.515 108.237 17.246Q108.237 17.351 108.143 17.375L105.694 17.375L105.694 17.445Q105.694 17.855 105.815 18.211Q105.936 18.566 106.207 18.783Q106.479 19 106.909 19Q107.272 19 107.569 18.771Q107.866 18.543 107.967 18.191Q107.975 18.144 108.061 18.129L108.143 18.129Q108.237 18.156 108.237 18.238Q108.237 18.246 108.229 18.277Q108.166 18.504 108.028 18.687Q107.889 18.871 107.698 19.004Q107.506 19.137 107.288 19.207Q107.069 19.277 106.831 19.277Q106.459 19.277 106.122 19.140Q105.784 19.004 105.516 18.752Q105.249 18.500 105.098 18.160Q104.948 17.820 104.948 17.445M105.702 17.137L107.663 17.137Q107.663 16.832 107.561 16.541Q107.459 16.250 107.243 16.068Q107.026 15.887 106.709 15.887Q106.409 15.887 106.178 16.074Q105.948 16.262 105.825 16.553Q105.702 16.844 105.702 17.137\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-116.789 -82.58)\">\u003Cpath d=\"M111.606 19.191L111.606 17.969Q111.606 17.941 111.638 17.910Q111.669 17.879 111.692 17.879L111.798 17.879Q111.868 17.879 111.884 17.941Q111.946 18.262 112.085 18.502Q112.223 18.742 112.456 18.883Q112.688 19.023 112.997 19.023Q113.235 19.023 113.444 18.963Q113.653 18.902 113.790 18.754Q113.927 18.605 113.927 18.359Q113.927 18.105 113.716 17.939Q113.505 17.773 113.235 17.719L112.614 17.605Q112.208 17.527 111.907 17.271Q111.606 17.015 111.606 16.640Q111.606 16.273 111.807 16.051Q112.009 15.828 112.333 15.730Q112.657 15.633 112.997 15.633Q113.462 15.633 113.759 15.840L113.981 15.656Q114.005 15.633 114.036 15.633L114.087 15.633Q114.118 15.633 114.145 15.660Q114.173 15.687 114.173 15.719L114.173 16.703Q114.173 16.734 114.147 16.763Q114.122 16.793 114.087 16.793L113.981 16.793Q113.946 16.793 113.919 16.765Q113.891 16.738 113.891 16.703Q113.891 16.304 113.639 16.084Q113.388 15.863 112.989 15.863Q112.634 15.863 112.350 15.986Q112.067 16.109 112.067 16.414Q112.067 16.633 112.268 16.765Q112.470 16.898 112.716 16.941L113.341 17.054Q113.770 17.144 114.079 17.441Q114.388 17.738 114.388 18.152Q114.388 18.722 113.989 19Q113.591 19.277 112.997 19.277Q112.446 19.277 112.095 18.941L111.798 19.254Q111.774 19.277 111.739 19.277L111.692 19.277Q111.669 19.277 111.638 19.246Q111.606 19.215 111.606 19.191M114.915 17.504Q114.915 17 115.171 16.568Q115.427 16.137 115.862 15.885Q116.298 15.633 116.798 15.633Q117.184 15.633 117.526 15.777Q117.868 15.922 118.130 16.183Q118.391 16.445 118.534 16.781Q118.677 17.117 118.677 17.504Q118.677 17.996 118.413 18.406Q118.149 18.816 117.720 19.047Q117.290 19.277 116.798 19.277Q116.305 19.277 115.872 19.045Q115.438 18.812 115.177 18.404Q114.915 17.996 114.915 17.504M116.798 19Q117.255 19 117.507 18.777Q117.759 18.554 117.846 18.203Q117.934 17.851 117.934 17.406Q117.934 16.976 117.841 16.638Q117.747 16.301 117.493 16.094Q117.239 15.887 116.798 15.887Q116.149 15.887 115.905 16.303Q115.661 16.719 115.661 17.406Q115.661 17.851 115.749 18.203Q115.837 18.554 116.089 18.777Q116.341 19 116.798 19M119.845 18.246L119.845 16.504Q119.845 16.289 119.782 16.193Q119.720 16.097 119.600 16.076Q119.481 16.054 119.235 16.054L119.235 15.758L120.481 15.672L120.481 18.222L120.481 18.246Q120.481 18.558 120.536 18.720Q120.591 18.883 120.741 18.953Q120.891 19.023 121.212 19.023Q121.641 19.023 121.915 18.685Q122.188 18.347 122.188 17.902L122.188 16.504Q122.188 16.289 122.126 16.193Q122.063 16.097 121.944 16.076Q121.825 16.054 121.579 16.054L121.579 15.758L122.825 15.672L122.825 18.457Q122.825 18.668 122.888 18.763Q122.950 18.859 123.069 18.881Q123.188 18.902 123.434 18.902L123.434 19.199L122.212 19.277L122.212 18.656Q122.044 18.945 121.763 19.111Q121.481 19.277 121.161 19.277Q119.845 19.277 119.845 18.246M125.888 19.199L123.907 19.199L123.907 18.902Q124.177 18.902 124.345 18.857Q124.513 18.812 124.513 18.640L124.513 16.504Q124.513 16.289 124.450 16.193Q124.388 16.097 124.270 16.076Q124.153 16.054 123.907 16.054L123.907 15.758L125.075 15.672L125.075 16.457Q125.153 16.246 125.305 16.060Q125.458 15.875 125.657 15.773Q125.856 15.672 126.083 15.672Q126.329 15.672 126.520 15.816Q126.712 15.961 126.712 16.191Q126.712 16.347 126.606 16.457Q126.501 16.566 126.345 16.566Q126.188 16.566 126.079 16.457Q125.970 16.347 125.970 16.191Q125.970 16.031 126.075 15.926Q125.751 15.926 125.536 16.154Q125.321 16.383 125.225 16.722Q125.130 17.062 125.130 17.367L125.130 18.640Q125.130 18.808 125.356 18.855Q125.583 18.902 125.888 18.902L125.888 19.199M127.235 17.472Q127.235 16.976 127.485 16.551Q127.735 16.125 128.155 15.879Q128.575 15.633 129.075 15.633Q129.614 15.633 130.005 15.758Q130.395 15.883 130.395 16.297Q130.395 16.402 130.345 16.494Q130.294 16.586 130.202 16.637Q130.110 16.687 130.001 16.687Q129.895 16.687 129.804 16.637Q129.712 16.586 129.661 16.494Q129.610 16.402 129.610 16.297Q129.610 16.074 129.778 15.969Q129.555 15.910 129.083 15.910Q128.786 15.910 128.571 16.049Q128.356 16.187 128.225 16.418Q128.095 16.648 128.036 16.918Q127.977 17.187 127.977 17.472Q127.977 17.867 128.110 18.217Q128.243 18.566 128.514 18.783Q128.786 19 129.184 19Q129.559 19 129.835 18.783Q130.110 18.566 130.212 18.207Q130.227 18.144 130.290 18.144L130.395 18.144Q130.430 18.144 130.456 18.172Q130.481 18.199 130.481 18.238L130.481 18.262Q130.348 18.742 129.964 19.010Q129.579 19.277 129.075 19.277Q128.712 19.277 128.378 19.140Q128.044 19.004 127.784 18.754Q127.524 18.504 127.380 18.168Q127.235 17.832 127.235 17.472M130.970 17.445Q130.970 16.965 131.202 16.549Q131.434 16.133 131.845 15.883Q132.255 15.633 132.731 15.633Q133.462 15.633 133.860 16.074Q134.259 16.515 134.259 17.246Q134.259 17.351 134.165 17.375L131.716 17.375L131.716 17.445Q131.716 17.855 131.837 18.211Q131.958 18.566 132.229 18.783Q132.501 19 132.930 19Q133.294 19 133.591 18.771Q133.887 18.543 133.989 18.191Q133.997 18.144 134.083 18.129L134.165 18.129Q134.259 18.156 134.259 18.238Q134.259 18.246 134.251 18.277Q134.188 18.504 134.050 18.687Q133.911 18.871 133.720 19.004Q133.528 19.137 133.309 19.207Q133.091 19.277 132.852 19.277Q132.481 19.277 132.143 19.140Q131.805 19.004 131.538 18.752Q131.270 18.500 131.120 18.160Q130.970 17.820 130.970 17.445M131.723 17.137L133.684 17.137Q133.684 16.832 133.583 16.541Q133.481 16.250 133.264 16.068Q133.048 15.887 132.731 15.887Q132.430 15.887 132.200 16.074Q131.970 16.262 131.846 16.553Q131.723 16.844 131.723 17.137\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M-62.07-29.17h96.74v-25.608h-96.74Z\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-105.657 -59.423)\">\u003Cpath d=\"M73.360 19.172L72.379 16.673Q72.318 16.530 72.200 16.495Q72.082 16.461 71.866 16.461L71.866 16.181L73.346 16.181L73.346 16.461Q72.967 16.461 72.967 16.622Q72.967 16.632 72.981 16.673L73.695 18.505L74.368 16.800Q74.338 16.728 74.338 16.700Q74.338 16.673 74.310 16.673Q74.249 16.526 74.131 16.494Q74.013 16.461 73.801 16.461L73.801 16.181L75.199 16.181L75.199 16.461Q74.823 16.461 74.823 16.622Q74.823 16.653 74.830 16.673L75.585 18.611L76.272 16.861Q76.293 16.810 76.293 16.755Q76.293 16.615 76.180 16.538Q76.067 16.461 75.927 16.461L75.927 16.181L77.147 16.181L77.147 16.461Q76.942 16.461 76.787 16.567Q76.631 16.673 76.559 16.861L75.654 19.172Q75.619 19.267 75.507 19.267L75.438 19.267Q75.329 19.267 75.291 19.172L74.509 17.169L73.722 19.172Q73.688 19.267 73.575 19.267L73.507 19.267Q73.398 19.267 73.360 19.172\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-105.657 -59.423)\">\u003Cpath d=\"M77.424 17.664Q77.424 17.343 77.549 17.054Q77.674 16.765 77.900 16.542Q78.125 16.318 78.421 16.198Q78.716 16.078 79.034 16.078Q79.362 16.078 79.624 16.178Q79.885 16.277 80.061 16.459Q80.237 16.642 80.331 16.900Q80.425 17.158 80.425 17.490Q80.425 17.582 80.343 17.603L78.088 17.603L78.088 17.664Q78.088 18.252 78.371 18.635Q78.655 19.018 79.222 19.018Q79.544 19.018 79.812 18.825Q80.080 18.632 80.169 18.317Q80.176 18.276 80.251 18.262L80.343 18.262Q80.425 18.286 80.425 18.358Q80.425 18.365 80.419 18.392Q80.306 18.789 79.935 19.028Q79.564 19.267 79.140 19.267Q78.703 19.267 78.303 19.059Q77.903 18.850 77.664 18.483Q77.424 18.116 77.424 17.664M78.094 17.394L79.909 17.394Q79.909 17.117 79.812 16.865Q79.714 16.612 79.516 16.456Q79.318 16.301 79.034 16.301Q78.757 16.301 78.544 16.459Q78.330 16.618 78.212 16.873Q78.094 17.128 78.094 17.394M82.630 19.199L81.078 19.199L81.078 18.919Q81.304 18.919 81.453 18.885Q81.601 18.850 81.601 18.710L81.601 16.861Q81.601 16.673 81.553 16.589Q81.505 16.506 81.408 16.487Q81.311 16.468 81.099 16.468L81.099 16.188L82.155 16.113L82.155 18.710Q82.155 18.850 82.286 18.885Q82.418 18.919 82.630 18.919L82.630 19.199M81.359 14.892Q81.359 14.721 81.482 14.602Q81.605 14.482 81.776 14.482Q81.943 14.482 82.066 14.602Q82.189 14.721 82.189 14.892Q82.189 15.067 82.066 15.190Q81.943 15.313 81.776 15.313Q81.605 15.313 81.482 15.190Q81.359 15.067 81.359 14.892M83.235 19.732Q83.235 19.486 83.432 19.302Q83.628 19.117 83.884 19.038Q83.748 18.926 83.676 18.765Q83.604 18.604 83.604 18.423Q83.604 18.102 83.816 17.856Q83.481 17.558 83.481 17.148Q83.481 16.687 83.871 16.400Q84.260 16.113 84.739 16.113Q85.211 16.113 85.546 16.359Q85.720 16.205 85.930 16.123Q86.140 16.041 86.369 16.041Q86.533 16.041 86.655 16.148Q86.776 16.256 86.776 16.420Q86.776 16.516 86.704 16.588Q86.632 16.659 86.540 16.659Q86.441 16.659 86.371 16.586Q86.301 16.512 86.301 16.413Q86.301 16.359 86.315 16.328L86.321 16.314Q86.328 16.294 86.337 16.283Q86.345 16.273 86.349 16.266Q85.993 16.266 85.706 16.489Q85.993 16.782 85.993 17.148Q85.993 17.463 85.809 17.695Q85.624 17.928 85.335 18.056Q85.047 18.184 84.739 18.184Q84.537 18.184 84.346 18.134Q84.154 18.085 83.977 17.975Q83.884 18.102 83.884 18.245Q83.884 18.427 84.013 18.562Q84.141 18.697 84.325 18.697L84.958 18.697Q85.405 18.697 85.775 18.768Q86.144 18.840 86.403 19.069Q86.663 19.298 86.663 19.732Q86.663 20.053 86.368 20.255Q86.072 20.457 85.669 20.546Q85.265 20.635 84.951 20.635Q84.633 20.635 84.230 20.546Q83.826 20.457 83.531 20.255Q83.235 20.053 83.235 19.732M83.690 19.732Q83.690 19.961 83.908 20.110Q84.127 20.259 84.419 20.327Q84.712 20.395 84.951 20.395Q85.115 20.395 85.323 20.359Q85.532 20.324 85.739 20.243Q85.945 20.163 86.077 20.035Q86.209 19.907 86.209 19.732Q86.209 19.380 85.828 19.286Q85.446 19.192 84.944 19.192L84.325 19.192Q84.086 19.192 83.888 19.343Q83.690 19.493 83.690 19.732M84.739 17.945Q85.405 17.945 85.405 17.148Q85.405 16.348 84.739 16.348Q84.069 16.348 84.069 17.148Q84.069 17.945 84.739 17.945M88.940 19.199L87.306 19.199L87.306 18.919Q87.535 18.919 87.683 18.885Q87.832 18.850 87.832 18.710L87.832 15.091Q87.832 14.821 87.724 14.759Q87.617 14.698 87.306 14.698L87.306 14.417L88.386 14.342L88.386 16.728Q88.492 16.543 88.670 16.401Q88.847 16.260 89.056 16.186Q89.264 16.113 89.490 16.113Q89.996 16.113 90.279 16.336Q90.563 16.560 90.563 17.056L90.563 18.710Q90.563 18.847 90.712 18.883Q90.860 18.919 91.086 18.919L91.086 19.199L89.456 19.199L89.456 18.919Q89.685 18.919 89.833 18.885Q89.982 18.850 89.982 18.710L89.982 17.070Q89.982 16.735 89.862 16.535Q89.743 16.335 89.428 16.335Q89.158 16.335 88.924 16.471Q88.690 16.608 88.552 16.842Q88.413 17.076 88.413 17.350L88.413 18.710Q88.413 18.847 88.564 18.883Q88.714 18.919 88.940 18.919\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-105.657 -59.423)\">\u003Cpath d=\"M91.999 18.358L91.999 16.461L91.360 16.461L91.360 16.239Q91.678 16.239 91.895 16.029Q92.112 15.819 92.212 15.509Q92.313 15.200 92.313 14.892L92.580 14.892L92.580 16.181L93.657 16.181L93.657 16.461L92.580 16.461L92.580 18.345Q92.580 18.621 92.684 18.820Q92.788 19.018 93.048 19.018Q93.205 19.018 93.311 18.914Q93.417 18.809 93.467 18.656Q93.516 18.502 93.516 18.345L93.516 17.931L93.783 17.931L93.783 18.358Q93.783 18.584 93.684 18.794Q93.585 19.004 93.400 19.136Q93.216 19.267 92.987 19.267Q92.549 19.267 92.274 19.030Q91.999 18.792 91.999 18.358M94.552 17.664Q94.552 17.343 94.677 17.054Q94.802 16.765 95.027 16.542Q95.253 16.318 95.548 16.198Q95.844 16.078 96.162 16.078Q96.490 16.078 96.752 16.178Q97.013 16.277 97.189 16.459Q97.365 16.642 97.459 16.900Q97.553 17.158 97.553 17.490Q97.553 17.582 97.471 17.603L95.215 17.603L95.215 17.664Q95.215 18.252 95.499 18.635Q95.783 19.018 96.350 19.018Q96.671 19.018 96.939 18.825Q97.208 18.632 97.297 18.317Q97.304 18.276 97.379 18.262L97.471 18.262Q97.553 18.286 97.553 18.358Q97.553 18.365 97.546 18.392Q97.433 18.789 97.063 19.028Q96.692 19.267 96.268 19.267Q95.830 19.267 95.430 19.059Q95.031 18.850 94.791 18.483Q94.552 18.116 94.552 17.664M95.222 17.394L97.037 17.394Q97.037 17.117 96.939 16.865Q96.842 16.612 96.644 16.456Q96.446 16.301 96.162 16.301Q95.885 16.301 95.671 16.459Q95.458 16.618 95.340 16.873Q95.222 17.128 95.222 17.394M98.141 17.688Q98.141 17.350 98.281 17.059Q98.421 16.769 98.666 16.555Q98.910 16.342 99.214 16.227Q99.518 16.113 99.843 16.113Q100.113 16.113 100.376 16.212Q100.639 16.311 100.831 16.489L100.831 15.091Q100.831 14.821 100.723 14.759Q100.616 14.698 100.304 14.698L100.304 14.417L101.381 14.342L101.381 18.526Q101.381 18.714 101.436 18.797Q101.491 18.881 101.591 18.900Q101.692 18.919 101.908 18.919L101.908 19.199L100.800 19.267L100.800 18.850Q100.383 19.267 99.758 19.267Q99.327 19.267 98.954 19.055Q98.582 18.844 98.361 18.483Q98.141 18.122 98.141 17.688M99.816 19.045Q100.024 19.045 100.210 18.973Q100.397 18.902 100.551 18.765Q100.704 18.628 100.800 18.450L100.800 16.841Q100.715 16.694 100.569 16.574Q100.424 16.454 100.255 16.395Q100.086 16.335 99.905 16.335Q99.344 16.335 99.076 16.724Q98.807 17.114 98.807 17.695Q98.807 18.266 99.042 18.656Q99.276 19.045 99.816 19.045\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-105.657 -59.423)\">\u003Cpath d=\"M107.496 19.199L105.292 19.199L105.292 18.919Q106.051 18.919 106.051 18.710L106.051 14.909Q106.051 14.698 105.292 14.698L105.292 14.417L107.496 14.417L107.496 14.698Q106.741 14.698 106.741 14.909L106.741 18.710Q106.741 18.919 107.496 18.919L107.496 19.199M108.324 19.261L108.324 17.688Q108.324 17.661 108.349 17.635Q108.375 17.610 108.402 17.610L108.515 17.610Q108.542 17.610 108.566 17.637Q108.590 17.664 108.590 17.688Q108.590 18.033 108.722 18.297Q108.853 18.560 109.082 18.729Q109.311 18.898 109.614 18.979Q109.916 19.059 110.258 19.059Q110.525 19.059 110.761 18.931Q110.996 18.803 111.142 18.580Q111.287 18.358 111.287 18.092Q111.287 17.869 111.181 17.673Q111.075 17.476 110.894 17.341Q110.713 17.206 110.487 17.155L109.458 16.923Q109.147 16.851 108.888 16.665Q108.628 16.478 108.476 16.207Q108.324 15.935 108.324 15.620Q108.324 15.234 108.537 14.927Q108.751 14.619 109.098 14.448Q109.445 14.277 109.824 14.277Q110.053 14.277 110.282 14.330Q110.511 14.383 110.709 14.491Q110.908 14.598 111.061 14.762L111.355 14.322Q111.379 14.277 111.420 14.277L111.468 14.277Q111.499 14.277 111.521 14.303Q111.543 14.328 111.543 14.356L111.543 15.931Q111.543 15.952 111.519 15.979Q111.495 16.007 111.468 16.007L111.355 16.007Q111.294 16.007 111.280 15.931Q111.239 15.518 111.058 15.198Q110.877 14.879 110.566 14.704Q110.255 14.530 109.824 14.530Q109.575 14.530 109.335 14.641Q109.096 14.752 108.946 14.950Q108.795 15.149 108.795 15.412Q108.795 15.624 108.903 15.805Q109.011 15.986 109.187 16.106Q109.363 16.225 109.571 16.266L110.600 16.495Q110.918 16.567 111.184 16.772Q111.451 16.977 111.603 17.271Q111.755 17.565 111.755 17.897Q111.755 18.290 111.550 18.625Q111.345 18.960 111 19.149Q110.655 19.339 110.258 19.339Q109.838 19.339 109.458 19.226Q109.079 19.114 108.809 18.864L108.515 19.298Q108.488 19.339 108.450 19.339L108.402 19.339Q108.375 19.339 108.349 19.314Q108.324 19.288 108.324 19.261\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M-62.07 2.127h96.74V-23.48h-96.74Z\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-124.06 -28.125)\">\u003Cpath d=\"M72.499 18.358L72.499 16.461L71.860 16.461L71.860 16.239Q72.178 16.239 72.395 16.029Q72.612 15.819 72.712 15.509Q72.813 15.200 72.813 14.892L73.080 14.892L73.080 16.181L74.157 16.181L74.157 16.461L73.080 16.461L73.080 18.345Q73.080 18.621 73.184 18.820Q73.288 19.018 73.548 19.018Q73.705 19.018 73.811 18.914Q73.917 18.809 73.967 18.656Q74.016 18.502 74.016 18.345L74.016 17.931L74.283 17.931L74.283 18.358Q74.283 18.584 74.184 18.794Q74.085 19.004 73.900 19.136Q73.716 19.267 73.487 19.267Q73.049 19.267 72.774 19.030Q72.499 18.792 72.499 18.358M76.843 19.199L75.107 19.199L75.107 18.919Q75.336 18.919 75.484 18.885Q75.633 18.850 75.633 18.710L75.633 16.861Q75.633 16.591 75.525 16.530Q75.418 16.468 75.107 16.468L75.107 16.188L76.136 16.113L76.136 16.820Q76.265 16.512 76.508 16.313Q76.751 16.113 77.069 16.113Q77.287 16.113 77.458 16.237Q77.629 16.362 77.629 16.574Q77.629 16.711 77.530 16.810Q77.431 16.909 77.298 16.909Q77.161 16.909 77.062 16.810Q76.963 16.711 76.963 16.574Q76.963 16.434 77.062 16.335Q76.771 16.335 76.571 16.531Q76.371 16.728 76.279 17.022Q76.187 17.316 76.187 17.596L76.187 18.710Q76.187 18.919 76.843 18.919L76.843 19.199M78.173 17.664Q78.173 17.343 78.297 17.054Q78.422 16.765 78.648 16.542Q78.873 16.318 79.169 16.198Q79.465 16.078 79.783 16.078Q80.111 16.078 80.372 16.178Q80.634 16.277 80.810 16.459Q80.986 16.642 81.080 16.900Q81.174 17.158 81.174 17.490Q81.174 17.582 81.092 17.603L78.836 17.603L78.836 17.664Q78.836 18.252 79.119 18.635Q79.403 19.018 79.971 19.018Q80.292 19.018 80.560 18.825Q80.828 18.632 80.917 18.317Q80.924 18.276 80.999 18.262L81.092 18.262Q81.174 18.286 81.174 18.358Q81.174 18.365 81.167 18.392Q81.054 18.789 80.683 19.028Q80.312 19.267 79.888 19.267Q79.451 19.267 79.051 19.059Q78.651 18.850 78.412 18.483Q78.173 18.116 78.173 17.664M78.843 17.394L80.658 17.394Q80.658 17.117 80.560 16.865Q80.463 16.612 80.264 16.456Q80.066 16.301 79.783 16.301Q79.506 16.301 79.292 16.459Q79.078 16.618 78.960 16.873Q78.843 17.128 78.843 17.394M81.721 17.664Q81.721 17.343 81.845 17.054Q81.970 16.765 82.196 16.542Q82.421 16.318 82.717 16.198Q83.012 16.078 83.330 16.078Q83.658 16.078 83.920 16.178Q84.181 16.277 84.357 16.459Q84.533 16.642 84.627 16.900Q84.721 17.158 84.721 17.490Q84.721 17.582 84.639 17.603L82.384 17.603L82.384 17.664Q82.384 18.252 82.667 18.635Q82.951 19.018 83.518 19.018Q83.840 19.018 84.108 18.825Q84.376 18.632 84.465 18.317Q84.472 18.276 84.547 18.262L84.639 18.262Q84.721 18.286 84.721 18.358Q84.721 18.365 84.715 18.392Q84.602 18.789 84.231 19.028Q83.860 19.267 83.436 19.267Q82.999 19.267 82.599 19.059Q82.199 18.850 81.960 18.483Q81.721 18.116 81.721 17.664M82.390 17.394L84.205 17.394Q84.205 17.117 84.108 16.865Q84.011 16.612 83.812 16.456Q83.614 16.301 83.330 16.301Q83.054 16.301 82.840 16.459Q82.626 16.618 82.508 16.873Q82.390 17.128 82.390 17.394M87.193 17.945L85.135 17.945L85.135 17.442L87.193 17.442L87.193 17.945M88.803 19.199L88.536 19.199L88.536 15.091Q88.536 14.821 88.428 14.759Q88.321 14.698 88.010 14.698L88.010 14.417L89.090 14.342L89.090 16.512Q89.298 16.321 89.584 16.217Q89.869 16.113 90.166 16.113Q90.484 16.113 90.782 16.234Q91.079 16.355 91.301 16.571Q91.523 16.786 91.650 17.071Q91.776 17.357 91.776 17.688Q91.776 18.133 91.537 18.497Q91.298 18.861 90.905 19.064Q90.512 19.267 90.067 19.267Q89.872 19.267 89.683 19.211Q89.493 19.155 89.332 19.050Q89.172 18.946 89.032 18.785L88.803 19.199M89.117 16.854L89.117 18.471Q89.254 18.731 89.495 18.888Q89.736 19.045 90.012 19.045Q90.306 19.045 90.518 18.938Q90.730 18.830 90.864 18.638Q90.997 18.447 91.055 18.208Q91.113 17.969 91.113 17.688Q91.113 17.329 91.019 17.025Q90.925 16.721 90.698 16.528Q90.471 16.335 90.105 16.335Q89.804 16.335 89.537 16.471Q89.271 16.608 89.117 16.854M92.470 18.471Q92.470 18.139 92.694 17.912Q92.918 17.685 93.261 17.557Q93.605 17.428 93.977 17.376Q94.350 17.323 94.654 17.323L94.654 17.070Q94.654 16.865 94.546 16.685Q94.439 16.506 94.258 16.403Q94.076 16.301 93.868 16.301Q93.461 16.301 93.225 16.393Q93.314 16.430 93.360 16.514Q93.407 16.598 93.407 16.700Q93.407 16.796 93.360 16.875Q93.314 16.953 93.234 16.998Q93.154 17.042 93.065 17.042Q92.914 17.042 92.814 16.945Q92.713 16.847 92.713 16.700Q92.713 16.078 93.868 16.078Q94.080 16.078 94.329 16.142Q94.579 16.205 94.781 16.324Q94.982 16.444 95.109 16.629Q95.235 16.813 95.235 17.056L95.235 18.632Q95.235 18.748 95.297 18.844Q95.358 18.939 95.471 18.939Q95.580 18.939 95.645 18.845Q95.710 18.751 95.710 18.632L95.710 18.184L95.977 18.184L95.977 18.632Q95.977 18.902 95.750 19.067Q95.522 19.233 95.242 19.233Q95.033 19.233 94.897 19.079Q94.760 18.926 94.736 18.710Q94.589 18.977 94.307 19.122Q94.025 19.267 93.700 19.267Q93.424 19.267 93.140 19.192Q92.856 19.117 92.663 18.938Q92.470 18.758 92.470 18.471M93.085 18.471Q93.085 18.645 93.186 18.775Q93.287 18.905 93.442 18.975Q93.598 19.045 93.762 19.045Q93.981 19.045 94.189 18.948Q94.398 18.850 94.526 18.669Q94.654 18.488 94.654 18.262L94.654 17.534Q94.329 17.534 93.964 17.625Q93.598 17.716 93.342 17.928Q93.085 18.139 93.085 18.471M96.394 17.688Q96.394 17.360 96.529 17.059Q96.664 16.759 96.900 16.538Q97.136 16.318 97.440 16.198Q97.744 16.078 98.069 16.078Q98.575 16.078 98.923 16.181Q99.272 16.283 99.272 16.659Q99.272 16.806 99.174 16.907Q99.077 17.008 98.930 17.008Q98.776 17.008 98.677 16.909Q98.578 16.810 98.578 16.659Q98.578 16.471 98.718 16.379Q98.516 16.328 98.075 16.328Q97.720 16.328 97.491 16.524Q97.262 16.721 97.161 17.030Q97.060 17.340 97.060 17.688Q97.060 18.037 97.187 18.343Q97.313 18.649 97.568 18.833Q97.823 19.018 98.178 19.018Q98.400 19.018 98.585 18.934Q98.769 18.850 98.904 18.695Q99.039 18.539 99.097 18.331Q99.111 18.276 99.166 18.276L99.279 18.276Q99.309 18.276 99.332 18.300Q99.354 18.324 99.354 18.358L99.354 18.379Q99.268 18.666 99.080 18.864Q98.892 19.062 98.627 19.165Q98.363 19.267 98.069 19.267Q97.638 19.267 97.250 19.061Q96.862 18.854 96.628 18.491Q96.394 18.129 96.394 17.688\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-124.06 -28.125)\">\u003Cpath d=\"M101.368 19.199L99.785 19.199L99.785 18.919Q100.014 18.919 100.163 18.885Q100.311 18.850 100.311 18.710L100.311 15.091Q100.311 14.821 100.204 14.759Q100.096 14.698 99.785 14.698L99.785 14.417L100.865 14.342L100.865 17.630L101.850 16.861Q102.055 16.724 102.055 16.574Q102.055 16.530 102.014 16.495Q101.973 16.461 101.928 16.461L101.928 16.181L103.292 16.181L103.292 16.461Q102.803 16.461 102.284 16.861L101.727 17.295L102.704 18.519Q102.906 18.765 103.039 18.842Q103.172 18.919 103.459 18.919L103.459 19.199L102.027 19.199L102.027 18.919Q102.215 18.919 102.215 18.806Q102.215 18.710 102.061 18.519L101.327 17.610L100.845 17.989L100.845 18.710Q100.845 18.847 100.993 18.883Q101.142 18.919 101.368 18.919L101.368 19.199M104.546 18.365L104.546 16.861Q104.546 16.591 104.439 16.530Q104.331 16.468 104.020 16.468L104.020 16.188L105.127 16.113L105.127 18.345L105.127 18.365Q105.127 18.645 105.179 18.789Q105.230 18.932 105.372 18.989Q105.514 19.045 105.801 19.045Q106.054 19.045 106.259 18.905Q106.464 18.765 106.580 18.539Q106.696 18.314 106.696 18.064L106.696 16.861Q106.696 16.591 106.589 16.530Q106.481 16.468 106.170 16.468L106.170 16.188L107.277 16.113L107.277 18.526Q107.277 18.717 107.330 18.799Q107.383 18.881 107.484 18.900Q107.585 18.919 107.800 18.919L107.800 19.199L106.724 19.267L106.724 18.703Q106.614 18.885 106.469 19.008Q106.324 19.131 106.137 19.199Q105.951 19.267 105.749 19.267Q104.546 19.267 104.546 18.365M110.032 20.556L108.402 20.556L108.402 20.276Q108.631 20.276 108.780 20.241Q108.928 20.207 108.928 20.067L108.928 16.721Q108.928 16.550 108.791 16.509Q108.655 16.468 108.402 16.468L108.402 16.188L109.482 16.113L109.482 16.519Q109.704 16.318 109.991 16.215Q110.278 16.113 110.586 16.113Q111.013 16.113 111.377 16.326Q111.741 16.540 111.955 16.904Q112.168 17.268 112.168 17.688Q112.168 18.133 111.929 18.497Q111.690 18.861 111.297 19.064Q110.904 19.267 110.459 19.267Q110.193 19.267 109.945 19.167Q109.697 19.066 109.509 18.885L109.509 20.067Q109.509 20.204 109.658 20.240Q109.807 20.276 110.032 20.276L110.032 20.556M109.509 16.868L109.509 18.478Q109.643 18.731 109.885 18.888Q110.128 19.045 110.405 19.045Q110.733 19.045 110.986 18.844Q111.239 18.642 111.372 18.324Q111.505 18.006 111.505 17.688Q111.505 17.459 111.440 17.230Q111.375 17.001 111.247 16.803Q111.119 16.605 110.924 16.485Q110.729 16.366 110.497 16.366Q110.203 16.366 109.935 16.495Q109.666 16.625 109.509 16.868M113.204 18.779Q113.204 18.611 113.327 18.488Q113.450 18.365 113.624 18.365Q113.792 18.365 113.915 18.488Q114.038 18.611 114.038 18.779Q114.038 18.953 113.915 19.076Q113.792 19.199 113.624 19.199Q113.450 19.199 113.327 19.076Q113.204 18.953 113.204 18.779M113.204 16.595Q113.204 16.427 113.327 16.304Q113.450 16.181 113.624 16.181Q113.792 16.181 113.915 16.304Q114.038 16.427 114.038 16.595Q114.038 16.769 113.915 16.892Q113.792 17.015 113.624 17.015Q113.450 17.015 113.327 16.892Q113.204 16.769 113.204 16.595\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-124.06 -28.125)\">\u003Cpath d=\"M120.307 19.199L118.673 19.199L118.673 18.919Q118.902 18.919 119.051 18.885Q119.200 18.850 119.200 18.710L119.200 16.861Q119.200 16.591 119.092 16.530Q118.984 16.468 118.673 16.468L118.673 16.188L119.733 16.113L119.733 16.762Q119.904 16.454 120.208 16.283Q120.512 16.113 120.857 16.113Q121.363 16.113 121.647 16.336Q121.931 16.560 121.931 17.056L121.931 18.710Q121.931 18.847 122.079 18.883Q122.228 18.919 122.454 18.919L122.454 19.199L120.823 19.199L120.823 18.919Q121.052 18.919 121.201 18.885Q121.350 18.850 121.350 18.710L121.350 17.070Q121.350 16.735 121.230 16.535Q121.110 16.335 120.796 16.335Q120.526 16.335 120.292 16.471Q120.058 16.608 119.919 16.842Q119.781 17.076 119.781 17.350L119.781 18.710Q119.781 18.847 119.931 18.883Q120.082 18.919 120.307 18.919L120.307 19.199M123 17.716Q123 17.374 123.135 17.075Q123.270 16.776 123.510 16.552Q123.749 16.328 124.067 16.203Q124.385 16.078 124.716 16.078Q125.161 16.078 125.561 16.294Q125.960 16.509 126.195 16.887Q126.429 17.264 126.429 17.716Q126.429 18.057 126.287 18.341Q126.145 18.625 125.901 18.832Q125.656 19.038 125.347 19.153Q125.038 19.267 124.716 19.267Q124.286 19.267 123.884 19.066Q123.482 18.864 123.241 18.512Q123 18.160 123 17.716M124.716 19.018Q125.318 19.018 125.542 18.640Q125.766 18.262 125.766 17.630Q125.766 17.018 125.531 16.659Q125.297 16.301 124.716 16.301Q123.664 16.301 123.664 17.630Q123.664 18.262 123.889 18.640Q124.115 19.018 124.716 19.018\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-124.06 -28.125)\">\u003Cpath d=\"M131.472 19.199L129.736 19.199L129.736 18.919Q129.965 18.919 130.114 18.885Q130.262 18.850 130.262 18.710L130.262 16.861Q130.262 16.591 130.155 16.530Q130.047 16.468 129.736 16.468L129.736 16.188L130.765 16.113L130.765 16.820Q130.895 16.512 131.137 16.313Q131.380 16.113 131.698 16.113Q131.917 16.113 132.088 16.237Q132.259 16.362 132.259 16.574Q132.259 16.711 132.159 16.810Q132.060 16.909 131.927 16.909Q131.790 16.909 131.691 16.810Q131.592 16.711 131.592 16.574Q131.592 16.434 131.691 16.335Q131.401 16.335 131.201 16.531Q131.001 16.728 130.908 17.022Q130.816 17.316 130.816 17.596L130.816 18.710Q130.816 18.919 131.472 18.919L131.472 19.199M132.901 18.471Q132.901 18.139 133.125 17.912Q133.349 17.685 133.692 17.557Q134.036 17.428 134.408 17.376Q134.781 17.323 135.085 17.323L135.085 17.070Q135.085 16.865 134.978 16.685Q134.870 16.506 134.689 16.403Q134.508 16.301 134.299 16.301Q133.892 16.301 133.657 16.393Q133.745 16.430 133.792 16.514Q133.838 16.598 133.838 16.700Q133.838 16.796 133.792 16.875Q133.745 16.953 133.665 16.998Q133.585 17.042 133.496 17.042Q133.345 17.042 133.245 16.945Q133.144 16.847 133.144 16.700Q133.144 16.078 134.299 16.078Q134.511 16.078 134.761 16.142Q135.010 16.205 135.212 16.324Q135.413 16.444 135.540 16.629Q135.666 16.813 135.666 17.056L135.666 18.632Q135.666 18.748 135.728 18.844Q135.789 18.939 135.902 18.939Q136.012 18.939 136.076 18.845Q136.141 18.751 136.141 18.632L136.141 18.184L136.408 18.184L136.408 18.632Q136.408 18.902 136.181 19.067Q135.953 19.233 135.673 19.233Q135.465 19.233 135.328 19.079Q135.191 18.926 135.167 18.710Q135.020 18.977 134.738 19.122Q134.456 19.267 134.132 19.267Q133.855 19.267 133.571 19.192Q133.287 19.117 133.094 18.938Q132.901 18.758 132.901 18.471M133.516 18.471Q133.516 18.645 133.617 18.775Q133.718 18.905 133.874 18.975Q134.029 19.045 134.193 19.045Q134.412 19.045 134.620 18.948Q134.829 18.850 134.957 18.669Q135.085 18.488 135.085 18.262L135.085 17.534Q134.761 17.534 134.395 17.625Q134.029 17.716 133.773 17.928Q133.516 18.139 133.516 18.471M137.351 18.358L137.351 16.461L136.712 16.461L136.712 16.239Q137.030 16.239 137.247 16.029Q137.464 15.819 137.565 15.509Q137.666 15.200 137.666 14.892L137.932 14.892L137.932 16.181L139.009 16.181L139.009 16.461L137.932 16.461L137.932 18.345Q137.932 18.621 138.037 18.820Q138.141 19.018 138.401 19.018Q138.558 19.018 138.664 18.914Q138.770 18.809 138.819 18.656Q138.869 18.502 138.869 18.345L138.869 17.931L139.136 17.931L139.136 18.358Q139.136 18.584 139.036 18.794Q138.937 19.004 138.753 19.136Q138.568 19.267 138.339 19.267Q137.902 19.267 137.627 19.030Q137.351 18.792 137.351 18.358M141.562 19.199L140.011 19.199L140.011 18.919Q140.236 18.919 140.385 18.885Q140.533 18.850 140.533 18.710L140.533 16.861Q140.533 16.673 140.486 16.589Q140.438 16.506 140.340 16.487Q140.243 16.468 140.031 16.468L140.031 16.188L141.087 16.113L141.087 18.710Q141.087 18.850 141.219 18.885Q141.350 18.919 141.562 18.919L141.562 19.199M140.291 14.892Q140.291 14.721 140.414 14.602Q140.537 14.482 140.708 14.482Q140.875 14.482 140.998 14.602Q141.121 14.721 141.121 14.892Q141.121 15.067 140.998 15.190Q140.875 15.313 140.708 15.313Q140.537 15.313 140.414 15.190Q140.291 15.067 140.291 14.892M142.167 17.716Q142.167 17.374 142.302 17.075Q142.437 16.776 142.677 16.552Q142.916 16.328 143.234 16.203Q143.552 16.078 143.883 16.078Q144.327 16.078 144.727 16.294Q145.127 16.509 145.361 16.887Q145.595 17.264 145.595 17.716Q145.595 18.057 145.454 18.341Q145.312 18.625 145.067 18.832Q144.823 19.038 144.514 19.153Q144.204 19.267 143.883 19.267Q143.452 19.267 143.051 19.066Q142.649 18.864 142.408 18.512Q142.167 18.160 142.167 17.716M143.883 19.018Q144.485 19.018 144.709 18.640Q144.932 18.262 144.932 17.630Q144.932 17.018 144.698 16.659Q144.464 16.301 143.883 16.301Q142.830 16.301 142.830 17.630Q142.830 18.262 143.056 18.640Q143.282 19.018 143.883 19.018M146.190 19.192L146.190 18.129Q146.190 18.105 146.218 18.078Q146.245 18.051 146.269 18.051L146.378 18.051Q146.443 18.051 146.457 18.109Q146.553 18.543 146.799 18.794Q147.045 19.045 147.458 19.045Q147.800 19.045 148.053 18.912Q148.306 18.779 148.306 18.471Q148.306 18.314 148.212 18.199Q148.118 18.085 147.980 18.016Q147.841 17.948 147.674 17.910L147.093 17.811Q146.737 17.743 146.464 17.522Q146.190 17.302 146.190 16.960Q146.190 16.711 146.301 16.536Q146.412 16.362 146.599 16.263Q146.785 16.164 147 16.121Q147.216 16.078 147.458 16.078Q147.872 16.078 148.152 16.260L148.367 16.085Q148.378 16.082 148.385 16.080Q148.391 16.078 148.402 16.078L148.453 16.078Q148.480 16.078 148.504 16.102Q148.528 16.126 148.528 16.154L148.528 17.001Q148.528 17.022 148.504 17.049Q148.480 17.076 148.453 17.076L148.340 17.076Q148.313 17.076 148.287 17.051Q148.262 17.025 148.262 17.001Q148.262 16.765 148.156 16.601Q148.050 16.437 147.867 16.355Q147.684 16.273 147.451 16.273Q147.123 16.273 146.867 16.376Q146.611 16.478 146.611 16.755Q146.611 16.950 146.793 17.059Q146.976 17.169 147.205 17.210L147.780 17.316Q148.026 17.364 148.239 17.492Q148.453 17.620 148.590 17.823Q148.726 18.027 148.726 18.276Q148.726 18.789 148.361 19.028Q147.995 19.267 147.458 19.267Q146.963 19.267 146.631 18.973L146.365 19.247Q146.344 19.267 146.317 19.267L146.269 19.267Q146.245 19.267 146.218 19.240Q146.190 19.213 146.190 19.192\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M-62.07 33.425h96.74V7.818h-96.74Z\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-121.637 3.173)\">\u003Cpath d=\"M71.972 17.688Q71.972 17.360 72.107 17.059Q72.242 16.759 72.478 16.538Q72.714 16.318 73.018 16.198Q73.323 16.078 73.647 16.078Q74.153 16.078 74.502 16.181Q74.850 16.283 74.850 16.659Q74.850 16.806 74.753 16.907Q74.656 17.008 74.509 17.008Q74.355 17.008 74.256 16.909Q74.157 16.810 74.157 16.659Q74.157 16.471 74.297 16.379Q74.095 16.328 73.654 16.328Q73.299 16.328 73.070 16.524Q72.841 16.721 72.740 17.030Q72.639 17.340 72.639 17.688Q72.639 18.037 72.765 18.343Q72.892 18.649 73.147 18.833Q73.401 19.018 73.757 19.018Q73.979 19.018 74.163 18.934Q74.348 18.850 74.483 18.695Q74.618 18.539 74.676 18.331Q74.690 18.276 74.744 18.276L74.857 18.276Q74.888 18.276 74.910 18.300Q74.932 18.324 74.932 18.358L74.932 18.379Q74.847 18.666 74.659 18.864Q74.471 19.062 74.206 19.165Q73.941 19.267 73.647 19.267Q73.217 19.267 72.829 19.061Q72.441 18.854 72.207 18.491Q71.972 18.129 71.972 17.688M75.479 17.716Q75.479 17.374 75.614 17.075Q75.749 16.776 75.989 16.552Q76.228 16.328 76.546 16.203Q76.864 16.078 77.195 16.078Q77.639 16.078 78.039 16.294Q78.439 16.509 78.673 16.887Q78.908 17.264 78.908 17.716Q78.908 18.057 78.766 18.341Q78.624 18.625 78.379 18.832Q78.135 19.038 77.826 19.153Q77.516 19.267 77.195 19.267Q76.764 19.267 76.363 19.066Q75.961 18.864 75.720 18.512Q75.479 18.160 75.479 17.716M77.195 19.018Q77.797 19.018 78.021 18.640Q78.244 18.262 78.244 17.630Q78.244 17.018 78.010 16.659Q77.776 16.301 77.195 16.301Q76.142 16.301 76.142 17.630Q76.142 18.262 76.368 18.640Q76.594 19.018 77.195 19.018M81.184 19.199L79.550 19.199L79.550 18.919Q79.779 18.919 79.928 18.885Q80.076 18.850 80.076 18.710L80.076 16.861Q80.076 16.591 79.969 16.530Q79.861 16.468 79.550 16.468L79.550 16.188L80.610 16.113L80.610 16.762Q80.781 16.454 81.085 16.283Q81.389 16.113 81.734 16.113Q82.240 16.113 82.524 16.336Q82.807 16.560 82.807 17.056L82.807 18.710Q82.807 18.847 82.956 18.883Q83.105 18.919 83.330 18.919L83.330 19.199L81.700 19.199L81.700 18.919Q81.929 18.919 82.078 18.885Q82.226 18.850 82.226 18.710L82.226 17.070Q82.226 16.735 82.107 16.535Q81.987 16.335 81.673 16.335Q81.403 16.335 81.168 16.471Q80.934 16.608 80.796 16.842Q80.658 17.076 80.658 17.350L80.658 18.710Q80.658 18.847 80.808 18.883Q80.958 18.919 81.184 18.919L81.184 19.199M83.918 19.192L83.918 18.129Q83.918 18.105 83.946 18.078Q83.973 18.051 83.997 18.051L84.106 18.051Q84.171 18.051 84.185 18.109Q84.281 18.543 84.527 18.794Q84.773 19.045 85.186 19.045Q85.528 19.045 85.781 18.912Q86.034 18.779 86.034 18.471Q86.034 18.314 85.940 18.199Q85.846 18.085 85.708 18.016Q85.569 17.948 85.402 17.910L84.821 17.811Q84.465 17.743 84.192 17.522Q83.918 17.302 83.918 16.960Q83.918 16.711 84.029 16.536Q84.140 16.362 84.327 16.263Q84.513 16.164 84.728 16.121Q84.944 16.078 85.186 16.078Q85.600 16.078 85.880 16.260L86.096 16.085Q86.106 16.082 86.113 16.080Q86.119 16.078 86.130 16.078L86.181 16.078Q86.208 16.078 86.232 16.102Q86.256 16.126 86.256 16.154L86.256 17.001Q86.256 17.022 86.232 17.049Q86.208 17.076 86.181 17.076L86.068 17.076Q86.041 17.076 86.015 17.051Q85.990 17.025 85.990 17.001Q85.990 16.765 85.884 16.601Q85.778 16.437 85.595 16.355Q85.412 16.273 85.179 16.273Q84.851 16.273 84.595 16.376Q84.339 16.478 84.339 16.755Q84.339 16.950 84.522 17.059Q84.704 17.169 84.933 17.210L85.508 17.316Q85.754 17.364 85.967 17.492Q86.181 17.620 86.318 17.823Q86.454 18.027 86.454 18.276Q86.454 18.789 86.089 19.028Q85.723 19.267 85.186 19.267Q84.691 19.267 84.359 18.973L84.093 19.247Q84.072 19.267 84.045 19.267L83.997 19.267Q83.973 19.267 83.946 19.240Q83.918 19.213 83.918 19.192M87.610 18.358L87.610 16.461L86.971 16.461L86.971 16.239Q87.288 16.239 87.505 16.029Q87.722 15.819 87.823 15.509Q87.924 15.200 87.924 14.892L88.191 14.892L88.191 16.181L89.267 16.181L89.267 16.461L88.191 16.461L88.191 18.345Q88.191 18.621 88.295 18.820Q88.399 19.018 88.659 19.018Q88.816 19.018 88.922 18.914Q89.028 18.809 89.078 18.656Q89.127 18.502 89.127 18.345L89.127 17.931L89.394 17.931L89.394 18.358Q89.394 18.584 89.295 18.794Q89.196 19.004 89.011 19.136Q88.826 19.267 88.597 19.267Q88.160 19.267 87.885 19.030Q87.610 18.792 87.610 18.358M91.954 19.199L90.218 19.199L90.218 18.919Q90.447 18.919 90.595 18.885Q90.744 18.850 90.744 18.710L90.744 16.861Q90.744 16.591 90.636 16.530Q90.529 16.468 90.218 16.468L90.218 16.188L91.246 16.113L91.246 16.820Q91.376 16.512 91.619 16.313Q91.862 16.113 92.179 16.113Q92.398 16.113 92.569 16.237Q92.740 16.362 92.740 16.574Q92.740 16.711 92.641 16.810Q92.542 16.909 92.408 16.909Q92.272 16.909 92.173 16.810Q92.074 16.711 92.074 16.574Q92.074 16.434 92.173 16.335Q91.882 16.335 91.682 16.531Q91.482 16.728 91.390 17.022Q91.298 17.316 91.298 17.596L91.298 18.710Q91.298 18.919 91.954 18.919L91.954 19.199M93.383 18.471Q93.383 18.139 93.606 17.912Q93.830 17.685 94.174 17.557Q94.517 17.428 94.890 17.376Q95.262 17.323 95.567 17.323L95.567 17.070Q95.567 16.865 95.459 16.685Q95.351 16.506 95.170 16.403Q94.989 16.301 94.781 16.301Q94.374 16.301 94.138 16.393Q94.227 16.430 94.273 16.514Q94.319 16.598 94.319 16.700Q94.319 16.796 94.273 16.875Q94.227 16.953 94.147 16.998Q94.066 17.042 93.977 17.042Q93.827 17.042 93.726 16.945Q93.625 16.847 93.625 16.700Q93.625 16.078 94.781 16.078Q94.992 16.078 95.242 16.142Q95.491 16.205 95.693 16.324Q95.895 16.444 96.021 16.629Q96.148 16.813 96.148 17.056L96.148 18.632Q96.148 18.748 96.209 18.844Q96.271 18.939 96.384 18.939Q96.493 18.939 96.558 18.845Q96.623 18.751 96.623 18.632L96.623 18.184L96.889 18.184L96.889 18.632Q96.889 18.902 96.662 19.067Q96.435 19.233 96.155 19.233Q95.946 19.233 95.809 19.079Q95.673 18.926 95.649 18.710Q95.502 18.977 95.220 19.122Q94.938 19.267 94.613 19.267Q94.336 19.267 94.053 19.192Q93.769 19.117 93.576 18.938Q93.383 18.758 93.383 18.471M93.998 18.471Q93.998 18.645 94.099 18.775Q94.200 18.905 94.355 18.975Q94.511 19.045 94.675 19.045Q94.893 19.045 95.102 18.948Q95.310 18.850 95.439 18.669Q95.567 18.488 95.567 18.262L95.567 17.534Q95.242 17.534 94.876 17.625Q94.511 17.716 94.254 17.928Q93.998 18.139 93.998 18.471M98.923 19.199L97.371 19.199L97.371 18.919Q97.597 18.919 97.746 18.885Q97.894 18.850 97.894 18.710L97.894 16.861Q97.894 16.673 97.846 16.589Q97.799 16.506 97.701 16.487Q97.604 16.468 97.392 16.468L97.392 16.188L98.448 16.113L98.448 18.710Q98.448 18.850 98.580 18.885Q98.711 18.919 98.923 18.919L98.923 19.199M97.652 14.892Q97.652 14.721 97.775 14.602Q97.898 14.482 98.069 14.482Q98.236 14.482 98.359 14.602Q98.482 14.721 98.482 14.892Q98.482 15.067 98.359 15.190Q98.236 15.313 98.069 15.313Q97.898 15.313 97.775 15.190Q97.652 15.067 97.652 14.892M101.251 19.199L99.617 19.199L99.617 18.919Q99.846 18.919 99.995 18.885Q100.143 18.850 100.143 18.710L100.143 16.861Q100.143 16.591 100.036 16.530Q99.928 16.468 99.617 16.468L99.617 16.188L100.677 16.113L100.677 16.762Q100.847 16.454 101.152 16.283Q101.456 16.113 101.801 16.113Q102.307 16.113 102.591 16.336Q102.874 16.560 102.874 17.056L102.874 18.710Q102.874 18.847 103.023 18.883Q103.172 18.919 103.397 18.919L103.397 19.199L101.767 19.199L101.767 18.919Q101.996 18.919 102.145 18.885Q102.293 18.850 102.293 18.710L102.293 17.070Q102.293 16.735 102.174 16.535Q102.054 16.335 101.740 16.335Q101.470 16.335 101.235 16.471Q101.001 16.608 100.863 16.842Q100.724 17.076 100.724 17.350L100.724 18.710Q100.724 18.847 100.875 18.883Q101.025 18.919 101.251 18.919\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-121.637 3.173)\">\u003Cpath d=\"M107.237 18.358L107.237 16.461L106.598 16.461L106.598 16.239Q106.916 16.239 107.133 16.029Q107.350 15.819 107.450 15.509Q107.551 15.200 107.551 14.892L107.818 14.892L107.818 16.181L108.895 16.181L108.895 16.461L107.818 16.461L107.818 18.345Q107.818 18.621 107.922 18.820Q108.026 19.018 108.286 19.018Q108.443 19.018 108.549 18.914Q108.655 18.809 108.705 18.656Q108.754 18.502 108.754 18.345L108.754 17.931L109.021 17.931L109.021 18.358Q109.021 18.584 108.922 18.794Q108.823 19.004 108.638 19.136Q108.454 19.267 108.225 19.267Q107.787 19.267 107.512 19.030Q107.237 18.792 107.237 18.358M109.889 18.471Q109.889 18.139 110.113 17.912Q110.337 17.685 110.680 17.557Q111.024 17.428 111.396 17.376Q111.769 17.323 112.073 17.323L112.073 17.070Q112.073 16.865 111.966 16.685Q111.858 16.506 111.677 16.403Q111.496 16.301 111.287 16.301Q110.880 16.301 110.645 16.393Q110.733 16.430 110.780 16.514Q110.826 16.598 110.826 16.700Q110.826 16.796 110.780 16.875Q110.733 16.953 110.653 16.998Q110.573 17.042 110.484 17.042Q110.334 17.042 110.233 16.945Q110.132 16.847 110.132 16.700Q110.132 16.078 111.287 16.078Q111.499 16.078 111.749 16.142Q111.998 16.205 112.200 16.324Q112.401 16.444 112.528 16.629Q112.654 16.813 112.654 17.056L112.654 18.632Q112.654 18.748 112.716 18.844Q112.777 18.939 112.890 18.939Q113 18.939 113.064 18.845Q113.129 18.751 113.129 18.632L113.129 18.184L113.396 18.184L113.396 18.632Q113.396 18.902 113.169 19.067Q112.941 19.233 112.661 19.233Q112.453 19.233 112.316 19.079Q112.179 18.926 112.155 18.710Q112.008 18.977 111.726 19.122Q111.444 19.267 111.120 19.267Q110.843 19.267 110.559 19.192Q110.275 19.117 110.082 18.938Q109.889 18.758 109.889 18.471M110.504 18.471Q110.504 18.645 110.605 18.775Q110.706 18.905 110.862 18.975Q111.017 19.045 111.181 19.045Q111.400 19.045 111.608 18.948Q111.817 18.850 111.945 18.669Q112.073 18.488 112.073 18.262L112.073 17.534Q111.749 17.534 111.383 17.625Q111.017 17.716 110.761 17.928Q110.504 18.139 110.504 18.471M115.563 19.199L113.827 19.199L113.827 18.919Q114.056 18.919 114.204 18.885Q114.353 18.850 114.353 18.710L114.353 16.861Q114.353 16.591 114.245 16.530Q114.138 16.468 113.827 16.468L113.827 16.188L114.855 16.113L114.855 16.820Q114.985 16.512 115.228 16.313Q115.471 16.113 115.789 16.113Q116.007 16.113 116.178 16.237Q116.349 16.362 116.349 16.574Q116.349 16.711 116.250 16.810Q116.151 16.909 116.018 16.909Q115.881 16.909 115.782 16.810Q115.683 16.711 115.683 16.574Q115.683 16.434 115.782 16.335Q115.491 16.335 115.291 16.531Q115.091 16.728 114.999 17.022Q114.907 17.316 114.907 17.596L114.907 18.710Q114.907 18.919 115.563 18.919L115.563 19.199M116.893 19.732Q116.893 19.486 117.089 19.302Q117.286 19.117 117.542 19.038Q117.405 18.926 117.334 18.765Q117.262 18.604 117.262 18.423Q117.262 18.102 117.474 17.856Q117.139 17.558 117.139 17.148Q117.139 16.687 117.528 16.400Q117.918 16.113 118.396 16.113Q118.868 16.113 119.203 16.359Q119.377 16.205 119.588 16.123Q119.798 16.041 120.027 16.041Q120.191 16.041 120.312 16.148Q120.434 16.256 120.434 16.420Q120.434 16.516 120.362 16.588Q120.290 16.659 120.198 16.659Q120.099 16.659 120.029 16.586Q119.959 16.512 119.959 16.413Q119.959 16.359 119.972 16.328L119.979 16.314Q119.986 16.294 119.994 16.283Q120.003 16.273 120.006 16.266Q119.651 16.266 119.364 16.489Q119.651 16.782 119.651 17.148Q119.651 17.463 119.466 17.695Q119.282 17.928 118.993 18.056Q118.704 18.184 118.396 18.184Q118.195 18.184 118.003 18.134Q117.812 18.085 117.634 17.975Q117.542 18.102 117.542 18.245Q117.542 18.427 117.670 18.562Q117.798 18.697 117.983 18.697L118.615 18.697Q119.063 18.697 119.432 18.768Q119.801 18.840 120.061 19.069Q120.321 19.298 120.321 19.732Q120.321 20.053 120.025 20.255Q119.729 20.457 119.326 20.546Q118.923 20.635 118.608 20.635Q118.291 20.635 117.887 20.546Q117.484 20.457 117.188 20.255Q116.893 20.053 116.893 19.732M117.347 19.732Q117.347 19.961 117.566 20.110Q117.785 20.259 118.077 20.327Q118.369 20.395 118.608 20.395Q118.772 20.395 118.981 20.359Q119.189 20.324 119.396 20.243Q119.603 20.163 119.735 20.035Q119.866 19.907 119.866 19.732Q119.866 19.380 119.485 19.286Q119.104 19.192 118.602 19.192L117.983 19.192Q117.744 19.192 117.545 19.343Q117.347 19.493 117.347 19.732M118.396 17.945Q119.063 17.945 119.063 17.148Q119.063 16.348 118.396 16.348Q117.727 16.348 117.727 17.148Q117.727 17.945 118.396 17.945M120.875 17.664Q120.875 17.343 120.999 17.054Q121.124 16.765 121.350 16.542Q121.575 16.318 121.871 16.198Q122.167 16.078 122.484 16.078Q122.813 16.078 123.074 16.178Q123.335 16.277 123.511 16.459Q123.688 16.642 123.781 16.900Q123.875 17.158 123.875 17.490Q123.875 17.582 123.793 17.603L121.538 17.603L121.538 17.664Q121.538 18.252 121.821 18.635Q122.105 19.018 122.672 19.018Q122.994 19.018 123.262 18.825Q123.530 18.632 123.619 18.317Q123.626 18.276 123.701 18.262L123.793 18.262Q123.875 18.286 123.875 18.358Q123.875 18.365 123.869 18.392Q123.756 18.789 123.385 19.028Q123.014 19.267 122.590 19.267Q122.153 19.267 121.753 19.059Q121.353 18.850 121.114 18.483Q120.875 18.116 120.875 17.664M121.544 17.394L123.359 17.394Q123.359 17.117 123.262 16.865Q123.165 16.612 122.966 16.456Q122.768 16.301 122.484 16.301Q122.208 16.301 121.994 16.459Q121.780 16.618 121.662 16.873Q121.544 17.128 121.544 17.394M124.990 18.358L124.990 16.461L124.351 16.461L124.351 16.239Q124.668 16.239 124.886 16.029Q125.103 15.819 125.203 15.509Q125.304 15.200 125.304 14.892L125.571 14.892L125.571 16.181L126.647 16.181L126.647 16.461L125.571 16.461L125.571 18.345Q125.571 18.621 125.675 18.820Q125.779 19.018 126.039 19.018Q126.196 19.018 126.302 18.914Q126.408 18.809 126.458 18.656Q126.507 18.502 126.507 18.345L126.507 17.931L126.774 17.931L126.774 18.358Q126.774 18.584 126.675 18.794Q126.576 19.004 126.391 19.136Q126.207 19.267 125.978 19.267Q125.540 19.267 125.265 19.030Q124.990 18.792 124.990 18.358\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-121.637 3.173)\">\u003Cpath d=\"M130.834 18.358L130.834 16.461L130.195 16.461L130.195 16.239Q130.513 16.239 130.730 16.029Q130.947 15.819 131.047 15.509Q131.148 15.200 131.148 14.892L131.415 14.892L131.415 16.181L132.492 16.181L132.492 16.461L131.415 16.461L131.415 18.345Q131.415 18.621 131.519 18.820Q131.623 19.018 131.883 19.018Q132.040 19.018 132.146 18.914Q132.252 18.809 132.302 18.656Q132.351 18.502 132.351 18.345L132.351 17.931L132.618 17.931L132.618 18.358Q132.618 18.584 132.519 18.794Q132.420 19.004 132.235 19.136Q132.051 19.267 131.822 19.267Q131.384 19.267 131.109 19.030Q130.834 18.792 130.834 18.358M133.387 17.716Q133.387 17.374 133.522 17.075Q133.657 16.776 133.896 16.552Q134.136 16.328 134.453 16.203Q134.771 16.078 135.103 16.078Q135.547 16.078 135.947 16.294Q136.347 16.509 136.581 16.887Q136.815 17.264 136.815 17.716Q136.815 18.057 136.673 18.341Q136.532 18.625 136.287 18.832Q136.043 19.038 135.733 19.153Q135.424 19.267 135.103 19.267Q134.672 19.267 134.271 19.066Q133.869 18.864 133.628 18.512Q133.387 18.160 133.387 17.716M135.103 19.018Q135.704 19.018 135.928 18.640Q136.152 18.262 136.152 17.630Q136.152 17.018 135.918 16.659Q135.684 16.301 135.103 16.301Q134.050 16.301 134.050 17.630Q134.050 18.262 134.276 18.640Q134.501 19.018 135.103 19.018\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-121.637 3.173)\">\u003Cpath d=\"M140.920 19.199L140.653 19.199L140.653 15.091Q140.653 14.821 140.546 14.759Q140.438 14.698 140.127 14.698L140.127 14.417L141.207 14.342L141.207 16.512Q141.416 16.321 141.701 16.217Q141.987 16.113 142.284 16.113Q142.602 16.113 142.899 16.234Q143.196 16.355 143.419 16.571Q143.641 16.786 143.767 17.071Q143.894 17.357 143.894 17.688Q143.894 18.133 143.654 18.497Q143.415 18.861 143.022 19.064Q142.629 19.267 142.185 19.267Q141.990 19.267 141.800 19.211Q141.611 19.155 141.450 19.050Q141.289 18.946 141.149 18.785L140.920 19.199M141.235 16.854L141.235 18.471Q141.371 18.731 141.612 18.888Q141.853 19.045 142.130 19.045Q142.424 19.045 142.636 18.938Q142.848 18.830 142.981 18.638Q143.114 18.447 143.173 18.208Q143.231 17.969 143.231 17.688Q143.231 17.329 143.137 17.025Q143.043 16.721 142.815 16.528Q142.588 16.335 142.222 16.335Q141.922 16.335 141.655 16.471Q141.388 16.608 141.235 16.854\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(47.313 -82.58)\">\u003Cpath d=\"M71.994 18.367Q71.994 17.883 72.396 17.588Q72.799 17.293 73.349 17.174Q73.900 17.054 74.392 17.054L74.392 16.765Q74.392 16.539 74.277 16.332Q74.162 16.125 73.965 16.006Q73.767 15.887 73.537 15.887Q73.111 15.887 72.826 15.992Q72.896 16.019 72.943 16.074Q72.990 16.129 73.015 16.199Q73.041 16.269 73.041 16.344Q73.041 16.449 72.990 16.541Q72.939 16.633 72.847 16.683Q72.756 16.734 72.650 16.734Q72.545 16.734 72.453 16.683Q72.361 16.633 72.310 16.541Q72.260 16.449 72.260 16.344Q72.260 15.926 72.648 15.779Q73.037 15.633 73.537 15.633Q73.869 15.633 74.222 15.763Q74.576 15.894 74.804 16.148Q75.033 16.402 75.033 16.750L75.033 18.551Q75.033 18.683 75.105 18.793Q75.178 18.902 75.306 18.902Q75.431 18.902 75.500 18.797Q75.568 18.691 75.568 18.551L75.568 18.039L75.849 18.039L75.849 18.551Q75.849 18.754 75.732 18.912Q75.615 19.070 75.433 19.154Q75.252 19.238 75.049 19.238Q74.818 19.238 74.666 19.066Q74.513 18.894 74.482 18.664Q74.322 18.945 74.013 19.111Q73.705 19.277 73.353 19.277Q72.842 19.277 72.418 19.054Q71.994 18.832 71.994 18.367M72.681 18.367Q72.681 18.652 72.908 18.838Q73.135 19.023 73.428 19.023Q73.674 19.023 73.898 18.906Q74.123 18.789 74.258 18.586Q74.392 18.383 74.392 18.129L74.392 17.297Q74.127 17.297 73.842 17.351Q73.556 17.406 73.285 17.535Q73.013 17.664 72.847 17.871Q72.681 18.078 72.681 18.367M77.056 19.199L76.775 19.199L76.775 14.480Q76.775 14.265 76.713 14.170Q76.650 14.074 76.533 14.053Q76.416 14.031 76.170 14.031L76.170 13.734L77.392 13.648L77.392 16.137Q77.869 15.672 78.568 15.672Q79.049 15.672 79.457 15.916Q79.865 16.160 80.101 16.574Q80.338 16.988 80.338 17.472Q80.338 17.847 80.189 18.176Q80.041 18.504 79.771 18.756Q79.502 19.008 79.158 19.142Q78.814 19.277 78.455 19.277Q78.135 19.277 77.836 19.129Q77.537 18.980 77.330 18.719L77.056 19.199M77.416 16.527L77.416 18.367Q77.568 18.664 77.828 18.844Q78.088 19.023 78.400 19.023Q78.826 19.023 79.094 18.804Q79.361 18.586 79.476 18.240Q79.592 17.894 79.592 17.472Q79.592 16.824 79.344 16.375Q79.096 15.926 78.498 15.926Q78.162 15.926 77.873 16.084Q77.584 16.242 77.416 16.527M80.904 19.191L80.904 17.969Q80.904 17.941 80.935 17.910Q80.967 17.879 80.990 17.879L81.096 17.879Q81.166 17.879 81.181 17.941Q81.244 18.262 81.383 18.502Q81.521 18.742 81.754 18.883Q81.986 19.023 82.295 19.023Q82.533 19.023 82.742 18.963Q82.951 18.902 83.088 18.754Q83.224 18.605 83.224 18.359Q83.224 18.105 83.013 17.939Q82.803 17.773 82.533 17.719L81.912 17.605Q81.506 17.527 81.205 17.271Q80.904 17.015 80.904 16.640Q80.904 16.273 81.105 16.051Q81.306 15.828 81.631 15.730Q81.955 15.633 82.295 15.633Q82.760 15.633 83.056 15.840L83.279 15.656Q83.303 15.633 83.334 15.633L83.385 15.633Q83.416 15.633 83.443 15.660Q83.471 15.687 83.471 15.719L83.471 16.703Q83.471 16.734 83.445 16.763Q83.420 16.793 83.385 16.793L83.279 16.793Q83.244 16.793 83.217 16.765Q83.189 16.738 83.189 16.703Q83.189 16.304 82.937 16.084Q82.685 15.863 82.287 15.863Q81.931 15.863 81.648 15.986Q81.365 16.109 81.365 16.414Q81.365 16.633 81.566 16.765Q81.767 16.898 82.013 16.941L82.638 17.054Q83.068 17.144 83.377 17.441Q83.685 17.738 83.685 18.152Q83.685 18.722 83.287 19Q82.888 19.277 82.295 19.277Q81.744 19.277 81.392 18.941L81.096 19.254Q81.072 19.277 81.037 19.277L80.990 19.277Q80.967 19.277 80.935 19.246Q80.904 19.215 80.904 19.191M84.213 17.504Q84.213 17 84.469 16.568Q84.724 16.137 85.160 15.885Q85.596 15.633 86.096 15.633Q86.482 15.633 86.824 15.777Q87.166 15.922 87.428 16.183Q87.689 16.445 87.832 16.781Q87.974 17.117 87.974 17.504Q87.974 17.996 87.711 18.406Q87.447 18.816 87.017 19.047Q86.588 19.277 86.096 19.277Q85.603 19.277 85.170 19.045Q84.736 18.812 84.474 18.404Q84.213 17.996 84.213 17.504M86.096 19Q86.553 19 86.804 18.777Q87.056 18.554 87.144 18.203Q87.232 17.851 87.232 17.406Q87.232 16.976 87.138 16.638Q87.045 16.301 86.791 16.094Q86.537 15.887 86.096 15.887Q85.447 15.887 85.203 16.303Q84.959 16.719 84.959 17.406Q84.959 17.851 85.047 18.203Q85.135 18.554 85.387 18.777Q85.638 19 86.096 19M90.467 19.199L88.486 19.199L88.486 18.902Q88.756 18.902 88.924 18.857Q89.092 18.812 89.092 18.640L89.092 16.504Q89.092 16.289 89.029 16.193Q88.967 16.097 88.849 16.076Q88.732 16.054 88.486 16.054L88.486 15.758L89.654 15.672L89.654 16.457Q89.732 16.246 89.885 16.060Q90.037 15.875 90.236 15.773Q90.435 15.672 90.662 15.672Q90.908 15.672 91.099 15.816Q91.291 15.961 91.291 16.191Q91.291 16.347 91.185 16.457Q91.080 16.566 90.924 16.566Q90.767 16.566 90.658 16.457Q90.549 16.347 90.549 16.191Q90.549 16.031 90.654 15.926Q90.330 15.926 90.115 16.154Q89.900 16.383 89.804 16.722Q89.709 17.062 89.709 17.367L89.709 18.640Q89.709 18.808 89.935 18.855Q90.162 18.902 90.467 18.902L90.467 19.199M92.685 19.199L92.404 19.199L92.404 14.480Q92.404 14.265 92.342 14.170Q92.279 14.074 92.162 14.053Q92.045 14.031 91.799 14.031L91.799 13.734L93.021 13.648L93.021 16.137Q93.498 15.672 94.197 15.672Q94.678 15.672 95.086 15.916Q95.494 16.160 95.730 16.574Q95.967 16.988 95.967 17.472Q95.967 17.847 95.818 18.176Q95.670 18.504 95.400 18.756Q95.131 19.008 94.787 19.142Q94.443 19.277 94.084 19.277Q93.763 19.277 93.465 19.129Q93.166 18.980 92.959 18.719L92.685 19.199M93.045 16.527L93.045 18.367Q93.197 18.664 93.457 18.844Q93.717 19.023 94.029 19.023Q94.455 19.023 94.722 18.804Q94.990 18.586 95.105 18.240Q95.221 17.894 95.221 17.472Q95.221 16.824 94.972 16.375Q94.724 15.926 94.127 15.926Q93.791 15.926 93.502 16.084Q93.213 16.242 93.045 16.527\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(47.313 -82.58)\">\u003Cpath d=\"M99.437 18.367Q99.437 17.883 99.839 17.588Q100.242 17.293 100.792 17.174Q101.343 17.054 101.835 17.054L101.835 16.765Q101.835 16.539 101.720 16.332Q101.605 16.125 101.408 16.006Q101.210 15.887 100.980 15.887Q100.554 15.887 100.269 15.992Q100.339 16.019 100.386 16.074Q100.433 16.129 100.458 16.199Q100.484 16.269 100.484 16.344Q100.484 16.449 100.433 16.541Q100.382 16.633 100.290 16.683Q100.199 16.734 100.093 16.734Q99.988 16.734 99.896 16.683Q99.804 16.633 99.753 16.541Q99.703 16.449 99.703 16.344Q99.703 15.926 100.091 15.779Q100.480 15.633 100.980 15.633Q101.312 15.633 101.665 15.763Q102.019 15.894 102.247 16.148Q102.476 16.402 102.476 16.750L102.476 18.551Q102.476 18.683 102.548 18.793Q102.621 18.902 102.749 18.902Q102.874 18.902 102.943 18.797Q103.011 18.691 103.011 18.551L103.011 18.039L103.292 18.039L103.292 18.551Q103.292 18.754 103.175 18.912Q103.058 19.070 102.876 19.154Q102.695 19.238 102.492 19.238Q102.261 19.238 102.109 19.066Q101.956 18.894 101.925 18.664Q101.765 18.945 101.456 19.111Q101.148 19.277 100.796 19.277Q100.285 19.277 99.861 19.054Q99.437 18.832 99.437 18.367M100.124 18.367Q100.124 18.652 100.351 18.838Q100.578 19.023 100.871 19.023Q101.117 19.023 101.341 18.906Q101.566 18.789 101.701 18.586Q101.835 18.383 101.835 18.129L101.835 17.297Q101.570 17.297 101.285 17.351Q100.999 17.406 100.728 17.535Q100.456 17.664 100.290 17.871Q100.124 18.078 100.124 18.367M105.652 19.199L103.667 19.199L103.667 18.902Q103.941 18.902 104.109 18.855Q104.277 18.808 104.277 18.640L104.277 16.047L103.636 16.047L103.636 15.750L104.277 15.750L104.277 14.816Q104.277 14.551 104.394 14.314Q104.511 14.078 104.705 13.914Q104.898 13.750 105.146 13.658Q105.394 13.566 105.660 13.566Q105.945 13.566 106.169 13.724Q106.394 13.883 106.394 14.160Q106.394 14.316 106.288 14.426Q106.183 14.535 106.019 14.535Q105.863 14.535 105.753 14.426Q105.644 14.316 105.644 14.160Q105.644 13.953 105.804 13.847Q105.706 13.824 105.613 13.824Q105.382 13.824 105.210 13.980Q105.038 14.137 104.953 14.373Q104.867 14.609 104.867 14.832L104.867 15.750L105.835 15.750L105.835 16.047L104.890 16.047L104.890 18.640Q104.890 18.808 105.117 18.855Q105.343 18.902 105.652 18.902L105.652 19.199M106.804 18.238L106.804 16.047L106.101 16.047L106.101 15.793Q106.456 15.793 106.699 15.560Q106.941 15.328 107.052 14.980Q107.163 14.633 107.163 14.277L107.445 14.277L107.445 15.750L108.621 15.750L108.621 16.047L107.445 16.047L107.445 18.222Q107.445 18.543 107.564 18.771Q107.683 19 107.964 19Q108.144 19 108.261 18.877Q108.378 18.754 108.431 18.574Q108.484 18.394 108.484 18.222L108.484 17.750L108.765 17.750L108.765 18.238Q108.765 18.492 108.660 18.732Q108.554 18.972 108.357 19.125Q108.160 19.277 107.902 19.277Q107.585 19.277 107.333 19.154Q107.081 19.031 106.943 18.797Q106.804 18.562 106.804 18.238M109.484 17.445Q109.484 16.965 109.716 16.549Q109.949 16.133 110.359 15.883Q110.769 15.633 111.246 15.633Q111.976 15.633 112.374 16.074Q112.773 16.515 112.773 17.246Q112.773 17.351 112.679 17.375L110.230 17.375L110.230 17.445Q110.230 17.855 110.351 18.211Q110.472 18.566 110.744 18.783Q111.015 19 111.445 19Q111.808 19 112.105 18.771Q112.402 18.543 112.503 18.191Q112.511 18.144 112.597 18.129L112.679 18.129Q112.773 18.156 112.773 18.238Q112.773 18.246 112.765 18.277Q112.703 18.504 112.564 18.687Q112.425 18.871 112.234 19.004Q112.042 19.137 111.824 19.207Q111.605 19.277 111.367 19.277Q110.996 19.277 110.658 19.140Q110.320 19.004 110.052 18.752Q109.785 18.500 109.634 18.160Q109.484 17.820 109.484 17.445M110.238 17.137L112.199 17.137Q112.199 16.832 112.097 16.541Q111.996 16.250 111.779 16.068Q111.562 15.887 111.246 15.887Q110.945 15.887 110.714 16.074Q110.484 16.262 110.361 16.553Q110.238 16.844 110.238 17.137M115.269 19.199L113.288 19.199L113.288 18.902Q113.558 18.902 113.726 18.857Q113.894 18.812 113.894 18.640L113.894 16.504Q113.894 16.289 113.831 16.193Q113.769 16.097 113.652 16.076Q113.535 16.054 113.288 16.054L113.288 15.758L114.456 15.672L114.456 16.457Q114.535 16.246 114.687 16.060Q114.839 15.875 115.038 15.773Q115.238 15.672 115.464 15.672Q115.710 15.672 115.902 15.816Q116.093 15.961 116.093 16.191Q116.093 16.347 115.988 16.457Q115.882 16.566 115.726 16.566Q115.570 16.566 115.460 16.457Q115.351 16.347 115.351 16.191Q115.351 16.031 115.456 15.926Q115.132 15.926 114.917 16.154Q114.703 16.383 114.607 16.722Q114.511 17.062 114.511 17.367L114.511 18.640Q114.511 18.808 114.738 18.855Q114.964 18.902 115.269 18.902\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(47.313 -82.58)\">\u003Cpath d=\"M120.041 18.238L120.041 16.047L119.338 16.047L119.338 15.793Q119.694 15.793 119.936 15.560Q120.178 15.328 120.289 14.980Q120.401 14.633 120.401 14.277L120.682 14.277L120.682 15.750L121.858 15.750L121.858 16.047L120.682 16.047L120.682 18.222Q120.682 18.543 120.801 18.771Q120.920 19 121.201 19Q121.381 19 121.498 18.877Q121.615 18.754 121.668 18.574Q121.721 18.394 121.721 18.222L121.721 17.750L122.002 17.750L122.002 18.238Q122.002 18.492 121.897 18.732Q121.791 18.972 121.594 19.125Q121.397 19.277 121.139 19.277Q120.823 19.277 120.571 19.154Q120.319 19.031 120.180 18.797Q120.041 18.562 120.041 18.238M124.651 19.199L122.795 19.199L122.795 18.902Q123.069 18.902 123.237 18.855Q123.405 18.808 123.405 18.640L123.405 14.480Q123.405 14.265 123.342 14.170Q123.280 14.074 123.160 14.053Q123.041 14.031 122.795 14.031L122.795 13.734L124.018 13.648L124.018 16.351Q124.143 16.140 124.330 15.990Q124.518 15.840 124.744 15.756Q124.971 15.672 125.217 15.672Q126.385 15.672 126.385 16.750L126.385 18.640Q126.385 18.808 126.555 18.855Q126.725 18.902 126.994 18.902L126.994 19.199L125.139 19.199L125.139 18.902Q125.412 18.902 125.580 18.855Q125.748 18.808 125.748 18.640L125.748 16.765Q125.748 16.383 125.627 16.154Q125.506 15.926 125.155 15.926Q124.842 15.926 124.588 16.088Q124.334 16.250 124.188 16.519Q124.041 16.789 124.041 17.086L124.041 18.640Q124.041 18.808 124.211 18.855Q124.381 18.902 124.651 18.902L124.651 19.199M127.440 17.445Q127.440 16.965 127.672 16.549Q127.905 16.133 128.315 15.883Q128.725 15.633 129.201 15.633Q129.932 15.633 130.330 16.074Q130.729 16.515 130.729 17.246Q130.729 17.351 130.635 17.375L128.186 17.375L128.186 17.445Q128.186 17.855 128.307 18.211Q128.428 18.566 128.699 18.783Q128.971 19 129.401 19Q129.764 19 130.061 18.771Q130.358 18.543 130.459 18.191Q130.467 18.144 130.553 18.129L130.635 18.129Q130.729 18.156 130.729 18.238Q130.729 18.246 130.721 18.277Q130.658 18.504 130.520 18.687Q130.381 18.871 130.190 19.004Q129.998 19.137 129.780 19.207Q129.561 19.277 129.323 19.277Q128.951 19.277 128.614 19.140Q128.276 19.004 128.008 18.752Q127.740 18.500 127.590 18.160Q127.440 17.820 127.440 17.445M128.194 17.137L130.155 17.137Q130.155 16.832 130.053 16.541Q129.951 16.250 129.735 16.068Q129.518 15.887 129.201 15.887Q128.901 15.887 128.670 16.074Q128.440 16.262 128.317 16.553Q128.194 16.844 128.194 17.137\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(47.313 -82.58)\">\u003Cpath d=\"M136.122 19.199L134.137 19.199L134.137 18.902Q134.411 18.902 134.579 18.855Q134.747 18.808 134.747 18.640L134.747 16.047L134.106 16.047L134.106 15.750L134.747 15.750L134.747 14.816Q134.747 14.551 134.864 14.314Q134.981 14.078 135.174 13.914Q135.368 13.750 135.616 13.658Q135.864 13.566 136.130 13.566Q136.415 13.566 136.639 13.724Q136.864 13.883 136.864 14.160Q136.864 14.316 136.758 14.426Q136.653 14.535 136.489 14.535Q136.333 14.535 136.223 14.426Q136.114 14.316 136.114 14.160Q136.114 13.953 136.274 13.847Q136.176 13.824 136.083 13.824Q135.852 13.824 135.680 13.980Q135.508 14.137 135.422 14.373Q135.337 14.609 135.337 14.832L135.337 15.750L136.305 15.750L136.305 16.047L135.360 16.047L135.360 18.640Q135.360 18.808 135.587 18.855Q135.813 18.902 136.122 18.902L136.122 19.199M136.747 18.367Q136.747 17.883 137.149 17.588Q137.551 17.293 138.102 17.174Q138.653 17.054 139.145 17.054L139.145 16.765Q139.145 16.539 139.030 16.332Q138.915 16.125 138.717 16.006Q138.520 15.887 138.290 15.887Q137.864 15.887 137.579 15.992Q137.649 16.019 137.696 16.074Q137.743 16.129 137.768 16.199Q137.794 16.269 137.794 16.344Q137.794 16.449 137.743 16.541Q137.692 16.633 137.600 16.683Q137.508 16.734 137.403 16.734Q137.297 16.734 137.206 16.683Q137.114 16.633 137.063 16.541Q137.012 16.449 137.012 16.344Q137.012 15.926 137.401 15.779Q137.790 15.633 138.290 15.633Q138.622 15.633 138.975 15.763Q139.329 15.894 139.557 16.148Q139.786 16.402 139.786 16.750L139.786 18.551Q139.786 18.683 139.858 18.793Q139.930 18.902 140.059 18.902Q140.184 18.902 140.253 18.797Q140.321 18.691 140.321 18.551L140.321 18.039L140.602 18.039L140.602 18.551Q140.602 18.754 140.485 18.912Q140.368 19.070 140.186 19.154Q140.005 19.238 139.801 19.238Q139.571 19.238 139.419 19.066Q139.266 18.894 139.235 18.664Q139.075 18.945 138.766 19.111Q138.458 19.277 138.106 19.277Q137.594 19.277 137.171 19.054Q136.747 18.832 136.747 18.367M137.434 18.367Q137.434 18.652 137.661 18.838Q137.887 19.023 138.180 19.023Q138.426 19.023 138.651 18.906Q138.876 18.789 139.010 18.586Q139.145 18.383 139.145 18.129L139.145 17.297Q138.880 17.297 138.594 17.351Q138.309 17.406 138.038 17.535Q137.766 17.664 137.600 17.871Q137.434 18.078 137.434 18.367M140.938 17.472Q140.938 16.976 141.188 16.551Q141.438 16.125 141.858 15.879Q142.278 15.633 142.778 15.633Q143.317 15.633 143.708 15.758Q144.098 15.883 144.098 16.297Q144.098 16.402 144.047 16.494Q143.997 16.586 143.905 16.637Q143.813 16.687 143.704 16.687Q143.598 16.687 143.506 16.637Q143.415 16.586 143.364 16.494Q143.313 16.402 143.313 16.297Q143.313 16.074 143.481 15.969Q143.258 15.910 142.786 15.910Q142.489 15.910 142.274 16.049Q142.059 16.187 141.928 16.418Q141.797 16.648 141.739 16.918Q141.680 17.187 141.680 17.472Q141.680 17.867 141.813 18.217Q141.946 18.566 142.217 18.783Q142.489 19 142.887 19Q143.262 19 143.538 18.783Q143.813 18.566 143.915 18.207Q143.930 18.144 143.993 18.144L144.098 18.144Q144.133 18.144 144.159 18.172Q144.184 18.199 144.184 18.238L144.184 18.262Q144.051 18.742 143.667 19.010Q143.282 19.277 142.778 19.277Q142.415 19.277 142.081 19.140Q141.747 19.004 141.487 18.754Q141.227 18.504 141.083 18.168Q140.938 17.832 140.938 17.472M145.297 18.238L145.297 16.047L144.594 16.047L144.594 15.793Q144.950 15.793 145.192 15.560Q145.434 15.328 145.546 14.980Q145.657 14.633 145.657 14.277L145.938 14.277L145.938 15.750L147.114 15.750L147.114 16.047L145.938 16.047L145.938 18.222Q145.938 18.543 146.057 18.771Q146.176 19 146.458 19Q146.637 19 146.755 18.877Q146.872 18.754 146.924 18.574Q146.977 18.394 146.977 18.222L146.977 17.750L147.258 17.750L147.258 18.238Q147.258 18.492 147.153 18.732Q147.047 18.972 146.850 19.125Q146.653 19.277 146.395 19.277Q146.079 19.277 145.827 19.154Q145.575 19.031 145.436 18.797Q145.297 18.562 145.297 18.238\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-warn)\">\u003Cpath fill=\"none\" d=\"M108.646-29.17h96.74v-25.608h-96.74Z\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(44.636 -59.423)\">\u003Cpath d=\"M73.654 19.199L72.020 19.199L72.020 18.919Q72.249 18.919 72.398 18.885Q72.547 18.850 72.547 18.710L72.547 16.861Q72.547 16.591 72.439 16.530Q72.331 16.468 72.020 16.468L72.020 16.188L73.080 16.113L73.080 16.762Q73.251 16.454 73.555 16.283Q73.859 16.113 74.204 16.113Q74.604 16.113 74.881 16.253Q75.158 16.393 75.243 16.741Q75.411 16.448 75.710 16.280Q76.009 16.113 76.354 16.113Q76.860 16.113 77.144 16.336Q77.428 16.560 77.428 17.056L77.428 18.710Q77.428 18.847 77.576 18.883Q77.725 18.919 77.950 18.919L77.950 19.199L76.320 19.199L76.320 18.919Q76.546 18.919 76.696 18.883Q76.846 18.847 76.846 18.710L76.846 17.070Q76.846 16.735 76.727 16.535Q76.607 16.335 76.293 16.335Q76.023 16.335 75.789 16.471Q75.554 16.608 75.416 16.842Q75.278 17.076 75.278 17.350L75.278 18.710Q75.278 18.847 75.426 18.883Q75.575 18.919 75.801 18.919L75.801 19.199L74.170 19.199L74.170 18.919Q74.399 18.919 74.548 18.885Q74.697 18.850 74.697 18.710L74.697 17.070Q74.697 16.735 74.577 16.535Q74.457 16.335 74.143 16.335Q73.873 16.335 73.639 16.471Q73.405 16.608 73.266 16.842Q73.128 17.076 73.128 17.350L73.128 18.710Q73.128 18.847 73.278 18.883Q73.429 18.919 73.654 18.919L73.654 19.199M78.497 17.716Q78.497 17.374 78.632 17.075Q78.767 16.776 79.007 16.552Q79.246 16.328 79.564 16.203Q79.882 16.078 80.213 16.078Q80.658 16.078 81.057 16.294Q81.457 16.509 81.691 16.887Q81.926 17.264 81.926 17.716Q81.926 18.057 81.784 18.341Q81.642 18.625 81.398 18.832Q81.153 19.038 80.844 19.153Q80.534 19.267 80.213 19.267Q79.783 19.267 79.381 19.066Q78.979 18.864 78.738 18.512Q78.497 18.160 78.497 17.716M80.213 19.018Q80.815 19.018 81.039 18.640Q81.262 18.262 81.262 17.630Q81.262 17.018 81.028 16.659Q80.794 16.301 80.213 16.301Q79.160 16.301 79.160 17.630Q79.160 18.262 79.386 18.640Q79.612 19.018 80.213 19.018M84.202 19.199L82.568 19.199L82.568 18.919Q82.797 18.919 82.946 18.885Q83.095 18.850 83.095 18.710L83.095 16.861Q83.095 16.591 82.987 16.530Q82.879 16.468 82.568 16.468L82.568 16.188L83.628 16.113L83.628 16.762Q83.799 16.454 84.103 16.283Q84.407 16.113 84.752 16.113Q85.152 16.113 85.429 16.253Q85.706 16.393 85.791 16.741Q85.959 16.448 86.258 16.280Q86.557 16.113 86.902 16.113Q87.408 16.113 87.692 16.336Q87.975 16.560 87.975 17.056L87.975 18.710Q87.975 18.847 88.124 18.883Q88.273 18.919 88.498 18.919L88.498 19.199L86.868 19.199L86.868 18.919Q87.094 18.919 87.244 18.883Q87.394 18.847 87.394 18.710L87.394 17.070Q87.394 16.735 87.275 16.535Q87.155 16.335 86.841 16.335Q86.571 16.335 86.336 16.471Q86.102 16.608 85.964 16.842Q85.825 17.076 85.825 17.350L85.825 18.710Q85.825 18.847 85.974 18.883Q86.123 18.919 86.348 18.919L86.348 19.199L84.718 19.199L84.718 18.919Q84.947 18.919 85.096 18.885Q85.244 18.850 85.244 18.710L85.244 17.070Q85.244 16.735 85.125 16.535Q85.005 16.335 84.691 16.335Q84.421 16.335 84.187 16.471Q83.952 16.608 83.814 16.842Q83.676 17.076 83.676 17.350L83.676 18.710Q83.676 18.847 83.826 18.883Q83.976 18.919 84.202 18.919L84.202 19.199M89.045 17.664Q89.045 17.343 89.170 17.054Q89.295 16.765 89.520 16.542Q89.746 16.318 90.042 16.198Q90.337 16.078 90.655 16.078Q90.983 16.078 91.245 16.178Q91.506 16.277 91.682 16.459Q91.858 16.642 91.952 16.900Q92.046 17.158 92.046 17.490Q92.046 17.582 91.964 17.603L89.708 17.603L89.708 17.664Q89.708 18.252 89.992 18.635Q90.276 19.018 90.843 19.018Q91.164 19.018 91.433 18.825Q91.701 18.632 91.790 18.317Q91.797 18.276 91.872 18.262L91.964 18.262Q92.046 18.286 92.046 18.358Q92.046 18.365 92.039 18.392Q91.927 18.789 91.556 19.028Q91.185 19.267 90.761 19.267Q90.324 19.267 89.924 19.059Q89.524 18.850 89.284 18.483Q89.045 18.116 89.045 17.664M89.715 17.394L91.530 17.394Q91.530 17.117 91.433 16.865Q91.335 16.612 91.137 16.456Q90.939 16.301 90.655 16.301Q90.378 16.301 90.165 16.459Q89.951 16.618 89.833 16.873Q89.715 17.128 89.715 17.394M94.316 19.199L92.682 19.199L92.682 18.919Q92.911 18.919 93.060 18.885Q93.208 18.850 93.208 18.710L93.208 16.861Q93.208 16.591 93.101 16.530Q92.993 16.468 92.682 16.468L92.682 16.188L93.741 16.113L93.741 16.762Q93.912 16.454 94.217 16.283Q94.521 16.113 94.866 16.113Q95.372 16.113 95.656 16.336Q95.939 16.560 95.939 17.056L95.939 18.710Q95.939 18.847 96.088 18.883Q96.237 18.919 96.462 18.919L96.462 19.199L94.832 19.199L94.832 18.919Q95.061 18.919 95.210 18.885Q95.358 18.850 95.358 18.710L95.358 17.070Q95.358 16.735 95.239 16.535Q95.119 16.335 94.804 16.335Q94.534 16.335 94.300 16.471Q94.066 16.608 93.928 16.842Q93.789 17.076 93.789 17.350L93.789 18.710Q93.789 18.847 93.940 18.883Q94.090 18.919 94.316 18.919\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(44.636 -59.423)\">\u003Cpath d=\"M97.381 18.358L97.381 16.461L96.742 16.461L96.742 16.239Q97.060 16.239 97.277 16.029Q97.494 15.819 97.594 15.509Q97.695 15.200 97.695 14.892L97.962 14.892L97.962 16.181L99.039 16.181L99.039 16.461L97.962 16.461L97.962 18.345Q97.962 18.621 98.066 18.820Q98.170 19.018 98.430 19.018Q98.587 19.018 98.693 18.914Q98.799 18.809 98.849 18.656Q98.898 18.502 98.898 18.345L98.898 17.931L99.165 17.931L99.165 18.358Q99.165 18.584 99.066 18.794Q98.967 19.004 98.782 19.136Q98.598 19.267 98.369 19.267Q97.931 19.267 97.656 19.030Q97.381 18.792 97.381 18.358M100.549 18.365L100.549 16.861Q100.549 16.591 100.442 16.530Q100.334 16.468 100.023 16.468L100.023 16.188L101.130 16.113L101.130 18.345L101.130 18.365Q101.130 18.645 101.182 18.789Q101.233 18.932 101.375 18.989Q101.517 19.045 101.804 19.045Q102.057 19.045 102.262 18.905Q102.467 18.765 102.583 18.539Q102.699 18.314 102.699 18.064L102.699 16.861Q102.699 16.591 102.592 16.530Q102.484 16.468 102.173 16.468L102.173 16.188L103.280 16.113L103.280 18.526Q103.280 18.717 103.333 18.799Q103.386 18.881 103.487 18.900Q103.588 18.919 103.803 18.919L103.803 19.199L102.727 19.267L102.727 18.703Q102.617 18.885 102.472 19.008Q102.327 19.131 102.140 19.199Q101.954 19.267 101.752 19.267Q100.549 19.267 100.549 18.365M106.073 19.199L104.439 19.199L104.439 18.919Q104.668 18.919 104.817 18.885Q104.965 18.850 104.965 18.710L104.965 16.861Q104.965 16.591 104.858 16.530Q104.750 16.468 104.439 16.468L104.439 16.188L105.498 16.113L105.498 16.762Q105.669 16.454 105.974 16.283Q106.278 16.113 106.623 16.113Q107.023 16.113 107.300 16.253Q107.577 16.393 107.662 16.741Q107.830 16.448 108.129 16.280Q108.428 16.113 108.773 16.113Q109.279 16.113 109.562 16.336Q109.846 16.560 109.846 17.056L109.846 18.710Q109.846 18.847 109.995 18.883Q110.144 18.919 110.369 18.919L110.369 19.199L108.739 19.199L108.739 18.919Q108.964 18.919 109.115 18.883Q109.265 18.847 109.265 18.710L109.265 17.070Q109.265 16.735 109.145 16.535Q109.026 16.335 108.711 16.335Q108.441 16.335 108.207 16.471Q107.973 16.608 107.835 16.842Q107.696 17.076 107.696 17.350L107.696 18.710Q107.696 18.847 107.845 18.883Q107.994 18.919 108.219 18.919L108.219 19.199L106.589 19.199L106.589 18.919Q106.818 18.919 106.967 18.885Q107.115 18.850 107.115 18.710L107.115 17.070Q107.115 16.735 106.996 16.535Q106.876 16.335 106.561 16.335Q106.291 16.335 106.057 16.471Q105.823 16.608 105.685 16.842Q105.546 17.076 105.546 17.350L105.546 18.710Q105.546 18.847 105.697 18.883Q105.847 18.919 106.073 18.919\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(44.636 -59.423)\">\u003Cpath d=\"M113.835 20.782Q113.835 20.764 113.848 20.717L116.504 14.055Q116.559 13.949 116.665 13.949Q116.730 13.949 116.781 14Q116.832 14.052 116.832 14.116Q116.832 14.140 116.831 14.152Q116.829 14.164 116.825 14.181L114.173 20.843Q114.101 20.949 114.013 20.949Q113.944 20.949 113.889 20.898Q113.835 20.846 113.835 20.782\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(44.636 -59.423)\">\u003Cpath d=\"M120.399 18.471Q120.399 18.139 120.622 17.912Q120.846 17.685 121.190 17.557Q121.533 17.428 121.906 17.376Q122.278 17.323 122.583 17.323L122.583 17.070Q122.583 16.865 122.475 16.685Q122.367 16.506 122.186 16.403Q122.005 16.301 121.797 16.301Q121.390 16.301 121.154 16.393Q121.243 16.430 121.289 16.514Q121.335 16.598 121.335 16.700Q121.335 16.796 121.289 16.875Q121.243 16.953 121.162 16.998Q121.082 17.042 120.993 17.042Q120.843 17.042 120.742 16.945Q120.641 16.847 120.641 16.700Q120.641 16.078 121.797 16.078Q122.008 16.078 122.258 16.142Q122.507 16.205 122.709 16.324Q122.911 16.444 123.037 16.629Q123.164 16.813 123.164 17.056L123.164 18.632Q123.164 18.748 123.225 18.844Q123.287 18.939 123.400 18.939Q123.509 18.939 123.574 18.845Q123.639 18.751 123.639 18.632L123.639 18.184L123.905 18.184L123.905 18.632Q123.905 18.902 123.678 19.067Q123.451 19.233 123.171 19.233Q122.962 19.233 122.825 19.079Q122.689 18.926 122.665 18.710Q122.518 18.977 122.236 19.122Q121.954 19.267 121.629 19.267Q121.352 19.267 121.068 19.192Q120.785 19.117 120.592 18.938Q120.399 18.758 120.399 18.471M121.014 18.471Q121.014 18.645 121.115 18.775Q121.215 18.905 121.371 18.975Q121.526 19.045 121.691 19.045Q121.909 19.045 122.118 18.948Q122.326 18.850 122.454 18.669Q122.583 18.488 122.583 18.262L122.583 17.534Q122.258 17.534 121.892 17.625Q121.526 17.716 121.270 17.928Q121.014 18.139 121.014 18.471\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(44.636 -59.423)\">\u003Cpath d=\"M125.701 19.172L124.573 16.673Q124.501 16.526 124.371 16.494Q124.241 16.461 124.012 16.461L124.012 16.181L125.526 16.181L125.526 16.461Q125.174 16.461 125.174 16.608Q125.174 16.653 125.185 16.673L126.049 18.591L126.829 16.861Q126.863 16.793 126.863 16.714Q126.863 16.601 126.779 16.531Q126.695 16.461 126.576 16.461L126.576 16.181L127.772 16.181L127.772 16.461Q127.553 16.461 127.382 16.564Q127.212 16.666 127.123 16.861L126.087 19.172Q126.039 19.267 125.933 19.267L125.855 19.267Q125.749 19.267 125.701 19.172\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(44.636 -59.423)\">\u003Cpath d=\"M128.056 17.664Q128.056 17.343 128.181 17.054Q128.306 16.765 128.532 16.542Q128.757 16.318 129.053 16.198Q129.348 16.078 129.666 16.078Q129.994 16.078 130.256 16.178Q130.517 16.277 130.693 16.459Q130.869 16.642 130.963 16.900Q131.057 17.158 131.057 17.490Q131.057 17.582 130.975 17.603L128.720 17.603L128.720 17.664Q128.720 18.252 129.003 18.635Q129.287 19.018 129.854 19.018Q130.176 19.018 130.444 18.825Q130.712 18.632 130.801 18.317Q130.808 18.276 130.883 18.262L130.975 18.262Q131.057 18.286 131.057 18.358Q131.057 18.365 131.051 18.392Q130.938 18.789 130.567 19.028Q130.196 19.267 129.772 19.267Q129.335 19.267 128.935 19.059Q128.535 18.850 128.296 18.483Q128.056 18.116 128.056 17.664M128.726 17.394L130.541 17.394Q130.541 17.117 130.444 16.865Q130.346 16.612 130.148 16.456Q129.950 16.301 129.666 16.301Q129.389 16.301 129.176 16.459Q128.962 16.618 128.844 16.873Q128.726 17.128 128.726 17.394M133.395 19.199L131.659 19.199L131.659 18.919Q131.888 18.919 132.037 18.885Q132.185 18.850 132.185 18.710L132.185 16.861Q132.185 16.591 132.078 16.530Q131.970 16.468 131.659 16.468L131.659 16.188L132.688 16.113L132.688 16.820Q132.818 16.512 133.060 16.313Q133.303 16.113 133.621 16.113Q133.840 16.113 134.011 16.237Q134.181 16.362 134.181 16.574Q134.181 16.711 134.082 16.810Q133.983 16.909 133.850 16.909Q133.713 16.909 133.614 16.810Q133.515 16.711 133.515 16.574Q133.515 16.434 133.614 16.335Q133.324 16.335 133.124 16.531Q132.924 16.728 132.831 17.022Q132.739 17.316 132.739 17.596L132.739 18.710Q132.739 18.919 133.395 18.919L133.395 19.199M134.824 18.471Q134.824 18.139 135.048 17.912Q135.272 17.685 135.615 17.557Q135.959 17.428 136.331 17.376Q136.704 17.323 137.008 17.323L137.008 17.070Q137.008 16.865 136.900 16.685Q136.793 16.506 136.612 16.403Q136.430 16.301 136.222 16.301Q135.815 16.301 135.579 16.393Q135.668 16.430 135.714 16.514Q135.761 16.598 135.761 16.700Q135.761 16.796 135.714 16.875Q135.668 16.953 135.588 16.998Q135.508 17.042 135.419 17.042Q135.268 17.042 135.168 16.945Q135.067 16.847 135.067 16.700Q135.067 16.078 136.222 16.078Q136.434 16.078 136.683 16.142Q136.933 16.205 137.135 16.324Q137.336 16.444 137.463 16.629Q137.589 16.813 137.589 17.056L137.589 18.632Q137.589 18.748 137.651 18.844Q137.712 18.939 137.825 18.939Q137.934 18.939 137.999 18.845Q138.064 18.751 138.064 18.632L138.064 18.184L138.331 18.184L138.331 18.632Q138.331 18.902 138.104 19.067Q137.876 19.233 137.596 19.233Q137.387 19.233 137.251 19.079Q137.114 18.926 137.090 18.710Q136.943 18.977 136.661 19.122Q136.379 19.267 136.054 19.267Q135.778 19.267 135.494 19.192Q135.210 19.117 135.017 18.938Q134.824 18.758 134.824 18.471M135.439 18.471Q135.439 18.645 135.540 18.775Q135.641 18.905 135.796 18.975Q135.952 19.045 136.116 19.045Q136.335 19.045 136.543 18.948Q136.752 18.850 136.880 18.669Q137.008 18.488 137.008 18.262L137.008 17.534Q136.683 17.534 136.318 17.625Q135.952 17.716 135.696 17.928Q135.439 18.139 135.439 18.471M138.707 19.732Q138.707 19.486 138.903 19.302Q139.100 19.117 139.356 19.038Q139.220 18.926 139.148 18.765Q139.076 18.604 139.076 18.423Q139.076 18.102 139.288 17.856Q138.953 17.558 138.953 17.148Q138.953 16.687 139.343 16.400Q139.732 16.113 140.211 16.113Q140.682 16.113 141.017 16.359Q141.192 16.205 141.402 16.123Q141.612 16.041 141.841 16.041Q142.005 16.041 142.127 16.148Q142.248 16.256 142.248 16.420Q142.248 16.516 142.176 16.588Q142.104 16.659 142.012 16.659Q141.913 16.659 141.843 16.586Q141.773 16.512 141.773 16.413Q141.773 16.359 141.786 16.328L141.793 16.314Q141.800 16.294 141.809 16.283Q141.817 16.273 141.821 16.266Q141.465 16.266 141.178 16.489Q141.465 16.782 141.465 17.148Q141.465 17.463 141.281 17.695Q141.096 17.928 140.807 18.056Q140.518 18.184 140.211 18.184Q140.009 18.184 139.818 18.134Q139.626 18.085 139.449 17.975Q139.356 18.102 139.356 18.245Q139.356 18.427 139.484 18.562Q139.613 18.697 139.797 18.697L140.429 18.697Q140.877 18.697 141.246 18.768Q141.616 18.840 141.875 19.069Q142.135 19.298 142.135 19.732Q142.135 20.053 141.839 20.255Q141.544 20.457 141.140 20.546Q140.737 20.635 140.423 20.635Q140.105 20.635 139.701 20.546Q139.298 20.457 139.002 20.255Q138.707 20.053 138.707 19.732M139.161 19.732Q139.161 19.961 139.380 20.110Q139.599 20.259 139.891 20.327Q140.183 20.395 140.423 20.395Q140.587 20.395 140.795 20.359Q141.004 20.324 141.210 20.243Q141.417 20.163 141.549 20.035Q141.680 19.907 141.680 19.732Q141.680 19.380 141.299 19.286Q140.918 19.192 140.416 19.192L139.797 19.192Q139.558 19.192 139.360 19.343Q139.161 19.493 139.161 19.732M140.211 17.945Q140.877 17.945 140.877 17.148Q140.877 16.348 140.211 16.348Q139.541 16.348 139.541 17.148Q139.541 17.945 140.211 17.945M144.346 19.199L142.795 19.199L142.795 18.919Q143.020 18.919 143.169 18.885Q143.318 18.850 143.318 18.710L143.318 16.861Q143.318 16.673 143.270 16.589Q143.222 16.506 143.125 16.487Q143.027 16.468 142.815 16.468L142.815 16.188L143.871 16.113L143.871 18.710Q143.871 18.850 144.003 18.885Q144.135 18.919 144.346 18.919L144.346 19.199M143.075 14.892Q143.075 14.721 143.198 14.602Q143.321 14.482 143.492 14.482Q143.659 14.482 143.783 14.602Q143.906 14.721 143.906 14.892Q143.906 15.067 143.783 15.190Q143.659 15.313 143.492 15.313Q143.321 15.313 143.198 15.190Q143.075 15.067 143.075 14.892M146.674 19.199L145.040 19.199L145.040 18.919Q145.269 18.919 145.418 18.885Q145.567 18.850 145.567 18.710L145.567 16.861Q145.567 16.591 145.459 16.530Q145.351 16.468 145.040 16.468L145.040 16.188L146.100 16.113L146.100 16.762Q146.271 16.454 146.575 16.283Q146.879 16.113 147.224 16.113Q147.730 16.113 148.014 16.336Q148.298 16.560 148.298 17.056L148.298 18.710Q148.298 18.847 148.446 18.883Q148.595 18.919 148.821 18.919L148.821 19.199L147.190 19.199L147.190 18.919Q147.419 18.919 147.568 18.885Q147.717 18.850 147.717 18.710L147.717 17.070Q147.717 16.735 147.597 16.535Q147.477 16.335 147.163 16.335Q146.893 16.335 146.659 16.471Q146.425 16.608 146.286 16.842Q146.148 17.076 146.148 17.350L146.148 18.710Q146.148 18.847 146.298 18.883Q146.449 18.919 146.674 18.919L146.674 19.199M149.367 19.732Q149.367 19.486 149.564 19.302Q149.761 19.117 150.017 19.038Q149.880 18.926 149.808 18.765Q149.737 18.604 149.737 18.423Q149.737 18.102 149.949 17.856Q149.614 17.558 149.614 17.148Q149.614 16.687 150.003 16.400Q150.393 16.113 150.871 16.113Q151.343 16.113 151.678 16.359Q151.852 16.205 152.063 16.123Q152.273 16.041 152.502 16.041Q152.666 16.041 152.787 16.148Q152.908 16.256 152.908 16.420Q152.908 16.516 152.837 16.588Q152.765 16.659 152.673 16.659Q152.574 16.659 152.503 16.586Q152.433 16.512 152.433 16.413Q152.433 16.359 152.447 16.328L152.454 16.314Q152.461 16.294 152.469 16.283Q152.478 16.273 152.481 16.266Q152.126 16.266 151.839 16.489Q152.126 16.782 152.126 17.148Q152.126 17.463 151.941 17.695Q151.757 17.928 151.468 18.056Q151.179 18.184 150.871 18.184Q150.670 18.184 150.478 18.134Q150.287 18.085 150.109 17.975Q150.017 18.102 150.017 18.245Q150.017 18.427 150.145 18.562Q150.273 18.697 150.458 18.697L151.090 18.697Q151.538 18.697 151.907 18.768Q152.276 18.840 152.536 19.069Q152.796 19.298 152.796 19.732Q152.796 20.053 152.500 20.255Q152.204 20.457 151.801 20.546Q151.398 20.635 151.083 20.635Q150.765 20.635 150.362 20.546Q149.959 20.457 149.663 20.255Q149.367 20.053 149.367 19.732M149.822 19.732Q149.822 19.961 150.041 20.110Q150.260 20.259 150.552 20.327Q150.844 20.395 151.083 20.395Q151.247 20.395 151.456 20.359Q151.664 20.324 151.871 20.243Q152.078 20.163 152.210 20.035Q152.341 19.907 152.341 19.732Q152.341 19.380 151.960 19.286Q151.579 19.192 151.076 19.192L150.458 19.192Q150.219 19.192 150.020 19.343Q149.822 19.493 149.822 19.732M150.871 17.945Q151.538 17.945 151.538 17.148Q151.538 16.348 150.871 16.348Q150.201 16.348 150.201 17.148Q150.201 17.945 150.871 17.945\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-warn)\">\u003Cpath fill=\"none\" d=\"M108.646 2.127h96.74V-23.48h-96.74Z\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(52.807 -28.125)\">\u003Cpath d=\"M72.031 18.471Q72.031 18.139 72.254 17.912Q72.478 17.685 72.822 17.557Q73.165 17.428 73.538 17.376Q73.910 17.323 74.215 17.323L74.215 17.070Q74.215 16.865 74.107 16.685Q73.999 16.506 73.818 16.403Q73.637 16.301 73.429 16.301Q73.022 16.301 72.786 16.393Q72.875 16.430 72.921 16.514Q72.967 16.598 72.967 16.700Q72.967 16.796 72.921 16.875Q72.875 16.953 72.794 16.998Q72.714 17.042 72.625 17.042Q72.475 17.042 72.374 16.945Q72.273 16.847 72.273 16.700Q72.273 16.078 73.429 16.078Q73.640 16.078 73.890 16.142Q74.139 16.205 74.341 16.324Q74.543 16.444 74.669 16.629Q74.796 16.813 74.796 17.056L74.796 18.632Q74.796 18.748 74.857 18.844Q74.919 18.939 75.032 18.939Q75.141 18.939 75.206 18.845Q75.271 18.751 75.271 18.632L75.271 18.184L75.537 18.184L75.537 18.632Q75.537 18.902 75.310 19.067Q75.083 19.233 74.803 19.233Q74.594 19.233 74.457 19.079Q74.321 18.926 74.297 18.710Q74.150 18.977 73.868 19.122Q73.586 19.267 73.261 19.267Q72.984 19.267 72.700 19.192Q72.417 19.117 72.224 18.938Q72.031 18.758 72.031 18.471M72.646 18.471Q72.646 18.645 72.747 18.775Q72.847 18.905 73.003 18.975Q73.158 19.045 73.323 19.045Q73.541 19.045 73.750 18.948Q73.958 18.850 74.086 18.669Q74.215 18.488 74.215 18.262L74.215 17.534Q73.890 17.534 73.524 17.625Q73.158 17.716 72.902 17.928Q72.646 18.139 72.646 18.471M75.954 17.688Q75.954 17.350 76.095 17.059Q76.235 16.769 76.479 16.555Q76.723 16.342 77.028 16.227Q77.332 16.113 77.657 16.113Q77.927 16.113 78.190 16.212Q78.453 16.311 78.644 16.489L78.644 15.091Q78.644 14.821 78.537 14.759Q78.429 14.698 78.118 14.698L78.118 14.417L79.195 14.342L79.195 18.526Q79.195 18.714 79.249 18.797Q79.304 18.881 79.405 18.900Q79.506 18.919 79.721 18.919L79.721 19.199L78.614 19.267L78.614 18.850Q78.197 19.267 77.571 19.267Q77.140 19.267 76.768 19.055Q76.395 18.844 76.175 18.483Q75.954 18.122 75.954 17.688M77.629 19.045Q77.838 19.045 78.024 18.973Q78.210 18.902 78.364 18.765Q78.518 18.628 78.614 18.450L78.614 16.841Q78.528 16.694 78.383 16.574Q78.238 16.454 78.068 16.395Q77.899 16.335 77.718 16.335Q77.158 16.335 76.889 16.724Q76.621 17.114 76.621 17.695Q76.621 18.266 76.855 18.656Q77.089 19.045 77.629 19.045M80.429 18.471Q80.429 18.139 80.652 17.912Q80.876 17.685 81.220 17.557Q81.563 17.428 81.936 17.376Q82.308 17.323 82.613 17.323L82.613 17.070Q82.613 16.865 82.505 16.685Q82.397 16.506 82.216 16.403Q82.035 16.301 81.826 16.301Q81.420 16.301 81.184 16.393Q81.273 16.430 81.319 16.514Q81.365 16.598 81.365 16.700Q81.365 16.796 81.319 16.875Q81.273 16.953 81.192 16.998Q81.112 17.042 81.023 17.042Q80.873 17.042 80.772 16.945Q80.671 16.847 80.671 16.700Q80.671 16.078 81.826 16.078Q82.038 16.078 82.288 16.142Q82.537 16.205 82.739 16.324Q82.941 16.444 83.067 16.629Q83.194 16.813 83.194 17.056L83.194 18.632Q83.194 18.748 83.255 18.844Q83.317 18.939 83.429 18.939Q83.539 18.939 83.604 18.845Q83.669 18.751 83.669 18.632L83.669 18.184L83.935 18.184L83.935 18.632Q83.935 18.902 83.708 19.067Q83.481 19.233 83.200 19.233Q82.992 19.233 82.855 19.079Q82.719 18.926 82.695 18.710Q82.548 18.977 82.266 19.122Q81.984 19.267 81.659 19.267Q81.382 19.267 81.098 19.192Q80.815 19.117 80.622 18.938Q80.429 18.758 80.429 18.471M81.044 18.471Q81.044 18.645 81.145 18.775Q81.245 18.905 81.401 18.975Q81.556 19.045 81.721 19.045Q81.939 19.045 82.148 18.948Q82.356 18.850 82.484 18.669Q82.613 18.488 82.613 18.262L82.613 17.534Q82.288 17.534 81.922 17.625Q81.556 17.716 81.300 17.928Q81.044 18.139 81.044 18.471M85.996 20.556L84.366 20.556L84.366 20.276Q84.595 20.276 84.744 20.241Q84.892 20.207 84.892 20.067L84.892 16.721Q84.892 16.550 84.756 16.509Q84.619 16.468 84.366 16.468L84.366 16.188L85.446 16.113L85.446 16.519Q85.668 16.318 85.955 16.215Q86.242 16.113 86.550 16.113Q86.977 16.113 87.341 16.326Q87.705 16.540 87.919 16.904Q88.133 17.268 88.133 17.688Q88.133 18.133 87.893 18.497Q87.654 18.861 87.261 19.064Q86.868 19.267 86.424 19.267Q86.157 19.267 85.909 19.167Q85.661 19.066 85.473 18.885L85.473 20.067Q85.473 20.204 85.622 20.240Q85.771 20.276 85.996 20.276L85.996 20.556M85.473 16.868L85.473 18.478Q85.607 18.731 85.849 18.888Q86.092 19.045 86.369 19.045Q86.697 19.045 86.950 18.844Q87.203 18.642 87.336 18.324Q87.470 18.006 87.470 17.688Q87.470 17.459 87.405 17.230Q87.340 17.001 87.211 16.803Q87.083 16.605 86.888 16.485Q86.694 16.366 86.461 16.366Q86.167 16.366 85.899 16.495Q85.631 16.625 85.473 16.868M89.295 18.358L89.295 16.461L88.656 16.461L88.656 16.239Q88.973 16.239 89.190 16.029Q89.408 15.819 89.508 15.509Q89.609 15.200 89.609 14.892L89.876 14.892L89.876 16.181L90.952 16.181L90.952 16.461L89.876 16.461L89.876 18.345Q89.876 18.621 89.980 18.820Q90.084 19.018 90.344 19.018Q90.501 19.018 90.607 18.914Q90.713 18.809 90.763 18.656Q90.812 18.502 90.812 18.345L90.812 17.931L91.079 17.931L91.079 18.358Q91.079 18.584 90.980 18.794Q90.881 19.004 90.696 19.136Q90.512 19.267 90.283 19.267Q89.845 19.267 89.570 19.030Q89.295 18.792 89.295 18.358M93.506 19.199L91.954 19.199L91.954 18.919Q92.179 18.919 92.328 18.885Q92.477 18.850 92.477 18.710L92.477 16.861Q92.477 16.673 92.429 16.589Q92.381 16.506 92.284 16.487Q92.186 16.468 91.974 16.468L91.974 16.188L93.031 16.113L93.031 18.710Q93.031 18.850 93.162 18.885Q93.294 18.919 93.506 18.919L93.506 19.199M92.234 14.892Q92.234 14.721 92.357 14.602Q92.480 14.482 92.651 14.482Q92.819 14.482 92.942 14.602Q93.065 14.721 93.065 14.892Q93.065 15.067 92.942 15.190Q92.819 15.313 92.651 15.313Q92.480 15.313 92.357 15.190Q92.234 15.067 92.234 14.892M95.741 19.172L94.613 16.673Q94.541 16.526 94.411 16.494Q94.282 16.461 94.053 16.461L94.053 16.181L95.567 16.181L95.567 16.461Q95.215 16.461 95.215 16.608Q95.215 16.653 95.225 16.673L96.090 18.591L96.869 16.861Q96.903 16.793 96.903 16.714Q96.903 16.601 96.819 16.531Q96.736 16.461 96.616 16.461L96.616 16.181L97.812 16.181L97.812 16.461Q97.594 16.461 97.423 16.564Q97.252 16.666 97.163 16.861L96.127 19.172Q96.079 19.267 95.973 19.267L95.895 19.267Q95.789 19.267 95.741 19.172\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(52.807 -28.125)\">\u003Cpath d=\"M98.112 17.664Q98.112 17.343 98.237 17.054Q98.362 16.765 98.588 16.542Q98.813 16.318 99.109 16.198Q99.404 16.078 99.722 16.078Q100.050 16.078 100.312 16.178Q100.573 16.277 100.749 16.459Q100.925 16.642 101.019 16.900Q101.113 17.158 101.113 17.490Q101.113 17.582 101.031 17.603L98.776 17.603L98.776 17.664Q98.776 18.252 99.059 18.635Q99.343 19.018 99.910 19.018Q100.232 19.018 100.500 18.825Q100.768 18.632 100.857 18.317Q100.864 18.276 100.939 18.262L101.031 18.262Q101.113 18.286 101.113 18.358Q101.113 18.365 101.107 18.392Q100.994 18.789 100.623 19.028Q100.252 19.267 99.828 19.267Q99.391 19.267 98.991 19.059Q98.591 18.850 98.352 18.483Q98.112 18.116 98.112 17.664M98.782 17.394L100.597 17.394Q100.597 17.117 100.500 16.865Q100.402 16.612 100.204 16.456Q100.006 16.301 99.722 16.301Q99.445 16.301 99.232 16.459Q99.018 16.618 98.900 16.873Q98.782 17.128 98.782 17.394\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(52.807 -28.125)\">\u003Cpath d=\"M104.403 19.192L104.403 18.129Q104.403 18.105 104.431 18.078Q104.458 18.051 104.482 18.051L104.591 18.051Q104.656 18.051 104.670 18.109Q104.766 18.543 105.012 18.794Q105.258 19.045 105.672 19.045Q106.013 19.045 106.266 18.912Q106.519 18.779 106.519 18.471Q106.519 18.314 106.425 18.199Q106.331 18.085 106.193 18.016Q106.054 17.948 105.887 17.910L105.306 17.811Q104.950 17.743 104.677 17.522Q104.403 17.302 104.403 16.960Q104.403 16.711 104.515 16.536Q104.626 16.362 104.812 16.263Q104.998 16.164 105.214 16.121Q105.429 16.078 105.672 16.078Q106.085 16.078 106.365 16.260L106.581 16.085Q106.591 16.082 106.598 16.080Q106.605 16.078 106.615 16.078L106.666 16.078Q106.693 16.078 106.717 16.102Q106.741 16.126 106.741 16.154L106.741 17.001Q106.741 17.022 106.717 17.049Q106.693 17.076 106.666 17.076L106.553 17.076Q106.526 17.076 106.500 17.051Q106.475 17.025 106.475 17.001Q106.475 16.765 106.369 16.601Q106.263 16.437 106.080 16.355Q105.897 16.273 105.665 16.273Q105.337 16.273 105.080 16.376Q104.824 16.478 104.824 16.755Q104.824 16.950 105.007 17.059Q105.190 17.169 105.419 17.210L105.993 17.316Q106.239 17.364 106.453 17.492Q106.666 17.620 106.803 17.823Q106.940 18.027 106.940 18.276Q106.940 18.789 106.574 19.028Q106.208 19.267 105.672 19.267Q105.176 19.267 104.844 18.973L104.578 19.247Q104.557 19.267 104.530 19.267L104.482 19.267Q104.458 19.267 104.431 19.240Q104.403 19.213 104.403 19.192M108.095 18.358L108.095 16.461L107.456 16.461L107.456 16.239Q107.774 16.239 107.991 16.029Q108.208 15.819 108.308 15.509Q108.409 15.200 108.409 14.892L108.676 14.892L108.676 16.181L109.753 16.181L109.753 16.461L108.676 16.461L108.676 18.345Q108.676 18.621 108.780 18.820Q108.884 19.018 109.144 19.018Q109.301 19.018 109.407 18.914Q109.513 18.809 109.563 18.656Q109.612 18.502 109.612 18.345L109.612 17.931L109.879 17.931L109.879 18.358Q109.879 18.584 109.780 18.794Q109.681 19.004 109.496 19.136Q109.312 19.267 109.083 19.267Q108.645 19.267 108.370 19.030Q108.095 18.792 108.095 18.358M110.648 17.664Q110.648 17.343 110.773 17.054Q110.898 16.765 111.123 16.542Q111.349 16.318 111.644 16.198Q111.940 16.078 112.258 16.078Q112.586 16.078 112.848 16.178Q113.109 16.277 113.285 16.459Q113.461 16.642 113.555 16.900Q113.649 17.158 113.649 17.490Q113.649 17.582 113.567 17.603L111.311 17.603L111.311 17.664Q111.311 18.252 111.595 18.635Q111.879 19.018 112.446 19.018Q112.767 19.018 113.036 18.825Q113.304 18.632 113.393 18.317Q113.400 18.276 113.475 18.262L113.567 18.262Q113.649 18.286 113.649 18.358Q113.649 18.365 113.642 18.392Q113.529 18.789 113.159 19.028Q112.788 19.267 112.364 19.267Q111.926 19.267 111.526 19.059Q111.127 18.850 110.887 18.483Q110.648 18.116 110.648 17.664M111.318 17.394L113.133 17.394Q113.133 17.117 113.036 16.865Q112.938 16.612 112.740 16.456Q112.542 16.301 112.258 16.301Q111.981 16.301 111.767 16.459Q111.554 16.618 111.436 16.873Q111.318 17.128 111.318 17.394M115.881 20.556L114.251 20.556L114.251 20.276Q114.480 20.276 114.628 20.241Q114.777 20.207 114.777 20.067L114.777 16.721Q114.777 16.550 114.640 16.509Q114.504 16.468 114.251 16.468L114.251 16.188L115.331 16.113L115.331 16.519Q115.553 16.318 115.840 16.215Q116.127 16.113 116.435 16.113Q116.862 16.113 117.226 16.326Q117.590 16.540 117.804 16.904Q118.017 17.268 118.017 17.688Q118.017 18.133 117.778 18.497Q117.539 18.861 117.146 19.064Q116.753 19.267 116.308 19.267Q116.042 19.267 115.794 19.167Q115.546 19.066 115.358 18.885L115.358 20.067Q115.358 20.204 115.507 20.240Q115.655 20.276 115.881 20.276L115.881 20.556M115.358 16.868L115.358 18.478Q115.491 18.731 115.734 18.888Q115.977 19.045 116.254 19.045Q116.582 19.045 116.835 18.844Q117.088 18.642 117.221 18.324Q117.354 18.006 117.354 17.688Q117.354 17.459 117.289 17.230Q117.224 17.001 117.096 16.803Q116.968 16.605 116.773 16.485Q116.578 16.366 116.346 16.366Q116.052 16.366 115.784 16.495Q115.515 16.625 115.358 16.868\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(52.807 -28.125)\">\u003Cpath d=\"M121.362 19.192L121.362 18.129Q121.362 18.105 121.390 18.078Q121.417 18.051 121.441 18.051L121.550 18.051Q121.615 18.051 121.629 18.109Q121.725 18.543 121.971 18.794Q122.217 19.045 122.631 19.045Q122.972 19.045 123.225 18.912Q123.478 18.779 123.478 18.471Q123.478 18.314 123.384 18.199Q123.290 18.085 123.152 18.016Q123.013 17.948 122.846 17.910L122.265 17.811Q121.909 17.743 121.636 17.522Q121.362 17.302 121.362 16.960Q121.362 16.711 121.474 16.536Q121.585 16.362 121.771 16.263Q121.957 16.164 122.173 16.121Q122.388 16.078 122.631 16.078Q123.044 16.078 123.324 16.260L123.540 16.085Q123.550 16.082 123.557 16.080Q123.564 16.078 123.574 16.078L123.625 16.078Q123.652 16.078 123.676 16.102Q123.700 16.126 123.700 16.154L123.700 17.001Q123.700 17.022 123.676 17.049Q123.652 17.076 123.625 17.076L123.512 17.076Q123.485 17.076 123.459 17.051Q123.434 17.025 123.434 17.001Q123.434 16.765 123.328 16.601Q123.222 16.437 123.039 16.355Q122.856 16.273 122.624 16.273Q122.296 16.273 122.039 16.376Q121.783 16.478 121.783 16.755Q121.783 16.950 121.966 17.059Q122.149 17.169 122.378 17.210L122.952 17.316Q123.198 17.364 123.412 17.492Q123.625 17.620 123.762 17.823Q123.899 18.027 123.899 18.276Q123.899 18.789 123.533 19.028Q123.167 19.267 122.631 19.267Q122.135 19.267 121.803 18.973L121.537 19.247Q121.516 19.267 121.489 19.267L121.441 19.267Q121.417 19.267 121.390 19.240Q121.362 19.213 121.362 19.192M126.144 19.199L124.592 19.199L124.592 18.919Q124.818 18.919 124.967 18.885Q125.115 18.850 125.115 18.710L125.115 16.861Q125.115 16.673 125.068 16.589Q125.020 16.506 124.922 16.487Q124.825 16.468 124.613 16.468L124.613 16.188L125.669 16.113L125.669 18.710Q125.669 18.850 125.801 18.885Q125.932 18.919 126.144 18.919L126.144 19.199M124.873 14.892Q124.873 14.721 124.996 14.602Q125.119 14.482 125.290 14.482Q125.457 14.482 125.580 14.602Q125.703 14.721 125.703 14.892Q125.703 15.067 125.580 15.190Q125.457 15.313 125.290 15.313Q125.119 15.313 124.996 15.190Q124.873 15.067 124.873 14.892M129.521 19.199L126.848 19.199Q126.804 19.199 126.777 19.172Q126.749 19.144 126.749 19.100L126.749 19.032Q126.749 18.991 126.777 18.960L128.885 16.407L128.246 16.407Q127.822 16.407 127.590 16.473Q127.358 16.540 127.238 16.750Q127.118 16.960 127.118 17.381L126.855 17.381L126.937 16.181L129.528 16.181Q129.569 16.181 129.598 16.208Q129.627 16.236 129.627 16.280L129.627 16.328Q129.627 16.372 129.603 16.400L127.498 18.946L128.178 18.946Q128.506 18.946 128.723 18.910Q128.940 18.874 129.108 18.724Q129.248 18.587 129.301 18.363Q129.354 18.139 129.381 17.811L129.648 17.811L129.521 19.199M130.297 17.664Q130.297 17.343 130.422 17.054Q130.547 16.765 130.772 16.542Q130.998 16.318 131.293 16.198Q131.589 16.078 131.907 16.078Q132.235 16.078 132.496 16.178Q132.758 16.277 132.934 16.459Q133.110 16.642 133.204 16.900Q133.298 17.158 133.298 17.490Q133.298 17.582 133.216 17.603L130.960 17.603L130.960 17.664Q130.960 18.252 131.244 18.635Q131.527 19.018 132.095 19.018Q132.416 19.018 132.684 18.825Q132.953 18.632 133.042 18.317Q133.048 18.276 133.124 18.262L133.216 18.262Q133.298 18.286 133.298 18.358Q133.298 18.365 133.291 18.392Q133.178 18.789 132.808 19.028Q132.437 19.267 132.013 19.267Q131.575 19.267 131.175 19.059Q130.776 18.850 130.536 18.483Q130.297 18.116 130.297 17.664M130.967 17.394L132.782 17.394Q132.782 17.117 132.684 16.865Q132.587 16.612 132.389 16.456Q132.191 16.301 131.907 16.301Q131.630 16.301 131.416 16.459Q131.203 16.618 131.085 16.873Q130.967 17.128 130.967 17.394M133.886 19.192L133.886 18.129Q133.886 18.105 133.913 18.078Q133.941 18.051 133.965 18.051L134.074 18.051Q134.139 18.051 134.152 18.109Q134.248 18.543 134.494 18.794Q134.740 19.045 135.154 19.045Q135.496 19.045 135.749 18.912Q136.002 18.779 136.002 18.471Q136.002 18.314 135.908 18.199Q135.814 18.085 135.675 18.016Q135.537 17.948 135.369 17.910L134.788 17.811Q134.433 17.743 134.159 17.522Q133.886 17.302 133.886 16.960Q133.886 16.711 133.997 16.536Q134.108 16.362 134.294 16.263Q134.481 16.164 134.696 16.121Q134.911 16.078 135.154 16.078Q135.568 16.078 135.848 16.260L136.063 16.085Q136.073 16.082 136.080 16.080Q136.087 16.078 136.097 16.078L136.149 16.078Q136.176 16.078 136.200 16.102Q136.224 16.126 136.224 16.154L136.224 17.001Q136.224 17.022 136.200 17.049Q136.176 17.076 136.149 17.076L136.036 17.076Q136.008 17.076 135.983 17.051Q135.957 17.025 135.957 17.001Q135.957 16.765 135.851 16.601Q135.745 16.437 135.562 16.355Q135.380 16.273 135.147 16.273Q134.819 16.273 134.563 16.376Q134.306 16.478 134.306 16.755Q134.306 16.950 134.489 17.059Q134.672 17.169 134.901 17.210L135.475 17.316Q135.721 17.364 135.935 17.492Q136.149 17.620 136.285 17.823Q136.422 18.027 136.422 18.276Q136.422 18.789 136.056 19.028Q135.691 19.267 135.154 19.267Q134.658 19.267 134.327 18.973L134.060 19.247Q134.040 19.267 134.012 19.267L133.965 19.267Q133.941 19.267 133.913 19.240Q133.886 19.213 133.886 19.192\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(-31.837 49.43)\">\u003Cpath d=\"M79.313 9.699L77.481 9.699L77.481 9.402Q77.755 9.402 77.923 9.355Q78.091 9.308 78.091 9.140L78.091 4.980Q78.091 4.765 78.028 4.670Q77.966 4.574 77.847 4.553Q77.727 4.531 77.481 4.531L77.481 4.234L78.704 4.148L78.704 9.140Q78.704 9.308 78.872 9.355Q79.040 9.402 79.313 9.402L79.313 9.699M79.759 8.004Q79.759 7.500 80.015 7.068Q80.270 6.636 80.706 6.385Q81.141 6.133 81.641 6.133Q82.028 6.133 82.370 6.277Q82.712 6.422 82.974 6.683Q83.235 6.945 83.378 7.281Q83.520 7.617 83.520 8.004Q83.520 8.496 83.257 8.906Q82.993 9.316 82.563 9.547Q82.134 9.777 81.641 9.777Q81.149 9.777 80.716 9.545Q80.282 9.312 80.020 8.904Q79.759 8.496 79.759 8.004M81.641 9.500Q82.099 9.500 82.350 9.277Q82.602 9.054 82.690 8.703Q82.778 8.351 82.778 7.906Q82.778 7.476 82.684 7.138Q82.591 6.801 82.337 6.594Q82.083 6.386 81.641 6.386Q80.993 6.386 80.749 6.803Q80.505 7.219 80.505 7.906Q80.505 8.351 80.593 8.703Q80.681 9.054 80.932 9.277Q81.184 9.500 81.641 9.500\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-31.837 49.43)\">\u003Cpath d=\"M85.361 9.668L84.291 6.812Q84.225 6.633 84.094 6.590Q83.963 6.547 83.705 6.547L83.705 6.250L85.385 6.250L85.385 6.547Q84.935 6.547 84.935 6.746Q84.939 6.761 84.941 6.779Q84.943 6.797 84.943 6.812L85.736 8.906L86.447 6.996Q86.412 6.902 86.412 6.857Q86.412 6.812 86.377 6.812Q86.310 6.633 86.180 6.590Q86.049 6.547 85.795 6.547L85.795 6.250L87.385 6.250L87.385 6.547Q86.935 6.547 86.935 6.746Q86.939 6.765 86.941 6.783Q86.943 6.801 86.943 6.812L87.775 9.027L88.529 7.027Q88.553 6.969 88.553 6.898Q88.553 6.738 88.416 6.642Q88.279 6.547 88.111 6.547L88.111 6.250L89.498 6.250L89.498 6.547Q89.264 6.547 89.086 6.674Q88.908 6.801 88.826 7.027L87.842 9.668Q87.787 9.777 87.674 9.777L87.615 9.777Q87.502 9.777 87.459 9.668L86.600 7.394L85.744 9.668Q85.705 9.777 85.584 9.777L85.529 9.777Q85.416 9.777 85.361 9.668\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-31.837 49.43)\">\u003Cpath d=\"M89.677 7.945Q89.677 7.465 89.910 7.049Q90.142 6.633 90.552 6.383Q90.962 6.133 91.439 6.133Q92.169 6.133 92.568 6.574Q92.966 7.015 92.966 7.746Q92.966 7.851 92.873 7.875L90.423 7.875L90.423 7.945Q90.423 8.355 90.544 8.711Q90.666 9.066 90.937 9.283Q91.209 9.500 91.638 9.500Q92.001 9.500 92.298 9.271Q92.595 9.043 92.697 8.691Q92.705 8.644 92.791 8.629L92.873 8.629Q92.966 8.656 92.966 8.738Q92.966 8.746 92.959 8.777Q92.896 9.004 92.757 9.187Q92.619 9.371 92.427 9.504Q92.236 9.636 92.017 9.707Q91.798 9.777 91.560 9.777Q91.189 9.777 90.851 9.640Q90.513 9.504 90.246 9.252Q89.978 9 89.828 8.660Q89.677 8.320 89.677 7.945M90.431 7.636L92.392 7.636Q92.392 7.332 92.291 7.041Q92.189 6.750 91.972 6.568Q91.755 6.386 91.439 6.386Q91.138 6.386 90.908 6.574Q90.677 6.761 90.554 7.053Q90.431 7.344 90.431 7.636M95.462 9.699L93.482 9.699L93.482 9.402Q93.751 9.402 93.919 9.357Q94.087 9.312 94.087 9.140L94.087 7.004Q94.087 6.789 94.025 6.693Q93.962 6.597 93.845 6.576Q93.728 6.554 93.482 6.554L93.482 6.258L94.650 6.172L94.650 6.957Q94.728 6.746 94.880 6.560Q95.033 6.375 95.232 6.273Q95.431 6.172 95.658 6.172Q95.904 6.172 96.095 6.316Q96.287 6.461 96.287 6.691Q96.287 6.847 96.181 6.957Q96.076 7.066 95.919 7.066Q95.763 7.066 95.654 6.957Q95.544 6.847 95.544 6.691Q95.544 6.531 95.650 6.426Q95.326 6.426 95.111 6.654Q94.896 6.883 94.800 7.222Q94.705 7.562 94.705 7.867L94.705 9.140Q94.705 9.308 94.931 9.355Q95.158 9.402 95.462 9.402L95.462 9.699M98.880 8.250L96.626 8.250L96.626 7.699L98.880 7.699L98.880 8.250M101.400 9.668L100.177 6.812Q100.095 6.636 99.951 6.592Q99.806 6.547 99.537 6.547L99.537 6.250L101.248 6.250L101.248 6.547Q100.826 6.547 100.826 6.730Q100.826 6.765 100.841 6.812L101.787 9.004L102.626 7.027Q102.666 6.949 102.666 6.859Q102.666 6.719 102.560 6.633Q102.455 6.547 102.314 6.547L102.314 6.250L103.666 6.250L103.666 6.547Q103.142 6.547 102.927 7.027L101.802 9.668Q101.740 9.777 101.634 9.777L101.568 9.777Q101.455 9.777 101.400 9.668\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-31.837 49.43)\">\u003Cpath d=\"M103.713 8.867Q103.713 8.383 104.115 8.088Q104.518 7.793 105.068 7.674Q105.619 7.554 106.111 7.554L106.111 7.265Q106.111 7.039 105.996 6.832Q105.881 6.625 105.684 6.506Q105.486 6.386 105.256 6.386Q104.830 6.386 104.545 6.492Q104.615 6.519 104.662 6.574Q104.709 6.629 104.734 6.699Q104.760 6.769 104.760 6.844Q104.760 6.949 104.709 7.041Q104.658 7.133 104.566 7.183Q104.475 7.234 104.369 7.234Q104.264 7.234 104.172 7.183Q104.080 7.133 104.029 7.041Q103.979 6.949 103.979 6.844Q103.979 6.426 104.367 6.279Q104.756 6.133 105.256 6.133Q105.588 6.133 105.941 6.263Q106.295 6.394 106.523 6.648Q106.752 6.902 106.752 7.250L106.752 9.051Q106.752 9.183 106.824 9.293Q106.897 9.402 107.025 9.402Q107.150 9.402 107.219 9.297Q107.287 9.191 107.287 9.051L107.287 8.539L107.568 8.539L107.568 9.051Q107.568 9.254 107.451 9.412Q107.334 9.570 107.152 9.654Q106.971 9.738 106.768 9.738Q106.537 9.738 106.385 9.566Q106.232 9.394 106.201 9.164Q106.041 9.445 105.732 9.611Q105.424 9.777 105.072 9.777Q104.561 9.777 104.137 9.554Q103.713 9.332 103.713 8.867M104.400 8.867Q104.400 9.152 104.627 9.338Q104.854 9.523 105.147 9.523Q105.393 9.523 105.617 9.406Q105.842 9.289 105.977 9.086Q106.111 8.883 106.111 8.629L106.111 7.797Q105.846 7.797 105.561 7.851Q105.275 7.906 105.004 8.035Q104.732 8.164 104.566 8.371Q104.400 8.578 104.400 8.867M109.869 9.699L107.889 9.699L107.889 9.402Q108.158 9.402 108.326 9.357Q108.494 9.312 108.494 9.140L108.494 7.004Q108.494 6.789 108.432 6.693Q108.369 6.597 108.252 6.576Q108.135 6.554 107.889 6.554L107.889 6.258L109.057 6.172L109.057 6.957Q109.135 6.746 109.287 6.560Q109.439 6.375 109.639 6.273Q109.838 6.172 110.064 6.172Q110.311 6.172 110.502 6.316Q110.693 6.461 110.693 6.691Q110.693 6.847 110.588 6.957Q110.482 7.066 110.326 7.066Q110.170 7.066 110.061 6.957Q109.951 6.847 109.951 6.691Q109.951 6.531 110.057 6.426Q109.732 6.426 109.518 6.654Q109.303 6.883 109.207 7.222Q109.111 7.562 109.111 7.867L109.111 9.140Q109.111 9.308 109.338 9.355Q109.564 9.402 109.869 9.402L109.869 9.699M113.033 9.699L111.256 9.699L111.256 9.402Q111.529 9.402 111.697 9.355Q111.865 9.308 111.865 9.140L111.865 7.004Q111.865 6.789 111.809 6.693Q111.752 6.597 111.639 6.576Q111.525 6.554 111.279 6.554L111.279 6.258L112.479 6.172L112.479 9.140Q112.479 9.308 112.625 9.355Q112.772 9.402 113.033 9.402L113.033 9.699M111.592 4.777Q111.592 4.586 111.727 4.455Q111.861 4.324 112.057 4.324Q112.178 4.324 112.281 4.386Q112.385 4.449 112.447 4.553Q112.510 4.656 112.510 4.777Q112.510 4.972 112.379 5.107Q112.248 5.242 112.057 5.242Q111.857 5.242 111.725 5.109Q111.592 4.976 111.592 4.777M113.631 8.867Q113.631 8.383 114.033 8.088Q114.436 7.793 114.986 7.674Q115.537 7.554 116.029 7.554L116.029 7.265Q116.029 7.039 115.914 6.832Q115.799 6.625 115.602 6.506Q115.404 6.386 115.174 6.386Q114.748 6.386 114.463 6.492Q114.533 6.519 114.580 6.574Q114.627 6.629 114.652 6.699Q114.678 6.769 114.678 6.844Q114.678 6.949 114.627 7.041Q114.576 7.133 114.484 7.183Q114.393 7.234 114.287 7.234Q114.182 7.234 114.090 7.183Q113.998 7.133 113.947 7.041Q113.897 6.949 113.897 6.844Q113.897 6.426 114.285 6.279Q114.674 6.133 115.174 6.133Q115.506 6.133 115.859 6.263Q116.213 6.394 116.441 6.648Q116.670 6.902 116.670 7.250L116.670 9.051Q116.670 9.183 116.742 9.293Q116.814 9.402 116.943 9.402Q117.068 9.402 117.137 9.297Q117.205 9.191 117.205 9.051L117.205 8.539L117.486 8.539L117.486 9.051Q117.486 9.254 117.369 9.412Q117.252 9.570 117.070 9.654Q116.889 9.738 116.686 9.738Q116.455 9.738 116.303 9.566Q116.150 9.394 116.119 9.164Q115.959 9.445 115.650 9.611Q115.342 9.777 114.990 9.777Q114.479 9.777 114.055 9.554Q113.631 9.332 113.631 8.867M114.318 8.867Q114.318 9.152 114.545 9.338Q114.772 9.523 115.064 9.523Q115.311 9.523 115.535 9.406Q115.760 9.289 115.895 9.086Q116.029 8.883 116.029 8.629L116.029 7.797Q115.764 7.797 115.479 7.851Q115.193 7.906 114.922 8.035Q114.650 8.164 114.484 8.371Q114.318 8.578 114.318 8.867M119.709 9.699L117.854 9.699L117.854 9.402Q118.127 9.402 118.295 9.355Q118.463 9.308 118.463 9.140L118.463 7.004Q118.463 6.789 118.400 6.693Q118.338 6.597 118.219 6.576Q118.100 6.554 117.854 6.554L117.854 6.258L119.045 6.172L119.045 6.906Q119.158 6.691 119.352 6.523Q119.545 6.355 119.783 6.263Q120.022 6.172 120.275 6.172Q121.443 6.172 121.443 7.250L121.443 9.140Q121.443 9.308 121.613 9.355Q121.783 9.402 122.053 9.402L122.053 9.699L120.197 9.699L120.197 9.402Q120.471 9.402 120.639 9.355Q120.807 9.308 120.807 9.140L120.807 7.265Q120.807 6.883 120.686 6.654Q120.564 6.426 120.213 6.426Q119.900 6.426 119.647 6.588Q119.393 6.750 119.246 7.019Q119.100 7.289 119.100 7.586L119.100 9.140Q119.100 9.308 119.270 9.355Q119.439 9.402 119.709 9.402L119.709 9.699M122.541 7.972Q122.541 7.476 122.791 7.051Q123.041 6.625 123.461 6.379Q123.881 6.133 124.381 6.133Q124.920 6.133 125.311 6.258Q125.701 6.383 125.701 6.797Q125.701 6.902 125.650 6.994Q125.600 7.086 125.508 7.136Q125.416 7.187 125.307 7.187Q125.201 7.187 125.109 7.136Q125.018 7.086 124.967 6.994Q124.916 6.902 124.916 6.797Q124.916 6.574 125.084 6.469Q124.861 6.410 124.389 6.410Q124.092 6.410 123.877 6.549Q123.662 6.687 123.531 6.918Q123.400 7.148 123.342 7.418Q123.283 7.687 123.283 7.972Q123.283 8.367 123.416 8.717Q123.549 9.066 123.820 9.283Q124.092 9.500 124.490 9.500Q124.865 9.500 125.141 9.283Q125.416 9.066 125.518 8.707Q125.533 8.644 125.596 8.644L125.701 8.644Q125.736 8.644 125.762 8.672Q125.787 8.699 125.787 8.738L125.787 8.761Q125.654 9.242 125.270 9.510Q124.885 9.777 124.381 9.777Q124.018 9.777 123.684 9.640Q123.350 9.504 123.090 9.254Q122.830 9.004 122.686 8.668Q122.541 8.332 122.541 7.972M126.275 7.945Q126.275 7.465 126.508 7.049Q126.740 6.633 127.150 6.383Q127.561 6.133 128.037 6.133Q128.768 6.133 129.166 6.574Q129.565 7.015 129.565 7.746Q129.565 7.851 129.471 7.875L127.022 7.875L127.022 7.945Q127.022 8.355 127.143 8.711Q127.264 9.066 127.535 9.283Q127.807 9.500 128.236 9.500Q128.600 9.500 128.897 9.271Q129.193 9.043 129.295 8.691Q129.303 8.644 129.389 8.629L129.471 8.629Q129.565 8.656 129.565 8.738Q129.565 8.746 129.557 8.777Q129.494 9.004 129.356 9.187Q129.217 9.371 129.025 9.504Q128.834 9.636 128.615 9.707Q128.397 9.777 128.158 9.777Q127.787 9.777 127.449 9.640Q127.111 9.504 126.844 9.252Q126.576 9 126.426 8.660Q126.275 8.320 126.275 7.945M127.029 7.636L128.990 7.636Q128.990 7.332 128.889 7.041Q128.787 6.750 128.570 6.568Q128.354 6.386 128.037 6.386Q127.736 6.386 127.506 6.574Q127.275 6.761 127.152 7.053Q127.029 7.344 127.029 7.636\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-31.837 49.43)\">\u003Cpath d=\"M71.896 17.504Q71.896 17 72.152 16.568Q72.408 16.137 72.844 15.885Q73.279 15.633 73.779 15.633Q74.166 15.633 74.508 15.777Q74.849 15.922 75.111 16.183Q75.373 16.445 75.515 16.781Q75.658 17.117 75.658 17.504Q75.658 17.996 75.394 18.406Q75.131 18.816 74.701 19.047Q74.271 19.277 73.779 19.277Q73.287 19.277 72.853 19.045Q72.420 18.812 72.158 18.404Q71.896 17.996 71.896 17.504M73.779 19Q74.236 19 74.488 18.777Q74.740 18.554 74.828 18.203Q74.916 17.851 74.916 17.406Q74.916 16.976 74.822 16.638Q74.728 16.301 74.474 16.094Q74.221 15.887 73.779 15.887Q73.131 15.887 72.887 16.303Q72.642 16.719 72.642 17.406Q72.642 17.851 72.730 18.203Q72.818 18.554 73.070 18.777Q73.322 19 73.779 19M78.209 19.199L76.224 19.199L76.224 18.902Q76.498 18.902 76.666 18.855Q76.834 18.808 76.834 18.640L76.834 16.047L76.193 16.047L76.193 15.750L76.834 15.750L76.834 14.816Q76.834 14.551 76.951 14.314Q77.068 14.078 77.262 13.914Q77.455 13.750 77.703 13.658Q77.951 13.566 78.217 13.566Q78.502 13.566 78.726 13.724Q78.951 13.883 78.951 14.160Q78.951 14.316 78.846 14.426Q78.740 14.535 78.576 14.535Q78.420 14.535 78.310 14.426Q78.201 14.316 78.201 14.160Q78.201 13.953 78.361 13.847Q78.263 13.824 78.170 13.824Q77.939 13.824 77.767 13.980Q77.596 14.137 77.510 14.373Q77.424 14.609 77.424 14.832L77.424 15.750L78.392 15.750L78.392 16.047L77.447 16.047L77.447 18.640Q77.447 18.808 77.674 18.855Q77.900 18.902 78.209 18.902\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-31.837 49.43)\">\u003Cpath d=\"M81.442 19.199L79.457 19.199L79.457 18.902Q79.731 18.902 79.899 18.855Q80.067 18.808 80.067 18.640L80.067 16.047L79.426 16.047L79.426 15.750L80.067 15.750L80.067 14.816Q80.067 14.551 80.184 14.314Q80.301 14.078 80.494 13.914Q80.688 13.750 80.936 13.658Q81.184 13.566 81.450 13.566Q81.735 13.566 81.959 13.724Q82.184 13.883 82.184 14.160Q82.184 14.316 82.078 14.426Q81.973 14.535 81.809 14.535Q81.653 14.535 81.543 14.426Q81.434 14.316 81.434 14.160Q81.434 13.953 81.594 13.847Q81.496 13.824 81.403 13.824Q81.172 13.824 81 13.980Q80.828 14.137 80.742 14.373Q80.657 14.609 80.657 14.832L80.657 15.750L81.625 15.750L81.625 16.047L80.680 16.047L80.680 18.640Q80.680 18.808 80.907 18.855Q81.133 18.902 81.442 18.902L81.442 19.199M84.082 17.750L81.828 17.750L81.828 17.199L84.082 17.199L84.082 17.750M86.684 20.750L84.828 20.750L84.828 20.457Q85.098 20.457 85.266 20.412Q85.434 20.367 85.434 20.191L85.434 16.367Q85.434 16.160 85.278 16.107Q85.121 16.054 84.828 16.054L84.828 15.758L86.051 15.672L86.051 16.137Q86.282 15.914 86.596 15.793Q86.910 15.672 87.250 15.672Q87.723 15.672 88.127 15.918Q88.532 16.164 88.764 16.580Q88.996 16.996 88.996 17.472Q88.996 17.847 88.848 18.176Q88.700 18.504 88.430 18.756Q88.160 19.008 87.817 19.142Q87.473 19.277 87.114 19.277Q86.825 19.277 86.553 19.156Q86.282 19.035 86.075 18.824L86.075 20.191Q86.075 20.367 86.242 20.412Q86.410 20.457 86.684 20.457L86.684 20.750M86.075 16.535L86.075 18.375Q86.227 18.664 86.489 18.844Q86.750 19.023 87.059 19.023Q87.344 19.023 87.567 18.885Q87.789 18.746 87.942 18.515Q88.094 18.285 88.172 18.013Q88.250 17.742 88.250 17.472Q88.250 17.140 88.125 16.783Q88 16.426 87.752 16.189Q87.504 15.953 87.157 15.953Q86.832 15.953 86.537 16.109Q86.242 16.265 86.075 16.535\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-31.837 49.43)\">\u003Cpath d=\"M89.764 17.504Q89.764 17 90.020 16.568Q90.276 16.137 90.712 15.885Q91.147 15.633 91.647 15.633Q92.034 15.633 92.376 15.777Q92.717 15.922 92.979 16.183Q93.241 16.445 93.383 16.781Q93.526 17.117 93.526 17.504Q93.526 17.996 93.262 18.406Q92.999 18.816 92.569 19.047Q92.139 19.277 91.647 19.277Q91.155 19.277 90.721 19.045Q90.288 18.812 90.026 18.404Q89.764 17.996 89.764 17.504M91.647 19Q92.104 19 92.356 18.777Q92.608 18.554 92.696 18.203Q92.784 17.851 92.784 17.406Q92.784 16.976 92.690 16.638Q92.596 16.301 92.342 16.094Q92.088 15.887 91.647 15.887Q90.999 15.887 90.755 16.303Q90.510 16.719 90.510 17.406Q90.510 17.851 90.598 18.203Q90.686 18.554 90.938 18.777Q91.190 19 91.647 19M95.924 19.199L94.092 19.199L94.092 18.902Q94.366 18.902 94.534 18.855Q94.702 18.808 94.702 18.640L94.702 14.480Q94.702 14.265 94.639 14.170Q94.577 14.074 94.458 14.053Q94.338 14.031 94.092 14.031L94.092 13.734L95.315 13.648L95.315 18.640Q95.315 18.808 95.483 18.855Q95.651 18.902 95.924 18.902L95.924 19.199M98.229 19.199L96.452 19.199L96.452 18.902Q96.725 18.902 96.893 18.855Q97.061 18.808 97.061 18.640L97.061 16.504Q97.061 16.289 97.005 16.193Q96.948 16.097 96.835 16.076Q96.721 16.054 96.475 16.054L96.475 15.758L97.674 15.672L97.674 18.640Q97.674 18.808 97.821 18.855Q97.967 18.902 98.229 18.902L98.229 19.199M96.788 14.277Q96.788 14.086 96.922 13.955Q97.057 13.824 97.253 13.824Q97.374 13.824 97.477 13.887Q97.581 13.949 97.643 14.053Q97.706 14.156 97.706 14.277Q97.706 14.472 97.575 14.607Q97.444 14.742 97.253 14.742Q97.053 14.742 96.921 14.609Q96.788 14.476 96.788 14.277M98.772 17.472Q98.772 16.976 99.022 16.551Q99.272 16.125 99.692 15.879Q100.112 15.633 100.612 15.633Q101.151 15.633 101.542 15.758Q101.932 15.883 101.932 16.297Q101.932 16.402 101.881 16.494Q101.831 16.586 101.739 16.637Q101.647 16.687 101.538 16.687Q101.432 16.687 101.340 16.637Q101.249 16.586 101.198 16.494Q101.147 16.402 101.147 16.297Q101.147 16.074 101.315 15.969Q101.092 15.910 100.620 15.910Q100.323 15.910 100.108 16.049Q99.893 16.187 99.762 16.418Q99.631 16.648 99.573 16.918Q99.514 17.187 99.514 17.472Q99.514 17.867 99.647 18.217Q99.780 18.566 100.051 18.783Q100.323 19 100.721 19Q101.096 19 101.372 18.783Q101.647 18.566 101.749 18.207Q101.764 18.144 101.827 18.144L101.932 18.144Q101.967 18.144 101.993 18.172Q102.018 18.199 102.018 18.238L102.018 18.262Q101.885 18.742 101.501 19.010Q101.116 19.277 100.612 19.277Q100.249 19.277 99.915 19.140Q99.581 19.004 99.321 18.754Q99.061 18.504 98.917 18.168Q98.772 17.832 98.772 17.472M102.924 20.496Q103.038 20.574 103.213 20.574Q103.503 20.574 103.723 20.361Q103.944 20.148 104.069 19.847L104.358 19.199L103.085 16.312Q103.003 16.137 102.858 16.092Q102.713 16.047 102.444 16.047L102.444 15.750L104.163 15.750L104.163 16.047Q103.741 16.047 103.741 16.230Q103.741 16.242 103.756 16.312L104.694 18.437L105.526 16.527Q105.565 16.437 105.565 16.359Q105.565 16.219 105.463 16.133Q105.362 16.047 105.221 16.047L105.221 15.750L106.573 15.750L106.573 16.047Q106.319 16.047 106.126 16.172Q105.932 16.297 105.827 16.527L104.381 19.847Q104.268 20.101 104.102 20.324Q103.936 20.547 103.708 20.689Q103.479 20.832 103.213 20.832Q102.917 20.832 102.676 20.640Q102.436 20.449 102.436 20.160Q102.436 20.004 102.542 19.902Q102.647 19.801 102.796 19.801Q102.901 19.801 102.981 19.847Q103.061 19.894 103.108 19.972Q103.155 20.051 103.155 20.160Q103.155 20.281 103.094 20.369Q103.034 20.457 102.924 20.496\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-31.837 49.43)\">\u003Cpath d=\"M110.518 18.246L110.518 16.504Q110.518 16.289 110.455 16.193Q110.393 16.097 110.274 16.076Q110.155 16.054 109.909 16.054L109.909 15.758L111.155 15.672L111.155 18.222L111.155 18.246Q111.155 18.558 111.209 18.720Q111.264 18.883 111.414 18.953Q111.565 19.023 111.885 19.023Q112.315 19.023 112.588 18.685Q112.862 18.347 112.862 17.902L112.862 16.504Q112.862 16.289 112.799 16.193Q112.737 16.097 112.617 16.076Q112.498 16.054 112.252 16.054L112.252 15.758L113.498 15.672L113.498 18.457Q113.498 18.668 113.561 18.763Q113.623 18.859 113.742 18.881Q113.862 18.902 114.108 18.902L114.108 19.199L112.885 19.277L112.885 18.656Q112.717 18.945 112.436 19.111Q112.155 19.277 111.834 19.277Q110.518 19.277 110.518 18.246M116.436 20.750L114.580 20.750L114.580 20.457Q114.850 20.457 115.018 20.412Q115.186 20.367 115.186 20.191L115.186 16.367Q115.186 16.160 115.030 16.107Q114.873 16.054 114.580 16.054L114.580 15.758L115.803 15.672L115.803 16.137Q116.034 15.914 116.348 15.793Q116.662 15.672 117.002 15.672Q117.475 15.672 117.879 15.918Q118.284 16.164 118.516 16.580Q118.748 16.996 118.748 17.472Q118.748 17.847 118.600 18.176Q118.451 18.504 118.182 18.756Q117.912 19.008 117.569 19.142Q117.225 19.277 116.866 19.277Q116.576 19.277 116.305 19.156Q116.034 19.035 115.826 18.824L115.826 20.191Q115.826 20.367 115.994 20.412Q116.162 20.457 116.436 20.457L116.436 20.750M115.826 16.535L115.826 18.375Q115.979 18.664 116.241 18.844Q116.502 19.023 116.811 19.023Q117.096 19.023 117.319 18.885Q117.541 18.746 117.694 18.515Q117.846 18.285 117.924 18.013Q118.002 17.742 118.002 17.472Q118.002 17.140 117.877 16.783Q117.752 16.426 117.504 16.189Q117.256 15.953 116.909 15.953Q116.584 15.953 116.289 16.109Q115.994 16.265 115.826 16.535\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-31.837 49.43)\">\u003Cpath d=\"M121.332 19.277Q120.851 19.277 120.443 19.033Q120.035 18.789 119.797 18.375Q119.558 17.961 119.558 17.472Q119.558 16.980 119.816 16.564Q120.074 16.148 120.506 15.910Q120.937 15.672 121.429 15.672Q122.050 15.672 122.500 16.109L122.500 14.480Q122.500 14.265 122.437 14.170Q122.375 14.074 122.257 14.053Q122.140 14.031 121.894 14.031L121.894 13.734L123.117 13.648L123.117 18.457Q123.117 18.668 123.179 18.763Q123.242 18.859 123.359 18.881Q123.476 18.902 123.726 18.902L123.726 19.199L122.476 19.277L122.476 18.793Q122.011 19.277 121.332 19.277M121.398 19.023Q121.738 19.023 122.031 18.832Q122.324 18.640 122.476 18.344L122.476 16.512Q122.328 16.238 122.066 16.082Q121.804 15.926 121.492 15.926Q120.867 15.926 120.584 16.373Q120.300 16.820 120.300 17.480Q120.300 18.125 120.552 18.574Q120.804 19.023 121.398 19.023M124.332 18.367Q124.332 17.883 124.734 17.588Q125.136 17.293 125.687 17.174Q126.238 17.054 126.730 17.054L126.730 16.765Q126.730 16.539 126.615 16.332Q126.500 16.125 126.302 16.006Q126.105 15.887 125.875 15.887Q125.449 15.887 125.164 15.992Q125.234 16.019 125.281 16.074Q125.328 16.129 125.353 16.199Q125.379 16.269 125.379 16.344Q125.379 16.449 125.328 16.541Q125.277 16.633 125.185 16.683Q125.093 16.734 124.988 16.734Q124.882 16.734 124.791 16.683Q124.699 16.633 124.648 16.541Q124.597 16.449 124.597 16.344Q124.597 15.926 124.986 15.779Q125.375 15.633 125.875 15.633Q126.207 15.633 126.560 15.763Q126.914 15.894 127.142 16.148Q127.371 16.402 127.371 16.750L127.371 18.551Q127.371 18.683 127.443 18.793Q127.515 18.902 127.644 18.902Q127.769 18.902 127.838 18.797Q127.906 18.691 127.906 18.551L127.906 18.039L128.187 18.039L128.187 18.551Q128.187 18.754 128.070 18.912Q127.953 19.070 127.771 19.154Q127.590 19.238 127.386 19.238Q127.156 19.238 127.004 19.066Q126.851 18.894 126.820 18.664Q126.660 18.945 126.351 19.111Q126.043 19.277 125.691 19.277Q125.179 19.277 124.756 19.054Q124.332 18.832 124.332 18.367M125.019 18.367Q125.019 18.652 125.246 18.838Q125.472 19.023 125.765 19.023Q126.011 19.023 126.236 18.906Q126.461 18.789 126.595 18.586Q126.730 18.383 126.730 18.129L126.730 17.297Q126.465 17.297 126.179 17.351Q125.894 17.406 125.623 17.535Q125.351 17.664 125.185 17.871Q125.019 18.078 125.019 18.367M129.105 18.238L129.105 16.047L128.402 16.047L128.402 15.793Q128.757 15.793 129 15.560Q129.242 15.328 129.353 14.980Q129.464 14.633 129.464 14.277L129.746 14.277L129.746 15.750L130.922 15.750L130.922 16.047L129.746 16.047L129.746 18.222Q129.746 18.543 129.865 18.771Q129.984 19 130.265 19Q130.445 19 130.562 18.877Q130.679 18.754 130.732 18.574Q130.785 18.394 130.785 18.222L130.785 17.750L131.066 17.750L131.066 18.238Q131.066 18.492 130.961 18.732Q130.855 18.972 130.658 19.125Q130.461 19.277 130.203 19.277Q129.886 19.277 129.634 19.154Q129.382 19.031 129.244 18.797Q129.105 18.562 129.105 18.238M131.785 17.445Q131.785 16.965 132.017 16.549Q132.250 16.133 132.660 15.883Q133.070 15.633 133.547 15.633Q134.277 15.633 134.675 16.074Q135.074 16.515 135.074 17.246Q135.074 17.351 134.980 17.375L132.531 17.375L132.531 17.445Q132.531 17.855 132.652 18.211Q132.773 18.566 133.045 18.783Q133.316 19 133.746 19Q134.109 19 134.406 18.771Q134.703 18.543 134.804 18.191Q134.812 18.144 134.898 18.129L134.980 18.129Q135.074 18.156 135.074 18.238Q135.074 18.246 135.066 18.277Q135.004 18.504 134.865 18.687Q134.726 18.871 134.535 19.004Q134.343 19.137 134.125 19.207Q133.906 19.277 133.668 19.277Q133.297 19.277 132.959 19.140Q132.621 19.004 132.353 18.752Q132.086 18.500 131.935 18.160Q131.785 17.820 131.785 17.445M132.539 17.137L134.500 17.137Q134.500 16.832 134.398 16.541Q134.297 16.250 134.080 16.068Q133.863 15.887 133.547 15.887Q133.246 15.887 133.015 16.074Q132.785 16.262 132.662 16.553Q132.539 16.844 132.539 17.137\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"m-3.016-28.97 63.674 77.465\"\u002F>\u003Cpath stroke=\"none\" d=\"m61.928 50.04-.796-3.488-.474 1.943-1.997.089\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M1.594 2.327 56.21 48.744\"\u002F>\u003Cpath stroke=\"none\" d=\"m57.734 50.04-1.402-3.292-.121 1.996-1.95.442\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M13.198 33.625 45.372 49.17\"\u002F>\u003Cpath stroke=\"none\" d=\"m47.172 50.04-2.185-2.833.385 1.963-1.777.918\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cpath fill=\"none\" d=\"M146.332-28.97 82.657 48.494\"\u002F>\u003Cpath stroke=\"none\" d=\"m81.387 50.04 3.268-1.456-1.998-.09-.474-1.942\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cpath fill=\"none\" d=\"M141.722 2.327 87.105 48.744\"\u002F>\u003Cpath stroke=\"none\" d=\"m85.581 50.04 3.475-.854-1.95-.442-.123-1.996\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">Five levers on off-policy variance. Some cut the variance at its source (weighted IS, tree-backup, constrained targets); others absorb it after the fact (momentum\u002Faveraging, adaptive step sizes). They are complementary and can be combined.\u003C\u002Ffigcaption>",1785117790071]