[{"data":1,"prerenderedAt":23497},["ShallowReactive",2],{"lesson:\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces":3,"course-wordcounts":17713,"ref-card-index":18625,"nav:reinforcement-learning":23310,"tikz:c67d14c9d8c46168d50c7733e51a247b825199092766c585dd3270254372e121":23491,"tikz:fcfde34023cf4df69cb83817bae4904b9f5c9c74232ac9c68883650811acd6c0":23492,"tikz:f2c9601fc1e7ddc99af75ebc80b986d2cc125b05a0574b5f180a1bafee2bbd91":23493,"tikz:c563d4b33f0957098a59629f0a669ca61616fa3df664122e260cdcba705ca6e5":23494,"tikz:60fc2f3c6edd0292d65c0108af959e2364e0ea28fbbe80060b4543ca4710f58c":23495,"tikz:8151b32aed0919140db9d01c8d7224dd476915e132332fa82a4f3e99bef4bdf5":23496},{"id":4,"title":5,"blurb":6,"body":7,"brief":17682,"category":17683,"description":17684,"draft":17685,"extension":17686,"meta":17687,"module":17691,"navigation":17692,"path":17693,"practice":17694,"rawbody":17695,"readingTime":17696,"seo":17701,"sources":17702,"status":17707,"stem":17708,"summary":17709,"topics":17710,"__hash__":17712},"course\u002F07.reinforcement-learning\u002F03.approximation\u002F13.off-policy-eligibility-traces.md","Off-Policy Eligibility Traces","",{"type":8,"value":9,"toc":17674},"minimark",[10,202,341,346,844,1397,1650,1654,1697,2214,3189,3413,3610,3665,4247,4471,5097,5100,5953,6140,6144,6228,6569,6641,7456,7919,7922,9267,9270,9751,9754,10352,10517,11579,11601,12006,12391,12394,12413,12865,12868,13392,13678,13682,13734,13737,13756,13837,14655,14658,15171,15687,16466,16469,16473,16624,16665],[11,12,13,14,19,20,45,46,118,119,134,135,150,151,166,167,182,183,201],"p",{},"The ",[15,16,18],"a",{"href":17},"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces","eligibility-traces lesson","\ncompleted the on-policy case: the ",[21,22,25],"span",{"className":23},[24],"katex",[21,26,30],{"className":27,"ariaHidden":29},[28],"katex-html","true",[21,31,34,39],{"className":32},[33],"base",[21,35],{"className":36,"style":38},[37],"strut","height:0.6944em;",[21,40,44],{"className":41},[42,43],"mord","mathnormal","λ","-return as a forward\nview, the trace vector ",[21,47,49],{"className":48},[24],[21,50,52],{"className":51,"ariaHidden":29},[28],[21,53,55,59],{"className":54},[33],[21,56],{"className":57,"style":58},[37],"height:0.5944em;vertical-align:-0.15em;",[21,60,62,67],{"className":61},[42],[21,63,66],{"className":64},[42,65],"mathbf","z",[21,68,71],{"className":69},[70],"msupsub",[21,72,76,109],{"className":73},[74,75],"vlist-t","vlist-t2",[21,77,80,104],{"className":78},[79],"vlist-r",[21,81,85],{"className":82,"style":84},[83],"vlist","height:0.2806em;",[21,86,88,93],{"style":87},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[21,89],{"className":90,"style":92},[91],"pstrut","height:2.7em;",[21,94,100],{"className":95},[96,97,98,99],"sizing","reset-size6","size3","mtight",[21,101,103],{"className":102},[42,43,99],"t",[21,105,108],{"className":106},[107],"vlist-s","​",[21,110,112],{"className":111},[79],[21,113,116],{"className":114,"style":115},[83],"height:0.15em;",[21,117],{}," as a backward view, TD(",[21,120,122],{"className":121},[24],[21,123,125],{"className":124,"ariaHidden":29},[28],[21,126,128,131],{"className":127},[33],[21,129],{"className":130,"style":38},[37],[21,132,44],{"className":133},[42,43],") making\nthe two nearly agree, true online TD(",[21,136,138],{"className":137},[24],[21,139,141],{"className":140,"ariaHidden":29},[28],[21,142,144,147],{"className":143},[33],[21,145],{"className":146,"style":38},[37],[21,148,44],{"className":149},[42,43],") making them agree exactly, and\nSarsa(",[21,152,154],{"className":153},[24],[21,155,157],{"className":156,"ariaHidden":29},[28],[21,158,160,163],{"className":159},[33],[21,161],{"className":162,"style":38},[37],[21,164,44],{"className":165},[42,43],") lifting the whole apparatus to on-policy control. Throughout,\n",[21,168,170],{"className":169},[24],[21,171,173],{"className":172,"ariaHidden":29},[28],[21,174,176,179],{"className":175},[33],[21,177],{"className":178,"style":38},[37],[21,180,44],{"className":181},[42,43]," and ",[21,184,186],{"className":185},[24],[21,187,189],{"className":188,"ariaHidden":29},[28],[21,190,192,196],{"className":191},[33],[21,193],{"className":194,"style":195},[37],"height:0.625em;vertical-align:-0.1944em;",[21,197,200],{"className":198,"style":199},[42,43],"margin-right:0.0556em;","γ"," were fixed constants and every step was generated by the\npolicy being evaluated.",[11,203,204,205,182,220,235,236,240,241,259,260,276,277,330,331,335,336,340],{},"This lesson removes both restrictions. First ",[21,206,208],{"className":207},[24],[21,209,211],{"className":210,"ariaHidden":29},[28],[21,212,214,217],{"className":213},[33],[21,215],{"className":216,"style":38},[37],[21,218,44],{"className":219},[42,43],[21,221,223],{"className":222},[24],[21,224,226],{"className":225,"ariaHidden":29},[28],[21,227,229,232],{"className":228},[33],[21,230],{"className":231,"style":195},[37],[21,233,200],{"className":234,"style":199},[42,43]," become\n",[237,238,239],"em",{},"functions of state and action",", which turns out to unify the episodic and\ncontinuing settings and absorb termination into the discount. Then we learn about\na target policy ",[21,242,244],{"className":243},[24],[21,245,247],{"className":246,"ariaHidden":29},[28],[21,248,250,254],{"className":249},[33],[21,251],{"className":252,"style":253},[37],"height:0.4306em;",[21,255,258],{"className":256,"style":257},[42,43],"margin-right:0.0359em;","π"," from behavior policy ",[21,261,263],{"className":262},[24],[21,264,266],{"className":265,"ariaHidden":29},[28],[21,267,269,272],{"className":268},[33],[21,270],{"className":271,"style":38},[37],[21,273,275],{"className":274},[42,43],"b",", which brings the per-decision\nimportance ratio ",[21,278,280],{"className":279},[24],[21,281,283],{"className":282,"ariaHidden":29},[28],[21,284,286,289],{"className":285},[33],[21,287],{"className":288,"style":195},[37],[21,290,292,296],{"className":291},[42],[21,293,295],{"className":294},[42,43],"ρ",[21,297,299],{"className":298},[70],[21,300,302,322],{"className":301},[74,75],[21,303,305,319],{"className":304},[79],[21,306,308],{"className":307,"style":84},[83],[21,309,310,313],{"style":87},[21,311],{"className":312,"style":92},[91],[21,314,316],{"className":315},[96,97,98,99],[21,317,103],{"className":318},[42,43,99],[21,320,108],{"className":321},[107],[21,323,325],{"className":324},[79],[21,326,328],{"className":327,"style":115},[83],[21,329],{}," into the trace, with a control variate to limit the\nvariance. The combination — traces, off-policy data, and a parameterized value\nfunction — is the\n",[15,332,334],{"href":333},"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad","deadly triad","\nitself, and convergence is no longer guaranteed. Naming the methods that restore\nstability is the job of the\n",[15,337,339],{"href":338},"\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces","next lesson","; this\none builds the returns and traces they all rest on.",[342,343,345],"h2",{"id":344},"variable-λ-and-γ","Variable λ and γ",[11,347,348,349,359,360,467,468,636,637,703,704,822,823,827,828,843],{},"To state the final algorithms in their most general form, promote the two\nconstants to functions.",[350,351,352],"sup",{},[15,353,358],{"href":354,"ariaDescribedBy":355,"dataFootnoteRef":6,"id":357},"#user-content-fn-sb-varlg",[356],"footnote-label","user-content-fnref-sb-varlg","1"," Let ",[21,361,363],{"className":362},[24],[21,364,366,388,413,434],{"className":365,"ariaHidden":29},[28],[21,367,369,372,375,380,385],{"className":368},[33],[21,370],{"className":371,"style":38},[37],[21,373,44],{"className":374},[42,43],[21,376],{"className":377,"style":379},[378],"mspace","margin-right:0.2778em;",[21,381,384],{"className":382},[383],"mrel",":",[21,386],{"className":387,"style":379},[378],[21,389,391,395,401,405,410],{"className":390},[33],[21,392],{"className":393,"style":394},[37],"height:0.7667em;vertical-align:-0.0833em;",[21,396,400],{"className":397,"style":399},[42,398],"mathcal","margin-right:0.075em;","S",[21,402],{"className":403,"style":404},[378],"margin-right:0.2222em;",[21,406,409],{"className":407},[408],"mbin","×",[21,411],{"className":412,"style":404},[378],[21,414,416,420,424,427,431],{"className":415},[33],[21,417],{"className":418,"style":419},[37],"height:0.6833em;",[21,421,423],{"className":422},[42,398],"A",[21,425],{"className":426,"style":379},[378],[21,428,430],{"className":429},[383],"→",[21,432],{"className":433,"style":379},[378],[21,435,437,441,446,450,455,459,462],{"className":436},[33],[21,438],{"className":439,"style":440},[37],"height:1em;vertical-align:-0.25em;",[21,442,445],{"className":443},[444],"mopen","[",[21,447,449],{"className":448},[42],"0",[21,451,454],{"className":452},[453],"mpunct",",",[21,456],{"className":457,"style":458},[378],"margin-right:0.1667em;",[21,460,358],{"className":461},[42],[21,463,466],{"className":464},[465],"mclose","]"," depend on state and action, ",[21,469,471],{"className":470},[24],[21,472,474,531],{"className":473,"ariaHidden":29},[28],[21,475,477,481,521,524,528],{"className":476},[33],[21,478],{"className":479,"style":480},[37],"height:0.8444em;vertical-align:-0.15em;",[21,482,484,487],{"className":483},[42],[21,485,44],{"className":486},[42,43],[21,488,490],{"className":489},[70],[21,491,493,513],{"className":492},[74,75],[21,494,496,510],{"className":495},[79],[21,497,499],{"className":498,"style":84},[83],[21,500,501,504],{"style":87},[21,502],{"className":503,"style":92},[91],[21,505,507],{"className":506},[96,97,98,99],[21,508,103],{"className":509},[42,43,99],[21,511,108],{"className":512},[107],[21,514,516],{"className":515},[79],[21,517,519],{"className":518,"style":115},[83],[21,520],{},[21,522],{"className":523,"style":379},[378],[21,525,527],{"className":526},[383],"≐",[21,529],{"className":530,"style":379},[378],[21,532,534,537,540,544,586,589,592,632],{"className":533},[33],[21,535],{"className":536,"style":440},[37],[21,538,44],{"className":539},[42,43],[21,541,543],{"className":542},[444],"(",[21,545,547,551],{"className":546},[42],[21,548,400],{"className":549,"style":550},[42,43],"margin-right:0.0576em;",[21,552,554],{"className":553},[70],[21,555,557,578],{"className":556},[74,75],[21,558,560,575],{"className":559},[79],[21,561,563],{"className":562,"style":84},[83],[21,564,566,569],{"style":565},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[21,567],{"className":568,"style":92},[91],[21,570,572],{"className":571},[96,97,98,99],[21,573,103],{"className":574},[42,43,99],[21,576,108],{"className":577},[107],[21,579,581],{"className":580},[79],[21,582,584],{"className":583,"style":115},[83],[21,585],{},[21,587,454],{"className":588},[453],[21,590],{"className":591,"style":458},[378],[21,593,595,598],{"className":594},[42],[21,596,423],{"className":597},[42,43],[21,599,601],{"className":600},[70],[21,602,604,624],{"className":603},[74,75],[21,605,607,621],{"className":606},[79],[21,608,610],{"className":609,"style":84},[83],[21,611,612,615],{"style":87},[21,613],{"className":614,"style":92},[91],[21,616,618],{"className":617},[96,97,98,99],[21,619,103],{"className":620},[42,43,99],[21,622,108],{"className":623},[107],[21,625,627],{"className":626},[79],[21,628,630],{"className":629,"style":115},[83],[21,631],{},[21,633,635],{"className":634},[465],")",", and\nlet ",[21,638,640],{"className":639},[24],[21,641,643,661,679],{"className":642,"ariaHidden":29},[28],[21,644,646,649,652,655,658],{"className":645},[33],[21,647],{"className":648,"style":195},[37],[21,650,200],{"className":651,"style":199},[42,43],[21,653],{"className":654,"style":379},[378],[21,656,384],{"className":657},[383],[21,659],{"className":660,"style":379},[378],[21,662,664,667,670,673,676],{"className":663},[33],[21,665],{"className":666,"style":419},[37],[21,668,400],{"className":669,"style":399},[42,398],[21,671],{"className":672,"style":379},[378],[21,674,430],{"className":675},[383],[21,677],{"className":678,"style":379},[378],[21,680,682,685,688,691,694,697,700],{"className":681},[33],[21,683],{"className":684,"style":440},[37],[21,686,445],{"className":687},[444],[21,689,449],{"className":690},[42],[21,692,454],{"className":693},[453],[21,695],{"className":696,"style":458},[378],[21,698,358],{"className":699},[42],[21,701,466],{"className":702},[465]," depend on state, ",[21,705,707],{"className":706},[24],[21,708,710,767],{"className":709,"ariaHidden":29},[28],[21,711,713,717,758,761,764],{"className":712},[33],[21,714],{"className":715,"style":716},[37],"height:0.8674em;vertical-align:-0.1944em;",[21,718,720,723],{"className":719},[42],[21,721,200],{"className":722,"style":199},[42,43],[21,724,726],{"className":725},[70],[21,727,729,750],{"className":728},[74,75],[21,730,732,747],{"className":731},[79],[21,733,735],{"className":734,"style":84},[83],[21,736,738,741],{"style":737},"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;",[21,739],{"className":740,"style":92},[91],[21,742,744],{"className":743},[96,97,98,99],[21,745,103],{"className":746},[42,43,99],[21,748,108],{"className":749},[107],[21,751,753],{"className":752},[79],[21,754,756],{"className":755,"style":115},[83],[21,757],{},[21,759],{"className":760,"style":379},[378],[21,762,527],{"className":763},[383],[21,765],{"className":766,"style":379},[378],[21,768,770,773,776,779,819],{"className":769},[33],[21,771],{"className":772,"style":440},[37],[21,774,200],{"className":775,"style":199},[42,43],[21,777,543],{"className":778},[444],[21,780,782,785],{"className":781},[42],[21,783,400],{"className":784,"style":550},[42,43],[21,786,788],{"className":787},[70],[21,789,791,811],{"className":790},[74,75],[21,792,794,808],{"className":793},[79],[21,795,797],{"className":796,"style":84},[83],[21,798,799,802],{"style":565},[21,800],{"className":801,"style":92},[91],[21,803,805],{"className":804},[96,97,98,99],[21,806,103],{"className":807},[42,43,99],[21,809,108],{"className":810},[107],[21,812,814],{"className":813},[79],[21,815,817],{"className":816,"style":115},[83],[21,818],{},[21,820,635],{"className":821},[465],". The ",[824,825,826],"strong",{},"termination function"," ",[21,829,831],{"className":830},[24],[21,832,834],{"className":833,"ariaHidden":29},[28],[21,835,837,840],{"className":836},[33],[21,838],{"className":839,"style":195},[37],[21,841,200],{"className":842,"style":199},[42,43]," is the significant one,\nbecause it changes the return itself — the random variable whose expectation we\nare estimating. The return is now defined by a running product of per-step\ndiscounts,",[21,845,848],{"className":846},[847],"katex-display",[21,849,851],{"className":850},[24],[21,852,854,917,988,1108],{"className":853,"ariaHidden":29},[28],[21,855,857,861,902,905,908,911,914],{"className":856},[33],[21,858],{"className":859,"style":860},[37],"height:0.8333em;vertical-align:-0.15em;",[21,862,864,868],{"className":863},[42],[21,865,867],{"className":866},[42,43],"G",[21,869,871],{"className":870},[70],[21,872,874,894],{"className":873},[74,75],[21,875,877,891],{"className":876},[79],[21,878,880],{"className":879,"style":84},[83],[21,881,882,885],{"style":87},[21,883],{"className":884,"style":92},[91],[21,886,888],{"className":887},[96,97,98,99],[21,889,103],{"className":890},[42,43,99],[21,892,108],{"className":893},[107],[21,895,897],{"className":896},[79],[21,898,900],{"className":899,"style":115},[83],[21,901],{},[21,903],{"className":904,"style":379},[378],[21,906],{"className":907,"style":379},[378],[21,909,527],{"className":910},[383],[21,912],{"className":913,"style":379},[378],[21,915],{"className":916,"style":379},[378],[21,918,920,924,979,982,985],{"className":919},[33],[21,921],{"className":922,"style":923},[37],"height:0.8917em;vertical-align:-0.2083em;",[21,925,927,932],{"className":926},[42],[21,928,931],{"className":929,"style":930},[42,43],"margin-right:0.0077em;","R",[21,933,935],{"className":934},[70],[21,936,938,970],{"className":937},[74,75],[21,939,941,967],{"className":940},[79],[21,942,945],{"className":943,"style":944},[83],"height:0.3011em;",[21,946,948,951],{"style":947},"top:-2.55em;margin-left:-0.0077em;margin-right:0.05em;",[21,949],{"className":950,"style":92},[91],[21,952,954],{"className":953},[96,97,98,99],[21,955,957,960,964],{"className":956},[42,99],[21,958,103],{"className":959},[42,43,99],[21,961,963],{"className":962},[408,99],"+",[21,965,358],{"className":966},[42,99],[21,968,108],{"className":969},[107],[21,971,973],{"className":972},[79],[21,974,977],{"className":975,"style":976},[83],"height:0.2083em;",[21,978],{},[21,980],{"className":981,"style":404},[378],[21,983,963],{"className":984},[408],[21,986],{"className":987,"style":404},[378],[21,989,991,994,1043,1092,1095,1098,1102,1105],{"className":990},[33],[21,992],{"className":993,"style":923},[37],[21,995,997,1000],{"className":996},[42],[21,998,200],{"className":999,"style":199},[42,43],[21,1001,1003],{"className":1002},[70],[21,1004,1006,1035],{"className":1005},[74,75],[21,1007,1009,1032],{"className":1008},[79],[21,1010,1012],{"className":1011,"style":944},[83],[21,1013,1014,1017],{"style":737},[21,1015],{"className":1016,"style":92},[91],[21,1018,1020],{"className":1019},[96,97,98,99],[21,1021,1023,1026,1029],{"className":1022},[42,99],[21,1024,103],{"className":1025},[42,43,99],[21,1027,963],{"className":1028},[408,99],[21,1030,358],{"className":1031},[42,99],[21,1033,108],{"className":1034},[107],[21,1036,1038],{"className":1037},[79],[21,1039,1041],{"className":1040,"style":976},[83],[21,1042],{},[21,1044,1046,1049],{"className":1045},[42],[21,1047,867],{"className":1048},[42,43],[21,1050,1052],{"className":1051},[70],[21,1053,1055,1084],{"className":1054},[74,75],[21,1056,1058,1081],{"className":1057},[79],[21,1059,1061],{"className":1060,"style":944},[83],[21,1062,1063,1066],{"style":87},[21,1064],{"className":1065,"style":92},[91],[21,1067,1069],{"className":1068},[96,97,98,99],[21,1070,1072,1075,1078],{"className":1071},[42,99],[21,1073,103],{"className":1074},[42,43,99],[21,1076,963],{"className":1077},[408,99],[21,1079,358],{"className":1080},[42,99],[21,1082,108],{"className":1083},[107],[21,1085,1087],{"className":1086},[79],[21,1088,1090],{"className":1089,"style":976},[83],[21,1091],{},[21,1093],{"className":1094,"style":379},[378],[21,1096],{"className":1097,"style":379},[378],[21,1099,1101],{"className":1100},[383],"=",[21,1103],{"className":1104,"style":379},[378],[21,1106],{"className":1107,"style":379},[378],[21,1109,1111,1115,1196,1199,1341,1344,1394],{"className":1110},[33],[21,1112],{"className":1113,"style":1114},[37],"height:3.1721em;vertical-align:-1.336em;",[21,1116,1120],{"className":1117},[1118,1119],"mop","op-limits",[21,1121,1123,1187],{"className":1122},[74,75],[21,1124,1126,1184],{"className":1125},[79],[21,1127,1130,1154,1168],{"className":1128,"style":1129},[83],"height:1.6514em;",[21,1131,1133,1137],{"style":1132},"top:-1.8479em;margin-left:0em;",[21,1134],{"className":1135,"style":1136},[91],"height:3.05em;",[21,1138,1140],{"className":1139},[96,97,98,99],[21,1141,1143,1148,1151],{"className":1142},[42,99],[21,1144,1147],{"className":1145,"style":1146},[42,43,99],"margin-right:0.0315em;","k",[21,1149,1101],{"className":1150},[383,99],[21,1152,103],{"className":1153},[42,43,99],[21,1155,1157,1160],{"style":1156},"top:-3.05em;",[21,1158],{"className":1159,"style":1136},[91],[21,1161,1162],{},[21,1163,1167],{"className":1164},[1118,1165,1166],"op-symbol","large-op","∑",[21,1169,1171,1174],{"style":1170},"top:-4.3em;margin-left:0em;",[21,1172],{"className":1173,"style":1136},[91],[21,1175,1177],{"className":1176},[96,97,98,99],[21,1178,1180],{"className":1179},[42,99],[21,1181,1183],{"className":1182},[42,99],"∞",[21,1185,108],{"className":1186},[107],[21,1188,1190],{"className":1189},[79],[21,1191,1194],{"className":1192,"style":1193},[83],"height:1.3021em;",[21,1195],{},[21,1197],{"className":1198,"style":458},[378],[21,1200,1203,1213,1291,1294,1335],{"className":1201},[1202],"minner",[21,1204,1208],{"className":1205,"style":1207},[444,1206],"delimcenter","top:0em;",[21,1209,543],{"className":1210},[1211,1212],"delimsizing","size4",[21,1214,1216],{"className":1215},[1118,1119],[21,1217,1219,1282],{"className":1218},[74,75],[21,1220,1222,1279],{"className":1221},[79],[21,1223,1226,1254,1265],{"className":1224,"style":1225},[83],"height:1.8361em;",[21,1227,1229,1232],{"style":1228},"top:-1.8723em;margin-left:0em;",[21,1230],{"className":1231,"style":1136},[91],[21,1233,1235],{"className":1234},[96,97,98,99],[21,1236,1238,1242,1245,1248,1251],{"className":1237},[42,99],[21,1239,1241],{"className":1240},[42,43,99],"i",[21,1243,1101],{"className":1244},[383,99],[21,1246,103],{"className":1247},[42,43,99],[21,1249,963],{"className":1250},[408,99],[21,1252,358],{"className":1253},[42,99],[21,1255,1256,1259],{"style":1156},[21,1257],{"className":1258,"style":1136},[91],[21,1260,1261],{},[21,1262,1264],{"className":1263},[1118,1165,1166],"∏",[21,1266,1267,1270],{"style":1170},[21,1268],{"className":1269,"style":1136},[91],[21,1271,1273],{"className":1272},[96,97,98,99],[21,1274,1276],{"className":1275},[42,99],[21,1277,1147],{"className":1278,"style":1146},[42,43,99],[21,1280,108],{"className":1281},[107],[21,1283,1285],{"className":1284},[79],[21,1286,1289],{"className":1287,"style":1288},[83],"height:1.336em;",[21,1290],{},[21,1292],{"className":1293,"style":458},[378],[21,1295,1297,1300],{"className":1296},[42],[21,1298,200],{"className":1299,"style":199},[42,43],[21,1301,1303],{"className":1302},[70],[21,1304,1306,1327],{"className":1305},[74,75],[21,1307,1309,1324],{"className":1308},[79],[21,1310,1313],{"className":1311,"style":1312},[83],"height:0.3117em;",[21,1314,1315,1318],{"style":737},[21,1316],{"className":1317,"style":92},[91],[21,1319,1321],{"className":1320},[96,97,98,99],[21,1322,1241],{"className":1323},[42,43,99],[21,1325,108],{"className":1326},[107],[21,1328,1330],{"className":1329},[79],[21,1331,1333],{"className":1332,"style":115},[83],[21,1334],{},[21,1336,1338],{"className":1337,"style":1207},[465,1206],[21,1339,635],{"className":1340},[1211,1212],[21,1342],{"className":1343,"style":458},[378],[21,1345,1347,1350],{"className":1346},[42],[21,1348,931],{"className":1349,"style":930},[42,43],[21,1351,1353],{"className":1352},[70],[21,1354,1356,1386],{"className":1355},[74,75],[21,1357,1359,1383],{"className":1358},[79],[21,1360,1363],{"className":1361,"style":1362},[83],"height:0.3361em;",[21,1364,1365,1368],{"style":947},[21,1366],{"className":1367,"style":92},[91],[21,1369,1371],{"className":1370},[96,97,98,99],[21,1372,1374,1377,1380],{"className":1373},[42,99],[21,1375,1147],{"className":1376,"style":1146},[42,43,99],[21,1378,963],{"className":1379},[408,99],[21,1381,358],{"className":1382},[42,99],[21,1384,108],{"className":1385},[107],[21,1387,1389],{"className":1388},[79],[21,1390,1392],{"className":1391,"style":976},[83],[21,1393],{},[21,1395,454],{"className":1396},[453],[11,1398,1399,1400,1544,1545,1561,1562,1605,1606,1649],{},"where we require ",[21,1401,1403],{"className":1402},[24],[21,1404,1406,1534],{"className":1405,"ariaHidden":29},[28],[21,1407,1409,1413,1482,1485,1525,1528,1531],{"className":1408},[33],[21,1410],{"className":1411,"style":1412},[37],"height:1.104em;vertical-align:-0.2997em;",[21,1414,1416,1421],{"className":1415},[1118],[21,1417,1264],{"className":1418,"style":1420},[1118,1165,1419],"small-op","position:relative;top:0em;",[21,1422,1424],{"className":1423},[70],[21,1425,1427,1473],{"className":1426},[74,75],[21,1428,1430,1470],{"className":1429},[79],[21,1431,1434,1455],{"className":1432,"style":1433},[83],"height:0.8043em;",[21,1435,1437,1440],{"style":1436},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[21,1438],{"className":1439,"style":92},[91],[21,1441,1443],{"className":1442},[96,97,98,99],[21,1444,1446,1449,1452],{"className":1445},[42,99],[21,1447,1147],{"className":1448,"style":1146},[42,43,99],[21,1450,1101],{"className":1451},[383,99],[21,1453,103],{"className":1454},[42,43,99],[21,1456,1458,1461],{"style":1457},"top:-3.2029em;margin-right:0.05em;",[21,1459],{"className":1460,"style":92},[91],[21,1462,1464],{"className":1463},[96,97,98,99],[21,1465,1467],{"className":1466},[42,99],[21,1468,1183],{"className":1469},[42,99],[21,1471,108],{"className":1472},[107],[21,1474,1476],{"className":1475},[79],[21,1477,1480],{"className":1478,"style":1479},[83],"height:0.2997em;",[21,1481],{},[21,1483],{"className":1484,"style":458},[378],[21,1486,1488,1491],{"className":1487},[42],[21,1489,200],{"className":1490,"style":199},[42,43],[21,1492,1494],{"className":1493},[70],[21,1495,1497,1517],{"className":1496},[74,75],[21,1498,1500,1514],{"className":1499},[79],[21,1501,1503],{"className":1502,"style":1362},[83],[21,1504,1505,1508],{"style":737},[21,1506],{"className":1507,"style":92},[91],[21,1509,1511],{"className":1510},[96,97,98,99],[21,1512,1147],{"className":1513,"style":1146},[42,43,99],[21,1515,108],{"className":1516},[107],[21,1518,1520],{"className":1519},[79],[21,1521,1523],{"className":1522,"style":115},[83],[21,1524],{},[21,1526],{"className":1527,"style":379},[378],[21,1529,1101],{"className":1530},[383],[21,1532],{"className":1533,"style":379},[378],[21,1535,1537,1541],{"className":1536},[33],[21,1538],{"className":1539,"style":1540},[37],"height:0.6444em;",[21,1542,449],{"className":1543},[42]," with probability one for\nevery ",[21,1546,1548],{"className":1547},[24],[21,1549,1551],{"className":1550,"ariaHidden":29},[28],[21,1552,1554,1558],{"className":1553},[33],[21,1555],{"className":1556,"style":1557},[37],"height:0.6151em;",[21,1559,103],{"className":1560},[42,43],", so the sums are finite. This single definition lets the episodic\nsetting be presented as one uninterrupted stream of\nexperience — no special terminal states, no start distribution, no explicit\ntermination time. A former terminal state becomes an ordinary state at which\n",[21,1563,1565],{"className":1564},[24],[21,1566,1568,1596],{"className":1567,"ariaHidden":29},[28],[21,1569,1571,1574,1577,1580,1584,1587,1590,1593],{"className":1570},[33],[21,1572],{"className":1573,"style":440},[37],[21,1575,200],{"className":1576,"style":199},[42,43],[21,1578,543],{"className":1579},[444],[21,1581,1583],{"className":1582},[42,43],"s",[21,1585,635],{"className":1586},[465],[21,1588],{"className":1589,"style":379},[378],[21,1591,1101],{"className":1592},[383],[21,1594],{"className":1595,"style":379},[378],[21,1597,1599,1602],{"className":1598},[33],[21,1600],{"className":1601,"style":1540},[37],[21,1603,449],{"className":1604},[42]," and which transitions to the start distribution; setting\n",[21,1607,1609],{"className":1608},[24],[21,1610,1612,1640],{"className":1611,"ariaHidden":29},[28],[21,1613,1615,1618,1621,1624,1628,1631,1634,1637],{"className":1614},[33],[21,1616],{"className":1617,"style":440},[37],[21,1619,200],{"className":1620,"style":199},[42,43],[21,1622,543],{"className":1623},[444],[21,1625,1627],{"className":1626},[42],"⋅",[21,1629,635],{"className":1630},[465],[21,1632],{"className":1633,"style":379},[378],[21,1635,1101],{"className":1636},[383],[21,1638],{"className":1639,"style":379},[378],[21,1641,1643,1646],{"className":1642},[33],[21,1644],{"className":1645,"style":1540},[37],[21,1647,358],{"className":1648},[42]," everywhere else recovers the classical episodic case exactly.",[1651,1652],"tikz-figure",{"hash":1653},"c67d14c9d8c46168d50c7733e51a247b825199092766c585dd3270254372e121",[11,1655,1656,1657,1660,1661,1696],{},"State-dependent termination also reaches beyond ordinary episodes to ",[237,1658,1659],{},"pseudo\ntermination",": predicting some quantity as if the stream ended here, without\nactually interrupting the Markov process. A discounted return is one such\nquantity, so this one construction unifies episodic, discounted-continuing, and\npseudo-terminating prediction. (The undiscounted-continuing case, ",[21,1662,1664],{"className":1663},[24],[21,1665,1667,1687],{"className":1666,"ariaHidden":29},[28],[21,1668,1670,1674,1677,1680,1684],{"className":1669},[33],[21,1671],{"className":1672,"style":1673},[37],"height:0.6582em;vertical-align:-0.1944em;",[21,1675,200],{"className":1676,"style":199},[42,43],[21,1678],{"className":1679,"style":379},[378],[21,1681,1683],{"className":1682},[383],"≡",[21,1685],{"className":1686,"style":379},[378],[21,1688,1690,1693],{"className":1689},[33],[21,1691],{"className":1692,"style":1540},[37],[21,1694,358],{"className":1695},[42]," with no termination, still needs its own treatment via average reward.)",[1698,1699,1701],"callout",{"type":1700},"definition",[11,1702,1703,1706,1707,1773,1774,1890,1891,2111,2112,2170,2171,2213],{},[824,1704,1705],{},"Definition (Termination function)."," A map ",[21,1708,1710],{"className":1709},[24],[21,1711,1713,1731,1749],{"className":1712,"ariaHidden":29},[28],[21,1714,1716,1719,1722,1725,1728],{"className":1715},[33],[21,1717],{"className":1718,"style":195},[37],[21,1720,200],{"className":1721,"style":199},[42,43],[21,1723],{"className":1724,"style":379},[378],[21,1726,384],{"className":1727},[383],[21,1729],{"className":1730,"style":379},[378],[21,1732,1734,1737,1740,1743,1746],{"className":1733},[33],[21,1735],{"className":1736,"style":419},[37],[21,1738,400],{"className":1739,"style":399},[42,398],[21,1741],{"className":1742,"style":379},[378],[21,1744,430],{"className":1745},[383],[21,1747],{"className":1748,"style":379},[378],[21,1750,1752,1755,1758,1761,1764,1767,1770],{"className":1751},[33],[21,1753],{"className":1754,"style":440},[37],[21,1756,445],{"className":1757},[444],[21,1759,449],{"className":1760},[42],[21,1762,454],{"className":1763},[453],[21,1765],{"className":1766,"style":458},[378],[21,1768,358],{"className":1769},[42],[21,1771,466],{"className":1772},[465],"\nwith ",[21,1775,1777],{"className":1776},[24],[21,1778,1780,1835],{"className":1779,"ariaHidden":29},[28],[21,1781,1783,1786,1826,1829,1832],{"className":1782},[33],[21,1784],{"className":1785,"style":716},[37],[21,1787,1789,1792],{"className":1788},[42],[21,1790,200],{"className":1791,"style":199},[42,43],[21,1793,1795],{"className":1794},[70],[21,1796,1798,1818],{"className":1797},[74,75],[21,1799,1801,1815],{"className":1800},[79],[21,1802,1804],{"className":1803,"style":84},[83],[21,1805,1806,1809],{"style":737},[21,1807],{"className":1808,"style":92},[91],[21,1810,1812],{"className":1811},[96,97,98,99],[21,1813,103],{"className":1814},[42,43,99],[21,1816,108],{"className":1817},[107],[21,1819,1821],{"className":1820},[79],[21,1822,1824],{"className":1823,"style":115},[83],[21,1825],{},[21,1827],{"className":1828,"style":379},[378],[21,1830,527],{"className":1831},[383],[21,1833],{"className":1834,"style":379},[378],[21,1836,1838,1841,1844,1847,1887],{"className":1837},[33],[21,1839],{"className":1840,"style":440},[37],[21,1842,200],{"className":1843,"style":199},[42,43],[21,1845,543],{"className":1846},[444],[21,1848,1850,1853],{"className":1849},[42],[21,1851,400],{"className":1852,"style":550},[42,43],[21,1854,1856],{"className":1855},[70],[21,1857,1859,1879],{"className":1858},[74,75],[21,1860,1862,1876],{"className":1861},[79],[21,1863,1865],{"className":1864,"style":84},[83],[21,1866,1867,1870],{"style":565},[21,1868],{"className":1869,"style":92},[91],[21,1871,1873],{"className":1872},[96,97,98,99],[21,1874,103],{"className":1875},[42,43,99],[21,1877,108],{"className":1878},[107],[21,1880,1882],{"className":1881},[79],[21,1883,1885],{"className":1884,"style":115},[83],[21,1886],{},[21,1888,635],{"className":1889},[465]," that sets the per-step continuation\nprobability. The return ",[21,1892,1894],{"className":1893},[24],[21,1895,1897,1952,2016],{"className":1896,"ariaHidden":29},[28],[21,1898,1900,1903,1943,1946,1949],{"className":1899},[33],[21,1901],{"className":1902,"style":860},[37],[21,1904,1906,1909],{"className":1905},[42],[21,1907,867],{"className":1908},[42,43],[21,1910,1912],{"className":1911},[70],[21,1913,1915,1935],{"className":1914},[74,75],[21,1916,1918,1932],{"className":1917},[79],[21,1919,1921],{"className":1920,"style":84},[83],[21,1922,1923,1926],{"style":87},[21,1924],{"className":1925,"style":92},[91],[21,1927,1929],{"className":1928},[96,97,98,99],[21,1930,103],{"className":1931},[42,43,99],[21,1933,108],{"className":1934},[107],[21,1936,1938],{"className":1937},[79],[21,1939,1941],{"className":1940,"style":115},[83],[21,1942],{},[21,1944],{"className":1945,"style":379},[378],[21,1947,1101],{"className":1948},[383],[21,1950],{"className":1951,"style":379},[378],[21,1953,1955,1958,2007,2010,2013],{"className":1954},[33],[21,1956],{"className":1957,"style":923},[37],[21,1959,1961,1964],{"className":1960},[42],[21,1962,931],{"className":1963,"style":930},[42,43],[21,1965,1967],{"className":1966},[70],[21,1968,1970,1999],{"className":1969},[74,75],[21,1971,1973,1996],{"className":1972},[79],[21,1974,1976],{"className":1975,"style":944},[83],[21,1977,1978,1981],{"style":947},[21,1979],{"className":1980,"style":92},[91],[21,1982,1984],{"className":1983},[96,97,98,99],[21,1985,1987,1990,1993],{"className":1986},[42,99],[21,1988,103],{"className":1989},[42,43,99],[21,1991,963],{"className":1992},[408,99],[21,1994,358],{"className":1995},[42,99],[21,1997,108],{"className":1998},[107],[21,2000,2002],{"className":2001},[79],[21,2003,2005],{"className":2004,"style":976},[83],[21,2006],{},[21,2008],{"className":2009,"style":404},[378],[21,2011,963],{"className":2012},[408],[21,2014],{"className":2015,"style":404},[378],[21,2017,2019,2022,2071],{"className":2018},[33],[21,2020],{"className":2021,"style":923},[37],[21,2023,2025,2028],{"className":2024},[42],[21,2026,200],{"className":2027,"style":199},[42,43],[21,2029,2031],{"className":2030},[70],[21,2032,2034,2063],{"className":2033},[74,75],[21,2035,2037,2060],{"className":2036},[79],[21,2038,2040],{"className":2039,"style":944},[83],[21,2041,2042,2045],{"style":737},[21,2043],{"className":2044,"style":92},[91],[21,2046,2048],{"className":2047},[96,97,98,99],[21,2049,2051,2054,2057],{"className":2050},[42,99],[21,2052,103],{"className":2053},[42,43,99],[21,2055,963],{"className":2056},[408,99],[21,2058,358],{"className":2059},[42,99],[21,2061,108],{"className":2062},[107],[21,2064,2066],{"className":2065},[79],[21,2067,2069],{"className":2068,"style":976},[83],[21,2070],{},[21,2072,2074,2077],{"className":2073},[42],[21,2075,867],{"className":2076},[42,43],[21,2078,2080],{"className":2079},[70],[21,2081,2083,2103],{"className":2082},[74,75],[21,2084,2086,2100],{"className":2085},[79],[21,2087,2089],{"className":2088,"style":84},[83],[21,2090,2091,2094],{"style":87},[21,2092],{"className":2093,"style":92},[91],[21,2095,2097],{"className":2096},[96,97,98,99],[21,2098,103],{"className":2099},[42,43,99],[21,2101,108],{"className":2102},[107],[21,2104,2106],{"className":2105},[79],[21,2107,2109],{"className":2108,"style":115},[83],[21,2110],{}," discounts future\nrewards by the running product ",[21,2113,2115],{"className":2114},[24],[21,2116,2118],{"className":2117,"ariaHidden":29},[28],[21,2119,2121,2124,2127,2130],{"className":2120},[33],[21,2122],{"className":2123,"style":440},[37],[21,2125,1264],{"className":2126,"style":1420},[1118,1165,1419],[21,2128],{"className":2129,"style":458},[378],[21,2131,2133,2136],{"className":2132},[42],[21,2134,200],{"className":2135,"style":199},[42,43],[21,2137,2139],{"className":2138},[70],[21,2140,2142,2162],{"className":2141},[74,75],[21,2143,2145,2159],{"className":2144},[79],[21,2146,2148],{"className":2147,"style":1312},[83],[21,2149,2150,2153],{"style":737},[21,2151],{"className":2152,"style":92},[91],[21,2154,2156],{"className":2155},[96,97,98,99],[21,2157,1241],{"className":2158},[42,43,99],[21,2160,108],{"className":2161},[107],[21,2163,2165],{"className":2164},[79],[21,2166,2168],{"className":2167,"style":115},[83],[21,2169],{},"; a state with ",[21,2172,2174],{"className":2173},[24],[21,2175,2177,2204],{"className":2176,"ariaHidden":29},[28],[21,2178,2180,2183,2186,2189,2192,2195,2198,2201],{"className":2179},[33],[21,2181],{"className":2182,"style":440},[37],[21,2184,200],{"className":2185,"style":199},[42,43],[21,2187,543],{"className":2188},[444],[21,2190,1583],{"className":2191},[42,43],[21,2193,635],{"className":2194},[465],[21,2196],{"className":2197,"style":379},[378],[21,2199,1101],{"className":2200},[383],[21,2202],{"className":2203,"style":379},[378],[21,2205,2207,2210],{"className":2206},[33],[21,2208],{"className":2209,"style":1540},[37],[21,2211,449],{"className":2212},[42],"\nterminates the return there. It unifies episodic, continuing, and\npseudo-terminating prediction in one stream of experience.",[11,2215,2216,2219,2220,2449,2450,2484,2485,2519,2520,2563,2564,2625,2626,2641,2642,2703,2704,2641,2784,2625,2845,2974,2975,3188],{},[824,2217,2218],{},"A worked return."," Take rewards ",[21,2221,2223],{"className":2222},[24],[21,2224,2226,2412],{"className":2225,"ariaHidden":29},[28],[21,2227,2229,2232,2281,2284,2287,2337,2340,2343,2393,2396,2399,2403,2406,2409],{"className":2228},[33],[21,2230],{"className":2231,"style":923},[37],[21,2233,2235,2238],{"className":2234},[42],[21,2236,931],{"className":2237,"style":930},[42,43],[21,2239,2241],{"className":2240},[70],[21,2242,2244,2273],{"className":2243},[74,75],[21,2245,2247,2270],{"className":2246},[79],[21,2248,2250],{"className":2249,"style":944},[83],[21,2251,2252,2255],{"style":947},[21,2253],{"className":2254,"style":92},[91],[21,2256,2258],{"className":2257},[96,97,98,99],[21,2259,2261,2264,2267],{"className":2260},[42,99],[21,2262,103],{"className":2263},[42,43,99],[21,2265,963],{"className":2266},[408,99],[21,2268,358],{"className":2269},[42,99],[21,2271,108],{"className":2272},[107],[21,2274,2276],{"className":2275},[79],[21,2277,2279],{"className":2278,"style":976},[83],[21,2280],{},[21,2282,454],{"className":2283},[453],[21,2285],{"className":2286,"style":458},[378],[21,2288,2290,2293],{"className":2289},[42],[21,2291,931],{"className":2292,"style":930},[42,43],[21,2294,2296],{"className":2295},[70],[21,2297,2299,2329],{"className":2298},[74,75],[21,2300,2302,2326],{"className":2301},[79],[21,2303,2305],{"className":2304,"style":944},[83],[21,2306,2307,2310],{"style":947},[21,2308],{"className":2309,"style":92},[91],[21,2311,2313],{"className":2312},[96,97,98,99],[21,2314,2316,2319,2322],{"className":2315},[42,99],[21,2317,103],{"className":2318},[42,43,99],[21,2320,963],{"className":2321},[408,99],[21,2323,2325],{"className":2324},[42,99],"2",[21,2327,108],{"className":2328},[107],[21,2330,2332],{"className":2331},[79],[21,2333,2335],{"className":2334,"style":976},[83],[21,2336],{},[21,2338,454],{"className":2339},[453],[21,2341],{"className":2342,"style":458},[378],[21,2344,2346,2349],{"className":2345},[42],[21,2347,931],{"className":2348,"style":930},[42,43],[21,2350,2352],{"className":2351},[70],[21,2353,2355,2385],{"className":2354},[74,75],[21,2356,2358,2382],{"className":2357},[79],[21,2359,2361],{"className":2360,"style":944},[83],[21,2362,2363,2366],{"style":947},[21,2364],{"className":2365,"style":92},[91],[21,2367,2369],{"className":2368},[96,97,98,99],[21,2370,2372,2375,2378],{"className":2371},[42,99],[21,2373,103],{"className":2374},[42,43,99],[21,2376,963],{"className":2377},[408,99],[21,2379,2381],{"className":2380},[42,99],"3",[21,2383,108],{"className":2384},[107],[21,2386,2388],{"className":2387},[79],[21,2389,2391],{"className":2390,"style":976},[83],[21,2392],{},[21,2394,454],{"className":2395},[453],[21,2397],{"className":2398,"style":458},[378],[21,2400,2402],{"className":2401},[1202],"…",[21,2404],{"className":2405,"style":379},[378],[21,2407,1101],{"className":2408},[383],[21,2410],{"className":2411,"style":379},[378],[21,2413,2415,2419,2422,2425,2428,2431,2434,2437,2440,2443,2446],{"className":2414},[33],[21,2416],{"className":2417,"style":2418},[37],"height:0.8389em;vertical-align:-0.1944em;",[21,2420,358],{"className":2421},[42],[21,2423,454],{"className":2424},[453],[21,2426],{"className":2427,"style":458},[378],[21,2429,358],{"className":2430},[42],[21,2432,454],{"className":2433},[453],[21,2435],{"className":2436,"style":458},[378],[21,2438,358],{"className":2439},[42],[21,2441,454],{"className":2442},[453],[21,2444],{"className":2445,"style":458},[378],[21,2447,2402],{"className":2448},[1202]," and a per-state discount that is ",[21,2451,2453],{"className":2452},[24],[21,2454,2456,2474],{"className":2455,"ariaHidden":29},[28],[21,2457,2459,2462,2465,2468,2471],{"className":2458},[33],[21,2460],{"className":2461,"style":195},[37],[21,2463,200],{"className":2464,"style":199},[42,43],[21,2466],{"className":2467,"style":379},[378],[21,2469,1101],{"className":2470},[383],[21,2472],{"className":2473,"style":379},[378],[21,2475,2477,2480],{"className":2476},[33],[21,2478],{"className":2479,"style":1540},[37],[21,2481,2483],{"className":2482},[42],"0.9"," everywhere except a goal\nstate reached at step ",[21,2486,2488],{"className":2487},[24],[21,2489,2491,2510],{"className":2490,"ariaHidden":29},[28],[21,2492,2494,2498,2501,2504,2507],{"className":2493},[33],[21,2495],{"className":2496,"style":2497},[37],"height:0.6984em;vertical-align:-0.0833em;",[21,2499,103],{"className":2500},[42,43],[21,2502],{"className":2503,"style":404},[378],[21,2505,963],{"className":2506},[408],[21,2508],{"className":2509,"style":404},[378],[21,2511,2513,2516],{"className":2512},[33],[21,2514],{"className":2515,"style":1540},[37],[21,2517,2381],{"className":2518},[42],", where ",[21,2521,2523],{"className":2522},[24],[21,2524,2526,2554],{"className":2525,"ariaHidden":29},[28],[21,2527,2529,2532,2535,2538,2542,2545,2548,2551],{"className":2528},[33],[21,2530],{"className":2531,"style":440},[37],[21,2533,200],{"className":2534,"style":199},[42,43],[21,2536,543],{"className":2537},[444],[21,2539,2541],{"className":2540,"style":257},[42,43],"g",[21,2543,635],{"className":2544},[465],[21,2546],{"className":2547,"style":379},[378],[21,2549,1101],{"className":2550},[383],[21,2552],{"className":2553,"style":379},[378],[21,2555,2557,2560],{"className":2556},[33],[21,2558],{"className":2559,"style":1540},[37],[21,2561,449],{"className":2562},[42],". The running product truncates\nthe sum the moment it hits the zero: the weight on ",[21,2565,2567],{"className":2566},[24],[21,2568,2570],{"className":2569,"ariaHidden":29},[28],[21,2571,2573,2576],{"className":2572},[33],[21,2574],{"className":2575,"style":923},[37],[21,2577,2579,2582],{"className":2578},[42],[21,2580,931],{"className":2581,"style":930},[42,43],[21,2583,2585],{"className":2584},[70],[21,2586,2588,2617],{"className":2587},[74,75],[21,2589,2591,2614],{"className":2590},[79],[21,2592,2594],{"className":2593,"style":944},[83],[21,2595,2596,2599],{"style":947},[21,2597],{"className":2598,"style":92},[91],[21,2600,2602],{"className":2601},[96,97,98,99],[21,2603,2605,2608,2611],{"className":2604},[42,99],[21,2606,103],{"className":2607},[42,43,99],[21,2609,963],{"className":2610},[408,99],[21,2612,358],{"className":2613},[42,99],[21,2615,108],{"className":2616},[107],[21,2618,2620],{"className":2619},[79],[21,2621,2623],{"className":2622,"style":976},[83],[21,2624],{}," is ",[21,2627,2629],{"className":2628},[24],[21,2630,2632],{"className":2631,"ariaHidden":29},[28],[21,2633,2635,2638],{"className":2634},[33],[21,2636],{"className":2637,"style":1540},[37],[21,2639,358],{"className":2640},[42],", on ",[21,2643,2645],{"className":2644},[24],[21,2646,2648],{"className":2647,"ariaHidden":29},[28],[21,2649,2651,2654],{"className":2650},[33],[21,2652],{"className":2653,"style":923},[37],[21,2655,2657,2660],{"className":2656},[42],[21,2658,931],{"className":2659,"style":930},[42,43],[21,2661,2663],{"className":2662},[70],[21,2664,2666,2695],{"className":2665},[74,75],[21,2667,2669,2692],{"className":2668},[79],[21,2670,2672],{"className":2671,"style":944},[83],[21,2673,2674,2677],{"style":947},[21,2675],{"className":2676,"style":92},[91],[21,2678,2680],{"className":2679},[96,97,98,99],[21,2681,2683,2686,2689],{"className":2682},[42,99],[21,2684,103],{"className":2685},[42,43,99],[21,2687,963],{"className":2688},[408,99],[21,2690,2325],{"className":2691},[42,99],[21,2693,108],{"className":2694},[107],[21,2696,2698],{"className":2697},[79],[21,2699,2701],{"className":2700,"style":976},[83],[21,2702],{}," is\n",[21,2705,2707],{"className":2706},[24],[21,2708,2710,2775],{"className":2709,"ariaHidden":29},[28],[21,2711,2713,2717,2766,2769,2772],{"className":2712},[33],[21,2714],{"className":2715,"style":2716},[37],"height:0.6389em;vertical-align:-0.2083em;",[21,2718,2720,2723],{"className":2719},[42],[21,2721,200],{"className":2722,"style":199},[42,43],[21,2724,2726],{"className":2725},[70],[21,2727,2729,2758],{"className":2728},[74,75],[21,2730,2732,2755],{"className":2731},[79],[21,2733,2735],{"className":2734,"style":944},[83],[21,2736,2737,2740],{"style":737},[21,2738],{"className":2739,"style":92},[91],[21,2741,2743],{"className":2742},[96,97,98,99],[21,2744,2746,2749,2752],{"className":2745},[42,99],[21,2747,103],{"className":2748},[42,43,99],[21,2750,963],{"className":2751},[408,99],[21,2753,358],{"className":2754},[42,99],[21,2756,108],{"className":2757},[107],[21,2759,2761],{"className":2760},[79],[21,2762,2764],{"className":2763,"style":976},[83],[21,2765],{},[21,2767],{"className":2768,"style":379},[378],[21,2770,1101],{"className":2771},[383],[21,2773],{"className":2774,"style":379},[378],[21,2776,2778,2781],{"className":2777},[33],[21,2779],{"className":2780,"style":1540},[37],[21,2782,2483],{"className":2783},[42],[21,2785,2787],{"className":2786},[24],[21,2788,2790],{"className":2789,"ariaHidden":29},[28],[21,2791,2793,2796],{"className":2792},[33],[21,2794],{"className":2795,"style":923},[37],[21,2797,2799,2802],{"className":2798},[42],[21,2800,931],{"className":2801,"style":930},[42,43],[21,2803,2805],{"className":2804},[70],[21,2806,2808,2837],{"className":2807},[74,75],[21,2809,2811,2834],{"className":2810},[79],[21,2812,2814],{"className":2813,"style":944},[83],[21,2815,2816,2819],{"style":947},[21,2817],{"className":2818,"style":92},[91],[21,2820,2822],{"className":2821},[96,97,98,99],[21,2823,2825,2828,2831],{"className":2824},[42,99],[21,2826,103],{"className":2827},[42,43,99],[21,2829,963],{"className":2830},[408,99],[21,2832,2381],{"className":2833},[42,99],[21,2835,108],{"className":2836},[107],[21,2838,2840],{"className":2839},[79],[21,2841,2843],{"className":2842,"style":976},[83],[21,2844],{},[21,2846,2848],{"className":2847},[24],[21,2849,2851,2964],{"className":2850,"ariaHidden":29},[28],[21,2852,2854,2857,2906,2955,2958,2961],{"className":2853},[33],[21,2855],{"className":2856,"style":2716},[37],[21,2858,2860,2863],{"className":2859},[42],[21,2861,200],{"className":2862,"style":199},[42,43],[21,2864,2866],{"className":2865},[70],[21,2867,2869,2898],{"className":2868},[74,75],[21,2870,2872,2895],{"className":2871},[79],[21,2873,2875],{"className":2874,"style":944},[83],[21,2876,2877,2880],{"style":737},[21,2878],{"className":2879,"style":92},[91],[21,2881,2883],{"className":2882},[96,97,98,99],[21,2884,2886,2889,2892],{"className":2885},[42,99],[21,2887,103],{"className":2888},[42,43,99],[21,2890,963],{"className":2891},[408,99],[21,2893,358],{"className":2894},[42,99],[21,2896,108],{"className":2897},[107],[21,2899,2901],{"className":2900},[79],[21,2902,2904],{"className":2903,"style":976},[83],[21,2905],{},[21,2907,2909,2912],{"className":2908},[42],[21,2910,200],{"className":2911,"style":199},[42,43],[21,2913,2915],{"className":2914},[70],[21,2916,2918,2947],{"className":2917},[74,75],[21,2919,2921,2944],{"className":2920},[79],[21,2922,2924],{"className":2923,"style":944},[83],[21,2925,2926,2929],{"style":737},[21,2927],{"className":2928,"style":92},[91],[21,2930,2932],{"className":2931},[96,97,98,99],[21,2933,2935,2938,2941],{"className":2934},[42,99],[21,2936,103],{"className":2937},[42,43,99],[21,2939,963],{"className":2940},[408,99],[21,2942,2325],{"className":2943},[42,99],[21,2945,108],{"className":2946},[107],[21,2948,2950],{"className":2949},[79],[21,2951,2953],{"className":2952,"style":976},[83],[21,2954],{},[21,2956],{"className":2957,"style":379},[378],[21,2959,1101],{"className":2960},[383],[21,2962],{"className":2963,"style":379},[378],[21,2965,2967,2970],{"className":2966},[33],[21,2968],{"className":2969,"style":1540},[37],[21,2971,2973],{"className":2972},[42],"0.81",", and every\nlater reward is multiplied by ",[21,2976,2978],{"className":2977},[24],[21,2979,2981,3143,3161,3179],{"className":2980,"ariaHidden":29},[28],[21,2982,2984,2987,3036,3085,3134,3137,3140],{"className":2983},[33],[21,2985],{"className":2986,"style":2716},[37],[21,2988,2990,2993],{"className":2989},[42],[21,2991,200],{"className":2992,"style":199},[42,43],[21,2994,2996],{"className":2995},[70],[21,2997,2999,3028],{"className":2998},[74,75],[21,3000,3002,3025],{"className":3001},[79],[21,3003,3005],{"className":3004,"style":944},[83],[21,3006,3007,3010],{"style":737},[21,3008],{"className":3009,"style":92},[91],[21,3011,3013],{"className":3012},[96,97,98,99],[21,3014,3016,3019,3022],{"className":3015},[42,99],[21,3017,103],{"className":3018},[42,43,99],[21,3020,963],{"className":3021},[408,99],[21,3023,358],{"className":3024},[42,99],[21,3026,108],{"className":3027},[107],[21,3029,3031],{"className":3030},[79],[21,3032,3034],{"className":3033,"style":976},[83],[21,3035],{},[21,3037,3039,3042],{"className":3038},[42],[21,3040,200],{"className":3041,"style":199},[42,43],[21,3043,3045],{"className":3044},[70],[21,3046,3048,3077],{"className":3047},[74,75],[21,3049,3051,3074],{"className":3050},[79],[21,3052,3054],{"className":3053,"style":944},[83],[21,3055,3056,3059],{"style":737},[21,3057],{"className":3058,"style":92},[91],[21,3060,3062],{"className":3061},[96,97,98,99],[21,3063,3065,3068,3071],{"className":3064},[42,99],[21,3066,103],{"className":3067},[42,43,99],[21,3069,963],{"className":3070},[408,99],[21,3072,2325],{"className":3073},[42,99],[21,3075,108],{"className":3076},[107],[21,3078,3080],{"className":3079},[79],[21,3081,3083],{"className":3082,"style":976},[83],[21,3084],{},[21,3086,3088,3091],{"className":3087},[42],[21,3089,200],{"className":3090,"style":199},[42,43],[21,3092,3094],{"className":3093},[70],[21,3095,3097,3126],{"className":3096},[74,75],[21,3098,3100,3123],{"className":3099},[79],[21,3101,3103],{"className":3102,"style":944},[83],[21,3104,3105,3108],{"style":737},[21,3106],{"className":3107,"style":92},[91],[21,3109,3111],{"className":3110},[96,97,98,99],[21,3112,3114,3117,3120],{"className":3113},[42,99],[21,3115,103],{"className":3116},[42,43,99],[21,3118,963],{"className":3119},[408,99],[21,3121,2381],{"className":3122},[42,99],[21,3124,108],{"className":3125},[107],[21,3127,3129],{"className":3128},[79],[21,3130,3132],{"className":3131,"style":976},[83],[21,3133],{},[21,3135],{"className":3136,"style":379},[378],[21,3138,1101],{"className":3139},[383],[21,3141],{"className":3142,"style":379},[378],[21,3144,3146,3149,3152,3155,3158],{"className":3145},[33],[21,3147],{"className":3148,"style":1540},[37],[21,3150,2973],{"className":3151},[42],[21,3153],{"className":3154,"style":404},[378],[21,3156,1627],{"className":3157},[408],[21,3159],{"className":3160,"style":404},[378],[21,3162,3164,3167,3170,3173,3176],{"className":3163},[33],[21,3165],{"className":3166,"style":1540},[37],[21,3168,449],{"className":3169},[42],[21,3171],{"className":3172,"style":379},[378],[21,3174,1101],{"className":3175},[383],[21,3177],{"className":3178,"style":379},[378],[21,3180,3182,3185],{"className":3181},[33],[21,3183],{"className":3184,"style":1540},[37],[21,3186,449],{"className":3187},[42],". So",[21,3190,3192],{"className":3191},[847],[21,3193,3195],{"className":3194},[24],[21,3196,3198,3253,3272,3290,3308,3326,3344,3362,3380,3400],{"className":3197,"ariaHidden":29},[28],[21,3199,3201,3204,3244,3247,3250],{"className":3200},[33],[21,3202],{"className":3203,"style":860},[37],[21,3205,3207,3210],{"className":3206},[42],[21,3208,867],{"className":3209},[42,43],[21,3211,3213],{"className":3212},[70],[21,3214,3216,3236],{"className":3215},[74,75],[21,3217,3219,3233],{"className":3218},[79],[21,3220,3222],{"className":3221,"style":84},[83],[21,3223,3224,3227],{"style":87},[21,3225],{"className":3226,"style":92},[91],[21,3228,3230],{"className":3229},[96,97,98,99],[21,3231,103],{"className":3232},[42,43,99],[21,3234,108],{"className":3235},[107],[21,3237,3239],{"className":3238},[79],[21,3240,3242],{"className":3241,"style":115},[83],[21,3243],{},[21,3245],{"className":3246,"style":379},[378],[21,3248,1101],{"className":3249},[383],[21,3251],{"className":3252,"style":379},[378],[21,3254,3256,3260,3263,3266,3269],{"className":3255},[33],[21,3257],{"className":3258,"style":3259},[37],"height:0.7278em;vertical-align:-0.0833em;",[21,3261,358],{"className":3262},[42],[21,3264],{"className":3265,"style":404},[378],[21,3267,963],{"className":3268},[408],[21,3270],{"className":3271,"style":404},[378],[21,3273,3275,3278,3281,3284,3287],{"className":3274},[33],[21,3276],{"className":3277,"style":1540},[37],[21,3279,2483],{"className":3280},[42],[21,3282],{"className":3283,"style":404},[378],[21,3285,1627],{"className":3286},[408],[21,3288],{"className":3289,"style":404},[378],[21,3291,3293,3296,3299,3302,3305],{"className":3292},[33],[21,3294],{"className":3295,"style":3259},[37],[21,3297,358],{"className":3298},[42],[21,3300],{"className":3301,"style":404},[378],[21,3303,963],{"className":3304},[408],[21,3306],{"className":3307,"style":404},[378],[21,3309,3311,3314,3317,3320,3323],{"className":3310},[33],[21,3312],{"className":3313,"style":1540},[37],[21,3315,2973],{"className":3316},[42],[21,3318],{"className":3319,"style":404},[378],[21,3321,1627],{"className":3322},[408],[21,3324],{"className":3325,"style":404},[378],[21,3327,3329,3332,3335,3338,3341],{"className":3328},[33],[21,3330],{"className":3331,"style":3259},[37],[21,3333,358],{"className":3334},[42],[21,3336],{"className":3337,"style":404},[378],[21,3339,963],{"className":3340},[408],[21,3342],{"className":3343,"style":404},[378],[21,3345,3347,3350,3353,3356,3359],{"className":3346},[33],[21,3348],{"className":3349,"style":3259},[37],[21,3351,449],{"className":3352},[42],[21,3354],{"className":3355,"style":404},[378],[21,3357,963],{"className":3358},[408],[21,3360],{"className":3361,"style":404},[378],[21,3363,3365,3368,3371,3374,3377],{"className":3364},[33],[21,3366],{"className":3367,"style":3259},[37],[21,3369,449],{"className":3370},[42],[21,3372],{"className":3373,"style":404},[378],[21,3375,963],{"className":3376},[408],[21,3378],{"className":3379,"style":404},[378],[21,3381,3383,3387,3391,3394,3397],{"className":3382},[33],[21,3384],{"className":3385,"style":3386},[37],"height:0.3669em;",[21,3388,3390],{"className":3389},[1202],"⋯",[21,3392],{"className":3393,"style":379},[378],[21,3395,1101],{"className":3396},[383],[21,3398],{"className":3399,"style":379},[378],[21,3401,3403,3406,3410],{"className":3402},[33],[21,3404],{"className":3405,"style":2418},[37],[21,3407,3409],{"className":3408},[42],"2.71",[21,3411,454],{"className":3412},[453],[11,3414,3415,3416,3458,3459,3501,3502,3609],{},"a finite return produced with no special terminal state and no episode boundary —\nthe goal's ",[21,3417,3419],{"className":3418},[24],[21,3420,3422,3449],{"className":3421,"ariaHidden":29},[28],[21,3423,3425,3428,3431,3434,3437,3440,3443,3446],{"className":3424},[33],[21,3426],{"className":3427,"style":440},[37],[21,3429,200],{"className":3430,"style":199},[42,43],[21,3432,543],{"className":3433},[444],[21,3435,2541],{"className":3436,"style":257},[42,43],[21,3438,635],{"className":3439},[465],[21,3441],{"className":3442,"style":379},[378],[21,3444,1101],{"className":3445},[383],[21,3447],{"className":3448,"style":379},[378],[21,3450,3452,3455],{"className":3451},[33],[21,3453],{"className":3454,"style":1540},[37],[21,3456,449],{"className":3457},[42]," truncates the sum. Set ",[21,3460,3462],{"className":3461},[24],[21,3463,3465,3492],{"className":3464,"ariaHidden":29},[28],[21,3466,3468,3471,3474,3477,3480,3483,3486,3489],{"className":3467},[33],[21,3469],{"className":3470,"style":440},[37],[21,3472,200],{"className":3473,"style":199},[42,43],[21,3475,543],{"className":3476},[444],[21,3478,2541],{"className":3479,"style":257},[42,43],[21,3481,635],{"className":3482},[465],[21,3484],{"className":3485,"style":379},[378],[21,3487,1101],{"className":3488},[383],[21,3490],{"className":3491,"style":379},[378],[21,3493,3495,3498],{"className":3494},[33],[21,3496],{"className":3497,"style":1540},[37],[21,3499,358],{"className":3500},[42]," instead and the\nsame stream keeps accumulating ",[21,3503,3505],{"className":3504},[24],[21,3506,3508,3526,3544,3562,3581,3599],{"className":3507,"ariaHidden":29},[28],[21,3509,3511,3514,3517,3520,3523],{"className":3510},[33],[21,3512],{"className":3513,"style":3259},[37],[21,3515,358],{"className":3516},[42],[21,3518],{"className":3519,"style":404},[378],[21,3521,963],{"className":3522},[408],[21,3524],{"className":3525,"style":404},[378],[21,3527,3529,3532,3535,3538,3541],{"className":3528},[33],[21,3530],{"className":3531,"style":3259},[37],[21,3533,2483],{"className":3534},[42],[21,3536],{"className":3537,"style":404},[378],[21,3539,963],{"className":3540},[408],[21,3542],{"className":3543,"style":404},[378],[21,3545,3547,3550,3553,3556,3559],{"className":3546},[33],[21,3548],{"className":3549,"style":3259},[37],[21,3551,2973],{"className":3552},[42],[21,3554],{"className":3555,"style":404},[378],[21,3557,963],{"className":3558},[408],[21,3560],{"className":3561,"style":404},[378],[21,3563,3565,3568,3572,3575,3578],{"className":3564},[33],[21,3566],{"className":3567,"style":3259},[37],[21,3569,3571],{"className":3570},[42],"0.729",[21,3573],{"className":3574,"style":404},[378],[21,3576,963],{"className":3577},[408],[21,3579],{"className":3580,"style":404},[378],[21,3582,3584,3587,3590,3593,3596],{"className":3583},[33],[21,3585],{"className":3586,"style":3386},[37],[21,3588,3390],{"className":3589},[1202],[21,3591],{"className":3592,"style":379},[378],[21,3594,1101],{"className":3595},[383],[21,3597],{"className":3598,"style":379},[378],[21,3600,3602,3605],{"className":3601},[33],[21,3603],{"className":3604,"style":1540},[37],[21,3606,3608],{"className":3607},[42],"10",", the ordinary\ndiscounted return of a continuing task. One definition covers both cases.",[11,3611,3612,3613,3628,3629,3632,3633,3648,3649,3664],{},"Making ",[21,3614,3616],{"className":3615},[24],[21,3617,3619],{"className":3618,"ariaHidden":29},[28],[21,3620,3622,3625],{"className":3621},[33],[21,3623],{"className":3624,"style":38},[37],[21,3626,44],{"className":3627},[42,43]," variable is a change of ",[237,3630,3631],{},"solution strategy",", not of the problem:\nit sets, state by state, how much to bootstrap. The generalization touches the\n",[21,3634,3636],{"className":3635},[24],[21,3637,3639],{"className":3638,"ariaHidden":29},[28],[21,3640,3642,3645],{"className":3641},[33],[21,3643],{"className":3644,"style":38},[37],[21,3646,44],{"className":3647},[42,43],"-return, which now recurses. In the state-based case, writing the\nsuperscript ",[21,3650,3652],{"className":3651},[24],[21,3653,3655],{"className":3654,"ariaHidden":29},[28],[21,3656,3658,3661],{"className":3657},[33],[21,3659],{"className":3660,"style":253},[37],[21,3662,1583],{"className":3663},[42,43]," to mark that it bootstraps from state values,",[21,3666,3668],{"className":3667},[847],[21,3669,3671],{"className":3670},[24],[21,3672,3674,3757,3821,3900,4112],{"className":3673,"ariaHidden":29},[28],[21,3675,3677,3681,3742,3745,3748,3751,3754],{"className":3676},[33],[21,3678],{"className":3679,"style":3680},[37],"height:1.1461em;vertical-align:-0.247em;",[21,3682,3684,3687],{"className":3683},[42],[21,3685,867],{"className":3686},[42,43],[21,3688,3690],{"className":3689},[70],[21,3691,3693,3733],{"className":3692},[74,75],[21,3694,3696,3730],{"className":3695},[79],[21,3697,3700,3712],{"className":3698,"style":3699},[83],"height:0.8991em;",[21,3701,3703,3706],{"style":3702},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[21,3704],{"className":3705,"style":92},[91],[21,3707,3709],{"className":3708},[96,97,98,99],[21,3710,103],{"className":3711},[42,43,99],[21,3713,3715,3718],{"style":3714},"top:-3.113em;margin-right:0.05em;",[21,3716],{"className":3717,"style":92},[91],[21,3719,3721],{"className":3720},[96,97,98,99],[21,3722,3724,3727],{"className":3723},[42,99],[21,3725,44],{"className":3726},[42,43,99],[21,3728,1583],{"className":3729},[42,43,99],[21,3731,108],{"className":3732},[107],[21,3734,3736],{"className":3735},[79],[21,3737,3740],{"className":3738,"style":3739},[83],"height:0.247em;",[21,3741],{},[21,3743],{"className":3744,"style":379},[378],[21,3746],{"className":3747,"style":379},[378],[21,3749,527],{"className":3750},[383],[21,3752],{"className":3753,"style":379},[378],[21,3755],{"className":3756,"style":379},[378],[21,3758,3760,3763,3812,3815,3818],{"className":3759},[33],[21,3761],{"className":3762,"style":923},[37],[21,3764,3766,3769],{"className":3765},[42],[21,3767,931],{"className":3768,"style":930},[42,43],[21,3770,3772],{"className":3771},[70],[21,3773,3775,3804],{"className":3774},[74,75],[21,3776,3778,3801],{"className":3777},[79],[21,3779,3781],{"className":3780,"style":944},[83],[21,3782,3783,3786],{"style":947},[21,3784],{"className":3785,"style":92},[91],[21,3787,3789],{"className":3788},[96,97,98,99],[21,3790,3792,3795,3798],{"className":3791},[42,99],[21,3793,103],{"className":3794},[42,43,99],[21,3796,963],{"className":3797},[408,99],[21,3799,358],{"className":3800},[42,99],[21,3802,108],{"className":3803},[107],[21,3805,3807],{"className":3806},[79],[21,3808,3810],{"className":3809,"style":976},[83],[21,3811],{},[21,3813],{"className":3814,"style":404},[378],[21,3816,963],{"className":3817},[408],[21,3819],{"className":3820,"style":404},[378],[21,3822,3824,3828,3877,3884,3887,3890,3893,3897],{"className":3823},[33],[21,3825],{"className":3826,"style":3827},[37],"height:1.8em;vertical-align:-0.65em;",[21,3829,3831,3834],{"className":3830},[42],[21,3832,200],{"className":3833,"style":199},[42,43],[21,3835,3837],{"className":3836},[70],[21,3838,3840,3869],{"className":3839},[74,75],[21,3841,3843,3866],{"className":3842},[79],[21,3844,3846],{"className":3845,"style":944},[83],[21,3847,3848,3851],{"style":737},[21,3849],{"className":3850,"style":92},[91],[21,3852,3854],{"className":3853},[96,97,98,99],[21,3855,3857,3860,3863],{"className":3856},[42,99],[21,3858,103],{"className":3859},[42,43,99],[21,3861,963],{"className":3862},[408,99],[21,3864,358],{"className":3865},[42,99],[21,3867,108],{"className":3868},[107],[21,3870,3872],{"className":3871},[79],[21,3873,3875],{"className":3874,"style":976},[83],[21,3876],{},[21,3878,3880],{"className":3879},[42],[21,3881,543],{"className":3882},[1211,3883],"size2",[21,3885,543],{"className":3886},[444],[21,3888,358],{"className":3889},[42],[21,3891],{"className":3892,"style":404},[378],[21,3894,3896],{"className":3895},[408],"−",[21,3898],{"className":3899,"style":404},[378],[21,3901,3903,3906,3955,3958,3961,3999,4002,4051,4054,4057,4100,4103,4106,4109],{"className":3902},[33],[21,3904],{"className":3905,"style":440},[37],[21,3907,3909,3912],{"className":3908},[42],[21,3910,44],{"className":3911},[42,43],[21,3913,3915],{"className":3914},[70],[21,3916,3918,3947],{"className":3917},[74,75],[21,3919,3921,3944],{"className":3920},[79],[21,3922,3924],{"className":3923,"style":944},[83],[21,3925,3926,3929],{"style":87},[21,3927],{"className":3928,"style":92},[91],[21,3930,3932],{"className":3931},[96,97,98,99],[21,3933,3935,3938,3941],{"className":3934},[42,99],[21,3936,103],{"className":3937},[42,43,99],[21,3939,963],{"className":3940},[408,99],[21,3942,358],{"className":3943},[42,99],[21,3945,108],{"className":3946},[107],[21,3948,3950],{"className":3949},[79],[21,3951,3953],{"className":3952,"style":976},[83],[21,3954],{},[21,3956,635],{"className":3957},[465],[21,3959],{"className":3960,"style":458},[378],[21,3962,3965],{"className":3963},[42,3964],"accent",[21,3966,3968],{"className":3967},[74],[21,3969,3971],{"className":3970},[79],[21,3972,3974,3985],{"className":3973,"style":38},[83],[21,3975,3977,3981],{"style":3976},"top:-3em;",[21,3978],{"className":3979,"style":3980},[91],"height:3em;",[21,3982,3984],{"className":3983,"style":257},[42,43],"v",[21,3986,3987,3990],{"style":3976},[21,3988],{"className":3989,"style":3980},[91],[21,3991,3995],{"className":3992,"style":3994},[3993],"accent-body","left:-0.2222em;",[21,3996,3998],{"className":3997},[42],"^",[21,4000,543],{"className":4001},[444],[21,4003,4005,4008],{"className":4004},[42],[21,4006,400],{"className":4007,"style":550},[42,43],[21,4009,4011],{"className":4010},[70],[21,4012,4014,4043],{"className":4013},[74,75],[21,4015,4017,4040],{"className":4016},[79],[21,4018,4020],{"className":4019,"style":944},[83],[21,4021,4022,4025],{"style":565},[21,4023],{"className":4024,"style":92},[91],[21,4026,4028],{"className":4027},[96,97,98,99],[21,4029,4031,4034,4037],{"className":4030},[42,99],[21,4032,103],{"className":4033},[42,43,99],[21,4035,963],{"className":4036},[408,99],[21,4038,358],{"className":4039},[42,99],[21,4041,108],{"className":4042},[107],[21,4044,4046],{"className":4045},[79],[21,4047,4049],{"className":4048,"style":976},[83],[21,4050],{},[21,4052,454],{"className":4053},[453],[21,4055],{"className":4056,"style":458},[378],[21,4058,4060,4065],{"className":4059},[42],[21,4061,4064],{"className":4062,"style":4063},[42,65],"margin-right:0.016em;","w",[21,4066,4068],{"className":4067},[70],[21,4069,4071,4092],{"className":4070},[74,75],[21,4072,4074,4089],{"className":4073},[79],[21,4075,4077],{"className":4076,"style":84},[83],[21,4078,4080,4083],{"style":4079},"top:-2.55em;margin-left:-0.016em;margin-right:0.05em;",[21,4081],{"className":4082,"style":92},[91],[21,4084,4086],{"className":4085},[96,97,98,99],[21,4087,103],{"className":4088},[42,43,99],[21,4090,108],{"className":4091},[107],[21,4093,4095],{"className":4094},[79],[21,4096,4098],{"className":4097,"style":115},[83],[21,4099],{},[21,4101,635],{"className":4102},[465],[21,4104],{"className":4105,"style":404},[378],[21,4107,963],{"className":4108},[408],[21,4110],{"className":4111,"style":404},[378],[21,4113,4115,4118,4167,4170,4237,4243],{"className":4114},[33],[21,4116],{"className":4117,"style":3827},[37],[21,4119,4121,4124],{"className":4120},[42],[21,4122,44],{"className":4123},[42,43],[21,4125,4127],{"className":4126},[70],[21,4128,4130,4159],{"className":4129},[74,75],[21,4131,4133,4156],{"className":4132},[79],[21,4134,4136],{"className":4135,"style":944},[83],[21,4137,4138,4141],{"style":87},[21,4139],{"className":4140,"style":92},[91],[21,4142,4144],{"className":4143},[96,97,98,99],[21,4145,4147,4150,4153],{"className":4146},[42,99],[21,4148,103],{"className":4149},[42,43,99],[21,4151,963],{"className":4152},[408,99],[21,4154,358],{"className":4155},[42,99],[21,4157,108],{"className":4158},[107],[21,4160,4162],{"className":4161},[79],[21,4163,4165],{"className":4164,"style":976},[83],[21,4166],{},[21,4168],{"className":4169,"style":458},[378],[21,4171,4173,4176],{"className":4172},[42],[21,4174,867],{"className":4175},[42,43],[21,4177,4179],{"className":4178},[70],[21,4180,4182,4228],{"className":4181},[74,75],[21,4183,4185,4225],{"className":4184},[79],[21,4186,4188,4208],{"className":4187,"style":3699},[83],[21,4189,4190,4193],{"style":3702},[21,4191],{"className":4192,"style":92},[91],[21,4194,4196],{"className":4195},[96,97,98,99],[21,4197,4199,4202,4205],{"className":4198},[42,99],[21,4200,103],{"className":4201},[42,43,99],[21,4203,963],{"className":4204},[408,99],[21,4206,358],{"className":4207},[42,99],[21,4209,4210,4213],{"style":3714},[21,4211],{"className":4212,"style":92},[91],[21,4214,4216],{"className":4215},[96,97,98,99],[21,4217,4219,4222],{"className":4218},[42,99],[21,4220,44],{"className":4221},[42,43,99],[21,4223,1583],{"className":4224},[42,43,99],[21,4226,108],{"className":4227},[107],[21,4229,4231],{"className":4230},[79],[21,4232,4235],{"className":4233,"style":4234},[83],"height:0.3053em;",[21,4236],{},[21,4238,4240],{"className":4239},[42],[21,4241,635],{"className":4242},[1211,3883],[21,4244,4246],{"className":4245},[42],".",[11,4248,4249,4250,4311,4312,4392,4393,4454,4455,4470],{},"Read it left to right: the return is the first reward, plus (to the extent we are\nnot terminating, ",[21,4251,4253],{"className":4252},[24],[21,4254,4256],{"className":4255,"ariaHidden":29},[28],[21,4257,4259,4262],{"className":4258},[33],[21,4260],{"className":4261,"style":2716},[37],[21,4263,4265,4268],{"className":4264},[42],[21,4266,200],{"className":4267,"style":199},[42,43],[21,4269,4271],{"className":4270},[70],[21,4272,4274,4303],{"className":4273},[74,75],[21,4275,4277,4300],{"className":4276},[79],[21,4278,4280],{"className":4279,"style":944},[83],[21,4281,4282,4285],{"style":737},[21,4283],{"className":4284,"style":92},[91],[21,4286,4288],{"className":4287},[96,97,98,99],[21,4289,4291,4294,4297],{"className":4290},[42,99],[21,4292,103],{"className":4293},[42,43,99],[21,4295,963],{"className":4296},[408,99],[21,4298,358],{"className":4299},[42,99],[21,4301,108],{"className":4302},[107],[21,4304,4306],{"className":4305},[79],[21,4307,4309],{"className":4308,"style":976},[83],[21,4310],{},") a second term that splits by the degree of\nbootstrapping — a fraction ",[21,4313,4315],{"className":4314},[24],[21,4316,4318,4336],{"className":4317,"ariaHidden":29},[28],[21,4319,4321,4324,4327,4330,4333],{"className":4320},[33],[21,4322],{"className":4323,"style":3259},[37],[21,4325,358],{"className":4326},[42],[21,4328],{"className":4329,"style":404},[378],[21,4331,3896],{"className":4332},[408],[21,4334],{"className":4335,"style":404},[378],[21,4337,4339,4343],{"className":4338},[33],[21,4340],{"className":4341,"style":4342},[37],"height:0.9028em;vertical-align:-0.2083em;",[21,4344,4346,4349],{"className":4345},[42],[21,4347,44],{"className":4348},[42,43],[21,4350,4352],{"className":4351},[70],[21,4353,4355,4384],{"className":4354},[74,75],[21,4356,4358,4381],{"className":4357},[79],[21,4359,4361],{"className":4360,"style":944},[83],[21,4362,4363,4366],{"style":87},[21,4364],{"className":4365,"style":92},[91],[21,4367,4369],{"className":4368},[96,97,98,99],[21,4370,4372,4375,4378],{"className":4371},[42,99],[21,4373,103],{"className":4374},[42,43,99],[21,4376,963],{"className":4377},[408,99],[21,4379,358],{"className":4380},[42,99],[21,4382,108],{"className":4383},[107],[21,4385,4387],{"className":4386},[79],[21,4388,4390],{"className":4389,"style":976},[83],[21,4391],{}," takes the estimated value at the next\nstate, and the complementary fraction ",[21,4394,4396],{"className":4395},[24],[21,4397,4399],{"className":4398,"ariaHidden":29},[28],[21,4400,4402,4405],{"className":4401},[33],[21,4403],{"className":4404,"style":4342},[37],[21,4406,4408,4411],{"className":4407},[42],[21,4409,44],{"className":4410},[42,43],[21,4412,4414],{"className":4413},[70],[21,4415,4417,4446],{"className":4416},[74,75],[21,4418,4420,4443],{"className":4419},[79],[21,4421,4423],{"className":4422,"style":944},[83],[21,4424,4425,4428],{"style":87},[21,4426],{"className":4427,"style":92},[91],[21,4429,4431],{"className":4430},[96,97,98,99],[21,4432,4434,4437,4440],{"className":4433},[42,99],[21,4435,103],{"className":4436},[42,43,99],[21,4438,963],{"className":4439},[408,99],[21,4441,358],{"className":4442},[42,99],[21,4444,108],{"className":4445},[107],[21,4447,4449],{"className":4448},[79],[21,4450,4452],{"className":4451,"style":976},[83],[21,4453],{}," recurses into the\n",[21,4456,4458],{"className":4457},[24],[21,4459,4461],{"className":4460,"ariaHidden":29},[28],[21,4462,4464,4467],{"className":4463},[33],[21,4465],{"className":4466,"style":38},[37],[21,4468,44],{"className":4469},[42,43],"-return of the next step. The action-based return has a Sarsa form,",[21,4472,4474],{"className":4473},[847],[21,4475,4477],{"className":4476},[24],[21,4478,4480,4556,4620,4696,4967],{"className":4479,"ariaHidden":29},[28],[21,4481,4483,4486,4541,4544,4547,4550,4553],{"className":4482},[33],[21,4484],{"className":4485,"style":3680},[37],[21,4487,4489,4492],{"className":4488},[42],[21,4490,867],{"className":4491},[42,43],[21,4493,4495],{"className":4494},[70],[21,4496,4498,4533],{"className":4497},[74,75],[21,4499,4501,4530],{"className":4500},[79],[21,4502,4504,4515],{"className":4503,"style":3699},[83],[21,4505,4506,4509],{"style":3702},[21,4507],{"className":4508,"style":92},[91],[21,4510,4512],{"className":4511},[96,97,98,99],[21,4513,103],{"className":4514},[42,43,99],[21,4516,4517,4520],{"style":3714},[21,4518],{"className":4519,"style":92},[91],[21,4521,4523],{"className":4522},[96,97,98,99],[21,4524,4526],{"className":4525},[42,99],[21,4527,4529],{"className":4528},[42,43,99],"λa",[21,4531,108],{"className":4532},[107],[21,4534,4536],{"className":4535},[79],[21,4537,4539],{"className":4538,"style":3739},[83],[21,4540],{},[21,4542],{"className":4543,"style":379},[378],[21,4545],{"className":4546,"style":379},[378],[21,4548,527],{"className":4549},[383],[21,4551],{"className":4552,"style":379},[378],[21,4554],{"className":4555,"style":379},[378],[21,4557,4559,4562,4611,4614,4617],{"className":4558},[33],[21,4560],{"className":4561,"style":923},[37],[21,4563,4565,4568],{"className":4564},[42],[21,4566,931],{"className":4567,"style":930},[42,43],[21,4569,4571],{"className":4570},[70],[21,4572,4574,4603],{"className":4573},[74,75],[21,4575,4577,4600],{"className":4576},[79],[21,4578,4580],{"className":4579,"style":944},[83],[21,4581,4582,4585],{"style":947},[21,4583],{"className":4584,"style":92},[91],[21,4586,4588],{"className":4587},[96,97,98,99],[21,4589,4591,4594,4597],{"className":4590},[42,99],[21,4592,103],{"className":4593},[42,43,99],[21,4595,963],{"className":4596},[408,99],[21,4598,358],{"className":4599},[42,99],[21,4601,108],{"className":4602},[107],[21,4604,4606],{"className":4605},[79],[21,4607,4609],{"className":4608,"style":976},[83],[21,4610],{},[21,4612],{"className":4613,"style":404},[378],[21,4615,963],{"className":4616},[408],[21,4618],{"className":4619,"style":404},[378],[21,4621,4623,4626,4675,4681,4684,4687,4690,4693],{"className":4622},[33],[21,4624],{"className":4625,"style":3827},[37],[21,4627,4629,4632],{"className":4628},[42],[21,4630,200],{"className":4631,"style":199},[42,43],[21,4633,4635],{"className":4634},[70],[21,4636,4638,4667],{"className":4637},[74,75],[21,4639,4641,4664],{"className":4640},[79],[21,4642,4644],{"className":4643,"style":944},[83],[21,4645,4646,4649],{"style":737},[21,4647],{"className":4648,"style":92},[91],[21,4650,4652],{"className":4651},[96,97,98,99],[21,4653,4655,4658,4661],{"className":4654},[42,99],[21,4656,103],{"className":4657},[42,43,99],[21,4659,963],{"className":4660},[408,99],[21,4662,358],{"className":4663},[42,99],[21,4665,108],{"className":4666},[107],[21,4668,4670],{"className":4669},[79],[21,4671,4673],{"className":4672,"style":976},[83],[21,4674],{},[21,4676,4678],{"className":4677},[42],[21,4679,543],{"className":4680},[1211,3883],[21,4682,543],{"className":4683},[444],[21,4685,358],{"className":4686},[42],[21,4688],{"className":4689,"style":404},[378],[21,4691,3896],{"className":4692},[408],[21,4694],{"className":4695,"style":404},[378],[21,4697,4699,4702,4751,4754,4757,4802,4805,4854,4857,4860,4909,4912,4915,4955,4958,4961,4964],{"className":4698},[33],[21,4700],{"className":4701,"style":440},[37],[21,4703,4705,4708],{"className":4704},[42],[21,4706,44],{"className":4707},[42,43],[21,4709,4711],{"className":4710},[70],[21,4712,4714,4743],{"className":4713},[74,75],[21,4715,4717,4740],{"className":4716},[79],[21,4718,4720],{"className":4719,"style":944},[83],[21,4721,4722,4725],{"style":87},[21,4723],{"className":4724,"style":92},[91],[21,4726,4728],{"className":4727},[96,97,98,99],[21,4729,4731,4734,4737],{"className":4730},[42,99],[21,4732,103],{"className":4733},[42,43,99],[21,4735,963],{"className":4736},[408,99],[21,4738,358],{"className":4739},[42,99],[21,4741,108],{"className":4742},[107],[21,4744,4746],{"className":4745},[79],[21,4747,4749],{"className":4748,"style":976},[83],[21,4750],{},[21,4752,635],{"className":4753},[465],[21,4755],{"className":4756,"style":458},[378],[21,4758,4760],{"className":4759},[42,3964],[21,4761,4763,4793],{"className":4762},[74,75],[21,4764,4766,4790],{"className":4765},[79],[21,4767,4769,4778],{"className":4768,"style":38},[83],[21,4770,4771,4774],{"style":3976},[21,4772],{"className":4773,"style":3980},[91],[21,4775,4777],{"className":4776,"style":257},[42,43],"q",[21,4779,4780,4783],{"style":3976},[21,4781],{"className":4782,"style":3980},[91],[21,4784,4787],{"className":4785,"style":4786},[3993],"left:-0.1667em;",[21,4788,3998],{"className":4789},[42],[21,4791,108],{"className":4792},[107],[21,4794,4796],{"className":4795},[79],[21,4797,4800],{"className":4798,"style":4799},[83],"height:0.1944em;",[21,4801],{},[21,4803,543],{"className":4804},[444],[21,4806,4808,4811],{"className":4807},[42],[21,4809,400],{"className":4810,"style":550},[42,43],[21,4812,4814],{"className":4813},[70],[21,4815,4817,4846],{"className":4816},[74,75],[21,4818,4820,4843],{"className":4819},[79],[21,4821,4823],{"className":4822,"style":944},[83],[21,4824,4825,4828],{"style":565},[21,4826],{"className":4827,"style":92},[91],[21,4829,4831],{"className":4830},[96,97,98,99],[21,4832,4834,4837,4840],{"className":4833},[42,99],[21,4835,103],{"className":4836},[42,43,99],[21,4838,963],{"className":4839},[408,99],[21,4841,358],{"className":4842},[42,99],[21,4844,108],{"className":4845},[107],[21,4847,4849],{"className":4848},[79],[21,4850,4852],{"className":4851,"style":976},[83],[21,4853],{},[21,4855,454],{"className":4856},[453],[21,4858],{"className":4859,"style":458},[378],[21,4861,4863,4866],{"className":4862},[42],[21,4864,423],{"className":4865},[42,43],[21,4867,4869],{"className":4868},[70],[21,4870,4872,4901],{"className":4871},[74,75],[21,4873,4875,4898],{"className":4874},[79],[21,4876,4878],{"className":4877,"style":944},[83],[21,4879,4880,4883],{"style":87},[21,4881],{"className":4882,"style":92},[91],[21,4884,4886],{"className":4885},[96,97,98,99],[21,4887,4889,4892,4895],{"className":4888},[42,99],[21,4890,103],{"className":4891},[42,43,99],[21,4893,963],{"className":4894},[408,99],[21,4896,358],{"className":4897},[42,99],[21,4899,108],{"className":4900},[107],[21,4902,4904],{"className":4903},[79],[21,4905,4907],{"className":4906,"style":976},[83],[21,4908],{},[21,4910,454],{"className":4911},[453],[21,4913],{"className":4914,"style":458},[378],[21,4916,4918,4921],{"className":4917},[42],[21,4919,4064],{"className":4920,"style":4063},[42,65],[21,4922,4924],{"className":4923},[70],[21,4925,4927,4947],{"className":4926},[74,75],[21,4928,4930,4944],{"className":4929},[79],[21,4931,4933],{"className":4932,"style":84},[83],[21,4934,4935,4938],{"style":4079},[21,4936],{"className":4937,"style":92},[91],[21,4939,4941],{"className":4940},[96,97,98,99],[21,4942,103],{"className":4943},[42,43,99],[21,4945,108],{"className":4946},[107],[21,4948,4950],{"className":4949},[79],[21,4951,4953],{"className":4952,"style":115},[83],[21,4954],{},[21,4956,635],{"className":4957},[465],[21,4959],{"className":4960,"style":404},[378],[21,4962,963],{"className":4963},[408],[21,4965],{"className":4966,"style":404},[378],[21,4968,4970,4973,5022,5025,5088,5094],{"className":4969},[33],[21,4971],{"className":4972,"style":3827},[37],[21,4974,4976,4979],{"className":4975},[42],[21,4977,44],{"className":4978},[42,43],[21,4980,4982],{"className":4981},[70],[21,4983,4985,5014],{"className":4984},[74,75],[21,4986,4988,5011],{"className":4987},[79],[21,4989,4991],{"className":4990,"style":944},[83],[21,4992,4993,4996],{"style":87},[21,4994],{"className":4995,"style":92},[91],[21,4997,4999],{"className":4998},[96,97,98,99],[21,5000,5002,5005,5008],{"className":5001},[42,99],[21,5003,103],{"className":5004},[42,43,99],[21,5006,963],{"className":5007},[408,99],[21,5009,358],{"className":5010},[42,99],[21,5012,108],{"className":5013},[107],[21,5015,5017],{"className":5016},[79],[21,5018,5020],{"className":5019,"style":976},[83],[21,5021],{},[21,5023],{"className":5024,"style":458},[378],[21,5026,5028,5031],{"className":5027},[42],[21,5029,867],{"className":5030},[42,43],[21,5032,5034],{"className":5033},[70],[21,5035,5037,5080],{"className":5036},[74,75],[21,5038,5040,5077],{"className":5039},[79],[21,5041,5043,5063],{"className":5042,"style":3699},[83],[21,5044,5045,5048],{"style":3702},[21,5046],{"className":5047,"style":92},[91],[21,5049,5051],{"className":5050},[96,97,98,99],[21,5052,5054,5057,5060],{"className":5053},[42,99],[21,5055,103],{"className":5056},[42,43,99],[21,5058,963],{"className":5059},[408,99],[21,5061,358],{"className":5062},[42,99],[21,5064,5065,5068],{"style":3714},[21,5066],{"className":5067,"style":92},[91],[21,5069,5071],{"className":5070},[96,97,98,99],[21,5072,5074],{"className":5073},[42,99],[21,5075,4529],{"className":5076},[42,43,99],[21,5078,108],{"className":5079},[107],[21,5081,5083],{"className":5082},[79],[21,5084,5086],{"className":5085,"style":4234},[83],[21,5087],{},[21,5089,5091],{"className":5090},[42],[21,5092,635],{"className":5093},[1211,3883],[21,5095,454],{"className":5096},[453],[11,5098,5099],{},"and an Expected-Sarsa form that bootstraps from the expected value under the\ntarget policy,",[21,5101,5103],{"className":5102},[847],[21,5104,5106],{"className":5105},[24],[21,5107,5109,5184,5248,5324,5524,5753,5829],{"className":5108,"ariaHidden":29},[28],[21,5110,5112,5115,5169,5172,5175,5178,5181],{"className":5111},[33],[21,5113],{"className":5114,"style":3680},[37],[21,5116,5118,5121],{"className":5117},[42],[21,5119,867],{"className":5120},[42,43],[21,5122,5124],{"className":5123},[70],[21,5125,5127,5161],{"className":5126},[74,75],[21,5128,5130,5158],{"className":5129},[79],[21,5131,5133,5144],{"className":5132,"style":3699},[83],[21,5134,5135,5138],{"style":3702},[21,5136],{"className":5137,"style":92},[91],[21,5139,5141],{"className":5140},[96,97,98,99],[21,5142,103],{"className":5143},[42,43,99],[21,5145,5146,5149],{"style":3714},[21,5147],{"className":5148,"style":92},[91],[21,5150,5152],{"className":5151},[96,97,98,99],[21,5153,5155],{"className":5154},[42,99],[21,5156,4529],{"className":5157},[42,43,99],[21,5159,108],{"className":5160},[107],[21,5162,5164],{"className":5163},[79],[21,5165,5167],{"className":5166,"style":3739},[83],[21,5168],{},[21,5170],{"className":5171,"style":379},[378],[21,5173],{"className":5174,"style":379},[378],[21,5176,527],{"className":5177},[383],[21,5179],{"className":5180,"style":379},[378],[21,5182],{"className":5183,"style":379},[378],[21,5185,5187,5190,5239,5242,5245],{"className":5186},[33],[21,5188],{"className":5189,"style":923},[37],[21,5191,5193,5196],{"className":5192},[42],[21,5194,931],{"className":5195,"style":930},[42,43],[21,5197,5199],{"className":5198},[70],[21,5200,5202,5231],{"className":5201},[74,75],[21,5203,5205,5228],{"className":5204},[79],[21,5206,5208],{"className":5207,"style":944},[83],[21,5209,5210,5213],{"style":947},[21,5211],{"className":5212,"style":92},[91],[21,5214,5216],{"className":5215},[96,97,98,99],[21,5217,5219,5222,5225],{"className":5218},[42,99],[21,5220,103],{"className":5221},[42,43,99],[21,5223,963],{"className":5224},[408,99],[21,5226,358],{"className":5227},[42,99],[21,5229,108],{"className":5230},[107],[21,5232,5234],{"className":5233},[79],[21,5235,5237],{"className":5236,"style":976},[83],[21,5238],{},[21,5240],{"className":5241,"style":404},[378],[21,5243,963],{"className":5244},[408],[21,5246],{"className":5247,"style":404},[378],[21,5249,5251,5254,5303,5309,5312,5315,5318,5321],{"className":5250},[33],[21,5252],{"className":5253,"style":3827},[37],[21,5255,5257,5260],{"className":5256},[42],[21,5258,200],{"className":5259,"style":199},[42,43],[21,5261,5263],{"className":5262},[70],[21,5264,5266,5295],{"className":5265},[74,75],[21,5267,5269,5292],{"className":5268},[79],[21,5270,5272],{"className":5271,"style":944},[83],[21,5273,5274,5277],{"style":737},[21,5275],{"className":5276,"style":92},[91],[21,5278,5280],{"className":5279},[96,97,98,99],[21,5281,5283,5286,5289],{"className":5282},[42,99],[21,5284,103],{"className":5285},[42,43,99],[21,5287,963],{"className":5288},[408,99],[21,5290,358],{"className":5291},[42,99],[21,5293,108],{"className":5294},[107],[21,5296,5298],{"className":5297},[79],[21,5299,5301],{"className":5300,"style":976},[83],[21,5302],{},[21,5304,5306],{"className":5305},[42],[21,5307,543],{"className":5308},[1211,3883],[21,5310,543],{"className":5311},[444],[21,5313,358],{"className":5314},[42],[21,5316],{"className":5317,"style":404},[378],[21,5319,3896],{"className":5320},[408],[21,5322],{"className":5323,"style":404},[378],[21,5325,5327,5331,5380,5383,5386,5460,5463,5512,5515,5518,5521],{"className":5326},[33],[21,5328],{"className":5329,"style":5330},[37],"height:1.0701em;vertical-align:-0.25em;",[21,5332,5334,5337],{"className":5333},[42],[21,5335,44],{"className":5336},[42,43],[21,5338,5340],{"className":5339},[70],[21,5341,5343,5372],{"className":5342},[74,75],[21,5344,5346,5369],{"className":5345},[79],[21,5347,5349],{"className":5348,"style":944},[83],[21,5350,5351,5354],{"style":87},[21,5352],{"className":5353,"style":92},[91],[21,5355,5357],{"className":5356},[96,97,98,99],[21,5358,5360,5363,5366],{"className":5359},[42,99],[21,5361,103],{"className":5362},[42,43,99],[21,5364,963],{"className":5365},[408,99],[21,5367,358],{"className":5368},[42,99],[21,5370,108],{"className":5371},[107],[21,5373,5375],{"className":5374},[79],[21,5376,5378],{"className":5377,"style":976},[83],[21,5379],{},[21,5381,635],{"className":5382},[465],[21,5384],{"className":5385,"style":458},[378],[21,5387,5389,5425],{"className":5388},[42],[21,5390,5392],{"className":5391},[42,3964],[21,5393,5395],{"className":5394},[74],[21,5396,5398],{"className":5397},[79],[21,5399,5402,5411],{"className":5400,"style":5401},[83],"height:0.8201em;",[21,5403,5404,5407],{"style":3976},[21,5405],{"className":5406,"style":3980},[91],[21,5408,5410],{"className":5409,"style":404},[42,43],"V",[21,5412,5414,5417],{"style":5413},"top:-3.2523em;",[21,5415],{"className":5416,"style":3980},[91],[21,5418,5421],{"className":5419,"style":5420},[3993],"left:-0.25em;",[21,5422,5424],{"className":5423},[42],"ˉ",[21,5426,5428],{"className":5427},[70],[21,5429,5431,5452],{"className":5430},[74,75],[21,5432,5434,5449],{"className":5433},[79],[21,5435,5437],{"className":5436,"style":84},[83],[21,5438,5440,5443],{"style":5439},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[21,5441],{"className":5442,"style":92},[91],[21,5444,5446],{"className":5445},[96,97,98,99],[21,5447,103],{"className":5448},[42,43,99],[21,5450,108],{"className":5451},[107],[21,5453,5455],{"className":5454},[79],[21,5456,5458],{"className":5457,"style":115},[83],[21,5459],{},[21,5461,543],{"className":5462},[444],[21,5464,5466,5469],{"className":5465},[42],[21,5467,400],{"className":5468,"style":550},[42,43],[21,5470,5472],{"className":5471},[70],[21,5473,5475,5504],{"className":5474},[74,75],[21,5476,5478,5501],{"className":5477},[79],[21,5479,5481],{"className":5480,"style":944},[83],[21,5482,5483,5486],{"style":565},[21,5484],{"className":5485,"style":92},[91],[21,5487,5489],{"className":5488},[96,97,98,99],[21,5490,5492,5495,5498],{"className":5491},[42,99],[21,5493,103],{"className":5494},[42,43,99],[21,5496,963],{"className":5497},[408,99],[21,5499,358],{"className":5500},[42,99],[21,5502,108],{"className":5503},[107],[21,5505,5507],{"className":5506},[79],[21,5508,5510],{"className":5509,"style":976},[83],[21,5511],{},[21,5513,635],{"className":5514},[465],[21,5516],{"className":5517,"style":404},[378],[21,5519,963],{"className":5520},[408],[21,5522],{"className":5523,"style":404},[378],[21,5525,5527,5530,5579,5582,5645,5651,5654,5658,5661,5729,5732,5735,5738,5741,5744,5747,5750],{"className":5526},[33],[21,5528],{"className":5529,"style":3827},[37],[21,5531,5533,5536],{"className":5532},[42],[21,5534,44],{"className":5535},[42,43],[21,5537,5539],{"className":5538},[70],[21,5540,5542,5571],{"className":5541},[74,75],[21,5543,5545,5568],{"className":5544},[79],[21,5546,5548],{"className":5547,"style":944},[83],[21,5549,5550,5553],{"style":87},[21,5551],{"className":5552,"style":92},[91],[21,5554,5556],{"className":5555},[96,97,98,99],[21,5557,5559,5562,5565],{"className":5558},[42,99],[21,5560,103],{"className":5561},[42,43,99],[21,5563,963],{"className":5564},[408,99],[21,5566,358],{"className":5567},[42,99],[21,5569,108],{"className":5570},[107],[21,5572,5574],{"className":5573},[79],[21,5575,5577],{"className":5576,"style":976},[83],[21,5578],{},[21,5580],{"className":5581,"style":458},[378],[21,5583,5585,5588],{"className":5584},[42],[21,5586,867],{"className":5587},[42,43],[21,5589,5591],{"className":5590},[70],[21,5592,5594,5637],{"className":5593},[74,75],[21,5595,5597,5634],{"className":5596},[79],[21,5598,5600,5620],{"className":5599,"style":3699},[83],[21,5601,5602,5605],{"style":3702},[21,5603],{"className":5604,"style":92},[91],[21,5606,5608],{"className":5607},[96,97,98,99],[21,5609,5611,5614,5617],{"className":5610},[42,99],[21,5612,103],{"className":5613},[42,43,99],[21,5615,963],{"className":5616},[408,99],[21,5618,358],{"className":5619},[42,99],[21,5621,5622,5625],{"style":3714},[21,5623],{"className":5624,"style":92},[91],[21,5626,5628],{"className":5627},[96,97,98,99],[21,5629,5631],{"className":5630},[42,99],[21,5632,4529],{"className":5633},[42,43,99],[21,5635,108],{"className":5636},[107],[21,5638,5640],{"className":5639},[79],[21,5641,5643],{"className":5642,"style":4234},[83],[21,5644],{},[21,5646,5648],{"className":5647},[42],[21,5649,635],{"className":5650},[1211,3883],[21,5652,454],{"className":5653},[453],[21,5655],{"className":5656,"style":5657},[378],"margin-right:2em;",[21,5659],{"className":5660,"style":458},[378],[21,5662,5664,5695],{"className":5663},[42],[21,5665,5667],{"className":5666},[42,3964],[21,5668,5670],{"className":5669},[74],[21,5671,5673],{"className":5672},[79],[21,5674,5676,5684],{"className":5675,"style":5401},[83],[21,5677,5678,5681],{"style":3976},[21,5679],{"className":5680,"style":3980},[91],[21,5682,5410],{"className":5683,"style":404},[42,43],[21,5685,5686,5689],{"style":5413},[21,5687],{"className":5688,"style":3980},[91],[21,5690,5692],{"className":5691,"style":5420},[3993],[21,5693,5424],{"className":5694},[42],[21,5696,5698],{"className":5697},[70],[21,5699,5701,5721],{"className":5700},[74,75],[21,5702,5704,5718],{"className":5703},[79],[21,5705,5707],{"className":5706,"style":84},[83],[21,5708,5709,5712],{"style":5439},[21,5710],{"className":5711,"style":92},[91],[21,5713,5715],{"className":5714},[96,97,98,99],[21,5716,103],{"className":5717},[42,43,99],[21,5719,108],{"className":5720},[107],[21,5722,5724],{"className":5723},[79],[21,5725,5727],{"className":5726,"style":115},[83],[21,5728],{},[21,5730,543],{"className":5731},[444],[21,5733,1583],{"className":5734},[42,43],[21,5736,635],{"className":5737},[465],[21,5739],{"className":5740,"style":379},[378],[21,5742],{"className":5743,"style":379},[378],[21,5745,527],{"className":5746},[383],[21,5748],{"className":5749,"style":379},[378],[21,5751],{"className":5752,"style":379},[378],[21,5754,5756,5760,5807,5810,5813,5816,5819,5822,5826],{"className":5755},[33],[21,5757],{"className":5758,"style":5759},[37],"height:2.3em;vertical-align:-1.25em;",[21,5761,5763],{"className":5762},[1118,1119],[21,5764,5766,5798],{"className":5765},[74,75],[21,5767,5769,5795],{"className":5768},[79],[21,5770,5773,5785],{"className":5771,"style":5772},[83],"height:1.05em;",[21,5774,5776,5779],{"style":5775},"top:-1.9em;margin-left:0em;",[21,5777],{"className":5778,"style":1136},[91],[21,5780,5782],{"className":5781},[96,97,98,99],[21,5783,15],{"className":5784},[42,43,99],[21,5786,5787,5790],{"style":1156},[21,5788],{"className":5789,"style":1136},[91],[21,5791,5792],{},[21,5793,1167],{"className":5794},[1118,1165,1166],[21,5796,108],{"className":5797},[107],[21,5799,5801],{"className":5800},[79],[21,5802,5805],{"className":5803,"style":5804},[83],"height:1.25em;",[21,5806],{},[21,5808],{"className":5809,"style":458},[378],[21,5811,258],{"className":5812,"style":257},[42,43],[21,5814,543],{"className":5815},[444],[21,5817,15],{"className":5818},[42,43],[21,5820],{"className":5821,"style":379},[378],[21,5823,5825],{"className":5824},[383],"∣",[21,5827],{"className":5828,"style":379},[378],[21,5830,5832,5835,5838,5841,5844,5886,5889,5892,5895,5898,5901,5904,5907,5947,5950],{"className":5831},[33],[21,5833],{"className":5834,"style":440},[37],[21,5836,1583],{"className":5837},[42,43],[21,5839,635],{"className":5840},[465],[21,5842],{"className":5843,"style":458},[378],[21,5845,5847],{"className":5846},[42,3964],[21,5848,5850,5878],{"className":5849},[74,75],[21,5851,5853,5875],{"className":5852},[79],[21,5854,5856,5864],{"className":5855,"style":38},[83],[21,5857,5858,5861],{"style":3976},[21,5859],{"className":5860,"style":3980},[91],[21,5862,4777],{"className":5863,"style":257},[42,43],[21,5865,5866,5869],{"style":3976},[21,5867],{"className":5868,"style":3980},[91],[21,5870,5872],{"className":5871,"style":4786},[3993],[21,5873,3998],{"className":5874},[42],[21,5876,108],{"className":5877},[107],[21,5879,5881],{"className":5880},[79],[21,5882,5884],{"className":5883,"style":4799},[83],[21,5885],{},[21,5887,543],{"className":5888},[444],[21,5890,1583],{"className":5891},[42,43],[21,5893,454],{"className":5894},[453],[21,5896],{"className":5897,"style":458},[378],[21,5899,15],{"className":5900},[42,43],[21,5902,454],{"className":5903},[453],[21,5905],{"className":5906,"style":458},[378],[21,5908,5910,5913],{"className":5909},[42],[21,5911,4064],{"className":5912,"style":4063},[42,65],[21,5914,5916],{"className":5915},[70],[21,5917,5919,5939],{"className":5918},[74,75],[21,5920,5922,5936],{"className":5921},[79],[21,5923,5925],{"className":5924,"style":84},[83],[21,5926,5927,5930],{"style":4079},[21,5928],{"className":5929,"style":92},[91],[21,5931,5933],{"className":5932},[96,97,98,99],[21,5934,103],{"className":5935},[42,43,99],[21,5937,108],{"className":5938},[107],[21,5940,5942],{"className":5941},[79],[21,5943,5945],{"className":5944,"style":115},[83],[21,5946],{},[21,5948,635],{"className":5949},[465],[21,5951,4246],{"className":5952},[42],[11,5954,5955,5956,5981,5982,6052,6053,6123,6124,6139],{},"These recursions are the substrate for everything below. Constant ",[21,5957,5959],{"className":5958},[24],[21,5960,5962],{"className":5961,"ariaHidden":29},[28],[21,5963,5965,5969,5972,5975,5978],{"className":5964},[33],[21,5966],{"className":5967,"style":5968},[37],"height:0.8889em;vertical-align:-0.1944em;",[21,5970,44],{"className":5971},[42,43],[21,5973,454],{"className":5974},[453],[21,5976],{"className":5977,"style":458},[378],[21,5979,200],{"className":5980,"style":199},[42,43]," are the special case ",[21,5983,5985],{"className":5984},[24],[21,5986,5988,6043],{"className":5987,"ariaHidden":29},[28],[21,5989,5991,5994,6034,6037,6040],{"className":5990},[33],[21,5992],{"className":5993,"style":480},[37],[21,5995,5997,6000],{"className":5996},[42],[21,5998,44],{"className":5999},[42,43],[21,6001,6003],{"className":6002},[70],[21,6004,6006,6026],{"className":6005},[74,75],[21,6007,6009,6023],{"className":6008},[79],[21,6010,6012],{"className":6011,"style":84},[83],[21,6013,6014,6017],{"style":87},[21,6015],{"className":6016,"style":92},[91],[21,6018,6020],{"className":6019},[96,97,98,99],[21,6021,103],{"className":6022},[42,43,99],[21,6024,108],{"className":6025},[107],[21,6027,6029],{"className":6028},[79],[21,6030,6032],{"className":6031,"style":115},[83],[21,6033],{},[21,6035],{"className":6036,"style":379},[378],[21,6038,1683],{"className":6039},[383],[21,6041],{"className":6042,"style":379},[378],[21,6044,6046,6049],{"className":6045},[33],[21,6047],{"className":6048,"style":38},[37],[21,6050,44],{"className":6051},[42,43],", ",[21,6054,6056],{"className":6055},[24],[21,6057,6059,6114],{"className":6058,"ariaHidden":29},[28],[21,6060,6062,6065,6105,6108,6111],{"className":6061},[33],[21,6063],{"className":6064,"style":1673},[37],[21,6066,6068,6071],{"className":6067},[42],[21,6069,200],{"className":6070,"style":199},[42,43],[21,6072,6074],{"className":6073},[70],[21,6075,6077,6097],{"className":6076},[74,75],[21,6078,6080,6094],{"className":6079},[79],[21,6081,6083],{"className":6082,"style":84},[83],[21,6084,6085,6088],{"style":737},[21,6086],{"className":6087,"style":92},[91],[21,6089,6091],{"className":6090},[96,97,98,99],[21,6092,103],{"className":6093},[42,43,99],[21,6095,108],{"className":6096},[107],[21,6098,6100],{"className":6099},[79],[21,6101,6103],{"className":6102,"style":115},[83],[21,6104],{},[21,6106],{"className":6107,"style":379},[378],[21,6109,1683],{"className":6110},[383],[21,6112],{"className":6113,"style":379},[378],[21,6115,6117,6120],{"className":6116},[33],[21,6118],{"className":6119,"style":195},[37],[21,6121,200],{"className":6122,"style":199},[42,43],",\nwhich collapses each back to the ",[21,6125,6127],{"className":6126},[24],[21,6128,6130],{"className":6129,"ariaHidden":29},[28],[21,6131,6133,6136],{"className":6132},[33],[21,6134],{"className":6135,"style":38},[37],[21,6137,44],{"className":6138},[42,43],"-return of the previous lesson.",[342,6141,6143],{"id":6142},"off-policy-traces-with-control-variates","Off-policy traces with control variates",[11,6145,6146,6147,6162,6163,6178,6179,6195,6196,6199,6200,6215,6216,6219,6220,6227],{},"Now the data is off-policy: actions come from a behavior policy ",[21,6148,6150],{"className":6149},[24],[21,6151,6153],{"className":6152,"ariaHidden":29},[28],[21,6154,6156,6159],{"className":6155},[33],[21,6157],{"className":6158,"style":38},[37],[21,6160,275],{"className":6161},[42,43]," while we\nevaluate a target policy ",[21,6164,6166],{"className":6165},[24],[21,6167,6169],{"className":6168,"ariaHidden":29},[28],[21,6170,6172,6175],{"className":6171},[33],[21,6173],{"className":6174,"style":253},[37],[21,6176,258],{"className":6177,"style":257},[42,43],". For ",[21,6180,6182],{"className":6181},[24],[21,6183,6185],{"className":6184,"ariaHidden":29},[28],[21,6186,6188,6191],{"className":6187},[33],[21,6189],{"className":6190,"style":253},[37],[21,6192,6194],{"className":6193},[42,43],"n","-step methods, importance sampling was\napplied ",[237,6197,6198],{},"outside"," the return. That has no clean analogue for a full,\nnon-truncated ",[21,6201,6203],{"className":6202},[24],[21,6204,6206],{"className":6205,"ariaHidden":29},[28],[21,6207,6209,6212],{"className":6208},[33],[21,6210],{"className":6211,"style":38},[37],[21,6213,44],{"className":6214},[42,43],"-return, so we move directly to the modern construction:\nfold the per-decision importance ratio into the return itself, with a ",[824,6217,6218],{},"control\nvariate"," to hold the variance down.",[350,6221,6222],{},[15,6223,2325],{"href":6224,"ariaDescribedBy":6225,"dataFootnoteRef":6,"id":6226},"#user-content-fn-sb-cv",[356],"user-content-fnref-sb-cv"," The per-step ratio is the usual",[21,6229,6231],{"className":6230},[847],[21,6232,6234],{"className":6233},[24],[21,6235,6237,6298],{"className":6236,"ariaHidden":29},[28],[21,6238,6240,6243,6283,6286,6289,6292,6295],{"className":6239},[33],[21,6241],{"className":6242,"style":716},[37],[21,6244,6246,6249],{"className":6245},[42],[21,6247,295],{"className":6248},[42,43],[21,6250,6252],{"className":6251},[70],[21,6253,6255,6275],{"className":6254},[74,75],[21,6256,6258,6272],{"className":6257},[79],[21,6259,6261],{"className":6260,"style":84},[83],[21,6262,6263,6266],{"style":87},[21,6264],{"className":6265,"style":92},[91],[21,6267,6269],{"className":6268},[96,97,98,99],[21,6270,103],{"className":6271},[42,43,99],[21,6273,108],{"className":6274},[107],[21,6276,6278],{"className":6277},[79],[21,6279,6281],{"className":6280,"style":115},[83],[21,6282],{},[21,6284],{"className":6285,"style":379},[378],[21,6287],{"className":6288,"style":379},[378],[21,6290,527],{"className":6291},[383],[21,6293],{"className":6294,"style":379},[378],[21,6296],{"className":6297,"style":379},[378],[21,6299,6301,6305,6566],{"className":6300},[33],[21,6302],{"className":6303,"style":6304},[37],"height:2.363em;vertical-align:-0.936em;",[21,6306,6308,6312,6563],{"className":6307},[42],[21,6309],{"className":6310},[444,6311],"nulldelimiter",[21,6313,6316],{"className":6314},[6315],"mfrac",[21,6317,6319,6554],{"className":6318},[74,75],[21,6320,6322,6551],{"className":6321},[79],[21,6323,6326,6433,6444],{"className":6324,"style":6325},[83],"height:1.427em;",[21,6327,6329,6332],{"style":6328},"top:-2.314em;",[21,6330],{"className":6331,"style":3980},[91],[21,6333,6335,6338,6341,6381,6384,6387,6390,6430],{"className":6334},[42],[21,6336,275],{"className":6337},[42,43],[21,6339,543],{"className":6340},[444],[21,6342,6344,6347],{"className":6343},[42],[21,6345,423],{"className":6346},[42,43],[21,6348,6350],{"className":6349},[70],[21,6351,6353,6373],{"className":6352},[74,75],[21,6354,6356,6370],{"className":6355},[79],[21,6357,6359],{"className":6358,"style":84},[83],[21,6360,6361,6364],{"style":87},[21,6362],{"className":6363,"style":92},[91],[21,6365,6367],{"className":6366},[96,97,98,99],[21,6368,103],{"className":6369},[42,43,99],[21,6371,108],{"className":6372},[107],[21,6374,6376],{"className":6375},[79],[21,6377,6379],{"className":6378,"style":115},[83],[21,6380],{},[21,6382],{"className":6383,"style":379},[378],[21,6385,5825],{"className":6386},[383],[21,6388],{"className":6389,"style":379},[378],[21,6391,6393,6396],{"className":6392},[42],[21,6394,400],{"className":6395,"style":550},[42,43],[21,6397,6399],{"className":6398},[70],[21,6400,6402,6422],{"className":6401},[74,75],[21,6403,6405,6419],{"className":6404},[79],[21,6406,6408],{"className":6407,"style":84},[83],[21,6409,6410,6413],{"style":565},[21,6411],{"className":6412,"style":92},[91],[21,6414,6416],{"className":6415},[96,97,98,99],[21,6417,103],{"className":6418},[42,43,99],[21,6420,108],{"className":6421},[107],[21,6423,6425],{"className":6424},[79],[21,6426,6428],{"className":6427,"style":115},[83],[21,6429],{},[21,6431,635],{"className":6432},[465],[21,6434,6436,6439],{"style":6435},"top:-3.23em;",[21,6437],{"className":6438,"style":3980},[91],[21,6440],{"className":6441,"style":6443},[6442],"frac-line","border-bottom-width:0.04em;",[21,6445,6447,6450],{"style":6446},"top:-3.677em;",[21,6448],{"className":6449,"style":3980},[91],[21,6451,6453,6456,6459,6499,6502,6505,6508,6548],{"className":6452},[42],[21,6454,258],{"className":6455,"style":257},[42,43],[21,6457,543],{"className":6458},[444],[21,6460,6462,6465],{"className":6461},[42],[21,6463,423],{"className":6464},[42,43],[21,6466,6468],{"className":6467},[70],[21,6469,6471,6491],{"className":6470},[74,75],[21,6472,6474,6488],{"className":6473},[79],[21,6475,6477],{"className":6476,"style":84},[83],[21,6478,6479,6482],{"style":87},[21,6480],{"className":6481,"style":92},[91],[21,6483,6485],{"className":6484},[96,97,98,99],[21,6486,103],{"className":6487},[42,43,99],[21,6489,108],{"className":6490},[107],[21,6492,6494],{"className":6493},[79],[21,6495,6497],{"className":6496,"style":115},[83],[21,6498],{},[21,6500],{"className":6501,"style":379},[378],[21,6503,5825],{"className":6504},[383],[21,6506],{"className":6507,"style":379},[378],[21,6509,6511,6514],{"className":6510},[42],[21,6512,400],{"className":6513,"style":550},[42,43],[21,6515,6517],{"className":6516},[70],[21,6518,6520,6540],{"className":6519},[74,75],[21,6521,6523,6537],{"className":6522},[79],[21,6524,6526],{"className":6525,"style":84},[83],[21,6527,6528,6531],{"style":565},[21,6529],{"className":6530,"style":92},[91],[21,6532,6534],{"className":6533},[96,97,98,99],[21,6535,103],{"className":6536},[42,43,99],[21,6538,108],{"className":6539},[107],[21,6541,6543],{"className":6542},[79],[21,6544,6546],{"className":6545,"style":115},[83],[21,6547],{},[21,6549,635],{"className":6550},[465],[21,6552,108],{"className":6553},[107],[21,6555,6557],{"className":6556},[79],[21,6558,6561],{"className":6559,"style":6560},[83],"height:0.936em;",[21,6562],{},[21,6564],{"className":6565},[465,6311],[21,6567,4246],{"className":6568},[42],[11,6570,6571,6572,6587,6588,6640],{},"The state-based off-policy ",[21,6573,6575],{"className":6574},[24],[21,6576,6578],{"className":6577,"ariaHidden":29},[28],[21,6579,6581,6584],{"className":6580},[33],[21,6582],{"className":6583,"style":38},[37],[21,6585,44],{"className":6586},[42,43],"-return generalizes the recursion above by\nscaling the bootstrapped part by ",[21,6589,6591],{"className":6590},[24],[21,6592,6594],{"className":6593,"ariaHidden":29},[28],[21,6595,6597,6600],{"className":6596},[33],[21,6598],{"className":6599,"style":195},[37],[21,6601,6603,6606],{"className":6602},[42],[21,6604,295],{"className":6605},[42,43],[21,6607,6609],{"className":6608},[70],[21,6610,6612,6632],{"className":6611},[74,75],[21,6613,6615,6629],{"className":6614},[79],[21,6616,6618],{"className":6617,"style":84},[83],[21,6619,6620,6623],{"style":87},[21,6621],{"className":6622,"style":92},[91],[21,6624,6626],{"className":6625},[96,97,98,99],[21,6627,103],{"className":6628},[42,43,99],[21,6630,108],{"className":6631},[107],[21,6633,6635],{"className":6634},[79],[21,6636,6638],{"className":6637,"style":115},[83],[21,6639],{}," and adding a correction term that has\nzero mean but cancels much of the extra variance,",[21,6642,6644],{"className":6643},[847],[21,6645,6647],{"className":6646},[24],[21,6648,6650,6728,6838,6916,7115,7257,7278],{"className":6649,"ariaHidden":29},[28],[21,6651,6653,6656,6713,6716,6719,6722,6725],{"className":6652},[33],[21,6654],{"className":6655,"style":3680},[37],[21,6657,6659,6662],{"className":6658},[42],[21,6660,867],{"className":6661},[42,43],[21,6663,6665],{"className":6664},[70],[21,6666,6668,6705],{"className":6667},[74,75],[21,6669,6671,6702],{"className":6670},[79],[21,6672,6674,6685],{"className":6673,"style":3699},[83],[21,6675,6676,6679],{"style":3702},[21,6677],{"className":6678,"style":92},[91],[21,6680,6682],{"className":6681},[96,97,98,99],[21,6683,103],{"className":6684},[42,43,99],[21,6686,6687,6690],{"style":3714},[21,6688],{"className":6689,"style":92},[91],[21,6691,6693],{"className":6692},[96,97,98,99],[21,6694,6696,6699],{"className":6695},[42,99],[21,6697,44],{"className":6698},[42,43,99],[21,6700,1583],{"className":6701},[42,43,99],[21,6703,108],{"className":6704},[107],[21,6706,6708],{"className":6707},[79],[21,6709,6711],{"className":6710,"style":3739},[83],[21,6712],{},[21,6714],{"className":6715,"style":379},[378],[21,6717],{"className":6718,"style":379},[378],[21,6720,527],{"className":6721},[383],[21,6723],{"className":6724,"style":379},[378],[21,6726],{"className":6727,"style":379},[378],[21,6729,6731,6734,6774,6780,6829,6832,6835],{"className":6730},[33],[21,6732],{"className":6733,"style":3827},[37],[21,6735,6737,6740],{"className":6736},[42],[21,6738,295],{"className":6739},[42,43],[21,6741,6743],{"className":6742},[70],[21,6744,6746,6766],{"className":6745},[74,75],[21,6747,6749,6763],{"className":6748},[79],[21,6750,6752],{"className":6751,"style":84},[83],[21,6753,6754,6757],{"style":87},[21,6755],{"className":6756,"style":92},[91],[21,6758,6760],{"className":6759},[96,97,98,99],[21,6761,103],{"className":6762},[42,43,99],[21,6764,108],{"className":6765},[107],[21,6767,6769],{"className":6768},[79],[21,6770,6772],{"className":6771,"style":115},[83],[21,6773],{},[21,6775,6777],{"className":6776},[42],[21,6778,543],{"className":6779},[1211,3883],[21,6781,6783,6786],{"className":6782},[42],[21,6784,931],{"className":6785,"style":930},[42,43],[21,6787,6789],{"className":6788},[70],[21,6790,6792,6821],{"className":6791},[74,75],[21,6793,6795,6818],{"className":6794},[79],[21,6796,6798],{"className":6797,"style":944},[83],[21,6799,6800,6803],{"style":947},[21,6801],{"className":6802,"style":92},[91],[21,6804,6806],{"className":6805},[96,97,98,99],[21,6807,6809,6812,6815],{"className":6808},[42,99],[21,6810,103],{"className":6811},[42,43,99],[21,6813,963],{"className":6814},[408,99],[21,6816,358],{"className":6817},[42,99],[21,6819,108],{"className":6820},[107],[21,6822,6824],{"className":6823},[79],[21,6825,6827],{"className":6826,"style":976},[83],[21,6828],{},[21,6830],{"className":6831,"style":404},[378],[21,6833,963],{"className":6834},[408],[21,6836],{"className":6837,"style":404},[378],[21,6839,6841,6845,6894,6901,6904,6907,6910,6913],{"className":6840},[33],[21,6842],{"className":6843,"style":6844},[37],"height:1.2em;vertical-align:-0.35em;",[21,6846,6848,6851],{"className":6847},[42],[21,6849,200],{"className":6850,"style":199},[42,43],[21,6852,6854],{"className":6853},[70],[21,6855,6857,6886],{"className":6856},[74,75],[21,6858,6860,6883],{"className":6859},[79],[21,6861,6863],{"className":6862,"style":944},[83],[21,6864,6865,6868],{"style":737},[21,6866],{"className":6867,"style":92},[91],[21,6869,6871],{"className":6870},[96,97,98,99],[21,6872,6874,6877,6880],{"className":6873},[42,99],[21,6875,103],{"className":6876},[42,43,99],[21,6878,963],{"className":6879},[408,99],[21,6881,358],{"className":6882},[42,99],[21,6884,108],{"className":6885},[107],[21,6887,6889],{"className":6888},[79],[21,6890,6892],{"className":6891,"style":976},[83],[21,6893],{},[21,6895,6897],{"className":6896},[42],[21,6898,543],{"className":6899},[1211,6900],"size1",[21,6902,543],{"className":6903},[444],[21,6905,358],{"className":6906},[42],[21,6908],{"className":6909,"style":404},[378],[21,6911,3896],{"className":6912},[408],[21,6914],{"className":6915,"style":404},[378],[21,6917,6919,6922,6971,6974,7005,7008,7057,7060,7063,7103,7106,7109,7112],{"className":6918},[33],[21,6920],{"className":6921,"style":440},[37],[21,6923,6925,6928],{"className":6924},[42],[21,6926,44],{"className":6927},[42,43],[21,6929,6931],{"className":6930},[70],[21,6932,6934,6963],{"className":6933},[74,75],[21,6935,6937,6960],{"className":6936},[79],[21,6938,6940],{"className":6939,"style":944},[83],[21,6941,6942,6945],{"style":87},[21,6943],{"className":6944,"style":92},[91],[21,6946,6948],{"className":6947},[96,97,98,99],[21,6949,6951,6954,6957],{"className":6950},[42,99],[21,6952,103],{"className":6953},[42,43,99],[21,6955,963],{"className":6956},[408,99],[21,6958,358],{"className":6959},[42,99],[21,6961,108],{"className":6962},[107],[21,6964,6966],{"className":6965},[79],[21,6967,6969],{"className":6968,"style":976},[83],[21,6970],{},[21,6972,635],{"className":6973},[465],[21,6975,6977],{"className":6976},[42,3964],[21,6978,6980],{"className":6979},[74],[21,6981,6983],{"className":6982},[79],[21,6984,6986,6994],{"className":6985,"style":38},[83],[21,6987,6988,6991],{"style":3976},[21,6989],{"className":6990,"style":3980},[91],[21,6992,3984],{"className":6993,"style":257},[42,43],[21,6995,6996,6999],{"style":3976},[21,6997],{"className":6998,"style":3980},[91],[21,7000,7002],{"className":7001,"style":3994},[3993],[21,7003,3998],{"className":7004},[42],[21,7006,543],{"className":7007},[444],[21,7009,7011,7014],{"className":7010},[42],[21,7012,400],{"className":7013,"style":550},[42,43],[21,7015,7017],{"className":7016},[70],[21,7018,7020,7049],{"className":7019},[74,75],[21,7021,7023,7046],{"className":7022},[79],[21,7024,7026],{"className":7025,"style":944},[83],[21,7027,7028,7031],{"style":565},[21,7029],{"className":7030,"style":92},[91],[21,7032,7034],{"className":7033},[96,97,98,99],[21,7035,7037,7040,7043],{"className":7036},[42,99],[21,7038,103],{"className":7039},[42,43,99],[21,7041,963],{"className":7042},[408,99],[21,7044,358],{"className":7045},[42,99],[21,7047,108],{"className":7048},[107],[21,7050,7052],{"className":7051},[79],[21,7053,7055],{"className":7054,"style":976},[83],[21,7056],{},[21,7058,454],{"className":7059},[453],[21,7061],{"className":7062,"style":458},[378],[21,7064,7066,7069],{"className":7065},[42],[21,7067,4064],{"className":7068,"style":4063},[42,65],[21,7070,7072],{"className":7071},[70],[21,7073,7075,7095],{"className":7074},[74,75],[21,7076,7078,7092],{"className":7077},[79],[21,7079,7081],{"className":7080,"style":84},[83],[21,7082,7083,7086],{"style":4079},[21,7084],{"className":7085,"style":92},[91],[21,7087,7089],{"className":7088},[96,97,98,99],[21,7090,103],{"className":7091},[42,43,99],[21,7093,108],{"className":7094},[107],[21,7096,7098],{"className":7097},[79],[21,7099,7101],{"className":7100,"style":115},[83],[21,7102],{},[21,7104,635],{"className":7105},[465],[21,7107],{"className":7108,"style":404},[378],[21,7110,963],{"className":7111},[408],[21,7113],{"className":7114,"style":404},[378],[21,7116,7118,7121,7170,7236,7242,7248,7251,7254],{"className":7117},[33],[21,7119],{"className":7120,"style":3827},[37],[21,7122,7124,7127],{"className":7123},[42],[21,7125,44],{"className":7126},[42,43],[21,7128,7130],{"className":7129},[70],[21,7131,7133,7162],{"className":7132},[74,75],[21,7134,7136,7159],{"className":7135},[79],[21,7137,7139],{"className":7138,"style":944},[83],[21,7140,7141,7144],{"style":87},[21,7142],{"className":7143,"style":92},[91],[21,7145,7147],{"className":7146},[96,97,98,99],[21,7148,7150,7153,7156],{"className":7149},[42,99],[21,7151,103],{"className":7152},[42,43,99],[21,7154,963],{"className":7155},[408,99],[21,7157,358],{"className":7158},[42,99],[21,7160,108],{"className":7161},[107],[21,7163,7165],{"className":7164},[79],[21,7166,7168],{"className":7167,"style":976},[83],[21,7169],{},[21,7171,7173,7176],{"className":7172},[42],[21,7174,867],{"className":7175},[42,43],[21,7177,7179],{"className":7178},[70],[21,7180,7182,7228],{"className":7181},[74,75],[21,7183,7185,7225],{"className":7184},[79],[21,7186,7188,7208],{"className":7187,"style":3699},[83],[21,7189,7190,7193],{"style":3702},[21,7191],{"className":7192,"style":92},[91],[21,7194,7196],{"className":7195},[96,97,98,99],[21,7197,7199,7202,7205],{"className":7198},[42,99],[21,7200,103],{"className":7201},[42,43,99],[21,7203,963],{"className":7204},[408,99],[21,7206,358],{"className":7207},[42,99],[21,7209,7210,7213],{"style":3714},[21,7211],{"className":7212,"style":92},[91],[21,7214,7216],{"className":7215},[96,97,98,99],[21,7217,7219,7222],{"className":7218},[42,99],[21,7220,44],{"className":7221},[42,43,99],[21,7223,1583],{"className":7224},[42,43,99],[21,7226,108],{"className":7227},[107],[21,7229,7231],{"className":7230},[79],[21,7232,7234],{"className":7233,"style":4234},[83],[21,7235],{},[21,7237,7239],{"className":7238},[42],[21,7240,635],{"className":7241},[1211,6900],[21,7243,7245],{"className":7244},[42],[21,7246,635],{"className":7247},[1211,3883],[21,7249],{"className":7250,"style":404},[378],[21,7252,963],{"className":7253},[408],[21,7255],{"className":7256,"style":404},[378],[21,7258,7260,7263,7266,7269,7272,7275],{"className":7259},[33],[21,7261],{"className":7262,"style":440},[37],[21,7264,543],{"className":7265},[444],[21,7267,358],{"className":7268},[42],[21,7270],{"className":7271,"style":404},[378],[21,7273,3896],{"className":7274},[408],[21,7276],{"className":7277,"style":404},[378],[21,7279,7281,7284,7324,7327,7330,7361,7364,7404,7407,7410,7450,7453],{"className":7280},[33],[21,7282],{"className":7283,"style":440},[37],[21,7285,7287,7290],{"className":7286},[42],[21,7288,295],{"className":7289},[42,43],[21,7291,7293],{"className":7292},[70],[21,7294,7296,7316],{"className":7295},[74,75],[21,7297,7299,7313],{"className":7298},[79],[21,7300,7302],{"className":7301,"style":84},[83],[21,7303,7304,7307],{"style":87},[21,7305],{"className":7306,"style":92},[91],[21,7308,7310],{"className":7309},[96,97,98,99],[21,7311,103],{"className":7312},[42,43,99],[21,7314,108],{"className":7315},[107],[21,7317,7319],{"className":7318},[79],[21,7320,7322],{"className":7321,"style":115},[83],[21,7323],{},[21,7325,635],{"className":7326},[465],[21,7328],{"className":7329,"style":458},[378],[21,7331,7333],{"className":7332},[42,3964],[21,7334,7336],{"className":7335},[74],[21,7337,7339],{"className":7338},[79],[21,7340,7342,7350],{"className":7341,"style":38},[83],[21,7343,7344,7347],{"style":3976},[21,7345],{"className":7346,"style":3980},[91],[21,7348,3984],{"className":7349,"style":257},[42,43],[21,7351,7352,7355],{"style":3976},[21,7353],{"className":7354,"style":3980},[91],[21,7356,7358],{"className":7357,"style":3994},[3993],[21,7359,3998],{"className":7360},[42],[21,7362,543],{"className":7363},[444],[21,7365,7367,7370],{"className":7366},[42],[21,7368,400],{"className":7369,"style":550},[42,43],[21,7371,7373],{"className":7372},[70],[21,7374,7376,7396],{"className":7375},[74,75],[21,7377,7379,7393],{"className":7378},[79],[21,7380,7382],{"className":7381,"style":84},[83],[21,7383,7384,7387],{"style":565},[21,7385],{"className":7386,"style":92},[91],[21,7388,7390],{"className":7389},[96,97,98,99],[21,7391,103],{"className":7392},[42,43,99],[21,7394,108],{"className":7395},[107],[21,7397,7399],{"className":7398},[79],[21,7400,7402],{"className":7401,"style":115},[83],[21,7403],{},[21,7405,454],{"className":7406},[453],[21,7408],{"className":7409,"style":458},[378],[21,7411,7413,7416],{"className":7412},[42],[21,7414,4064],{"className":7415,"style":4063},[42,65],[21,7417,7419],{"className":7418},[70],[21,7420,7422,7442],{"className":7421},[74,75],[21,7423,7425,7439],{"className":7424},[79],[21,7426,7428],{"className":7427,"style":84},[83],[21,7429,7430,7433],{"style":4079},[21,7431],{"className":7432,"style":92},[91],[21,7434,7436],{"className":7435},[96,97,98,99],[21,7437,103],{"className":7438},[42,43,99],[21,7440,108],{"className":7441},[107],[21,7443,7445],{"className":7444},[79],[21,7446,7448],{"className":7447,"style":115},[83],[21,7449],{},[21,7451,635],{"className":7452},[465],[21,7454,4246],{"className":7455},[42],[11,7457,7458,7459,7658,7659,7730,7731,7849,7850,7902,7903,7918],{},"The second term ",[21,7460,7462],{"className":7461},[24],[21,7463,7465,7486],{"className":7464,"ariaHidden":29},[28],[21,7466,7468,7471,7474,7477,7480,7483],{"className":7467},[33],[21,7469],{"className":7470,"style":440},[37],[21,7472,543],{"className":7473},[444],[21,7475,358],{"className":7476},[42],[21,7478],{"className":7479,"style":404},[378],[21,7481,3896],{"className":7482},[408],[21,7484],{"className":7485,"style":404},[378],[21,7487,7489,7492,7532,7535,7566,7569,7609,7612,7615,7655],{"className":7488},[33],[21,7490],{"className":7491,"style":440},[37],[21,7493,7495,7498],{"className":7494},[42],[21,7496,295],{"className":7497},[42,43],[21,7499,7501],{"className":7500},[70],[21,7502,7504,7524],{"className":7503},[74,75],[21,7505,7507,7521],{"className":7506},[79],[21,7508,7510],{"className":7509,"style":84},[83],[21,7511,7512,7515],{"style":87},[21,7513],{"className":7514,"style":92},[91],[21,7516,7518],{"className":7517},[96,97,98,99],[21,7519,103],{"className":7520},[42,43,99],[21,7522,108],{"className":7523},[107],[21,7525,7527],{"className":7526},[79],[21,7528,7530],{"className":7529,"style":115},[83],[21,7531],{},[21,7533,635],{"className":7534},[465],[21,7536,7538],{"className":7537},[42,3964],[21,7539,7541],{"className":7540},[74],[21,7542,7544],{"className":7543},[79],[21,7545,7547,7555],{"className":7546,"style":38},[83],[21,7548,7549,7552],{"style":3976},[21,7550],{"className":7551,"style":3980},[91],[21,7553,3984],{"className":7554,"style":257},[42,43],[21,7556,7557,7560],{"style":3976},[21,7558],{"className":7559,"style":3980},[91],[21,7561,7563],{"className":7562,"style":3994},[3993],[21,7564,3998],{"className":7565},[42],[21,7567,543],{"className":7568},[444],[21,7570,7572,7575],{"className":7571},[42],[21,7573,400],{"className":7574,"style":550},[42,43],[21,7576,7578],{"className":7577},[70],[21,7579,7581,7601],{"className":7580},[74,75],[21,7582,7584,7598],{"className":7583},[79],[21,7585,7587],{"className":7586,"style":84},[83],[21,7588,7589,7592],{"style":565},[21,7590],{"className":7591,"style":92},[91],[21,7593,7595],{"className":7594},[96,97,98,99],[21,7596,103],{"className":7597},[42,43,99],[21,7599,108],{"className":7600},[107],[21,7602,7604],{"className":7603},[79],[21,7605,7607],{"className":7606,"style":115},[83],[21,7608],{},[21,7610,454],{"className":7611},[453],[21,7613],{"className":7614,"style":458},[378],[21,7616,7618,7621],{"className":7617},[42],[21,7619,4064],{"className":7620,"style":4063},[42,65],[21,7622,7624],{"className":7623},[70],[21,7625,7627,7647],{"className":7626},[74,75],[21,7628,7630,7644],{"className":7629},[79],[21,7631,7633],{"className":7632,"style":84},[83],[21,7634,7635,7638],{"style":4079},[21,7636],{"className":7637,"style":92},[91],[21,7639,7641],{"className":7640},[96,97,98,99],[21,7642,103],{"className":7643},[42,43,99],[21,7645,108],{"className":7646},[107],[21,7648,7650],{"className":7649},[79],[21,7651,7653],{"className":7652,"style":115},[83],[21,7654],{},[21,7656,635],{"className":7657},[465]," is the control variate: its\nexpectation over ",[21,7660,7662],{"className":7661},[24],[21,7663,7665,7721],{"className":7664,"ariaHidden":29},[28],[21,7666,7668,7671,7711,7714,7718],{"className":7667},[33],[21,7669],{"className":7670,"style":860},[37],[21,7672,7674,7677],{"className":7673},[42],[21,7675,423],{"className":7676},[42,43],[21,7678,7680],{"className":7679},[70],[21,7681,7683,7703],{"className":7682},[74,75],[21,7684,7686,7700],{"className":7685},[79],[21,7687,7689],{"className":7688,"style":84},[83],[21,7690,7691,7694],{"style":87},[21,7692],{"className":7693,"style":92},[91],[21,7695,7697],{"className":7696},[96,97,98,99],[21,7698,103],{"className":7699},[42,43,99],[21,7701,108],{"className":7702},[107],[21,7704,7706],{"className":7705},[79],[21,7707,7709],{"className":7708,"style":115},[83],[21,7710],{},[21,7712],{"className":7713,"style":379},[378],[21,7715,7717],{"className":7716},[383],"∼",[21,7719],{"className":7720,"style":379},[378],[21,7722,7724,7727],{"className":7723},[33],[21,7725],{"className":7726,"style":38},[37],[21,7728,275],{"className":7729},[42,43]," is zero (because ",[21,7732,7734],{"className":7733},[24],[21,7735,7737,7840],{"className":7736,"ariaHidden":29},[28],[21,7738,7740,7743,7785,7788,7828,7831,7834,7837],{"className":7739},[33],[21,7741],{"className":7742,"style":440},[37],[21,7744,7746,7751],{"className":7745},[42],[21,7747,7750],{"className":7748},[42,7749],"mathbb","E",[21,7752,7754],{"className":7753},[70],[21,7755,7757,7777],{"className":7756},[74,75],[21,7758,7760,7774],{"className":7759},[79],[21,7761,7763],{"className":7762,"style":1362},[83],[21,7764,7765,7768],{"style":87},[21,7766],{"className":7767,"style":92},[91],[21,7769,7771],{"className":7770},[96,97,98,99],[21,7772,275],{"className":7773},[42,43,99],[21,7775,108],{"className":7776},[107],[21,7778,7780],{"className":7779},[79],[21,7781,7783],{"className":7782,"style":115},[83],[21,7784],{},[21,7786,445],{"className":7787},[444],[21,7789,7791,7794],{"className":7790},[42],[21,7792,295],{"className":7793},[42,43],[21,7795,7797],{"className":7796},[70],[21,7798,7800,7820],{"className":7799},[74,75],[21,7801,7803,7817],{"className":7802},[79],[21,7804,7806],{"className":7805,"style":84},[83],[21,7807,7808,7811],{"style":87},[21,7809],{"className":7810,"style":92},[91],[21,7812,7814],{"className":7813},[96,97,98,99],[21,7815,103],{"className":7816},[42,43,99],[21,7818,108],{"className":7819},[107],[21,7821,7823],{"className":7822},[79],[21,7824,7826],{"className":7825,"style":115},[83],[21,7827],{},[21,7829,466],{"className":7830},[465],[21,7832],{"className":7833,"style":379},[378],[21,7835,1101],{"className":7836},[383],[21,7838],{"className":7839,"style":379},[378],[21,7841,7843,7846],{"className":7842},[33],[21,7844],{"className":7845,"style":1540},[37],[21,7847,358],{"className":7848},[42],"), so it\ndoes not bias the target, but when ",[21,7851,7853],{"className":7852},[24],[21,7854,7856],{"className":7855,"ariaHidden":29},[28],[21,7857,7859,7862],{"className":7858},[33],[21,7860],{"className":7861,"style":195},[37],[21,7863,7865,7868],{"className":7864},[42],[21,7866,295],{"className":7867},[42,43],[21,7869,7871],{"className":7870},[70],[21,7872,7874,7894],{"className":7873},[74,75],[21,7875,7877,7891],{"className":7876},[79],[21,7878,7880],{"className":7879,"style":84},[83],[21,7881,7882,7885],{"style":87},[21,7883],{"className":7884,"style":92},[91],[21,7886,7888],{"className":7887},[96,97,98,99],[21,7889,103],{"className":7890},[42,43,99],[21,7892,108],{"className":7893},[107],[21,7895,7897],{"className":7896},[79],[21,7898,7900],{"className":7899,"style":115},[83],[21,7901],{}," is far from ",[21,7904,7906],{"className":7905},[24],[21,7907,7909],{"className":7908,"ariaHidden":29},[28],[21,7910,7912,7915],{"className":7911},[33],[21,7913],{"className":7914,"style":1540},[37],[21,7916,358],{"className":7917},[42]," it pulls the return\nback toward the current estimate and dampens the swing.",[1651,7920],{"hash":7921},"fcfde34023cf4df69cb83817bae4904b9f5c9c74232ac9c68883650811acd6c0",[11,7923,7924,7927,7928,7980,7981,8116,8117,8252,8253,8343,8344,8360,8361,8468,8469,8559,8560,8579,8580,8595,8596,8745,8746,8765,8766,8769,8770,6052,8840,8919,8920,9052,9053,9068,9069,9084,9085,9100,9101,9173,9174,9177,9178,4246],{},[824,7925,7926],{},"A worked correction."," Suppose at ",[21,7929,7931],{"className":7930},[24],[21,7932,7934],{"className":7933,"ariaHidden":29},[28],[21,7935,7937,7940],{"className":7936},[33],[21,7938],{"className":7939,"style":860},[37],[21,7941,7943,7946],{"className":7942},[42],[21,7944,400],{"className":7945,"style":550},[42,43],[21,7947,7949],{"className":7948},[70],[21,7950,7952,7972],{"className":7951},[74,75],[21,7953,7955,7969],{"className":7954},[79],[21,7956,7958],{"className":7957,"style":84},[83],[21,7959,7960,7963],{"style":565},[21,7961],{"className":7962,"style":92},[91],[21,7964,7966],{"className":7965},[96,97,98,99],[21,7967,103],{"className":7968},[42,43,99],[21,7970,108],{"className":7971},[107],[21,7973,7975],{"className":7974},[79],[21,7976,7978],{"className":7977,"style":115},[83],[21,7979],{}," the target policy would take the sampled\naction with probability ",[21,7982,7984],{"className":7983},[24],[21,7985,7987,8048,8106],{"className":7986,"ariaHidden":29},[28],[21,7988,7990,7993,7996,7999,8039,8042,8045],{"className":7989},[33],[21,7991],{"className":7992,"style":440},[37],[21,7994,258],{"className":7995,"style":257},[42,43],[21,7997,543],{"className":7998},[444],[21,8000,8002,8005],{"className":8001},[42],[21,8003,423],{"className":8004},[42,43],[21,8006,8008],{"className":8007},[70],[21,8009,8011,8031],{"className":8010},[74,75],[21,8012,8014,8028],{"className":8013},[79],[21,8015,8017],{"className":8016,"style":84},[83],[21,8018,8019,8022],{"style":87},[21,8020],{"className":8021,"style":92},[91],[21,8023,8025],{"className":8024},[96,97,98,99],[21,8026,103],{"className":8027},[42,43,99],[21,8029,108],{"className":8030},[107],[21,8032,8034],{"className":8033},[79],[21,8035,8037],{"className":8036,"style":115},[83],[21,8038],{},[21,8040],{"className":8041,"style":379},[378],[21,8043,5825],{"className":8044},[383],[21,8046],{"className":8047,"style":379},[378],[21,8049,8051,8054,8094,8097,8100,8103],{"className":8050},[33],[21,8052],{"className":8053,"style":440},[37],[21,8055,8057,8060],{"className":8056},[42],[21,8058,400],{"className":8059,"style":550},[42,43],[21,8061,8063],{"className":8062},[70],[21,8064,8066,8086],{"className":8065},[74,75],[21,8067,8069,8083],{"className":8068},[79],[21,8070,8072],{"className":8071,"style":84},[83],[21,8073,8074,8077],{"style":565},[21,8075],{"className":8076,"style":92},[91],[21,8078,8080],{"className":8079},[96,97,98,99],[21,8081,103],{"className":8082},[42,43,99],[21,8084,108],{"className":8085},[107],[21,8087,8089],{"className":8088},[79],[21,8090,8092],{"className":8091,"style":115},[83],[21,8093],{},[21,8095,635],{"className":8096},[465],[21,8098],{"className":8099,"style":379},[378],[21,8101,1101],{"className":8102},[383],[21,8104],{"className":8105,"style":379},[378],[21,8107,8109,8112],{"className":8108},[33],[21,8110],{"className":8111,"style":1540},[37],[21,8113,8115],{"className":8114},[42],"0.8"," while the behavior policy took it\nwith ",[21,8118,8120],{"className":8119},[24],[21,8121,8123,8184,8242],{"className":8122,"ariaHidden":29},[28],[21,8124,8126,8129,8132,8135,8175,8178,8181],{"className":8125},[33],[21,8127],{"className":8128,"style":440},[37],[21,8130,275],{"className":8131},[42,43],[21,8133,543],{"className":8134},[444],[21,8136,8138,8141],{"className":8137},[42],[21,8139,423],{"className":8140},[42,43],[21,8142,8144],{"className":8143},[70],[21,8145,8147,8167],{"className":8146},[74,75],[21,8148,8150,8164],{"className":8149},[79],[21,8151,8153],{"className":8152,"style":84},[83],[21,8154,8155,8158],{"style":87},[21,8156],{"className":8157,"style":92},[91],[21,8159,8161],{"className":8160},[96,97,98,99],[21,8162,103],{"className":8163},[42,43,99],[21,8165,108],{"className":8166},[107],[21,8168,8170],{"className":8169},[79],[21,8171,8173],{"className":8172,"style":115},[83],[21,8174],{},[21,8176],{"className":8177,"style":379},[378],[21,8179,5825],{"className":8180},[383],[21,8182],{"className":8183,"style":379},[378],[21,8185,8187,8190,8230,8233,8236,8239],{"className":8186},[33],[21,8188],{"className":8189,"style":440},[37],[21,8191,8193,8196],{"className":8192},[42],[21,8194,400],{"className":8195,"style":550},[42,43],[21,8197,8199],{"className":8198},[70],[21,8200,8202,8222],{"className":8201},[74,75],[21,8203,8205,8219],{"className":8204},[79],[21,8206,8208],{"className":8207,"style":84},[83],[21,8209,8210,8213],{"style":565},[21,8211],{"className":8212,"style":92},[91],[21,8214,8216],{"className":8215},[96,97,98,99],[21,8217,103],{"className":8218},[42,43,99],[21,8220,108],{"className":8221},[107],[21,8223,8225],{"className":8224},[79],[21,8226,8228],{"className":8227,"style":115},[83],[21,8229],{},[21,8231,635],{"className":8232},[465],[21,8234],{"className":8235,"style":379},[378],[21,8237,1101],{"className":8238},[383],[21,8240],{"className":8241,"style":379},[378],[21,8243,8245,8248],{"className":8244},[33],[21,8246],{"className":8247,"style":1540},[37],[21,8249,8251],{"className":8250},[42],"0.2",", so ",[21,8254,8256],{"className":8255},[24],[21,8257,8259,8314,8333],{"className":8258,"ariaHidden":29},[28],[21,8260,8262,8265,8305,8308,8311],{"className":8261},[33],[21,8263],{"className":8264,"style":195},[37],[21,8266,8268,8271],{"className":8267},[42],[21,8269,295],{"className":8270},[42,43],[21,8272,8274],{"className":8273},[70],[21,8275,8277,8297],{"className":8276},[74,75],[21,8278,8280,8294],{"className":8279},[79],[21,8281,8283],{"className":8282,"style":84},[83],[21,8284,8285,8288],{"style":87},[21,8286],{"className":8287,"style":92},[91],[21,8289,8291],{"className":8290},[96,97,98,99],[21,8292,103],{"className":8293},[42,43,99],[21,8295,108],{"className":8296},[107],[21,8298,8300],{"className":8299},[79],[21,8301,8303],{"className":8302,"style":115},[83],[21,8304],{},[21,8306],{"className":8307,"style":379},[378],[21,8309,1101],{"className":8310},[383],[21,8312],{"className":8313,"style":379},[378],[21,8315,8317,8320,8324,8327,8330],{"className":8316},[33],[21,8318],{"className":8319,"style":440},[37],[21,8321,8323],{"className":8322},[42],"0.8\u002F0.2",[21,8325],{"className":8326,"style":379},[378],[21,8328,1101],{"className":8329},[383],[21,8331],{"className":8332,"style":379},[378],[21,8334,8336,8339],{"className":8335},[33],[21,8337],{"className":8338,"style":1540},[37],[21,8340,8342],{"className":8341},[42],"4",". Let the bracketed one-step\ntarget evaluate to ",[21,8345,8347],{"className":8346},[24],[21,8348,8350],{"className":8349,"ariaHidden":29},[28],[21,8351,8353,8356],{"className":8352},[33],[21,8354],{"className":8355,"style":1540},[37],[21,8357,8359],{"className":8358},[42],"12"," and the current estimate be ",[21,8362,8364],{"className":8363},[24],[21,8365,8367,8459],{"className":8366,"ariaHidden":29},[28],[21,8368,8370,8373,8404,8407,8447,8450,8453,8456],{"className":8369},[33],[21,8371],{"className":8372,"style":440},[37],[21,8374,8376],{"className":8375},[42,3964],[21,8377,8379],{"className":8378},[74],[21,8380,8382],{"className":8381},[79],[21,8383,8385,8393],{"className":8384,"style":38},[83],[21,8386,8387,8390],{"style":3976},[21,8388],{"className":8389,"style":3980},[91],[21,8391,3984],{"className":8392,"style":257},[42,43],[21,8394,8395,8398],{"style":3976},[21,8396],{"className":8397,"style":3980},[91],[21,8399,8401],{"className":8400,"style":3994},[3993],[21,8402,3998],{"className":8403},[42],[21,8405,543],{"className":8406},[444],[21,8408,8410,8413],{"className":8409},[42],[21,8411,400],{"className":8412,"style":550},[42,43],[21,8414,8416],{"className":8415},[70],[21,8417,8419,8439],{"className":8418},[74,75],[21,8420,8422,8436],{"className":8421},[79],[21,8423,8425],{"className":8424,"style":84},[83],[21,8426,8427,8430],{"style":565},[21,8428],{"className":8429,"style":92},[91],[21,8431,8433],{"className":8432},[96,97,98,99],[21,8434,103],{"className":8435},[42,43,99],[21,8437,108],{"className":8438},[107],[21,8440,8442],{"className":8441},[79],[21,8443,8445],{"className":8444,"style":115},[83],[21,8446],{},[21,8448,635],{"className":8449},[465],[21,8451],{"className":8452,"style":379},[378],[21,8454,1101],{"className":8455},[383],[21,8457],{"className":8458,"style":379},[378],[21,8460,8462,8465],{"className":8461},[33],[21,8463],{"className":8464,"style":1540},[37],[21,8466,3608],{"className":8467},[42],". Without the\ncontrol variate the return is ",[21,8470,8472],{"className":8471},[24],[21,8473,8475,8531,8549],{"className":8474,"ariaHidden":29},[28],[21,8476,8478,8482,8522,8525,8528],{"className":8477},[33],[21,8479],{"className":8480,"style":8481},[37],"height:0.6389em;vertical-align:-0.1944em;",[21,8483,8485,8488],{"className":8484},[42],[21,8486,295],{"className":8487},[42,43],[21,8489,8491],{"className":8490},[70],[21,8492,8494,8514],{"className":8493},[74,75],[21,8495,8497,8511],{"className":8496},[79],[21,8498,8500],{"className":8499,"style":84},[83],[21,8501,8502,8505],{"style":87},[21,8503],{"className":8504,"style":92},[91],[21,8506,8508],{"className":8507},[96,97,98,99],[21,8509,103],{"className":8510},[42,43,99],[21,8512,108],{"className":8513},[107],[21,8515,8517],{"className":8516},[79],[21,8518,8520],{"className":8519,"style":115},[83],[21,8521],{},[21,8523],{"className":8524,"style":404},[378],[21,8526,1627],{"className":8527},[408],[21,8529],{"className":8530,"style":404},[378],[21,8532,8534,8537,8540,8543,8546],{"className":8533},[33],[21,8535],{"className":8536,"style":1540},[37],[21,8538,8359],{"className":8539},[42],[21,8541],{"className":8542,"style":379},[378],[21,8544,1101],{"className":8545},[383],[21,8547],{"className":8548,"style":379},[378],[21,8550,8552,8555],{"className":8551},[33],[21,8553],{"className":8554,"style":1540},[37],[21,8556,8558],{"className":8557},[42],"48"," — a swing of ",[21,8561,8563],{"className":8562},[24],[21,8564,8566],{"className":8565,"ariaHidden":29},[28],[21,8567,8569,8572,8575],{"className":8568},[33],[21,8570],{"className":8571,"style":3259},[37],[21,8573,963],{"className":8574},[42],[21,8576,8578],{"className":8577},[42],"38"," above the\nbaseline from a single lucky-for-",[21,8581,8583],{"className":8582},[24],[21,8584,8586],{"className":8585,"ariaHidden":29},[28],[21,8587,8589,8592],{"className":8588},[33],[21,8590],{"className":8591,"style":253},[37],[21,8593,258],{"className":8594,"style":257},[42,43]," action. With the control variate it is\n",[21,8597,8599],{"className":8598},[24],[21,8600,8602,8620,8638,8659,8680,8698,8716,8735],{"className":8601,"ariaHidden":29},[28],[21,8603,8605,8608,8611,8614,8617],{"className":8604},[33],[21,8606],{"className":8607,"style":1540},[37],[21,8609,8342],{"className":8610},[42],[21,8612],{"className":8613,"style":404},[378],[21,8615,1627],{"className":8616},[408],[21,8618],{"className":8619,"style":404},[378],[21,8621,8623,8626,8629,8632,8635],{"className":8622},[33],[21,8624],{"className":8625,"style":3259},[37],[21,8627,8359],{"className":8628},[42],[21,8630],{"className":8631,"style":404},[378],[21,8633,963],{"className":8634},[408],[21,8636],{"className":8637,"style":404},[378],[21,8639,8641,8644,8647,8650,8653,8656],{"className":8640},[33],[21,8642],{"className":8643,"style":440},[37],[21,8645,543],{"className":8646},[444],[21,8648,358],{"className":8649},[42],[21,8651],{"className":8652,"style":404},[378],[21,8654,3896],{"className":8655},[408],[21,8657],{"className":8658,"style":404},[378],[21,8660,8662,8665,8668,8671,8674,8677],{"className":8661},[33],[21,8663],{"className":8664,"style":440},[37],[21,8666,8342],{"className":8667},[42],[21,8669,635],{"className":8670},[465],[21,8672],{"className":8673,"style":404},[378],[21,8675,1627],{"className":8676},[408],[21,8678],{"className":8679,"style":404},[378],[21,8681,8683,8686,8689,8692,8695],{"className":8682},[33],[21,8684],{"className":8685,"style":1540},[37],[21,8687,3608],{"className":8688},[42],[21,8690],{"className":8691,"style":379},[378],[21,8693,1101],{"className":8694},[383],[21,8696],{"className":8697,"style":379},[378],[21,8699,8701,8704,8707,8710,8713],{"className":8700},[33],[21,8702],{"className":8703,"style":3259},[37],[21,8705,8558],{"className":8706},[42],[21,8708],{"className":8709,"style":404},[378],[21,8711,3896],{"className":8712},[408],[21,8714],{"className":8715,"style":404},[378],[21,8717,8719,8722,8726,8729,8732],{"className":8718},[33],[21,8720],{"className":8721,"style":1540},[37],[21,8723,8725],{"className":8724},[42],"30",[21,8727],{"className":8728,"style":379},[378],[21,8730,1101],{"className":8731},[383],[21,8733],{"className":8734,"style":379},[378],[21,8736,8738,8741],{"className":8737},[33],[21,8739],{"className":8740,"style":1540},[37],[21,8742,8744],{"className":8743},[42],"18",", a swing of ",[21,8747,8749],{"className":8748},[24],[21,8750,8752],{"className":8751,"ariaHidden":29},[28],[21,8753,8755,8758,8761],{"className":8754},[33],[21,8756],{"className":8757,"style":3259},[37],[21,8759,963],{"className":8760},[42],[21,8762,8764],{"className":8763},[42],"8",". The correction has\nnot changed the ",[237,8767,8768],{},"expectation"," (over ",[21,8771,8773],{"className":8772},[24],[21,8774,8776,8831],{"className":8775,"ariaHidden":29},[28],[21,8777,8779,8782,8822,8825,8828],{"className":8778},[33],[21,8780],{"className":8781,"style":860},[37],[21,8783,8785,8788],{"className":8784},[42],[21,8786,423],{"className":8787},[42,43],[21,8789,8791],{"className":8790},[70],[21,8792,8794,8814],{"className":8793},[74,75],[21,8795,8797,8811],{"className":8796},[79],[21,8798,8800],{"className":8799,"style":84},[83],[21,8801,8802,8805],{"style":87},[21,8803],{"className":8804,"style":92},[91],[21,8806,8808],{"className":8807},[96,97,98,99],[21,8809,103],{"className":8810},[42,43,99],[21,8812,108],{"className":8813},[107],[21,8815,8817],{"className":8816},[79],[21,8818,8820],{"className":8819,"style":115},[83],[21,8821],{},[21,8823],{"className":8824,"style":379},[378],[21,8826,7717],{"className":8827},[383],[21,8829],{"className":8830,"style":379},[378],[21,8832,8834,8837],{"className":8833},[33],[21,8835],{"className":8836,"style":38},[37],[21,8838,275],{"className":8839},[42,43],[21,8841,8843],{"className":8842},[24],[21,8844,8846,8910],{"className":8845,"ariaHidden":29},[28],[21,8847,8849,8852,8855,8858,8898,8901,8904,8907],{"className":8848},[33],[21,8850],{"className":8851,"style":440},[37],[21,8853,7750],{"className":8854},[42,7749],[21,8856,445],{"className":8857},[444],[21,8859,8861,8864],{"className":8860},[42],[21,8862,295],{"className":8863},[42,43],[21,8865,8867],{"className":8866},[70],[21,8868,8870,8890],{"className":8869},[74,75],[21,8871,8873,8887],{"className":8872},[79],[21,8874,8876],{"className":8875,"style":84},[83],[21,8877,8878,8881],{"style":87},[21,8879],{"className":8880,"style":92},[91],[21,8882,8884],{"className":8883},[96,97,98,99],[21,8885,103],{"className":8886},[42,43,99],[21,8888,108],{"className":8889},[107],[21,8891,8893],{"className":8892},[79],[21,8894,8896],{"className":8895,"style":115},[83],[21,8897],{},[21,8899,466],{"className":8900},[465],[21,8902],{"className":8903,"style":379},[378],[21,8905,1101],{"className":8906},[383],[21,8908],{"className":8909,"style":379},[378],[21,8911,8913,8916],{"className":8912},[33],[21,8914],{"className":8915,"style":1540},[37],[21,8917,358],{"className":8918},[42]," so\n",[21,8921,8923],{"className":8922},[24],[21,8924,8926,8951,9043],{"className":8925,"ariaHidden":29},[28],[21,8927,8929,8932,8935,8939,8942,8945,8948],{"className":8928},[33],[21,8930],{"className":8931,"style":440},[37],[21,8933,7750],{"className":8934},[42,7749],[21,8936,8938],{"className":8937},[444],"[(",[21,8940,358],{"className":8941},[42],[21,8943],{"className":8944,"style":404},[378],[21,8946,3896],{"className":8947},[408],[21,8949],{"className":8950,"style":404},[378],[21,8952,8954,8957,8997,9000,9031,9034,9037,9040],{"className":8953},[33],[21,8955],{"className":8956,"style":440},[37],[21,8958,8960,8963],{"className":8959},[42],[21,8961,295],{"className":8962},[42,43],[21,8964,8966],{"className":8965},[70],[21,8967,8969,8989],{"className":8968},[74,75],[21,8970,8972,8986],{"className":8971},[79],[21,8973,8975],{"className":8974,"style":84},[83],[21,8976,8977,8980],{"style":87},[21,8978],{"className":8979,"style":92},[91],[21,8981,8983],{"className":8982},[96,97,98,99],[21,8984,103],{"className":8985},[42,43,99],[21,8987,108],{"className":8988},[107],[21,8990,8992],{"className":8991},[79],[21,8993,8995],{"className":8994,"style":115},[83],[21,8996],{},[21,8998,635],{"className":8999},[465],[21,9001,9003],{"className":9002},[42,3964],[21,9004,9006],{"className":9005},[74],[21,9007,9009],{"className":9008},[79],[21,9010,9012,9020],{"className":9011,"style":38},[83],[21,9013,9014,9017],{"style":3976},[21,9015],{"className":9016,"style":3980},[91],[21,9018,3984],{"className":9019,"style":257},[42,43],[21,9021,9022,9025],{"style":3976},[21,9023],{"className":9024,"style":3980},[91],[21,9026,9028],{"className":9027,"style":3994},[3993],[21,9029,3998],{"className":9030},[42],[21,9032,466],{"className":9033},[465],[21,9035],{"className":9036,"style":379},[378],[21,9038,1101],{"className":9039},[383],[21,9041],{"className":9042,"style":379},[378],[21,9044,9046,9049],{"className":9045},[33],[21,9047],{"className":9048,"style":1540},[37],[21,9050,449],{"className":9051},[42],"), but it has shrunk this particular sample's\nexcursion from ",[21,9054,9056],{"className":9055},[24],[21,9057,9059],{"className":9058,"ariaHidden":29},[28],[21,9060,9062,9065],{"className":9061},[33],[21,9063],{"className":9064,"style":1540},[37],[21,9066,8578],{"className":9067},[42]," to ",[21,9070,9072],{"className":9071},[24],[21,9073,9075],{"className":9074,"ariaHidden":29},[28],[21,9076,9078,9081],{"className":9077},[33],[21,9079],{"className":9080,"style":1540},[37],[21,9082,8764],{"className":9083},[42],". When instead the sampled action is one ",[21,9086,9088],{"className":9087},[24],[21,9089,9091],{"className":9090,"ariaHidden":29},[28],[21,9092,9094,9097],{"className":9093},[33],[21,9095],{"className":9096,"style":253},[37],[21,9098,258],{"className":9099,"style":257},[42,43]," dislikes,\n",[21,9102,9104],{"className":9103},[24],[21,9105,9107,9164],{"className":9106,"ariaHidden":29},[28],[21,9108,9110,9114,9154,9157,9161],{"className":9109},[33],[21,9111],{"className":9112,"style":9113},[37],"height:0.7335em;vertical-align:-0.1944em;",[21,9115,9117,9120],{"className":9116},[42],[21,9118,295],{"className":9119},[42,43],[21,9121,9123],{"className":9122},[70],[21,9124,9126,9146],{"className":9125},[74,75],[21,9127,9129,9143],{"className":9128},[79],[21,9130,9132],{"className":9131,"style":84},[83],[21,9133,9134,9137],{"style":87},[21,9135],{"className":9136,"style":92},[91],[21,9138,9140],{"className":9139},[96,97,98,99],[21,9141,103],{"className":9142},[42,43,99],[21,9144,108],{"className":9145},[107],[21,9147,9149],{"className":9148},[79],[21,9150,9152],{"className":9151,"style":115},[83],[21,9153],{},[21,9155],{"className":9156,"style":379},[378],[21,9158,9160],{"className":9159},[383],"\u003C",[21,9162],{"className":9163,"style":379},[378],[21,9165,9167,9170],{"className":9166},[33],[21,9168],{"className":9169,"style":1540},[37],[21,9171,358],{"className":9172},[42],", and the same term nudges the target back ",[237,9175,9176],{},"up"," toward the baseline —\neither way, toward ",[21,9179,9181],{"className":9180},[24],[21,9182,9184],{"className":9183,"ariaHidden":29},[28],[21,9185,9187,9190,9221,9224,9264],{"className":9186},[33],[21,9188],{"className":9189,"style":440},[37],[21,9191,9193],{"className":9192},[42,3964],[21,9194,9196],{"className":9195},[74],[21,9197,9199],{"className":9198},[79],[21,9200,9202,9210],{"className":9201,"style":38},[83],[21,9203,9204,9207],{"style":3976},[21,9205],{"className":9206,"style":3980},[91],[21,9208,3984],{"className":9209,"style":257},[42,43],[21,9211,9212,9215],{"style":3976},[21,9213],{"className":9214,"style":3980},[91],[21,9216,9218],{"className":9217,"style":3994},[3993],[21,9219,3998],{"className":9220},[42],[21,9222,543],{"className":9223},[444],[21,9225,9227,9230],{"className":9226},[42],[21,9228,400],{"className":9229,"style":550},[42,43],[21,9231,9233],{"className":9232},[70],[21,9234,9236,9256],{"className":9235},[74,75],[21,9237,9239,9253],{"className":9238},[79],[21,9240,9242],{"className":9241,"style":84},[83],[21,9243,9244,9247],{"style":565},[21,9245],{"className":9246,"style":92},[91],[21,9248,9250],{"className":9249},[96,97,98,99],[21,9251,103],{"className":9252},[42,43,99],[21,9254,108],{"className":9255},[107],[21,9257,9259],{"className":9258},[79],[21,9260,9262],{"className":9261,"style":115},[83],[21,9263],{},[21,9265,635],{"className":9266},[465],[11,9268,9269],{},"The value of this form is that the truncated version can be written, to a close\napproximation, as a sum of state-based TD errors, exactly as in the on-policy\ncase. With the per-decision TD error",[21,9271,9273],{"className":9272},[847],[21,9274,9276],{"className":9275},[24],[21,9277,9279,9356,9420,9619],{"className":9278,"ariaHidden":29},[28],[21,9280,9282,9286,9341,9344,9347,9350,9353],{"className":9281},[33],[21,9283],{"className":9284,"style":9285},[37],"height:0.9614em;vertical-align:-0.247em;",[21,9287,9289,9294],{"className":9288},[42],[21,9290,9293],{"className":9291,"style":9292},[42,43],"margin-right:0.0379em;","δ",[21,9295,9297],{"className":9296},[70],[21,9298,9300,9333],{"className":9299},[74,75],[21,9301,9303,9330],{"className":9302},[79],[21,9304,9307,9319],{"className":9305,"style":9306},[83],"height:0.7144em;",[21,9308,9310,9313],{"style":9309},"top:-2.453em;margin-left:-0.0379em;margin-right:0.05em;",[21,9311],{"className":9312,"style":92},[91],[21,9314,9316],{"className":9315},[96,97,98,99],[21,9317,103],{"className":9318},[42,43,99],[21,9320,9321,9324],{"style":3714},[21,9322],{"className":9323,"style":92},[91],[21,9325,9327],{"className":9326},[96,97,98,99],[21,9328,1583],{"className":9329},[42,43,99],[21,9331,108],{"className":9332},[107],[21,9334,9336],{"className":9335},[79],[21,9337,9339],{"className":9338,"style":3739},[83],[21,9340],{},[21,9342],{"className":9343,"style":379},[378],[21,9345],{"className":9346,"style":379},[378],[21,9348,527],{"className":9349},[383],[21,9351],{"className":9352,"style":379},[378],[21,9354],{"className":9355,"style":379},[378],[21,9357,9359,9362,9411,9414,9417],{"className":9358},[33],[21,9360],{"className":9361,"style":923},[37],[21,9363,9365,9368],{"className":9364},[42],[21,9366,931],{"className":9367,"style":930},[42,43],[21,9369,9371],{"className":9370},[70],[21,9372,9374,9403],{"className":9373},[74,75],[21,9375,9377,9400],{"className":9376},[79],[21,9378,9380],{"className":9379,"style":944},[83],[21,9381,9382,9385],{"style":947},[21,9383],{"className":9384,"style":92},[91],[21,9386,9388],{"className":9387},[96,97,98,99],[21,9389,9391,9394,9397],{"className":9390},[42,99],[21,9392,103],{"className":9393},[42,43,99],[21,9395,963],{"className":9396},[408,99],[21,9398,358],{"className":9399},[42,99],[21,9401,108],{"className":9402},[107],[21,9404,9406],{"className":9405},[79],[21,9407,9409],{"className":9408,"style":976},[83],[21,9410],{},[21,9412],{"className":9413,"style":404},[378],[21,9415,963],{"className":9416},[408],[21,9418],{"className":9419,"style":404},[378],[21,9421,9423,9426,9475,9478,9509,9512,9561,9564,9567,9607,9610,9613,9616],{"className":9422},[33],[21,9424],{"className":9425,"style":440},[37],[21,9427,9429,9432],{"className":9428},[42],[21,9430,200],{"className":9431,"style":199},[42,43],[21,9433,9435],{"className":9434},[70],[21,9436,9438,9467],{"className":9437},[74,75],[21,9439,9441,9464],{"className":9440},[79],[21,9442,9444],{"className":9443,"style":944},[83],[21,9445,9446,9449],{"style":737},[21,9447],{"className":9448,"style":92},[91],[21,9450,9452],{"className":9451},[96,97,98,99],[21,9453,9455,9458,9461],{"className":9454},[42,99],[21,9456,103],{"className":9457},[42,43,99],[21,9459,963],{"className":9460},[408,99],[21,9462,358],{"className":9463},[42,99],[21,9465,108],{"className":9466},[107],[21,9468,9470],{"className":9469},[79],[21,9471,9473],{"className":9472,"style":976},[83],[21,9474],{},[21,9476],{"className":9477,"style":458},[378],[21,9479,9481],{"className":9480},[42,3964],[21,9482,9484],{"className":9483},[74],[21,9485,9487],{"className":9486},[79],[21,9488,9490,9498],{"className":9489,"style":38},[83],[21,9491,9492,9495],{"style":3976},[21,9493],{"className":9494,"style":3980},[91],[21,9496,3984],{"className":9497,"style":257},[42,43],[21,9499,9500,9503],{"style":3976},[21,9501],{"className":9502,"style":3980},[91],[21,9504,9506],{"className":9505,"style":3994},[3993],[21,9507,3998],{"className":9508},[42],[21,9510,543],{"className":9511},[444],[21,9513,9515,9518],{"className":9514},[42],[21,9516,400],{"className":9517,"style":550},[42,43],[21,9519,9521],{"className":9520},[70],[21,9522,9524,9553],{"className":9523},[74,75],[21,9525,9527,9550],{"className":9526},[79],[21,9528,9530],{"className":9529,"style":944},[83],[21,9531,9532,9535],{"style":565},[21,9533],{"className":9534,"style":92},[91],[21,9536,9538],{"className":9537},[96,97,98,99],[21,9539,9541,9544,9547],{"className":9540},[42,99],[21,9542,103],{"className":9543},[42,43,99],[21,9545,963],{"className":9546},[408,99],[21,9548,358],{"className":9549},[42,99],[21,9551,108],{"className":9552},[107],[21,9554,9556],{"className":9555},[79],[21,9557,9559],{"className":9558,"style":976},[83],[21,9560],{},[21,9562,454],{"className":9563},[453],[21,9565],{"className":9566,"style":458},[378],[21,9568,9570,9573],{"className":9569},[42],[21,9571,4064],{"className":9572,"style":4063},[42,65],[21,9574,9576],{"className":9575},[70],[21,9577,9579,9599],{"className":9578},[74,75],[21,9580,9582,9596],{"className":9581},[79],[21,9583,9585],{"className":9584,"style":84},[83],[21,9586,9587,9590],{"style":4079},[21,9588],{"className":9589,"style":92},[91],[21,9591,9593],{"className":9592},[96,97,98,99],[21,9594,103],{"className":9595},[42,43,99],[21,9597,108],{"className":9598},[107],[21,9600,9602],{"className":9601},[79],[21,9603,9605],{"className":9604,"style":115},[83],[21,9606],{},[21,9608,635],{"className":9609},[465],[21,9611],{"className":9612,"style":404},[378],[21,9614,3896],{"className":9615},[408],[21,9617],{"className":9618,"style":404},[378],[21,9620,9622,9625,9656,9659,9699,9702,9705,9745,9748],{"className":9621},[33],[21,9623],{"className":9624,"style":440},[37],[21,9626,9628],{"className":9627},[42,3964],[21,9629,9631],{"className":9630},[74],[21,9632,9634],{"className":9633},[79],[21,9635,9637,9645],{"className":9636,"style":38},[83],[21,9638,9639,9642],{"style":3976},[21,9640],{"className":9641,"style":3980},[91],[21,9643,3984],{"className":9644,"style":257},[42,43],[21,9646,9647,9650],{"style":3976},[21,9648],{"className":9649,"style":3980},[91],[21,9651,9653],{"className":9652,"style":3994},[3993],[21,9654,3998],{"className":9655},[42],[21,9657,543],{"className":9658},[444],[21,9660,9662,9665],{"className":9661},[42],[21,9663,400],{"className":9664,"style":550},[42,43],[21,9666,9668],{"className":9667},[70],[21,9669,9671,9691],{"className":9670},[74,75],[21,9672,9674,9688],{"className":9673},[79],[21,9675,9677],{"className":9676,"style":84},[83],[21,9678,9679,9682],{"style":565},[21,9680],{"className":9681,"style":92},[91],[21,9683,9685],{"className":9684},[96,97,98,99],[21,9686,103],{"className":9687},[42,43,99],[21,9689,108],{"className":9690},[107],[21,9692,9694],{"className":9693},[79],[21,9695,9697],{"className":9696,"style":115},[83],[21,9698],{},[21,9700,454],{"className":9701},[453],[21,9703],{"className":9704,"style":458},[378],[21,9706,9708,9711],{"className":9707},[42],[21,9709,4064],{"className":9710,"style":4063},[42,65],[21,9712,9714],{"className":9713},[70],[21,9715,9717,9737],{"className":9716},[74,75],[21,9718,9720,9734],{"className":9719},[79],[21,9721,9723],{"className":9722,"style":84},[83],[21,9724,9725,9728],{"style":4079},[21,9726],{"className":9727,"style":92},[91],[21,9729,9731],{"className":9730},[96,97,98,99],[21,9732,103],{"className":9733},[42,43,99],[21,9735,108],{"className":9736},[107],[21,9738,9740],{"className":9739},[79],[21,9741,9743],{"className":9742,"style":115},[83],[21,9744],{},[21,9746,635],{"className":9747},[465],[21,9749,454],{"className":9750},[453],[11,9752,9753],{},"the return is approximately",[21,9755,9757],{"className":9756},[847],[21,9758,9760],{"className":9759},[24],[21,9761,9763,9842,9980],{"className":9762,"ariaHidden":29},[28],[21,9764,9766,9769,9826,9829,9832,9836,9839],{"className":9765},[33],[21,9767],{"className":9768,"style":3680},[37],[21,9770,9772,9775],{"className":9771},[42],[21,9773,867],{"className":9774},[42,43],[21,9776,9778],{"className":9777},[70],[21,9779,9781,9818],{"className":9780},[74,75],[21,9782,9784,9815],{"className":9783},[79],[21,9785,9787,9798],{"className":9786,"style":3699},[83],[21,9788,9789,9792],{"style":3702},[21,9790],{"className":9791,"style":92},[91],[21,9793,9795],{"className":9794},[96,97,98,99],[21,9796,103],{"className":9797},[42,43,99],[21,9799,9800,9803],{"style":3714},[21,9801],{"className":9802,"style":92},[91],[21,9804,9806],{"className":9805},[96,97,98,99],[21,9807,9809,9812],{"className":9808},[42,99],[21,9810,44],{"className":9811},[42,43,99],[21,9813,1583],{"className":9814},[42,43,99],[21,9816,108],{"className":9817},[107],[21,9819,9821],{"className":9820},[79],[21,9822,9824],{"className":9823,"style":3739},[83],[21,9825],{},[21,9827],{"className":9828,"style":379},[378],[21,9830],{"className":9831,"style":379},[378],[21,9833,9835],{"className":9834},[383],"≈",[21,9837],{"className":9838,"style":379},[378],[21,9840],{"className":9841,"style":379},[378],[21,9843,9845,9848,9879,9882,9922,9925,9928,9968,9971,9974,9977],{"className":9844},[33],[21,9846],{"className":9847,"style":440},[37],[21,9849,9851],{"className":9850},[42,3964],[21,9852,9854],{"className":9853},[74],[21,9855,9857],{"className":9856},[79],[21,9858,9860,9868],{"className":9859,"style":38},[83],[21,9861,9862,9865],{"style":3976},[21,9863],{"className":9864,"style":3980},[91],[21,9866,3984],{"className":9867,"style":257},[42,43],[21,9869,9870,9873],{"style":3976},[21,9871],{"className":9872,"style":3980},[91],[21,9874,9876],{"className":9875,"style":3994},[3993],[21,9877,3998],{"className":9878},[42],[21,9880,543],{"className":9881},[444],[21,9883,9885,9888],{"className":9884},[42],[21,9886,400],{"className":9887,"style":550},[42,43],[21,9889,9891],{"className":9890},[70],[21,9892,9894,9914],{"className":9893},[74,75],[21,9895,9897,9911],{"className":9896},[79],[21,9898,9900],{"className":9899,"style":84},[83],[21,9901,9902,9905],{"style":565},[21,9903],{"className":9904,"style":92},[91],[21,9906,9908],{"className":9907},[96,97,98,99],[21,9909,103],{"className":9910},[42,43,99],[21,9912,108],{"className":9913},[107],[21,9915,9917],{"className":9916},[79],[21,9918,9920],{"className":9919,"style":115},[83],[21,9921],{},[21,9923,454],{"className":9924},[453],[21,9926],{"className":9927,"style":458},[378],[21,9929,9931,9934],{"className":9930},[42],[21,9932,4064],{"className":9933,"style":4063},[42,65],[21,9935,9937],{"className":9936},[70],[21,9938,9940,9960],{"className":9939},[74,75],[21,9941,9943,9957],{"className":9942},[79],[21,9944,9946],{"className":9945,"style":84},[83],[21,9947,9948,9951],{"style":4079},[21,9949],{"className":9950,"style":92},[91],[21,9952,9954],{"className":9953},[96,97,98,99],[21,9955,103],{"className":9956},[42,43,99],[21,9958,108],{"className":9959},[107],[21,9961,9963],{"className":9962},[79],[21,9964,9966],{"className":9965,"style":115},[83],[21,9967],{},[21,9969,635],{"className":9970},[465],[21,9972],{"className":9973,"style":404},[378],[21,9975,963],{"className":9976},[408],[21,9978],{"className":9979,"style":404},[378],[21,9981,9983,9986,10026,10029,10096,10099,10150,10153,10226,10229,10269,10309,10349],{"className":9982},[33],[21,9984],{"className":9985,"style":1114},[37],[21,9987,9989,9992],{"className":9988},[42],[21,9990,295],{"className":9991},[42,43],[21,9993,9995],{"className":9994},[70],[21,9996,9998,10018],{"className":9997},[74,75],[21,9999,10001,10015],{"className":10000},[79],[21,10002,10004],{"className":10003,"style":84},[83],[21,10005,10006,10009],{"style":87},[21,10007],{"className":10008,"style":92},[91],[21,10010,10012],{"className":10011},[96,97,98,99],[21,10013,103],{"className":10014},[42,43,99],[21,10016,108],{"className":10017},[107],[21,10019,10021],{"className":10020},[79],[21,10022,10024],{"className":10023,"style":115},[83],[21,10025],{},[21,10027],{"className":10028,"style":458},[378],[21,10030,10032],{"className":10031},[1118,1119],[21,10033,10035,10088],{"className":10034},[74,75],[21,10036,10038,10085],{"className":10037},[79],[21,10039,10041,10061,10071],{"className":10040,"style":1129},[83],[21,10042,10043,10046],{"style":1132},[21,10044],{"className":10045,"style":1136},[91],[21,10047,10049],{"className":10048},[96,97,98,99],[21,10050,10052,10055,10058],{"className":10051},[42,99],[21,10053,1147],{"className":10054,"style":1146},[42,43,99],[21,10056,1101],{"className":10057},[383,99],[21,10059,103],{"className":10060},[42,43,99],[21,10062,10063,10066],{"style":1156},[21,10064],{"className":10065,"style":1136},[91],[21,10067,10068],{},[21,10069,1167],{"className":10070},[1118,1165,1166],[21,10072,10073,10076],{"style":1170},[21,10074],{"className":10075,"style":1136},[91],[21,10077,10079],{"className":10078},[96,97,98,99],[21,10080,10082],{"className":10081},[42,99],[21,10083,1183],{"className":10084},[42,99],[21,10086,108],{"className":10087},[107],[21,10089,10091],{"className":10090},[79],[21,10092,10094],{"className":10093,"style":1193},[83],[21,10095],{},[21,10097],{"className":10098,"style":458},[378],[21,10100,10102,10105],{"className":10101},[42],[21,10103,9293],{"className":10104,"style":9292},[42,43],[21,10106,10108],{"className":10107},[70],[21,10109,10111,10142],{"className":10110},[74,75],[21,10112,10114,10139],{"className":10113},[79],[21,10115,10117,10128],{"className":10116,"style":9306},[83],[21,10118,10119,10122],{"style":9309},[21,10120],{"className":10121,"style":92},[91],[21,10123,10125],{"className":10124},[96,97,98,99],[21,10126,1147],{"className":10127,"style":1146},[42,43,99],[21,10129,10130,10133],{"style":3714},[21,10131],{"className":10132,"style":92},[91],[21,10134,10136],{"className":10135},[96,97,98,99],[21,10137,1583],{"className":10138},[42,43,99],[21,10140,108],{"className":10141},[107],[21,10143,10145],{"className":10144},[79],[21,10146,10148],{"className":10147,"style":3739},[83],[21,10149],{},[21,10151],{"className":10152,"style":458},[378],[21,10154,10156],{"className":10155},[1118,1119],[21,10157,10159,10218],{"className":10158},[74,75],[21,10160,10162,10215],{"className":10161},[79],[21,10163,10165,10191,10201],{"className":10164,"style":1225},[83],[21,10166,10167,10170],{"style":1228},[21,10168],{"className":10169,"style":1136},[91],[21,10171,10173],{"className":10172},[96,97,98,99],[21,10174,10176,10179,10182,10185,10188],{"className":10175},[42,99],[21,10177,1241],{"className":10178},[42,43,99],[21,10180,1101],{"className":10181},[383,99],[21,10183,103],{"className":10184},[42,43,99],[21,10186,963],{"className":10187},[408,99],[21,10189,358],{"className":10190},[42,99],[21,10192,10193,10196],{"style":1156},[21,10194],{"className":10195,"style":1136},[91],[21,10197,10198],{},[21,10199,1264],{"className":10200},[1118,1165,1166],[21,10202,10203,10206],{"style":1170},[21,10204],{"className":10205,"style":1136},[91],[21,10207,10209],{"className":10208},[96,97,98,99],[21,10210,10212],{"className":10211},[42,99],[21,10213,1147],{"className":10214,"style":1146},[42,43,99],[21,10216,108],{"className":10217},[107],[21,10219,10221],{"className":10220},[79],[21,10222,10224],{"className":10223,"style":1288},[83],[21,10225],{},[21,10227],{"className":10228,"style":458},[378],[21,10230,10232,10235],{"className":10231},[42],[21,10233,200],{"className":10234,"style":199},[42,43],[21,10236,10238],{"className":10237},[70],[21,10239,10241,10261],{"className":10240},[74,75],[21,10242,10244,10258],{"className":10243},[79],[21,10245,10247],{"className":10246,"style":1312},[83],[21,10248,10249,10252],{"style":737},[21,10250],{"className":10251,"style":92},[91],[21,10253,10255],{"className":10254},[96,97,98,99],[21,10256,1241],{"className":10257},[42,43,99],[21,10259,108],{"className":10260},[107],[21,10262,10264],{"className":10263},[79],[21,10265,10267],{"className":10266,"style":115},[83],[21,10268],{},[21,10270,10272,10275],{"className":10271},[42],[21,10273,44],{"className":10274},[42,43],[21,10276,10278],{"className":10277},[70],[21,10279,10281,10301],{"className":10280},[74,75],[21,10282,10284,10298],{"className":10283},[79],[21,10285,10287],{"className":10286,"style":1312},[83],[21,10288,10289,10292],{"style":87},[21,10290],{"className":10291,"style":92},[91],[21,10293,10295],{"className":10294},[96,97,98,99],[21,10296,1241],{"className":10297},[42,43,99],[21,10299,108],{"className":10300},[107],[21,10302,10304],{"className":10303},[79],[21,10305,10307],{"className":10306,"style":115},[83],[21,10308],{},[21,10310,10312,10315],{"className":10311},[42],[21,10313,295],{"className":10314},[42,43],[21,10316,10318],{"className":10317},[70],[21,10319,10321,10341],{"className":10320},[74,75],[21,10322,10324,10338],{"className":10323},[79],[21,10325,10327],{"className":10326,"style":1312},[83],[21,10328,10329,10332],{"style":87},[21,10330],{"className":10331,"style":92},[91],[21,10333,10335],{"className":10334},[96,97,98,99],[21,10336,1241],{"className":10337},[42,43,99],[21,10339,108],{"className":10340},[107],[21,10342,10344],{"className":10343},[79],[21,10345,10347],{"className":10346,"style":115},[83],[21,10348],{},[21,10350,454],{"className":10351},[453],[11,10353,10354,10355,10358,10359,10497,10498,10516],{},"with the approximation becoming ",[237,10356,10357],{},"exact"," when the value function does not change.\nThe product ",[21,10360,10362],{"className":10361},[24],[21,10363,10365],{"className":10364,"ariaHidden":29},[28],[21,10366,10368,10371,10374,10377,10417,10457],{"className":10367},[33],[21,10369],{"className":10370,"style":440},[37],[21,10372,1264],{"className":10373,"style":1420},[1118,1165,1419],[21,10375],{"className":10376,"style":458},[378],[21,10378,10380,10383],{"className":10379},[42],[21,10381,200],{"className":10382,"style":199},[42,43],[21,10384,10386],{"className":10385},[70],[21,10387,10389,10409],{"className":10388},[74,75],[21,10390,10392,10406],{"className":10391},[79],[21,10393,10395],{"className":10394,"style":1312},[83],[21,10396,10397,10400],{"style":737},[21,10398],{"className":10399,"style":92},[91],[21,10401,10403],{"className":10402},[96,97,98,99],[21,10404,1241],{"className":10405},[42,43,99],[21,10407,108],{"className":10408},[107],[21,10410,10412],{"className":10411},[79],[21,10413,10415],{"className":10414,"style":115},[83],[21,10416],{},[21,10418,10420,10423],{"className":10419},[42],[21,10421,44],{"className":10422},[42,43],[21,10424,10426],{"className":10425},[70],[21,10427,10429,10449],{"className":10428},[74,75],[21,10430,10432,10446],{"className":10431},[79],[21,10433,10435],{"className":10434,"style":1312},[83],[21,10436,10437,10440],{"style":87},[21,10438],{"className":10439,"style":92},[91],[21,10441,10443],{"className":10442},[96,97,98,99],[21,10444,1241],{"className":10445},[42,43,99],[21,10447,108],{"className":10448},[107],[21,10450,10452],{"className":10451},[79],[21,10453,10455],{"className":10454,"style":115},[83],[21,10456],{},[21,10458,10460,10463],{"className":10459},[42],[21,10461,295],{"className":10462},[42,43],[21,10464,10466],{"className":10465},[70],[21,10467,10469,10489],{"className":10468},[74,75],[21,10470,10472,10486],{"className":10471},[79],[21,10473,10475],{"className":10474,"style":1312},[83],[21,10476,10477,10480],{"style":87},[21,10478],{"className":10479,"style":92},[91],[21,10481,10483],{"className":10482},[96,97,98,99],[21,10484,1241],{"className":10485},[42,43,99],[21,10487,108],{"className":10488},[107],[21,10490,10492],{"className":10491},[79],[21,10493,10495],{"className":10494,"style":115},[83],[21,10496],{}," is the eligibility-trace weight: it\ncompounds the same ",[21,10499,10501],{"className":10500},[24],[21,10502,10504],{"className":10503,"ariaHidden":29},[28],[21,10505,10507,10510,10513],{"className":10506},[33],[21,10508],{"className":10509,"style":5968},[37],[21,10511,200],{"className":10512,"style":199},[42,43],[21,10514,44],{"className":10515},[42,43]," decay as before, now multiplied by the\nimportance ratios along the way. The forward-view update",[21,10518,10520],{"className":10519},[847],[21,10521,10523],{"className":10522},[24],[21,10524,10526,10597,10653,10737,11014,11069],{"className":10525,"ariaHidden":29},[28],[21,10527,10529,10533,10582,10585,10588,10591,10594],{"className":10528},[33],[21,10530],{"className":10531,"style":10532},[37],"height:0.6528em;vertical-align:-0.2083em;",[21,10534,10536,10539],{"className":10535},[42],[21,10537,4064],{"className":10538,"style":4063},[42,65],[21,10540,10542],{"className":10541},[70],[21,10543,10545,10574],{"className":10544},[74,75],[21,10546,10548,10571],{"className":10547},[79],[21,10549,10551],{"className":10550,"style":944},[83],[21,10552,10553,10556],{"style":4079},[21,10554],{"className":10555,"style":92},[91],[21,10557,10559],{"className":10558},[96,97,98,99],[21,10560,10562,10565,10568],{"className":10561},[42,99],[21,10563,103],{"className":10564},[42,43,99],[21,10566,963],{"className":10567},[408,99],[21,10569,358],{"className":10570},[42,99],[21,10572,108],{"className":10573},[107],[21,10575,10577],{"className":10576},[79],[21,10578,10580],{"className":10579,"style":976},[83],[21,10581],{},[21,10583],{"className":10584,"style":379},[378],[21,10586],{"className":10587,"style":379},[378],[21,10589,1101],{"className":10590},[383],[21,10592],{"className":10593,"style":379},[378],[21,10595],{"className":10596,"style":379},[378],[21,10598,10600,10604,10644,10647,10650],{"className":10599},[33],[21,10601],{"className":10602,"style":10603},[37],"height:0.7333em;vertical-align:-0.15em;",[21,10605,10607,10610],{"className":10606},[42],[21,10608,4064],{"className":10609,"style":4063},[42,65],[21,10611,10613],{"className":10612},[70],[21,10614,10616,10636],{"className":10615},[74,75],[21,10617,10619,10633],{"className":10618},[79],[21,10620,10622],{"className":10621,"style":84},[83],[21,10623,10624,10627],{"style":4079},[21,10625],{"className":10626,"style":92},[91],[21,10628,10630],{"className":10629},[96,97,98,99],[21,10631,103],{"className":10632},[42,43,99],[21,10634,108],{"className":10635},[107],[21,10637,10639],{"className":10638},[79],[21,10640,10642],{"className":10641,"style":115},[83],[21,10643],{},[21,10645],{"className":10646,"style":404},[378],[21,10648,963],{"className":10649},[408],[21,10651],{"className":10652,"style":404},[378],[21,10654,10656,10660,10665,10671,10728,10731,10734],{"className":10655},[33],[21,10657],{"className":10658,"style":10659},[37],"height:1.2491em;vertical-align:-0.35em;",[21,10661,10664],{"className":10662,"style":10663},[42,43],"margin-right:0.0037em;","α",[21,10666,10668],{"className":10667},[42],[21,10669,543],{"className":10670},[1211,6900],[21,10672,10674,10677],{"className":10673},[42],[21,10675,867],{"className":10676},[42,43],[21,10678,10680],{"className":10679},[70],[21,10681,10683,10720],{"className":10682},[74,75],[21,10684,10686,10717],{"className":10685},[79],[21,10687,10689,10700],{"className":10688,"style":3699},[83],[21,10690,10691,10694],{"style":3702},[21,10692],{"className":10693,"style":92},[91],[21,10695,10697],{"className":10696},[96,97,98,99],[21,10698,103],{"className":10699},[42,43,99],[21,10701,10702,10705],{"style":3714},[21,10703],{"className":10704,"style":92},[91],[21,10706,10708],{"className":10707},[96,97,98,99],[21,10709,10711,10714],{"className":10710},[42,99],[21,10712,44],{"className":10713},[42,43,99],[21,10715,1583],{"className":10716},[42,43,99],[21,10718,108],{"className":10719},[107],[21,10721,10723],{"className":10722},[79],[21,10724,10726],{"className":10725,"style":3739},[83],[21,10727],{},[21,10729],{"className":10730,"style":404},[378],[21,10732,3896],{"className":10733},[408],[21,10735],{"className":10736,"style":404},[378],[21,10738,10740,10743,10774,10777,10817,10820,10823,10863,10866,10872,10876,10907,10910,10950,10953,10956,10996,10999,11002,11005,11008,11011],{"className":10739},[33],[21,10741],{"className":10742,"style":6844},[37],[21,10744,10746],{"className":10745},[42,3964],[21,10747,10749],{"className":10748},[74],[21,10750,10752],{"className":10751},[79],[21,10753,10755,10763],{"className":10754,"style":38},[83],[21,10756,10757,10760],{"style":3976},[21,10758],{"className":10759,"style":3980},[91],[21,10761,3984],{"className":10762,"style":257},[42,43],[21,10764,10765,10768],{"style":3976},[21,10766],{"className":10767,"style":3980},[91],[21,10769,10771],{"className":10770,"style":3994},[3993],[21,10772,3998],{"className":10773},[42],[21,10775,543],{"className":10776},[444],[21,10778,10780,10783],{"className":10779},[42],[21,10781,400],{"className":10782,"style":550},[42,43],[21,10784,10786],{"className":10785},[70],[21,10787,10789,10809],{"className":10788},[74,75],[21,10790,10792,10806],{"className":10791},[79],[21,10793,10795],{"className":10794,"style":84},[83],[21,10796,10797,10800],{"style":565},[21,10798],{"className":10799,"style":92},[91],[21,10801,10803],{"className":10802},[96,97,98,99],[21,10804,103],{"className":10805},[42,43,99],[21,10807,108],{"className":10808},[107],[21,10810,10812],{"className":10811},[79],[21,10813,10815],{"className":10814,"style":115},[83],[21,10816],{},[21,10818,454],{"className":10819},[453],[21,10821],{"className":10822,"style":458},[378],[21,10824,10826,10829],{"className":10825},[42],[21,10827,4064],{"className":10828,"style":4063},[42,65],[21,10830,10832],{"className":10831},[70],[21,10833,10835,10855],{"className":10834},[74,75],[21,10836,10838,10852],{"className":10837},[79],[21,10839,10841],{"className":10840,"style":84},[83],[21,10842,10843,10846],{"style":4079},[21,10844],{"className":10845,"style":92},[91],[21,10847,10849],{"className":10848},[96,97,98,99],[21,10850,103],{"className":10851},[42,43,99],[21,10853,108],{"className":10854},[107],[21,10856,10858],{"className":10857},[79],[21,10859,10861],{"className":10860,"style":115},[83],[21,10862],{},[21,10864,635],{"className":10865},[465],[21,10867,10869],{"className":10868},[42],[21,10870,635],{"className":10871},[1211,6900],[21,10873,10875],{"className":10874},[42],"∇",[21,10877,10879],{"className":10878},[42,3964],[21,10880,10882],{"className":10881},[74],[21,10883,10885],{"className":10884},[79],[21,10886,10888,10896],{"className":10887,"style":38},[83],[21,10889,10890,10893],{"style":3976},[21,10891],{"className":10892,"style":3980},[91],[21,10894,3984],{"className":10895,"style":257},[42,43],[21,10897,10898,10901],{"style":3976},[21,10899],{"className":10900,"style":3980},[91],[21,10902,10904],{"className":10903,"style":3994},[3993],[21,10905,3998],{"className":10906},[42],[21,10908,543],{"className":10909},[444],[21,10911,10913,10916],{"className":10912},[42],[21,10914,400],{"className":10915,"style":550},[42,43],[21,10917,10919],{"className":10918},[70],[21,10920,10922,10942],{"className":10921},[74,75],[21,10923,10925,10939],{"className":10924},[79],[21,10926,10928],{"className":10927,"style":84},[83],[21,10929,10930,10933],{"style":565},[21,10931],{"className":10932,"style":92},[91],[21,10934,10936],{"className":10935},[96,97,98,99],[21,10937,103],{"className":10938},[42,43,99],[21,10940,108],{"className":10941},[107],[21,10943,10945],{"className":10944},[79],[21,10946,10948],{"className":10947,"style":115},[83],[21,10949],{},[21,10951,454],{"className":10952},[453],[21,10954],{"className":10955,"style":458},[378],[21,10957,10959,10962],{"className":10958},[42],[21,10960,4064],{"className":10961,"style":4063},[42,65],[21,10963,10965],{"className":10964},[70],[21,10966,10968,10988],{"className":10967},[74,75],[21,10969,10971,10985],{"className":10970},[79],[21,10972,10974],{"className":10973,"style":84},[83],[21,10975,10976,10979],{"style":4079},[21,10977],{"className":10978,"style":92},[91],[21,10980,10982],{"className":10981},[96,97,98,99],[21,10983,103],{"className":10984},[42,43,99],[21,10986,108],{"className":10987},[107],[21,10989,10991],{"className":10990},[79],[21,10992,10994],{"className":10993,"style":115},[83],[21,10995],{},[21,10997,635],{"className":10998},[465],[21,11000],{"className":11001,"style":379},[378],[21,11003],{"className":11004,"style":379},[378],[21,11006,9835],{"className":11007},[383],[21,11009],{"className":11010,"style":379},[378],[21,11012],{"className":11013,"style":379},[378],[21,11015,11017,11020,11060,11063,11066],{"className":11016},[33],[21,11018],{"className":11019,"style":10603},[37],[21,11021,11023,11026],{"className":11022},[42],[21,11024,4064],{"className":11025,"style":4063},[42,65],[21,11027,11029],{"className":11028},[70],[21,11030,11032,11052],{"className":11031},[74,75],[21,11033,11035,11049],{"className":11034},[79],[21,11036,11038],{"className":11037,"style":84},[83],[21,11039,11040,11043],{"style":4079},[21,11041],{"className":11042,"style":92},[91],[21,11044,11046],{"className":11045},[96,97,98,99],[21,11047,103],{"className":11048},[42,43,99],[21,11050,108],{"className":11051},[107],[21,11053,11055],{"className":11054},[79],[21,11056,11058],{"className":11057,"style":115},[83],[21,11059],{},[21,11061],{"className":11062,"style":404},[378],[21,11064,963],{"className":11065},[408],[21,11067],{"className":11068,"style":404},[378],[21,11070,11072,11075,11078,11118,11124,11127,11194,11197,11248,11251,11324,11327,11367,11407,11447,11453,11456,11487,11490,11530,11533,11536,11576],{"className":11071},[33],[21,11073],{"className":11074,"style":1114},[37],[21,11076,10664],{"className":11077,"style":10663},[42,43],[21,11079,11081,11084],{"className":11080},[42],[21,11082,295],{"className":11083},[42,43],[21,11085,11087],{"className":11086},[70],[21,11088,11090,11110],{"className":11089},[74,75],[21,11091,11093,11107],{"className":11092},[79],[21,11094,11096],{"className":11095,"style":84},[83],[21,11097,11098,11101],{"style":87},[21,11099],{"className":11100,"style":92},[91],[21,11102,11104],{"className":11103},[96,97,98,99],[21,11105,103],{"className":11106},[42,43,99],[21,11108,108],{"className":11109},[107],[21,11111,11113],{"className":11112},[79],[21,11114,11116],{"className":11115,"style":115},[83],[21,11117],{},[21,11119,11121],{"className":11120},[42],[21,11122,543],{"className":11123},[1211,3883],[21,11125],{"className":11126,"style":458},[378],[21,11128,11130],{"className":11129},[1118,1119],[21,11131,11133,11186],{"className":11132},[74,75],[21,11134,11136,11183],{"className":11135},[79],[21,11137,11139,11159,11169],{"className":11138,"style":1129},[83],[21,11140,11141,11144],{"style":1132},[21,11142],{"className":11143,"style":1136},[91],[21,11145,11147],{"className":11146},[96,97,98,99],[21,11148,11150,11153,11156],{"className":11149},[42,99],[21,11151,1147],{"className":11152,"style":1146},[42,43,99],[21,11154,1101],{"className":11155},[383,99],[21,11157,103],{"className":11158},[42,43,99],[21,11160,11161,11164],{"style":1156},[21,11162],{"className":11163,"style":1136},[91],[21,11165,11166],{},[21,11167,1167],{"className":11168},[1118,1165,1166],[21,11170,11171,11174],{"style":1170},[21,11172],{"className":11173,"style":1136},[91],[21,11175,11177],{"className":11176},[96,97,98,99],[21,11178,11180],{"className":11179},[42,99],[21,11181,1183],{"className":11182},[42,99],[21,11184,108],{"className":11185},[107],[21,11187,11189],{"className":11188},[79],[21,11190,11192],{"className":11191,"style":1193},[83],[21,11193],{},[21,11195],{"className":11196,"style":458},[378],[21,11198,11200,11203],{"className":11199},[42],[21,11201,9293],{"className":11202,"style":9292},[42,43],[21,11204,11206],{"className":11205},[70],[21,11207,11209,11240],{"className":11208},[74,75],[21,11210,11212,11237],{"className":11211},[79],[21,11213,11215,11226],{"className":11214,"style":9306},[83],[21,11216,11217,11220],{"style":9309},[21,11218],{"className":11219,"style":92},[91],[21,11221,11223],{"className":11222},[96,97,98,99],[21,11224,1147],{"className":11225,"style":1146},[42,43,99],[21,11227,11228,11231],{"style":3714},[21,11229],{"className":11230,"style":92},[91],[21,11232,11234],{"className":11233},[96,97,98,99],[21,11235,1583],{"className":11236},[42,43,99],[21,11238,108],{"className":11239},[107],[21,11241,11243],{"className":11242},[79],[21,11244,11246],{"className":11245,"style":3739},[83],[21,11247],{},[21,11249],{"className":11250,"style":458},[378],[21,11252,11254],{"className":11253},[1118,1119],[21,11255,11257,11316],{"className":11256},[74,75],[21,11258,11260,11313],{"className":11259},[79],[21,11261,11263,11289,11299],{"className":11262,"style":1225},[83],[21,11264,11265,11268],{"style":1228},[21,11266],{"className":11267,"style":1136},[91],[21,11269,11271],{"className":11270},[96,97,98,99],[21,11272,11274,11277,11280,11283,11286],{"className":11273},[42,99],[21,11275,1241],{"className":11276},[42,43,99],[21,11278,1101],{"className":11279},[383,99],[21,11281,103],{"className":11282},[42,43,99],[21,11284,963],{"className":11285},[408,99],[21,11287,358],{"className":11288},[42,99],[21,11290,11291,11294],{"style":1156},[21,11292],{"className":11293,"style":1136},[91],[21,11295,11296],{},[21,11297,1264],{"className":11298},[1118,1165,1166],[21,11300,11301,11304],{"style":1170},[21,11302],{"className":11303,"style":1136},[91],[21,11305,11307],{"className":11306},[96,97,98,99],[21,11308,11310],{"className":11309},[42,99],[21,11311,1147],{"className":11312,"style":1146},[42,43,99],[21,11314,108],{"className":11315},[107],[21,11317,11319],{"className":11318},[79],[21,11320,11322],{"className":11321,"style":1288},[83],[21,11323],{},[21,11325],{"className":11326,"style":458},[378],[21,11328,11330,11333],{"className":11329},[42],[21,11331,200],{"className":11332,"style":199},[42,43],[21,11334,11336],{"className":11335},[70],[21,11337,11339,11359],{"className":11338},[74,75],[21,11340,11342,11356],{"className":11341},[79],[21,11343,11345],{"className":11344,"style":1312},[83],[21,11346,11347,11350],{"style":737},[21,11348],{"className":11349,"style":92},[91],[21,11351,11353],{"className":11352},[96,97,98,99],[21,11354,1241],{"className":11355},[42,43,99],[21,11357,108],{"className":11358},[107],[21,11360,11362],{"className":11361},[79],[21,11363,11365],{"className":11364,"style":115},[83],[21,11366],{},[21,11368,11370,11373],{"className":11369},[42],[21,11371,44],{"className":11372},[42,43],[21,11374,11376],{"className":11375},[70],[21,11377,11379,11399],{"className":11378},[74,75],[21,11380,11382,11396],{"className":11381},[79],[21,11383,11385],{"className":11384,"style":1312},[83],[21,11386,11387,11390],{"style":87},[21,11388],{"className":11389,"style":92},[91],[21,11391,11393],{"className":11392},[96,97,98,99],[21,11394,1241],{"className":11395},[42,43,99],[21,11397,108],{"className":11398},[107],[21,11400,11402],{"className":11401},[79],[21,11403,11405],{"className":11404,"style":115},[83],[21,11406],{},[21,11408,11410,11413],{"className":11409},[42],[21,11411,295],{"className":11412},[42,43],[21,11414,11416],{"className":11415},[70],[21,11417,11419,11439],{"className":11418},[74,75],[21,11420,11422,11436],{"className":11421},[79],[21,11423,11425],{"className":11424,"style":1312},[83],[21,11426,11427,11430],{"style":87},[21,11428],{"className":11429,"style":92},[91],[21,11431,11433],{"className":11432},[96,97,98,99],[21,11434,1241],{"className":11435},[42,43,99],[21,11437,108],{"className":11438},[107],[21,11440,11442],{"className":11441},[79],[21,11443,11445],{"className":11444,"style":115},[83],[21,11446],{},[21,11448,11450],{"className":11449},[42],[21,11451,635],{"className":11452},[1211,3883],[21,11454,10875],{"className":11455},[42],[21,11457,11459],{"className":11458},[42,3964],[21,11460,11462],{"className":11461},[74],[21,11463,11465],{"className":11464},[79],[21,11466,11468,11476],{"className":11467,"style":38},[83],[21,11469,11470,11473],{"style":3976},[21,11471],{"className":11472,"style":3980},[91],[21,11474,3984],{"className":11475,"style":257},[42,43],[21,11477,11478,11481],{"style":3976},[21,11479],{"className":11480,"style":3980},[91],[21,11482,11484],{"className":11483,"style":3994},[3993],[21,11485,3998],{"className":11486},[42],[21,11488,543],{"className":11489},[444],[21,11491,11493,11496],{"className":11492},[42],[21,11494,400],{"className":11495,"style":550},[42,43],[21,11497,11499],{"className":11498},[70],[21,11500,11502,11522],{"className":11501},[74,75],[21,11503,11505,11519],{"className":11504},[79],[21,11506,11508],{"className":11507,"style":84},[83],[21,11509,11510,11513],{"style":565},[21,11511],{"className":11512,"style":92},[91],[21,11514,11516],{"className":11515},[96,97,98,99],[21,11517,103],{"className":11518},[42,43,99],[21,11520,108],{"className":11521},[107],[21,11523,11525],{"className":11524},[79],[21,11526,11528],{"className":11527,"style":115},[83],[21,11529],{},[21,11531,454],{"className":11532},[453],[21,11534],{"className":11535,"style":458},[378],[21,11537,11539,11542],{"className":11538},[42],[21,11540,4064],{"className":11541,"style":4063},[42,65],[21,11543,11545],{"className":11544},[70],[21,11546,11548,11568],{"className":11547},[74,75],[21,11549,11551,11565],{"className":11550},[79],[21,11552,11554],{"className":11553,"style":84},[83],[21,11555,11556,11559],{"style":4079},[21,11557],{"className":11558,"style":92},[91],[21,11560,11562],{"className":11561},[96,97,98,99],[21,11563,103],{"className":11564},[42,43,99],[21,11566,108],{"className":11567},[107],[21,11569,11571],{"className":11570},[79],[21,11572,11574],{"className":11573,"style":115},[83],[21,11575],{},[21,11577,635],{"className":11578},[465],[11,11580,11581,11582,11597,11598,454],{},"already looks like a trace-based TD update: a product of past terms multiplying\npresent TD errors. Summing this forward-view update over time and applying the\nsummation-rule rearrangement (the same manipulation used for on-policy TD(",[21,11583,11585],{"className":11584},[24],[21,11586,11588],{"className":11587,"ariaHidden":29},[28],[21,11589,11591,11594],{"className":11590},[33],[21,11592],{"className":11593,"style":38},[37],[21,11595,44],{"className":11596},[42,43],"))\nconverts it into a backward view driven by a single accumulating trace. The trace\nthat falls out is the general ",[824,11599,11600],{},"accumulating trace with importance sampling",[21,11602,11604],{"className":11603},[847],[21,11605,11607],{"className":11606},[24],[21,11608,11610,11672,11865],{"className":11609,"ariaHidden":29},[28],[21,11611,11613,11617,11657,11660,11663,11666,11669],{"className":11612},[33],[21,11614],{"className":11615,"style":11616},[37],"height:0.823em;vertical-align:-0.15em;",[21,11618,11620,11623],{"className":11619},[42],[21,11621,66],{"className":11622},[42,65],[21,11624,11626],{"className":11625},[70],[21,11627,11629,11649],{"className":11628},[74,75],[21,11630,11632,11646],{"className":11631},[79],[21,11633,11635],{"className":11634,"style":84},[83],[21,11636,11637,11640],{"style":87},[21,11638],{"className":11639,"style":92},[91],[21,11641,11643],{"className":11642},[96,97,98,99],[21,11644,103],{"className":11645},[42,43,99],[21,11647,108],{"className":11648},[107],[21,11650,11652],{"className":11651},[79],[21,11653,11655],{"className":11654,"style":115},[83],[21,11656],{},[21,11658],{"className":11659,"style":379},[378],[21,11661],{"className":11662,"style":379},[378],[21,11664,527],{"className":11665},[383],[21,11667],{"className":11668,"style":379},[378],[21,11670],{"className":11671,"style":379},[378],[21,11673,11675,11678,11718,11724,11764,11804,11807,11856,11859,11862],{"className":11674},[33],[21,11676],{"className":11677,"style":6844},[37],[21,11679,11681,11684],{"className":11680},[42],[21,11682,295],{"className":11683},[42,43],[21,11685,11687],{"className":11686},[70],[21,11688,11690,11710],{"className":11689},[74,75],[21,11691,11693,11707],{"className":11692},[79],[21,11694,11696],{"className":11695,"style":84},[83],[21,11697,11698,11701],{"style":87},[21,11699],{"className":11700,"style":92},[91],[21,11702,11704],{"className":11703},[96,97,98,99],[21,11705,103],{"className":11706},[42,43,99],[21,11708,108],{"className":11709},[107],[21,11711,11713],{"className":11712},[79],[21,11714,11716],{"className":11715,"style":115},[83],[21,11717],{},[21,11719,11721],{"className":11720},[42],[21,11722,543],{"className":11723},[1211,6900],[21,11725,11727,11730],{"className":11726},[42],[21,11728,200],{"className":11729,"style":199},[42,43],[21,11731,11733],{"className":11732},[70],[21,11734,11736,11756],{"className":11735},[74,75],[21,11737,11739,11753],{"className":11738},[79],[21,11740,11742],{"className":11741,"style":84},[83],[21,11743,11744,11747],{"style":737},[21,11745],{"className":11746,"style":92},[91],[21,11748,11750],{"className":11749},[96,97,98,99],[21,11751,103],{"className":11752},[42,43,99],[21,11754,108],{"className":11755},[107],[21,11757,11759],{"className":11758},[79],[21,11760,11762],{"className":11761,"style":115},[83],[21,11763],{},[21,11765,11767,11770],{"className":11766},[42],[21,11768,44],{"className":11769},[42,43],[21,11771,11773],{"className":11772},[70],[21,11774,11776,11796],{"className":11775},[74,75],[21,11777,11779,11793],{"className":11778},[79],[21,11780,11782],{"className":11781,"style":84},[83],[21,11783,11784,11787],{"style":87},[21,11785],{"className":11786,"style":92},[91],[21,11788,11790],{"className":11789},[96,97,98,99],[21,11791,103],{"className":11792},[42,43,99],[21,11794,108],{"className":11795},[107],[21,11797,11799],{"className":11798},[79],[21,11800,11802],{"className":11801,"style":115},[83],[21,11803],{},[21,11805],{"className":11806,"style":458},[378],[21,11808,11810,11813],{"className":11809},[42],[21,11811,66],{"className":11812},[42,65],[21,11814,11816],{"className":11815},[70],[21,11817,11819,11848],{"className":11818},[74,75],[21,11820,11822,11845],{"className":11821},[79],[21,11823,11825],{"className":11824,"style":944},[83],[21,11826,11827,11830],{"style":87},[21,11828],{"className":11829,"style":92},[91],[21,11831,11833],{"className":11832},[96,97,98,99],[21,11834,11836,11839,11842],{"className":11835},[42,99],[21,11837,103],{"className":11838},[42,43,99],[21,11840,3896],{"className":11841},[408,99],[21,11843,358],{"className":11844},[42,99],[21,11846,108],{"className":11847},[107],[21,11849,11851],{"className":11850},[79],[21,11852,11854],{"className":11853,"style":976},[83],[21,11855],{},[21,11857],{"className":11858,"style":404},[378],[21,11860,963],{"className":11861},[408],[21,11863],{"className":11864,"style":404},[378],[21,11866,11868,11871,11874,11905,11908,11948,11951,11954,11994,11997,12003],{"className":11867},[33],[21,11869],{"className":11870,"style":6844},[37],[21,11872,10875],{"className":11873},[42],[21,11875,11877],{"className":11876},[42,3964],[21,11878,11880],{"className":11879},[74],[21,11881,11883],{"className":11882},[79],[21,11884,11886,11894],{"className":11885,"style":38},[83],[21,11887,11888,11891],{"style":3976},[21,11889],{"className":11890,"style":3980},[91],[21,11892,3984],{"className":11893,"style":257},[42,43],[21,11895,11896,11899],{"style":3976},[21,11897],{"className":11898,"style":3980},[91],[21,11900,11902],{"className":11901,"style":3994},[3993],[21,11903,3998],{"className":11904},[42],[21,11906,543],{"className":11907},[444],[21,11909,11911,11914],{"className":11910},[42],[21,11912,400],{"className":11913,"style":550},[42,43],[21,11915,11917],{"className":11916},[70],[21,11918,11920,11940],{"className":11919},[74,75],[21,11921,11923,11937],{"className":11922},[79],[21,11924,11926],{"className":11925,"style":84},[83],[21,11927,11928,11931],{"style":565},[21,11929],{"className":11930,"style":92},[91],[21,11932,11934],{"className":11933},[96,97,98,99],[21,11935,103],{"className":11936},[42,43,99],[21,11938,108],{"className":11939},[107],[21,11941,11943],{"className":11942},[79],[21,11944,11946],{"className":11945,"style":115},[83],[21,11947],{},[21,11949,454],{"className":11950},[453],[21,11952],{"className":11953,"style":458},[378],[21,11955,11957,11960],{"className":11956},[42],[21,11958,4064],{"className":11959,"style":4063},[42,65],[21,11961,11963],{"className":11962},[70],[21,11964,11966,11986],{"className":11965},[74,75],[21,11967,11969,11983],{"className":11968},[79],[21,11970,11972],{"className":11971,"style":84},[83],[21,11973,11974,11977],{"style":4079},[21,11975],{"className":11976,"style":92},[91],[21,11978,11980],{"className":11979},[96,97,98,99],[21,11981,103],{"className":11982},[42,43,99],[21,11984,108],{"className":11985},[107],[21,11987,11989],{"className":11988},[79],[21,11990,11992],{"className":11991,"style":115},[83],[21,11993],{},[21,11995,635],{"className":11996},[465],[21,11998,12000],{"className":11999},[42],[21,12001,635],{"className":12002},[1211,6900],[21,12004,454],{"className":12005},[453],[11,12007,12008,12009,12237,12238,12308,12309,12333,12334,12386,12387,12390],{},"paired with the usual semi-gradient update ",[21,12010,12012],{"className":12011},[24],[21,12013,12015,12079,12134],{"className":12014,"ariaHidden":29},[28],[21,12016,12018,12021,12070,12073,12076],{"className":12017},[33],[21,12019],{"className":12020,"style":10532},[37],[21,12022,12024,12027],{"className":12023},[42],[21,12025,4064],{"className":12026,"style":4063},[42,65],[21,12028,12030],{"className":12029},[70],[21,12031,12033,12062],{"className":12032},[74,75],[21,12034,12036,12059],{"className":12035},[79],[21,12037,12039],{"className":12038,"style":944},[83],[21,12040,12041,12044],{"style":4079},[21,12042],{"className":12043,"style":92},[91],[21,12045,12047],{"className":12046},[96,97,98,99],[21,12048,12050,12053,12056],{"className":12049},[42,99],[21,12051,103],{"className":12052},[42,43,99],[21,12054,963],{"className":12055},[408,99],[21,12057,358],{"className":12058},[42,99],[21,12060,108],{"className":12061},[107],[21,12063,12065],{"className":12064},[79],[21,12066,12068],{"className":12067,"style":976},[83],[21,12069],{},[21,12071],{"className":12072,"style":379},[378],[21,12074,1101],{"className":12075},[383],[21,12077],{"className":12078,"style":379},[378],[21,12080,12082,12085,12125,12128,12131],{"className":12081},[33],[21,12083],{"className":12084,"style":10603},[37],[21,12086,12088,12091],{"className":12087},[42],[21,12089,4064],{"className":12090,"style":4063},[42,65],[21,12092,12094],{"className":12093},[70],[21,12095,12097,12117],{"className":12096},[74,75],[21,12098,12100,12114],{"className":12099},[79],[21,12101,12103],{"className":12102,"style":84},[83],[21,12104,12105,12108],{"style":4079},[21,12106],{"className":12107,"style":92},[91],[21,12109,12111],{"className":12110},[96,97,98,99],[21,12112,103],{"className":12113},[42,43,99],[21,12115,108],{"className":12116},[107],[21,12118,12120],{"className":12119},[79],[21,12121,12123],{"className":12122,"style":115},[83],[21,12124],{},[21,12126],{"className":12127,"style":404},[378],[21,12129,963],{"className":12130},[408],[21,12132],{"className":12133,"style":404},[378],[21,12135,12137,12141,12144,12197],{"className":12136},[33],[21,12138],{"className":12139,"style":12140},[37],"height:0.9414em;vertical-align:-0.247em;",[21,12142,10664],{"className":12143,"style":10663},[42,43],[21,12145,12147,12150],{"className":12146},[42],[21,12148,9293],{"className":12149,"style":9292},[42,43],[21,12151,12153],{"className":12152},[70],[21,12154,12156,12189],{"className":12155},[74,75],[21,12157,12159,12186],{"className":12158},[79],[21,12160,12163,12174],{"className":12161,"style":12162},[83],"height:0.6644em;",[21,12164,12165,12168],{"style":9309},[21,12166],{"className":12167,"style":92},[91],[21,12169,12171],{"className":12170},[96,97,98,99],[21,12172,103],{"className":12173},[42,43,99],[21,12175,12177,12180],{"style":12176},"top:-3.063em;margin-right:0.05em;",[21,12178],{"className":12179,"style":92},[91],[21,12181,12183],{"className":12182},[96,97,98,99],[21,12184,1583],{"className":12185},[42,43,99],[21,12187,108],{"className":12188},[107],[21,12190,12192],{"className":12191},[79],[21,12193,12195],{"className":12194,"style":3739},[83],[21,12196],{},[21,12198,12200,12203],{"className":12199},[42],[21,12201,66],{"className":12202},[42,65],[21,12204,12206],{"className":12205},[70],[21,12207,12209,12229],{"className":12208},[74,75],[21,12210,12212,12226],{"className":12211},[79],[21,12213,12215],{"className":12214,"style":84},[83],[21,12216,12217,12220],{"style":87},[21,12218],{"className":12219,"style":92},[91],[21,12221,12223],{"className":12222},[96,97,98,99],[21,12224,103],{"className":12225},[42,43,99],[21,12227,108],{"className":12228},[107],[21,12230,12232],{"className":12231},[79],[21,12233,12235],{"className":12234,"style":115},[83],[21,12236],{},". In the on-policy case every ",[21,12239,12241],{"className":12240},[24],[21,12242,12244,12299],{"className":12243,"ariaHidden":29},[28],[21,12245,12247,12250,12290,12293,12296],{"className":12246},[33],[21,12248],{"className":12249,"style":195},[37],[21,12251,12253,12256],{"className":12252},[42],[21,12254,295],{"className":12255},[42,43],[21,12257,12259],{"className":12258},[70],[21,12260,12262,12282],{"className":12261},[74,75],[21,12263,12265,12279],{"className":12264},[79],[21,12266,12268],{"className":12267,"style":84},[83],[21,12269,12270,12273],{"style":87},[21,12271],{"className":12272,"style":92},[91],[21,12274,12276],{"className":12275},[96,97,98,99],[21,12277,103],{"className":12278},[42,43,99],[21,12280,108],{"className":12281},[107],[21,12283,12285],{"className":12284},[79],[21,12286,12288],{"className":12287,"style":115},[83],[21,12289],{},[21,12291],{"className":12292,"style":379},[378],[21,12294,1101],{"className":12295},[383],[21,12297],{"className":12298,"style":379},[378],[21,12300,12302,12305],{"className":12301},[33],[21,12303],{"className":12304,"style":1540},[37],[21,12306,358],{"className":12307},[42]," and this\nreduces to the ordinary accumulating trace (extended to variable ",[21,12310,12312],{"className":12311},[24],[21,12313,12315],{"className":12314,"ariaHidden":29},[28],[21,12316,12318,12321,12324,12327,12330],{"className":12317},[33],[21,12319],{"className":12320,"style":5968},[37],[21,12322,44],{"className":12323},[42,43],[21,12325,454],{"className":12326},[453],[21,12328],{"className":12329,"style":458},[378],[21,12331,200],{"className":12332,"style":199},[42,43],"); off-policy, the ",[21,12335,12337],{"className":12336},[24],[21,12338,12340],{"className":12339,"ariaHidden":29},[28],[21,12341,12343,12346],{"className":12342},[33],[21,12344],{"className":12345,"style":195},[37],[21,12347,12349,12352],{"className":12348},[42],[21,12350,295],{"className":12351},[42,43],[21,12353,12355],{"className":12354},[70],[21,12356,12358,12378],{"className":12357},[74,75],[21,12359,12361,12375],{"className":12360},[79],[21,12362,12364],{"className":12363,"style":84},[83],[21,12365,12366,12369],{"style":87},[21,12367],{"className":12368,"style":92},[91],[21,12370,12372],{"className":12371},[96,97,98,99],[21,12373,103],{"className":12374},[42,43,99],[21,12376,108],{"className":12377},[107],[21,12379,12381],{"className":12380},[79],[21,12382,12384],{"className":12383,"style":115},[83],[21,12385],{}," scaling makes the algorithm correct in\nexpectation. But as a semi-gradient method it is ",[237,12388,12389],{},"not"," guaranteed stable — the\nnext sections repair that.",[1651,12392],{"hash":12393},"f2c9601fc1e7ddc99af75ebc80b986d2cc125b05a0574b5f180a1bafee2bbd91",[11,12395,12396,12397,12412],{},"The identical derivation runs for action values. Starting from the Expected-Sarsa\nform of the general ",[21,12398,12400],{"className":12399},[24],[21,12401,12403],{"className":12402,"ariaHidden":29},[28],[21,12404,12406,12409],{"className":12405},[33],[21,12407],{"className":12408,"style":38},[37],[21,12410,44],{"className":12411},[42,43],"-return — which works out simpler than the Sarsa form\n— the same steps produce the action-value off-policy trace,",[21,12414,12416],{"className":12415},[847],[21,12417,12419],{"className":12418},[24],[21,12420,12422,12483,12673],{"className":12421,"ariaHidden":29},[28],[21,12423,12425,12428,12468,12471,12474,12477,12480],{"className":12424},[33],[21,12426],{"className":12427,"style":11616},[37],[21,12429,12431,12434],{"className":12430},[42],[21,12432,66],{"className":12433},[42,65],[21,12435,12437],{"className":12436},[70],[21,12438,12440,12460],{"className":12439},[74,75],[21,12441,12443,12457],{"className":12442},[79],[21,12444,12446],{"className":12445,"style":84},[83],[21,12447,12448,12451],{"style":87},[21,12449],{"className":12450,"style":92},[91],[21,12452,12454],{"className":12453},[96,97,98,99],[21,12455,103],{"className":12456},[42,43,99],[21,12458,108],{"className":12459},[107],[21,12461,12463],{"className":12462},[79],[21,12464,12466],{"className":12465,"style":115},[83],[21,12467],{},[21,12469],{"className":12470,"style":379},[378],[21,12472],{"className":12473,"style":379},[378],[21,12475,527],{"className":12476},[383],[21,12478],{"className":12479,"style":379},[378],[21,12481],{"className":12482,"style":379},[378],[21,12484,12486,12489,12529,12569,12572,12612,12615,12664,12667,12670],{"className":12485},[33],[21,12487],{"className":12488,"style":4342},[37],[21,12490,12492,12495],{"className":12491},[42],[21,12493,200],{"className":12494,"style":199},[42,43],[21,12496,12498],{"className":12497},[70],[21,12499,12501,12521],{"className":12500},[74,75],[21,12502,12504,12518],{"className":12503},[79],[21,12505,12507],{"className":12506,"style":84},[83],[21,12508,12509,12512],{"style":737},[21,12510],{"className":12511,"style":92},[91],[21,12513,12515],{"className":12514},[96,97,98,99],[21,12516,103],{"className":12517},[42,43,99],[21,12519,108],{"className":12520},[107],[21,12522,12524],{"className":12523},[79],[21,12525,12527],{"className":12526,"style":115},[83],[21,12528],{},[21,12530,12532,12535],{"className":12531},[42],[21,12533,44],{"className":12534},[42,43],[21,12536,12538],{"className":12537},[70],[21,12539,12541,12561],{"className":12540},[74,75],[21,12542,12544,12558],{"className":12543},[79],[21,12545,12547],{"className":12546,"style":84},[83],[21,12548,12549,12552],{"style":87},[21,12550],{"className":12551,"style":92},[91],[21,12553,12555],{"className":12554},[96,97,98,99],[21,12556,103],{"className":12557},[42,43,99],[21,12559,108],{"className":12560},[107],[21,12562,12564],{"className":12563},[79],[21,12565,12567],{"className":12566,"style":115},[83],[21,12568],{},[21,12570],{"className":12571,"style":458},[378],[21,12573,12575,12578],{"className":12574},[42],[21,12576,295],{"className":12577},[42,43],[21,12579,12581],{"className":12580},[70],[21,12582,12584,12604],{"className":12583},[74,75],[21,12585,12587,12601],{"className":12586},[79],[21,12588,12590],{"className":12589,"style":84},[83],[21,12591,12592,12595],{"style":87},[21,12593],{"className":12594,"style":92},[91],[21,12596,12598],{"className":12597},[96,97,98,99],[21,12599,103],{"className":12600},[42,43,99],[21,12602,108],{"className":12603},[107],[21,12605,12607],{"className":12606},[79],[21,12608,12610],{"className":12609,"style":115},[83],[21,12611],{},[21,12613],{"className":12614,"style":458},[378],[21,12616,12618,12621],{"className":12617},[42],[21,12619,66],{"className":12620},[42,65],[21,12622,12624],{"className":12623},[70],[21,12625,12627,12656],{"className":12626},[74,75],[21,12628,12630,12653],{"className":12629},[79],[21,12631,12633],{"className":12632,"style":944},[83],[21,12634,12635,12638],{"style":87},[21,12636],{"className":12637,"style":92},[91],[21,12639,12641],{"className":12640},[96,97,98,99],[21,12642,12644,12647,12650],{"className":12643},[42,99],[21,12645,103],{"className":12646},[42,43,99],[21,12648,3896],{"className":12649},[408,99],[21,12651,358],{"className":12652},[42,99],[21,12654,108],{"className":12655},[107],[21,12657,12659],{"className":12658},[79],[21,12660,12662],{"className":12661,"style":976},[83],[21,12663],{},[21,12665],{"className":12666,"style":404},[378],[21,12668,963],{"className":12669},[408],[21,12671],{"className":12672,"style":404},[378],[21,12674,12676,12679,12682,12724,12727,12767,12770,12773,12813,12816,12819,12859,12862],{"className":12675},[33],[21,12677],{"className":12678,"style":440},[37],[21,12680,10875],{"className":12681},[42],[21,12683,12685],{"className":12684},[42,3964],[21,12686,12688,12716],{"className":12687},[74,75],[21,12689,12691,12713],{"className":12690},[79],[21,12692,12694,12702],{"className":12693,"style":38},[83],[21,12695,12696,12699],{"style":3976},[21,12697],{"className":12698,"style":3980},[91],[21,12700,4777],{"className":12701,"style":257},[42,43],[21,12703,12704,12707],{"style":3976},[21,12705],{"className":12706,"style":3980},[91],[21,12708,12710],{"className":12709,"style":4786},[3993],[21,12711,3998],{"className":12712},[42],[21,12714,108],{"className":12715},[107],[21,12717,12719],{"className":12718},[79],[21,12720,12722],{"className":12721,"style":4799},[83],[21,12723],{},[21,12725,543],{"className":12726},[444],[21,12728,12730,12733],{"className":12729},[42],[21,12731,400],{"className":12732,"style":550},[42,43],[21,12734,12736],{"className":12735},[70],[21,12737,12739,12759],{"className":12738},[74,75],[21,12740,12742,12756],{"className":12741},[79],[21,12743,12745],{"className":12744,"style":84},[83],[21,12746,12747,12750],{"style":565},[21,12748],{"className":12749,"style":92},[91],[21,12751,12753],{"className":12752},[96,97,98,99],[21,12754,103],{"className":12755},[42,43,99],[21,12757,108],{"className":12758},[107],[21,12760,12762],{"className":12761},[79],[21,12763,12765],{"className":12764,"style":115},[83],[21,12766],{},[21,12768,454],{"className":12769},[453],[21,12771],{"className":12772,"style":458},[378],[21,12774,12776,12779],{"className":12775},[42],[21,12777,423],{"className":12778},[42,43],[21,12780,12782],{"className":12781},[70],[21,12783,12785,12805],{"className":12784},[74,75],[21,12786,12788,12802],{"className":12787},[79],[21,12789,12791],{"className":12790,"style":84},[83],[21,12792,12793,12796],{"style":87},[21,12794],{"className":12795,"style":92},[91],[21,12797,12799],{"className":12798},[96,97,98,99],[21,12800,103],{"className":12801},[42,43,99],[21,12803,108],{"className":12804},[107],[21,12806,12808],{"className":12807},[79],[21,12809,12811],{"className":12810,"style":115},[83],[21,12812],{},[21,12814,454],{"className":12815},[453],[21,12817],{"className":12818,"style":458},[378],[21,12820,12822,12825],{"className":12821},[42],[21,12823,4064],{"className":12824,"style":4063},[42,65],[21,12826,12828],{"className":12827},[70],[21,12829,12831,12851],{"className":12830},[74,75],[21,12832,12834,12848],{"className":12833},[79],[21,12835,12837],{"className":12836,"style":84},[83],[21,12838,12839,12842],{"style":4079},[21,12840],{"className":12841,"style":92},[91],[21,12843,12845],{"className":12844},[96,97,98,99],[21,12846,103],{"className":12847},[42,43,99],[21,12849,108],{"className":12850},[107],[21,12852,12854],{"className":12853},[79],[21,12855,12857],{"className":12856,"style":115},[83],[21,12858],{},[21,12860,635],{"className":12861},[465],[21,12863,454],{"className":12864},[453],[11,12866,12867],{},"with the expectation-form action-value TD error",[21,12869,12871],{"className":12870},[847],[21,12872,12874],{"className":12873},[24],[21,12875,12877,12949,13013,13203],{"className":12876,"ariaHidden":29},[28],[21,12878,12880,12883,12934,12937,12940,12943,12946],{"className":12879},[33],[21,12881],{"className":12882,"style":9285},[37],[21,12884,12886,12889],{"className":12885},[42],[21,12887,9293],{"className":12888,"style":9292},[42,43],[21,12890,12892],{"className":12891},[70],[21,12893,12895,12926],{"className":12894},[74,75],[21,12896,12898,12923],{"className":12897},[79],[21,12899,12901,12912],{"className":12900,"style":9306},[83],[21,12902,12903,12906],{"style":9309},[21,12904],{"className":12905,"style":92},[91],[21,12907,12909],{"className":12908},[96,97,98,99],[21,12910,103],{"className":12911},[42,43,99],[21,12913,12914,12917],{"style":3714},[21,12915],{"className":12916,"style":92},[91],[21,12918,12920],{"className":12919},[96,97,98,99],[21,12921,15],{"className":12922},[42,43,99],[21,12924,108],{"className":12925},[107],[21,12927,12929],{"className":12928},[79],[21,12930,12932],{"className":12931,"style":3739},[83],[21,12933],{},[21,12935],{"className":12936,"style":379},[378],[21,12938],{"className":12939,"style":379},[378],[21,12941,527],{"className":12942},[383],[21,12944],{"className":12945,"style":379},[378],[21,12947],{"className":12948,"style":379},[378],[21,12950,12952,12955,13004,13007,13010],{"className":12951},[33],[21,12953],{"className":12954,"style":923},[37],[21,12956,12958,12961],{"className":12957},[42],[21,12959,931],{"className":12960,"style":930},[42,43],[21,12962,12964],{"className":12963},[70],[21,12965,12967,12996],{"className":12966},[74,75],[21,12968,12970,12993],{"className":12969},[79],[21,12971,12973],{"className":12972,"style":944},[83],[21,12974,12975,12978],{"style":947},[21,12976],{"className":12977,"style":92},[91],[21,12979,12981],{"className":12980},[96,97,98,99],[21,12982,12984,12987,12990],{"className":12983},[42,99],[21,12985,103],{"className":12986},[42,43,99],[21,12988,963],{"className":12989},[408,99],[21,12991,358],{"className":12992},[42,99],[21,12994,108],{"className":12995},[107],[21,12997,12999],{"className":12998},[79],[21,13000,13002],{"className":13001,"style":976},[83],[21,13003],{},[21,13005],{"className":13006,"style":404},[378],[21,13008,963],{"className":13009},[408],[21,13011],{"className":13012,"style":404},[378],[21,13014,13016,13019,13068,13071,13139,13142,13191,13194,13197,13200],{"className":13015},[33],[21,13017],{"className":13018,"style":5330},[37],[21,13020,13022,13025],{"className":13021},[42],[21,13023,200],{"className":13024,"style":199},[42,43],[21,13026,13028],{"className":13027},[70],[21,13029,13031,13060],{"className":13030},[74,75],[21,13032,13034,13057],{"className":13033},[79],[21,13035,13037],{"className":13036,"style":944},[83],[21,13038,13039,13042],{"style":737},[21,13040],{"className":13041,"style":92},[91],[21,13043,13045],{"className":13044},[96,97,98,99],[21,13046,13048,13051,13054],{"className":13047},[42,99],[21,13049,103],{"className":13050},[42,43,99],[21,13052,963],{"className":13053},[408,99],[21,13055,358],{"className":13056},[42,99],[21,13058,108],{"className":13059},[107],[21,13061,13063],{"className":13062},[79],[21,13064,13066],{"className":13065,"style":976},[83],[21,13067],{},[21,13069],{"className":13070,"style":458},[378],[21,13072,13074,13105],{"className":13073},[42],[21,13075,13077],{"className":13076},[42,3964],[21,13078,13080],{"className":13079},[74],[21,13081,13083],{"className":13082},[79],[21,13084,13086,13094],{"className":13085,"style":5401},[83],[21,13087,13088,13091],{"style":3976},[21,13089],{"className":13090,"style":3980},[91],[21,13092,5410],{"className":13093,"style":404},[42,43],[21,13095,13096,13099],{"style":5413},[21,13097],{"className":13098,"style":3980},[91],[21,13100,13102],{"className":13101,"style":5420},[3993],[21,13103,5424],{"className":13104},[42],[21,13106,13108],{"className":13107},[70],[21,13109,13111,13131],{"className":13110},[74,75],[21,13112,13114,13128],{"className":13113},[79],[21,13115,13117],{"className":13116,"style":84},[83],[21,13118,13119,13122],{"style":5439},[21,13120],{"className":13121,"style":92},[91],[21,13123,13125],{"className":13124},[96,97,98,99],[21,13126,103],{"className":13127},[42,43,99],[21,13129,108],{"className":13130},[107],[21,13132,13134],{"className":13133},[79],[21,13135,13137],{"className":13136,"style":115},[83],[21,13138],{},[21,13140,543],{"className":13141},[444],[21,13143,13145,13148],{"className":13144},[42],[21,13146,400],{"className":13147,"style":550},[42,43],[21,13149,13151],{"className":13150},[70],[21,13152,13154,13183],{"className":13153},[74,75],[21,13155,13157,13180],{"className":13156},[79],[21,13158,13160],{"className":13159,"style":944},[83],[21,13161,13162,13165],{"style":565},[21,13163],{"className":13164,"style":92},[91],[21,13166,13168],{"className":13167},[96,97,98,99],[21,13169,13171,13174,13177],{"className":13170},[42,99],[21,13172,103],{"className":13173},[42,43,99],[21,13175,963],{"className":13176},[408,99],[21,13178,358],{"className":13179},[42,99],[21,13181,108],{"className":13182},[107],[21,13184,13186],{"className":13185},[79],[21,13187,13189],{"className":13188,"style":976},[83],[21,13190],{},[21,13192,635],{"className":13193},[465],[21,13195],{"className":13196,"style":404},[378],[21,13198,3896],{"className":13199},[408],[21,13201],{"className":13202,"style":404},[378],[21,13204,13206,13209,13251,13254,13294,13297,13300,13340,13343,13346,13386,13389],{"className":13205},[33],[21,13207],{"className":13208,"style":440},[37],[21,13210,13212],{"className":13211},[42,3964],[21,13213,13215,13243],{"className":13214},[74,75],[21,13216,13218,13240],{"className":13217},[79],[21,13219,13221,13229],{"className":13220,"style":38},[83],[21,13222,13223,13226],{"style":3976},[21,13224],{"className":13225,"style":3980},[91],[21,13227,4777],{"className":13228,"style":257},[42,43],[21,13230,13231,13234],{"style":3976},[21,13232],{"className":13233,"style":3980},[91],[21,13235,13237],{"className":13236,"style":4786},[3993],[21,13238,3998],{"className":13239},[42],[21,13241,108],{"className":13242},[107],[21,13244,13246],{"className":13245},[79],[21,13247,13249],{"className":13248,"style":4799},[83],[21,13250],{},[21,13252,543],{"className":13253},[444],[21,13255,13257,13260],{"className":13256},[42],[21,13258,400],{"className":13259,"style":550},[42,43],[21,13261,13263],{"className":13262},[70],[21,13264,13266,13286],{"className":13265},[74,75],[21,13267,13269,13283],{"className":13268},[79],[21,13270,13272],{"className":13271,"style":84},[83],[21,13273,13274,13277],{"style":565},[21,13275],{"className":13276,"style":92},[91],[21,13278,13280],{"className":13279},[96,97,98,99],[21,13281,103],{"className":13282},[42,43,99],[21,13284,108],{"className":13285},[107],[21,13287,13289],{"className":13288},[79],[21,13290,13292],{"className":13291,"style":115},[83],[21,13293],{},[21,13295,454],{"className":13296},[453],[21,13298],{"className":13299,"style":458},[378],[21,13301,13303,13306],{"className":13302},[42],[21,13304,423],{"className":13305},[42,43],[21,13307,13309],{"className":13308},[70],[21,13310,13312,13332],{"className":13311},[74,75],[21,13313,13315,13329],{"className":13314},[79],[21,13316,13318],{"className":13317,"style":84},[83],[21,13319,13320,13323],{"style":87},[21,13321],{"className":13322,"style":92},[91],[21,13324,13326],{"className":13325},[96,97,98,99],[21,13327,103],{"className":13328},[42,43,99],[21,13330,108],{"className":13331},[107],[21,13333,13335],{"className":13334},[79],[21,13336,13338],{"className":13337,"style":115},[83],[21,13339],{},[21,13341,454],{"className":13342},[453],[21,13344],{"className":13345,"style":458},[378],[21,13347,13349,13352],{"className":13348},[42],[21,13350,4064],{"className":13351,"style":4063},[42,65],[21,13353,13355],{"className":13354},[70],[21,13356,13358,13378],{"className":13357},[74,75],[21,13359,13361,13375],{"className":13360},[79],[21,13362,13364],{"className":13363,"style":84},[83],[21,13365,13366,13369],{"style":4079},[21,13367],{"className":13368,"style":92},[91],[21,13370,13372],{"className":13371},[96,97,98,99],[21,13373,103],{"className":13374},[42,43,99],[21,13376,108],{"className":13377},[107],[21,13379,13381],{"className":13380},[79],[21,13382,13384],{"className":13383,"style":115},[83],[21,13385],{},[21,13387,635],{"className":13388},[465],[21,13390,454],{"className":13391},[453],[11,13393,13394,13395,13620,13621,13636,13637,13661,13662,13677],{},"and the same weight update ",[21,13396,13398],{"className":13397},[24],[21,13399,13401,13465,13520],{"className":13400,"ariaHidden":29},[28],[21,13402,13404,13407,13456,13459,13462],{"className":13403},[33],[21,13405],{"className":13406,"style":10532},[37],[21,13408,13410,13413],{"className":13409},[42],[21,13411,4064],{"className":13412,"style":4063},[42,65],[21,13414,13416],{"className":13415},[70],[21,13417,13419,13448],{"className":13418},[74,75],[21,13420,13422,13445],{"className":13421},[79],[21,13423,13425],{"className":13424,"style":944},[83],[21,13426,13427,13430],{"style":4079},[21,13428],{"className":13429,"style":92},[91],[21,13431,13433],{"className":13432},[96,97,98,99],[21,13434,13436,13439,13442],{"className":13435},[42,99],[21,13437,103],{"className":13438},[42,43,99],[21,13440,963],{"className":13441},[408,99],[21,13443,358],{"className":13444},[42,99],[21,13446,108],{"className":13447},[107],[21,13449,13451],{"className":13450},[79],[21,13452,13454],{"className":13453,"style":976},[83],[21,13455],{},[21,13457],{"className":13458,"style":379},[378],[21,13460,1101],{"className":13461},[383],[21,13463],{"className":13464,"style":379},[378],[21,13466,13468,13471,13511,13514,13517],{"className":13467},[33],[21,13469],{"className":13470,"style":10603},[37],[21,13472,13474,13477],{"className":13473},[42],[21,13475,4064],{"className":13476,"style":4063},[42,65],[21,13478,13480],{"className":13479},[70],[21,13481,13483,13503],{"className":13482},[74,75],[21,13484,13486,13500],{"className":13485},[79],[21,13487,13489],{"className":13488,"style":84},[83],[21,13490,13491,13494],{"style":4079},[21,13492],{"className":13493,"style":92},[91],[21,13495,13497],{"className":13496},[96,97,98,99],[21,13498,103],{"className":13499},[42,43,99],[21,13501,108],{"className":13502},[107],[21,13504,13506],{"className":13505},[79],[21,13507,13509],{"className":13508,"style":115},[83],[21,13510],{},[21,13512],{"className":13513,"style":404},[378],[21,13515,963],{"className":13516},[408],[21,13518],{"className":13519,"style":404},[378],[21,13521,13523,13526,13529,13580],{"className":13522},[33],[21,13524],{"className":13525,"style":12140},[37],[21,13527,10664],{"className":13528,"style":10663},[42,43],[21,13530,13532,13535],{"className":13531},[42],[21,13533,9293],{"className":13534,"style":9292},[42,43],[21,13536,13538],{"className":13537},[70],[21,13539,13541,13572],{"className":13540},[74,75],[21,13542,13544,13569],{"className":13543},[79],[21,13545,13547,13558],{"className":13546,"style":12162},[83],[21,13548,13549,13552],{"style":9309},[21,13550],{"className":13551,"style":92},[91],[21,13553,13555],{"className":13554},[96,97,98,99],[21,13556,103],{"className":13557},[42,43,99],[21,13559,13560,13563],{"style":12176},[21,13561],{"className":13562,"style":92},[91],[21,13564,13566],{"className":13565},[96,97,98,99],[21,13567,15],{"className":13568},[42,43,99],[21,13570,108],{"className":13571},[107],[21,13573,13575],{"className":13574},[79],[21,13576,13578],{"className":13577,"style":3739},[83],[21,13579],{},[21,13581,13583,13586],{"className":13582},[42],[21,13584,66],{"className":13585},[42,65],[21,13587,13589],{"className":13588},[70],[21,13590,13592,13612],{"className":13591},[74,75],[21,13593,13595,13609],{"className":13594},[79],[21,13596,13598],{"className":13597,"style":84},[83],[21,13599,13600,13603],{"style":87},[21,13601],{"className":13602,"style":92},[91],[21,13604,13606],{"className":13605},[96,97,98,99],[21,13607,103],{"className":13608},[42,43,99],[21,13610,108],{"className":13611},[107],[21,13613,13615],{"className":13614},[79],[21,13616,13618],{"className":13617,"style":115},[83],[21,13619],{},". This is an efficient Expected-Sarsa(",[21,13622,13624],{"className":13623},[24],[21,13625,13627],{"className":13626,"ariaHidden":29},[28],[21,13628,13630,13633],{"className":13629},[33],[21,13631],{"className":13632,"style":38},[37],[21,13634,44],{"className":13635},[42,43],") that runs\non-policy or off-policy; with constant ",[21,13638,13640],{"className":13639},[24],[21,13641,13643],{"className":13642,"ariaHidden":29},[28],[21,13644,13646,13649,13652,13655,13658],{"className":13645},[33],[21,13647],{"className":13648,"style":5968},[37],[21,13650,44],{"className":13651},[42,43],[21,13653,454],{"className":13654},[453],[21,13656],{"className":13657,"style":458},[378],[21,13659,200],{"className":13660,"style":199},[42,43]," and the ordinary\nstate-action TD error it becomes exactly the Sarsa(",[21,13663,13665],{"className":13664},[24],[21,13666,13668],{"className":13667,"ariaHidden":29},[28],[21,13669,13671,13674],{"className":13670},[33],[21,13672],{"className":13673,"style":38},[37],[21,13675,44],{"className":13676},[42,43],") of the previous\nlesson. It is probably the best method of its kind — subject, still, to the\nstability caveat.",[342,13679,13681],{"id":13680},"watkinss-qλ-to-tree-backupλ","Watkins's Q(λ) to Tree-Backup(λ)",[11,13683,13684,13685,4246,13703,13710,13711,13729,13730,13733],{},"The oldest way to give Q-learning eligibility traces is ",[824,13686,13687,13688,635],{},"Watkins's Q(",[21,13689,13691],{"className":13690},[24],[21,13692,13694],{"className":13693,"ariaHidden":29},[28],[21,13695,13697,13700],{"className":13696},[33],[21,13698],{"className":13699,"style":38},[37],[21,13701,44],{"className":13702},[42,43],[350,13704,13705],{},[15,13706,2381],{"href":13707,"ariaDescribedBy":13708,"dataFootnoteRef":6,"id":13709},"#user-content-fn-sb-watkins",[356],"user-content-fnref-sb-watkins","\nQ-learning bootstraps from the greedy action, so its trace should reflect the\nfuture only as long as the agent keeps taking greedy actions. Watkins's rule is\nsimple: decay the trace by ",[21,13712,13714],{"className":13713},[24],[21,13715,13717],{"className":13716,"ariaHidden":29},[28],[21,13718,13720,13723,13726],{"className":13719},[33],[21,13721],{"className":13722,"style":5968},[37],[21,13724,200],{"className":13725,"style":199},[42,43],[21,13727,44],{"className":13728},[42,43]," in the usual way as long as greedy\nactions are taken, but the moment the behavior policy selects the ",[237,13731,13732],{},"first\nnon-greedy action",", cut the trace to zero. Everything after that first\nexploratory step is off the greedy path and irrelevant to the greedy target, so\nit receives no eligibility.",[1651,13735],{"hash":13736},"c563d4b33f0957098a59629f0a669ca61616fa3df664122e260cdcba705ca6e5",[11,13738,13739,13740,13755],{},"The cut is crude. If exploration is frequent — and early in learning it is — the\ntrace is repeatedly truncated, so Q(",[21,13741,13743],{"className":13742},[24],[21,13744,13746],{"className":13745,"ariaHidden":29},[28],[21,13747,13749,13752],{"className":13748},[33],[21,13750],{"className":13751,"style":38},[37],[21,13753,44],{"className":13754},[42,43],") rarely propagates credit more than\na step or two, and much of the benefit of traces is lost. The deeper problem is\nthat cutting on the first non-greedy action is a hard special case of a more\nprincipled operation, which removes the need for the cut entirely.",[11,13757,13758,13759,13777,13778,13793,13794,13801,13816,13817,13820,13821,13836],{},"The general version is ",[824,13760,13761,13762,635],{},"Tree-Backup(",[21,13763,13765],{"className":13764},[24],[21,13766,13768],{"className":13767,"ariaHidden":29},[28],[21,13769,13771,13774],{"className":13770},[33],[21,13772],{"className":13773,"style":38},[37],[21,13775,44],{"className":13776},[42,43],", abbreviated TB(",[21,13779,13781],{"className":13780},[24],[21,13782,13784],{"className":13783,"ariaHidden":29},[28],[21,13785,13787,13790],{"className":13786},[33],[21,13788],{"className":13789,"style":38},[37],[21,13791,44],{"className":13792},[42,43],").",[350,13795,13796],{},[15,13797,8342],{"href":13798,"ariaDescribedBy":13799,"dataFootnoteRef":6,"id":13800},"#user-content-fn-sb-tb",[356],"user-content-fnref-sb-tb",[21,13802,13804],{"className":13803},[24],[21,13805,13807],{"className":13806,"ariaHidden":29},[28],[21,13808,13810,13813],{"className":13809},[33],[21,13811],{"className":13812,"style":253},[37],[21,13814,6194],{"className":13815},[42,43],"-step Tree Backup already extended Expected Sarsa to arbitrary target policies\n",[237,13818,13819],{},"without any importance sampling","; TB(",[21,13822,13824],{"className":13823},[24],[21,13825,13827],{"className":13826,"ariaHidden":29},[28],[21,13828,13830,13833],{"className":13829},[33],[21,13831],{"className":13832,"style":38},[37],[21,13834,44],{"className":13835},[42,43],") is its eligibility-trace form, and\nit is arguably the true successor to Q-learning because it keeps that appealing\nabsence of importance sampling while applying to off-policy data. Starting from\nthe Expected-Sarsa recursion and expanding the bootstrapping case after the tree-backup\npattern, the return weights each action branch by its target-policy probability,",[21,13838,13840],{"className":13839},[847],[21,13841,13843],{"className":13842},[24],[21,13844,13846,13921,13985,14178,14300,14436],{"className":13845,"ariaHidden":29},[28],[21,13847,13849,13852,13906,13909,13912,13915,13918],{"className":13848},[33],[21,13850],{"className":13851,"style":3680},[37],[21,13853,13855,13858],{"className":13854},[42],[21,13856,867],{"className":13857},[42,43],[21,13859,13861],{"className":13860},[70],[21,13862,13864,13898],{"className":13863},[74,75],[21,13865,13867,13895],{"className":13866},[79],[21,13868,13870,13881],{"className":13869,"style":3699},[83],[21,13871,13872,13875],{"style":3702},[21,13873],{"className":13874,"style":92},[91],[21,13876,13878],{"className":13877},[96,97,98,99],[21,13879,103],{"className":13880},[42,43,99],[21,13882,13883,13886],{"style":3714},[21,13884],{"className":13885,"style":92},[91],[21,13887,13889],{"className":13888},[96,97,98,99],[21,13890,13892],{"className":13891},[42,99],[21,13893,4529],{"className":13894},[42,43,99],[21,13896,108],{"className":13897},[107],[21,13899,13901],{"className":13900},[79],[21,13902,13904],{"className":13903,"style":3739},[83],[21,13905],{},[21,13907],{"className":13908,"style":379},[378],[21,13910],{"className":13911,"style":379},[378],[21,13913,1101],{"className":13914},[383],[21,13916],{"className":13917,"style":379},[378],[21,13919],{"className":13920,"style":379},[378],[21,13922,13924,13927,13976,13979,13982],{"className":13923},[33],[21,13925],{"className":13926,"style":923},[37],[21,13928,13930,13933],{"className":13929},[42],[21,13931,931],{"className":13932,"style":930},[42,43],[21,13934,13936],{"className":13935},[70],[21,13937,13939,13968],{"className":13938},[74,75],[21,13940,13942,13965],{"className":13941},[79],[21,13943,13945],{"className":13944,"style":944},[83],[21,13946,13947,13950],{"style":947},[21,13948],{"className":13949,"style":92},[91],[21,13951,13953],{"className":13952},[96,97,98,99],[21,13954,13956,13959,13962],{"className":13955},[42,99],[21,13957,103],{"className":13958},[42,43,99],[21,13960,963],{"className":13961},[408,99],[21,13963,358],{"className":13964},[42,99],[21,13966,108],{"className":13967},[107],[21,13969,13971],{"className":13970},[79],[21,13972,13974],{"className":13973,"style":976},[83],[21,13975],{},[21,13977],{"className":13978,"style":404},[378],[21,13980,963],{"className":13981},[408],[21,13983],{"className":13984,"style":404},[378],[21,13986,13988,13991,14040,14046,14114,14117,14166,14169,14172,14175],{"className":13987},[33],[21,13989],{"className":13990,"style":3827},[37],[21,13992,13994,13997],{"className":13993},[42],[21,13995,200],{"className":13996,"style":199},[42,43],[21,13998,14000],{"className":13999},[70],[21,14001,14003,14032],{"className":14002},[74,75],[21,14004,14006,14029],{"className":14005},[79],[21,14007,14009],{"className":14008,"style":944},[83],[21,14010,14011,14014],{"style":737},[21,14012],{"className":14013,"style":92},[91],[21,14015,14017],{"className":14016},[96,97,98,99],[21,14018,14020,14023,14026],{"className":14019},[42,99],[21,14021,103],{"className":14022},[42,43,99],[21,14024,963],{"className":14025},[408,99],[21,14027,358],{"className":14028},[42,99],[21,14030,108],{"className":14031},[107],[21,14033,14035],{"className":14034},[79],[21,14036,14038],{"className":14037,"style":976},[83],[21,14039],{},[21,14041,14043],{"className":14042},[42],[21,14044,543],{"className":14045},[1211,3883],[21,14047,14049,14080],{"className":14048},[42],[21,14050,14052],{"className":14051},[42,3964],[21,14053,14055],{"className":14054},[74],[21,14056,14058],{"className":14057},[79],[21,14059,14061,14069],{"className":14060,"style":5401},[83],[21,14062,14063,14066],{"style":3976},[21,14064],{"className":14065,"style":3980},[91],[21,14067,5410],{"className":14068,"style":404},[42,43],[21,14070,14071,14074],{"style":5413},[21,14072],{"className":14073,"style":3980},[91],[21,14075,14077],{"className":14076,"style":5420},[3993],[21,14078,5424],{"className":14079},[42],[21,14081,14083],{"className":14082},[70],[21,14084,14086,14106],{"className":14085},[74,75],[21,14087,14089,14103],{"className":14088},[79],[21,14090,14092],{"className":14091,"style":84},[83],[21,14093,14094,14097],{"style":5439},[21,14095],{"className":14096,"style":92},[91],[21,14098,14100],{"className":14099},[96,97,98,99],[21,14101,103],{"className":14102},[42,43,99],[21,14104,108],{"className":14105},[107],[21,14107,14109],{"className":14108},[79],[21,14110,14112],{"className":14111,"style":115},[83],[21,14113],{},[21,14115,543],{"className":14116},[444],[21,14118,14120,14123],{"className":14119},[42],[21,14121,400],{"className":14122,"style":550},[42,43],[21,14124,14126],{"className":14125},[70],[21,14127,14129,14158],{"className":14128},[74,75],[21,14130,14132,14155],{"className":14131},[79],[21,14133,14135],{"className":14134,"style":944},[83],[21,14136,14137,14140],{"style":565},[21,14138],{"className":14139,"style":92},[91],[21,14141,14143],{"className":14142},[96,97,98,99],[21,14144,14146,14149,14152],{"className":14145},[42,99],[21,14147,103],{"className":14148},[42,43,99],[21,14150,963],{"className":14151},[408,99],[21,14153,358],{"className":14154},[42,99],[21,14156,108],{"className":14157},[107],[21,14159,14161],{"className":14160},[79],[21,14162,14164],{"className":14163,"style":976},[83],[21,14165],{},[21,14167,635],{"className":14168},[465],[21,14170],{"className":14171,"style":404},[378],[21,14173,963],{"className":14174},[408],[21,14176],{"className":14177,"style":404},[378],[21,14179,14181,14184,14233,14236,14239,14242,14291,14294,14297],{"className":14180},[33],[21,14182],{"className":14183,"style":440},[37],[21,14185,14187,14190],{"className":14186},[42],[21,14188,44],{"className":14189},[42,43],[21,14191,14193],{"className":14192},[70],[21,14194,14196,14225],{"className":14195},[74,75],[21,14197,14199,14222],{"className":14198},[79],[21,14200,14202],{"className":14201,"style":944},[83],[21,14203,14204,14207],{"style":87},[21,14205],{"className":14206,"style":92},[91],[21,14208,14210],{"className":14209},[96,97,98,99],[21,14211,14213,14216,14219],{"className":14212},[42,99],[21,14214,103],{"className":14215},[42,43,99],[21,14217,963],{"className":14218},[408,99],[21,14220,358],{"className":14221},[42,99],[21,14223,108],{"className":14224},[107],[21,14226,14228],{"className":14227},[79],[21,14229,14231],{"className":14230,"style":976},[83],[21,14232],{},[21,14234],{"className":14235,"style":458},[378],[21,14237,258],{"className":14238,"style":257},[42,43],[21,14240,543],{"className":14241},[444],[21,14243,14245,14248],{"className":14244},[42],[21,14246,423],{"className":14247},[42,43],[21,14249,14251],{"className":14250},[70],[21,14252,14254,14283],{"className":14253},[74,75],[21,14255,14257,14280],{"className":14256},[79],[21,14258,14260],{"className":14259,"style":944},[83],[21,14261,14262,14265],{"style":87},[21,14263],{"className":14264,"style":92},[91],[21,14266,14268],{"className":14267},[96,97,98,99],[21,14269,14271,14274,14277],{"className":14270},[42,99],[21,14272,103],{"className":14273},[42,43,99],[21,14275,963],{"className":14276},[408,99],[21,14278,358],{"className":14279},[42,99],[21,14281,108],{"className":14282},[107],[21,14284,14286],{"className":14285},[79],[21,14287,14289],{"className":14288,"style":976},[83],[21,14290],{},[21,14292],{"className":14293,"style":379},[378],[21,14295,5825],{"className":14296},[383],[21,14298],{"className":14299,"style":379},[378],[21,14301,14303,14306,14355,14358,14364,14427,14430,14433],{"className":14302},[33],[21,14304],{"className":14305,"style":10659},[37],[21,14307,14309,14312],{"className":14308},[42],[21,14310,400],{"className":14311,"style":550},[42,43],[21,14313,14315],{"className":14314},[70],[21,14316,14318,14347],{"className":14317},[74,75],[21,14319,14321,14344],{"className":14320},[79],[21,14322,14324],{"className":14323,"style":944},[83],[21,14325,14326,14329],{"style":565},[21,14327],{"className":14328,"style":92},[91],[21,14330,14332],{"className":14331},[96,97,98,99],[21,14333,14335,14338,14341],{"className":14334},[42,99],[21,14336,103],{"className":14337},[42,43,99],[21,14339,963],{"className":14340},[408,99],[21,14342,358],{"className":14343},[42,99],[21,14345,108],{"className":14346},[107],[21,14348,14350],{"className":14349},[79],[21,14351,14353],{"className":14352,"style":976},[83],[21,14354],{},[21,14356,635],{"className":14357},[465],[21,14359,14361],{"className":14360},[42],[21,14362,543],{"className":14363},[1211,6900],[21,14365,14367,14370],{"className":14366},[42],[21,14368,867],{"className":14369},[42,43],[21,14371,14373],{"className":14372},[70],[21,14374,14376,14419],{"className":14375},[74,75],[21,14377,14379,14416],{"className":14378},[79],[21,14380,14382,14402],{"className":14381,"style":3699},[83],[21,14383,14384,14387],{"style":3702},[21,14385],{"className":14386,"style":92},[91],[21,14388,14390],{"className":14389},[96,97,98,99],[21,14391,14393,14396,14399],{"className":14392},[42,99],[21,14394,103],{"className":14395},[42,43,99],[21,14397,963],{"className":14398},[408,99],[21,14400,358],{"className":14401},[42,99],[21,14403,14404,14407],{"style":3714},[21,14405],{"className":14406,"style":92},[91],[21,14408,14410],{"className":14409},[96,97,98,99],[21,14411,14413],{"className":14412},[42,99],[21,14414,4529],{"className":14415},[42,43,99],[21,14417,108],{"className":14418},[107],[21,14420,14422],{"className":14421},[79],[21,14423,14425],{"className":14424,"style":4234},[83],[21,14426],{},[21,14428],{"className":14429,"style":404},[378],[21,14431,3896],{"className":14432},[408],[21,14434],{"className":14435,"style":404},[378],[21,14437,14439,14442,14484,14487,14536,14539,14542,14591,14594,14597,14637,14640,14646,14652],{"className":14438},[33],[21,14440],{"className":14441,"style":3827},[37],[21,14443,14445],{"className":14444},[42,3964],[21,14446,14448,14476],{"className":14447},[74,75],[21,14449,14451,14473],{"className":14450},[79],[21,14452,14454,14462],{"className":14453,"style":38},[83],[21,14455,14456,14459],{"style":3976},[21,14457],{"className":14458,"style":3980},[91],[21,14460,4777],{"className":14461,"style":257},[42,43],[21,14463,14464,14467],{"style":3976},[21,14465],{"className":14466,"style":3980},[91],[21,14468,14470],{"className":14469,"style":4786},[3993],[21,14471,3998],{"className":14472},[42],[21,14474,108],{"className":14475},[107],[21,14477,14479],{"className":14478},[79],[21,14480,14482],{"className":14481,"style":4799},[83],[21,14483],{},[21,14485,543],{"className":14486},[444],[21,14488,14490,14493],{"className":14489},[42],[21,14491,400],{"className":14492,"style":550},[42,43],[21,14494,14496],{"className":14495},[70],[21,14497,14499,14528],{"className":14498},[74,75],[21,14500,14502,14525],{"className":14501},[79],[21,14503,14505],{"className":14504,"style":944},[83],[21,14506,14507,14510],{"style":565},[21,14508],{"className":14509,"style":92},[91],[21,14511,14513],{"className":14512},[96,97,98,99],[21,14514,14516,14519,14522],{"className":14515},[42,99],[21,14517,103],{"className":14518},[42,43,99],[21,14520,963],{"className":14521},[408,99],[21,14523,358],{"className":14524},[42,99],[21,14526,108],{"className":14527},[107],[21,14529,14531],{"className":14530},[79],[21,14532,14534],{"className":14533,"style":976},[83],[21,14535],{},[21,14537,454],{"className":14538},[453],[21,14540],{"className":14541,"style":458},[378],[21,14543,14545,14548],{"className":14544},[42],[21,14546,423],{"className":14547},[42,43],[21,14549,14551],{"className":14550},[70],[21,14552,14554,14583],{"className":14553},[74,75],[21,14555,14557,14580],{"className":14556},[79],[21,14558,14560],{"className":14559,"style":944},[83],[21,14561,14562,14565],{"style":87},[21,14563],{"className":14564,"style":92},[91],[21,14566,14568],{"className":14567},[96,97,98,99],[21,14569,14571,14574,14577],{"className":14570},[42,99],[21,14572,103],{"className":14573},[42,43,99],[21,14575,963],{"className":14576},[408,99],[21,14578,358],{"className":14579},[42,99],[21,14581,108],{"className":14582},[107],[21,14584,14586],{"className":14585},[79],[21,14587,14589],{"className":14588,"style":976},[83],[21,14590],{},[21,14592,454],{"className":14593},[453],[21,14595],{"className":14596,"style":458},[378],[21,14598,14600,14603],{"className":14599},[42],[21,14601,4064],{"className":14602,"style":4063},[42,65],[21,14604,14606],{"className":14605},[70],[21,14607,14609,14629],{"className":14608},[74,75],[21,14610,14612,14626],{"className":14611},[79],[21,14613,14615],{"className":14614,"style":84},[83],[21,14616,14617,14620],{"style":4079},[21,14618],{"className":14619,"style":92},[91],[21,14621,14623],{"className":14622},[96,97,98,99],[21,14624,103],{"className":14625},[42,43,99],[21,14627,108],{"className":14628},[107],[21,14630,14632],{"className":14631},[79],[21,14633,14635],{"className":14634,"style":115},[83],[21,14636],{},[21,14638,635],{"className":14639},[465],[21,14641,14643],{"className":14642},[42],[21,14644,635],{"className":14645},[1211,6900],[21,14647,14649],{"className":14648},[42],[21,14650,635],{"className":14651},[1211,3883],[21,14653,4246],{"className":14654},[42],[1651,14656],{"hash":14657},"60fc2f3c6edd0292d65c0108af959e2364e0ea28fbbe80060b4543ca4710f58c",[11,14659,14660,14661,15170],{},"Written approximately as a sum of TD errors, the return uses the expectation-based\nerror ",[21,14662,14664],{"className":14663},[24],[21,14665,14667,14733,14797,14984],{"className":14666,"ariaHidden":29},[28],[21,14668,14670,14673,14724,14727,14730],{"className":14669},[33],[21,14671],{"className":14672,"style":12140},[37],[21,14674,14676,14679],{"className":14675},[42],[21,14677,9293],{"className":14678,"style":9292},[42,43],[21,14680,14682],{"className":14681},[70],[21,14683,14685,14716],{"className":14684},[74,75],[21,14686,14688,14713],{"className":14687},[79],[21,14689,14691,14702],{"className":14690,"style":12162},[83],[21,14692,14693,14696],{"style":9309},[21,14694],{"className":14695,"style":92},[91],[21,14697,14699],{"className":14698},[96,97,98,99],[21,14700,103],{"className":14701},[42,43,99],[21,14703,14704,14707],{"style":12176},[21,14705],{"className":14706,"style":92},[91],[21,14708,14710],{"className":14709},[96,97,98,99],[21,14711,15],{"className":14712},[42,43,99],[21,14714,108],{"className":14715},[107],[21,14717,14719],{"className":14718},[79],[21,14720,14722],{"className":14721,"style":3739},[83],[21,14723],{},[21,14725],{"className":14726,"style":379},[378],[21,14728,1101],{"className":14729},[383],[21,14731],{"className":14732,"style":379},[378],[21,14734,14736,14739,14788,14791,14794],{"className":14735},[33],[21,14737],{"className":14738,"style":923},[37],[21,14740,14742,14745],{"className":14741},[42],[21,14743,931],{"className":14744,"style":930},[42,43],[21,14746,14748],{"className":14747},[70],[21,14749,14751,14780],{"className":14750},[74,75],[21,14752,14754,14777],{"className":14753},[79],[21,14755,14757],{"className":14756,"style":944},[83],[21,14758,14759,14762],{"style":947},[21,14760],{"className":14761,"style":92},[91],[21,14763,14765],{"className":14764},[96,97,98,99],[21,14766,14768,14771,14774],{"className":14767},[42,99],[21,14769,103],{"className":14770},[42,43,99],[21,14772,963],{"className":14773},[408,99],[21,14775,358],{"className":14776},[42,99],[21,14778,108],{"className":14779},[107],[21,14781,14783],{"className":14782},[79],[21,14784,14786],{"className":14785,"style":976},[83],[21,14787],{},[21,14789],{"className":14790,"style":404},[378],[21,14792,963],{"className":14793},[408],[21,14795],{"className":14796,"style":404},[378],[21,14798,14800,14803,14852,14920,14923,14972,14975,14978,14981],{"className":14799},[33],[21,14801],{"className":14802,"style":5330},[37],[21,14804,14806,14809],{"className":14805},[42],[21,14807,200],{"className":14808,"style":199},[42,43],[21,14810,14812],{"className":14811},[70],[21,14813,14815,14844],{"className":14814},[74,75],[21,14816,14818,14841],{"className":14817},[79],[21,14819,14821],{"className":14820,"style":944},[83],[21,14822,14823,14826],{"style":737},[21,14824],{"className":14825,"style":92},[91],[21,14827,14829],{"className":14828},[96,97,98,99],[21,14830,14832,14835,14838],{"className":14831},[42,99],[21,14833,103],{"className":14834},[42,43,99],[21,14836,963],{"className":14837},[408,99],[21,14839,358],{"className":14840},[42,99],[21,14842,108],{"className":14843},[107],[21,14845,14847],{"className":14846},[79],[21,14848,14850],{"className":14849,"style":976},[83],[21,14851],{},[21,14853,14855,14886],{"className":14854},[42],[21,14856,14858],{"className":14857},[42,3964],[21,14859,14861],{"className":14860},[74],[21,14862,14864],{"className":14863},[79],[21,14865,14867,14875],{"className":14866,"style":5401},[83],[21,14868,14869,14872],{"style":3976},[21,14870],{"className":14871,"style":3980},[91],[21,14873,5410],{"className":14874,"style":404},[42,43],[21,14876,14877,14880],{"style":5413},[21,14878],{"className":14879,"style":3980},[91],[21,14881,14883],{"className":14882,"style":5420},[3993],[21,14884,5424],{"className":14885},[42],[21,14887,14889],{"className":14888},[70],[21,14890,14892,14912],{"className":14891},[74,75],[21,14893,14895,14909],{"className":14894},[79],[21,14896,14898],{"className":14897,"style":84},[83],[21,14899,14900,14903],{"style":5439},[21,14901],{"className":14902,"style":92},[91],[21,14904,14906],{"className":14905},[96,97,98,99],[21,14907,103],{"className":14908},[42,43,99],[21,14910,108],{"className":14911},[107],[21,14913,14915],{"className":14914},[79],[21,14916,14918],{"className":14917,"style":115},[83],[21,14919],{},[21,14921,543],{"className":14922},[444],[21,14924,14926,14929],{"className":14925},[42],[21,14927,400],{"className":14928,"style":550},[42,43],[21,14930,14932],{"className":14931},[70],[21,14933,14935,14964],{"className":14934},[74,75],[21,14936,14938,14961],{"className":14937},[79],[21,14939,14941],{"className":14940,"style":944},[83],[21,14942,14943,14946],{"style":565},[21,14944],{"className":14945,"style":92},[91],[21,14947,14949],{"className":14948},[96,97,98,99],[21,14950,14952,14955,14958],{"className":14951},[42,99],[21,14953,103],{"className":14954},[42,43,99],[21,14956,963],{"className":14957},[408,99],[21,14959,358],{"className":14960},[42,99],[21,14962,108],{"className":14963},[107],[21,14965,14967],{"className":14966},[79],[21,14968,14970],{"className":14969,"style":976},[83],[21,14971],{},[21,14973,635],{"className":14974},[465],[21,14976],{"className":14977,"style":404},[378],[21,14979,3896],{"className":14980},[408],[21,14982],{"className":14983,"style":404},[378],[21,14985,14987,14990,15032,15035,15075,15078,15081,15121,15124,15127,15167],{"className":14986},[33],[21,14988],{"className":14989,"style":440},[37],[21,14991,14993],{"className":14992},[42,3964],[21,14994,14996,15024],{"className":14995},[74,75],[21,14997,14999,15021],{"className":14998},[79],[21,15000,15002,15010],{"className":15001,"style":38},[83],[21,15003,15004,15007],{"style":3976},[21,15005],{"className":15006,"style":3980},[91],[21,15008,4777],{"className":15009,"style":257},[42,43],[21,15011,15012,15015],{"style":3976},[21,15013],{"className":15014,"style":3980},[91],[21,15016,15018],{"className":15017,"style":4786},[3993],[21,15019,3998],{"className":15020},[42],[21,15022,108],{"className":15023},[107],[21,15025,15027],{"className":15026},[79],[21,15028,15030],{"className":15029,"style":4799},[83],[21,15031],{},[21,15033,543],{"className":15034},[444],[21,15036,15038,15041],{"className":15037},[42],[21,15039,400],{"className":15040,"style":550},[42,43],[21,15042,15044],{"className":15043},[70],[21,15045,15047,15067],{"className":15046},[74,75],[21,15048,15050,15064],{"className":15049},[79],[21,15051,15053],{"className":15052,"style":84},[83],[21,15054,15055,15058],{"style":565},[21,15056],{"className":15057,"style":92},[91],[21,15059,15061],{"className":15060},[96,97,98,99],[21,15062,103],{"className":15063},[42,43,99],[21,15065,108],{"className":15066},[107],[21,15068,15070],{"className":15069},[79],[21,15071,15073],{"className":15072,"style":115},[83],[21,15074],{},[21,15076,454],{"className":15077},[453],[21,15079],{"className":15080,"style":458},[378],[21,15082,15084,15087],{"className":15083},[42],[21,15085,423],{"className":15086},[42,43],[21,15088,15090],{"className":15089},[70],[21,15091,15093,15113],{"className":15092},[74,75],[21,15094,15096,15110],{"className":15095},[79],[21,15097,15099],{"className":15098,"style":84},[83],[21,15100,15101,15104],{"style":87},[21,15102],{"className":15103,"style":92},[91],[21,15105,15107],{"className":15106},[96,97,98,99],[21,15108,103],{"className":15109},[42,43,99],[21,15111,108],{"className":15112},[107],[21,15114,15116],{"className":15115},[79],[21,15117,15119],{"className":15118,"style":115},[83],[21,15120],{},[21,15122,454],{"className":15123},[453],[21,15125],{"className":15126,"style":458},[378],[21,15128,15130,15133],{"className":15129},[42],[21,15131,4064],{"className":15132,"style":4063},[42,65],[21,15134,15136],{"className":15135},[70],[21,15137,15139,15159],{"className":15138},[74,75],[21,15140,15142,15156],{"className":15141},[79],[21,15143,15145],{"className":15144,"style":84},[83],[21,15146,15147,15150],{"style":4079},[21,15148],{"className":15149,"style":92},[91],[21,15151,15153],{"className":15152},[96,97,98,99],[21,15154,103],{"className":15155},[42,43,99],[21,15157,108],{"className":15158},[107],[21,15160,15162],{"className":15161},[79],[21,15163,15165],{"className":15164,"style":115},[83],[21,15166],{},[21,15168,635],{"className":15169},[465]," and a trace weighted by the target-policy probability of each\nselected action,",[21,15172,15174],{"className":15173},[847],[21,15175,15177],{"className":15176},[24],[21,15178,15180,15241,15385,15495],{"className":15179,"ariaHidden":29},[28],[21,15181,15183,15186,15226,15229,15232,15235,15238],{"className":15182},[33],[21,15184],{"className":15185,"style":11616},[37],[21,15187,15189,15192],{"className":15188},[42],[21,15190,66],{"className":15191},[42,65],[21,15193,15195],{"className":15194},[70],[21,15196,15198,15218],{"className":15197},[74,75],[21,15199,15201,15215],{"className":15200},[79],[21,15202,15204],{"className":15203,"style":84},[83],[21,15205,15206,15209],{"style":87},[21,15207],{"className":15208,"style":92},[91],[21,15210,15212],{"className":15211},[96,97,98,99],[21,15213,103],{"className":15214},[42,43,99],[21,15216,108],{"className":15217},[107],[21,15219,15221],{"className":15220},[79],[21,15222,15224],{"className":15223,"style":115},[83],[21,15225],{},[21,15227],{"className":15228,"style":379},[378],[21,15230],{"className":15231,"style":379},[378],[21,15233,527],{"className":15234},[383],[21,15236],{"className":15237,"style":379},[378],[21,15239],{"className":15240,"style":379},[378],[21,15242,15244,15247,15287,15327,15330,15333,15336,15376,15379,15382],{"className":15243},[33],[21,15245],{"className":15246,"style":440},[37],[21,15248,15250,15253],{"className":15249},[42],[21,15251,200],{"className":15252,"style":199},[42,43],[21,15254,15256],{"className":15255},[70],[21,15257,15259,15279],{"className":15258},[74,75],[21,15260,15262,15276],{"className":15261},[79],[21,15263,15265],{"className":15264,"style":84},[83],[21,15266,15267,15270],{"style":737},[21,15268],{"className":15269,"style":92},[91],[21,15271,15273],{"className":15272},[96,97,98,99],[21,15274,103],{"className":15275},[42,43,99],[21,15277,108],{"className":15278},[107],[21,15280,15282],{"className":15281},[79],[21,15283,15285],{"className":15284,"style":115},[83],[21,15286],{},[21,15288,15290,15293],{"className":15289},[42],[21,15291,44],{"className":15292},[42,43],[21,15294,15296],{"className":15295},[70],[21,15297,15299,15319],{"className":15298},[74,75],[21,15300,15302,15316],{"className":15301},[79],[21,15303,15305],{"className":15304,"style":84},[83],[21,15306,15307,15310],{"style":87},[21,15308],{"className":15309,"style":92},[91],[21,15311,15313],{"className":15312},[96,97,98,99],[21,15314,103],{"className":15315},[42,43,99],[21,15317,108],{"className":15318},[107],[21,15320,15322],{"className":15321},[79],[21,15323,15325],{"className":15324,"style":115},[83],[21,15326],{},[21,15328],{"className":15329,"style":458},[378],[21,15331,258],{"className":15332,"style":257},[42,43],[21,15334,543],{"className":15335},[444],[21,15337,15339,15342],{"className":15338},[42],[21,15340,423],{"className":15341},[42,43],[21,15343,15345],{"className":15344},[70],[21,15346,15348,15368],{"className":15347},[74,75],[21,15349,15351,15365],{"className":15350},[79],[21,15352,15354],{"className":15353,"style":84},[83],[21,15355,15356,15359],{"style":87},[21,15357],{"className":15358,"style":92},[91],[21,15360,15362],{"className":15361},[96,97,98,99],[21,15363,103],{"className":15364},[42,43,99],[21,15366,108],{"className":15367},[107],[21,15369,15371],{"className":15370},[79],[21,15372,15374],{"className":15373,"style":115},[83],[21,15375],{},[21,15377],{"className":15378,"style":379},[378],[21,15380,5825],{"className":15381},[383],[21,15383],{"className":15384,"style":379},[378],[21,15386,15388,15391,15431,15434,15437,15486,15489,15492],{"className":15387},[33],[21,15389],{"className":15390,"style":440},[37],[21,15392,15394,15397],{"className":15393},[42],[21,15395,400],{"className":15396,"style":550},[42,43],[21,15398,15400],{"className":15399},[70],[21,15401,15403,15423],{"className":15402},[74,75],[21,15404,15406,15420],{"className":15405},[79],[21,15407,15409],{"className":15408,"style":84},[83],[21,15410,15411,15414],{"style":565},[21,15412],{"className":15413,"style":92},[91],[21,15415,15417],{"className":15416},[96,97,98,99],[21,15418,103],{"className":15419},[42,43,99],[21,15421,108],{"className":15422},[107],[21,15424,15426],{"className":15425},[79],[21,15427,15429],{"className":15428,"style":115},[83],[21,15430],{},[21,15432,635],{"className":15433},[465],[21,15435],{"className":15436,"style":458},[378],[21,15438,15440,15443],{"className":15439},[42],[21,15441,66],{"className":15442},[42,65],[21,15444,15446],{"className":15445},[70],[21,15447,15449,15478],{"className":15448},[74,75],[21,15450,15452,15475],{"className":15451},[79],[21,15453,15455],{"className":15454,"style":944},[83],[21,15456,15457,15460],{"style":87},[21,15458],{"className":15459,"style":92},[91],[21,15461,15463],{"className":15462},[96,97,98,99],[21,15464,15466,15469,15472],{"className":15465},[42,99],[21,15467,103],{"className":15468},[42,43,99],[21,15470,3896],{"className":15471},[408,99],[21,15473,358],{"className":15474},[42,99],[21,15476,108],{"className":15477},[107],[21,15479,15481],{"className":15480},[79],[21,15482,15484],{"className":15483,"style":976},[83],[21,15485],{},[21,15487],{"className":15488,"style":404},[378],[21,15490,963],{"className":15491},[408],[21,15493],{"className":15494,"style":404},[378],[21,15496,15498,15501,15504,15546,15549,15589,15592,15595,15635,15638,15641,15681,15684],{"className":15497},[33],[21,15499],{"className":15500,"style":440},[37],[21,15502,10875],{"className":15503},[42],[21,15505,15507],{"className":15506},[42,3964],[21,15508,15510,15538],{"className":15509},[74,75],[21,15511,15513,15535],{"className":15512},[79],[21,15514,15516,15524],{"className":15515,"style":38},[83],[21,15517,15518,15521],{"style":3976},[21,15519],{"className":15520,"style":3980},[91],[21,15522,4777],{"className":15523,"style":257},[42,43],[21,15525,15526,15529],{"style":3976},[21,15527],{"className":15528,"style":3980},[91],[21,15530,15532],{"className":15531,"style":4786},[3993],[21,15533,3998],{"className":15534},[42],[21,15536,108],{"className":15537},[107],[21,15539,15541],{"className":15540},[79],[21,15542,15544],{"className":15543,"style":4799},[83],[21,15545],{},[21,15547,543],{"className":15548},[444],[21,15550,15552,15555],{"className":15551},[42],[21,15553,400],{"className":15554,"style":550},[42,43],[21,15556,15558],{"className":15557},[70],[21,15559,15561,15581],{"className":15560},[74,75],[21,15562,15564,15578],{"className":15563},[79],[21,15565,15567],{"className":15566,"style":84},[83],[21,15568,15569,15572],{"style":565},[21,15570],{"className":15571,"style":92},[91],[21,15573,15575],{"className":15574},[96,97,98,99],[21,15576,103],{"className":15577},[42,43,99],[21,15579,108],{"className":15580},[107],[21,15582,15584],{"className":15583},[79],[21,15585,15587],{"className":15586,"style":115},[83],[21,15588],{},[21,15590,454],{"className":15591},[453],[21,15593],{"className":15594,"style":458},[378],[21,15596,15598,15601],{"className":15597},[42],[21,15599,423],{"className":15600},[42,43],[21,15602,15604],{"className":15603},[70],[21,15605,15607,15627],{"className":15606},[74,75],[21,15608,15610,15624],{"className":15609},[79],[21,15611,15613],{"className":15612,"style":84},[83],[21,15614,15615,15618],{"style":87},[21,15616],{"className":15617,"style":92},[91],[21,15619,15621],{"className":15620},[96,97,98,99],[21,15622,103],{"className":15623},[42,43,99],[21,15625,108],{"className":15626},[107],[21,15628,15630],{"className":15629},[79],[21,15631,15633],{"className":15632,"style":115},[83],[21,15634],{},[21,15636,454],{"className":15637},[453],[21,15639],{"className":15640,"style":458},[378],[21,15642,15644,15647],{"className":15643},[42],[21,15645,4064],{"className":15646,"style":4063},[42,65],[21,15648,15650],{"className":15649},[70],[21,15651,15653,15673],{"className":15652},[74,75],[21,15654,15656,15670],{"className":15655},[79],[21,15657,15659],{"className":15658,"style":84},[83],[21,15660,15661,15664],{"style":4079},[21,15662],{"className":15663,"style":92},[91],[21,15665,15667],{"className":15666},[96,97,98,99],[21,15668,103],{"className":15669},[42,43,99],[21,15671,108],{"className":15672},[107],[21,15674,15676],{"className":15675},[79],[21,15677,15679],{"className":15678,"style":115},[83],[21,15680],{},[21,15682,635],{"className":15683},[465],[21,15685,4246],{"className":15686},[42],[11,15688,15689,15690,15705,15706,15798,15799,13820,15931,15946,15947,16143,16144,16159,16160,16294,16295,16429,16430,16433,16434,16449,16450,16465],{},"Compare the three action-value traces. On-policy Sarsa(",[21,15691,15693],{"className":15692},[24],[21,15694,15696],{"className":15695,"ariaHidden":29},[28],[21,15697,15699,15702],{"className":15698},[33],[21,15700],{"className":15701,"style":38},[37],[21,15703,44],{"className":15704},[42,43],") decays by\n",[21,15707,15709],{"className":15708},[24],[21,15710,15712],{"className":15711,"ariaHidden":29},[28],[21,15713,15715,15718,15758],{"className":15714},[33],[21,15716],{"className":15717,"style":5968},[37],[21,15719,15721,15724],{"className":15720},[42],[21,15722,200],{"className":15723,"style":199},[42,43],[21,15725,15727],{"className":15726},[70],[21,15728,15730,15750],{"className":15729},[74,75],[21,15731,15733,15747],{"className":15732},[79],[21,15734,15736],{"className":15735,"style":84},[83],[21,15737,15738,15741],{"style":737},[21,15739],{"className":15740,"style":92},[91],[21,15742,15744],{"className":15743},[96,97,98,99],[21,15745,103],{"className":15746},[42,43,99],[21,15748,108],{"className":15749},[107],[21,15751,15753],{"className":15752},[79],[21,15754,15756],{"className":15755,"style":115},[83],[21,15757],{},[21,15759,15761,15764],{"className":15760},[42],[21,15762,44],{"className":15763},[42,43],[21,15765,15767],{"className":15766},[70],[21,15768,15770,15790],{"className":15769},[74,75],[21,15771,15773,15787],{"className":15772},[79],[21,15774,15776],{"className":15775,"style":84},[83],[21,15777,15778,15781],{"style":87},[21,15779],{"className":15780,"style":92},[91],[21,15782,15784],{"className":15783},[96,97,98,99],[21,15785,103],{"className":15786},[42,43,99],[21,15788,108],{"className":15789},[107],[21,15791,15793],{"className":15792},[79],[21,15794,15796],{"className":15795,"style":115},[83],[21,15797],{},"; the importance-sampling off-policy trace decays by ",[21,15800,15802],{"className":15801},[24],[21,15803,15805],{"className":15804,"ariaHidden":29},[28],[21,15806,15808,15811,15851,15891],{"className":15807},[33],[21,15809],{"className":15810,"style":5968},[37],[21,15812,15814,15817],{"className":15813},[42],[21,15815,200],{"className":15816,"style":199},[42,43],[21,15818,15820],{"className":15819},[70],[21,15821,15823,15843],{"className":15822},[74,75],[21,15824,15826,15840],{"className":15825},[79],[21,15827,15829],{"className":15828,"style":84},[83],[21,15830,15831,15834],{"style":737},[21,15832],{"className":15833,"style":92},[91],[21,15835,15837],{"className":15836},[96,97,98,99],[21,15838,103],{"className":15839},[42,43,99],[21,15841,108],{"className":15842},[107],[21,15844,15846],{"className":15845},[79],[21,15847,15849],{"className":15848,"style":115},[83],[21,15850],{},[21,15852,15854,15857],{"className":15853},[42],[21,15855,44],{"className":15856},[42,43],[21,15858,15860],{"className":15859},[70],[21,15861,15863,15883],{"className":15862},[74,75],[21,15864,15866,15880],{"className":15865},[79],[21,15867,15869],{"className":15868,"style":84},[83],[21,15870,15871,15874],{"style":87},[21,15872],{"className":15873,"style":92},[91],[21,15875,15877],{"className":15876},[96,97,98,99],[21,15878,103],{"className":15879},[42,43,99],[21,15881,108],{"className":15882},[107],[21,15884,15886],{"className":15885},[79],[21,15887,15889],{"className":15888,"style":115},[83],[21,15890],{},[21,15892,15894,15897],{"className":15893},[42],[21,15895,295],{"className":15896},[42,43],[21,15898,15900],{"className":15899},[70],[21,15901,15903,15923],{"className":15902},[74,75],[21,15904,15906,15920],{"className":15905},[79],[21,15907,15909],{"className":15908,"style":84},[83],[21,15910,15911,15914],{"style":87},[21,15912],{"className":15913,"style":92},[91],[21,15915,15917],{"className":15916},[96,97,98,99],[21,15918,103],{"className":15919},[42,43,99],[21,15921,108],{"className":15922},[107],[21,15924,15926],{"className":15925},[79],[21,15927,15929],{"className":15928,"style":115},[83],[21,15930],{},[21,15932,15934],{"className":15933},[24],[21,15935,15937],{"className":15936,"ariaHidden":29},[28],[21,15938,15940,15943],{"className":15939},[33],[21,15941],{"className":15942,"style":38},[37],[21,15944,44],{"className":15945},[42,43],") decays by ",[21,15948,15950],{"className":15949},[24],[21,15951,15953,16094],{"className":15952,"ariaHidden":29},[28],[21,15954,15956,15959,15999,16039,16042,16045,16085,16088,16091],{"className":15955},[33],[21,15957],{"className":15958,"style":440},[37],[21,15960,15962,15965],{"className":15961},[42],[21,15963,200],{"className":15964,"style":199},[42,43],[21,15966,15968],{"className":15967},[70],[21,15969,15971,15991],{"className":15970},[74,75],[21,15972,15974,15988],{"className":15973},[79],[21,15975,15977],{"className":15976,"style":84},[83],[21,15978,15979,15982],{"style":737},[21,15980],{"className":15981,"style":92},[91],[21,15983,15985],{"className":15984},[96,97,98,99],[21,15986,103],{"className":15987},[42,43,99],[21,15989,108],{"className":15990},[107],[21,15992,15994],{"className":15993},[79],[21,15995,15997],{"className":15996,"style":115},[83],[21,15998],{},[21,16000,16002,16005],{"className":16001},[42],[21,16003,44],{"className":16004},[42,43],[21,16006,16008],{"className":16007},[70],[21,16009,16011,16031],{"className":16010},[74,75],[21,16012,16014,16028],{"className":16013},[79],[21,16015,16017],{"className":16016,"style":84},[83],[21,16018,16019,16022],{"style":87},[21,16020],{"className":16021,"style":92},[91],[21,16023,16025],{"className":16024},[96,97,98,99],[21,16026,103],{"className":16027},[42,43,99],[21,16029,108],{"className":16030},[107],[21,16032,16034],{"className":16033},[79],[21,16035,16037],{"className":16036,"style":115},[83],[21,16038],{},[21,16040,258],{"className":16041,"style":257},[42,43],[21,16043,543],{"className":16044},[444],[21,16046,16048,16051],{"className":16047},[42],[21,16049,423],{"className":16050},[42,43],[21,16052,16054],{"className":16053},[70],[21,16055,16057,16077],{"className":16056},[74,75],[21,16058,16060,16074],{"className":16059},[79],[21,16061,16063],{"className":16062,"style":84},[83],[21,16064,16065,16068],{"style":87},[21,16066],{"className":16067,"style":92},[91],[21,16069,16071],{"className":16070},[96,97,98,99],[21,16072,103],{"className":16073},[42,43,99],[21,16075,108],{"className":16076},[107],[21,16078,16080],{"className":16079},[79],[21,16081,16083],{"className":16082,"style":115},[83],[21,16084],{},[21,16086],{"className":16087,"style":379},[378],[21,16089,5825],{"className":16090},[383],[21,16092],{"className":16093,"style":379},[378],[21,16095,16097,16100,16140],{"className":16096},[33],[21,16098],{"className":16099,"style":440},[37],[21,16101,16103,16106],{"className":16102},[42],[21,16104,400],{"className":16105,"style":550},[42,43],[21,16107,16109],{"className":16108},[70],[21,16110,16112,16132],{"className":16111},[74,75],[21,16113,16115,16129],{"className":16114},[79],[21,16116,16118],{"className":16117,"style":84},[83],[21,16119,16120,16123],{"style":565},[21,16121],{"className":16122,"style":92},[91],[21,16124,16126],{"className":16125},[96,97,98,99],[21,16127,103],{"className":16128},[42,43,99],[21,16130,108],{"className":16131},[107],[21,16133,16135],{"className":16134},[79],[21,16136,16138],{"className":16137,"style":115},[83],[21,16139],{},[21,16141,635],{"className":16142},[465],".\nWatkins's Q(",[21,16145,16147],{"className":16146},[24],[21,16148,16150],{"className":16149,"ariaHidden":29},[28],[21,16151,16153,16156],{"className":16152},[33],[21,16154],{"className":16155,"style":38},[37],[21,16157,44],{"className":16158},[42,43],") is the crude ancestor of that last form: taking a greedy\naction makes ",[21,16161,16163],{"className":16162},[24],[21,16164,16166,16227,16285],{"className":16165,"ariaHidden":29},[28],[21,16167,16169,16172,16175,16178,16218,16221,16224],{"className":16168},[33],[21,16170],{"className":16171,"style":440},[37],[21,16173,258],{"className":16174,"style":257},[42,43],[21,16176,543],{"className":16177},[444],[21,16179,16181,16184],{"className":16180},[42],[21,16182,423],{"className":16183},[42,43],[21,16185,16187],{"className":16186},[70],[21,16188,16190,16210],{"className":16189},[74,75],[21,16191,16193,16207],{"className":16192},[79],[21,16194,16196],{"className":16195,"style":84},[83],[21,16197,16198,16201],{"style":87},[21,16199],{"className":16200,"style":92},[91],[21,16202,16204],{"className":16203},[96,97,98,99],[21,16205,103],{"className":16206},[42,43,99],[21,16208,108],{"className":16209},[107],[21,16211,16213],{"className":16212},[79],[21,16214,16216],{"className":16215,"style":115},[83],[21,16217],{},[21,16219],{"className":16220,"style":379},[378],[21,16222,5825],{"className":16223},[383],[21,16225],{"className":16226,"style":379},[378],[21,16228,16230,16233,16273,16276,16279,16282],{"className":16229},[33],[21,16231],{"className":16232,"style":440},[37],[21,16234,16236,16239],{"className":16235},[42],[21,16237,400],{"className":16238,"style":550},[42,43],[21,16240,16242],{"className":16241},[70],[21,16243,16245,16265],{"className":16244},[74,75],[21,16246,16248,16262],{"className":16247},[79],[21,16249,16251],{"className":16250,"style":84},[83],[21,16252,16253,16256],{"style":565},[21,16254],{"className":16255,"style":92},[91],[21,16257,16259],{"className":16258},[96,97,98,99],[21,16260,103],{"className":16261},[42,43,99],[21,16263,108],{"className":16264},[107],[21,16266,16268],{"className":16267},[79],[21,16269,16271],{"className":16270,"style":115},[83],[21,16272],{},[21,16274,635],{"className":16275},[465],[21,16277],{"className":16278,"style":379},[378],[21,16280,1101],{"className":16281},[383],[21,16283],{"className":16284,"style":379},[378],[21,16286,16288,16291],{"className":16287},[33],[21,16289],{"className":16290,"style":1540},[37],[21,16292,358],{"className":16293},[42]," (no extra decay), and taking a non-greedy\naction makes ",[21,16296,16298],{"className":16297},[24],[21,16299,16301,16362,16420],{"className":16300,"ariaHidden":29},[28],[21,16302,16304,16307,16310,16313,16353,16356,16359],{"className":16303},[33],[21,16305],{"className":16306,"style":440},[37],[21,16308,258],{"className":16309,"style":257},[42,43],[21,16311,543],{"className":16312},[444],[21,16314,16316,16319],{"className":16315},[42],[21,16317,423],{"className":16318},[42,43],[21,16320,16322],{"className":16321},[70],[21,16323,16325,16345],{"className":16324},[74,75],[21,16326,16328,16342],{"className":16327},[79],[21,16329,16331],{"className":16330,"style":84},[83],[21,16332,16333,16336],{"style":87},[21,16334],{"className":16335,"style":92},[91],[21,16337,16339],{"className":16338},[96,97,98,99],[21,16340,103],{"className":16341},[42,43,99],[21,16343,108],{"className":16344},[107],[21,16346,16348],{"className":16347},[79],[21,16349,16351],{"className":16350,"style":115},[83],[21,16352],{},[21,16354],{"className":16355,"style":379},[378],[21,16357,5825],{"className":16358},[383],[21,16360],{"className":16361,"style":379},[378],[21,16363,16365,16368,16408,16411,16414,16417],{"className":16364},[33],[21,16366],{"className":16367,"style":440},[37],[21,16369,16371,16374],{"className":16370},[42],[21,16372,400],{"className":16373,"style":550},[42,43],[21,16375,16377],{"className":16376},[70],[21,16378,16380,16400],{"className":16379},[74,75],[21,16381,16383,16397],{"className":16382},[79],[21,16384,16386],{"className":16385,"style":84},[83],[21,16387,16388,16391],{"style":565},[21,16389],{"className":16390,"style":92},[91],[21,16392,16394],{"className":16393},[96,97,98,99],[21,16395,103],{"className":16396},[42,43,99],[21,16398,108],{"className":16399},[107],[21,16401,16403],{"className":16402},[79],[21,16404,16406],{"className":16405,"style":115},[83],[21,16407],{},[21,16409,635],{"className":16410},[465],[21,16412],{"className":16413,"style":379},[378],[21,16415,1101],{"className":16416},[383],[21,16418],{"className":16419,"style":379},[378],[21,16421,16423,16426],{"className":16422},[33],[21,16424],{"className":16425,"style":1540},[37],[21,16427,449],{"className":16428},[42]," (the trace vanishes) — which is precisely the\n",[4777,16431,16432],{},"cut on first non-greedy action"," rule, now recovered as a hard special case of the\nsmooth TB(",[21,16435,16437],{"className":16436},[24],[21,16438,16440],{"className":16439,"ariaHidden":29},[28],[21,16441,16443,16446],{"className":16442},[33],[21,16444],{"className":16445,"style":38},[37],[21,16447,44],{"className":16448},[42,43],") weighting. TB(",[21,16451,16453],{"className":16452},[24],[21,16454,16456],{"className":16455,"ariaHidden":29},[28],[21,16457,16459,16462],{"className":16458},[33],[21,16460],{"className":16461,"style":38},[37],[21,16463,44],{"className":16464},[42,43],"), together with the standard\nsemi-gradient update, is the tree-backup trace algorithm; like every semi-gradient\nmethod here, it is not guaranteed stable under off-policy data with a strong\nfunction approximator.",[1651,16467],{"hash":16468},"8151b32aed0919140db9d01c8d7224dd476915e132332fa82a4f3e99bef4bdf5",[342,16470,16472],{"id":16471},"where-this-leaves-us","Where this leaves us",[11,16474,16475,16476,182,16491,16506,16507,16559,16560,16575,16576,16591,16592,16607,16608,16623],{},"Two generalizations push the trace machinery to its most general form. Variable\n",[21,16477,16479],{"className":16478},[24],[21,16480,16482],{"className":16481,"ariaHidden":29},[28],[21,16483,16485,16488],{"className":16484},[33],[21,16486],{"className":16487,"style":38},[37],[21,16489,44],{"className":16490},[42,43],[21,16492,16494],{"className":16493},[24],[21,16495,16497],{"className":16496,"ariaHidden":29},[28],[21,16498,16500,16503],{"className":16499},[33],[21,16501],{"className":16502,"style":195},[37],[21,16504,200],{"className":16505,"style":199},[42,43]," turn the constants into functions, unify episodic and\ncontinuing tasks in a single stream, and fold termination into a state-dependent\ndiscount. Control variates fold the importance ratio ",[21,16508,16510],{"className":16509},[24],[21,16511,16513],{"className":16512,"ariaHidden":29},[28],[21,16514,16516,16519],{"className":16515},[33],[21,16517],{"className":16518,"style":195},[37],[21,16520,16522,16525],{"className":16521},[42],[21,16523,295],{"className":16524},[42,43],[21,16526,16528],{"className":16527},[70],[21,16529,16531,16551],{"className":16530},[74,75],[21,16532,16534,16548],{"className":16533},[79],[21,16535,16537],{"className":16536,"style":84},[83],[21,16538,16539,16542],{"style":87},[21,16540],{"className":16541,"style":92},[91],[21,16543,16545],{"className":16544},[96,97,98,99],[21,16546,103],{"className":16547},[42,43,99],[21,16549,108],{"className":16550},[107],[21,16552,16554],{"className":16553},[79],[21,16555,16557],{"className":16556,"style":115},[83],[21,16558],{}," into the trace and\ndampen the variance it brings, giving an efficient off-policy TD(",[21,16561,16563],{"className":16562},[24],[21,16564,16566],{"className":16565,"ariaHidden":29},[28],[21,16567,16569,16572],{"className":16568},[33],[21,16570],{"className":16571,"style":38},[37],[21,16573,44],{"className":16574},[42,43],") and\nExpected-Sarsa(",[21,16577,16579],{"className":16578},[24],[21,16580,16582],{"className":16581,"ariaHidden":29},[28],[21,16583,16585,16588],{"className":16584},[33],[21,16586],{"className":16587,"style":38},[37],[21,16589,44],{"className":16590},[42,43],"). Watkins's Q(",[21,16593,16595],{"className":16594},[24],[21,16596,16598],{"className":16597,"ariaHidden":29},[28],[21,16599,16601,16604],{"className":16600},[33],[21,16602],{"className":16603,"style":38},[37],[21,16605,44],{"className":16606},[42,43],") cuts the trace on the first\nnon-greedy action; Tree-Backup(",[21,16609,16611],{"className":16610},[24],[21,16612,16614],{"className":16613,"ariaHidden":29},[28],[21,16615,16617,16620],{"className":16616},[33],[21,16618],{"className":16619,"style":38},[37],[21,16621,44],{"className":16622},[42,43],") replaces that hard cut with a smooth\ntarget-policy weighting and needs no importance sampling at all.",[11,16625,16626,16627,16661,16662,4246],{},"All of it is still semi-gradient, though, so whenever ",[21,16628,16630],{"className":16629},[24],[21,16631,16633,16652],{"className":16632,"ariaHidden":29},[28],[21,16634,16636,16640,16643,16646,16649],{"className":16635},[33],[21,16637],{"className":16638,"style":16639},[37],"height:0.7335em;vertical-align:-0.0391em;",[21,16641,44],{"className":16642},[42,43],[21,16644],{"className":16645,"style":379},[378],[21,16647,9160],{"className":16648},[383],[21,16650],{"className":16651,"style":379},[378],[21,16653,16655,16658],{"className":16654},[33],[21,16656],{"className":16657,"style":1540},[37],[21,16659,358],{"className":16660},[42]," it bootstraps,\nand off-policy plus bootstrapping plus function approximation is the deadly triad —\nthe weights can diverge. The stabilizing methods, and the practical cost of running\ntraces, continue in\n",[15,16663,16664],{"href":338},"stable off-policy methods with traces",[16666,16667,16670,16675],"section",{"className":16668,"dataFootnotes":6},[16669],"footnotes",[342,16671,16674],{"className":16672,"id":356},[16673],"sr-only","Footnotes",[16676,16677,16678,17112,17208,17254],"ol",{},[16679,16680,16682,6052,16685,16688,16689,182,16704,16719,16720,182,16735,16750,16751,6052,16913,17029,17030,17088,17089,17104,17105],"li",{"id":16681},"user-content-fn-sb-varlg",[824,16683,16684],{},"Sutton & Barto",[237,16686,16687],{},"Reinforcement Learning: An Introduction"," (2nd ed.), §12.8 — Variable ",[21,16690,16692],{"className":16691},[24],[21,16693,16695],{"className":16694,"ariaHidden":29},[28],[21,16696,16698,16701],{"className":16697},[33],[21,16699],{"className":16700,"style":38},[37],[21,16702,44],{"className":16703},[42,43],[21,16705,16707],{"className":16706},[24],[21,16708,16710],{"className":16709,"ariaHidden":29},[28],[21,16711,16713,16716],{"className":16712},[33],[21,16714],{"className":16715,"style":195},[37],[21,16717,200],{"className":16718,"style":199},[42,43],": promoting ",[21,16721,16723],{"className":16722},[24],[21,16724,16726],{"className":16725,"ariaHidden":29},[28],[21,16727,16729,16732],{"className":16728},[33],[21,16730],{"className":16731,"style":38},[37],[21,16733,44],{"className":16734},[42,43],[21,16736,16738],{"className":16737},[24],[21,16739,16741],{"className":16740,"ariaHidden":29},[28],[21,16742,16744,16747],{"className":16743},[33],[21,16745],{"className":16746,"style":195},[37],[21,16748,200],{"className":16749,"style":199},[42,43]," to functions ",[21,16752,16754],{"className":16753},[24],[21,16755,16757,16812],{"className":16756,"ariaHidden":29},[28],[21,16758,16760,16763,16803,16806,16809],{"className":16759},[33],[21,16761],{"className":16762,"style":480},[37],[21,16764,16766,16769],{"className":16765},[42],[21,16767,44],{"className":16768},[42,43],[21,16770,16772],{"className":16771},[70],[21,16773,16775,16795],{"className":16774},[74,75],[21,16776,16778,16792],{"className":16777},[79],[21,16779,16781],{"className":16780,"style":84},[83],[21,16782,16783,16786],{"style":87},[21,16784],{"className":16785,"style":92},[91],[21,16787,16789],{"className":16788},[96,97,98,99],[21,16790,103],{"className":16791},[42,43,99],[21,16793,108],{"className":16794},[107],[21,16796,16798],{"className":16797},[79],[21,16799,16801],{"className":16800,"style":115},[83],[21,16802],{},[21,16804],{"className":16805,"style":379},[378],[21,16807,1101],{"className":16808},[383],[21,16810],{"className":16811,"style":379},[378],[21,16813,16815,16818,16821,16824,16864,16867,16870,16910],{"className":16814},[33],[21,16816],{"className":16817,"style":440},[37],[21,16819,44],{"className":16820},[42,43],[21,16822,543],{"className":16823},[444],[21,16825,16827,16830],{"className":16826},[42],[21,16828,400],{"className":16829,"style":550},[42,43],[21,16831,16833],{"className":16832},[70],[21,16834,16836,16856],{"className":16835},[74,75],[21,16837,16839,16853],{"className":16838},[79],[21,16840,16842],{"className":16841,"style":84},[83],[21,16843,16844,16847],{"style":565},[21,16845],{"className":16846,"style":92},[91],[21,16848,16850],{"className":16849},[96,97,98,99],[21,16851,103],{"className":16852},[42,43,99],[21,16854,108],{"className":16855},[107],[21,16857,16859],{"className":16858},[79],[21,16860,16862],{"className":16861,"style":115},[83],[21,16863],{},[21,16865,454],{"className":16866},[453],[21,16868],{"className":16869,"style":458},[378],[21,16871,16873,16876],{"className":16872},[42],[21,16874,423],{"className":16875},[42,43],[21,16877,16879],{"className":16878},[70],[21,16880,16882,16902],{"className":16881},[74,75],[21,16883,16885,16899],{"className":16884},[79],[21,16886,16888],{"className":16887,"style":84},[83],[21,16889,16890,16893],{"style":87},[21,16891],{"className":16892,"style":92},[91],[21,16894,16896],{"className":16895},[96,97,98,99],[21,16897,103],{"className":16898},[42,43,99],[21,16900,108],{"className":16901},[107],[21,16903,16905],{"className":16904},[79],[21,16906,16908],{"className":16907,"style":115},[83],[21,16909],{},[21,16911,635],{"className":16912},[465],[21,16914,16916],{"className":16915},[24],[21,16917,16919,16974],{"className":16918,"ariaHidden":29},[28],[21,16920,16922,16925,16965,16968,16971],{"className":16921},[33],[21,16923],{"className":16924,"style":195},[37],[21,16926,16928,16931],{"className":16927},[42],[21,16929,200],{"className":16930,"style":199},[42,43],[21,16932,16934],{"className":16933},[70],[21,16935,16937,16957],{"className":16936},[74,75],[21,16938,16940,16954],{"className":16939},[79],[21,16941,16943],{"className":16942,"style":84},[83],[21,16944,16945,16948],{"style":737},[21,16946],{"className":16947,"style":92},[91],[21,16949,16951],{"className":16950},[96,97,98,99],[21,16952,103],{"className":16953},[42,43,99],[21,16955,108],{"className":16956},[107],[21,16958,16960],{"className":16959},[79],[21,16961,16963],{"className":16962,"style":115},[83],[21,16964],{},[21,16966],{"className":16967,"style":379},[378],[21,16969,1101],{"className":16970},[383],[21,16972],{"className":16973,"style":379},[378],[21,16975,16977,16980,16983,16986,17026],{"className":16976},[33],[21,16978],{"className":16979,"style":440},[37],[21,16981,200],{"className":16982,"style":199},[42,43],[21,16984,543],{"className":16985},[444],[21,16987,16989,16992],{"className":16988},[42],[21,16990,400],{"className":16991,"style":550},[42,43],[21,16993,16995],{"className":16994},[70],[21,16996,16998,17018],{"className":16997},[74,75],[21,16999,17001,17015],{"className":17000},[79],[21,17002,17004],{"className":17003,"style":84},[83],[21,17005,17006,17009],{"style":565},[21,17007],{"className":17008,"style":92},[91],[21,17010,17012],{"className":17011},[96,97,98,99],[21,17013,103],{"className":17014},[42,43,99],[21,17016,108],{"className":17017},[107],[21,17019,17021],{"className":17020},[79],[21,17022,17024],{"className":17023,"style":115},[83],[21,17025],{},[21,17027,635],{"className":17028},[465],"; the generalized return (12.17) with the running product ",[21,17031,17033],{"className":17032},[24],[21,17034,17036],{"className":17035,"ariaHidden":29},[28],[21,17037,17039,17042,17045,17048],{"className":17038},[33],[21,17040],{"className":17041,"style":440},[37],[21,17043,1264],{"className":17044,"style":1420},[1118,1165,1419],[21,17046],{"className":17047,"style":458},[378],[21,17049,17051,17054],{"className":17050},[42],[21,17052,200],{"className":17053,"style":199},[42,43],[21,17055,17057],{"className":17056},[70],[21,17058,17060,17080],{"className":17059},[74,75],[21,17061,17063,17077],{"className":17062},[79],[21,17064,17066],{"className":17065,"style":1312},[83],[21,17067,17068,17071],{"style":737},[21,17069],{"className":17070,"style":92},[91],[21,17072,17074],{"className":17073},[96,97,98,99],[21,17075,1241],{"className":17076},[42,43,99],[21,17078,108],{"className":17079},[107],[21,17081,17083],{"className":17082},[79],[21,17084,17086],{"className":17085,"style":115},[83],[21,17087],{},"; the termination function unifying episodic, continuing, and pseudo-terminating prediction; and the recursive generalized ",[21,17090,17092],{"className":17091},[24],[21,17093,17095],{"className":17094,"ariaHidden":29},[28],[21,17096,17098,17101],{"className":17097},[33],[21,17099],{"className":17100,"style":38},[37],[21,17102,44],{"className":17103},[42,43],"-returns for states (12.18), Sarsa (12.19), and Expected Sarsa (12.20)–(12.21). ",[15,17106,17111],{"href":17107,"ariaLabel":17108,"className":17109,"dataFootnoteBackref":6},"#user-content-fnref-sb-varlg","Back to reference 1",[17110],"data-footnote-backref","↩",[16679,17113,17115,17117,17118,17170,17171,17186,17187,17202,17203],{"id":17114},"user-content-fn-sb-cv",[824,17116,16684],{},", §12.9 — Off-policy Traces with Control Variates: the per-decision importance ratio ",[21,17119,17121],{"className":17120},[24],[21,17122,17124],{"className":17123,"ariaHidden":29},[28],[21,17125,17127,17130],{"className":17126},[33],[21,17128],{"className":17129,"style":195},[37],[21,17131,17133,17136],{"className":17132},[42],[21,17134,295],{"className":17135},[42,43],[21,17137,17139],{"className":17138},[70],[21,17140,17142,17162],{"className":17141},[74,75],[21,17143,17145,17159],{"className":17144},[79],[21,17146,17148],{"className":17147,"style":84},[83],[21,17149,17150,17153],{"style":87},[21,17151],{"className":17152,"style":92},[91],[21,17154,17156],{"className":17155},[96,97,98,99],[21,17157,103],{"className":17158},[42,43,99],[21,17160,108],{"className":17161},[107],[21,17163,17165],{"className":17164},[79],[21,17166,17168],{"className":17167,"style":115},[83],[21,17169],{}," folded into the state-based ",[21,17172,17174],{"className":17173},[24],[21,17175,17177],{"className":17176,"ariaHidden":29},[28],[21,17178,17180,17183],{"className":17179},[33],[21,17181],{"className":17182,"style":38},[37],[21,17184,44],{"className":17185},[42,43],"-return with the control-variate term (12.22); the TD-error approximation (12.23)–(12.24); the general accumulating trace with importance sampling (12.25); and the action-value off-policy return (12.26)–(12.28) yielding the trace (12.29) and an Expected-Sarsa(",[21,17188,17190],{"className":17189},[24],[21,17191,17193],{"className":17192,"ariaHidden":29},[28],[21,17194,17196,17199],{"className":17195},[33],[21,17197],{"className":17198,"style":38},[37],[21,17200,44],{"className":17201},[42,43],"). ",[15,17204,17111],{"href":17205,"ariaLabel":17206,"className":17207,"dataFootnoteBackref":6},"#user-content-fnref-sb-cv","Back to reference 2",[17110],[16679,17209,17211,17213,17214,17229,17230,17248,17249],{"id":17210},"user-content-fn-sb-watkins",[824,17212,16684],{},", §12.10 — Watkins's Q(",[21,17215,17217],{"className":17216},[24],[21,17218,17220],{"className":17219,"ariaHidden":29},[28],[21,17221,17223,17226],{"className":17222},[33],[21,17224],{"className":17225,"style":38},[37],[21,17227,44],{"className":17228},[42,43],"): decay the trace by ",[21,17231,17233],{"className":17232},[24],[21,17234,17236],{"className":17235,"ariaHidden":29},[28],[21,17237,17239,17242,17245],{"className":17238},[33],[21,17240],{"className":17241,"style":5968},[37],[21,17243,200],{"className":17244,"style":199},[42,43],[21,17246,44],{"className":17247},[42,43]," while greedy actions are taken, cut it to zero on the first non-greedy action (backup diagram, Fig. 12.12). ",[15,17250,17111],{"href":17251,"ariaLabel":17252,"className":17253,"dataFootnoteBackref":6},"#user-content-fnref-sb-watkins","Back to reference 3",[17110],[16679,17255,17257,17259,17260,17275,17276,17291,17292,17307,17308,17668,17669],{"id":17256},"user-content-fn-sb-tb",[824,17258,16684],{},", §12.10 — Tree-Backup(",[21,17261,17263],{"className":17262},[24],[21,17264,17266],{"className":17265,"ariaHidden":29},[28],[21,17267,17269,17272],{"className":17268},[33],[21,17270],{"className":17271,"style":38},[37],[21,17273,44],{"className":17274},[42,43],"), TB(",[21,17277,17279],{"className":17278},[24],[21,17280,17282],{"className":17281,"ariaHidden":29},[28],[21,17283,17285,17288],{"className":17284},[33],[21,17286],{"className":17287,"style":38},[37],[21,17289,44],{"className":17290},[42,43],"): the eligibility-trace form of ",[21,17293,17295],{"className":17294},[24],[21,17296,17298],{"className":17297,"ariaHidden":29},[28],[21,17299,17301,17304],{"className":17300},[33],[21,17302],{"className":17303,"style":253},[37],[21,17305,6194],{"className":17306},[42,43],"-step Tree Backup, requiring no importance sampling; the target-policy-weighted return and the trace ",[21,17309,17311],{"className":17310},[24],[21,17312,17314,17369,17510,17617],{"className":17313,"ariaHidden":29},[28],[21,17315,17317,17320,17360,17363,17366],{"className":17316},[33],[21,17318],{"className":17319,"style":58},[37],[21,17321,17323,17326],{"className":17322},[42],[21,17324,66],{"className":17325},[42,65],[21,17327,17329],{"className":17328},[70],[21,17330,17332,17352],{"className":17331},[74,75],[21,17333,17335,17349],{"className":17334},[79],[21,17336,17338],{"className":17337,"style":84},[83],[21,17339,17340,17343],{"style":87},[21,17341],{"className":17342,"style":92},[91],[21,17344,17346],{"className":17345},[96,97,98,99],[21,17347,103],{"className":17348},[42,43,99],[21,17350,108],{"className":17351},[107],[21,17353,17355],{"className":17354},[79],[21,17356,17358],{"className":17357,"style":115},[83],[21,17359],{},[21,17361],{"className":17362,"style":379},[378],[21,17364,1101],{"className":17365},[383],[21,17367],{"className":17368,"style":379},[378],[21,17370,17372,17375,17415,17455,17458,17461,17501,17504,17507],{"className":17371},[33],[21,17373],{"className":17374,"style":440},[37],[21,17376,17378,17381],{"className":17377},[42],[21,17379,200],{"className":17380,"style":199},[42,43],[21,17382,17384],{"className":17383},[70],[21,17385,17387,17407],{"className":17386},[74,75],[21,17388,17390,17404],{"className":17389},[79],[21,17391,17393],{"className":17392,"style":84},[83],[21,17394,17395,17398],{"style":737},[21,17396],{"className":17397,"style":92},[91],[21,17399,17401],{"className":17400},[96,97,98,99],[21,17402,103],{"className":17403},[42,43,99],[21,17405,108],{"className":17406},[107],[21,17408,17410],{"className":17409},[79],[21,17411,17413],{"className":17412,"style":115},[83],[21,17414],{},[21,17416,17418,17421],{"className":17417},[42],[21,17419,44],{"className":17420},[42,43],[21,17422,17424],{"className":17423},[70],[21,17425,17427,17447],{"className":17426},[74,75],[21,17428,17430,17444],{"className":17429},[79],[21,17431,17433],{"className":17432,"style":84},[83],[21,17434,17435,17438],{"style":87},[21,17436],{"className":17437,"style":92},[91],[21,17439,17441],{"className":17440},[96,97,98,99],[21,17442,103],{"className":17443},[42,43,99],[21,17445,108],{"className":17446},[107],[21,17448,17450],{"className":17449},[79],[21,17451,17453],{"className":17452,"style":115},[83],[21,17454],{},[21,17456,258],{"className":17457,"style":257},[42,43],[21,17459,543],{"className":17460},[444],[21,17462,17464,17467],{"className":17463},[42],[21,17465,423],{"className":17466},[42,43],[21,17468,17470],{"className":17469},[70],[21,17471,17473,17493],{"className":17472},[74,75],[21,17474,17476,17490],{"className":17475},[79],[21,17477,17479],{"className":17478,"style":84},[83],[21,17480,17481,17484],{"style":87},[21,17482],{"className":17483,"style":92},[91],[21,17485,17487],{"className":17486},[96,97,98,99],[21,17488,103],{"className":17489},[42,43,99],[21,17491,108],{"className":17492},[107],[21,17494,17496],{"className":17495},[79],[21,17497,17499],{"className":17498,"style":115},[83],[21,17500],{},[21,17502],{"className":17503,"style":379},[378],[21,17505,5825],{"className":17506},[383],[21,17508],{"className":17509,"style":379},[378],[21,17511,17513,17516,17556,17559,17608,17611,17614],{"className":17512},[33],[21,17514],{"className":17515,"style":440},[37],[21,17517,17519,17522],{"className":17518},[42],[21,17520,400],{"className":17521,"style":550},[42,43],[21,17523,17525],{"className":17524},[70],[21,17526,17528,17548],{"className":17527},[74,75],[21,17529,17531,17545],{"className":17530},[79],[21,17532,17534],{"className":17533,"style":84},[83],[21,17535,17536,17539],{"style":565},[21,17537],{"className":17538,"style":92},[91],[21,17540,17542],{"className":17541},[96,97,98,99],[21,17543,103],{"className":17544},[42,43,99],[21,17546,108],{"className":17547},[107],[21,17549,17551],{"className":17550},[79],[21,17552,17554],{"className":17553,"style":115},[83],[21,17555],{},[21,17557,635],{"className":17558},[465],[21,17560,17562,17565],{"className":17561},[42],[21,17563,66],{"className":17564},[42,65],[21,17566,17568],{"className":17567},[70],[21,17569,17571,17600],{"className":17570},[74,75],[21,17572,17574,17597],{"className":17573},[79],[21,17575,17577],{"className":17576,"style":944},[83],[21,17578,17579,17582],{"style":87},[21,17580],{"className":17581,"style":92},[91],[21,17583,17585],{"className":17584},[96,97,98,99],[21,17586,17588,17591,17594],{"className":17587},[42,99],[21,17589,103],{"className":17590},[42,43,99],[21,17592,3896],{"className":17593},[408,99],[21,17595,358],{"className":17596},[42,99],[21,17598,108],{"className":17599},[107],[21,17601,17603],{"className":17602},[79],[21,17604,17606],{"className":17605,"style":976},[83],[21,17607],{},[21,17609],{"className":17610,"style":404},[378],[21,17612,963],{"className":17613},[408],[21,17615],{"className":17616,"style":404},[378],[21,17618,17620,17623,17626],{"className":17619},[33],[21,17621],{"className":17622,"style":5968},[37],[21,17624,10875],{"className":17625},[42],[21,17627,17629],{"className":17628},[42,3964],[21,17630,17632,17660],{"className":17631},[74,75],[21,17633,17635,17657],{"className":17634},[79],[21,17636,17638,17646],{"className":17637,"style":38},[83],[21,17639,17640,17643],{"style":3976},[21,17641],{"className":17642,"style":3980},[91],[21,17644,4777],{"className":17645,"style":257},[42,43],[21,17647,17648,17651],{"style":3976},[21,17649],{"className":17650,"style":3980},[91],[21,17652,17654],{"className":17653,"style":4786},[3993],[21,17655,3998],{"className":17656},[42],[21,17658,108],{"className":17659},[107],[21,17661,17663],{"className":17662},[79],[21,17664,17666],{"className":17665,"style":4799},[83],[21,17667],{}," (backup diagram, Fig. 12.13). ",[15,17670,17111],{"href":17671,"ariaLabel":17672,"className":17673,"dataFootnoteBackref":6},"#user-content-fnref-sb-tb","Back to reference 4",[17110],{"title":6,"searchDepth":17675,"depth":17675,"links":17676},2,[17677,17678,17679,17680,17681],{"id":344,"depth":17675,"text":345},{"id":6142,"depth":17675,"text":6143},{"id":13680,"depth":17675,"text":13681},{"id":16471,"depth":17675,"text":16472},{"id":356,"depth":17675,"text":16674},[],"computer-science","The eligibility-traces lesson\ncompleted the on-policy case: the λ-return as a forward\nview, the trace vector zt​ as a backward view, TD(λ) making\nthe two nearly agree, true online TD(λ) making them agree exactly, and\nSarsa(λ) lifting the whole apparatus to on-policy control. Throughout,\nλ and γ were fixed constants and every step was generated by the\npolicy being evaluated.",false,"md",{"moduleNumber":17688,"lessonNumber":17689,"order":17690},3,13,313,"Approximate Solution Methods",true,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces",[],"---\ntitle: Off-Policy Eligibility Traces\nmodule: Approximate Solution Methods\nmoduleNumber: 3\nlessonNumber: 13\norder: 313\nsummary: >\n  Eligibility traces meet off-policy learning and function approximation — the\n  corner where stability gets hard. We first let the bootstrapping and discounting\n  parameters vary with state, so a single generalized return covers episodic and\n  continuing tasks and folds termination into the discount. Then we fold the\n  per-decision importance ratio into the trace with a control-variate correction,\n  and build Watkins's Q(λ) and its importance-sampling-free successor\n  Tree-Backup(λ) — all correct in expectation, but still semi-gradient, so the\n  deadly triad and its fixes wait for the next lesson.\ntopics: [Approximation]\nsources:\n  - book: Sutton & Barto\n    ref: \"Ch. 12 — Eligibility Traces; §12.8 Variable λ and γ; §12.9 Off-policy Traces with Control Variates\"\n  - book: Sutton & Barto\n    ref: \"§12.10 Watkins's Q(λ) to Tree-Backup(λ)\"\n---\n\nThe [eligibility-traces lesson](\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces)\ncompleted the on-policy case: the $\\lambda$-return as a forward\nview, the trace vector $\\mathbf{z}_t$ as a backward view, TD($\\lambda$) making\nthe two nearly agree, true online TD($\\lambda$) making them agree exactly, and\nSarsa($\\lambda$) lifting the whole apparatus to on-policy control. Throughout,\n$\\lambda$ and $\\gamma$ were fixed constants and every step was generated by the\npolicy being evaluated.\n\nThis lesson removes both restrictions. First $\\lambda$ and $\\gamma$ become\n_functions of state and action_, which turns out to unify the episodic and\ncontinuing settings and absorb termination into the discount. Then we learn about\na target policy $\\pi$ from behavior policy $b$, which brings the per-decision\nimportance ratio $\\rho_t$ into the trace, with a control variate to limit the\nvariance. The combination — traces, off-policy data, and a parameterized value\nfunction — is the\n[deadly triad](\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad)\nitself, and convergence is no longer guaranteed. Naming the methods that restore\nstability is the job of the\n[next lesson](\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces); this\none builds the returns and traces they all rest on.\n\n## Variable λ and γ\n\nTo state the final algorithms in their most general form, promote the two\nconstants to functions.[^sb-varlg] Let $\\lambda : \\mathcal{S} \\times \\mathcal{A}\n\\to [0,1]$ depend on state and action, $\\lambda_t \\doteq \\lambda(S_t, A_t)$, and\nlet $\\gamma : \\mathcal{S} \\to [0,1]$ depend on state, $\\gamma_t \\doteq\n\\gamma(S_t)$. The **termination function** $\\gamma$ is the significant one,\nbecause it changes the return itself — the random variable whose expectation we\nare estimating. The return is now defined by a running product of per-step\ndiscounts,\n\n$$\nG_t \\;\\doteq\\; R_{t+1} + \\gamma_{t+1} G_{t+1}\n\\;=\\; \\sum_{k=t}^{\\infty} \\parens{\\prod_{i=t+1}^{k} \\gamma_i} R_{k+1},\n$$\n\nwhere we require $\\prod_{k=t}^{\\infty} \\gamma_k = 0$ with probability one for\nevery $t$, so the sums are finite. This single definition lets the episodic\nsetting be presented as one uninterrupted stream of\nexperience — no special terminal states, no start distribution, no explicit\ntermination time. A former terminal state becomes an ordinary state at which\n$\\gamma(s) = 0$ and which transitions to the start distribution; setting\n$\\gamma(\\cdot) = 1$ everywhere else recovers the classical episodic case exactly.\n\n$$\n% caption: State-dependent termination $\\gamma(s)$ unifies the settings. Top: the\n% classic episodic view, with a special terminal state and a reset. Bottom: the\n% same experience as one continuing stream where the goal state has $\\gamma=0$ and\n% jumps back to the start; a bootstrapping state has $\\gamma \\approx 1$. Setting\n% $\\gamma \\equiv 1$ off the goal recovers the episodic case; a constant $\\gamma$\n% recovers ordinary discounting.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize,\n  st\u002F.style={circle, draw, minimum size=7mm, inner sep=0pt},\n  term\u002F.style={rectangle, draw, fill=black!18, minimum size=7mm, inner sep=0pt}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  % ---- top: episodic view ----\n  \\node[anchor=west, font=\\scriptsize] at (-1.6,2.0) {episodic:};\n  \\node[st] (a0) at (0,2.0) {s0};\n  \\node[st] (a1) at (1.6,2.0) {s1};\n  \\node[st] (a2) at (3.2,2.0) {s2};\n  \\node[term] (aT) at (4.8,2.0) {T};\n  \\draw[->, acc, thick] (a0) -- (a1);\n  \\draw[->, acc, thick] (a1) -- (a2);\n  \\draw[->, acc, thick] (a2) -- (aT);\n  \\node[anchor=south, font=\\scriptsize, text=black] at (4.8,2.4) {terminal};\n  \\draw[->, red, thick, dashed] (aT) to[bend right=42] (a0);\n  \\node[anchor=north, font=\\scriptsize, text=red] at (2.4,1.35) {reset to start};\n  % ---- bottom: continuing view ----\n  \\node[anchor=west, font=\\scriptsize] at (-1.6,0.0) {continuing:};\n  \\node[st] (b0) at (0,0.0) {s0};\n  \\node[st] (b1) at (1.6,0.0) {s1};\n  \\node[st] (b2) at (3.2,0.0) {s2};\n  \\node[st, draw=red, text=red, thick] (bg) at (4.8,0.0) {g};\n  \\draw[->, acc, thick] (b0) -- (b1);\n  \\draw[->, acc, thick] (b1) -- (b2);\n  \\draw[->, acc, thick] (b2) -- (bg);\n  \\node[anchor=south, font=\\scriptsize, text=red] at (4.8,0.4) {gamma(g)=0};\n  \\node[anchor=north, font=\\scriptsize, text=black] at (1.6,-0.42) {gamma approx 1};\n  \\draw[->, red, thick] (bg) to[bend right=42] (b0);\n\\end{tikzpicture}\n$$\n\nState-dependent termination also reaches beyond ordinary episodes to _pseudo\ntermination_: predicting some quantity as if the stream ended here, without\nactually interrupting the Markov process. A discounted return is one such\nquantity, so this one construction unifies episodic, discounted-continuing, and\npseudo-terminating prediction. (The undiscounted-continuing case, $\\gamma \\equiv\n1$ with no termination, still needs its own treatment via average reward.)\n\n> **Definition (Termination function).** A map $\\gamma : \\mathcal{S} \\to [0,1]$\n> with $\\gamma_t \\doteq \\gamma(S_t)$ that sets the per-step continuation\n> probability. The return $G_t = R_{t+1} + \\gamma_{t+1} G_t$ discounts future\n> rewards by the running product $\\prod \\gamma_i$; a state with $\\gamma(s)=0$\n> terminates the return there. It unifies episodic, continuing, and\n> pseudo-terminating prediction in one stream of experience.\n\n**A worked return.** Take rewards $R_{t+1}, R_{t+2}, R_{t+3}, \\ldots = 1, 1, 1,\n\\ldots$ and a per-state discount that is $\\gamma = 0.9$ everywhere except a goal\nstate reached at step $t+3$, where $\\gamma(g) = 0$. The running product truncates\nthe sum the moment it hits the zero: the weight on $R_{t+1}$ is $1$, on $R_{t+2}$ is\n$\\gamma_{t+1} = 0.9$, on $R_{t+3}$ is $\\gamma_{t+1}\\gamma_{t+2} = 0.81$, and every\nlater reward is multiplied by $\\gamma_{t+1}\\gamma_{t+2}\\gamma_{t+3} = 0.81 \\cdot 0 =\n0$. So\n\n$$\nG_t = 1 + 0.9\\cdot 1 + 0.81\\cdot 1 + 0 + 0 + \\cdots = 2.71,\n$$\n\na finite return produced with no special terminal state and no episode boundary —\nthe goal's $\\gamma(g) = 0$ truncates the sum. Set $\\gamma(g) = 1$ instead and the\nsame stream keeps accumulating $1 + 0.9 + 0.81 + 0.729 + \\cdots = 10$, the ordinary\ndiscounted return of a continuing task. One definition covers both cases.\n\nMaking $\\lambda$ variable is a change of _solution strategy_, not of the problem:\nit sets, state by state, how much to bootstrap. The generalization touches the\n$\\lambda$-return, which now recurses. In the state-based case, writing the\nsuperscript $s$ to mark that it bootstraps from state values,\n\n$$\nG_t^{\\lambda s} \\;\\doteq\\; R_{t+1} + \\gamma_{t+1}\\Big( (1-\\lambda_{t+1})\\,\\hat v(S_{t+1}, \\mathbf{w}_t) + \\lambda_{t+1}\\, G_{t+1}^{\\lambda s} \\Big).\n$$\n\nRead it left to right: the return is the first reward, plus (to the extent we are\nnot terminating, $\\gamma_{t+1}$) a second term that splits by the degree of\nbootstrapping — a fraction $1-\\lambda_{t+1}$ takes the estimated value at the next\nstate, and the complementary fraction $\\lambda_{t+1}$ recurses into the\n$\\lambda$-return of the next step. The action-based return has a Sarsa form,\n\n$$\nG_t^{\\lambda a} \\;\\doteq\\; R_{t+1} + \\gamma_{t+1}\\Big( (1-\\lambda_{t+1})\\,\\hat q(S_{t+1}, A_{t+1}, \\mathbf{w}_t) + \\lambda_{t+1}\\, G_{t+1}^{\\lambda a} \\Big),\n$$\n\nand an Expected-Sarsa form that bootstraps from the expected value under the\ntarget policy,\n\n$$\nG_t^{\\lambda a} \\;\\doteq\\; R_{t+1} + \\gamma_{t+1}\\Big( (1-\\lambda_{t+1})\\,\\bar V_t(S_{t+1}) + \\lambda_{t+1}\\, G_{t+1}^{\\lambda a} \\Big),\n\\qquad\n\\bar V_t(s) \\;\\doteq\\; \\sum_a \\pi(a \\mid s)\\, \\hat q(s, a, \\mathbf{w}_t).\n$$\n\nThese recursions are the substrate for everything below. Constant $\\lambda,\n\\gamma$ are the special case $\\lambda_t \\equiv \\lambda$, $\\gamma_t \\equiv \\gamma$,\nwhich collapses each back to the $\\lambda$-return of the previous lesson.\n\n## Off-policy traces with control variates\n\nNow the data is off-policy: actions come from a behavior policy $b$ while we\nevaluate a target policy $\\pi$. For $n$-step methods, importance sampling was\napplied _outside_ the return. That has no clean analogue for a full,\nnon-truncated $\\lambda$-return, so we move directly to the modern construction:\nfold the per-decision importance ratio into the return itself, with a **control\nvariate** to hold the variance down.[^sb-cv] The per-step ratio is the usual\n\n$$\n\\rho_t \\;\\doteq\\; \\frac{\\pi(A_t \\mid S_t)}{b(A_t \\mid S_t)}.\n$$\n\nThe state-based off-policy $\\lambda$-return generalizes the recursion above by\nscaling the bootstrapped part by $\\rho_t$ and adding a correction term that has\nzero mean but cancels much of the extra variance,\n\n$$\nG_t^{\\lambda s} \\;\\doteq\\; \\rho_t\\Big( R_{t+1} + \\gamma_{t+1}\\big( (1-\\lambda_{t+1})\\hat v(S_{t+1},\\mathbf{w}_t) + \\lambda_{t+1} G_{t+1}^{\\lambda s} \\big) \\Big) + (1-\\rho_t)\\,\\hat v(S_t, \\mathbf{w}_t).\n$$\n\nThe second term $(1-\\rho_t)\\hat v(S_t, \\mathbf{w}_t)$ is the control variate: its\nexpectation over $A_t \\sim b$ is zero (because $\\mathbb{E}_b[\\rho_t] = 1$), so it\ndoes not bias the target, but when $\\rho_t$ is far from $1$ it pulls the return\nback toward the current estimate and dampens the swing.\n\n$$\n% caption: The control variate at work. Without it (top), a large importance\n% ratio $\\rho_t$ scales the whole target and the estimate whipsaws far from the\n% baseline $\\hat v(S_t)$. With the correction $(1-\\rho_t)\\hat v(S_t)$ (bottom),\n% the same ratios produce a target that stays near the baseline; the correction\n% has zero mean, so no bias is added.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  % ---- top: no control variate ----\n  \\draw[black, dashed] (0,2.0) -- (7.2,2.0);\n  \\node[anchor=west, font=\\scriptsize, text=black] at (7.25,2.0) {baseline v(St)};\n  \\node[anchor=east, font=\\scriptsize] at (-0.1,2.0) {no CV:};\n  \\foreach \\x\u002F\\y in {0.7\u002F3.2, 1.7\u002F0.9, 2.7\u002F2.9, 3.7\u002F1.0, 4.7\u002F3.3, 5.7\u002F1.1} {\n    \\draw[red, thick] (\\x,2.0) -- (\\x,\\y);\n    \\fill[red] (\\x,\\y) circle (1.6pt);\n  }\n  % ---- bottom: with control variate ----\n  \\draw[black, dashed] (0,-1.2) -- (7.2,-1.2);\n  \\node[anchor=west, font=\\scriptsize, text=black] at (7.25,-1.2) {baseline v(St)};\n  \\node[anchor=east, font=\\scriptsize] at (-0.1,-1.2) {with CV:};\n  \\foreach \\x\u002F\\y in {0.7\u002F-0.75, 1.7\u002F-1.5, 2.7\u002F-0.85, 3.7\u002F-1.45, 4.7\u002F-0.7, 5.7\u002F-1.4} {\n    \\draw[acc, thick] (\\x,-1.2) -- (\\x,\\y);\n    \\fill[acc] (\\x,\\y) circle (1.6pt);\n  }\n  \\node[anchor=south, font=\\scriptsize, text=black] at (3.6,3.35) {high variance};\n  \\node[anchor=north, font=\\scriptsize, text=black] at (3.6,-1.65) {variance reduced};\n\\end{tikzpicture}\n$$\n\n**A worked correction.** Suppose at $S_t$ the target policy would take the sampled\naction with probability $\\pi(A_t \\mid S_t) = 0.8$ while the behavior policy took it\nwith $b(A_t \\mid S_t) = 0.2$, so $\\rho_t = 0.8\u002F0.2 = 4$. Let the bracketed one-step\ntarget evaluate to $12$ and the current estimate be $\\hat v(S_t) = 10$. Without the\ncontrol variate the return is $\\rho_t \\cdot 12 = 48$ — a swing of $+38$ above the\nbaseline from a single lucky-for-$\\pi$ action. With the control variate it is\n$4\\cdot 12 + (1 - 4)\\cdot 10 = 48 - 30 = 18$, a swing of $+8$. The correction has\nnot changed the _expectation_ (over $A_t \\sim b$, $\\mathbb{E}[\\rho_t] = 1$ so\n$\\mathbb{E}[(1-\\rho_t)\\hat v] = 0$), but it has shrunk this particular sample's\nexcursion from $38$ to $8$. When instead the sampled action is one $\\pi$ dislikes,\n$\\rho_t \u003C 1$, and the same term nudges the target back _up_ toward the baseline —\neither way, toward $\\hat v(S_t)$.\n\nThe value of this form is that the truncated version can be written, to a close\napproximation, as a sum of state-based TD errors, exactly as in the on-policy\ncase. With the per-decision TD error\n\n$$\n\\delta_t^s \\;\\doteq\\; R_{t+1} + \\gamma_{t+1}\\, \\hat v(S_{t+1}, \\mathbf{w}_t) - \\hat v(S_t, \\mathbf{w}_t),\n$$\n\nthe return is approximately\n\n$$\nG_t^{\\lambda s} \\;\\approx\\; \\hat v(S_t, \\mathbf{w}_t) + \\rho_t \\sum_{k=t}^{\\infty} \\delta_k^s \\prod_{i=t+1}^{k} \\gamma_i \\lambda_i \\rho_i,\n$$\n\nwith the approximation becoming _exact_ when the value function does not change.\nThe product $\\prod \\gamma_i \\lambda_i \\rho_i$ is the eligibility-trace weight: it\ncompounds the same $\\gamma \\lambda$ decay as before, now multiplied by the\nimportance ratios along the way. The forward-view update\n\n$$\n\\mathbf{w}_{t+1} \\;=\\; \\mathbf{w}_t + \\alpha \\big( G_t^{\\lambda s} - \\hat v(S_t, \\mathbf{w}_t) \\big) \\nabla \\hat v(S_t, \\mathbf{w}_t)\n\\;\\approx\\; \\mathbf{w}_t + \\alpha \\rho_t \\Big( \\sum_{k=t}^{\\infty} \\delta_k^s \\prod_{i=t+1}^{k} \\gamma_i \\lambda_i \\rho_i \\Big) \\nabla \\hat v(S_t, \\mathbf{w}_t)\n$$\n\nalready looks like a trace-based TD update: a product of past terms multiplying\npresent TD errors. Summing this forward-view update over time and applying the\nsummation-rule rearrangement (the same manipulation used for on-policy TD($\\lambda$))\nconverts it into a backward view driven by a single accumulating trace. The trace\nthat falls out is the general **accumulating trace with importance sampling**,\n\n$$\n\\mathbf{z}_t \\;\\doteq\\; \\rho_t \\big( \\gamma_t \\lambda_t\\, \\mathbf{z}_{t-1} + \\nabla \\hat v(S_t, \\mathbf{w}_t) \\big),\n$$\n\npaired with the usual semi-gradient update $\\mathbf{w}_{t+1} = \\mathbf{w}_t +\n\\alpha \\delta_t^s \\mathbf{z}_t$. In the on-policy case every $\\rho_t = 1$ and this\nreduces to the ordinary accumulating trace (extended to variable $\\lambda,\n\\gamma$); off-policy, the $\\rho_t$ scaling makes the algorithm correct in\nexpectation. But as a semi-gradient method it is _not_ guaranteed stable — the\nnext sections repair that.\n\n$$\n% caption: Where the importance ratio enters the trace. The trace at $S_t$ decays\n% the previous trace by $\\gamma_t \\lambda_t$, adds the new gradient, and scales the\n% whole thing by $\\rho_t = \\pi(A_t \\mid S_t) \u002F b(A_t \\mid S_t)$. When $\\pi$ agrees\n% with $b$, $\\rho_t = 1$ and this is the on-policy accumulating trace; a target\n% action that $b$ rarely takes gives $\\rho_t \\gg 1$ and amplifies the trace.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize,\n  box\u002F.style={draw, minimum width=22mm, minimum height=10mm, align=center, font=\\scriptsize}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[box] (prev)  at (0,0)    {prev trace\\\\z(t-1)};\n  \\node[box] (decay) at (3.2,0)  {decay by\\\\gamma(t) lambda(t)};\n  \\node[box] (grad)  at (3.2,-1.9) {add gradient\\\\grad v(St)};\n  \\node[box, draw=acc, text=acc, thick] (scale) at (6.6,0) {scale by\\\\rho(t)};\n  \\node[box] (out)   at (9.8,0)  {new trace\\\\z(t)};\n  \\draw[->, thick] (prev) -- (decay);\n  \\draw[->, thick] (grad) -- (decay);\n  \\draw[->, acc, thick] (decay) -- (scale);\n  \\draw[->, acc, thick] (scale) -- (out);\n  \\node[anchor=north, font=\\scriptsize, text=black] at (6.6,-0.75) {rho = pi \u002F b};\n\\end{tikzpicture}\n$$\n\nThe identical derivation runs for action values. Starting from the Expected-Sarsa\nform of the general $\\lambda$-return — which works out simpler than the Sarsa form\n— the same steps produce the action-value off-policy trace,\n\n$$\n\\mathbf{z}_t \\;\\doteq\\; \\gamma_t \\lambda_t\\, \\rho_t\\, \\mathbf{z}_{t-1} + \\nabla \\hat q(S_t, A_t, \\mathbf{w}_t),\n$$\n\nwith the expectation-form action-value TD error\n\n$$\n\\delta_t^a \\;\\doteq\\; R_{t+1} + \\gamma_{t+1}\\, \\bar V_t(S_{t+1}) - \\hat q(S_t, A_t, \\mathbf{w}_t),\n$$\n\nand the same weight update $\\mathbf{w}_{t+1} = \\mathbf{w}_t + \\alpha \\delta_t^a\n\\mathbf{z}_t$. This is an efficient Expected-Sarsa($\\lambda$) that runs\non-policy or off-policy; with constant $\\lambda, \\gamma$ and the ordinary\nstate-action TD error it becomes exactly the Sarsa($\\lambda$) of the previous\nlesson. It is probably the best method of its kind — subject, still, to the\nstability caveat.\n\n## Watkins's Q(λ) to Tree-Backup(λ)\n\nThe oldest way to give Q-learning eligibility traces is **Watkins's Q($\\lambda$)**.[^sb-watkins]\nQ-learning bootstraps from the greedy action, so its trace should reflect the\nfuture only as long as the agent keeps taking greedy actions. Watkins's rule is\nsimple: decay the trace by $\\gamma\\lambda$ in the usual way as long as greedy\nactions are taken, but the moment the behavior policy selects the _first\nnon-greedy action_, cut the trace to zero. Everything after that first\nexploratory step is off the greedy path and irrelevant to the greedy target, so\nit receives no eligibility.\n\n$$\n% caption: Backup diagram for Watkins's Q(lambda). Component backups of increasing\n% length are weighted (1-lambda), (1-lambda)lambda, (1-lambda)lambda^2, ..., as in\n% any lambda-return; the series ends either at episode termination (weight\n% lambda^{T-t-1}, the box) or at the first non-greedy action, whichever comes\n% first. After that first non-greedy branch the trace is cut.\n\\begin{tikzpicture}[>=stealth, font=\\scriptsize,\n  sn\u002F.style={circle, draw, minimum size=3.2mm, inner sep=0pt},\n  an\u002F.style={circle, draw, fill=black, minimum size=1.6mm, inner sep=0pt},\n  term\u002F.style={rectangle, draw, fill=black!18, minimum size=4mm, inner sep=0pt}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  \\node[anchor=south, font=\\small] at (3.6,3.7) {Watkins's Q(lambda)};\n  % column 1: 1-step\n  \\node[an] (a1) at (0,3.3) {};\n  \\node[sn] (s1) at (0,2.7) {};\n  \\draw (a1) -- (s1);\n  \\node[anchor=north, font=\\scriptsize] at (0,2.55) {1-lambda};\n  % column 2: 2-step\n  \\node[an] (a2) at (1.2,3.3) {};\n  \\node[sn] (s2a) at (1.2,2.7) {};\n  \\node[an] (a2b) at (1.2,2.2) {};\n  \\node[sn] (s2b) at (1.2,1.6) {};\n  \\draw (a2) -- (s2a) -- (a2b) -- (s2b);\n  \\node[anchor=north, font=\\scriptsize] at (1.2,1.45) {(1-lambda)lambda};\n  % column 3: 3-step\n  \\node[an] (a3) at (2.4,3.3) {};\n  \\node[sn] (s3a) at (2.4,2.7) {};\n  \\node[an] (a3b) at (2.4,2.2) {};\n  \\node[sn] (s3b) at (2.4,1.6) {};\n  \\node[an] (a3c) at (2.4,1.1) {};\n  \\node[sn] (s3c) at (2.4,0.5) {};\n  \\draw (a3) -- (s3a) -- (a3b) -- (s3b) -- (a3c) -- (s3c);\n  \\node[anchor=north, font=\\scriptsize] at (2.4,0.35) {(1-lambda)lambda lambda};\n  % dots\n  \\node at (3.5,2.0) {...};\n  % OR\n  \\node[font=\\small] at (4.5,2.4) {OR};\n  % termination column\n  \\node[an] (t1) at (5.4,3.3) {};\n  \\node[sn] (t2) at (5.4,2.7) {};\n  \\node[an] (t3) at (5.4,2.2) {};\n  \\node[sn] (t4) at (5.4,1.6) {};\n  \\node[term] (tT) at (5.4,0.9) {};\n  \\draw (t1) -- (t2) -- (t3) -- (t4) -- (tT);\n  \\node[anchor=east, font=\\scriptsize] at (5.1,0.9) {lambda to the T-t-1};\n  % right: cut at first non-greedy\n  \\node[an] (r1) at (7.0,3.3) {};\n  \\node[sn] (r2) at (7.0,2.7) {};\n  \\node[an] (r3) at (7.0,2.2) {};\n  \\node[sn] (r4) at (7.0,1.6) {};\n  \\node[anchor=west, font=\\scriptsize] at (7.15,3.3) {St, At};\n  \\node[anchor=west, font=\\scriptsize] at (7.15,2.7) {St+1};\n  \\node[anchor=west, font=\\scriptsize] at (7.15,1.6) {St+2};\n  \\draw (r1) -- (r2) -- (r3) -- (r4);\n  % branch: first non-greedy action\n  \\node[an] (rb1) at (6.7,1.0) {};\n  \\node[an] (rb2) at (7.0,1.0) {};\n  \\node[an, draw=red, fill=red] (rb3) at (7.3,1.0) {};\n  \\draw (r4) -- (rb1);  \\draw (r4) -- (rb2);  \\draw[red] (r4) -- (rb3);\n  \\node[anchor=west, font=\\scriptsize, text=red] at (7.5,1.0) {f\\\u002First non-greedy};\n  \\node[anchor=north, font=\\scriptsize] at (7.0,0.75) {lambda to the n-1};\n\\end{tikzpicture}\n$$\n\nThe cut is crude. If exploration is frequent — and early in learning it is — the\ntrace is repeatedly truncated, so Q($\\lambda$) rarely propagates credit more than\na step or two, and much of the benefit of traces is lost. The deeper problem is\nthat cutting on the first non-greedy action is a hard special case of a more\nprincipled operation, which removes the need for the cut entirely.\n\nThe general version is **Tree-Backup($\\lambda$)**, abbreviated TB($\\lambda$).[^sb-tb]\n$n$-step Tree Backup already extended Expected Sarsa to arbitrary target policies\n_without any importance sampling_; TB($\\lambda$) is its eligibility-trace form, and\nit is arguably the true successor to Q-learning because it keeps that appealing\nabsence of importance sampling while applying to off-policy data. Starting from\nthe Expected-Sarsa recursion and expanding the bootstrapping case after the tree-backup\npattern, the return weights each action branch by its target-policy probability,\n\n$$\nG_t^{\\lambda a} \\;=\\; R_{t+1} + \\gamma_{t+1}\\Big( \\bar V_t(S_{t+1}) + \\lambda_{t+1}\\, \\pi(A_{t+1} \\mid S_{t+1})\\big( G_{t+1}^{\\lambda a} - \\hat q(S_{t+1}, A_{t+1}, \\mathbf{w}_t) \\big) \\Big).\n$$\n\n$$\n% caption: Backup diagram for Tree-Backup(lambda). Every backup fans out to all\n% actions at each level: the selected action (solid, continuing down) plus the\n% unselected actions (leaf dots), each weighted by its target-policy probability\n% pi(a|s). No importance sampling is needed — the target policy enters only\n% through these branch weights. Component lengths are weighted 1-lambda,\n% (1-lambda)lambda, ... exactly as before, terminating at the box.\n\\begin{tikzpicture}[>=stealth, font=\\scriptsize,\n  sn\u002F.style={circle, draw, minimum size=3.2mm, inner sep=0pt},\n  an\u002F.style={circle, draw, fill=black, minimum size=1.6mm, inner sep=0pt},\n  leaf\u002F.style={circle, draw, fill=black, minimum size=1.2mm, inner sep=0pt},\n  term\u002F.style={rectangle, draw, fill=black!18, minimum size=4mm, inner sep=0pt}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[anchor=south, font=\\small] at (3.6,3.9) {Tree Backup(lambda)};\n  % helper for a fan of leaves at a state node given coords\n  % column 1: 1-step (root fans out)\n  \\node[an] (c1a) at (0,3.4) {};\n  \\node[sn] (c1s) at (0,2.8) {};\n  \\draw (c1a) -- (c1s);\n  \\node[leaf] at (-0.35,2.35) {}; \\node[leaf] at (0.35,2.35) {};\n  \\draw (c1s) -- (-0.35,2.35);  \\draw (c1s) -- (0.35,2.35);\n  \\node[anchor=north, font=\\scriptsize] at (0,1.95) {1-lambda};\n  % column 2: 2-step\n  \\node[an] (c2a) at (1.4,3.4) {};\n  \\node[sn] (c2s1) at (1.4,2.8) {};\n  \\node[leaf] at (1.05,2.4) {}; \\node[leaf] at (1.75,2.4) {};\n  \\draw (c2s1) -- (1.05,2.4); \\draw (c2s1) -- (1.75,2.4);\n  \\node[an] (c2a2) at (1.4,2.3) {};\n  \\node[sn] (c2s2) at (1.4,1.7) {};\n  \\node[leaf] at (1.05,1.3) {}; \\node[leaf] at (1.75,1.3) {};\n  \\draw (c2s2) -- (1.05,1.3); \\draw (c2s2) -- (1.75,1.3);\n  \\draw (c2a) -- (c2s1) -- (c2a2) -- (c2s2);\n  \\node[anchor=north, font=\\scriptsize] at (1.4,0.95) {(1-lambda)lambda};\n  % dots\n  \\node at (3.1,2.4) {...};\n  % termination column\n  \\node[an] (t1) at (4.6,3.4) {};\n  \\node[sn] (t2) at (4.6,2.8) {};\n  \\node[leaf] at (4.25,2.4) {}; \\node[leaf] at (4.95,2.4) {};\n  \\draw (t2) -- (4.25,2.4); \\draw (t2) -- (4.95,2.4);\n  \\node[an] (t3) at (4.6,2.3) {};\n  \\node[sn] (t4) at (4.6,1.7) {};\n  \\node[leaf] at (4.25,1.35) {}; \\node[leaf] at (4.95,1.35) {};\n  \\draw (t4) -- (4.25,1.35); \\draw (t4) -- (4.95,1.35);\n  \\node[term] (tT) at (4.6,0.85) {};\n  \\draw (t1) -- (t2) -- (t3) -- (t4) -- (tT);\n  \\node[anchor=north, font=\\scriptsize] at (4.6,0.6) {lambda to the T-t-1};\n  % right labels\n  \\node[anchor=west, font=\\scriptsize] at (5.4,3.4) {St, At};\n  \\node[anchor=west, font=\\scriptsize] at (5.4,2.8) {St+1};\n  \\node[anchor=west, font=\\scriptsize] at (5.4,1.7) {St+2};\n  \\node[anchor=west, font=\\scriptsize] at (5.4,1.0) {ST};\n  % legend\n  \\node[an] (lg1) at (0.2,0.2) {};\n  \\node[anchor=west, font=\\scriptsize] at (0.35,0.2) {selected action};\n  \\node[leaf] (lg2) at (3.0,0.2) {};\n  \\node[anchor=west, font=\\scriptsize] at (3.15,0.2) {unselected, weight pi(a given s)};\n\\end{tikzpicture}\n$$\n\nWritten approximately as a sum of TD errors, the return uses the expectation-based\nerror $\\delta_t^a = R_{t+1} + \\gamma_{t+1}\\bar V_t(S_{t+1}) - \\hat q(S_t, A_t,\n\\mathbf{w}_t)$ and a trace weighted by the target-policy probability of each\nselected action,\n\n$$\n\\mathbf{z}_t \\;\\doteq\\; \\gamma_t \\lambda_t\\, \\pi(A_t \\mid S_t)\\, \\mathbf{z}_{t-1} + \\nabla \\hat q(S_t, A_t, \\mathbf{w}_t).\n$$\n\nCompare the three action-value traces. On-policy Sarsa($\\lambda$) decays by\n$\\gamma_t \\lambda_t$; the importance-sampling off-policy trace decays by $\\gamma_t\n\\lambda_t \\rho_t$; TB($\\lambda$) decays by $\\gamma_t \\lambda_t \\pi(A_t \\mid S_t)$.\nWatkins's Q($\\lambda$) is the crude ancestor of that last form: taking a greedy\naction makes $\\pi(A_t \\mid S_t) = 1$ (no extra decay), and taking a non-greedy\naction makes $\\pi(A_t \\mid S_t) = 0$ (the trace vanishes) — which is precisely the\n\"cut on first non-greedy action\" rule, now recovered as a hard special case of the\nsmooth TB($\\lambda$) weighting. TB($\\lambda$), together with the standard\nsemi-gradient update, is the tree-backup trace algorithm; like every semi-gradient\nmethod here, it is not guaranteed stable under off-policy data with a strong\nfunction approximator.\n\n$$\n% caption: The three action-value traces share the base decay $\\gamma_t \\lambda_t$\n% and differ only in the off-policy factor. Sarsa($\\lambda$) uses none; the\n% importance-sampling trace multiplies by $\\rho_t$, which can spike far above 1;\n% TB($\\lambda$) multiplies by $\\pi(A_t\\mid S_t) \\in [0,1]$, which only ever damps\n% the trace. Watkins's Q($\\lambda$) is TB with a hard 0\u002F1 gate.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize,\n  row\u002F.style={draw, minimum width=30mm, minimum height=9mm, align=center, font=\\scriptsize}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  \\node[row] (a) at (0,2.0) {Sarsa(lambda)};\n  \\node[row] (b) at (0,0.7) {IS o\\\u002Ff\\\u002Ff-policy};\n  \\node[row, draw=acc, text=acc] (c) at (0,-0.6) {Tree-Backup(lambda)};\n  \\node[anchor=west, font=\\scriptsize] at (2.3,2.0) {decay = gamma lambda};\n  \\node[anchor=west, font=\\scriptsize] at (2.3,0.7) {decay = gamma lambda rho};\n  \\node[anchor=west, font=\\scriptsize, text=acc] at (2.3,-0.6) {decay = gamma lambda pi(A:S)};\n  \\node[red, anchor=west, font=\\scriptsize] at (7.0,0.7) {rho can be big: variance};\n  \\node[acc, anchor=west, font=\\scriptsize] at (7.0,-0.6) {pi in 0..1: only damps};\n\\end{tikzpicture}\n$$\n\n## Where this leaves us\n\nTwo generalizations push the trace machinery to its most general form. Variable\n$\\lambda$ and $\\gamma$ turn the constants into functions, unify episodic and\ncontinuing tasks in a single stream, and fold termination into a state-dependent\ndiscount. Control variates fold the importance ratio $\\rho_t$ into the trace and\ndampen the variance it brings, giving an efficient off-policy TD($\\lambda$) and\nExpected-Sarsa($\\lambda$). Watkins's Q($\\lambda$) cuts the trace on the first\nnon-greedy action; Tree-Backup($\\lambda$) replaces that hard cut with a smooth\ntarget-policy weighting and needs no importance sampling at all.\n\nAll of it is still semi-gradient, though, so whenever $\\lambda \u003C 1$ it bootstraps,\nand off-policy plus bootstrapping plus function approximation is the deadly triad —\nthe weights can diverge. The stabilizing methods, and the practical cost of running\ntraces, continue in\n[stable off-policy methods with traces](\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces).\n\n[^sb-varlg]: **Sutton & Barto**, _Reinforcement Learning: An Introduction_ (2nd ed.), §12.8 — Variable $\\lambda$ and $\\gamma$: promoting $\\lambda$ and $\\gamma$ to functions $\\lambda_t = \\lambda(S_t, A_t)$, $\\gamma_t = \\gamma(S_t)$; the generalized return (12.17) with the running product $\\prod \\gamma_i$; the termination function unifying episodic, continuing, and pseudo-terminating prediction; and the recursive generalized $\\lambda$-returns for states (12.18), Sarsa (12.19), and Expected Sarsa (12.20)–(12.21).\n[^sb-cv]: **Sutton & Barto**, §12.9 — Off-policy Traces with Control Variates: the per-decision importance ratio $\\rho_t$ folded into the state-based $\\lambda$-return with the control-variate term (12.22); the TD-error approximation (12.23)–(12.24); the general accumulating trace with importance sampling (12.25); and the action-value off-policy return (12.26)–(12.28) yielding the trace (12.29) and an Expected-Sarsa($\\lambda$).\n[^sb-watkins]: **Sutton & Barto**, §12.10 — Watkins's Q($\\lambda$): decay the trace by $\\gamma\\lambda$ while greedy actions are taken, cut it to zero on the first non-greedy action (backup diagram, Fig. 12.12).\n[^sb-tb]: **Sutton & Barto**, §12.10 — Tree-Backup($\\lambda$), TB($\\lambda$): the eligibility-trace form of $n$-step Tree Backup, requiring no importance sampling; the target-policy-weighted return and the trace $\\mathbf{z}_t = \\gamma_t \\lambda_t \\pi(A_t\\mid S_t)\\mathbf{z}_{t-1} + \\nabla\\hat q$ (backup diagram, Fig. 12.13).\n",{"text":17697,"minutes":17698,"time":17699,"words":17700},"9 min read",8.905,534300,1781,{"title":5,"description":17684},[17703,17705],{"book":16684,"ref":17704},"Ch. 12 — Eligibility Traces; §12.8 Variable λ and γ; §12.9 Off-policy Traces with Control Variates",{"book":16684,"ref":17706},"§12.10 Watkins's Q(λ) to Tree-Backup(λ)","available","07.reinforcement-learning\u002F03.approximation\u002F13.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",[17711],"Approximation","Sekcd6N5gHyldVPdvaL0vwT2M3eIFwZxo8sOGqEbX-w",{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":17714,"\u002Falgorithms\u002Ffoundations\u002Fproof-techniques":17715,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":17716,"\u002Falgorithms\u002Ffoundations\u002Fgrowth-rates-and-loop-analysis":17717,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":17718,"\u002Falgorithms\u002Ffoundations\u002Famortized-analysis":17719,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":17720,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":17721,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":17722,"\u002Falgorithms\u002Fdivide-and-conquer\u002Ffast-multiplication":17723,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":17724,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":17725,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":17726,"\u002Falgorithms\u002Fsorting\u002Fexternal-sorting":17727,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":17728,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":17729,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":17730,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":17731,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":17732,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":17733,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":17734,"\u002Falgorithms\u002Fdata-structures\u002Fspatial-data-structures":17735,"\u002Falgorithms\u002Fdata-structures\u002Fskip-lists-and-probabilistic-structures":17736,"\u002Falgorithms\u002Fdata-structures\u002Fb-trees":17737,"\u002Falgorithms\u002Fdata-structures\u002Fdata-stream-algorithms":17738,"\u002Falgorithms\u002Fdata-structures\u002Fstreaming-sketches":17739,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":17740,"\u002Falgorithms\u002Fsequences\u002Fprefix-sums":17741,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":17742,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":17743,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":17744,"\u002Falgorithms\u002Fsequences\u002Fkmp-and-z-function":17745,"\u002Falgorithms\u002Fsequences\u002Ftries":17746,"\u002Falgorithms\u002Fsequences\u002Fsuffix-arrays-and-aho-corasick":17747,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":17748,"\u002Falgorithms\u002Fgraphs\u002Fdepth-first-search":17749,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":17750,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":17751,"\u002Falgorithms\u002Fgraphs\u002Fkruskal-and-prim":17752,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":17753,"\u002Falgorithms\u002Fgraphs\u002Fall-pairs-and-negative-weights":17754,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":17755,"\u002Falgorithms\u002Fgraphs\u002Fmax-flow-min-cut":17756,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":17757,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":17758,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":17759,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":17760,"\u002Falgorithms\u002Fgraphs\u002Fbipartite-matching":17761,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":17762,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":17763,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":17764,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":17765,"\u002Falgorithms\u002Fgreedy\u002Fstable-matching":17766,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":17767,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":17768,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":17769,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":17770,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":17771,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":17772,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":17773,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":17774,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":17775,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":17776,"\u002Falgorithms\u002Fdynamic-programming\u002Fdigit-and-probability-dp":17777,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":17778,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":17779,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":17780,"\u002Falgorithms\u002Fbacktracking\u002Fgraph-backtracking":17781,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":17782,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":17783,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":17784,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":17785,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmatrix-exponentiation":17786,"\u002Falgorithms\u002Fmathematical-algorithms\u002Ffast-fourier-transform":17787,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fgradient-descent":17788,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":17789,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":17790,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":17791,"\u002Falgorithms\u002Fcomputational-geometry\u002Fpolygons-and-proximity":17792,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":17793,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":17794,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":17795,"\u002Falgorithms\u002Fintractability\u002Fapproximation-algorithms":17796,"\u002Falgorithms":17797,"\u002Fcalculus\u002Flimits-and-continuity\u002Ffunctions-and-models":17798,"\u002Fcalculus\u002Flimits-and-continuity\u002Fthe-limit-of-a-function":17799,"\u002Fcalculus\u002Flimits-and-continuity\u002Flimit-laws-and-the-precise-definition":17800,"\u002Fcalculus\u002Flimits-and-continuity\u002Fcontinuity":17801,"\u002Fcalculus\u002Fderivatives\u002Fthe-derivative-and-rates-of-change":17802,"\u002Fcalculus\u002Fderivatives\u002Fdifferentiation-rules-and-the-chain-rule":17803,"\u002Fcalculus\u002Fderivatives\u002Fimplicit-differentiation-and-related-rates":17804,"\u002Fcalculus\u002Fderivatives\u002Flinear-approximations-and-differentials":17805,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fextrema-and-the-mean-value-theorem":17806,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fhow-derivatives-shape-a-graph":17807,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fcurve-sketching-and-optimization":17808,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fnewtons-method-and-antiderivatives":17809,"\u002Fcalculus\u002Fintegrals\u002Farea-and-the-definite-integral":17810,"\u002Fcalculus\u002Fintegrals\u002Fthe-fundamental-theorem-of-calculus":17811,"\u002Fcalculus\u002Fintegrals\u002Fthe-substitution-rule":17812,"\u002Fcalculus\u002Fapplications-of-integration\u002Fareas-and-volumes":17813,"\u002Fcalculus\u002Fapplications-of-integration\u002Fwork-average-value-and-arc-length":17814,"\u002Fcalculus\u002Fapplications-of-integration\u002Fphysics-economics-and-probability":17815,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Finverse-functions-logarithms-and-exponentials":17816,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Fgrowth-decay-inverse-trig-and-hyperbolic-functions":17817,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Flhospitals-rule":17818,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fintegration-by-parts":17819,"\u002Fcalculus\u002Ftechniques-of-integration\u002Ftrigonometric-integrals-and-substitution":17820,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fpartial-fractions-and-integration-strategy":17821,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fapproximate-and-improper-integrals":17822,"\u002Fcalculus\u002Fparametric-and-polar\u002Fparametric-curves-and-their-calculus":17823,"\u002Fcalculus\u002Fparametric-and-polar\u002Fpolar-coordinates":17824,"\u002Fcalculus\u002Fparametric-and-polar\u002Fconic-sections":17825,"\u002Fcalculus\u002Fsequences-and-series\u002Fsequences":17826,"\u002Fcalculus\u002Fsequences-and-series\u002Fseries-and-the-integral-test":17827,"\u002Fcalculus\u002Fsequences-and-series\u002Fthe-convergence-tests":17828,"\u002Fcalculus\u002Fsequences-and-series\u002Fpower-series":17829,"\u002Fcalculus\u002Fsequences-and-series\u002Ftaylor-and-maclaurin-series":17830,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvectors-and-the-dot-product":17831,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fthe-cross-product-lines-and-planes":17812,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fcylinders-and-quadric-surfaces":17832,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvector-functions-and-space-curves":17833,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Farc-length-curvature-and-motion":17834,"\u002Fcalculus\u002Fpartial-derivatives\u002Ffunctions-of-several-variables":17802,"\u002Fcalculus\u002Fpartial-derivatives\u002Fpartial-derivatives":17835,"\u002Fcalculus\u002Fpartial-derivatives\u002Ftangent-planes-and-the-chain-rule":17836,"\u002Fcalculus\u002Fpartial-derivatives\u002Fdirectional-derivatives-and-the-gradient":17837,"\u002Fcalculus\u002Fpartial-derivatives\u002Foptimization-and-lagrange-multipliers":17838,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fdouble-integrals":17839,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Ftriple-integrals-and-coordinate-systems":17840,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fvector-fields-and-line-integrals":17841,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fgreens-theorem-curl-and-divergence":17842,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fsurface-integrals":17843,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fstokes-and-the-divergence-theorem":17844,"\u002Fcalculus":17845,"\u002Fmechanics\u002Ffoundations\u002Fmeasurement-and-dimensions":17846,"\u002Fmechanics\u002Ffoundations\u002Fvector-algebra":17847,"\u002Fmechanics\u002Fkinematics\u002Fone-dimensional-motion":17848,"\u002Fmechanics\u002Fkinematics\u002Fmotion-graphs":17849,"\u002Fmechanics\u002Fkinematics\u002Fprojectile-motion":17850,"\u002Fmechanics\u002Fkinematics\u002Frelative-motion":17851,"\u002Fmechanics\u002Fkinematics\u002Fcircular-motion":17852,"\u002Fmechanics\u002Fdynamics\u002Fnewtons-laws":17853,"\u002Fmechanics\u002Fdynamics\u002Ffree-body-diagrams":17854,"\u002Fmechanics\u002Fdynamics\u002Ffriction-and-curved-motion":17855,"\u002Fmechanics\u002Fdynamics\u002Fnumerical-dynamics":17856,"\u002Fmechanics\u002Fdynamics\u002Fcenter-of-mass-systems":17857,"\u002Fmechanics\u002Fenergy\u002Fwork-and-kinetic-energy":17858,"\u002Fmechanics\u002Fenergy\u002Fpotential-energy":17859,"\u002Fmechanics\u002Fenergy\u002Fmultiparticle-work":17860,"\u002Fmechanics\u002Fenergy\u002Fmass-energy-and-binding":17861,"\u002Fmechanics\u002Fenergy\u002Fphotons-and-quantization":17862,"\u002Fmechanics\u002Fmomentum\u002Fmomentum-and-collisions":17863,"\u002Fmechanics\u002Fmomentum\u002Fcenter-of-mass-collisions":17864,"\u002Fmechanics\u002Fmomentum\u002Frocket-propulsion":17865,"\u002Fmechanics\u002Frotation\u002Frotational-inertia":17866,"\u002Fmechanics\u002Frotation\u002Frotational-dynamics":17867,"\u002Fmechanics\u002Frotation\u002Frolling-motion":17868,"\u002Fmechanics\u002Frotation\u002Fangular-momentum":17869,"\u002Fmechanics\u002Frotation\u002Frolling-resistance":17870,"\u002Fmechanics\u002Frotation\u002Fgyroscopic-precession":17871,"\u002Fmechanics\u002Fgravity-and-matter\u002Fkeplerian-orbits":17872,"\u002Fmechanics\u002Fgravity-and-matter\u002Fgravitational-fields":17873,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstatic-equilibrium":17874,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-statics":17875,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-flow":17876,"\u002Fmechanics\u002Fgravity-and-matter\u002Forbital-motion":17877,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstress-and-elasticity":17878,"\u002Fmechanics\u002Foscillations-waves\u002Fdamped-oscillators":17879,"\u002Fmechanics\u002Foscillations-waves\u002Ftravelling-waves":17880,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-superposition":17881,"\u002Fmechanics\u002Foscillations-waves\u002Fstanding-waves":17882,"\u002Fmechanics\u002Foscillations-waves\u002Fsound-waves":17883,"\u002Fmechanics\u002Foscillations-waves\u002Fdoppler-effect":17884,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-packets":17885,"\u002Fmechanics\u002Foscillations-waves\u002Fbeats-and-coupling":17886,"\u002Fmechanics\u002Foscillations-waves\u002Fsimple-harmonic-motion":17887,"\u002Fmechanics\u002Foscillations-waves\u002Fpendulum-motion":17888,"\u002Fmechanics\u002Foscillations-waves\u002Fdriven-oscillators":17889,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-boundaries":17890,"\u002Fmechanics\u002Fthermodynamics\u002Fkinetic-theory-of-ideal-gases":17891,"\u002Fmechanics\u002Fthermodynamics\u002Ffirst-law-of-thermodynamics":17892,"\u002Fmechanics\u002Fthermodynamics\u002Fentropy-and-the-second-law":17893,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-processes":17894,"\u002Fmechanics\u002Fthermodynamics\u002Fphase-changes":17895,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-machines":17896,"\u002Fmechanics":17897,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcharge-and-conductors":17898,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcoulombs-law":17899,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-and-force":17900,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-maps":17901,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-dipoles":17902,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fcontinuous-charge-fields":17903,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fgauss-law-and-conductors":17904,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpoint-charge-potential":17905,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpotential-gradients-and-equipotentials":17906,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Felectrostatic-energy-and-pressure":17907,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Flaplace-boundary-problems":17908,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fcontinuous-charge-potentials":17909,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitance-fundamentals":17886,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-networks":17910,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-energy-and-force":17911,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fdielectric-polarization-and-breakdown":17912,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fcurrent-and-resistance":17882,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fkirchhoff-network-analysis":17747,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Frc-transients":17913,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-trajectories":17873,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fhall-effect":17914,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-force-on-conductors":17915,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-dipoles":17916,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmass-spectrometry":17917,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmoving-charge-fields":17918,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fbiot-savart-law":17919,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fcircular-current-loops":17920,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Famperes-law":17921,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fgauss-law-for-magnetism":17922,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmagnetic-materials":17847,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-flux":17923,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Ffaradays-law":17924,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Flenzs-law":17925,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmotional-emf":17926,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Feddy-currents":17927,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fself-inductance":17928,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-energy":17929,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Frl-circuits":17930,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-fundamentals":17865,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Freactance":17864,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Frlc-resonance":17931,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-power":17932,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Ftransformers":17933,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdisplacement-current":17934,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-waves":17935,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-momentum":17936,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdipole-radiation":17937,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fpolarization":17938,"\u002Felectricity-and-magnetism\u002Foptics\u002Freflection-and-refraction":17939,"\u002Felectricity-and-magnetism\u002Foptics\u002Fthin-lenses":17891,"\u002Felectricity-and-magnetism\u002Foptics\u002Fspherical-mirrors":17889,"\u002Felectricity-and-magnetism":17940,"\u002Flinear-algebra\u002Flinear-systems\u002Fsystems-and-echelon-forms":17941,"\u002Flinear-algebra\u002Flinear-systems\u002Fvector-and-matrix-equations":17942,"\u002Flinear-algebra\u002Flinear-systems\u002Fsolution-sets-and-applications":17943,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-independence":17944,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-transformations":17945,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-operations":17946,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-inverse-and-invertibility":17947,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fpartitioned-matrices-and-lu":17948,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fsubspaces-dimension-rank":17949,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fapplications-leontief-and-graphics":17799,"\u002Flinear-algebra\u002Fdeterminants\u002Fdeterminants-and-cofactors":17950,"\u002Flinear-algebra\u002Fdeterminants\u002Fproperties-of-determinants":17951,"\u002Flinear-algebra\u002Fdeterminants\u002Fcramer-volume-and-area":17803,"\u002Flinear-algebra\u002Fvector-spaces\u002Fvector-spaces-and-subspaces":17952,"\u002Flinear-algebra\u002Fvector-spaces\u002Fnull-and-column-spaces":17953,"\u002Flinear-algebra\u002Fvector-spaces\u002Fbases-and-independent-sets":17954,"\u002Flinear-algebra\u002Fvector-spaces\u002Fcoordinate-systems":17955,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdimension-and-rank":17956,"\u002Flinear-algebra\u002Fvector-spaces\u002Fchange-of-basis":17957,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdifference-equations-and-markov":17958,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-eigenvalues":17959,"\u002Flinear-algebra\u002Feigenvalues\u002Fthe-characteristic-equation":17960,"\u002Flinear-algebra\u002Feigenvalues\u002Fdiagonalization":17961,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-linear-transformations":17962,"\u002Flinear-algebra\u002Feigenvalues\u002Fcomplex-eigenvalues":17963,"\u002Flinear-algebra\u002Feigenvalues\u002Fdynamical-systems":17964,"\u002Flinear-algebra\u002Feigenvalues\u002Fpower-method":17965,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-length-orthogonality":17966,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Forthogonal-sets-and-projections":17967,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fgram-schmidt-and-qr":17968,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-problems":17969,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-applications":17970,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-spaces":17971,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fdiagonalizing-symmetric-matrices":17838,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fquadratic-forms":17972,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fconstrained-optimization":17973,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsingular-value-decomposition":17974,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsvd-applications-pca-imaging":17975,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-thinking-and-matrix-computation":17976,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Flu-and-cholesky":17977,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fconditioning-and-floating-point":17978,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fstability-and-error-analysis":17979,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fqr-and-numerical-least-squares":17980,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-eigenvalues-and-svd":17981,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-combinations":17982,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-independence-and-barycentric-coordinates":17983,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fconvex-combinations-and-convex-sets":17984,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fhyperplanes-and-polytopes":17985,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fcurves-and-surfaces":17986,"\u002Flinear-algebra":17987,"\u002Ftheory-of-computation":17988,"\u002Fcomputer-architecture\u002Ffoundations\u002Fbits-bytes-and-words":17989,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-representation":17990,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-arithmetic":17991,"\u002Fcomputer-architecture\u002Ffoundations\u002Ffloating-point":17992,"\u002Fcomputer-architecture\u002Ffoundations\u002Fboolean-algebra-and-bit-manipulation":17993,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fthe-machines-view":17994,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fdata-movement":17995,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farithmetic-and-logic":17996,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fcontrol-flow":17997,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fprocedures":17998,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farrays-structs-and-alignment":17999,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fmemory-layout-and-buffer-overflows":18000,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fwhat-an-isa-is":18001,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Finstruction-formats-and-operands":18002,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Faddressing-modes":18003,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fthe-y86-64-instruction-set":18004,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fy86-64-programming":18005,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Ftransistors-gates-and-boolean-functions":18006,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fcombinational-logic-and-hcl":18007,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmultiplexers-decoders-and-the-alu":18008,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmemory-elements-latches-flip-flops-and-clocking":18009,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fregister-files-and-random-access-memory":18010,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-fetch-decode-execute-cycle":18011,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-seq-stages":18012,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fcontrol-logic-and-sequencing":18013,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fassembling-seq":18014,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Ftracing-a-program":18015,"\u002Fcomputer-architecture\u002Fpipelining\u002Fpipelining-principles":18016,"\u002Fcomputer-architecture\u002Fpipelining\u002Ffrom-seq-to-pipe":18017,"\u002Fcomputer-architecture\u002Fpipelining\u002Fdata-hazards-stalling-and-forwarding":18018,"\u002Fcomputer-architecture\u002Fpipelining\u002Fcontrol-hazards-and-branch-prediction":18019,"\u002Fcomputer-architecture\u002Fpipelining\u002Fthe-complete-pipe-processor":18020,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fstorage-technologies-and-the-latency-gap":18021,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Flocality":18022,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-memories-direct-mapped":18023,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fset-associative-and-write-policies":18024,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-performance-and-cache-friendly-code":18025,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Faddress-spaces-and-translation":18026,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fpage-tables-and-page-faults":18027,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fthe-tlb-and-multi-level-page-tables":18028,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Fexceptional-control-flow":18029,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Finterrupts-and-the-kernel":18030,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fprocesses-threads-and-parallelism":18031,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fhardware-multithreading":18032,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fcache-coherence":18033,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmemory-consistency-and-synchronization":18034,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmulticore-organization":18035,"\u002Fcomputer-architecture\u002Fcapstone\u002Fthe-whole-machine":18036,"\u002Fcomputer-architecture\u002Fcapstone\u002Fassembling-a-complete-cpu":18037,"\u002Fcomputer-architecture":17988,"\u002Fdifferential-equations\u002Ffoundations\u002Fmodels-and-direction-fields":18038,"\u002Fdifferential-equations\u002Ffoundations\u002Fclassification-and-terminology":18039,"\u002Fdifferential-equations\u002Ffirst-order\u002Flinear-first-order-integrating-factors":18040,"\u002Fdifferential-equations\u002Ffirst-order\u002Fseparable-and-exact":17803,"\u002Fdifferential-equations\u002Ffirst-order\u002Fmodeling-first-order":18041,"\u002Fdifferential-equations\u002Ffirst-order\u002Fautonomous-and-population-dynamics":17802,"\u002Fdifferential-equations\u002Ffirst-order\u002Fexistence-uniqueness-euler":17809,"\u002Fdifferential-equations\u002Ffirst-order\u002Ffirst-order-difference-equations":18042,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhomogeneous-constant-coefficients":18043,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fcomplex-and-repeated-roots":17843,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fnonhomogeneous-undetermined-coefficients":18044,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fvariation-of-parameters":18045,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fmechanical-electrical-vibrations":18046,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhigher-order-linear":18047,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fpower-series-ordinary-points":18048,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fregular-singular-frobenius":18049,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fbessel-and-special-functions":18050,"\u002Fdifferential-equations\u002Flaplace\u002Flaplace-definition-ivps":18051,"\u002Fdifferential-equations\u002Flaplace\u002Fstep-impulse-convolution":18052,"\u002Fdifferential-equations\u002Fsystems\u002Fmatrices-eigenvalues-review":18053,"\u002Fdifferential-equations\u002Fsystems\u002Fconstant-coefficient-systems-phase-portraits":18054,"\u002Fdifferential-equations\u002Fsystems\u002Frepeated-eigenvalues-fundamental-matrices":18055,"\u002Fdifferential-equations\u002Fnumerical\u002Feuler-and-runge-kutta":18050,"\u002Fdifferential-equations\u002Fnumerical\u002Fmultistep-systems-stability":18056,"\u002Fdifferential-equations\u002Fnonlinear\u002Fphase-plane-autonomous-stability":18057,"\u002Fdifferential-equations\u002Fnonlinear\u002Flocally-linear-and-liapunov":18058,"\u002Fdifferential-equations\u002Fnonlinear\u002Fcompeting-species-predator-prey-limit-cycles":18059,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Ffourier-series":18060,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fheat-wave-laplace-equations":18061,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fsturm-liouville":18062,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fcalculus-of-variations":18063,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fhistorical-notes":18064,"\u002Fdifferential-equations":18065,"\u002Frelativity\u002Ffoundations\u002Fspecial-relativity-postulates":18066,"\u002Frelativity\u002Ffoundations\u002Florentz-transformation-spacetime":18067,"\u002Frelativity\u002Ffoundations\u002Ftime-dilation-length-contraction":18068,"\u002Frelativity\u002Ffoundations\u002Frelativistic-momentum-energy":18069,"\u002Frelativity\u002Ffoundations\u002Fgeneral-relativity":17948,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fminkowski-spacetime-and-the-interval":18070,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Ffour-vectors-and-index-notation":18071,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fthe-lorentz-group-and-rapidity":18072,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fdoppler-aberration-and-appearance":18073,"\u002Frelativity\u002Frelativistic-dynamics\u002Ffour-momentum-force-and-accelerated-motion":18074,"\u002Frelativity\u002Frelativistic-dynamics\u002Fparticle-decays-and-two-body-kinematics":18075,"\u002Frelativity\u002Frelativistic-dynamics\u002Fcollisions-thresholds-and-the-cm-frame":17818,"\u002Frelativity\u002Frelativistic-dynamics\u002Fmandelstam-variables-and-invariants":18076,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ffour-current-and-the-four-potential":18077,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fthe-electromagnetic-field-tensor":18078,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ftransformation-of-electric-and-magnetic-fields":18079,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fcovariant-maxwell-and-the-stress-energy-tensor":18080,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-equivalence-principle-formalized":18081,"\u002Frelativity\u002Fcurved-spacetime\u002Fmanifolds-vectors-and-the-metric":18082,"\u002Frelativity\u002Fcurved-spacetime\u002Fcovariant-derivative-and-christoffel-symbols":18083,"\u002Frelativity\u002Fcurved-spacetime\u002Fgeodesics-and-the-geodesic-equation":18084,"\u002Frelativity\u002Fcurved-spacetime\u002Fcurvature-riemann-and-geodesic-deviation":18085,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-einstein-field-equations":18040,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fthe-schwarzschild-metric":18086,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fgeodesics-and-orbits-in-schwarzschild":18087,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Flight-bending-and-null-geodesics":18088,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fperihelion-precession-of-mercury":18089,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fdeflection-of-light-and-gravitational-lensing":18090,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fgravitational-redshift-and-shapiro-delay":17969,"\u002Frelativity\u002Ftests-of-general-relativity\u002Frelativity-in-technology-gps":18091,"\u002Frelativity\u002Fblack-holes\u002Fhorizons-and-coordinate-singularities":18092,"\u002Frelativity\u002Fblack-holes\u002Frotating-and-charged-black-holes":17979,"\u002Frelativity\u002Fblack-holes\u002Fblack-hole-thermodynamics":18093,"\u002Frelativity\u002Fgravitational-waves\u002Flinearized-gravity-and-wave-solutions":18094,"\u002Frelativity\u002Fgravitational-waves\u002Fgeneration-and-the-quadrupole-formula":18095,"\u002Frelativity\u002Fgravitational-waves\u002Fdetection-ligo-and-the-first-events":18096,"\u002Frelativity\u002Fcosmological-bridge\u002Fthe-cosmological-principle-and-flrw-metric":18097,"\u002Frelativity\u002Fcosmological-bridge\u002Ffriedmann-equations-and-cosmic-dynamics":18098,"\u002Frelativity":18099,"\u002Fphysical-computing":17988,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fblackbody-radiation-and-the-planck-quantum":18100,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-photoelectric-effect-and-the-photon":18079,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fx-rays-and-the-compton-effect":18101,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-old-quantum-theory-bohr-and-sommerfeld":18102,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fde-broglie-waves-and-electron-diffraction":18103,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fwave-packets-and-the-probability-interpretation":18104,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fthe-uncertainty-principle":18105,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-schrodinger-equation-in-one-dimension":18106,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-free-particle-and-wave-packet-dynamics":18107,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fparticle-in-infinite-and-finite-square-wells":18055,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Foperators-expectation-values-and-the-harmonic-oscillator":17979,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-dirac-delta-potential":18108,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fbarrier-penetration-and-quantum-tunneling":18109,"\u002Fquantum-mechanics\u002Fformalism\u002Fhilbert-space-and-dirac-notation":18110,"\u002Fquantum-mechanics\u002Fformalism\u002Fobservables-hermitian-operators-and-eigenvalues":18111,"\u002Fquantum-mechanics\u002Fformalism\u002Fthe-postulates-and-quantum-measurement":18107,"\u002Fquantum-mechanics\u002Fformalism\u002Fposition-momentum-and-continuous-spectra":18112,"\u002Fquantum-mechanics\u002Fformalism\u002Fcommutators-and-the-generalized-uncertainty-principle":18087,"\u002Fquantum-mechanics\u002Fformalism\u002Ftime-evolution-schrodinger-and-heisenberg-pictures":17840,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fladder-operators-and-the-number-states":18113,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fcoherent-and-squeezed-states":18114,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fsymmetries-generators-and-conservation-laws":17820,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fparity-time-reversal-and-discrete-symmetries":18115,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Forbital-angular-momentum-and-spherical-harmonics":18116,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Fthe-angular-momentum-algebra":18117,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Faddition-of-angular-momenta-and-clebsch-gordan":18118,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-schrodinger-equation-in-three-dimensions":18119,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-hydrogen-atom":18120,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-isotropic-oscillator-and-hidden-symmetry":18121,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-half-pauli-matrices-and-stern-gerlach":18122,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-in-a-magnetic-field-precession-and-resonance":18123,"\u002Fquantum-mechanics\u002Fspin\u002Ftwo-level-systems-and-the-bloch-sphere":18079,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fidentical-particles-and-exchange-symmetry":18124,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fthe-pauli-principle-atoms-and-the-periodic-table":18125,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ftime-independent-perturbation-theory":18126,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ffine-structure-and-the-real-hydrogen-atom":18113,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-zeeman-and-stark-effects":17801,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-variational-method":18127,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-wkb-approximation":18128,"\u002Fquantum-mechanics":18129,"\u002Freal-analysis\u002Ffoundations\u002Fsets-logic-functions":18063,"\u002Freal-analysis\u002Ffoundations\u002Fordered-fields-completeness":18130,"\u002Freal-analysis\u002Ffoundations\u002Fabsolute-value-bounds":18131,"\u002Freal-analysis\u002Ffoundations\u002Fintervals-uncountability":17952,"\u002Freal-analysis\u002Fsequences-series\u002Fsequences-limits":18132,"\u002Freal-analysis\u002Fsequences-series\u002Flimit-laws-monotone":17838,"\u002Freal-analysis\u002Fsequences-series\u002Flimsup-bolzano-weierstrass":18133,"\u002Freal-analysis\u002Fsequences-series\u002Fcauchy-completeness":18134,"\u002Freal-analysis\u002Fsequences-series\u002Fseries-convergence":17975,"\u002Freal-analysis\u002Fsequences-series\u002Fabsolute-conditional-rearrangement":18085,"\u002Freal-analysis\u002Fmetric-spaces\u002Fmetric-spaces-norms":18135,"\u002Freal-analysis\u002Fmetric-spaces\u002Fopen-closed-sets":18136,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconvergence-completeness":18137,"\u002Freal-analysis\u002Fmetric-spaces\u002Fcompactness":18138,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconnectedness":18139,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-of-functions":18112,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuous-functions":18140,"\u002Freal-analysis\u002Fcontinuity\u002Fevt-ivt":17945,"\u002Freal-analysis\u002Fcontinuity\u002Funiform-continuity":18141,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuity-metric-spaces":18142,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-infinity-monotone":17799,"\u002Freal-analysis\u002Fdifferentiation\u002Fthe-derivative":18143,"\u002Freal-analysis\u002Fdifferentiation\u002Fmean-value-theorem":18144,"\u002Freal-analysis\u002Fdifferentiation\u002Ftaylors-theorem":18101,"\u002Freal-analysis\u002Fdifferentiation\u002Finverse-function-1d":17828,"\u002Freal-analysis\u002Friemann-integration\u002Fdarboux-integral":17979,"\u002Freal-analysis\u002Friemann-integration\u002Fintegrability-classes":18145,"\u002Freal-analysis\u002Friemann-integration\u002Fproperties-of-the-integral":18146,"\u002Freal-analysis\u002Friemann-integration\u002Ffundamental-theorem":17964,"\u002Freal-analysis\u002Friemann-integration\u002Flog-exp-improper":18089,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpointwise-uniform-convergence":18147,"\u002Freal-analysis\u002Ffunction-sequences\u002Finterchange-of-limits":18148,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpower-series-weierstrass":18149,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpicard-ode":17984,"\u002Freal-analysis\u002Fseveral-variables\u002Fdifferentiability-rn":18150,"\u002Freal-analysis\u002Fseveral-variables\u002Fgradient-chain-rule":18151,"\u002Freal-analysis\u002Fseveral-variables\u002Fhigher-derivatives-taylor-extrema":18152,"\u002Freal-analysis\u002Fseveral-variables\u002Finverse-implicit-theorems":18152,"\u002Freal-analysis\u002Fseveral-variables\u002Fmultiple-integrals":18153,"\u002Freal-analysis":18154,"\u002Fabstract-algebra\u002Ffoundations\u002Fsets-functions-relations":18155,"\u002Fabstract-algebra\u002Ffoundations\u002Fintegers-and-modular-arithmetic":18156,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fgroup-axioms-and-first-examples":18157,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fdihedral-and-symmetric-groups":18158,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fmatrix-and-quaternion-groups":18159,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fhomomorphisms-and-group-actions":18160,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fsubgroups-and-substructures":18161,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcyclic-groups":18162,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fgeneration-and-subgroup-lattices":18163,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcosets-lagrange-and-normal-subgroups":18164,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fisomorphism-theorems":18128,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcomposition-series-and-the-alternating-group":18165,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Factions-and-cayleys-theorem":18157,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fconjugation-and-the-class-equation":18074,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fsylow-theorems":18166,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fautomorphisms-and-simple-groups":18167,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fdirect-products-and-finite-abelian-groups":18168,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fsemidirect-products":18169,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fnilpotent-and-solvable-groups":18170,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fclassifying-small-groups":18171,"\u002Fabstract-algebra\u002Fring-theory\u002Frings-definitions-and-examples":18172,"\u002Fabstract-algebra\u002Fring-theory\u002Fideals-quotients-and-homomorphisms":18173,"\u002Fabstract-algebra\u002Fring-theory\u002Ffractions-and-the-chinese-remainder-theorem":18167,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Feuclidean-domains-pids-ufds":18174,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fpolynomial-rings-over-fields":18143,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fgauss-lemma-and-unique-factorization":18175,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Firreducibility-criteria-and-groebner":17700,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fintroduction-to-modules":18176,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ffree-modules-and-direct-sums":18177,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ftensor-products-and-exact-sequences":18178,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fvector-spaces-and-linear-maps":18179,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fstructure-theorem-over-pids":18180,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Frational-canonical-form":18181,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fjordan-canonical-form":18182,"\u002Fabstract-algebra\u002Ffield-theory\u002Ffield-extensions-and-algebraic-elements":18183,"\u002Fabstract-algebra\u002Ffield-theory\u002Fstraightedge-and-compass-constructions":17817,"\u002Fabstract-algebra\u002Ffield-theory\u002Fsplitting-fields-and-algebraic-closure":18184,"\u002Fabstract-algebra\u002Ffield-theory\u002Fseparable-and-cyclotomic-extensions":18185,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fthe-galois-correspondence":18059,"\u002Fabstract-algebra\u002Fgalois-theory\u002Ffinite-fields":18186,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fcyclotomic-and-abelian-extensions":18187,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fgalois-groups-of-polynomials":18116,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fsolvability-by-radicals-and-the-quintic":18187,"\u002Fabstract-algebra\u002Fcapstone\u002Fcommutative-algebra-and-algebraic-geometry":18188,"\u002Fabstract-algebra\u002Fcapstone\u002Frepresentation-and-character-theory":18189,"\u002Fabstract-algebra":18190,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fatomic-spectra-rutherford":18191,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-model-hydrogen":18192,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fx-ray-spectra-franck-hertz":18193,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-sommerfeld-old-quantum-theory":18194,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fold-quantum-theory-limits-wkb":18195,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fschrodinger-3d-hydrogen":18108,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fhydrogen-wave-functions":18196,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fradial-equation-in-full":18197,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fsymmetry-degeneracy-runge-lenz":18198,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fexpectation-values-virial":18199,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fquantum-defects-alkali-spectra":18200,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Frydberg-atoms":18201,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Frelativistic-kinetic-correction":18202,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fspin-orbit-thomas-precession":17942,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdarwin-term-fine-structure-formula":18071,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdirac-equation-hydrogen":17817,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Flamb-shift-qed":18203,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fhyperfine-structure-21cm":17801,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fnuclear-effects-isotope-shift":18204,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fperiodic-table-atomic-spectra":18205,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fcentral-field-self-consistent":17839,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fidentical-particles-hartree-fock":18133,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhelium-two-electron-atom":18206,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fls-jj-coupling-term-symbols":18207,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhund-rules-ground-terms":18208,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fzeeman-effect":18209,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fpaschen-back-intermediate":18210,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fstark-effect-polarizability":18211,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Ftime-dependent-perturbation-golden-rule":18212,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fdipole-approximation-einstein-coefficients":18213,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fselection-rules-forbidden-transitions":18214,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Flifetimes-and-line-shapes":18215,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Flaser-principles":18216,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fspectroscopy-techniques":18217,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fline-catalog-nist-asd":18218,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Flaser-cooling-doppler":18219,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fsub-doppler-trapping":18164,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fbose-einstein-condensation":18220,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Foptical-clocks-precision":18221,"\u002Fatomic-physics":18222,"\u002Fdatabases":17988,"\u002Fcategory-theory\u002Ffoundations\u002Fwhat-is-a-category":18223,"\u002Fcategory-theory\u002Ffoundations\u002Fexamples-of-categories":18224,"\u002Fcategory-theory\u002Ffoundations\u002Fspecial-morphisms":18225,"\u002Fcategory-theory\u002Ffoundations\u002Ffunctors":18155,"\u002Fcategory-theory\u002Ffoundations\u002Fnatural-transformations":18226,"\u002Fcategory-theory\u002Ffoundations\u002Fsize-and-set-theory":18227,"\u002Fcategory-theory\u002Funiversal-properties\u002Funiversal-properties":18228,"\u002Fcategory-theory\u002Funiversal-properties\u002Fproducts-and-coproducts":18229,"\u002Fcategory-theory\u002Funiversal-properties\u002Fconstructions-on-categories":18230,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Frepresentable-functors":18231,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-lemma":18232,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-consequences":18233,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits":18234,"\u002Fcategory-theory\u002Flimits-colimits\u002Fproducts-equalizers-pullbacks":18235,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcolimits":18236,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcomputing-limits":18237,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits-and-functors":18238,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions":18239,"\u002Fcategory-theory\u002Fadjunctions\u002Funits-and-counits":18240,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions-via-universal-arrows":18241,"\u002Fcategory-theory\u002Fadjunctions\u002Ffree-forgetful-adjunctions":18242,"\u002Fcategory-theory\u002Fadjoints-limits\u002Flimits-via-adjoints":18243,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fpresheaf-limits-colimits":18244,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoints-preserve-limits":18245,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoint-functor-theorem":18236,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fmonads":18246,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-eilenberg-moore":18247,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fkleisli-and-programming":18248,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-for-endofunctors":18249,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Fcartesian-closed-categories":18250,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Flambda-calculus-correspondence":18195,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Ffixed-points-and-recursion":18251,"\u002Fcategory-theory":18252,"\u002Fdeep-learning\u002Fmathematical-background\u002Flinear-algebra-for-deep-learning":18253,"\u002Fdeep-learning\u002Fmathematical-background\u002Fprobability-and-information-theory":18254,"\u002Fdeep-learning\u002Fmathematical-background\u002Fnumerical-computation":18255,"\u002Fdeep-learning\u002Fmathematical-background\u002Fcalculus":18256,"\u002Fdeep-learning\u002Ffoundations\u002Fwhat-is-deep-learning":18257,"\u002Fdeep-learning\u002Ffoundations\u002Fmachine-learning-refresher":18258,"\u002Fdeep-learning\u002Ffoundations\u002Flinear-models-and-the-perceptron":18217,"\u002Fdeep-learning\u002Fneural-networks\u002Fthe-multilayer-perceptron":18259,"\u002Fdeep-learning\u002Fneural-networks\u002Factivation-functions":18260,"\u002Fdeep-learning\u002Fneural-networks\u002Funiversal-approximation":18261,"\u002Fdeep-learning\u002Fneural-networks\u002Fbackpropagation":18262,"\u002Fdeep-learning\u002Fneural-networks\u002Floss-functions-and-output-units":18263,"\u002Fdeep-learning\u002Foptimization\u002Fgradient-descent-and-sgd":18264,"\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods":18265,"\u002Fdeep-learning\u002Foptimization\u002Finitialization":18266,"\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape":18267,"\u002Fdeep-learning\u002Foptimization\u002Fsecond-order-and-approximate-methods":18268,"\u002Fdeep-learning\u002Fregularization\u002Fregularization-overview":18269,"\u002Fdeep-learning\u002Fregularization\u002Fdropout-and-data-augmentation":18270,"\u002Fdeep-learning\u002Fregularization\u002Fearly-stopping-and-parameter-sharing":18271,"\u002Fdeep-learning\u002Fregularization\u002Fnormalization":18272,"\u002Fdeep-learning\u002Farchitectures\u002Fconvolutional-networks":18273,"\u002Fdeep-learning\u002Farchitectures\u002Fcnn-architectures":18274,"\u002Fdeep-learning\u002Farchitectures\u002Frecurrent-networks":18275,"\u002Fdeep-learning\u002Farchitectures\u002Flstm-and-gru":18276,"\u002Fdeep-learning\u002Farchitectures\u002Fattention-and-transformers":18277,"\u002Fdeep-learning\u002Farchitectures\u002Fthe-transformer-architecture":18278,"\u002Fdeep-learning\u002Farchitectures\u002Ftransformers-in-practice":18279,"\u002Fdeep-learning\u002Farchitectures\u002Fgraph-neural-networks":18280,"\u002Fdeep-learning\u002Farchitectures\u002Fstate-space-models":18281,"\u002Fdeep-learning\u002Ftheory\u002Fgeneralization-theory":18282,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-robustness":18283,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-defenses":18284,"\u002Fdeep-learning\u002Ftheory\u002Fbayesian-and-ensemble-methods":18285,"\u002Fdeep-learning\u002Ftheory\u002Fdeep-equilibrium-models":18229,"\u002Fdeep-learning\u002Fgenerative-models\u002Flinear-factor-models":18286,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoencoders":18287,"\u002Fdeep-learning\u002Fgenerative-models\u002Fvariational-autoencoders":18288,"\u002Fdeep-learning\u002Fgenerative-models\u002Fgenerative-adversarial-networks":18289,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoregressive-and-normalizing-flows":18290,"\u002Fdeep-learning\u002Fgenerative-models\u002Fenergy-based-and-boltzmann-machines":18291,"\u002Fdeep-learning\u002Fgenerative-models\u002Fdiffusion-and-score-based-models":18292,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fstructured-probabilistic-models":17782,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fmonte-carlo-and-mcmc":18293,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fapproximate-inference":18294,"\u002Fdeep-learning\u002Fpractical\u002Fpractical-methodology":18018,"\u002Fdeep-learning\u002Fpractical\u002Fhyperparameters-and-debugging":18295,"\u002Fdeep-learning\u002Fpractical\u002Frepresentation-learning":18296,"\u002Fdeep-learning\u002Fpractical\u002Ftransfer-learning":18297,"\u002Fdeep-learning\u002Fpractical\u002Fapplications":18298,"\u002Fdeep-learning\u002Fpractical\u002Fmodel-compression-and-distillation":18299,"\u002Fdeep-learning\u002Fpractical\u002Fmeta-learning-and-few-shot":18300,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models":18301,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment":18302,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fseq2seq-pretraining-and-bart":18303,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ftext-to-text-transfer-and-conditional-generation":18304,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fspeech-and-audio-models":18305,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fself-supervised-speech-and-synthesis":18306,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents":17991,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fagent-memory-retrieval-and-orchestration":18307,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmixture-of-experts":18308,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmultimodal-models":18309,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ffusion-and-vision-language-models":18310,"\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning":18028,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fmodel-free-prediction-and-control":18311,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fdeep-q-networks":18312,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic":18313,"\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback":18314,"\u002Fdeep-learning":17988,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fequilibrium-state-variables-zeroth-law":18315,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Ffirst-law-heat-and-work":18113,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fsecond-law-entropy-and-the-carnot-bound":18316,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fthermodynamic-potentials-and-maxwell-relations":18317,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fstability-response-functions-and-the-third-law":18318,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fclassical-statistics-and-equipartition":18319,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fphase-space-and-liouvilles-theorem":18320,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fensembles-and-the-equal-probability-postulate":18321,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fstatistical-entropy-boltzmann-and-gibbs":18322,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fmicrocanonical-ensemble-and-entropy":18323,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fequilibrium-conditions-temperature-pressure-chemical-potential":18160,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fideal-gas-phase-space-and-the-sackur-tetrode-entropy":18324,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Ftwo-state-systems-paramagnets-and-negative-temperature":18325,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fcanonical-ensemble-and-the-boltzmann-distribution":18326,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fpartition-function-and-the-helmholtz-free-energy":18047,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fenergy-fluctuations-and-ensemble-equivalence":17840,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fthe-einstein-solid-and-harmonic-systems":18327,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fparamagnetism-and-the-schottky-anomaly":18328,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fideal-gas-partition-function-and-the-gibbs-paradox":18329,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fequipartition-and-the-virial-theorem":17974,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fmolecular-gases-rotation-and-vibration":18330,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fgrand-canonical-ensemble-and-the-grand-partition-function":18331,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fchemical-potential-fugacity-and-number-fluctuations":18052,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fensemble-summary-and-the-thermodynamic-web":17979,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fquantum-statistics-bose-einstein-and-fermi-dirac":18332,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fderiving-the-quantum-distributions":17948,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fthe-classical-limit-and-quantum-concentration":17818,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fideal-quantum-gases-general-framework":17834,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-and-the-fermion-gas":18333,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthe-photon-gas-and-plancks-radiation-law":18334,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fblackbody-thermodynamics-and-radiation-pressure":18335,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fphonons-and-the-debye-model":17966,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-derived":18336,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthermodynamics-of-the-bose-gas-and-superfluidity":17828,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fthe-ideal-fermi-gas-at-zero-temperature":18337,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fsommerfeld-expansion-and-electrons-in-metals":18338,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":18339,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fneutron-stars-and-nuclear-matter":18340,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-cluster-expansion-and-virial-coefficients":18162,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-van-der-waals-gas-and-liquid-gas-coexistence":18341,"\u002Fstatistical-mechanics\u002Finteractions\u002Fquantum-gases-with-interactions-and-exchange":18342,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fphases-coexistence-and-classification":18343,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-ising-model-and-exact-solutions":18344,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fmean-field-theory-and-the-weiss-model":18108,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fcritical-exponents-and-landau-theory":18345,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-renormalization-group-idea":18086,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fthermodynamic-fluctuations-and-response":18346,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fbrownian-motion-and-the-langevin-equation":17947,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Flinear-response-and-the-fluctuation-dissipation-theorem":18347,"\u002Fstatistical-mechanics":18348,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fbonding-mechanisms":18349,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fmolecular-orbitals-and-h2-plus":17813,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fhydrogen-molecule-and-exchange":18073,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fvan-der-waals-forces":18350,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Frotational-vibrational-spectra":18351,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Fanharmonicity-and-rovibrational-structure":18352,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Framan-and-electronic-bands":18353,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Flasers-and-masers":18354,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fstructure-of-solids":18355,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fbravais-lattices-and-crystal-systems":17946,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Freciprocal-lattice-and-brillouin-zones":18356,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fdiffraction-and-structure-factors":18357,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonon-dispersion":18358,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonons-quantization-and-dos":18359,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fdebye-einstein-heat-capacity":18089,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fanharmonicity-and-thermal-transport":18360,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ffree-electron-gas-and-conduction":18361,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fsommerfeld-model-and-heat-capacity":18362,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ftransport-and-the-hall-effect":18363,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fscreening-and-plasmons":18364,"\u002Fcondensed-matter\u002Fband-theory\u002Fblochs-theorem-and-energy-bands":18144,"\u002Fcondensed-matter\u002Fband-theory\u002Fnearly-free-electron-model":18088,"\u002Fcondensed-matter\u002Fband-theory\u002Ftight-binding-method":18365,"\u002Fcondensed-matter\u002Fband-theory\u002Ffermi-surfaces-and-semiclassical-dynamics":18366,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fsemiconductor-bands-and-junctions":18367,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fintrinsic-and-extrinsic-semiconductors":18368,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fcarrier-transport-and-recombination":18079,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fthe-pn-junction":18369,"\u002Fcondensed-matter\u002Fsemiconductors\u002Ftransistors-and-optoelectronics":18370,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fdielectrics-and-polarization":18315,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fferroelectrics-and-piezoelectrics":18215,"\u002Fcondensed-matter\u002Fmagnetism\u002Fdiamagnetism-and-paramagnetism":17943,"\u002Fcondensed-matter\u002Fmagnetism\u002Fexchange-and-ferromagnetism":18371,"\u002Fcondensed-matter\u002Fmagnetism\u002Fantiferromagnetism-and-domains":17809,"\u002Fcondensed-matter\u002Fmagnetism\u002Fspin-waves-and-magnons":18372,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fsuperconductivity-phenomenology":18373,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Flondon-theory-and-the-meissner-effect":17954,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fginzburg-landau-theory":18374,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fbcs-theory":18208,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fjosephson-and-high-tc":18375,"\u002Fcondensed-matter\u002Fnanostructures\u002Fquantum-wells-wires-and-dots":17801,"\u002Fcondensed-matter\u002Fnanostructures\u002Finteger-quantum-hall-effect":18376,"\u002Fcondensed-matter\u002Fnanostructures\u002Ffractional-quantum-hall-and-topology":17812,"\u002Fcondensed-matter\u002Fnanostructures\u002Fgraphene-and-dirac-materials":18377,"\u002Fcondensed-matter":18129,"\u002Flogic\u002Ffoundations\u002Flogic-as-a-mathematical-model":18378,"\u002Flogic\u002Fsentential-logic\u002Fformal-languages-and-well-formed-formulas":18379,"\u002Flogic\u002Fsentential-logic\u002Ftruth-assignments-and-tautologies":18380,"\u002Flogic\u002Fsentential-logic\u002Funique-readability-and-parsing":18381,"\u002Flogic\u002Fsentential-logic\u002Finduction-and-recursion":17826,"\u002Flogic\u002Fsentential-logic\u002Fexpressive-completeness-and-normal-forms":18382,"\u002Flogic\u002Fsentential-logic\u002Fboolean-circuits":18383,"\u002Flogic\u002Fsentential-logic\u002Fcompactness-and-effectiveness":17826,"\u002Flogic\u002Ffirst-order-languages\u002Ffirst-order-languages":18384,"\u002Flogic\u002Ffirst-order-languages\u002Fstructures-truth-and-satisfaction":18238,"\u002Flogic\u002Ffirst-order-languages\u002Fdefinability-and-elementary-equivalence":18385,"\u002Flogic\u002Ffirst-order-languages\u002Fterms-substitution-and-parsing":18386,"\u002Flogic\u002Fdeductive-calculus\u002Fa-deductive-calculus":18387,"\u002Flogic\u002Fdeductive-calculus\u002Fdeduction-theorem-and-derived-rules":18385,"\u002Flogic\u002Fdeductive-calculus\u002Fsoundness":18388,"\u002Flogic\u002Fdeductive-calculus\u002Fcompleteness-and-consistency":18389,"\u002Flogic\u002Fmodels-and-theories\u002Fcompactness-and-lowenheim-skolem":18390,"\u002Flogic\u002Fmodels-and-theories\u002Ftheories-elementary-classes-and-categoricity":18391,"\u002Flogic\u002Fmodels-and-theories\u002Finterpretations-between-theories":18392,"\u002Flogic\u002Fmodels-and-theories\u002Fnonstandard-analysis":18393,"\u002Flogic\u002Farithmetic-and-definability\u002Fdefinability-in-arithmetic":18394,"\u002Flogic\u002Farithmetic-and-definability\u002Fnatural-numbers-with-successor":18395,"\u002Flogic\u002Farithmetic-and-definability\u002Fpresburger-and-reducts":18316,"\u002Flogic\u002Farithmetic-and-definability\u002Fa-subtheory-and-representability":18396,"\u002Flogic\u002Fincompleteness\u002Farithmetization-of-syntax":18389,"\u002Flogic\u002Fincompleteness\u002Fincompleteness-and-undecidability":18397,"\u002Flogic\u002Fincompleteness\u002Fsecond-incompleteness-theorem":18398,"\u002Flogic\u002Fcomputability-and-representability\u002Frecursive-functions":18032,"\u002Flogic\u002Fcomputability-and-representability\u002Frepresenting-exponentiation":18399,"\u002Flogic\u002Fsecond-order-logic\u002Fsecond-order-languages":18216,"\u002Flogic\u002Fsecond-order-logic\u002Fskolem-functions-and-many-sorted-logic":18400,"\u002Flogic\u002Fsecond-order-logic\u002Fgeneral-structures":18401,"\u002Flogic":18402,"\u002Freinforcement-learning\u002Ffoundations\u002Fwhat-is-reinforcement-learning":18403,"\u002Freinforcement-learning\u002Ffoundations\u002Fa-brief-history-of-rl":18404,"\u002Freinforcement-learning\u002Ffoundations\u002Fmulti-armed-bandits":18018,"\u002Freinforcement-learning\u002Ffoundations\u002Fbandit-exploration-algorithms":18405,"\u002Freinforcement-learning\u002Ffoundations\u002Fmarkov-decision-processes":18406,"\u002Freinforcement-learning\u002Ffoundations\u002Fvalue-functions-and-optimality":18407,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdynamic-programming":18408,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdp-async-and-gpi":18398,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods":18409,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-off-policy":18410,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftemporal-difference-learning":18411,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftd-control-sarsa-and-q-learning":18309,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping":18412,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-off-policy-methods":18413,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-and-learning":18414,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-focusing-and-decision-time":18415,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdecision-time-planning":18416,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-tree-search":18035,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-prediction":18417,"\u002Freinforcement-learning\u002Fapproximation\u002Ffeature-construction-and-nonlinear":18418,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-control":18419,"\u002Freinforcement-learning\u002Fapproximation\u002Faverage-reward-control":18420,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad":18225,"\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td":18421,"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces":18397,"\u002Freinforcement-learning\u002Fapproximation\u002Ftrue-online-and-sarsa-lambda":18422,"\u002Freinforcement-learning\u002Fapproximation\u002Fpolicy-gradient-methods":18423,"\u002Freinforcement-learning\u002Fapproximation\u002Factor-critic-and-continuous-actions":18424,"\u002Freinforcement-learning\u002Fapproximation\u002Fleast-squares-and-memory-based-methods":18046,"\u002Freinforcement-learning\u002Fapproximation\u002Fmemory-and-kernel-methods":18425,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces":17700,"\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces":18426,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks":18427,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdqn-improvements":18008,"\u002Freinforcement-learning\u002Fdeep-rl\u002Factor-critic-and-ppo":18428,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fppo-and-continuous-control":18429,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fcase-studies":18430,"\u002Freinforcement-learning\u002Fdeep-rl\u002Frl-beyond-games":18431,"\u002Freinforcement-learning\u002Fdeep-rl\u002Ffrontiers":18432,"\u002Freinforcement-learning\u002Fdeep-rl\u002Freward-design-and-open-problems":18285,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow":18433,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow-part-2":18434,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control":18435,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control-part-2":18320,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl":18436,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl-part-2":18437,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration":18438,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration-part-2":18035,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl":18102,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2":18439,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl":18440,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl-part-2":18441,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl":18442,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl-part-2":18443,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl":18444,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl-part-2":18445,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Frlhf-and-language-models":18446,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps":18447,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps-part-2":18448,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl":18449,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl-part-2":18450,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmeta-rl-and-generalization":18451,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fpsychology-of-reinforcement":18452,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Finstrumental-conditioning-and-control":18453,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-and-td-error":18454,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-in-the-brain":18455,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fanimal-learning-and-cognition":18456,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fcognitive-maps-and-planning":18457,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fneuroscience-of-reinforcement":18458,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fseveral-learning-systems":18459,"\u002Freinforcement-learning":17988,"\u002Fartificial-intelligence\u002Ffoundations\u002Fwhat-is-ai":18460,"\u002Fartificial-intelligence\u002Ffoundations\u002Ffoundations-of-ai":18461,"\u002Fartificial-intelligence\u002Ffoundations\u002Fintelligent-agents":18462,"\u002Fartificial-intelligence\u002Ffoundations\u002Fagent-architectures":18463,"\u002Fartificial-intelligence\u002Fsearch\u002Funinformed-search":18464,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-strategies-compared":18465,"\u002Fartificial-intelligence\u002Fsearch\u002Finformed-search":18466,"\u002Fartificial-intelligence\u002Fsearch\u002Fheuristic-functions":18467,"\u002Fartificial-intelligence\u002Fsearch\u002Flocal-search":18468,"\u002Fartificial-intelligence\u002Fsearch\u002Fpopulation-and-continuous-search":18469,"\u002Fartificial-intelligence\u002Fsearch\u002Fadversarial-search":18470,"\u002Fartificial-intelligence\u002Fsearch\u002Fgames-of-chance-and-imperfect-information":18471,"\u002Fartificial-intelligence\u002Fsearch\u002Fconstraint-satisfaction":18472,"\u002Fartificial-intelligence\u002Fsearch\u002Fcsp-search-and-structure":18314,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-under-uncertainty":18170,"\u002Fartificial-intelligence\u002Fsearch\u002Fbelief-state-and-online-search":18473,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-logic":18474,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-inference":18475,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic":18476,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic-in-use":18477,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Finference-and-resolution":18478,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-resolution":18296,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fclassical-planning":18479,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-graphs-and-graphplan":18480,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-in-the-real-world":18481,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-under-uncertainty":18482,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fknowledge-representation":18483,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Freasoning-systems-and-defaults":18484,"\u002Fartificial-intelligence\u002Funcertainty\u002Fprobability-and-bayes":18485,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayes-rule-and-naive-bayes":18486,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayesian-networks":18487,"\u002Fartificial-intelligence\u002Funcertainty\u002Finference-in-bayesian-networks":18488,"\u002Fartificial-intelligence\u002Funcertainty\u002Freasoning-over-time":18489,"\u002Fartificial-intelligence\u002Funcertainty\u002Ftracking-and-data-association":18490,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmaking-decisions":18228,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmarkov-decision-processes":18480,"\u002Fartificial-intelligence\u002Funcertainty\u002Fdecision-networks-and-game-theory":18491,"\u002Fartificial-intelligence\u002Funcertainty\u002Fgame-theory-and-mechanism-design":17769,"\u002Fartificial-intelligence\u002Flearning\u002Flearning-from-examples":18492,"\u002Fartificial-intelligence\u002Flearning\u002Ftheory-and-model-families":18493,"\u002Fartificial-intelligence\u002Flearning\u002Fprobabilistic-learning":18494,"\u002Fartificial-intelligence\u002Flearning\u002Fexpectation-maximization":18495,"\u002Fartificial-intelligence\u002Flearning\u002Freinforcement-learning":18496,"\u002Fartificial-intelligence\u002Flearning\u002Fgeneralization-and-policy-search":18277,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-in-learning":18497,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-based-learning-methods":18498,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fvision-and-perception":18499,"\u002Fartificial-intelligence\u002Ffrontiers\u002Freconstructing-the-3d-world":18500,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobotics":18501,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobot-planning-and-control":18502,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnatural-language-in-ai":18503,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnlp-grammar-translation-and-speech":18504,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fphilosophy-and-future":18505,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fai-ethics-and-future":18506,"\u002Fartificial-intelligence":17988,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-constituents-nuclide-chart":18244,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-size-charge-distributions":18507,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-masses-binding-energy":17805,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fsemi-empirical-mass-formula":17803,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-moments-multipoles":18338,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnuclear-force-shell-overview":18508,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fthe-deuteron":17831,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnucleon-nucleon-scattering":18141,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fmeson-theory-isospin":18509,"\u002Fnuclear-physics\u002Fnuclear-models\u002Ffermi-gas-model":18510,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fliquid-drop-collective-coordinates":18511,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fshell-model-single-particle":18200,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fcollective-model-rotations-vibrations":18512,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-law-modes":18513,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-kinetics-equilibrium":18514,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-decay-gamow-theory":18399,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-fine-structure-hindrance":18515,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fbeta-decay-energetics-neutrino":18516,"\u002Fnuclear-physics\u002Fbeta-decay\u002Ffermi-theory-beta-decay":17815,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fweak-interaction-parity-violation":18044,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fdouble-beta-decay-neutrino-mass":18517,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fgamma-multipole-radiation":18145,"\u002Fnuclear-physics\u002Fgamma-decay\u002Finternal-conversion-isomers":18518,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fangular-correlations-mossbauer":18519,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Freaction-kinematics-cross-sections":18040,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fcompound-nucleus-resonances":18128,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fdirect-reactions-optical-model":18520,"\u002Fnuclear-physics\u002Ffission\u002Ffission-barrier-dynamics":18521,"\u002Fnuclear-physics\u002Ffission\u002Fchain-reactions-reactor-physics":18522,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Ffusion-reactions-confinement":17810,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fstellar-nucleosynthesis":18184,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fbig-bang-nucleosynthesis":18047,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fcharged-particle-stopping-power":18523,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fphoton-neutron-interactions":18095,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fradiation-detectors":18159,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fdosimetry-radiation-biology":18524,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fnuclear-applications-dating-medicine":18525,"\u002Fnuclear-physics":18526,"\u002Fnatural-language-processing\u002Ffoundations\u002Fwhat-is-nlp":18527,"\u002Fnatural-language-processing\u002Ffoundations\u002Fregex-and-text-normalization":18528,"\u002Fnatural-language-processing\u002Ffoundations\u002Fminimum-edit-distance":18166,"\u002Fnatural-language-processing\u002Ffoundations\u002Fn-gram-language-models":18529,"\u002Fnatural-language-processing\u002Ffoundations\u002Fsmoothing-and-backoff":18530,"\u002Fnatural-language-processing\u002Fclassification\u002Fnaive-bayes-and-sentiment":18531,"\u002Fnatural-language-processing\u002Fclassification\u002Fevaluating-classifiers":18014,"\u002Fnatural-language-processing\u002Fclassification\u002Flogistic-regression":18532,"\u002Fnatural-language-processing\u002Fclassification\u002Fsentiment-and-affect-lexicons":18533,"\u002Fnatural-language-processing\u002Fsemantics\u002Fvector-semantics-and-embeddings":18310,"\u002Fnatural-language-processing\u002Fsemantics\u002Fstatic-word-embeddings":18534,"\u002Fnatural-language-processing\u002Fsemantics\u002Fneural-language-models":18479,"\u002Fnatural-language-processing\u002Fsequences\u002Fsequence-labeling":18535,"\u002Fnatural-language-processing\u002Fsequences\u002Fcrfs-and-neural-taggers":18536,"\u002Fnatural-language-processing\u002Fsequences\u002Frnns-and-lstms":18537,"\u002Fnatural-language-processing\u002Ftransformers\u002Ftransformers-and-attention":18538,"\u002Fnatural-language-processing\u002Ftransformers\u002Fthe-transformer-architecture":18539,"\u002Fnatural-language-processing\u002Ftransformers\u002Flarge-language-models":18540,"\u002Fnatural-language-processing\u002Ftransformers\u002Fllm-pretraining-and-scaling":18541,"\u002Fnatural-language-processing\u002Ftransformers\u002Ffine-tuning-and-prompting":17992,"\u002Fnatural-language-processing\u002Ftransformers\u002Fprompting-and-alignment":18542,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-parsing":18543,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcky-scoring-and-evaluation":18485,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdependency-parsing":18544,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fgraph-based-and-neural-dependency-parsing":18545,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fword-senses-and-wsd":18546,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fwsd-in-practice-and-induction":18547,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-roles-and-information-extraction":18548,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Frelations-events-and-templates":18549,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoreference-and-discourse":18550,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoherence-and-discourse-structure":18551,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Flogical-semantics":18409,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcompositional-semantics-and-description-logics":18552,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-parsing":18553,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fneural-semantic-parsing":18554,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Finformation-extraction":18555,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftimes-events-and-templates":18556,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdiscourse-coherence":18557,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fentity-based-and-global-coherence":18558,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-grammars":18559,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftreebanks-and-lexicalized-grammars":18560,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation":18561,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation-decoding-and-evaluation":18562,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering":18563,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering-knowledge-and-llms":18263,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-and-chatbots":18430,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-systems-and-assistants":18020,"\u002Fnatural-language-processing\u002Fapplications\u002Ftext-summarization":18564,"\u002Fnatural-language-processing\u002Fapplications\u002Fabstractive-summarization-and-evaluation":18565,"\u002Fnatural-language-processing\u002Fspeech\u002Fphonetics":18566,"\u002Fnatural-language-processing\u002Fspeech\u002Facoustic-phonetics":18567,"\u002Fnatural-language-processing\u002Fspeech\u002Fautomatic-speech-recognition":18277,"\u002Fnatural-language-processing\u002Fspeech\u002Fasr-evaluation-and-applications":18568,"\u002Fnatural-language-processing":17988,"\u002Fparticle-physics\u002Ffoundations\u002Fhistorical-overview-particle-zoo":18569,"\u002Fparticle-physics\u002Ffoundations\u002Fparticle-physics-basic-concepts":18185,"\u002Fparticle-physics\u002Ffoundations\u002Ffundamental-interactions-force-carriers":18570,"\u002Fparticle-physics\u002Funits-kinematics\u002Fnatural-units-and-scales":18330,"\u002Fparticle-physics\u002Funits-kinematics\u002Ffour-vectors-invariant-mass":18571,"\u002Fparticle-physics\u002Funits-kinematics\u002Fdecay-scattering-kinematics-mandelstam":18572,"\u002Fparticle-physics\u002Funits-kinematics\u002Fcross-sections-golden-rule":18520,"\u002Fparticle-physics\u002Fsymmetries\u002Fconservation-laws-symmetries":18573,"\u002Fparticle-physics\u002Fsymmetries\u002Fdiscrete-symmetries-cpt":18574,"\u002Fparticle-physics\u002Fsymmetries\u002Fparity-violation-weak":18136,"\u002Fparticle-physics\u002Fsymmetries\u002Fsu2-su3-flavor-symmetry":17838,"\u002Fparticle-physics\u002Fquark-model\u002Feightfold-way-su3":18575,"\u002Fparticle-physics\u002Fquark-model\u002Fmeson-spectroscopy":18090,"\u002Fparticle-physics\u002Fquark-model\u002Fbaryon-spectroscopy":18576,"\u002Fparticle-physics\u002Fquark-model\u002Fcolor-confinement-exotics":18123,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fklein-gordon-equation":18577,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fdirac-equation-spinors":18376,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fantiparticles-hole-theory":18578,"\u002Fparticle-physics\u002Fqed\u002Ffeynman-rules-qed":18579,"\u002Fparticle-physics\u002Fqed\u002Fqed-tree-processes":18345,"\u002Fparticle-physics\u002Fqed\u002Frenormalization-running-coupling":18580,"\u002Fparticle-physics\u002Fqed\u002Felectron-g-2":17809,"\u002Fparticle-physics\u002Fweak-interaction\u002Fva-structure-weak":18581,"\u002Fparticle-physics\u002Fweak-interaction\u002Fw-z-bosons-decays":18582,"\u002Fparticle-physics\u002Fweak-interaction\u002Fckm-matrix":18583,"\u002Fparticle-physics\u002Fweak-interaction\u002Fcp-violation-kaons-b-mesons":18127,"\u002Fparticle-physics\u002Fqcd\u002Fcolor-su3-gluons":18342,"\u002Fparticle-physics\u002Fqcd\u002Fasymptotic-freedom-confinement":18584,"\u002Fparticle-physics\u002Fqcd\u002Fdeep-inelastic-scattering-partons":18585,"\u002Fparticle-physics\u002Fqcd\u002Fjets-hadronization":18201,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Felectroweak-su2-u1":18586,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fspontaneous-symmetry-breaking":18246,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-mechanism":18587,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-boson-discovery":18588,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fstandard-model":18136,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-oscillations":18169,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-mass-pmns":18589,"\u002Fparticle-physics\u002Fneutrinos\u002Fdirac-majorana-experiments":18590,"\u002Fparticle-physics\u002Fexperiment\u002Faccelerators-luminosity":18591,"\u002Fparticle-physics\u002Fexperiment\u002Fdetectors-subsystems":18224,"\u002Fparticle-physics\u002Fexperiment\u002Fhow-discoveries-are-made":18592,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fbeyond-standard-model":17966,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fgrand-unified-theories":18593,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fsupersymmetry":18069,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fhierarchy-problem-naturalness":18594,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fdark-matter-candidates":18595,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fmatter-antimatter-open-questions":18392,"\u002Fparticle-physics":18596,"\u002Fastrophysics-cosmology\u002Forientation\u002Fthe-sun-and-stars":18186,"\u002Fastrophysics-cosmology\u002Forientation\u002Fstellar-death-final-states":18340,"\u002Fastrophysics-cosmology\u002Forientation\u002Fgalaxies-and-cosmology":18597,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fmagnitudes-fluxes-and-the-distance-modulus":18126,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fstellar-spectra-and-spectral-classification":18598,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Ftelescopes-and-detectors-across-the-spectrum":18185,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fthe-cosmic-distance-ladder":18098,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fblackbody-radiation-and-specific-intensity":18599,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fradiative-transfer-and-the-transfer-equation":18600,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fspectral-line-formation-and-broadening":18601,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fopacity-and-the-rosseland-mean":18602,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fhydrostatic-equilibrium-and-the-virial-theorem":18603,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equations-of-stellar-structure":18391,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equation-of-state-and-polytropes":18322,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-standard-solar-model":18604,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fthermonuclear-reaction-rates-and-the-gamow-peak":18605,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhydrogen-burning-pp-chains-and-cno":18606,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhelium-burning-and-the-triple-alpha-process":18607,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fadvanced-burning-and-neutron-capture-nucleosynthesis":18519,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fphases-of-the-interstellar-medium":18608,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fmolecular-clouds-and-gravitational-collapse":18082,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fprotostars-and-the-pre-main-sequence":18145,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-main-sequence-and-its-structure":18127,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fpost-main-sequence-low-mass-evolution":18609,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-evolution-of-massive-stars":18210,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fstellar-pulsation-and-the-instability-strip":17827,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":18610,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fcore-collapse-supernovae":18611,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fthermonuclear-supernovae-type-ia":18341,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fneutron-stars-and-pulsars":18612,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fblack-holes-schwarzschild-and-kerr":18613,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fbinary-systems-and-mass-transfer":18614,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Faccreting-compact-objects":18615,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fgravitational-waves-from-inspiraling-binaries":17808,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fmultimessenger-astronomy-and-gamma-ray-bursts":18616,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fthe-milky-way":18617,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-morphology-and-classification":18069,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-rotation-curves-and-dark-matter":18618,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Factive-galactic-nuclei-and-supermassive-black-holes":18619,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-clusters-and-large-scale-structure":18334,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-expanding-universe-and-hubbles-law":17990,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-frw-metric-and-cosmological-redshift":18620,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-friedmann-equations-and-cosmic-dynamics":18621,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fcosmological-models-and-distances":18246,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fdark-energy-and-the-accelerating-universe":18232,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fthe-thermal-history-of-the-universe":18244,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fbig-bang-nucleosynthesis":18341,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Frecombination-and-the-cosmic-microwave-background":18622,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcmb-anisotropies-and-cosmological-parameters":17807,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcosmic-inflation":17999,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fstructure-formation-and-the-growth-of-perturbations":18623,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fdark-matter-dark-energy-and-open-questions":18086,"\u002Fastrophysics-cosmology":18190,"\u002Fcolophon":18624,"\u002F":17988},4250,4808,3626,2682,4109,4786,3878,3875,3751,3415,4067,3153,3000,4042,5461,5808,3961,3749,4327,5067,4246,4655,4154,5436,2640,4003,3601,2158,4331,4189,2273,3252,4633,4964,4172,3131,5524,3160,4031,2309,4207,3226,2648,4842,5340,3307,5701,4977,4039,2615,3472,4460,3848,4075,4400,3382,3010,3602,3737,3740,3707,3922,5191,4043,3804,4542,4214,5062,2850,4361,3443,3627,4044,3766,4140,3860,4006,5199,4334,5234,3651,5509,5680,153,1375,1073,1093,1125,1146,1014,1132,876,1541,1189,1173,984,1402,1301,950,1268,1063,1107,1408,1161,925,1012,866,964,1090,1142,1085,1020,1207,973,980,728,764,1225,1329,796,929,801,878,774,1044,1488,1175,1130,890,814,870,154,4073,5140,4961,5127,4870,5382,5195,4955,5369,4501,5576,3824,4132,4289,4307,4570,3403,5084,5105,5201,5116,5341,5175,5368,5188,5211,5499,5155,4981,5125,5415,5255,5304,5130,5167,5552,5164,5094,5239,5036,5190,5004,5099,5035,5159,5088,5026,4937,5023,5264,5244,133,5114,5078,5043,5312,5170,5342,5139,5151,5049,5212,5013,5068,5079,5102,5121,5081,5029,5379,5854,5110,2139,3798,5055,5364,4984,4935,4895,4972,5289,5112,5156,4987,5031,5025,5149,5302,5042,5002,4979,4922,4960,5279,126,1877,1180,1129,907,958,1112,1300,1053,1250,1181,1241,1234,966,1050,734,1190,484,1082,926,733,761,571,607,798,804,952,977,731,784,645,771,1017,742,1004,1000,1562,1254,1288,1101,1011,1486,1061,856,992,1169,988,137,0,2037,1782,2384,2254,2123,2332,1643,1714,2089,1751,1367,1660,2511,1998,1892,1854,1791,2438,2487,1917,2375,2525,2266,1845,2275,1810,1631,2310,2166,2233,2113,2505,2347,2672,2112,2473,2592,2380,3013,2513,3256,3218,2194,2173,2205,2326,2081,3342,3152,1799,1670,1027,960,1095,1291,986,897,1209,1055,1817,1801,1593,1465,1196,1464,1201,1230,1435,1684,1461,1926,1500,1409,1284,1774,1869,162,1487,1122,1188,1001,1351,982,1005,979,1325,1046,943,1279,824,1008,989,1798,1277,1025,987,1043,1211,1074,981,939,1002,739,1139,1108,1013,1070,978,1458,1317,157,1357,1077,2355,1116,1037,1178,1637,1314,1109,1056,1702,1474,1071,1158,832,993,1404,1024,1068,1339,1106,1264,1248,913,1848,1328,1633,1224,1143,135,1378,959,1028,998,911,1527,1203,1266,1483,1165,990,938,965,1257,1418,1099,942,1352,956,1035,1398,1003,1094,1292,138,1721,1827,1449,1354,1148,1184,1285,1281,1213,1290,1271,1252,1274,1778,1591,1503,1437,1571,1584,1957,1117,1648,1342,1667,1510,1965,1607,1365,1849,1259,1303,1356,1238,2208,1564,173,1671,1286,1227,1638,1529,668,1078,918,709,865,880,940,1534,1015,874,922,841,794,1194,822,1105,1658,1359,1296,1438,1921,1844,1570,1429,1324,1400,140,1787,1558,1654,1492,1747,2224,2002,2009,1323,1349,1785,1573,1722,1829,1353,1548,1552,1583,1624,1585,1245,1364,1514,1343,1397,1355,2211,1481,1770,160,2388,2293,2256,2552,2569,2478,2039,2496,2578,2814,2519,2461,2587,2492,2714,3278,2654,3050,2447,2849,2238,2369,2061,2214,2602,2563,2186,2985,2749,3364,2038,2282,2409,2126,2573,2206,2176,2268,2182,2402,2705,2633,2414,2213,2801,3313,3410,3195,1952,2017,1509,2537,2645,2027,2415,2838,2356,1906,3184,2950,2807,2954,1683,1316,1034,1138,1763,1822,1705,1246,1701,1097,1104,1187,1032,1083,1228,916,1489,1033,1652,997,692,837,1023,888,864,1089,1231,1214,1675,1156,1075,1520,1309,139,1205,1051,735,1123,1072,915,567,768,825,1253,983,1007,762,1058,861,862,971,1208,1149,1145,1029,1084,927,810,838,857,807,936,949,2321,1622,1069,1113,1057,854,1958,1528,1618,2049,1432,1679,1796,1685,1346,1275,1476,1505,1610,2018,1599,1215,1838,1909,132,3902,2215,2240,3266,3208,3073,2454,2969,2451,1875,2728,1884,2371,2516,2842,1690,1904,2346,3146,1386,2607,1966,2668,1665,2885,1606,2577,3074,2869,2403,2433,2082,1939,1587,2460,2747,2032,2642,1619,3123,1993,2090,2339,3829,1737,2622,2340,2322,3828,4409,2305,3411,2510,4527,3030,3569,3043,2457,1946,2277,2044,2909,1693,1945,2093,2399,2115,2898,2742,2242,3895,3378,3376,2769,2223,3062,3262,2651,2949,2768,3128,2423,1977,2087,2866,3388,2830,2210,2489,2884,3945,2099,2713,3402,1692,2931,4195,3989,3206,4391,3004,3704,3494,2902,999,881,901,919,748,869,1018,1045,1049,1333,954,1092,1019,976,1771,1480,1396,953,1026,161,3533,2495,1818,3007,2595,3427,3537,2216,1895,2304,3396,1739,2073,1962,2203,1767,2666,2264,2276,2852,1807,3735,1560,4144,1669,1676,1972,2418,3291,1525,2040,2766,2337,2220,2800,3001,2078,1759,2836,1896,2026,1758,1543,1047,896,946,1060,1384,1482,815,1414,1322,1440,1240,1468,1098,1133,847,1009,1381,1052,1191,1258,1370,1712,1441,1199,957,1079,150,1262,1417,1368,1219,1136,1064,1463,1636,1059,931,1115,1736,1174,1376,1363,1411,1247,1746,1313,1299,1617,1102,1076,1495,1265,1193,1263,80,{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":18626,"\u002Falgorithms\u002Ffoundations\u002Fproof-techniques":18631,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":18635,"\u002Falgorithms\u002Ffoundations\u002Fgrowth-rates-and-loop-analysis":18639,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":18643,"\u002Falgorithms\u002Ffoundations\u002Famortized-analysis":18647,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":18651,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":18656,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":18660,"\u002Falgorithms\u002Fdivide-and-conquer\u002Ffast-multiplication":18664,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":18668,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":18673,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":18677,"\u002Falgorithms\u002Fsorting\u002Fexternal-sorting":18681,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":18685,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":18690,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":18694,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":18698,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":18702,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":18706,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":18710,"\u002Falgorithms\u002Fdata-structures\u002Fspatial-data-structures":18714,"\u002Falgorithms\u002Fdata-structures\u002Fskip-lists-and-probabilistic-structures":18718,"\u002Falgorithms\u002Fdata-structures\u002Fb-trees":18722,"\u002Falgorithms\u002Fdata-structures\u002Fdata-stream-algorithms":18726,"\u002Falgorithms\u002Fdata-structures\u002Fstreaming-sketches":18730,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":18734,"\u002Falgorithms\u002Fsequences\u002Fprefix-sums":18739,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":18743,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":18747,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":18751,"\u002Falgorithms\u002Fsequences\u002Fkmp-and-z-function":18755,"\u002Falgorithms\u002Fsequences\u002Ftries":18759,"\u002Falgorithms\u002Fsequences\u002Fsuffix-arrays-and-aho-corasick":18763,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":18767,"\u002Falgorithms\u002Fgraphs\u002Fdepth-first-search":18772,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":18776,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":18780,"\u002Falgorithms\u002Fgraphs\u002Fkruskal-and-prim":18784,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":18788,"\u002Falgorithms\u002Fgraphs\u002Fall-pairs-and-negative-weights":18792,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":18796,"\u002Falgorithms\u002Fgraphs\u002Fmax-flow-min-cut":18800,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":18804,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":18808,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":18812,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":18816,"\u002Falgorithms\u002Fgraphs\u002Fbipartite-matching":18820,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":18824,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":18829,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":18833,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":18837,"\u002Falgorithms\u002Fgreedy\u002Fstable-matching":18841,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":18845,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":18850,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":18854,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":18858,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":18862,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":18866,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":18870,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":18874,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":18878,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":18882,"\u002Falgorithms\u002Fdynamic-programming\u002Fdigit-and-probability-dp":18886,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":18890,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":18895,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":18899,"\u002Falgorithms\u002Fbacktracking\u002Fgraph-backtracking":18903,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":18907,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":18912,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":18916,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":18920,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmatrix-exponentiation":18924,"\u002Falgorithms\u002Fmathematical-algorithms\u002Ffast-fourier-transform":18928,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fgradient-descent":18932,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":18936,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":18941,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":18945,"\u002Falgorithms\u002Fcomputational-geometry\u002Fpolygons-and-proximity":18949,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":18953,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":18958,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":18962,"\u002Falgorithms\u002Fintractability\u002Fapproximation-algorithms":18966,"\u002Falgorithms":18970,"\u002Fcalculus\u002Flimits-and-continuity\u002Ffunctions-and-models":18973,"\u002Fcalculus\u002Flimits-and-continuity\u002Fthe-limit-of-a-function":18978,"\u002Fcalculus\u002Flimits-and-continuity\u002Flimit-laws-and-the-precise-definition":18982,"\u002Fcalculus\u002Flimits-and-continuity\u002Fcontinuity":18986,"\u002Fcalculus\u002Fderivatives\u002Fthe-derivative-and-rates-of-change":18990,"\u002Fcalculus\u002Fderivatives\u002Fdifferentiation-rules-and-the-chain-rule":18995,"\u002Fcalculus\u002Fderivatives\u002Fimplicit-differentiation-and-related-rates":18999,"\u002Fcalculus\u002Fderivatives\u002Flinear-approximations-and-differentials":19003,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fextrema-and-the-mean-value-theorem":19007,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fhow-derivatives-shape-a-graph":19012,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fcurve-sketching-and-optimization":19016,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fnewtons-method-and-antiderivatives":19020,"\u002Fcalculus\u002Fintegrals\u002Farea-and-the-definite-integral":19024,"\u002Fcalculus\u002Fintegrals\u002Fthe-fundamental-theorem-of-calculus":19029,"\u002Fcalculus\u002Fintegrals\u002Fthe-substitution-rule":19033,"\u002Fcalculus\u002Fapplications-of-integration\u002Fareas-and-volumes":19037,"\u002Fcalculus\u002Fapplications-of-integration\u002Fwork-average-value-and-arc-length":19042,"\u002Fcalculus\u002Fapplications-of-integration\u002Fphysics-economics-and-probability":19046,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Finverse-functions-logarithms-and-exponentials":19050,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Fgrowth-decay-inverse-trig-and-hyperbolic-functions":19055,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Flhospitals-rule":19059,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fintegration-by-parts":19063,"\u002Fcalculus\u002Ftechniques-of-integration\u002Ftrigonometric-integrals-and-substitution":19068,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fpartial-fractions-and-integration-strategy":19072,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fapproximate-and-improper-integrals":19076,"\u002Fcalculus\u002Fparametric-and-polar\u002Fparametric-curves-and-their-calculus":19080,"\u002Fcalculus\u002Fparametric-and-polar\u002Fpolar-coordinates":19085,"\u002Fcalculus\u002Fparametric-and-polar\u002Fconic-sections":19089,"\u002Fcalculus\u002Fsequences-and-series\u002Fsequences":19093,"\u002Fcalculus\u002Fsequences-and-series\u002Fseries-and-the-integral-test":19098,"\u002Fcalculus\u002Fsequences-and-series\u002Fthe-convergence-tests":19102,"\u002Fcalculus\u002Fsequences-and-series\u002Fpower-series":19106,"\u002Fcalculus\u002Fsequences-and-series\u002Ftaylor-and-maclaurin-series":19110,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvectors-and-the-dot-product":19114,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fthe-cross-product-lines-and-planes":19119,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fcylinders-and-quadric-surfaces":19123,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvector-functions-and-space-curves":19127,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Farc-length-curvature-and-motion":19131,"\u002Fcalculus\u002Fpartial-derivatives\u002Ffunctions-of-several-variables":19135,"\u002Fcalculus\u002Fpartial-derivatives\u002Fpartial-derivatives":19140,"\u002Fcalculus\u002Fpartial-derivatives\u002Ftangent-planes-and-the-chain-rule":19143,"\u002Fcalculus\u002Fpartial-derivatives\u002Fdirectional-derivatives-and-the-gradient":19147,"\u002Fcalculus\u002Fpartial-derivatives\u002Foptimization-and-lagrange-multipliers":19151,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fdouble-integrals":19155,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Ftriple-integrals-and-coordinate-systems":19160,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fvector-fields-and-line-integrals":19164,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fgreens-theorem-curl-and-divergence":19168,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fsurface-integrals":19172,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fstokes-and-the-divergence-theorem":19176,"\u002Fcalculus":19180,"\u002Fmechanics\u002Ffoundations\u002Fmeasurement-and-dimensions":19183,"\u002Fmechanics\u002Ffoundations\u002Fvector-algebra":19187,"\u002Fmechanics\u002Fkinematics\u002Fone-dimensional-motion":19191,"\u002Fmechanics\u002Fkinematics\u002Fmotion-graphs":19196,"\u002Fmechanics\u002Fkinematics\u002Fprojectile-motion":19200,"\u002Fmechanics\u002Fkinematics\u002Frelative-motion":19204,"\u002Fmechanics\u002Fkinematics\u002Fcircular-motion":19208,"\u002Fmechanics\u002Fdynamics\u002Fnewtons-laws":19212,"\u002Fmechanics\u002Fdynamics\u002Ffree-body-diagrams":19217,"\u002Fmechanics\u002Fdynamics\u002Ffriction-and-curved-motion":19221,"\u002Fmechanics\u002Fdynamics\u002Fnumerical-dynamics":19225,"\u002Fmechanics\u002Fdynamics\u002Fcenter-of-mass-systems":19229,"\u002Fmechanics\u002Fenergy\u002Fwork-and-kinetic-energy":19233,"\u002Fmechanics\u002Fenergy\u002Fpotential-energy":19238,"\u002Fmechanics\u002Fenergy\u002Fmultiparticle-work":19242,"\u002Fmechanics\u002Fenergy\u002Fmass-energy-and-binding":19246,"\u002Fmechanics\u002Fenergy\u002Fphotons-and-quantization":19250,"\u002Fmechanics\u002Fmomentum\u002Fmomentum-and-collisions":19254,"\u002Fmechanics\u002Fmomentum\u002Fcenter-of-mass-collisions":19259,"\u002Fmechanics\u002Fmomentum\u002Frocket-propulsion":19263,"\u002Fmechanics\u002Frotation\u002Frotational-inertia":19267,"\u002Fmechanics\u002Frotation\u002Frotational-dynamics":19272,"\u002Fmechanics\u002Frotation\u002Frolling-motion":19276,"\u002Fmechanics\u002Frotation\u002Fangular-momentum":19280,"\u002Fmechanics\u002Frotation\u002Frolling-resistance":19284,"\u002Fmechanics\u002Frotation\u002Fgyroscopic-precession":19288,"\u002Fmechanics\u002Fgravity-and-matter\u002Fkeplerian-orbits":19292,"\u002Fmechanics\u002Fgravity-and-matter\u002Fgravitational-fields":19297,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstatic-equilibrium":19301,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-statics":19305,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-flow":19309,"\u002Fmechanics\u002Fgravity-and-matter\u002Forbital-motion":19313,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstress-and-elasticity":19317,"\u002Fmechanics\u002Foscillations-waves\u002Fdamped-oscillators":19321,"\u002Fmechanics\u002Foscillations-waves\u002Ftravelling-waves":19326,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-superposition":19330,"\u002Fmechanics\u002Foscillations-waves\u002Fstanding-waves":19334,"\u002Fmechanics\u002Foscillations-waves\u002Fsound-waves":19338,"\u002Fmechanics\u002Foscillations-waves\u002Fdoppler-effect":19342,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-packets":19346,"\u002Fmechanics\u002Foscillations-waves\u002Fbeats-and-coupling":19350,"\u002Fmechanics\u002Foscillations-waves\u002Fsimple-harmonic-motion":19354,"\u002Fmechanics\u002Foscillations-waves\u002Fpendulum-motion":19358,"\u002Fmechanics\u002Foscillations-waves\u002Fdriven-oscillators":19362,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-boundaries":19366,"\u002Fmechanics\u002Fthermodynamics\u002Fkinetic-theory-of-ideal-gases":19370,"\u002Fmechanics\u002Fthermodynamics\u002Ffirst-law-of-thermodynamics":19375,"\u002Fmechanics\u002Fthermodynamics\u002Fentropy-and-the-second-law":19379,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-processes":19383,"\u002Fmechanics\u002Fthermodynamics\u002Fphase-changes":19387,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-machines":19391,"\u002Fmechanics":19395,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcharge-and-conductors":19398,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcoulombs-law":19403,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-and-force":19407,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-maps":19411,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-dipoles":19415,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fcontinuous-charge-fields":19419,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fgauss-law-and-conductors":19424,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpoint-charge-potential":19428,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpotential-gradients-and-equipotentials":19433,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Felectrostatic-energy-and-pressure":19437,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Flaplace-boundary-problems":19441,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fcontinuous-charge-potentials":19445,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitance-fundamentals":19449,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-networks":19454,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-energy-and-force":19458,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fdielectric-polarization-and-breakdown":19462,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fcurrent-and-resistance":19466,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fkirchhoff-network-analysis":19471,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Frc-transients":19475,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-trajectories":19479,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fhall-effect":19484,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-force-on-conductors":19488,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-dipoles":19492,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmass-spectrometry":19496,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmoving-charge-fields":19500,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fbiot-savart-law":19505,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fcircular-current-loops":19509,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Famperes-law":19513,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fgauss-law-for-magnetism":19517,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmagnetic-materials":19521,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-flux":19525,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Ffaradays-law":19530,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Flenzs-law":19534,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmotional-emf":19538,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Feddy-currents":19542,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fself-inductance":19546,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-energy":19550,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Frl-circuits":19554,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-fundamentals":19558,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Freactance":19563,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Frlc-resonance":19567,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-power":19571,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Ftransformers":19575,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdisplacement-current":19579,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-waves":19584,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-momentum":19588,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdipole-radiation":19592,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fpolarization":19596,"\u002Felectricity-and-magnetism\u002Foptics\u002Freflection-and-refraction":19600,"\u002Felectricity-and-magnetism\u002Foptics\u002Fthin-lenses":19605,"\u002Felectricity-and-magnetism\u002Foptics\u002Fspherical-mirrors":19609,"\u002Felectricity-and-magnetism":19613,"\u002Flinear-algebra\u002Flinear-systems\u002Fsystems-and-echelon-forms":19616,"\u002Flinear-algebra\u002Flinear-systems\u002Fvector-and-matrix-equations":19621,"\u002Flinear-algebra\u002Flinear-systems\u002Fsolution-sets-and-applications":19625,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-independence":19629,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-transformations":19633,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-operations":19637,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-inverse-and-invertibility":19642,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fpartitioned-matrices-and-lu":19646,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fsubspaces-dimension-rank":19650,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fapplications-leontief-and-graphics":19654,"\u002Flinear-algebra\u002Fdeterminants\u002Fdeterminants-and-cofactors":19658,"\u002Flinear-algebra\u002Fdeterminants\u002Fproperties-of-determinants":19663,"\u002Flinear-algebra\u002Fdeterminants\u002Fcramer-volume-and-area":19667,"\u002Flinear-algebra\u002Fvector-spaces\u002Fvector-spaces-and-subspaces":19671,"\u002Flinear-algebra\u002Fvector-spaces\u002Fnull-and-column-spaces":19676,"\u002Flinear-algebra\u002Fvector-spaces\u002Fbases-and-independent-sets":19680,"\u002Flinear-algebra\u002Fvector-spaces\u002Fcoordinate-systems":19684,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdimension-and-rank":19688,"\u002Flinear-algebra\u002Fvector-spaces\u002Fchange-of-basis":19692,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdifference-equations-and-markov":19696,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-eigenvalues":19700,"\u002Flinear-algebra\u002Feigenvalues\u002Fthe-characteristic-equation":19705,"\u002Flinear-algebra\u002Feigenvalues\u002Fdiagonalization":19709,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-linear-transformations":19713,"\u002Flinear-algebra\u002Feigenvalues\u002Fcomplex-eigenvalues":19717,"\u002Flinear-algebra\u002Feigenvalues\u002Fdynamical-systems":19721,"\u002Flinear-algebra\u002Feigenvalues\u002Fpower-method":19725,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-length-orthogonality":19729,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Forthogonal-sets-and-projections":19734,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fgram-schmidt-and-qr":19738,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-problems":19742,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-applications":19746,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-spaces":19750,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fdiagonalizing-symmetric-matrices":19754,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fquadratic-forms":19759,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fconstrained-optimization":19763,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsingular-value-decomposition":19767,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsvd-applications-pca-imaging":19771,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-thinking-and-matrix-computation":19775,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Flu-and-cholesky":19780,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fconditioning-and-floating-point":19784,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fstability-and-error-analysis":19788,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fqr-and-numerical-least-squares":19792,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-eigenvalues-and-svd":19796,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-combinations":19800,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-independence-and-barycentric-coordinates":19805,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fconvex-combinations-and-convex-sets":19809,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fhyperplanes-and-polytopes":19813,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fcurves-and-surfaces":19817,"\u002Flinear-algebra":19821,"\u002Ftheory-of-computation":19824,"\u002Fcomputer-architecture\u002Ffoundations\u002Fbits-bytes-and-words":19827,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-representation":19831,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-arithmetic":19835,"\u002Fcomputer-architecture\u002Ffoundations\u002Ffloating-point":19839,"\u002Fcomputer-architecture\u002Ffoundations\u002Fboolean-algebra-and-bit-manipulation":19843,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fthe-machines-view":19847,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fdata-movement":19852,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farithmetic-and-logic":19856,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fcontrol-flow":19860,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fprocedures":19864,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farrays-structs-and-alignment":19868,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fmemory-layout-and-buffer-overflows":19872,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fwhat-an-isa-is":19876,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Finstruction-formats-and-operands":19881,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Faddressing-modes":19885,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fthe-y86-64-instruction-set":19889,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fy86-64-programming":19893,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Ftransistors-gates-and-boolean-functions":19897,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fcombinational-logic-and-hcl":19902,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmultiplexers-decoders-and-the-alu":19906,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmemory-elements-latches-flip-flops-and-clocking":19910,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fregister-files-and-random-access-memory":19914,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-fetch-decode-execute-cycle":19918,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-seq-stages":19923,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fcontrol-logic-and-sequencing":19927,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fassembling-seq":19931,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Ftracing-a-program":19935,"\u002Fcomputer-architecture\u002Fpipelining\u002Fpipelining-principles":19939,"\u002Fcomputer-architecture\u002Fpipelining\u002Ffrom-seq-to-pipe":19944,"\u002Fcomputer-architecture\u002Fpipelining\u002Fdata-hazards-stalling-and-forwarding":19948,"\u002Fcomputer-architecture\u002Fpipelining\u002Fcontrol-hazards-and-branch-prediction":19952,"\u002Fcomputer-architecture\u002Fpipelining\u002Fthe-complete-pipe-processor":19956,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fstorage-technologies-and-the-latency-gap":19960,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Flocality":19965,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-memories-direct-mapped":19969,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fset-associative-and-write-policies":19973,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-performance-and-cache-friendly-code":19977,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Faddress-spaces-and-translation":19981,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fpage-tables-and-page-faults":19986,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fthe-tlb-and-multi-level-page-tables":19990,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Fexceptional-control-flow":19994,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Finterrupts-and-the-kernel":19999,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fprocesses-threads-and-parallelism":20003,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fhardware-multithreading":20008,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fcache-coherence":20012,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmemory-consistency-and-synchronization":20016,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmulticore-organization":20020,"\u002Fcomputer-architecture\u002Fcapstone\u002Fthe-whole-machine":20024,"\u002Fcomputer-architecture\u002Fcapstone\u002Fassembling-a-complete-cpu":20029,"\u002Fcomputer-architecture":20033,"\u002Fdifferential-equations\u002Ffoundations\u002Fmodels-and-direction-fields":20036,"\u002Fdifferential-equations\u002Ffoundations\u002Fclassification-and-terminology":20040,"\u002Fdifferential-equations\u002Ffirst-order\u002Flinear-first-order-integrating-factors":20044,"\u002Fdifferential-equations\u002Ffirst-order\u002Fseparable-and-exact":20049,"\u002Fdifferential-equations\u002Ffirst-order\u002Fmodeling-first-order":20053,"\u002Fdifferential-equations\u002Ffirst-order\u002Fautonomous-and-population-dynamics":20057,"\u002Fdifferential-equations\u002Ffirst-order\u002Fexistence-uniqueness-euler":20061,"\u002Fdifferential-equations\u002Ffirst-order\u002Ffirst-order-difference-equations":20065,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhomogeneous-constant-coefficients":20069,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fcomplex-and-repeated-roots":20074,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fnonhomogeneous-undetermined-coefficients":20078,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fvariation-of-parameters":20082,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fmechanical-electrical-vibrations":20086,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhigher-order-linear":20090,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fpower-series-ordinary-points":20094,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fregular-singular-frobenius":20099,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fbessel-and-special-functions":20103,"\u002Fdifferential-equations\u002Flaplace\u002Flaplace-definition-ivps":20107,"\u002Fdifferential-equations\u002Flaplace\u002Fstep-impulse-convolution":20112,"\u002Fdifferential-equations\u002Fsystems\u002Fmatrices-eigenvalues-review":20116,"\u002Fdifferential-equations\u002Fsystems\u002Fconstant-coefficient-systems-phase-portraits":20121,"\u002Fdifferential-equations\u002Fsystems\u002Frepeated-eigenvalues-fundamental-matrices":20125,"\u002Fdifferential-equations\u002Fnumerical\u002Feuler-and-runge-kutta":20129,"\u002Fdifferential-equations\u002Fnumerical\u002Fmultistep-systems-stability":20134,"\u002Fdifferential-equations\u002Fnonlinear\u002Fphase-plane-autonomous-stability":20138,"\u002Fdifferential-equations\u002Fnonlinear\u002Flocally-linear-and-liapunov":20143,"\u002Fdifferential-equations\u002Fnonlinear\u002Fcompeting-species-predator-prey-limit-cycles":20147,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Ffourier-series":20151,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fheat-wave-laplace-equations":20156,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fsturm-liouville":20160,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fcalculus-of-variations":20164,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fhistorical-notes":20169,"\u002Fdifferential-equations":20173,"\u002Frelativity\u002Ffoundations\u002Fspecial-relativity-postulates":20176,"\u002Frelativity\u002Ffoundations\u002Florentz-transformation-spacetime":20181,"\u002Frelativity\u002Ffoundations\u002Ftime-dilation-length-contraction":20185,"\u002Frelativity\u002Ffoundations\u002Frelativistic-momentum-energy":20189,"\u002Frelativity\u002Ffoundations\u002Fgeneral-relativity":20193,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fminkowski-spacetime-and-the-interval":20197,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Ffour-vectors-and-index-notation":20202,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fthe-lorentz-group-and-rapidity":20206,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fdoppler-aberration-and-appearance":20210,"\u002Frelativity\u002Frelativistic-dynamics\u002Ffour-momentum-force-and-accelerated-motion":20214,"\u002Frelativity\u002Frelativistic-dynamics\u002Fparticle-decays-and-two-body-kinematics":20219,"\u002Frelativity\u002Frelativistic-dynamics\u002Fcollisions-thresholds-and-the-cm-frame":20223,"\u002Frelativity\u002Frelativistic-dynamics\u002Fmandelstam-variables-and-invariants":20227,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ffour-current-and-the-four-potential":20231,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fthe-electromagnetic-field-tensor":20236,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ftransformation-of-electric-and-magnetic-fields":20240,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fcovariant-maxwell-and-the-stress-energy-tensor":20244,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-equivalence-principle-formalized":20248,"\u002Frelativity\u002Fcurved-spacetime\u002Fmanifolds-vectors-and-the-metric":20253,"\u002Frelativity\u002Fcurved-spacetime\u002Fcovariant-derivative-and-christoffel-symbols":20257,"\u002Frelativity\u002Fcurved-spacetime\u002Fgeodesics-and-the-geodesic-equation":20261,"\u002Frelativity\u002Fcurved-spacetime\u002Fcurvature-riemann-and-geodesic-deviation":20265,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-einstein-field-equations":20269,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fthe-schwarzschild-metric":20273,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fgeodesics-and-orbits-in-schwarzschild":20278,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Flight-bending-and-null-geodesics":20282,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fperihelion-precession-of-mercury":20286,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fdeflection-of-light-and-gravitational-lensing":20291,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fgravitational-redshift-and-shapiro-delay":20295,"\u002Frelativity\u002Ftests-of-general-relativity\u002Frelativity-in-technology-gps":20299,"\u002Frelativity\u002Fblack-holes\u002Fhorizons-and-coordinate-singularities":20303,"\u002Frelativity\u002Fblack-holes\u002Frotating-and-charged-black-holes":20308,"\u002Frelativity\u002Fblack-holes\u002Fblack-hole-thermodynamics":20312,"\u002Frelativity\u002Fgravitational-waves\u002Flinearized-gravity-and-wave-solutions":20316,"\u002Frelativity\u002Fgravitational-waves\u002Fgeneration-and-the-quadrupole-formula":20321,"\u002Frelativity\u002Fgravitational-waves\u002Fdetection-ligo-and-the-first-events":20325,"\u002Frelativity\u002Fcosmological-bridge\u002Fthe-cosmological-principle-and-flrw-metric":20329,"\u002Frelativity\u002Fcosmological-bridge\u002Ffriedmann-equations-and-cosmic-dynamics":20334,"\u002Frelativity":20338,"\u002Fphysical-computing":20341,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fblackbody-radiation-and-the-planck-quantum":20344,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-photoelectric-effect-and-the-photon":20349,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fx-rays-and-the-compton-effect":20353,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-old-quantum-theory-bohr-and-sommerfeld":20357,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fde-broglie-waves-and-electron-diffraction":20361,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fwave-packets-and-the-probability-interpretation":20366,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fthe-uncertainty-principle":20370,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-schrodinger-equation-in-one-dimension":20374,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-free-particle-and-wave-packet-dynamics":20379,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fparticle-in-infinite-and-finite-square-wells":20383,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Foperators-expectation-values-and-the-harmonic-oscillator":20387,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-dirac-delta-potential":20391,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fbarrier-penetration-and-quantum-tunneling":20395,"\u002Fquantum-mechanics\u002Fformalism\u002Fhilbert-space-and-dirac-notation":20399,"\u002Fquantum-mechanics\u002Fformalism\u002Fobservables-hermitian-operators-and-eigenvalues":20404,"\u002Fquantum-mechanics\u002Fformalism\u002Fthe-postulates-and-quantum-measurement":20408,"\u002Fquantum-mechanics\u002Fformalism\u002Fposition-momentum-and-continuous-spectra":20412,"\u002Fquantum-mechanics\u002Fformalism\u002Fcommutators-and-the-generalized-uncertainty-principle":20416,"\u002Fquantum-mechanics\u002Fformalism\u002Ftime-evolution-schrodinger-and-heisenberg-pictures":20420,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fladder-operators-and-the-number-states":20424,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fcoherent-and-squeezed-states":20429,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fsymmetries-generators-and-conservation-laws":20433,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fparity-time-reversal-and-discrete-symmetries":20437,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Forbital-angular-momentum-and-spherical-harmonics":20441,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Fthe-angular-momentum-algebra":20445,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Faddition-of-angular-momenta-and-clebsch-gordan":20449,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-schrodinger-equation-in-three-dimensions":20453,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-hydrogen-atom":20458,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-isotropic-oscillator-and-hidden-symmetry":20462,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-half-pauli-matrices-and-stern-gerlach":20466,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-in-a-magnetic-field-precession-and-resonance":20471,"\u002Fquantum-mechanics\u002Fspin\u002Ftwo-level-systems-and-the-bloch-sphere":20475,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fidentical-particles-and-exchange-symmetry":20479,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fthe-pauli-principle-atoms-and-the-periodic-table":20484,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ftime-independent-perturbation-theory":20488,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ffine-structure-and-the-real-hydrogen-atom":20493,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-zeeman-and-stark-effects":20497,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-variational-method":20501,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-wkb-approximation":20505,"\u002Fquantum-mechanics":20509,"\u002Freal-analysis\u002Ffoundations\u002Fsets-logic-functions":20512,"\u002Freal-analysis\u002Ffoundations\u002Fordered-fields-completeness":20517,"\u002Freal-analysis\u002Ffoundations\u002Fabsolute-value-bounds":20521,"\u002Freal-analysis\u002Ffoundations\u002Fintervals-uncountability":20525,"\u002Freal-analysis\u002Fsequences-series\u002Fsequences-limits":20529,"\u002Freal-analysis\u002Fsequences-series\u002Flimit-laws-monotone":20534,"\u002Freal-analysis\u002Fsequences-series\u002Flimsup-bolzano-weierstrass":20538,"\u002Freal-analysis\u002Fsequences-series\u002Fcauchy-completeness":20542,"\u002Freal-analysis\u002Fsequences-series\u002Fseries-convergence":20546,"\u002Freal-analysis\u002Fsequences-series\u002Fabsolute-conditional-rearrangement":20550,"\u002Freal-analysis\u002Fmetric-spaces\u002Fmetric-spaces-norms":20554,"\u002Freal-analysis\u002Fmetric-spaces\u002Fopen-closed-sets":20559,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconvergence-completeness":20563,"\u002Freal-analysis\u002Fmetric-spaces\u002Fcompactness":20567,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconnectedness":20571,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-of-functions":20575,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuous-functions":20579,"\u002Freal-analysis\u002Fcontinuity\u002Fevt-ivt":20583,"\u002Freal-analysis\u002Fcontinuity\u002Funiform-continuity":20587,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuity-metric-spaces":20591,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-infinity-monotone":20595,"\u002Freal-analysis\u002Fdifferentiation\u002Fthe-derivative":20599,"\u002Freal-analysis\u002Fdifferentiation\u002Fmean-value-theorem":20604,"\u002Freal-analysis\u002Fdifferentiation\u002Ftaylors-theorem":20608,"\u002Freal-analysis\u002Fdifferentiation\u002Finverse-function-1d":20612,"\u002Freal-analysis\u002Friemann-integration\u002Fdarboux-integral":20616,"\u002Freal-analysis\u002Friemann-integration\u002Fintegrability-classes":20621,"\u002Freal-analysis\u002Friemann-integration\u002Fproperties-of-the-integral":20625,"\u002Freal-analysis\u002Friemann-integration\u002Ffundamental-theorem":20629,"\u002Freal-analysis\u002Friemann-integration\u002Flog-exp-improper":20632,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpointwise-uniform-convergence":20636,"\u002Freal-analysis\u002Ffunction-sequences\u002Finterchange-of-limits":20641,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpower-series-weierstrass":20645,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpicard-ode":20649,"\u002Freal-analysis\u002Fseveral-variables\u002Fdifferentiability-rn":20653,"\u002Freal-analysis\u002Fseveral-variables\u002Fgradient-chain-rule":20658,"\u002Freal-analysis\u002Fseveral-variables\u002Fhigher-derivatives-taylor-extrema":20662,"\u002Freal-analysis\u002Fseveral-variables\u002Finverse-implicit-theorems":20666,"\u002Freal-analysis\u002Fseveral-variables\u002Fmultiple-integrals":20670,"\u002Freal-analysis":20674,"\u002Fabstract-algebra\u002Ffoundations\u002Fsets-functions-relations":20677,"\u002Fabstract-algebra\u002Ffoundations\u002Fintegers-and-modular-arithmetic":20681,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fgroup-axioms-and-first-examples":20685,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fdihedral-and-symmetric-groups":20690,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fmatrix-and-quaternion-groups":20694,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fhomomorphisms-and-group-actions":20698,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fsubgroups-and-substructures":20702,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcyclic-groups":20707,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fgeneration-and-subgroup-lattices":20711,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcosets-lagrange-and-normal-subgroups":20715,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fisomorphism-theorems":20719,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcomposition-series-and-the-alternating-group":20723,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Factions-and-cayleys-theorem":20727,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fconjugation-and-the-class-equation":20732,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fsylow-theorems":20736,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fautomorphisms-and-simple-groups":20740,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fdirect-products-and-finite-abelian-groups":20744,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fsemidirect-products":20749,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fnilpotent-and-solvable-groups":20753,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fclassifying-small-groups":20757,"\u002Fabstract-algebra\u002Fring-theory\u002Frings-definitions-and-examples":20761,"\u002Fabstract-algebra\u002Fring-theory\u002Fideals-quotients-and-homomorphisms":20766,"\u002Fabstract-algebra\u002Fring-theory\u002Ffractions-and-the-chinese-remainder-theorem":20770,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Feuclidean-domains-pids-ufds":20774,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fpolynomial-rings-over-fields":20779,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fgauss-lemma-and-unique-factorization":20783,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Firreducibility-criteria-and-groebner":20787,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fintroduction-to-modules":20791,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ffree-modules-and-direct-sums":20796,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ftensor-products-and-exact-sequences":20800,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fvector-spaces-and-linear-maps":20804,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fstructure-theorem-over-pids":20808,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Frational-canonical-form":20813,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fjordan-canonical-form":20817,"\u002Fabstract-algebra\u002Ffield-theory\u002Ffield-extensions-and-algebraic-elements":20821,"\u002Fabstract-algebra\u002Ffield-theory\u002Fstraightedge-and-compass-constructions":20826,"\u002Fabstract-algebra\u002Ffield-theory\u002Fsplitting-fields-and-algebraic-closure":20830,"\u002Fabstract-algebra\u002Ffield-theory\u002Fseparable-and-cyclotomic-extensions":20834,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fthe-galois-correspondence":20838,"\u002Fabstract-algebra\u002Fgalois-theory\u002Ffinite-fields":20843,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fcyclotomic-and-abelian-extensions":20847,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fgalois-groups-of-polynomials":20851,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fsolvability-by-radicals-and-the-quintic":20855,"\u002Fabstract-algebra\u002Fcapstone\u002Fcommutative-algebra-and-algebraic-geometry":20859,"\u002Fabstract-algebra\u002Fcapstone\u002Frepresentation-and-character-theory":20864,"\u002Fabstract-algebra":20868,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fatomic-spectra-rutherford":20871,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-model-hydrogen":20876,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fx-ray-spectra-franck-hertz":20880,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-sommerfeld-old-quantum-theory":20884,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fold-quantum-theory-limits-wkb":20888,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fschrodinger-3d-hydrogen":20892,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fhydrogen-wave-functions":20897,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fradial-equation-in-full":20901,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fsymmetry-degeneracy-runge-lenz":20905,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fexpectation-values-virial":20909,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fquantum-defects-alkali-spectra":20913,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Frydberg-atoms":20917,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Frelativistic-kinetic-correction":20921,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fspin-orbit-thomas-precession":20926,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdarwin-term-fine-structure-formula":20930,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdirac-equation-hydrogen":20934,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Flamb-shift-qed":20938,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fhyperfine-structure-21cm":20943,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fnuclear-effects-isotope-shift":20947,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fperiodic-table-atomic-spectra":20951,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fcentral-field-self-consistent":20956,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fidentical-particles-hartree-fock":20960,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhelium-two-electron-atom":20964,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fls-jj-coupling-term-symbols":20968,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhund-rules-ground-terms":20972,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fzeeman-effect":20976,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fpaschen-back-intermediate":20981,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fstark-effect-polarizability":20985,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Ftime-dependent-perturbation-golden-rule":20989,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fdipole-approximation-einstein-coefficients":20994,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fselection-rules-forbidden-transitions":20998,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Flifetimes-and-line-shapes":21002,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Flaser-principles":21006,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fspectroscopy-techniques":21011,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fline-catalog-nist-asd":21015,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Flaser-cooling-doppler":21019,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fsub-doppler-trapping":21024,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fbose-einstein-condensation":21028,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Foptical-clocks-precision":21032,"\u002Fatomic-physics":21036,"\u002Fdatabases":21039,"\u002Fcategory-theory\u002Ffoundations\u002Fwhat-is-a-category":21042,"\u002Fcategory-theory\u002Ffoundations\u002Fexamples-of-categories":21046,"\u002Fcategory-theory\u002Ffoundations\u002Fspecial-morphisms":21050,"\u002Fcategory-theory\u002Ffoundations\u002Ffunctors":21054,"\u002Fcategory-theory\u002Ffoundations\u002Fnatural-transformations":21058,"\u002Fcategory-theory\u002Ffoundations\u002Fsize-and-set-theory":21062,"\u002Fcategory-theory\u002Funiversal-properties\u002Funiversal-properties":21066,"\u002Fcategory-theory\u002Funiversal-properties\u002Fproducts-and-coproducts":21071,"\u002Fcategory-theory\u002Funiversal-properties\u002Fconstructions-on-categories":21075,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Frepresentable-functors":21079,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-lemma":21084,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-consequences":21088,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits":21092,"\u002Fcategory-theory\u002Flimits-colimits\u002Fproducts-equalizers-pullbacks":21097,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcolimits":21101,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcomputing-limits":21105,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits-and-functors":21109,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions":21113,"\u002Fcategory-theory\u002Fadjunctions\u002Funits-and-counits":21118,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions-via-universal-arrows":21122,"\u002Fcategory-theory\u002Fadjunctions\u002Ffree-forgetful-adjunctions":21126,"\u002Fcategory-theory\u002Fadjoints-limits\u002Flimits-via-adjoints":21130,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fpresheaf-limits-colimits":21135,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoints-preserve-limits":21139,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoint-functor-theorem":21143,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fmonads":21147,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-eilenberg-moore":21152,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fkleisli-and-programming":21156,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-for-endofunctors":21160,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Fcartesian-closed-categories":21164,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Flambda-calculus-correspondence":21169,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Ffixed-points-and-recursion":21173,"\u002Fcategory-theory":21177,"\u002Fdeep-learning\u002Fmathematical-background\u002Flinear-algebra-for-deep-learning":21180,"\u002Fdeep-learning\u002Fmathematical-background\u002Fprobability-and-information-theory":21184,"\u002Fdeep-learning\u002Fmathematical-background\u002Fnumerical-computation":21188,"\u002Fdeep-learning\u002Fmathematical-background\u002Fcalculus":21192,"\u002Fdeep-learning\u002Ffoundations\u002Fwhat-is-deep-learning":21195,"\u002Fdeep-learning\u002Ffoundations\u002Fmachine-learning-refresher":21199,"\u002Fdeep-learning\u002Ffoundations\u002Flinear-models-and-the-perceptron":21203,"\u002Fdeep-learning\u002Fneural-networks\u002Fthe-multilayer-perceptron":21207,"\u002Fdeep-learning\u002Fneural-networks\u002Factivation-functions":21212,"\u002Fdeep-learning\u002Fneural-networks\u002Funiversal-approximation":21216,"\u002Fdeep-learning\u002Fneural-networks\u002Fbackpropagation":21220,"\u002Fdeep-learning\u002Fneural-networks\u002Floss-functions-and-output-units":21224,"\u002Fdeep-learning\u002Foptimization\u002Fgradient-descent-and-sgd":21228,"\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods":21233,"\u002Fdeep-learning\u002Foptimization\u002Finitialization":21237,"\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape":21241,"\u002Fdeep-learning\u002Foptimization\u002Fsecond-order-and-approximate-methods":21245,"\u002Fdeep-learning\u002Fregularization\u002Fregularization-overview":21249,"\u002Fdeep-learning\u002Fregularization\u002Fdropout-and-data-augmentation":21254,"\u002Fdeep-learning\u002Fregularization\u002Fearly-stopping-and-parameter-sharing":21258,"\u002Fdeep-learning\u002Fregularization\u002Fnormalization":21262,"\u002Fdeep-learning\u002Farchitectures\u002Fconvolutional-networks":21266,"\u002Fdeep-learning\u002Farchitectures\u002Fcnn-architectures":21271,"\u002Fdeep-learning\u002Farchitectures\u002Frecurrent-networks":21275,"\u002Fdeep-learning\u002Farchitectures\u002Flstm-and-gru":21279,"\u002Fdeep-learning\u002Farchitectures\u002Fattention-and-transformers":21283,"\u002Fdeep-learning\u002Farchitectures\u002Fthe-transformer-architecture":21287,"\u002Fdeep-learning\u002Farchitectures\u002Ftransformers-in-practice":21291,"\u002Fdeep-learning\u002Farchitectures\u002Fgraph-neural-networks":21295,"\u002Fdeep-learning\u002Farchitectures\u002Fstate-space-models":21299,"\u002Fdeep-learning\u002Ftheory\u002Fgeneralization-theory":21303,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-robustness":21308,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-defenses":21312,"\u002Fdeep-learning\u002Ftheory\u002Fbayesian-and-ensemble-methods":21316,"\u002Fdeep-learning\u002Ftheory\u002Fdeep-equilibrium-models":21320,"\u002Fdeep-learning\u002Fgenerative-models\u002Flinear-factor-models":21324,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoencoders":21329,"\u002Fdeep-learning\u002Fgenerative-models\u002Fvariational-autoencoders":21333,"\u002Fdeep-learning\u002Fgenerative-models\u002Fgenerative-adversarial-networks":21337,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoregressive-and-normalizing-flows":21341,"\u002Fdeep-learning\u002Fgenerative-models\u002Fenergy-based-and-boltzmann-machines":21345,"\u002Fdeep-learning\u002Fgenerative-models\u002Fdiffusion-and-score-based-models":21349,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fstructured-probabilistic-models":21353,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fmonte-carlo-and-mcmc":21358,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fapproximate-inference":21362,"\u002Fdeep-learning\u002Fpractical\u002Fpractical-methodology":21366,"\u002Fdeep-learning\u002Fpractical\u002Fhyperparameters-and-debugging":21371,"\u002Fdeep-learning\u002Fpractical\u002Frepresentation-learning":21375,"\u002Fdeep-learning\u002Fpractical\u002Ftransfer-learning":21379,"\u002Fdeep-learning\u002Fpractical\u002Fapplications":21383,"\u002Fdeep-learning\u002Fpractical\u002Fmodel-compression-and-distillation":21387,"\u002Fdeep-learning\u002Fpractical\u002Fmeta-learning-and-few-shot":21391,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models":21395,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment":21400,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fseq2seq-pretraining-and-bart":21404,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ftext-to-text-transfer-and-conditional-generation":21408,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fspeech-and-audio-models":21412,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fself-supervised-speech-and-synthesis":21416,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents":21420,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fagent-memory-retrieval-and-orchestration":21424,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmixture-of-experts":21428,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmultimodal-models":21432,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ffusion-and-vision-language-models":21436,"\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning":21440,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fmodel-free-prediction-and-control":21445,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fdeep-q-networks":21449,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic":21453,"\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback":21457,"\u002Fdeep-learning":21461,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fequilibrium-state-variables-zeroth-law":21464,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Ffirst-law-heat-and-work":21468,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fsecond-law-entropy-and-the-carnot-bound":21472,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fthermodynamic-potentials-and-maxwell-relations":21476,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fstability-response-functions-and-the-third-law":21480,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fclassical-statistics-and-equipartition":21484,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fphase-space-and-liouvilles-theorem":21489,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fensembles-and-the-equal-probability-postulate":21493,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fstatistical-entropy-boltzmann-and-gibbs":21497,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fmicrocanonical-ensemble-and-entropy":21501,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fequilibrium-conditions-temperature-pressure-chemical-potential":21506,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fideal-gas-phase-space-and-the-sackur-tetrode-entropy":21510,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Ftwo-state-systems-paramagnets-and-negative-temperature":21514,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fcanonical-ensemble-and-the-boltzmann-distribution":21518,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fpartition-function-and-the-helmholtz-free-energy":21523,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fenergy-fluctuations-and-ensemble-equivalence":21527,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fthe-einstein-solid-and-harmonic-systems":21531,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fparamagnetism-and-the-schottky-anomaly":21535,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fideal-gas-partition-function-and-the-gibbs-paradox":21539,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fequipartition-and-the-virial-theorem":21544,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fmolecular-gases-rotation-and-vibration":21548,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fgrand-canonical-ensemble-and-the-grand-partition-function":21552,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fchemical-potential-fugacity-and-number-fluctuations":21557,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fensemble-summary-and-the-thermodynamic-web":21561,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fquantum-statistics-bose-einstein-and-fermi-dirac":21565,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fderiving-the-quantum-distributions":21570,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fthe-classical-limit-and-quantum-concentration":21574,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fideal-quantum-gases-general-framework":21578,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-and-the-fermion-gas":21582,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthe-photon-gas-and-plancks-radiation-law":21587,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fblackbody-thermodynamics-and-radiation-pressure":21591,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fphonons-and-the-debye-model":21595,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-derived":21599,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthermodynamics-of-the-bose-gas-and-superfluidity":21603,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fthe-ideal-fermi-gas-at-zero-temperature":21607,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fsommerfeld-expansion-and-electrons-in-metals":21612,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":21616,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fneutron-stars-and-nuclear-matter":21620,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-cluster-expansion-and-virial-coefficients":21624,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-van-der-waals-gas-and-liquid-gas-coexistence":21629,"\u002Fstatistical-mechanics\u002Finteractions\u002Fquantum-gases-with-interactions-and-exchange":21633,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fphases-coexistence-and-classification":21637,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-ising-model-and-exact-solutions":21642,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fmean-field-theory-and-the-weiss-model":21646,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fcritical-exponents-and-landau-theory":21650,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-renormalization-group-idea":21654,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fthermodynamic-fluctuations-and-response":21658,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fbrownian-motion-and-the-langevin-equation":21663,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Flinear-response-and-the-fluctuation-dissipation-theorem":21667,"\u002Fstatistical-mechanics":21671,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fbonding-mechanisms":21674,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fmolecular-orbitals-and-h2-plus":21679,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fhydrogen-molecule-and-exchange":21683,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fvan-der-waals-forces":21687,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Frotational-vibrational-spectra":21691,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Fanharmonicity-and-rovibrational-structure":21696,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Framan-and-electronic-bands":21700,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Flasers-and-masers":21704,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fstructure-of-solids":21708,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fbravais-lattices-and-crystal-systems":21713,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Freciprocal-lattice-and-brillouin-zones":21717,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fdiffraction-and-structure-factors":21721,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonon-dispersion":21725,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonons-quantization-and-dos":21730,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fdebye-einstein-heat-capacity":21734,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fanharmonicity-and-thermal-transport":21738,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ffree-electron-gas-and-conduction":21742,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fsommerfeld-model-and-heat-capacity":21747,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ftransport-and-the-hall-effect":21751,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fscreening-and-plasmons":21755,"\u002Fcondensed-matter\u002Fband-theory\u002Fblochs-theorem-and-energy-bands":21759,"\u002Fcondensed-matter\u002Fband-theory\u002Fnearly-free-electron-model":21764,"\u002Fcondensed-matter\u002Fband-theory\u002Ftight-binding-method":21768,"\u002Fcondensed-matter\u002Fband-theory\u002Ffermi-surfaces-and-semiclassical-dynamics":21772,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fsemiconductor-bands-and-junctions":21776,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fintrinsic-and-extrinsic-semiconductors":21781,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fcarrier-transport-and-recombination":21785,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fthe-pn-junction":21789,"\u002Fcondensed-matter\u002Fsemiconductors\u002Ftransistors-and-optoelectronics":21793,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fdielectrics-and-polarization":21797,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fferroelectrics-and-piezoelectrics":21802,"\u002Fcondensed-matter\u002Fmagnetism\u002Fdiamagnetism-and-paramagnetism":21806,"\u002Fcondensed-matter\u002Fmagnetism\u002Fexchange-and-ferromagnetism":21811,"\u002Fcondensed-matter\u002Fmagnetism\u002Fantiferromagnetism-and-domains":21815,"\u002Fcondensed-matter\u002Fmagnetism\u002Fspin-waves-and-magnons":21819,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fsuperconductivity-phenomenology":21823,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Flondon-theory-and-the-meissner-effect":21828,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fginzburg-landau-theory":21832,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fbcs-theory":21836,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fjosephson-and-high-tc":21840,"\u002Fcondensed-matter\u002Fnanostructures\u002Fquantum-wells-wires-and-dots":21844,"\u002Fcondensed-matter\u002Fnanostructures\u002Finteger-quantum-hall-effect":21849,"\u002Fcondensed-matter\u002Fnanostructures\u002Ffractional-quantum-hall-and-topology":21853,"\u002Fcondensed-matter\u002Fnanostructures\u002Fgraphene-and-dirac-materials":21857,"\u002Fcondensed-matter":21861,"\u002Flogic\u002Ffoundations\u002Flogic-as-a-mathematical-model":21864,"\u002Flogic\u002Fsentential-logic\u002Fformal-languages-and-well-formed-formulas":21868,"\u002Flogic\u002Fsentential-logic\u002Ftruth-assignments-and-tautologies":21873,"\u002Flogic\u002Fsentential-logic\u002Funique-readability-and-parsing":21877,"\u002Flogic\u002Fsentential-logic\u002Finduction-and-recursion":21881,"\u002Flogic\u002Fsentential-logic\u002Fexpressive-completeness-and-normal-forms":21885,"\u002Flogic\u002Fsentential-logic\u002Fboolean-circuits":21889,"\u002Flogic\u002Fsentential-logic\u002Fcompactness-and-effectiveness":21893,"\u002Flogic\u002Ffirst-order-languages\u002Ffirst-order-languages":21897,"\u002Flogic\u002Ffirst-order-languages\u002Fstructures-truth-and-satisfaction":21902,"\u002Flogic\u002Ffirst-order-languages\u002Fdefinability-and-elementary-equivalence":21906,"\u002Flogic\u002Ffirst-order-languages\u002Fterms-substitution-and-parsing":21910,"\u002Flogic\u002Fdeductive-calculus\u002Fa-deductive-calculus":21914,"\u002Flogic\u002Fdeductive-calculus\u002Fdeduction-theorem-and-derived-rules":21919,"\u002Flogic\u002Fdeductive-calculus\u002Fsoundness":21923,"\u002Flogic\u002Fdeductive-calculus\u002Fcompleteness-and-consistency":21927,"\u002Flogic\u002Fmodels-and-theories\u002Fcompactness-and-lowenheim-skolem":21931,"\u002Flogic\u002Fmodels-and-theories\u002Ftheories-elementary-classes-and-categoricity":21936,"\u002Flogic\u002Fmodels-and-theories\u002Finterpretations-between-theories":21940,"\u002Flogic\u002Fmodels-and-theories\u002Fnonstandard-analysis":21944,"\u002Flogic\u002Farithmetic-and-definability\u002Fdefinability-in-arithmetic":21948,"\u002Flogic\u002Farithmetic-and-definability\u002Fnatural-numbers-with-successor":21953,"\u002Flogic\u002Farithmetic-and-definability\u002Fpresburger-and-reducts":21957,"\u002Flogic\u002Farithmetic-and-definability\u002Fa-subtheory-and-representability":21961,"\u002Flogic\u002Fincompleteness\u002Farithmetization-of-syntax":21965,"\u002Flogic\u002Fincompleteness\u002Fincompleteness-and-undecidability":21970,"\u002Flogic\u002Fincompleteness\u002Fsecond-incompleteness-theorem":21974,"\u002Flogic\u002Fcomputability-and-representability\u002Frecursive-functions":21978,"\u002Flogic\u002Fcomputability-and-representability\u002Frepresenting-exponentiation":21983,"\u002Flogic\u002Fsecond-order-logic\u002Fsecond-order-languages":21987,"\u002Flogic\u002Fsecond-order-logic\u002Fskolem-functions-and-many-sorted-logic":21992,"\u002Flogic\u002Fsecond-order-logic\u002Fgeneral-structures":21996,"\u002Flogic":22000,"\u002Freinforcement-learning\u002Ffoundations\u002Fwhat-is-reinforcement-learning":22003,"\u002Freinforcement-learning\u002Ffoundations\u002Fa-brief-history-of-rl":22007,"\u002Freinforcement-learning\u002Ffoundations\u002Fmulti-armed-bandits":22011,"\u002Freinforcement-learning\u002Ffoundations\u002Fbandit-exploration-algorithms":22015,"\u002Freinforcement-learning\u002Ffoundations\u002Fmarkov-decision-processes":22019,"\u002Freinforcement-learning\u002Ffoundations\u002Fvalue-functions-and-optimality":22023,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdynamic-programming":22027,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdp-async-and-gpi":22031,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods":22035,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-off-policy":22039,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftemporal-difference-learning":22043,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftd-control-sarsa-and-q-learning":22047,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping":22051,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-off-policy-methods":22055,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-and-learning":22059,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-focusing-and-decision-time":22063,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdecision-time-planning":22067,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-tree-search":22071,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-prediction":22075,"\u002Freinforcement-learning\u002Fapproximation\u002Ffeature-construction-and-nonlinear":22079,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-control":22083,"\u002Freinforcement-learning\u002Fapproximation\u002Faverage-reward-control":22087,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad":22091,"\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td":22094,"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces":22098,"\u002Freinforcement-learning\u002Fapproximation\u002Ftrue-online-and-sarsa-lambda":22101,"\u002Freinforcement-learning\u002Fapproximation\u002Fpolicy-gradient-methods":22105,"\u002Freinforcement-learning\u002Fapproximation\u002Factor-critic-and-continuous-actions":22109,"\u002Freinforcement-learning\u002Fapproximation\u002Fleast-squares-and-memory-based-methods":22113,"\u002Freinforcement-learning\u002Fapproximation\u002Fmemory-and-kernel-methods":22117,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces":22121,"\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces":22122,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks":22125,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdqn-improvements":22129,"\u002Freinforcement-learning\u002Fdeep-rl\u002Factor-critic-and-ppo":22133,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fppo-and-continuous-control":22137,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fcase-studies":22141,"\u002Freinforcement-learning\u002Fdeep-rl\u002Frl-beyond-games":22145,"\u002Freinforcement-learning\u002Fdeep-rl\u002Ffrontiers":22149,"\u002Freinforcement-learning\u002Fdeep-rl\u002Freward-design-and-open-problems":22153,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow":22157,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow-part-2":22162,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control":22166,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control-part-2":22170,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl":22174,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl-part-2":22178,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration":22182,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration-part-2":22186,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl":22190,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2":22194,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl":22198,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl-part-2":22202,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl":22206,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl-part-2":22210,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl":22214,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl-part-2":22218,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Frlhf-and-language-models":22222,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps":22226,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps-part-2":22230,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl":22234,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl-part-2":22238,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmeta-rl-and-generalization":22242,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fpsychology-of-reinforcement":22246,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Finstrumental-conditioning-and-control":22251,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-and-td-error":22255,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-in-the-brain":22259,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fanimal-learning-and-cognition":22263,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fcognitive-maps-and-planning":22267,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fneuroscience-of-reinforcement":22271,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fseveral-learning-systems":22275,"\u002Freinforcement-learning":22279,"\u002Fartificial-intelligence\u002Ffoundations\u002Fwhat-is-ai":22281,"\u002Fartificial-intelligence\u002Ffoundations\u002Ffoundations-of-ai":22285,"\u002Fartificial-intelligence\u002Ffoundations\u002Fintelligent-agents":22289,"\u002Fartificial-intelligence\u002Ffoundations\u002Fagent-architectures":22293,"\u002Fartificial-intelligence\u002Fsearch\u002Funinformed-search":22297,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-strategies-compared":22302,"\u002Fartificial-intelligence\u002Fsearch\u002Finformed-search":22306,"\u002Fartificial-intelligence\u002Fsearch\u002Fheuristic-functions":22310,"\u002Fartificial-intelligence\u002Fsearch\u002Flocal-search":22314,"\u002Fartificial-intelligence\u002Fsearch\u002Fpopulation-and-continuous-search":22318,"\u002Fartificial-intelligence\u002Fsearch\u002Fadversarial-search":22322,"\u002Fartificial-intelligence\u002Fsearch\u002Fgames-of-chance-and-imperfect-information":22326,"\u002Fartificial-intelligence\u002Fsearch\u002Fconstraint-satisfaction":22330,"\u002Fartificial-intelligence\u002Fsearch\u002Fcsp-search-and-structure":22334,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-under-uncertainty":22338,"\u002Fartificial-intelligence\u002Fsearch\u002Fbelief-state-and-online-search":22342,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-logic":22346,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-inference":22351,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic":22355,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic-in-use":22359,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Finference-and-resolution":22363,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-resolution":22367,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fclassical-planning":22371,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-graphs-and-graphplan":22375,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-in-the-real-world":22379,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-under-uncertainty":22383,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fknowledge-representation":22387,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Freasoning-systems-and-defaults":22391,"\u002Fartificial-intelligence\u002Funcertainty\u002Fprobability-and-bayes":22395,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayes-rule-and-naive-bayes":22400,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayesian-networks":22404,"\u002Fartificial-intelligence\u002Funcertainty\u002Finference-in-bayesian-networks":22408,"\u002Fartificial-intelligence\u002Funcertainty\u002Freasoning-over-time":22412,"\u002Fartificial-intelligence\u002Funcertainty\u002Ftracking-and-data-association":22416,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmaking-decisions":22420,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmarkov-decision-processes":22424,"\u002Fartificial-intelligence\u002Funcertainty\u002Fdecision-networks-and-game-theory":22427,"\u002Fartificial-intelligence\u002Funcertainty\u002Fgame-theory-and-mechanism-design":22431,"\u002Fartificial-intelligence\u002Flearning\u002Flearning-from-examples":22435,"\u002Fartificial-intelligence\u002Flearning\u002Ftheory-and-model-families":22440,"\u002Fartificial-intelligence\u002Flearning\u002Fprobabilistic-learning":22444,"\u002Fartificial-intelligence\u002Flearning\u002Fexpectation-maximization":22448,"\u002Fartificial-intelligence\u002Flearning\u002Freinforcement-learning":22452,"\u002Fartificial-intelligence\u002Flearning\u002Fgeneralization-and-policy-search":22455,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-in-learning":22459,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-based-learning-methods":22463,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fvision-and-perception":22467,"\u002Fartificial-intelligence\u002Ffrontiers\u002Freconstructing-the-3d-world":22472,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobotics":22476,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobot-planning-and-control":22480,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnatural-language-in-ai":22484,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnlp-grammar-translation-and-speech":22488,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fphilosophy-and-future":22492,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fai-ethics-and-future":22496,"\u002Fartificial-intelligence":22500,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-constituents-nuclide-chart":22503,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-size-charge-distributions":22508,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-masses-binding-energy":22512,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fsemi-empirical-mass-formula":22516,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-moments-multipoles":22520,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnuclear-force-shell-overview":22524,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fthe-deuteron":22529,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnucleon-nucleon-scattering":22533,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fmeson-theory-isospin":22537,"\u002Fnuclear-physics\u002Fnuclear-models\u002Ffermi-gas-model":22541,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fliquid-drop-collective-coordinates":22546,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fshell-model-single-particle":22550,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fcollective-model-rotations-vibrations":22554,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-law-modes":22558,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-kinetics-equilibrium":22563,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-decay-gamow-theory":22567,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-fine-structure-hindrance":22572,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fbeta-decay-energetics-neutrino":22576,"\u002Fnuclear-physics\u002Fbeta-decay\u002Ffermi-theory-beta-decay":22581,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fweak-interaction-parity-violation":22585,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fdouble-beta-decay-neutrino-mass":22589,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fgamma-multipole-radiation":22593,"\u002Fnuclear-physics\u002Fgamma-decay\u002Finternal-conversion-isomers":22598,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fangular-correlations-mossbauer":22602,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Freaction-kinematics-cross-sections":22606,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fcompound-nucleus-resonances":22611,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fdirect-reactions-optical-model":22615,"\u002Fnuclear-physics\u002Ffission\u002Ffission-barrier-dynamics":22619,"\u002Fnuclear-physics\u002Ffission\u002Fchain-reactions-reactor-physics":22624,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Ffusion-reactions-confinement":22628,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fstellar-nucleosynthesis":22633,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fbig-bang-nucleosynthesis":22637,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fcharged-particle-stopping-power":22641,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fphoton-neutron-interactions":22646,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fradiation-detectors":22650,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fdosimetry-radiation-biology":22654,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fnuclear-applications-dating-medicine":22658,"\u002Fnuclear-physics":22662,"\u002Fnatural-language-processing\u002Ffoundations\u002Fwhat-is-nlp":22665,"\u002Fnatural-language-processing\u002Ffoundations\u002Fregex-and-text-normalization":22669,"\u002Fnatural-language-processing\u002Ffoundations\u002Fminimum-edit-distance":22673,"\u002Fnatural-language-processing\u002Ffoundations\u002Fn-gram-language-models":22677,"\u002Fnatural-language-processing\u002Ffoundations\u002Fsmoothing-and-backoff":22681,"\u002Fnatural-language-processing\u002Fclassification\u002Fnaive-bayes-and-sentiment":22685,"\u002Fnatural-language-processing\u002Fclassification\u002Fevaluating-classifiers":22690,"\u002Fnatural-language-processing\u002Fclassification\u002Flogistic-regression":22694,"\u002Fnatural-language-processing\u002Fclassification\u002Fsentiment-and-affect-lexicons":22698,"\u002Fnatural-language-processing\u002Fsemantics\u002Fvector-semantics-and-embeddings":22702,"\u002Fnatural-language-processing\u002Fsemantics\u002Fstatic-word-embeddings":22707,"\u002Fnatural-language-processing\u002Fsemantics\u002Fneural-language-models":22711,"\u002Fnatural-language-processing\u002Fsequences\u002Fsequence-labeling":22715,"\u002Fnatural-language-processing\u002Fsequences\u002Fcrfs-and-neural-taggers":22719,"\u002Fnatural-language-processing\u002Fsequences\u002Frnns-and-lstms":22723,"\u002Fnatural-language-processing\u002Ftransformers\u002Ftransformers-and-attention":22727,"\u002Fnatural-language-processing\u002Ftransformers\u002Fthe-transformer-architecture":22731,"\u002Fnatural-language-processing\u002Ftransformers\u002Flarge-language-models":22734,"\u002Fnatural-language-processing\u002Ftransformers\u002Fllm-pretraining-and-scaling":22737,"\u002Fnatural-language-processing\u002Ftransformers\u002Ffine-tuning-and-prompting":22741,"\u002Fnatural-language-processing\u002Ftransformers\u002Fprompting-and-alignment":22745,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-parsing":22749,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcky-scoring-and-evaluation":22754,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdependency-parsing":22758,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fgraph-based-and-neural-dependency-parsing":22762,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fword-senses-and-wsd":22766,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fwsd-in-practice-and-induction":22770,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-roles-and-information-extraction":22774,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Frelations-events-and-templates":22778,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoreference-and-discourse":22782,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoherence-and-discourse-structure":22786,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Flogical-semantics":22790,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcompositional-semantics-and-description-logics":22794,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-parsing":22798,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fneural-semantic-parsing":22802,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Finformation-extraction":22806,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftimes-events-and-templates":22810,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdiscourse-coherence":22814,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fentity-based-and-global-coherence":22818,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-grammars":22822,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftreebanks-and-lexicalized-grammars":22826,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation":22830,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation-decoding-and-evaluation":22834,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering":22838,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering-knowledge-and-llms":22842,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-and-chatbots":22846,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-systems-and-assistants":22850,"\u002Fnatural-language-processing\u002Fapplications\u002Ftext-summarization":22854,"\u002Fnatural-language-processing\u002Fapplications\u002Fabstractive-summarization-and-evaluation":22858,"\u002Fnatural-language-processing\u002Fspeech\u002Fphonetics":22862,"\u002Fnatural-language-processing\u002Fspeech\u002Facoustic-phonetics":22867,"\u002Fnatural-language-processing\u002Fspeech\u002Fautomatic-speech-recognition":22871,"\u002Fnatural-language-processing\u002Fspeech\u002Fasr-evaluation-and-applications":22875,"\u002Fnatural-language-processing":22879,"\u002Fparticle-physics\u002Ffoundations\u002Fhistorical-overview-particle-zoo":22882,"\u002Fparticle-physics\u002Ffoundations\u002Fparticle-physics-basic-concepts":22886,"\u002Fparticle-physics\u002Ffoundations\u002Ffundamental-interactions-force-carriers":22890,"\u002Fparticle-physics\u002Funits-kinematics\u002Fnatural-units-and-scales":22894,"\u002Fparticle-physics\u002Funits-kinematics\u002Ffour-vectors-invariant-mass":22899,"\u002Fparticle-physics\u002Funits-kinematics\u002Fdecay-scattering-kinematics-mandelstam":22903,"\u002Fparticle-physics\u002Funits-kinematics\u002Fcross-sections-golden-rule":22907,"\u002Fparticle-physics\u002Fsymmetries\u002Fconservation-laws-symmetries":22911,"\u002Fparticle-physics\u002Fsymmetries\u002Fdiscrete-symmetries-cpt":22916,"\u002Fparticle-physics\u002Fsymmetries\u002Fparity-violation-weak":22920,"\u002Fparticle-physics\u002Fsymmetries\u002Fsu2-su3-flavor-symmetry":22924,"\u002Fparticle-physics\u002Fquark-model\u002Feightfold-way-su3":22928,"\u002Fparticle-physics\u002Fquark-model\u002Fmeson-spectroscopy":22933,"\u002Fparticle-physics\u002Fquark-model\u002Fbaryon-spectroscopy":22937,"\u002Fparticle-physics\u002Fquark-model\u002Fcolor-confinement-exotics":22941,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fklein-gordon-equation":22945,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fdirac-equation-spinors":22950,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fantiparticles-hole-theory":22954,"\u002Fparticle-physics\u002Fqed\u002Ffeynman-rules-qed":22958,"\u002Fparticle-physics\u002Fqed\u002Fqed-tree-processes":22963,"\u002Fparticle-physics\u002Fqed\u002Frenormalization-running-coupling":22967,"\u002Fparticle-physics\u002Fqed\u002Felectron-g-2":22971,"\u002Fparticle-physics\u002Fweak-interaction\u002Fva-structure-weak":22975,"\u002Fparticle-physics\u002Fweak-interaction\u002Fw-z-bosons-decays":22980,"\u002Fparticle-physics\u002Fweak-interaction\u002Fckm-matrix":22984,"\u002Fparticle-physics\u002Fweak-interaction\u002Fcp-violation-kaons-b-mesons":22988,"\u002Fparticle-physics\u002Fqcd\u002Fcolor-su3-gluons":22992,"\u002Fparticle-physics\u002Fqcd\u002Fasymptotic-freedom-confinement":22997,"\u002Fparticle-physics\u002Fqcd\u002Fdeep-inelastic-scattering-partons":23001,"\u002Fparticle-physics\u002Fqcd\u002Fjets-hadronization":23005,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Felectroweak-su2-u1":23009,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fspontaneous-symmetry-breaking":23014,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-mechanism":23018,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-boson-discovery":23022,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fstandard-model":23026,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-oscillations":23030,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-mass-pmns":23035,"\u002Fparticle-physics\u002Fneutrinos\u002Fdirac-majorana-experiments":23039,"\u002Fparticle-physics\u002Fexperiment\u002Faccelerators-luminosity":23043,"\u002Fparticle-physics\u002Fexperiment\u002Fdetectors-subsystems":23048,"\u002Fparticle-physics\u002Fexperiment\u002Fhow-discoveries-are-made":23052,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fbeyond-standard-model":23056,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fgrand-unified-theories":23060,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fsupersymmetry":23064,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fhierarchy-problem-naturalness":23068,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fdark-matter-candidates":23072,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fmatter-antimatter-open-questions":23076,"\u002Fparticle-physics":23080,"\u002Fastrophysics-cosmology\u002Forientation\u002Fthe-sun-and-stars":23083,"\u002Fastrophysics-cosmology\u002Forientation\u002Fstellar-death-final-states":23088,"\u002Fastrophysics-cosmology\u002Forientation\u002Fgalaxies-and-cosmology":23092,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fmagnitudes-fluxes-and-the-distance-modulus":23096,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fstellar-spectra-and-spectral-classification":23101,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Ftelescopes-and-detectors-across-the-spectrum":23105,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fthe-cosmic-distance-ladder":23109,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fblackbody-radiation-and-specific-intensity":23113,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fradiative-transfer-and-the-transfer-equation":23118,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fspectral-line-formation-and-broadening":23122,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fopacity-and-the-rosseland-mean":23126,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fhydrostatic-equilibrium-and-the-virial-theorem":23130,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equations-of-stellar-structure":23135,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equation-of-state-and-polytropes":23139,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-standard-solar-model":23143,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fthermonuclear-reaction-rates-and-the-gamow-peak":23147,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhydrogen-burning-pp-chains-and-cno":23152,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhelium-burning-and-the-triple-alpha-process":23156,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fadvanced-burning-and-neutron-capture-nucleosynthesis":23160,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fphases-of-the-interstellar-medium":23164,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fmolecular-clouds-and-gravitational-collapse":23169,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fprotostars-and-the-pre-main-sequence":23173,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-main-sequence-and-its-structure":23177,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fpost-main-sequence-low-mass-evolution":23182,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-evolution-of-massive-stars":23186,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fstellar-pulsation-and-the-instability-strip":23190,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":23194,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fcore-collapse-supernovae":23198,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fthermonuclear-supernovae-type-ia":23202,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fneutron-stars-and-pulsars":23206,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fblack-holes-schwarzschild-and-kerr":23210,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fbinary-systems-and-mass-transfer":23214,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Faccreting-compact-objects":23219,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fgravitational-waves-from-inspiraling-binaries":23223,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fmultimessenger-astronomy-and-gamma-ray-bursts":23227,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fthe-milky-way":23231,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-morphology-and-classification":23236,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-rotation-curves-and-dark-matter":23240,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Factive-galactic-nuclei-and-supermassive-black-holes":23244,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-clusters-and-large-scale-structure":23248,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-expanding-universe-and-hubbles-law":23252,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-frw-metric-and-cosmological-redshift":23257,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-friedmann-equations-and-cosmic-dynamics":23261,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fcosmological-models-and-distances":23264,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fdark-energy-and-the-accelerating-universe":23268,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fthe-thermal-history-of-the-universe":23272,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fbig-bang-nucleosynthesis":23277,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Frecombination-and-the-cosmic-microwave-background":23281,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcmb-anisotropies-and-cosmological-parameters":23285,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcosmic-inflation":23289,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fstructure-formation-and-the-growth-of-perturbations":23293,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fdark-matter-dark-energy-and-open-questions":23297,"\u002Fastrophysics-cosmology":23301,"\u002Fcolophon":23304,"\u002F":23307},{"path":18627,"title":18628,"module":18629,"summary":18630},"\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":18632,"title":18633,"module":18629,"summary":18634},"\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":18636,"title":18637,"module":18629,"summary":18638},"\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":18640,"title":18641,"module":18629,"summary":18642},"\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":18644,"title":18645,"module":18629,"summary":18646},"\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":18648,"title":18649,"module":18629,"summary":18650},"\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":18652,"title":18653,"module":18654,"summary":18655},"\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":18657,"title":18658,"module":18654,"summary":18659},"\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":18661,"title":18662,"module":18654,"summary":18663},"\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":18665,"title":18666,"module":18654,"summary":18667},"\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":18669,"title":18670,"module":18671,"summary":18672},"\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":18674,"title":18675,"module":18671,"summary":18676},"\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":18678,"title":18679,"module":18671,"summary":18680},"\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":18682,"title":18683,"module":18671,"summary":18684},"\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":18686,"title":18687,"module":18688,"summary":18689},"\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":18691,"title":18692,"module":18688,"summary":18693},"\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":18695,"title":18696,"module":18688,"summary":18697},"\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":18699,"title":18700,"module":18688,"summary":18701},"\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":18703,"title":18704,"module":18688,"summary":18705},"\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":18707,"title":18708,"module":18688,"summary":18709},"\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":18711,"title":18712,"module":18688,"summary":18713},"\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":18715,"title":18716,"module":18688,"summary":18717},"\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":18719,"title":18720,"module":18688,"summary":18721},"\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":18723,"title":18724,"module":18688,"summary":18725},"\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":18727,"title":18728,"module":18688,"summary":18729},"\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":18731,"title":18732,"module":18688,"summary":18733},"\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":18735,"title":18736,"module":18737,"summary":18738},"\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":18740,"title":18741,"module":18737,"summary":18742},"\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":18744,"title":18745,"module":18737,"summary":18746},"\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":18748,"title":18749,"module":18737,"summary":18750},"\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":18752,"title":18753,"module":18737,"summary":18754},"\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":18756,"title":18757,"module":18737,"summary":18758},"\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":18760,"title":18761,"module":18737,"summary":18762},"\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":18764,"title":18765,"module":18737,"summary":18766},"\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":18768,"title":18769,"module":18770,"summary":18771},"\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":18773,"title":18774,"module":18770,"summary":18775},"\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":18777,"title":18778,"module":18770,"summary":18779},"\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":18781,"title":18782,"module":18770,"summary":18783},"\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":18785,"title":18786,"module":18770,"summary":18787},"\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":18789,"title":18790,"module":18770,"summary":18791},"\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":18793,"title":18794,"module":18770,"summary":18795},"\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":18797,"title":18798,"module":18770,"summary":18799},"\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":18801,"title":18802,"module":18770,"summary":18803},"\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":18805,"title":18806,"module":18770,"summary":18807},"\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":18809,"title":18810,"module":18770,"summary":18811},"\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":18813,"title":18814,"module":18770,"summary":18815},"\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":18817,"title":18818,"module":18770,"summary":18819},"\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":18821,"title":18822,"module":18770,"summary":18823},"\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":18825,"title":18826,"module":18827,"summary":18828},"\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":18830,"title":18831,"module":18827,"summary":18832},"\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":18834,"title":18835,"module":18827,"summary":18836},"\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":18838,"title":18839,"module":18827,"summary":18840},"\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":18842,"title":18843,"module":18827,"summary":18844},"\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":18846,"title":18847,"module":18848,"summary":18849},"\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":18851,"title":18852,"module":18848,"summary":18853},"\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":18855,"title":18856,"module":18848,"summary":18857},"\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":18859,"title":18860,"module":18848,"summary":18861},"\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":18863,"title":18864,"module":18848,"summary":18865},"\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":18867,"title":18868,"module":18848,"summary":18869},"\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":18871,"title":18872,"module":18848,"summary":18873},"\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":18875,"title":18876,"module":18848,"summary":18877},"\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":18879,"title":18880,"module":18848,"summary":18881},"\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":18883,"title":18884,"module":18848,"summary":18885},"\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":18887,"title":18888,"module":18848,"summary":18889},"\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":18891,"title":18892,"module":18893,"summary":18894},"\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":18896,"title":18897,"module":18893,"summary":18898},"\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":18900,"title":18901,"module":18893,"summary":18902},"\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":18904,"title":18905,"module":18893,"summary":18906},"\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":18908,"title":18909,"module":18910,"summary":18911},"\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":18913,"title":18914,"module":18910,"summary":18915},"\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":18917,"title":18918,"module":18910,"summary":18919},"\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":18921,"title":18922,"module":18910,"summary":18923},"\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":18925,"title":18926,"module":18910,"summary":18927},"\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":18929,"title":18930,"module":18910,"summary":18931},"\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":18933,"title":18934,"module":18910,"summary":18935},"\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":18937,"title":18938,"module":18939,"summary":18940},"\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":18942,"title":18943,"module":18939,"summary":18944},"\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":18946,"title":18947,"module":18939,"summary":18948},"\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":18950,"title":18951,"module":18939,"summary":18952},"\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":18954,"title":18955,"module":18956,"summary":18957},"\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":18959,"title":18960,"module":18956,"summary":18961},"\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":18963,"title":18964,"module":18956,"summary":18965},"\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":18967,"title":18968,"module":18956,"summary":18969},"\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":18971,"title":18972,"module":6,"summary":6},"\u002Falgorithms","Algorithms",{"path":18974,"title":18975,"module":18976,"summary":18977},"\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":18979,"title":18980,"module":18976,"summary":18981},"\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":18983,"title":18984,"module":18976,"summary":18985},"\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":18987,"title":18988,"module":18976,"summary":18989},"\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":18991,"title":18992,"module":18993,"summary":18994},"\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":18996,"title":18997,"module":18993,"summary":18998},"\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":19000,"title":19001,"module":18993,"summary":19002},"\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":19004,"title":19005,"module":18993,"summary":19006},"\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":19008,"title":19009,"module":19010,"summary":19011},"\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":19013,"title":19014,"module":19010,"summary":19015},"\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":19017,"title":19018,"module":19010,"summary":19019},"\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":19021,"title":19022,"module":19010,"summary":19023},"\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":19025,"title":19026,"module":19027,"summary":19028},"\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":19030,"title":19031,"module":19027,"summary":19032},"\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":19034,"title":19035,"module":19027,"summary":19036},"\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":19038,"title":19039,"module":19040,"summary":19041},"\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":19043,"title":19044,"module":19040,"summary":19045},"\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":19047,"title":19048,"module":19040,"summary":19049},"\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":19051,"title":19052,"module":19053,"summary":19054},"\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":19056,"title":19057,"module":19053,"summary":19058},"\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":19060,"title":19061,"module":19053,"summary":19062},"\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":19064,"title":19065,"module":19066,"summary":19067},"\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":19069,"title":19070,"module":19066,"summary":19071},"\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":19073,"title":19074,"module":19066,"summary":19075},"\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":19077,"title":19078,"module":19066,"summary":19079},"\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":19081,"title":19082,"module":19083,"summary":19084},"\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":19086,"title":19087,"module":19083,"summary":19088},"\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":19090,"title":19091,"module":19083,"summary":19092},"\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":19094,"title":19095,"module":19096,"summary":19097},"\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":19099,"title":19100,"module":19096,"summary":19101},"\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":19103,"title":19104,"module":19096,"summary":19105},"\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":19107,"title":19108,"module":19096,"summary":19109},"\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":19111,"title":19112,"module":19096,"summary":19113},"\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":19115,"title":19116,"module":19117,"summary":19118},"\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":19120,"title":19121,"module":19117,"summary":19122},"\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":19124,"title":19125,"module":19117,"summary":19126},"\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":19128,"title":19129,"module":19117,"summary":19130},"\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":19132,"title":19133,"module":19117,"summary":19134},"\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":19136,"title":19137,"module":19138,"summary":19139},"\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":19141,"title":19138,"module":19138,"summary":19142},"\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":19144,"title":19145,"module":19138,"summary":19146},"\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":19148,"title":19149,"module":19138,"summary":19150},"\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":19152,"title":19153,"module":19138,"summary":19154},"\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":19156,"title":19157,"module":19158,"summary":19159},"\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":19161,"title":19162,"module":19158,"summary":19163},"\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":19165,"title":19166,"module":19158,"summary":19167},"\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":19169,"title":19170,"module":19158,"summary":19171},"\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":19173,"title":19174,"module":19158,"summary":19175},"\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":19177,"title":19178,"module":19158,"summary":19179},"\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":19181,"title":19182,"module":6,"summary":6},"\u002Fcalculus","Calculus",{"path":19184,"title":19185,"module":18629,"summary":19186},"\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":19188,"title":19189,"module":18629,"summary":19190},"\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":19192,"title":19193,"module":19194,"summary":19195},"\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":19197,"title":19198,"module":19194,"summary":19199},"\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":19201,"title":19202,"module":19194,"summary":19203},"\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":19205,"title":19206,"module":19194,"summary":19207},"\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":19209,"title":19210,"module":19194,"summary":19211},"\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":19213,"title":19214,"module":19215,"summary":19216},"\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":19218,"title":19219,"module":19215,"summary":19220},"\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":19222,"title":19223,"module":19215,"summary":19224},"\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":19226,"title":19227,"module":19215,"summary":19228},"\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":19230,"title":19231,"module":19215,"summary":19232},"\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":19234,"title":19235,"module":19236,"summary":19237},"\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":19239,"title":19240,"module":19236,"summary":19241},"\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":19243,"title":19244,"module":19236,"summary":19245},"\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":19247,"title":19248,"module":19236,"summary":19249},"\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":19251,"title":19252,"module":19236,"summary":19253},"\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":19255,"title":19256,"module":19257,"summary":19258},"\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":19260,"title":19261,"module":19257,"summary":19262},"\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":19264,"title":19265,"module":19257,"summary":19266},"\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":19268,"title":19269,"module":19270,"summary":19271},"\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":19273,"title":19274,"module":19270,"summary":19275},"\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":19277,"title":19278,"module":19270,"summary":19279},"\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":19281,"title":19282,"module":19270,"summary":19283},"\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":19285,"title":19286,"module":19270,"summary":19287},"\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":19289,"title":19290,"module":19270,"summary":19291},"\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":19293,"title":19294,"module":19295,"summary":19296},"\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":19298,"title":19299,"module":19295,"summary":19300},"\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":19302,"title":19303,"module":19295,"summary":19304},"\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":19306,"title":19307,"module":19295,"summary":19308},"\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":19310,"title":19311,"module":19295,"summary":19312},"\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":19314,"title":19315,"module":19295,"summary":19316},"\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":19318,"title":19319,"module":19295,"summary":19320},"\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":19322,"title":19323,"module":19324,"summary":19325},"\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":19327,"title":19328,"module":19324,"summary":19329},"\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":19331,"title":19332,"module":19324,"summary":19333},"\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":19335,"title":19336,"module":19324,"summary":19337},"\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":19339,"title":19340,"module":19324,"summary":19341},"\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":19343,"title":19344,"module":19324,"summary":19345},"\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":19347,"title":19348,"module":19324,"summary":19349},"\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":19351,"title":19352,"module":19324,"summary":19353},"\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":19355,"title":19356,"module":19324,"summary":19357},"\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":19359,"title":19360,"module":19324,"summary":19361},"\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":19363,"title":19364,"module":19324,"summary":19365},"\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":19367,"title":19368,"module":19324,"summary":19369},"\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":19371,"title":19372,"module":19373,"summary":19374},"\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":19376,"title":19377,"module":19373,"summary":19378},"\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":19380,"title":19381,"module":19373,"summary":19382},"\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":19384,"title":19385,"module":19373,"summary":19386},"\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":19388,"title":19389,"module":19373,"summary":19390},"\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":19392,"title":19393,"module":19373,"summary":19394},"\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":19396,"title":19397,"module":6,"summary":6},"\u002Fmechanics","Mechanics & Dynamics",{"path":19399,"title":19400,"module":19401,"summary":19402},"\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":19404,"title":19405,"module":19401,"summary":19406},"\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":19408,"title":19409,"module":19401,"summary":19410},"\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":19412,"title":19413,"module":19401,"summary":19414},"\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":19416,"title":19417,"module":19401,"summary":19418},"\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":19420,"title":19421,"module":19422,"summary":19423},"\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":19425,"title":19426,"module":19422,"summary":19427},"\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":19429,"title":19430,"module":19431,"summary":19432},"\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":19434,"title":19435,"module":19431,"summary":19436},"\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":19438,"title":19439,"module":19431,"summary":19440},"\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":19442,"title":19443,"module":19431,"summary":19444},"\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":19446,"title":19447,"module":19431,"summary":19448},"\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":19450,"title":19451,"module":19452,"summary":19453},"\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":19455,"title":19456,"module":19452,"summary":19457},"\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":19459,"title":19460,"module":19452,"summary":19461},"\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":19463,"title":19464,"module":19452,"summary":19465},"\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":19467,"title":19468,"module":19469,"summary":19470},"\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":19472,"title":19473,"module":19469,"summary":19474},"\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":19476,"title":19477,"module":19469,"summary":19478},"\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":19480,"title":19481,"module":19482,"summary":19483},"\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":19485,"title":19486,"module":19482,"summary":19487},"\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":19489,"title":19490,"module":19482,"summary":19491},"\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":19493,"title":19494,"module":19482,"summary":19495},"\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":19497,"title":19498,"module":19482,"summary":19499},"\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":19501,"title":19502,"module":19503,"summary":19504},"\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":19506,"title":19507,"module":19503,"summary":19508},"\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":19510,"title":19511,"module":19503,"summary":19512},"\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":19514,"title":19515,"module":19503,"summary":19516},"\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":19518,"title":19519,"module":19503,"summary":19520},"\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":19522,"title":19523,"module":19503,"summary":19524},"\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":19526,"title":19527,"module":19528,"summary":19529},"\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":19531,"title":19532,"module":19528,"summary":19533},"\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":19535,"title":19536,"module":19528,"summary":19537},"\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":19539,"title":19540,"module":19528,"summary":19541},"\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":19543,"title":19544,"module":19528,"summary":19545},"\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":19547,"title":19548,"module":19528,"summary":19549},"\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":19551,"title":19552,"module":19528,"summary":19553},"\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":19555,"title":19556,"module":19528,"summary":19557},"\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":19559,"title":19560,"module":19561,"summary":19562},"\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":19564,"title":19565,"module":19561,"summary":19566},"\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":19568,"title":19569,"module":19561,"summary":19570},"\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":19572,"title":19573,"module":19561,"summary":19574},"\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":19576,"title":19577,"module":19561,"summary":19578},"\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":19580,"title":19581,"module":19582,"summary":19583},"\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":19585,"title":19586,"module":19582,"summary":19587},"\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":19589,"title":19590,"module":19582,"summary":19591},"\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":19593,"title":19594,"module":19582,"summary":19595},"\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":19597,"title":19598,"module":19582,"summary":19599},"\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":19601,"title":19602,"module":19603,"summary":19604},"\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":19606,"title":19607,"module":19603,"summary":19608},"\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":19610,"title":19611,"module":19603,"summary":19612},"\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":19614,"title":19615,"module":6,"summary":6},"\u002Felectricity-and-magnetism","Electricity & Magnetism",{"path":19617,"title":19618,"module":19619,"summary":19620},"\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":19622,"title":19623,"module":19619,"summary":19624},"\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":19626,"title":19627,"module":19619,"summary":19628},"\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":19630,"title":19631,"module":19619,"summary":19632},"\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":19634,"title":19635,"module":19619,"summary":19636},"\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":19638,"title":19639,"module":19640,"summary":19641},"\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":19643,"title":19644,"module":19640,"summary":19645},"\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":19647,"title":19648,"module":19640,"summary":19649},"\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":19651,"title":19652,"module":19640,"summary":19653},"\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":19655,"title":19656,"module":19640,"summary":19657},"\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":19659,"title":19660,"module":19661,"summary":19662},"\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":19664,"title":19665,"module":19661,"summary":19666},"\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":19668,"title":19669,"module":19661,"summary":19670},"\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":19672,"title":19673,"module":19674,"summary":19675},"\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":19677,"title":19678,"module":19674,"summary":19679},"\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":19681,"title":19682,"module":19674,"summary":19683},"\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":19685,"title":19686,"module":19674,"summary":19687},"\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":19689,"title":19690,"module":19674,"summary":19691},"\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":19693,"title":19694,"module":19674,"summary":19695},"\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":19697,"title":19698,"module":19674,"summary":19699},"\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":19701,"title":19702,"module":19703,"summary":19704},"\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":19706,"title":19707,"module":19703,"summary":19708},"\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":19710,"title":19711,"module":19703,"summary":19712},"\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":19714,"title":19715,"module":19703,"summary":19716},"\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":19718,"title":19719,"module":19703,"summary":19720},"\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":19722,"title":19723,"module":19703,"summary":19724},"\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":19726,"title":19727,"module":19703,"summary":19728},"\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":19730,"title":19731,"module":19732,"summary":19733},"\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":19735,"title":19736,"module":19732,"summary":19737},"\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":19739,"title":19740,"module":19732,"summary":19741},"\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":19743,"title":19744,"module":19732,"summary":19745},"\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":19747,"title":19748,"module":19732,"summary":19749},"\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":19751,"title":19752,"module":19732,"summary":19753},"\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":19755,"title":19756,"module":19757,"summary":19758},"\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":19760,"title":19761,"module":19757,"summary":19762},"\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":19764,"title":19765,"module":19757,"summary":19766},"\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":19768,"title":19769,"module":19757,"summary":19770},"\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":19772,"title":19773,"module":19757,"summary":19774},"\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":19776,"title":19777,"module":19778,"summary":19779},"\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":19781,"title":19782,"module":19778,"summary":19783},"\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":19785,"title":19786,"module":19778,"summary":19787},"\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":19789,"title":19790,"module":19778,"summary":19791},"\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":19793,"title":19794,"module":19778,"summary":19795},"\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":19797,"title":19798,"module":19778,"summary":19799},"\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":19801,"title":19802,"module":19803,"summary":19804},"\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":19806,"title":19807,"module":19803,"summary":19808},"\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":19810,"title":19811,"module":19803,"summary":19812},"\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":19814,"title":19815,"module":19803,"summary":19816},"\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":19818,"title":19819,"module":19803,"summary":19820},"\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":19822,"title":19823,"module":6,"summary":6},"\u002Flinear-algebra","Linear Algebra",{"path":19825,"title":19826,"module":6,"summary":6},"\u002Ftheory-of-computation","Theory of Computation",{"path":19828,"title":19829,"module":18629,"summary":19830},"\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":19832,"title":19833,"module":18629,"summary":19834},"\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":19836,"title":19837,"module":18629,"summary":19838},"\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":19840,"title":19841,"module":18629,"summary":19842},"\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":19844,"title":19845,"module":18629,"summary":19846},"\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":19848,"title":19849,"module":19850,"summary":19851},"\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":19853,"title":19854,"module":19850,"summary":19855},"\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":19857,"title":19858,"module":19850,"summary":19859},"\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":19861,"title":19862,"module":19850,"summary":19863},"\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":19865,"title":19866,"module":19850,"summary":19867},"\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":19869,"title":19870,"module":19850,"summary":19871},"\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":19873,"title":19874,"module":19850,"summary":19875},"\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":19877,"title":19878,"module":19879,"summary":19880},"\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":19882,"title":19883,"module":19879,"summary":19884},"\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":19886,"title":19887,"module":19879,"summary":19888},"\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":19890,"title":19891,"module":19879,"summary":19892},"\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":19894,"title":19895,"module":19879,"summary":19896},"\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":19898,"title":19899,"module":19900,"summary":19901},"\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":19903,"title":19904,"module":19900,"summary":19905},"\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":19907,"title":19908,"module":19900,"summary":19909},"\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":19911,"title":19912,"module":19900,"summary":19913},"\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":19915,"title":19916,"module":19900,"summary":19917},"\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":19919,"title":19920,"module":19921,"summary":19922},"\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":19924,"title":19925,"module":19921,"summary":19926},"\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":19928,"title":19929,"module":19921,"summary":19930},"\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":19932,"title":19933,"module":19921,"summary":19934},"\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":19936,"title":19937,"module":19921,"summary":19938},"\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":19940,"title":19941,"module":19942,"summary":19943},"\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":19945,"title":19946,"module":19942,"summary":19947},"\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":19949,"title":19950,"module":19942,"summary":19951},"\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":19953,"title":19954,"module":19942,"summary":19955},"\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":19957,"title":19958,"module":19942,"summary":19959},"\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":19961,"title":19962,"module":19963,"summary":19964},"\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":19966,"title":19967,"module":19963,"summary":19968},"\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":19970,"title":19971,"module":19963,"summary":19972},"\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":19974,"title":19975,"module":19963,"summary":19976},"\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":19978,"title":19979,"module":19963,"summary":19980},"\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":19982,"title":19983,"module":19984,"summary":19985},"\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":19987,"title":19988,"module":19984,"summary":19989},"\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":19991,"title":19992,"module":19984,"summary":19993},"\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":19995,"title":19996,"module":19997,"summary":19998},"\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":20000,"title":20001,"module":19997,"summary":20002},"\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":20004,"title":20005,"module":20006,"summary":20007},"\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":20009,"title":20010,"module":20006,"summary":20011},"\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":20013,"title":20014,"module":20006,"summary":20015},"\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":20017,"title":20018,"module":20006,"summary":20019},"\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":20021,"title":20022,"module":20006,"summary":20023},"\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":20025,"title":20026,"module":20027,"summary":20028},"\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":20030,"title":20031,"module":20027,"summary":20032},"\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":20034,"title":20035,"module":6,"summary":6},"\u002Fcomputer-architecture","Computer Architecture",{"path":20037,"title":20038,"module":18629,"summary":20039},"\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":20041,"title":20042,"module":18629,"summary":20043},"\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":20045,"title":20046,"module":20047,"summary":20048},"\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":20050,"title":20051,"module":20047,"summary":20052},"\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":20054,"title":20055,"module":20047,"summary":20056},"\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":20058,"title":20059,"module":20047,"summary":20060},"\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":20062,"title":20063,"module":20047,"summary":20064},"\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":20066,"title":20067,"module":20047,"summary":20068},"\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":20070,"title":20071,"module":20072,"summary":20073},"\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":20075,"title":20076,"module":20072,"summary":20077},"\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":20079,"title":20080,"module":20072,"summary":20081},"\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":20083,"title":20084,"module":20072,"summary":20085},"\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":20087,"title":20088,"module":20072,"summary":20089},"\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":20091,"title":20092,"module":20072,"summary":20093},"\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":20095,"title":20096,"module":20097,"summary":20098},"\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":20100,"title":20101,"module":20097,"summary":20102},"\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":20104,"title":20105,"module":20097,"summary":20106},"\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":20108,"title":20109,"module":20110,"summary":20111},"\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":20113,"title":20114,"module":20110,"summary":20115},"\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":20117,"title":20118,"module":20119,"summary":20120},"\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":20122,"title":20123,"module":20119,"summary":20124},"\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":20126,"title":20127,"module":20119,"summary":20128},"\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":20130,"title":20131,"module":20132,"summary":20133},"\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":20135,"title":20136,"module":20132,"summary":20137},"\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":20139,"title":20140,"module":20141,"summary":20142},"\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":20144,"title":20145,"module":20141,"summary":20146},"\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":20148,"title":20149,"module":20141,"summary":20150},"\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":20152,"title":20153,"module":20154,"summary":20155},"\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":20157,"title":20158,"module":20154,"summary":20159},"\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":20161,"title":20162,"module":20154,"summary":20163},"\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":20165,"title":20166,"module":20167,"summary":20168},"\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":20170,"title":20171,"module":20167,"summary":20172},"\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":20174,"title":20175,"module":6,"summary":6},"\u002Fdifferential-equations","Differential Equations",{"path":20177,"title":20178,"module":20179,"summary":20180},"\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":20182,"title":20183,"module":20179,"summary":20184},"\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":20186,"title":20187,"module":20179,"summary":20188},"\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":20190,"title":20191,"module":20179,"summary":20192},"\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":20194,"title":20195,"module":20179,"summary":20196},"\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":20198,"title":20199,"module":20200,"summary":20201},"\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":20203,"title":20204,"module":20200,"summary":20205},"\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":20207,"title":20208,"module":20200,"summary":20209},"\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":20211,"title":20212,"module":20200,"summary":20213},"\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":20215,"title":20216,"module":20217,"summary":20218},"\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":20220,"title":20221,"module":20217,"summary":20222},"\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":20224,"title":20225,"module":20217,"summary":20226},"\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":20228,"title":20229,"module":20217,"summary":20230},"\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":20232,"title":20233,"module":20234,"summary":20235},"\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":20237,"title":20238,"module":20234,"summary":20239},"\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":20241,"title":20242,"module":20234,"summary":20243},"\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":20245,"title":20246,"module":20234,"summary":20247},"\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":20249,"title":20250,"module":20251,"summary":20252},"\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":20254,"title":20255,"module":20251,"summary":20256},"\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":20258,"title":20259,"module":20251,"summary":20260},"\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":20262,"title":20263,"module":20251,"summary":20264},"\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":20266,"title":20267,"module":20251,"summary":20268},"\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":20270,"title":20271,"module":20251,"summary":20272},"\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":20274,"title":20275,"module":20276,"summary":20277},"\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":20279,"title":20280,"module":20276,"summary":20281},"\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":20283,"title":20284,"module":20276,"summary":20285},"\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":20287,"title":20288,"module":20289,"summary":20290},"\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":20292,"title":20293,"module":20289,"summary":20294},"\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":20296,"title":20297,"module":20289,"summary":20298},"\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":20300,"title":20301,"module":20289,"summary":20302},"\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":20304,"title":20305,"module":20306,"summary":20307},"\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":20309,"title":20310,"module":20306,"summary":20311},"\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":20313,"title":20314,"module":20306,"summary":20315},"\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":20317,"title":20318,"module":20319,"summary":20320},"\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":20322,"title":20323,"module":20319,"summary":20324},"\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":20326,"title":20327,"module":20319,"summary":20328},"\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":20330,"title":20331,"module":20332,"summary":20333},"\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":20335,"title":20336,"module":20332,"summary":20337},"\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":20339,"title":20340,"module":6,"summary":6},"\u002Frelativity","Relativity",{"path":20342,"title":20343,"module":6,"summary":6},"\u002Fphysical-computing","Physical Computing",{"path":20345,"title":20346,"module":20347,"summary":20348},"\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":20350,"title":20351,"module":20347,"summary":20352},"\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":20354,"title":20355,"module":20347,"summary":20356},"\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":20358,"title":20359,"module":20347,"summary":20360},"\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":20362,"title":20363,"module":20364,"summary":20365},"\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":20367,"title":20368,"module":20364,"summary":20369},"\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":20371,"title":20372,"module":20364,"summary":20373},"\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":20375,"title":20376,"module":20377,"summary":20378},"\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":20380,"title":20381,"module":20377,"summary":20382},"\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":20384,"title":20385,"module":20377,"summary":20386},"\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":20388,"title":20389,"module":20377,"summary":20390},"\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":20392,"title":20393,"module":20377,"summary":20394},"\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":20396,"title":20397,"module":20377,"summary":20398},"\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":20400,"title":20401,"module":20402,"summary":20403},"\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":20405,"title":20406,"module":20402,"summary":20407},"\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":20409,"title":20410,"module":20402,"summary":20411},"\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":20413,"title":20414,"module":20402,"summary":20415},"\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":20417,"title":20418,"module":20402,"summary":20419},"\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":20421,"title":20422,"module":20402,"summary":20423},"\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":20425,"title":20426,"module":20427,"summary":20428},"\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":20430,"title":20431,"module":20427,"summary":20432},"\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":20434,"title":20435,"module":20427,"summary":20436},"\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":20438,"title":20439,"module":20427,"summary":20440},"\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":20442,"title":20443,"module":19282,"summary":20444},"\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":20446,"title":20447,"module":19282,"summary":20448},"\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":20450,"title":20451,"module":19282,"summary":20452},"\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":20454,"title":20455,"module":20456,"summary":20457},"\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":20459,"title":20460,"module":20456,"summary":20461},"\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":20463,"title":20464,"module":20456,"summary":20465},"\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":20467,"title":20468,"module":20469,"summary":20470},"\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":20472,"title":20473,"module":20469,"summary":20474},"\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":20476,"title":20477,"module":20469,"summary":20478},"\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":20480,"title":20481,"module":20482,"summary":20483},"\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":20485,"title":20486,"module":20482,"summary":20487},"\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":20489,"title":20490,"module":20491,"summary":20492},"\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":20494,"title":20495,"module":20491,"summary":20496},"\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":20498,"title":20499,"module":20491,"summary":20500},"\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":20502,"title":20503,"module":20491,"summary":20504},"\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":20506,"title":20507,"module":20491,"summary":20508},"\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":20510,"title":20511,"module":6,"summary":6},"\u002Fquantum-mechanics","Quantum Mechanics",{"path":20513,"title":20514,"module":20515,"summary":20516},"\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":20518,"title":20519,"module":20515,"summary":20520},"\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":20522,"title":20523,"module":20515,"summary":20524},"\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":20526,"title":20527,"module":20515,"summary":20528},"\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":20530,"title":20531,"module":20532,"summary":20533},"\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":20535,"title":20536,"module":20532,"summary":20537},"\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":20539,"title":20540,"module":20532,"summary":20541},"\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":20543,"title":20544,"module":20532,"summary":20545},"\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":20547,"title":20548,"module":20532,"summary":20549},"\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":20551,"title":20552,"module":20532,"summary":20553},"\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":20555,"title":20556,"module":20557,"summary":20558},"\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":20560,"title":20561,"module":20557,"summary":20562},"\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":20564,"title":20565,"module":20557,"summary":20566},"\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":20568,"title":20569,"module":20557,"summary":20570},"\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":20572,"title":20573,"module":20557,"summary":20574},"\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":20576,"title":20577,"module":18976,"summary":20578},"\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":20580,"title":20581,"module":18976,"summary":20582},"\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":20584,"title":20585,"module":18976,"summary":20586},"\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":20588,"title":20589,"module":18976,"summary":20590},"\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":20592,"title":20593,"module":18976,"summary":20594},"\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":20596,"title":20597,"module":18976,"summary":20598},"\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":20600,"title":20601,"module":20602,"summary":20603},"\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":20605,"title":20606,"module":20602,"summary":20607},"\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":20609,"title":20610,"module":20602,"summary":20611},"\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":20613,"title":20614,"module":20602,"summary":20615},"\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":20617,"title":20618,"module":20619,"summary":20620},"\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":20622,"title":20623,"module":20619,"summary":20624},"\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":20626,"title":20627,"module":20619,"summary":20628},"\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":20630,"title":19031,"module":20619,"summary":20631},"\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":20633,"title":20634,"module":20619,"summary":20635},"\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":20637,"title":20638,"module":20639,"summary":20640},"\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":20642,"title":20643,"module":20639,"summary":20644},"\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":20646,"title":20647,"module":20639,"summary":20648},"\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":20650,"title":20651,"module":20639,"summary":20652},"\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":20654,"title":20655,"module":20656,"summary":20657},"\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":20659,"title":20660,"module":20656,"summary":20661},"\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":20663,"title":20664,"module":20656,"summary":20665},"\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":20667,"title":20668,"module":20656,"summary":20669},"\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":20671,"title":20672,"module":20656,"summary":20673},"\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":20675,"title":20676,"module":6,"summary":6},"\u002Freal-analysis","Real Analysis",{"path":20678,"title":20679,"module":18629,"summary":20680},"\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":20682,"title":20683,"module":18629,"summary":20684},"\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":20686,"title":20687,"module":20688,"summary":20689},"\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":20691,"title":20692,"module":20688,"summary":20693},"\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":20695,"title":20696,"module":20688,"summary":20697},"\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":20699,"title":20700,"module":20688,"summary":20701},"\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":20703,"title":20704,"module":20705,"summary":20706},"\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":20708,"title":20709,"module":20705,"summary":20710},"\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":20712,"title":20713,"module":20705,"summary":20714},"\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":20716,"title":20717,"module":20705,"summary":20718},"\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":20720,"title":20721,"module":20705,"summary":20722},"\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":20724,"title":20725,"module":20705,"summary":20726},"\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":20728,"title":20729,"module":20730,"summary":20731},"\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":20733,"title":20734,"module":20730,"summary":20735},"\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":20737,"title":20738,"module":20730,"summary":20739},"\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":20741,"title":20742,"module":20730,"summary":20743},"\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":20745,"title":20746,"module":20747,"summary":20748},"\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":20750,"title":20751,"module":20747,"summary":20752},"\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":20754,"title":20755,"module":20747,"summary":20756},"\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":20758,"title":20759,"module":20747,"summary":20760},"\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":20762,"title":20763,"module":20764,"summary":20765},"\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":20767,"title":20768,"module":20764,"summary":20769},"\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":20771,"title":20772,"module":20764,"summary":20773},"\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":20775,"title":20776,"module":20777,"summary":20778},"\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":20780,"title":20781,"module":20777,"summary":20782},"\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":20784,"title":20785,"module":20777,"summary":20786},"\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":20788,"title":20789,"module":20777,"summary":20790},"\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":20792,"title":20793,"module":20794,"summary":20795},"\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":20797,"title":20798,"module":20794,"summary":20799},"\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":20801,"title":20802,"module":20794,"summary":20803},"\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":20805,"title":20806,"module":20794,"summary":20807},"\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":20809,"title":20810,"module":20811,"summary":20812},"\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":20814,"title":20815,"module":20811,"summary":20816},"\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":20818,"title":20819,"module":20811,"summary":20820},"\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":20822,"title":20823,"module":20824,"summary":20825},"\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":20827,"title":20828,"module":20824,"summary":20829},"\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":20831,"title":20832,"module":20824,"summary":20833},"\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":20835,"title":20836,"module":20824,"summary":20837},"\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":20839,"title":20840,"module":20841,"summary":20842},"\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":20844,"title":20845,"module":20841,"summary":20846},"\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":20848,"title":20849,"module":20841,"summary":20850},"\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":20852,"title":20853,"module":20841,"summary":20854},"\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":20856,"title":20857,"module":20841,"summary":20858},"\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":20860,"title":20861,"module":20862,"summary":20863},"\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":20865,"title":20866,"module":20862,"summary":20867},"\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":20869,"title":20870,"module":6,"summary":6},"\u002Fabstract-algebra","Abstract Algebra",{"path":20872,"title":20873,"module":20874,"summary":20875},"\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":20877,"title":20878,"module":20874,"summary":20879},"\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":20881,"title":20882,"module":20874,"summary":20883},"\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":20885,"title":20886,"module":20874,"summary":20887},"\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":20889,"title":20890,"module":20874,"summary":20891},"\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":20893,"title":20894,"module":20895,"summary":20896},"\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":20898,"title":20899,"module":20895,"summary":20900},"\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":20902,"title":20903,"module":20895,"summary":20904},"\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":20906,"title":20907,"module":20895,"summary":20908},"\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":20910,"title":20911,"module":20895,"summary":20912},"\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":20914,"title":20915,"module":20895,"summary":20916},"\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":20918,"title":20919,"module":20895,"summary":20920},"\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":20922,"title":20923,"module":20924,"summary":20925},"\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":20927,"title":20928,"module":20924,"summary":20929},"\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":20931,"title":20932,"module":20924,"summary":20933},"\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":20935,"title":20936,"module":20924,"summary":20937},"\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":20939,"title":20940,"module":20941,"summary":20942},"\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":20944,"title":20945,"module":20941,"summary":20946},"\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":20948,"title":20949,"module":20941,"summary":20950},"\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":20952,"title":20953,"module":20954,"summary":20955},"\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":20957,"title":20958,"module":20954,"summary":20959},"\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":20961,"title":20962,"module":20954,"summary":20963},"\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":20965,"title":20966,"module":20954,"summary":20967},"\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":20969,"title":20970,"module":20954,"summary":20971},"\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":20973,"title":20974,"module":20954,"summary":20975},"\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":20977,"title":20978,"module":20979,"summary":20980},"\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":20982,"title":20983,"module":20979,"summary":20984},"\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":20986,"title":20987,"module":20979,"summary":20988},"\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":20990,"title":20991,"module":20992,"summary":20993},"\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":20995,"title":20996,"module":20992,"summary":20997},"\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":20999,"title":21000,"module":20992,"summary":21001},"\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":21003,"title":21004,"module":20992,"summary":21005},"\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":21007,"title":21008,"module":21009,"summary":21010},"\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":21012,"title":21013,"module":21009,"summary":21014},"\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":21016,"title":21017,"module":21009,"summary":21018},"\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":21020,"title":21021,"module":21022,"summary":21023},"\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":21025,"title":21026,"module":21022,"summary":21027},"\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":21029,"title":21030,"module":21022,"summary":21031},"\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":21033,"title":21034,"module":21022,"summary":21035},"\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":21037,"title":21038,"module":6,"summary":6},"\u002Fatomic-physics","Atomic Physics",{"path":21040,"title":21041,"module":6,"summary":6},"\u002Fdatabases","Databases",{"path":21043,"title":21044,"module":18629,"summary":21045},"\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":21047,"title":21048,"module":18629,"summary":21049},"\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":21051,"title":21052,"module":18629,"summary":21053},"\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":21055,"title":21056,"module":18629,"summary":21057},"\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":21059,"title":21060,"module":18629,"summary":21061},"\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":21063,"title":21064,"module":18629,"summary":21065},"\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":21067,"title":21068,"module":21069,"summary":21070},"\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":21072,"title":21073,"module":21069,"summary":21074},"\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":21076,"title":21077,"module":21069,"summary":21078},"\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":21080,"title":21081,"module":21082,"summary":21083},"\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":21085,"title":21086,"module":21082,"summary":21087},"\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":21089,"title":21090,"module":21082,"summary":21091},"\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":21093,"title":21094,"module":21095,"summary":21096},"\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":21098,"title":21099,"module":21095,"summary":21100},"\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":21102,"title":21103,"module":21095,"summary":21104},"\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":21106,"title":21107,"module":21095,"summary":21108},"\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":21110,"title":21111,"module":21095,"summary":21112},"\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":21114,"title":21115,"module":21116,"summary":21117},"\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":21119,"title":21120,"module":21116,"summary":21121},"\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":21123,"title":21124,"module":21116,"summary":21125},"\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":21127,"title":21128,"module":21116,"summary":21129},"\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":21131,"title":21132,"module":21133,"summary":21134},"\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":21136,"title":21137,"module":21133,"summary":21138},"\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":21140,"title":21141,"module":21133,"summary":21142},"\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":21144,"title":21145,"module":21133,"summary":21146},"\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":21148,"title":21149,"module":21150,"summary":21151},"\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":21153,"title":21154,"module":21150,"summary":21155},"\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":21157,"title":21158,"module":21150,"summary":21159},"\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":21161,"title":21162,"module":21150,"summary":21163},"\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":21165,"title":21166,"module":21167,"summary":21168},"\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":21170,"title":21171,"module":21167,"summary":21172},"\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":21174,"title":21175,"module":21167,"summary":21176},"\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":21178,"title":21179,"module":6,"summary":6},"\u002Fcategory-theory","Category Theory",{"path":21181,"title":19823,"module":21182,"summary":21183},"\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":21185,"title":21186,"module":21182,"summary":21187},"\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":21189,"title":21190,"module":21182,"summary":21191},"\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":21193,"title":19182,"module":21182,"summary":21194},"\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":21196,"title":21197,"module":18629,"summary":21198},"\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":21200,"title":21201,"module":18629,"summary":21202},"\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":21204,"title":21205,"module":18629,"summary":21206},"\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":21208,"title":21209,"module":21210,"summary":21211},"\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":21213,"title":21214,"module":21210,"summary":21215},"\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":21217,"title":21218,"module":21210,"summary":21219},"\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":21221,"title":21222,"module":21210,"summary":21223},"\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":21225,"title":21226,"module":21210,"summary":21227},"\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":21229,"title":21230,"module":21231,"summary":21232},"\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":21234,"title":21235,"module":21231,"summary":21236},"\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":21238,"title":21239,"module":21231,"summary":21240},"\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":21242,"title":21243,"module":21231,"summary":21244},"\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":21246,"title":21247,"module":21231,"summary":21248},"\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":21250,"title":21251,"module":21252,"summary":21253},"\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":21255,"title":21256,"module":21252,"summary":21257},"\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":21259,"title":21260,"module":21252,"summary":21261},"\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":21263,"title":21264,"module":21252,"summary":21265},"\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":21267,"title":21268,"module":21269,"summary":21270},"\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":21272,"title":21273,"module":21269,"summary":21274},"\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":21276,"title":21277,"module":21269,"summary":21278},"\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":21280,"title":21281,"module":21269,"summary":21282},"\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":21284,"title":21285,"module":21269,"summary":21286},"\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":21288,"title":21289,"module":21269,"summary":21290},"\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":21292,"title":21293,"module":21269,"summary":21294},"\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":21296,"title":21297,"module":21269,"summary":21298},"\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":21300,"title":21301,"module":21269,"summary":21302},"\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":21304,"title":21305,"module":21306,"summary":21307},"\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":21309,"title":21310,"module":21306,"summary":21311},"\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":21313,"title":21314,"module":21306,"summary":21315},"\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":21317,"title":21318,"module":21306,"summary":21319},"\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":21321,"title":21322,"module":21306,"summary":21323},"\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":21325,"title":21326,"module":21327,"summary":21328},"\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":21330,"title":21331,"module":21327,"summary":21332},"\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":21334,"title":21335,"module":21327,"summary":21336},"\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":21338,"title":21339,"module":21327,"summary":21340},"\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":21342,"title":21343,"module":21327,"summary":21344},"\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":21346,"title":21347,"module":21327,"summary":21348},"\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":21350,"title":21351,"module":21327,"summary":21352},"\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":21354,"title":21355,"module":21356,"summary":21357},"\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":21359,"title":21360,"module":21356,"summary":21361},"\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":21363,"title":21364,"module":21356,"summary":21365},"\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":21367,"title":21368,"module":21369,"summary":21370},"\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":21372,"title":21373,"module":21369,"summary":21374},"\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":21376,"title":21377,"module":21369,"summary":21378},"\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":21380,"title":21381,"module":21369,"summary":21382},"\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":21384,"title":21385,"module":21369,"summary":21386},"\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":21388,"title":21389,"module":21369,"summary":21390},"\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":21392,"title":21393,"module":21369,"summary":21394},"\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":21396,"title":21397,"module":21398,"summary":21399},"\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":21401,"title":21402,"module":21398,"summary":21403},"\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":21405,"title":21406,"module":21398,"summary":21407},"\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":21409,"title":21410,"module":21398,"summary":21411},"\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":21413,"title":21414,"module":21398,"summary":21415},"\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":21417,"title":21418,"module":21398,"summary":21419},"\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":21421,"title":21422,"module":21398,"summary":21423},"\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":21425,"title":21426,"module":21398,"summary":21427},"\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":21429,"title":21430,"module":21398,"summary":21431},"\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":21433,"title":21434,"module":21398,"summary":21435},"\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":21437,"title":21438,"module":21398,"summary":21439},"\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":21441,"title":21442,"module":21443,"summary":21444},"\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":21446,"title":21447,"module":21443,"summary":21448},"\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":21450,"title":21451,"module":21443,"summary":21452},"\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":21454,"title":21455,"module":21443,"summary":21456},"\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":21458,"title":21459,"module":21443,"summary":21460},"\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":21462,"title":21463,"module":6,"summary":6},"\u002Fdeep-learning","Deep Learning",{"path":21465,"title":21466,"module":19373,"summary":21467},"\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":21469,"title":21470,"module":19373,"summary":21471},"\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":21473,"title":21474,"module":19373,"summary":21475},"\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":21477,"title":21478,"module":19373,"summary":21479},"\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":21481,"title":21482,"module":19373,"summary":21483},"\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":21485,"title":21486,"module":21487,"summary":21488},"\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":21490,"title":21491,"module":21487,"summary":21492},"\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":21494,"title":21495,"module":21487,"summary":21496},"\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":21498,"title":21499,"module":21487,"summary":21500},"\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":21502,"title":21503,"module":21504,"summary":21505},"\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":21507,"title":21508,"module":21504,"summary":21509},"\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":21511,"title":21512,"module":21504,"summary":21513},"\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":21515,"title":21516,"module":21504,"summary":21517},"\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":21519,"title":21520,"module":21521,"summary":21522},"\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":21524,"title":21525,"module":21521,"summary":21526},"\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":21528,"title":21529,"module":21521,"summary":21530},"\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":21532,"title":21533,"module":21521,"summary":21534},"\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":21536,"title":21537,"module":21521,"summary":21538},"\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":21540,"title":21541,"module":21542,"summary":21543},"\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":21545,"title":21546,"module":21542,"summary":21547},"\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":21549,"title":21550,"module":21542,"summary":21551},"\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":21553,"title":21554,"module":21555,"summary":21556},"\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":21558,"title":21559,"module":21555,"summary":21560},"\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":21562,"title":21563,"module":21555,"summary":21564},"\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":21566,"title":21567,"module":21568,"summary":21569},"\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":21571,"title":21572,"module":21568,"summary":21573},"\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":21575,"title":21576,"module":21568,"summary":21577},"\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":21579,"title":21580,"module":21568,"summary":21581},"\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":21583,"title":21584,"module":21585,"summary":21586},"\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":21588,"title":21589,"module":21585,"summary":21590},"\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":21592,"title":21593,"module":21585,"summary":21594},"\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":21596,"title":21597,"module":21585,"summary":21598},"\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":21600,"title":21601,"module":21585,"summary":21602},"\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":21604,"title":21605,"module":21585,"summary":21606},"\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":21608,"title":21609,"module":21610,"summary":21611},"\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":21613,"title":21614,"module":21610,"summary":21615},"\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":21617,"title":21618,"module":21610,"summary":21619},"\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":21621,"title":21622,"module":21610,"summary":21623},"\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":21625,"title":21626,"module":21627,"summary":21628},"\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":21630,"title":21631,"module":21627,"summary":21632},"\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":21634,"title":21635,"module":21627,"summary":21636},"\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":21638,"title":21639,"module":21640,"summary":21641},"\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":21643,"title":21644,"module":21640,"summary":21645},"\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":21647,"title":21648,"module":21640,"summary":21649},"\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":21651,"title":21652,"module":21640,"summary":21653},"\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":21655,"title":21656,"module":21640,"summary":21657},"\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":21659,"title":21660,"module":21661,"summary":21662},"\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":21664,"title":21665,"module":21661,"summary":21666},"\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":21668,"title":21669,"module":21661,"summary":21670},"\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":21672,"title":21673,"module":6,"summary":6},"\u002Fstatistical-mechanics","Statistical Mechanics",{"path":21675,"title":21676,"module":21677,"summary":21678},"\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":21680,"title":21681,"module":21677,"summary":21682},"\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":21684,"title":21685,"module":21677,"summary":21686},"\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":21688,"title":21689,"module":21677,"summary":21690},"\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":21692,"title":21693,"module":21694,"summary":21695},"\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":21697,"title":21698,"module":21694,"summary":21699},"\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":21701,"title":21702,"module":21694,"summary":21703},"\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":21705,"title":21706,"module":21694,"summary":21707},"\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":21709,"title":21710,"module":21711,"summary":21712},"\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":21714,"title":21715,"module":21711,"summary":21716},"\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":21718,"title":21719,"module":21711,"summary":21720},"\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":21722,"title":21723,"module":21711,"summary":21724},"\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":21726,"title":21727,"module":21728,"summary":21729},"\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":21731,"title":21732,"module":21728,"summary":21733},"\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":21735,"title":21736,"module":21728,"summary":21737},"\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":21739,"title":21740,"module":21728,"summary":21741},"\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":21743,"title":21744,"module":21745,"summary":21746},"\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":21748,"title":21749,"module":21745,"summary":21750},"\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":21752,"title":21753,"module":21745,"summary":21754},"\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":21756,"title":21757,"module":21745,"summary":21758},"\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":21760,"title":21761,"module":21762,"summary":21763},"\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":21765,"title":21766,"module":21762,"summary":21767},"\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":21769,"title":21770,"module":21762,"summary":21771},"\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":21773,"title":21774,"module":21762,"summary":21775},"\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":21777,"title":21778,"module":21779,"summary":21780},"\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":21782,"title":21783,"module":21779,"summary":21784},"\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":21786,"title":21787,"module":21779,"summary":21788},"\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":21790,"title":21791,"module":21779,"summary":21792},"\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":21794,"title":21795,"module":21779,"summary":21796},"\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":21798,"title":21799,"module":21800,"summary":21801},"\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":21803,"title":21804,"module":21800,"summary":21805},"\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":21807,"title":21808,"module":21809,"summary":21810},"\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":21812,"title":21813,"module":21809,"summary":21814},"\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":21816,"title":21817,"module":21809,"summary":21818},"\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":21820,"title":21821,"module":21809,"summary":21822},"\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":21824,"title":21825,"module":21826,"summary":21827},"\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":21829,"title":21830,"module":21826,"summary":21831},"\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":21833,"title":21834,"module":21826,"summary":21835},"\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":21837,"title":21838,"module":21826,"summary":21839},"\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":21841,"title":21842,"module":21826,"summary":21843},"\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":21845,"title":21846,"module":21847,"summary":21848},"\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":21850,"title":21851,"module":21847,"summary":21852},"\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":21854,"title":21855,"module":21847,"summary":21856},"\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":21858,"title":21859,"module":21847,"summary":21860},"\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":21862,"title":21863,"module":6,"summary":6},"\u002Fcondensed-matter","Condensed Matter Physics",{"path":21865,"title":21866,"module":18629,"summary":21867},"\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":21869,"title":21870,"module":21871,"summary":21872},"\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":21874,"title":21875,"module":21871,"summary":21876},"\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":21878,"title":21879,"module":21871,"summary":21880},"\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":21882,"title":21883,"module":21871,"summary":21884},"\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":21886,"title":21887,"module":21871,"summary":21888},"\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":21890,"title":21891,"module":21871,"summary":21892},"\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":21894,"title":21895,"module":21871,"summary":21896},"\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":21898,"title":21899,"module":21900,"summary":21901},"\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":21903,"title":21904,"module":21900,"summary":21905},"\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":21907,"title":21908,"module":21900,"summary":21909},"\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":21911,"title":21912,"module":21900,"summary":21913},"\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":21915,"title":21916,"module":21917,"summary":21918},"\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":21920,"title":21921,"module":21917,"summary":21922},"\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":21924,"title":21925,"module":21917,"summary":21926},"\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":21928,"title":21929,"module":21917,"summary":21930},"\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":21932,"title":21933,"module":21934,"summary":21935},"\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":21937,"title":21938,"module":21934,"summary":21939},"\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":21941,"title":21942,"module":21934,"summary":21943},"\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":21945,"title":21946,"module":21934,"summary":21947},"\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":21949,"title":21950,"module":21951,"summary":21952},"\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":21954,"title":21955,"module":21951,"summary":21956},"\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":21958,"title":21959,"module":21951,"summary":21960},"\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":21962,"title":21963,"module":21951,"summary":21964},"\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":21966,"title":21967,"module":21968,"summary":21969},"\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":21971,"title":21972,"module":21968,"summary":21973},"\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":21975,"title":21976,"module":21968,"summary":21977},"\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":21979,"title":21980,"module":21981,"summary":21982},"\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":21984,"title":21985,"module":21981,"summary":21986},"\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":21988,"title":21989,"module":21990,"summary":21991},"\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":21993,"title":21994,"module":21990,"summary":21995},"\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":21997,"title":21998,"module":21990,"summary":21999},"\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":22001,"title":22002,"module":6,"summary":6},"\u002Flogic","Logic",{"path":22004,"title":22005,"module":18629,"summary":22006},"\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":22008,"title":22009,"module":18629,"summary":22010},"\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":22012,"title":22013,"module":18629,"summary":22014},"\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":22016,"title":22017,"module":18629,"summary":22018},"\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":22020,"title":22021,"module":18629,"summary":22022},"\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":22024,"title":22025,"module":18629,"summary":22026},"\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":22028,"title":18848,"module":22029,"summary":22030},"\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":22032,"title":22033,"module":22029,"summary":22034},"\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":22036,"title":22037,"module":22029,"summary":22038},"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods","Monte Carlo Methods","Monte Carlo methods learn value functions and optimal policies from complete sampled episodes, with no model of the environment: they simply average the returns that actually followed each state. We build prediction (first-visit and every-visit averaging), see why estimating action values forces the exploration question, and answer it two ways on-policy — exploring starts and epsilon-soft control. Throughout, Monte Carlo samples one whole trajectory to termination and never bootstraps.\n",{"path":22040,"title":22041,"module":22029,"summary":22042},"\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":22044,"title":22045,"module":22029,"summary":22046},"\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":22048,"title":22049,"module":22029,"summary":22050},"\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":22052,"title":22053,"module":22029,"summary":22054},"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping","n-Step Bootstrapping","Monte Carlo waits for the full return; one-step TD bootstraps after a single reward. Between them lies a whole spectrum, indexed by one integer n: look ahead n real rewards, then bootstrap from the value n steps out. The n-step return unifies the previous two lessons, and — on the random walk — an intermediate n beats both extremes. We build the n-step return, the n-step TD update, the backup-diagram spectrum, and n-step Sarsa for control.\n",{"path":22056,"title":22057,"module":22029,"summary":22058},"\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":22060,"title":22061,"module":22029,"summary":22062},"\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":22064,"title":22065,"module":22029,"summary":22066},"\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":22068,"title":22069,"module":22029,"summary":22070},"\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":22072,"title":22073,"module":22029,"summary":22074},"\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":22076,"title":22077,"module":17691,"summary":22078},"\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":22080,"title":22081,"module":17691,"summary":22082},"\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":22084,"title":22085,"module":17691,"summary":22086},"\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":22088,"title":22089,"module":17691,"summary":22090},"\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":333,"title":22092,"module":17691,"summary":22093},"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":22095,"title":22096,"module":17691,"summary":22097},"\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td","Value-Function Geometry and Gradient-TD Methods","Why does the deadly triad diverge, and how do you stop it? This lesson develops the geometry that explains the failure: value functions as vectors, the projection operator onto the representable subspace, and the split between the Bellman error, the value error, and the projected Bellman error: the three objectives have different minimizers. The projected Bellman error is the learnable one, and Gradient-TD methods (GTD2, TDC) do true stochastic gradient descent on it, staying stable even off-policy at $O(d)$ cost. Emphatic TD reweights states instead, and a survey of variance-reduction techniques closes the gap between stability and usable learning.\n",{"path":17,"title":22099,"module":17691,"summary":22100},"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":22102,"title":22103,"module":17691,"summary":22104},"\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":22106,"title":22107,"module":17691,"summary":22108},"\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":22110,"title":22111,"module":17691,"summary":22112},"\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":22114,"title":22115,"module":17691,"summary":22116},"\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":22118,"title":22119,"module":17691,"summary":22120},"\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":17693,"title":5,"module":17691,"summary":17709},{"path":338,"title":22123,"module":17691,"summary":22124},"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":22126,"title":21451,"module":22127,"summary":22128},"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks","Deep Reinforcement Learning","Deep Q-networks replace the linear value function with a neural network $Q(s,a;\\mathbf{w})$ and confront the fact that a nonlinear approximator, off-policy bootstrapping, and correlated online data — the deadly triad — make naive Q-learning diverge. DQN counters this empirically with two stabilizers: an experience replay buffer that decorrelates and reuses samples, and a periodically-frozen target network that fixes the bootstrap target. We derive the DQN loss and gradient, walk through the Atari convolutional architecture and its results, and then add the three refinements that define modern value-based deep RL — Double DQN, dueling networks, and prioritized experience replay.\n",{"path":22130,"title":22131,"module":22127,"summary":22132},"\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":22134,"title":22135,"module":22127,"summary":22136},"\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":22138,"title":22139,"module":22127,"summary":22140},"\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":22142,"title":22143,"module":22127,"summary":22144},"\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":22146,"title":22147,"module":22127,"summary":22148},"\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":22150,"title":22151,"module":22127,"summary":22152},"\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":22154,"title":22155,"module":22127,"summary":22156},"\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":22158,"title":22159,"module":22160,"summary":22161},"\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":22163,"title":22164,"module":22160,"summary":22165},"\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":22167,"title":22168,"module":22160,"summary":22169},"\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":22171,"title":22172,"module":22160,"summary":22173},"\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":22175,"title":22176,"module":22160,"summary":22177},"\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":22179,"title":22180,"module":22160,"summary":22181},"\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":22183,"title":22184,"module":22160,"summary":22185},"\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":22187,"title":22188,"module":22160,"summary":22189},"\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":22191,"title":22192,"module":22160,"summary":22193},"\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":22195,"title":22196,"module":22160,"summary":22197},"\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":22199,"title":22200,"module":22160,"summary":22201},"\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":22203,"title":22204,"module":22160,"summary":22205},"\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":22207,"title":22208,"module":22160,"summary":22209},"\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":22211,"title":22212,"module":22160,"summary":22213},"\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":22215,"title":22216,"module":22160,"summary":22217},"\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":22219,"title":22220,"module":22160,"summary":22221},"\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":22223,"title":22224,"module":22160,"summary":22225},"\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":22227,"title":22228,"module":22160,"summary":22229},"\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":22231,"title":22232,"module":22160,"summary":22233},"\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":22235,"title":22236,"module":22160,"summary":22237},"\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":22239,"title":22240,"module":22160,"summary":22241},"\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":22243,"title":22244,"module":22160,"summary":22245},"\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":22247,"title":22248,"module":22249,"summary":22250},"\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":22252,"title":22253,"module":22249,"summary":22254},"\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":22256,"title":22257,"module":22249,"summary":22258},"\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":22260,"title":22261,"module":22249,"summary":22262},"\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":22264,"title":22265,"module":22249,"summary":22266},"\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":22268,"title":22269,"module":22249,"summary":22270},"\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":22272,"title":22273,"module":22249,"summary":22274},"\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":22276,"title":22277,"module":22249,"summary":22278},"\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":22280,"title":21443,"module":6,"summary":6},"\u002Freinforcement-learning",{"path":22282,"title":22283,"module":18629,"summary":22284},"\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":22286,"title":22287,"module":18629,"summary":22288},"\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":22290,"title":22291,"module":18629,"summary":22292},"\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":22294,"title":22295,"module":18629,"summary":22296},"\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":22298,"title":22299,"module":22300,"summary":22301},"\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":22303,"title":22304,"module":22300,"summary":22305},"\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":22307,"title":22308,"module":22300,"summary":22309},"\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":22311,"title":22312,"module":22300,"summary":22313},"\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":22315,"title":22316,"module":22300,"summary":22317},"\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":22319,"title":22320,"module":22300,"summary":22321},"\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":22323,"title":22324,"module":22300,"summary":22325},"\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":22327,"title":22328,"module":22300,"summary":22329},"\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":22331,"title":22332,"module":22300,"summary":22333},"\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":22335,"title":22336,"module":22300,"summary":22337},"\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":22339,"title":22340,"module":22300,"summary":22341},"\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":22343,"title":22344,"module":22300,"summary":22345},"\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":22347,"title":22348,"module":22349,"summary":22350},"\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":22352,"title":22353,"module":22349,"summary":22354},"\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":22356,"title":22357,"module":22349,"summary":22358},"\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":22360,"title":22361,"module":22349,"summary":22362},"\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":22364,"title":22365,"module":22349,"summary":22366},"\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":22368,"title":22369,"module":22349,"summary":22370},"\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":22372,"title":22373,"module":22349,"summary":22374},"\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":22376,"title":22377,"module":22349,"summary":22378},"\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":22380,"title":22381,"module":22349,"summary":22382},"\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":22384,"title":22385,"module":22349,"summary":22386},"\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":22388,"title":22389,"module":22349,"summary":22390},"\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":22392,"title":22393,"module":22349,"summary":22394},"\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":22396,"title":22397,"module":22398,"summary":22399},"\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":22401,"title":22402,"module":22398,"summary":22403},"\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":22405,"title":22406,"module":22398,"summary":22407},"\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":22409,"title":22410,"module":22398,"summary":22411},"\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":22413,"title":22414,"module":22398,"summary":22415},"\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":22417,"title":22418,"module":22398,"summary":22419},"\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":22421,"title":22422,"module":22398,"summary":22423},"\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":22425,"title":22021,"module":22398,"summary":22426},"\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":22428,"title":22429,"module":22398,"summary":22430},"\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":22432,"title":22433,"module":22398,"summary":22434},"\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":22436,"title":22437,"module":22438,"summary":22439},"\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":22441,"title":22442,"module":22438,"summary":22443},"\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":22445,"title":22446,"module":22438,"summary":22447},"\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":22449,"title":22450,"module":22438,"summary":22451},"\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":22453,"title":21443,"module":22438,"summary":22454},"\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":22456,"title":22457,"module":22438,"summary":22458},"\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":22460,"title":22461,"module":22438,"summary":22462},"\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":22464,"title":22465,"module":22438,"summary":22466},"\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":22468,"title":22469,"module":22470,"summary":22471},"\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":22473,"title":22474,"module":22470,"summary":22475},"\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":22477,"title":22478,"module":22470,"summary":22479},"\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":22481,"title":22482,"module":22470,"summary":22483},"\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":22485,"title":22486,"module":22470,"summary":22487},"\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":22489,"title":22490,"module":22470,"summary":22491},"\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":22493,"title":22494,"module":22470,"summary":22495},"\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":22497,"title":22498,"module":22470,"summary":22499},"\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":22501,"title":22502,"module":6,"summary":6},"\u002Fartificial-intelligence","Artificial Intelligence",{"path":22504,"title":22505,"module":22506,"summary":22507},"\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":22509,"title":22510,"module":22506,"summary":22511},"\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":22513,"title":22514,"module":22506,"summary":22515},"\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":22517,"title":22518,"module":22506,"summary":22519},"\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":22521,"title":22522,"module":22506,"summary":22523},"\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":22525,"title":22526,"module":22527,"summary":22528},"\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":22530,"title":22531,"module":22527,"summary":22532},"\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":22534,"title":22535,"module":22527,"summary":22536},"\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":22538,"title":22539,"module":22527,"summary":22540},"\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":22542,"title":22543,"module":22544,"summary":22545},"\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":22547,"title":22548,"module":22544,"summary":22549},"\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":22551,"title":22552,"module":22544,"summary":22553},"\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":22555,"title":22556,"module":22544,"summary":22557},"\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":22559,"title":22560,"module":22561,"summary":22562},"\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":22564,"title":22565,"module":22561,"summary":22566},"\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":22568,"title":22569,"module":22570,"summary":22571},"\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":22573,"title":22574,"module":22570,"summary":22575},"\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":22577,"title":22578,"module":22579,"summary":22580},"\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":22582,"title":22583,"module":22579,"summary":22584},"\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":22586,"title":22587,"module":22579,"summary":22588},"\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":22590,"title":22591,"module":22579,"summary":22592},"\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":22594,"title":22595,"module":22596,"summary":22597},"\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":22599,"title":22600,"module":22596,"summary":22601},"\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":22603,"title":22604,"module":22596,"summary":22605},"\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":22607,"title":22608,"module":22609,"summary":22610},"\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":22612,"title":22613,"module":22609,"summary":22614},"\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":22616,"title":22617,"module":22609,"summary":22618},"\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":22620,"title":22621,"module":22622,"summary":22623},"\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":22625,"title":22626,"module":22622,"summary":22627},"\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":22629,"title":22630,"module":22631,"summary":22632},"\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":22634,"title":22635,"module":22631,"summary":22636},"\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":22638,"title":22639,"module":22631,"summary":22640},"\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":22642,"title":22643,"module":22644,"summary":22645},"\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":22647,"title":22648,"module":22644,"summary":22649},"\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":22651,"title":22652,"module":22644,"summary":22653},"\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":22655,"title":22656,"module":22644,"summary":22657},"\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":22659,"title":22660,"module":22644,"summary":22661},"\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":22663,"title":22664,"module":6,"summary":6},"\u002Fnuclear-physics","Nuclear Physics",{"path":22666,"title":22667,"module":18629,"summary":22668},"\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":22670,"title":22671,"module":18629,"summary":22672},"\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":22674,"title":22675,"module":18629,"summary":22676},"\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":22678,"title":22679,"module":18629,"summary":22680},"\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":22682,"title":22683,"module":18629,"summary":22684},"\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":22686,"title":22687,"module":22688,"summary":22689},"\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":22691,"title":22692,"module":22688,"summary":22693},"\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":22695,"title":22696,"module":22688,"summary":22697},"\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":22699,"title":22700,"module":22688,"summary":22701},"\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":22703,"title":22704,"module":22705,"summary":22706},"\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":22708,"title":22709,"module":22705,"summary":22710},"\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":22712,"title":22713,"module":22705,"summary":22714},"\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":22716,"title":22717,"module":19095,"summary":22718},"\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":22720,"title":22721,"module":19095,"summary":22722},"\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":22724,"title":22725,"module":19095,"summary":22726},"\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":22728,"title":22729,"module":19577,"summary":22730},"\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":22732,"title":21289,"module":19577,"summary":22733},"\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":22735,"title":21397,"module":19577,"summary":22736},"\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":22738,"title":22739,"module":19577,"summary":22740},"\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":22742,"title":22743,"module":19577,"summary":22744},"\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":22746,"title":22747,"module":19577,"summary":22748},"\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":22750,"title":22751,"module":22752,"summary":22753},"\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":22755,"title":22756,"module":22752,"summary":22757},"\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":22759,"title":22760,"module":22752,"summary":22761},"\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":22763,"title":22764,"module":22752,"summary":22765},"\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":22767,"title":22768,"module":22752,"summary":22769},"\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":22771,"title":22772,"module":22752,"summary":22773},"\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":22775,"title":22776,"module":22752,"summary":22777},"\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":22779,"title":22780,"module":22752,"summary":22781},"\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":22783,"title":22784,"module":22752,"summary":22785},"\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":22787,"title":22788,"module":22752,"summary":22789},"\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":22791,"title":22792,"module":22752,"summary":22793},"\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":22795,"title":22796,"module":22752,"summary":22797},"\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":22799,"title":22800,"module":22752,"summary":22801},"\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":22803,"title":22804,"module":22752,"summary":22805},"\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":22807,"title":22808,"module":22752,"summary":22809},"\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":22811,"title":22812,"module":22752,"summary":22813},"\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":22815,"title":22816,"module":22752,"summary":22817},"\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":22819,"title":22820,"module":22752,"summary":22821},"\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":22823,"title":22824,"module":22752,"summary":22825},"\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":22827,"title":22828,"module":22752,"summary":22829},"\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":22831,"title":22832,"module":21385,"summary":22833},"\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":22835,"title":22836,"module":21385,"summary":22837},"\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":22839,"title":22840,"module":21385,"summary":22841},"\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":22843,"title":22844,"module":21385,"summary":22845},"\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":22847,"title":22848,"module":21385,"summary":22849},"\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":22851,"title":22852,"module":21385,"summary":22853},"\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":22855,"title":22856,"module":21385,"summary":22857},"\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":22859,"title":22860,"module":21385,"summary":22861},"\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":22863,"title":22864,"module":22865,"summary":22866},"\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":22868,"title":22869,"module":22865,"summary":22870},"\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":22872,"title":22873,"module":22865,"summary":22874},"\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":22876,"title":22877,"module":22865,"summary":22878},"\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":22880,"title":22881,"module":6,"summary":6},"\u002Fnatural-language-processing","Natural Language Processing",{"path":22883,"title":22884,"module":18629,"summary":22885},"\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":22887,"title":22888,"module":18629,"summary":22889},"\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":22891,"title":22892,"module":18629,"summary":22893},"\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":22895,"title":22896,"module":22897,"summary":22898},"\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":22900,"title":22901,"module":22897,"summary":22902},"\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":22904,"title":22905,"module":22897,"summary":22906},"\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":22908,"title":22909,"module":22897,"summary":22910},"\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":22912,"title":22913,"module":22914,"summary":22915},"\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":22917,"title":22918,"module":22914,"summary":22919},"\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":22921,"title":22922,"module":22914,"summary":22923},"\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":22925,"title":22926,"module":22914,"summary":22927},"\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":22929,"title":22930,"module":22931,"summary":22932},"\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":22934,"title":22935,"module":22931,"summary":22936},"\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":22938,"title":22939,"module":22931,"summary":22940},"\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":22942,"title":22943,"module":22931,"summary":22944},"\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":22946,"title":22947,"module":22948,"summary":22949},"\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":22951,"title":22952,"module":22948,"summary":22953},"\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":22955,"title":22956,"module":22948,"summary":22957},"\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":22959,"title":22960,"module":22961,"summary":22962},"\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":22964,"title":22965,"module":22961,"summary":22966},"\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":22968,"title":22969,"module":22961,"summary":22970},"\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":22972,"title":22973,"module":22961,"summary":22974},"\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":22976,"title":22977,"module":22978,"summary":22979},"\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":22981,"title":22982,"module":22978,"summary":22983},"\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":22985,"title":22986,"module":22978,"summary":22987},"\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":22989,"title":22990,"module":22978,"summary":22991},"\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":22993,"title":22994,"module":22995,"summary":22996},"\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":22998,"title":22999,"module":22995,"summary":23000},"\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":23002,"title":23003,"module":22995,"summary":23004},"\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":23006,"title":23007,"module":22995,"summary":23008},"\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":23010,"title":23011,"module":23012,"summary":23013},"\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":23015,"title":23016,"module":23012,"summary":23017},"\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":23019,"title":23020,"module":23012,"summary":23021},"\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":23023,"title":23024,"module":23012,"summary":23025},"\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":23027,"title":23028,"module":23012,"summary":23029},"\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":23031,"title":23032,"module":23033,"summary":23034},"\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":23036,"title":23037,"module":23033,"summary":23038},"\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":23040,"title":23041,"module":23033,"summary":23042},"\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":23044,"title":23045,"module":23046,"summary":23047},"\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":23049,"title":23050,"module":23046,"summary":23051},"\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":23053,"title":23054,"module":23046,"summary":23055},"\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":23057,"title":23058,"module":23058,"summary":23059},"\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":23061,"title":23062,"module":23058,"summary":23063},"\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":23065,"title":23066,"module":23058,"summary":23067},"\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":23069,"title":23070,"module":23058,"summary":23071},"\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":23073,"title":23074,"module":23058,"summary":23075},"\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":23077,"title":23078,"module":23058,"summary":23079},"\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":23081,"title":23082,"module":6,"summary":6},"\u002Fparticle-physics","Particle Physics",{"path":23084,"title":23085,"module":23086,"summary":23087},"\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":23089,"title":23090,"module":23086,"summary":23091},"\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":23093,"title":23094,"module":23086,"summary":23095},"\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":23097,"title":23098,"module":23099,"summary":23100},"\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":23102,"title":23103,"module":23099,"summary":23104},"\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":23106,"title":23107,"module":23099,"summary":23108},"\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":23110,"title":23111,"module":23099,"summary":23112},"\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":23114,"title":23115,"module":23116,"summary":23117},"\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":23119,"title":23120,"module":23116,"summary":23121},"\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":23123,"title":23124,"module":23116,"summary":23125},"\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":23127,"title":23128,"module":23116,"summary":23129},"\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":23131,"title":23132,"module":23133,"summary":23134},"\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":23136,"title":23137,"module":23133,"summary":23138},"\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":23140,"title":23141,"module":23133,"summary":23142},"\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":23144,"title":23145,"module":23133,"summary":23146},"\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":23148,"title":23149,"module":23150,"summary":23151},"\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":23153,"title":23154,"module":23150,"summary":23155},"\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":23157,"title":23158,"module":23150,"summary":23159},"\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":23161,"title":23162,"module":23150,"summary":23163},"\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":23165,"title":23166,"module":23167,"summary":23168},"\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":23170,"title":23171,"module":23167,"summary":23172},"\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":23174,"title":23175,"module":23167,"summary":23176},"\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":23178,"title":23179,"module":23180,"summary":23181},"\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":23183,"title":23184,"module":23180,"summary":23185},"\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":23187,"title":23188,"module":23180,"summary":23189},"\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":23191,"title":23192,"module":23180,"summary":23193},"\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":23195,"title":21618,"module":23196,"summary":23197},"\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":23199,"title":23200,"module":23196,"summary":23201},"\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":23203,"title":23204,"module":23196,"summary":23205},"\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":23207,"title":23208,"module":23196,"summary":23209},"\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":23211,"title":23212,"module":23196,"summary":23213},"\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":23215,"title":23216,"module":23217,"summary":23218},"\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":23220,"title":23221,"module":23217,"summary":23222},"\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":23224,"title":23225,"module":23217,"summary":23226},"\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":23228,"title":23229,"module":23217,"summary":23230},"\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":23232,"title":23233,"module":23234,"summary":23235},"\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":23237,"title":23238,"module":23234,"summary":23239},"\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":23241,"title":23242,"module":23234,"summary":23243},"\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":23245,"title":23246,"module":23234,"summary":23247},"\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":23249,"title":23250,"module":23234,"summary":23251},"\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":23253,"title":23254,"module":23255,"summary":23256},"\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":23258,"title":23259,"module":23255,"summary":23260},"\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":23262,"title":20336,"module":23255,"summary":23263},"\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":23265,"title":23266,"module":23255,"summary":23267},"\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":23269,"title":23270,"module":23255,"summary":23271},"\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":23273,"title":23274,"module":23275,"summary":23276},"\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":23278,"title":23279,"module":23275,"summary":23280},"\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":23282,"title":23283,"module":23275,"summary":23284},"\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":23286,"title":23287,"module":23275,"summary":23288},"\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":23290,"title":23291,"module":23275,"summary":23292},"\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":23294,"title":23295,"module":23275,"summary":23296},"\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":23298,"title":23299,"module":23275,"summary":23300},"\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":23302,"title":23303,"module":6,"summary":6},"\u002Fastrophysics-cosmology","Astrophysics & Cosmology",{"path":23305,"title":23306,"module":6,"summary":6},"\u002Fcolophon","Colophon",{"path":23308,"title":23309,"module":6,"summary":6},"\u002F","Study Notes",[23311,23330,23364,23396,23416,23471],{"module":18629,"moduleNumber":23312,"slug":23313,"lessons":23314},1,"foundations",[23315,23317,23319,23321,23324,23327],{"title":22005,"path":22004,"lessonNumber":23312,"topics":23316,"summary":22006},[18629],{"title":22009,"path":22008,"lessonNumber":17675,"topics":23318,"summary":22010},[18629],{"title":22013,"path":22012,"lessonNumber":17688,"topics":23320,"summary":22014},[18629],{"title":22017,"path":22016,"lessonNumber":23322,"topics":23323,"summary":22018},4,[18629],{"title":22021,"path":22020,"lessonNumber":23325,"topics":23326,"summary":22022},5,[18629],{"title":22025,"path":22024,"lessonNumber":23328,"topics":23329,"summary":22026},6,[18629],{"module":22029,"moduleNumber":17675,"slug":23331,"lessons":23332},"tabular-methods",[23333,23336,23338,23340,23342,23344,23346,23349,23352,23355,23358,23361],{"title":18848,"path":22028,"lessonNumber":23312,"topics":23334,"summary":22030},[23335],"Tabular Methods",{"title":22033,"path":22032,"lessonNumber":17675,"topics":23337,"summary":22034},[23335],{"title":22037,"path":22036,"lessonNumber":17688,"topics":23339,"summary":22038},[23335],{"title":22041,"path":22040,"lessonNumber":23322,"topics":23341,"summary":22042},[23335],{"title":22045,"path":22044,"lessonNumber":23325,"topics":23343,"summary":22046},[23335],{"title":22049,"path":22048,"lessonNumber":23328,"topics":23345,"summary":22050},[23335],{"title":22053,"path":22052,"lessonNumber":23347,"topics":23348,"summary":22054},7,[23335],{"title":22057,"path":22056,"lessonNumber":23350,"topics":23351,"summary":22058},8,[23335],{"title":22061,"path":22060,"lessonNumber":23353,"topics":23354,"summary":22062},9,[23335],{"title":22065,"path":22064,"lessonNumber":23356,"topics":23357,"summary":22066},10,[23335],{"title":22069,"path":22068,"lessonNumber":23359,"topics":23360,"summary":22070},11,[23335],{"title":22073,"path":22072,"lessonNumber":23362,"topics":23363,"summary":22074},12,[23335],{"module":17691,"moduleNumber":17688,"slug":23365,"lessons":23366},"approximation",[23367,23369,23371,23373,23375,23377,23379,23381,23383,23385,23387,23389,23391,23393],{"title":22077,"path":22076,"lessonNumber":23312,"topics":23368,"summary":22078},[17711],{"title":22081,"path":22080,"lessonNumber":17675,"topics":23370,"summary":22082},[17711],{"title":22085,"path":22084,"lessonNumber":17688,"topics":23372,"summary":22086},[17711],{"title":22089,"path":22088,"lessonNumber":23322,"topics":23374,"summary":22090},[17711],{"title":22092,"path":333,"lessonNumber":23325,"topics":23376,"summary":22093},[17711],{"title":22096,"path":22095,"lessonNumber":23328,"topics":23378,"summary":22097},[17711],{"title":22099,"path":17,"lessonNumber":23347,"topics":23380,"summary":22100},[17711],{"title":22103,"path":22102,"lessonNumber":23350,"topics":23382,"summary":22104},[17711],{"title":22107,"path":22106,"lessonNumber":23353,"topics":23384,"summary":22108},[17711],{"title":22111,"path":22110,"lessonNumber":23356,"topics":23386,"summary":22112},[17711],{"title":22115,"path":22114,"lessonNumber":23359,"topics":23388,"summary":22116},[17711],{"title":22119,"path":22118,"lessonNumber":23362,"topics":23390,"summary":22120},[17711],{"title":5,"path":17693,"lessonNumber":17689,"topics":23392,"summary":17709},[17711],{"title":22123,"path":338,"lessonNumber":23394,"topics":23395,"summary":22124},14,[17711],{"module":22127,"moduleNumber":23322,"slug":23397,"lessons":23398},"deep-rl",[23399,23402,23404,23406,23408,23410,23412,23414],{"title":21451,"path":22126,"lessonNumber":23312,"topics":23400,"summary":22128},[23401],"Deep RL",{"title":22131,"path":22130,"lessonNumber":17675,"topics":23403,"summary":22132},[23401],{"title":22135,"path":22134,"lessonNumber":17688,"topics":23405,"summary":22136},[23401],{"title":22139,"path":22138,"lessonNumber":23322,"topics":23407,"summary":22140},[23401],{"title":22143,"path":22142,"lessonNumber":23325,"topics":23409,"summary":22144},[23401],{"title":22147,"path":22146,"lessonNumber":23328,"topics":23411,"summary":22148},[23401],{"title":22151,"path":22150,"lessonNumber":23347,"topics":23413,"summary":22152},[23401],{"title":22155,"path":22154,"lessonNumber":23350,"topics":23415,"summary":22156},[23401],{"module":22160,"moduleNumber":23325,"slug":23417,"lessons":23418},"modern-deep-rl",[23419,23421,23423,23425,23427,23429,23431,23433,23435,23437,23439,23441,23443,23445,23447,23450,23453,23456,23459,23462,23465,23468],{"title":22159,"path":22158,"lessonNumber":23312,"topics":23420,"summary":22161},[23401],{"title":22164,"path":22163,"lessonNumber":17675,"topics":23422,"summary":22165},[23401],{"title":22168,"path":22167,"lessonNumber":17688,"topics":23424,"summary":22169},[23401],{"title":22172,"path":22171,"lessonNumber":23322,"topics":23426,"summary":22173},[23401],{"title":22176,"path":22175,"lessonNumber":23325,"topics":23428,"summary":22177},[23401],{"title":22180,"path":22179,"lessonNumber":23328,"topics":23430,"summary":22181},[23401],{"title":22184,"path":22183,"lessonNumber":23347,"topics":23432,"summary":22185},[23401],{"title":22188,"path":22187,"lessonNumber":23350,"topics":23434,"summary":22189},[23401],{"title":22192,"path":22191,"lessonNumber":23353,"topics":23436,"summary":22193},[23401],{"title":22196,"path":22195,"lessonNumber":23356,"topics":23438,"summary":22197},[23401],{"title":22200,"path":22199,"lessonNumber":23359,"topics":23440,"summary":22201},[23401],{"title":22204,"path":22203,"lessonNumber":23362,"topics":23442,"summary":22205},[23401],{"title":22208,"path":22207,"lessonNumber":17689,"topics":23444,"summary":22209},[23401],{"title":22212,"path":22211,"lessonNumber":23394,"topics":23446,"summary":22213},[23401],{"title":22216,"path":22215,"lessonNumber":23448,"topics":23449,"summary":22217},15,[23401],{"title":22220,"path":22219,"lessonNumber":23451,"topics":23452,"summary":22221},16,[23401],{"title":22224,"path":22223,"lessonNumber":23454,"topics":23455,"summary":22225},17,[23401],{"title":22228,"path":22227,"lessonNumber":23457,"topics":23458,"summary":22229},18,[23401],{"title":22232,"path":22231,"lessonNumber":23460,"topics":23461,"summary":22233},19,[23401],{"title":22236,"path":22235,"lessonNumber":23463,"topics":23464,"summary":22237},20,[23401],{"title":22240,"path":22239,"lessonNumber":23466,"topics":23467,"summary":22241},21,[23401],{"title":22244,"path":22243,"lessonNumber":23469,"topics":23470,"summary":22245},22,[23401],{"module":22249,"moduleNumber":23328,"slug":23472,"lessons":23473},"minds-and-brains",[23474,23477,23479,23481,23483,23485,23487,23489],{"title":22248,"path":22247,"lessonNumber":23312,"topics":23475,"summary":22250},[23476],"Minds and Brains",{"title":22253,"path":22252,"lessonNumber":17675,"topics":23478,"summary":22254},[23476],{"title":22257,"path":22256,"lessonNumber":17688,"topics":23480,"summary":22258},[23476],{"title":22261,"path":22260,"lessonNumber":23322,"topics":23482,"summary":22262},[23476],{"title":22265,"path":22264,"lessonNumber":23325,"topics":23484,"summary":22266},[23476],{"title":22269,"path":22268,"lessonNumber":23328,"topics":23486,"summary":22270},[23476],{"title":22273,"path":22272,"lessonNumber":23347,"topics":23488,"summary":22274},[23476],{"title":22277,"path":22276,"lessonNumber":23350,"topics":23490,"summary":22278},[23476],"\u003Csvg style=\"width:100%;max-width:294.169px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 220.627 127.581\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-41.992 -55.155)\">\u003Cpath d=\"M-20.006 23.088Q-20.006 22.767-19.881 22.478Q-19.756 22.189-19.530 21.966Q-19.305 21.742-19.009 21.622Q-18.714 21.502-18.396 21.502Q-18.068 21.502-17.806 21.602Q-17.545 21.701-17.369 21.883Q-17.193 22.066-17.099 22.324Q-17.005 22.582-17.005 22.914Q-17.005 23.006-17.087 23.027L-19.342 23.027L-19.342 23.088Q-19.342 23.676-19.059 24.059Q-18.775 24.442-18.208 24.442Q-17.886 24.442-17.618 24.249Q-17.350 24.056-17.261 23.741Q-17.254 23.700-17.179 23.686L-17.087 23.686Q-17.005 23.710-17.005 23.782Q-17.005 23.789-17.011 23.816Q-17.124 24.213-17.495 24.452Q-17.866 24.691-18.290 24.691Q-18.727 24.691-19.127 24.483Q-19.527 24.274-19.766 23.907Q-20.006 23.540-20.006 23.088M-19.336 22.818L-17.521 22.818Q-17.521 22.541-17.618 22.289Q-17.716 22.036-17.914 21.880Q-18.112 21.725-18.396 21.725Q-18.673 21.725-18.886 21.883Q-19.100 22.042-19.218 22.297Q-19.336 22.552-19.336 22.818M-14.773 25.980L-16.403 25.980L-16.403 25.700Q-16.174 25.700-16.025 25.665Q-15.877 25.631-15.877 25.491L-15.877 22.145Q-15.877 21.974-16.013 21.933Q-16.150 21.892-16.403 21.892L-16.403 21.612L-15.323 21.537L-15.323 21.943Q-15.101 21.742-14.814 21.639Q-14.527 21.537-14.219 21.537Q-13.792 21.537-13.428 21.750Q-13.064 21.964-12.850 22.328Q-12.636 22.692-12.636 23.112Q-12.636 23.557-12.876 23.921Q-13.115 24.285-13.508 24.488Q-13.901 24.691-14.345 24.691Q-14.612 24.691-14.860 24.591Q-15.108 24.490-15.296 24.309L-15.296 25.491Q-15.296 25.628-15.147 25.664Q-14.998 25.700-14.773 25.700L-14.773 25.980M-15.296 22.292L-15.296 23.902Q-15.162 24.155-14.920 24.312Q-14.677 24.469-14.400 24.469Q-14.072 24.469-13.819 24.268Q-13.566 24.066-13.433 23.748Q-13.300 23.430-13.300 23.112Q-13.300 22.883-13.364 22.654Q-13.429 22.425-13.558 22.227Q-13.686 22.029-13.881 21.909Q-14.075 21.790-14.308 21.790Q-14.602 21.790-14.870 21.919Q-15.138 22.049-15.296 22.292M-10.384 24.623L-11.936 24.623L-11.936 24.343Q-11.710 24.343-11.561 24.309Q-11.413 24.274-11.413 24.134L-11.413 22.285Q-11.413 22.097-11.461 22.013Q-11.508 21.930-11.606 21.911Q-11.703 21.892-11.915 21.892L-11.915 21.612L-10.859 21.537L-10.859 24.134Q-10.859 24.274-10.727 24.309Q-10.596 24.343-10.384 24.343L-10.384 24.623M-11.655 20.316Q-11.655 20.145-11.532 20.026Q-11.409 19.906-11.238 19.906Q-11.071 19.906-10.948 20.026Q-10.825 20.145-10.825 20.316Q-10.825 20.491-10.948 20.614Q-11.071 20.737-11.238 20.737Q-11.409 20.737-11.532 20.614Q-11.655 20.491-11.655 20.316M-9.738 24.616L-9.738 23.553Q-9.738 23.529-9.711 23.502Q-9.683 23.475-9.659 23.475L-9.550 23.475Q-9.485 23.475-9.471 23.533Q-9.376 23.967-9.130 24.218Q-8.883 24.469-8.470 24.469Q-8.128 24.469-7.875 24.336Q-7.622 24.203-7.622 23.895Q-7.622 23.738-7.716 23.623Q-7.810 23.509-7.949 23.440Q-8.087 23.372-8.255 23.334L-8.836 23.235Q-9.191 23.167-9.465 22.946Q-9.738 22.726-9.738 22.384Q-9.738 22.135-9.627 21.960Q-9.516 21.786-9.330 21.687Q-9.143 21.588-8.928 21.545Q-8.713 21.502-8.470 21.502Q-8.056 21.502-7.776 21.684L-7.561 21.509Q-7.550 21.506-7.544 21.504Q-7.537 21.502-7.527 21.502L-7.475 21.502Q-7.448 21.502-7.424 21.526Q-7.400 21.550-7.400 21.578L-7.400 22.425Q-7.400 22.446-7.424 22.473Q-7.448 22.500-7.475 22.500L-7.588 22.500Q-7.615 22.500-7.641 22.475Q-7.667 22.449-7.667 22.425Q-7.667 22.189-7.773 22.025Q-7.879 21.861-8.061 21.779Q-8.244 21.697-8.477 21.697Q-8.805 21.697-9.061 21.800Q-9.318 21.902-9.318 22.179Q-9.318 22.374-9.135 22.483Q-8.952 22.593-8.723 22.634L-8.149 22.740Q-7.903 22.788-7.689 22.916Q-7.475 23.044-7.339 23.247Q-7.202 23.451-7.202 23.700Q-7.202 24.213-7.568 24.452Q-7.933 24.691-8.470 24.691Q-8.966 24.691-9.297 24.397L-9.564 24.671Q-9.584 24.691-9.612 24.691L-9.659 24.691Q-9.683 24.691-9.711 24.664Q-9.738 24.637-9.738 24.616M-6.614 23.140Q-6.614 22.798-6.479 22.499Q-6.344 22.200-6.105 21.976Q-5.865 21.752-5.548 21.627Q-5.230 21.502-4.898 21.502Q-4.454 21.502-4.054 21.718Q-3.654 21.933-3.420 22.311Q-3.186 22.688-3.186 23.140Q-3.186 23.481-3.328 23.765Q-3.469 24.049-3.714 24.256Q-3.958 24.462-4.268 24.577Q-4.577 24.691-4.898 24.691Q-5.329 24.691-5.730 24.490Q-6.132 24.288-6.373 23.936Q-6.614 23.584-6.614 23.140M-4.898 24.442Q-4.297 24.442-4.073 24.064Q-3.849 23.686-3.849 23.054Q-3.849 22.442-4.083 22.083Q-4.317 21.725-4.898 21.725Q-5.951 21.725-5.951 23.054Q-5.951 23.686-5.725 24.064Q-5.500 24.442-4.898 24.442\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-41.992 -55.155)\">\u003Cpath d=\"M-2.359 23.112Q-2.359 22.774-2.218 22.483Q-2.078 22.193-1.834 21.979Q-1.590 21.766-1.285 21.651Q-0.981 21.537-0.656 21.537Q-0.386 21.537-0.123 21.636Q0.140 21.735 0.331 21.913L0.331 20.515Q0.331 20.245 0.224 20.183Q0.116 20.122-0.195 20.122L-0.195 19.841L0.882 19.766L0.882 23.950Q0.882 24.138 0.936 24.221Q0.991 24.305 1.092 24.324Q1.193 24.343 1.408 24.343L1.408 24.623L0.301 24.691L0.301 24.274Q-0.116 24.691-0.742 24.691Q-1.173 24.691-1.545 24.479Q-1.918 24.268-2.138 23.907Q-2.359 23.546-2.359 23.112M-0.684 24.469Q-0.475 24.469-0.289 24.397Q-0.103 24.326 0.051 24.189Q0.205 24.052 0.301 23.874L0.301 22.265Q0.215 22.118 0.070 21.998Q-0.075 21.878-0.245 21.819Q-0.414 21.759-0.595 21.759Q-1.155 21.759-1.424 22.148Q-1.692 22.538-1.692 23.119Q-1.692 23.690-1.458 24.080Q-1.224 24.469-0.684 24.469M3.674 24.623L2.122 24.623L2.122 24.343Q2.348 24.343 2.497 24.309Q2.645 24.274 2.645 24.134L2.645 22.285Q2.645 22.097 2.598 22.013Q2.550 21.930 2.452 21.911Q2.355 21.892 2.143 21.892L2.143 21.612L3.199 21.537L3.199 24.134Q3.199 24.274 3.331 24.309Q3.462 24.343 3.674 24.343L3.674 24.623M2.403 20.316Q2.403 20.145 2.526 20.026Q2.649 19.906 2.820 19.906Q2.987 19.906 3.110 20.026Q3.233 20.145 3.233 20.316Q3.233 20.491 3.110 20.614Q2.987 20.737 2.820 20.737Q2.649 20.737 2.526 20.614Q2.403 20.491 2.403 20.316M4.320 23.112Q4.320 22.784 4.455 22.483Q4.590 22.183 4.826 21.962Q5.062 21.742 5.366 21.622Q5.670 21.502 5.995 21.502Q6.501 21.502 6.849 21.605Q7.198 21.707 7.198 22.083Q7.198 22.230 7.101 22.331Q7.003 22.432 6.856 22.432Q6.702 22.432 6.603 22.333Q6.504 22.234 6.504 22.083Q6.504 21.895 6.644 21.803Q6.443 21.752 6.002 21.752Q5.646 21.752 5.417 21.948Q5.188 22.145 5.087 22.454Q4.987 22.764 4.987 23.112Q4.987 23.461 5.113 23.767Q5.240 24.073 5.494 24.257Q5.749 24.442 6.104 24.442Q6.327 24.442 6.511 24.358Q6.696 24.274 6.831 24.119Q6.966 23.963 7.024 23.755Q7.037 23.700 7.092 23.700L7.205 23.700Q7.236 23.700 7.258 23.724Q7.280 23.748 7.280 23.782L7.280 23.803Q7.195 24.090 7.007 24.288Q6.819 24.486 6.554 24.589Q6.289 24.691 5.995 24.691Q5.564 24.691 5.176 24.485Q4.788 24.278 4.554 23.915Q4.320 23.553 4.320 23.112M8.268 24.203Q8.268 24.035 8.391 23.912Q8.514 23.789 8.688 23.789Q8.856 23.789 8.979 23.912Q9.102 24.035 9.102 24.203Q9.102 24.377 8.979 24.500Q8.856 24.623 8.688 24.623Q8.514 24.623 8.391 24.500Q8.268 24.377 8.268 24.203M8.268 22.019Q8.268 21.851 8.391 21.728Q8.514 21.605 8.688 21.605Q8.856 21.605 8.979 21.728Q9.102 21.851 9.102 22.019Q9.102 22.193 8.979 22.316Q8.856 22.439 8.688 22.439Q8.514 22.439 8.391 22.316Q8.268 22.193 8.268 22.019\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-10.32-32.283c0-5.5-4.459-9.958-9.959-9.958s-9.958 4.458-9.958 9.958 4.458 9.959 9.958 9.959 9.959-4.459 9.959-9.959Zm-9.959 0\"\u002F>\u003Cg transform=\"translate(-3.801 -54.328)\">\u003Cpath d=\"M-19.998 24.615L-19.998 23.393Q-19.998 23.365-19.966 23.334Q-19.935 23.303-19.912 23.303L-19.806 23.303Q-19.736 23.303-19.720 23.365Q-19.658 23.686-19.519 23.926Q-19.381 24.166-19.148 24.307Q-18.916 24.447-18.607 24.447Q-18.369 24.447-18.160 24.387Q-17.951 24.326-17.814 24.178Q-17.677 24.029-17.677 23.783Q-17.677 23.529-17.888 23.363Q-18.099 23.197-18.369 23.143L-18.990 23.029Q-19.396 22.951-19.697 22.695Q-19.998 22.439-19.998 22.064Q-19.998 21.697-19.797 21.475Q-19.595 21.252-19.271 21.154Q-18.947 21.057-18.607 21.057Q-18.142 21.057-17.845 21.264L-17.623 21.080Q-17.599 21.057-17.568 21.057L-17.517 21.057Q-17.486 21.057-17.459 21.084Q-17.431 21.111-17.431 21.143L-17.431 22.127Q-17.431 22.158-17.457 22.187Q-17.482 22.217-17.517 22.217L-17.623 22.217Q-17.658 22.217-17.685 22.189Q-17.713 22.162-17.713 22.127Q-17.713 21.728-17.965 21.508Q-18.216 21.287-18.615 21.287Q-18.970 21.287-19.254 21.410Q-19.537 21.533-19.537 21.838Q-19.537 22.057-19.336 22.189Q-19.134 22.322-18.888 22.365L-18.263 22.478Q-17.834 22.568-17.525 22.865Q-17.216 23.162-17.216 23.576Q-17.216 24.146-17.615 24.424Q-18.013 24.701-18.607 24.701Q-19.158 24.701-19.509 24.365L-19.806 24.678Q-19.830 24.701-19.865 24.701L-19.912 24.701Q-19.935 24.701-19.966 24.670Q-19.998 24.639-19.998 24.615M-14.806 24.791Q-15.509 24.791-15.910 24.391Q-16.310 23.990-16.455 23.381Q-16.599 22.771-16.599 22.072Q-16.599 21.549-16.529 21.086Q-16.459 20.623-16.265 20.211Q-16.072 19.799-15.715 19.551Q-15.357 19.303-14.806 19.303Q-14.256 19.303-13.898 19.551Q-13.541 19.799-13.349 20.209Q-13.158 20.619-13.088 21.088Q-13.017 21.557-13.017 22.072Q-13.017 22.771-13.160 23.379Q-13.302 23.986-13.703 24.389Q-14.103 24.791-14.806 24.791M-14.806 24.533Q-14.334 24.533-14.101 24.098Q-13.869 23.662-13.814 23.123Q-13.759 22.584-13.759 21.943Q-13.759 20.947-13.943 20.254Q-14.127 19.561-14.806 19.561Q-15.174 19.561-15.394 19.799Q-15.615 20.037-15.711 20.394Q-15.806 20.752-15.832 21.123Q-15.857 21.494-15.857 21.943Q-15.857 22.584-15.802 23.123Q-15.748 23.662-15.515 24.098Q-15.283 24.533-14.806 24.533\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M35.204-32.283c0-5.5-4.458-9.958-9.958-9.958s-9.959 4.458-9.959 9.958 4.459 9.959 9.959 9.959 9.958-4.459 9.958-9.959Zm-9.958 0\"\u002F>\u003Cg transform=\"translate(41.723 -54.328)\">\u003Cpath d=\"M-19.998 24.615L-19.998 23.393Q-19.998 23.365-19.966 23.334Q-19.935 23.303-19.912 23.303L-19.806 23.303Q-19.736 23.303-19.720 23.365Q-19.658 23.686-19.519 23.926Q-19.381 24.166-19.148 24.307Q-18.916 24.447-18.607 24.447Q-18.369 24.447-18.160 24.387Q-17.951 24.326-17.814 24.178Q-17.677 24.029-17.677 23.783Q-17.677 23.529-17.888 23.363Q-18.099 23.197-18.369 23.143L-18.990 23.029Q-19.396 22.951-19.697 22.695Q-19.998 22.439-19.998 22.064Q-19.998 21.697-19.797 21.475Q-19.595 21.252-19.271 21.154Q-18.947 21.057-18.607 21.057Q-18.142 21.057-17.845 21.264L-17.623 21.080Q-17.599 21.057-17.568 21.057L-17.517 21.057Q-17.486 21.057-17.459 21.084Q-17.431 21.111-17.431 21.143L-17.431 22.127Q-17.431 22.158-17.457 22.187Q-17.482 22.217-17.517 22.217L-17.623 22.217Q-17.658 22.217-17.685 22.189Q-17.713 22.162-17.713 22.127Q-17.713 21.728-17.965 21.508Q-18.216 21.287-18.615 21.287Q-18.970 21.287-19.254 21.410Q-19.537 21.533-19.537 21.838Q-19.537 22.057-19.336 22.189Q-19.134 22.322-18.888 22.365L-18.263 22.478Q-17.834 22.568-17.525 22.865Q-17.216 23.162-17.216 23.576Q-17.216 24.146-17.615 24.424Q-18.013 24.701-18.607 24.701Q-19.158 24.701-19.509 24.365L-19.806 24.678Q-19.830 24.701-19.865 24.701L-19.912 24.701Q-19.935 24.701-19.966 24.670Q-19.998 24.639-19.998 24.615M-13.334 24.623L-16.127 24.623L-16.127 24.326Q-15.064 24.326-15.064 24.064L-15.064 19.896Q-15.494 20.111-16.174 20.111L-16.174 19.814Q-15.154 19.814-14.638 19.303L-14.494 19.303Q-14.420 19.322-14.400 19.400L-14.400 24.064Q-14.400 24.326-13.334 24.326\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M80.728-32.283c0-5.5-4.458-9.958-9.958-9.958s-9.959 4.458-9.959 9.958 4.459 9.959 9.959 9.959 9.958-4.459 9.958-9.959Zm-9.958 0\"\u002F>\u003Cg transform=\"translate(87.247 -54.328)\">\u003Cpath d=\"M-19.998 24.615L-19.998 23.393Q-19.998 23.365-19.966 23.334Q-19.935 23.303-19.912 23.303L-19.806 23.303Q-19.736 23.303-19.720 23.365Q-19.658 23.686-19.519 23.926Q-19.381 24.166-19.148 24.307Q-18.916 24.447-18.607 24.447Q-18.369 24.447-18.160 24.387Q-17.951 24.326-17.814 24.178Q-17.677 24.029-17.677 23.783Q-17.677 23.529-17.888 23.363Q-18.099 23.197-18.369 23.143L-18.990 23.029Q-19.396 22.951-19.697 22.695Q-19.998 22.439-19.998 22.064Q-19.998 21.697-19.797 21.475Q-19.595 21.252-19.271 21.154Q-18.947 21.057-18.607 21.057Q-18.142 21.057-17.845 21.264L-17.623 21.080Q-17.599 21.057-17.568 21.057L-17.517 21.057Q-17.486 21.057-17.459 21.084Q-17.431 21.111-17.431 21.143L-17.431 22.127Q-17.431 22.158-17.457 22.187Q-17.482 22.217-17.517 22.217L-17.623 22.217Q-17.658 22.217-17.685 22.189Q-17.713 22.162-17.713 22.127Q-17.713 21.728-17.965 21.508Q-18.216 21.287-18.615 21.287Q-18.970 21.287-19.254 21.410Q-19.537 21.533-19.537 21.838Q-19.537 22.057-19.336 22.189Q-19.134 22.322-18.888 22.365L-18.263 22.478Q-17.834 22.568-17.525 22.865Q-17.216 23.162-17.216 23.576Q-17.216 24.146-17.615 24.424Q-18.013 24.701-18.607 24.701Q-19.158 24.701-19.509 24.365L-19.806 24.678Q-19.830 24.701-19.865 24.701L-19.912 24.701Q-19.935 24.701-19.966 24.670Q-19.998 24.639-19.998 24.615M-13.341 24.623L-16.502 24.623L-16.502 24.416Q-16.502 24.389-16.478 24.357L-15.127 22.959Q-14.748 22.572-14.500 22.283Q-14.252 21.994-14.078 21.637Q-13.904 21.279-13.904 20.889Q-13.904 20.541-14.037 20.248Q-14.170 19.955-14.424 19.777Q-14.677 19.600-15.033 19.600Q-15.392 19.600-15.683 19.795Q-15.974 19.990-16.119 20.318L-16.064 20.318Q-15.881 20.318-15.756 20.439Q-15.631 20.561-15.631 20.752Q-15.631 20.932-15.756 21.061Q-15.881 21.189-16.064 21.189Q-16.244 21.189-16.373 21.061Q-16.502 20.932-16.502 20.752Q-16.502 20.350-16.281 20.014Q-16.060 19.678-15.695 19.490Q-15.330 19.303-14.927 19.303Q-14.447 19.303-14.031 19.490Q-13.615 19.678-13.363 20.039Q-13.111 20.400-13.111 20.889Q-13.111 21.248-13.265 21.551Q-13.420 21.853-13.672 22.113Q-13.924 22.373-14.273 22.658Q-14.623 22.943-14.791 23.096L-15.720 23.936L-15.006 23.936Q-13.631 23.936-13.591 23.896Q-13.521 23.818-13.478 23.633Q-13.435 23.447-13.392 23.158L-13.111 23.158\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-neutral)\">\u003Cpath d=\"M106.336-22.324h19.917v-19.917h-19.917Z\"\u002F>\u003Cg transform=\"translate(133.504 -54.172)\">\u003Cpath d=\"M-15.693 24.623L-18.736 24.623L-18.736 24.326Q-17.974 24.326-17.791 24.287Q-17.748 24.275-17.699 24.242Q-17.650 24.209-17.625 24.166Q-17.599 24.123-17.599 24.064L-17.599 19.721Q-17.599 19.545-17.691 19.500Q-17.783 19.455-17.998 19.455L-18.392 19.455Q-19.088 19.455-19.377 19.744Q-19.525 19.893-19.588 20.213Q-19.650 20.533-19.693 21.006L-19.974 21.006L-19.814 19.158L-14.615 19.158L-14.455 21.006L-14.736 21.006Q-14.779 20.498-14.841 20.195Q-14.904 19.893-15.056 19.744Q-15.341 19.455-16.041 19.455L-16.431 19.455Q-16.646 19.455-16.738 19.498Q-16.830 19.541-16.830 19.721L-16.830 24.064Q-16.830 24.139-16.775 24.203Q-16.720 24.268-16.638 24.287Q-16.455 24.326-15.693 24.326\" 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)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-10.12-32.283h22.607\"\u002F>\u003Cpath stroke=\"none\" d=\"m15.087-32.283-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M35.404-32.283h22.608\"\u002F>\u003Cpath stroke=\"none\" d=\"m60.611-32.283-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M80.928-32.283h22.608\"\u002F>\u003Cpath stroke=\"none\" d=\"m106.136-32.283-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(121.92 -71.82)\">\u003Cpath d=\"M-19.438 23.782L-19.438 21.885L-20.077 21.885L-20.077 21.663Q-19.759 21.663-19.542 21.453Q-19.325 21.243-19.225 20.933Q-19.124 20.624-19.124 20.316L-18.857 20.316L-18.857 21.605L-17.780 21.605L-17.780 21.885L-18.857 21.885L-18.857 23.769Q-18.857 24.045-18.753 24.244Q-18.649 24.442-18.389 24.442Q-18.232 24.442-18.126 24.338Q-18.020 24.233-17.970 24.080Q-17.921 23.926-17.921 23.769L-17.921 23.355L-17.654 23.355L-17.654 23.782Q-17.654 24.008-17.753 24.218Q-17.852 24.428-18.037 24.560Q-18.221 24.691-18.450 24.691Q-18.888 24.691-19.163 24.454Q-19.438 24.216-19.438 23.782M-16.885 23.088Q-16.885 22.767-16.760 22.478Q-16.635 22.189-16.410 21.966Q-16.184 21.742-15.889 21.622Q-15.593 21.502-15.275 21.502Q-14.947 21.502-14.685 21.602Q-14.424 21.701-14.248 21.883Q-14.072 22.066-13.978 22.324Q-13.884 22.582-13.884 22.914Q-13.884 23.006-13.966 23.027L-16.222 23.027L-16.222 23.088Q-16.222 23.676-15.938 24.059Q-15.654 24.442-15.087 24.442Q-14.766 24.442-14.498 24.249Q-14.229 24.056-14.140 23.741Q-14.133 23.700-14.058 23.686L-13.966 23.686Q-13.884 23.710-13.884 23.782Q-13.884 23.789-13.891 23.816Q-14.004 24.213-14.374 24.452Q-14.745 24.691-15.169 24.691Q-15.607 24.691-16.007 24.483Q-16.406 24.274-16.646 23.907Q-16.885 23.540-16.885 23.088M-16.215 22.818L-14.400 22.818Q-14.400 22.541-14.498 22.289Q-14.595 22.036-14.793 21.880Q-14.991 21.725-15.275 21.725Q-15.552 21.725-15.766 21.883Q-15.979 22.042-16.097 22.297Q-16.215 22.552-16.215 22.818M-11.546 24.623L-13.282 24.623L-13.282 24.343Q-13.053 24.343-12.905 24.309Q-12.756 24.274-12.756 24.134L-12.756 22.285Q-12.756 22.015-12.864 21.954Q-12.971 21.892-13.282 21.892L-13.282 21.612L-12.254 21.537L-12.254 22.244Q-12.124 21.936-11.881 21.737Q-11.638 21.537-11.321 21.537Q-11.102 21.537-10.931 21.661Q-10.760 21.786-10.760 21.998Q-10.760 22.135-10.859 22.234Q-10.958 22.333-11.091 22.333Q-11.228 22.333-11.327 22.234Q-11.426 22.135-11.426 21.998Q-11.426 21.858-11.327 21.759Q-11.618 21.759-11.818 21.955Q-12.018 22.152-12.110 22.446Q-12.202 22.740-12.202 23.020L-12.202 24.134Q-12.202 24.343-11.546 24.343L-11.546 24.623M-8.494 24.623L-10.128 24.623L-10.128 24.343Q-9.899 24.343-9.750 24.309Q-9.601 24.274-9.601 24.134L-9.601 22.285Q-9.601 22.015-9.709 21.954Q-9.817 21.892-10.128 21.892L-10.128 21.612L-9.068 21.537L-9.068 22.186Q-8.897 21.878-8.593 21.707Q-8.289 21.537-7.944 21.537Q-7.544 21.537-7.267 21.677Q-6.990 21.817-6.904 22.165Q-6.737 21.872-6.438 21.704Q-6.139 21.537-5.794 21.537Q-5.288 21.537-5.004 21.760Q-4.720 21.984-4.720 22.480L-4.720 24.134Q-4.720 24.271-4.572 24.307Q-4.423 24.343-4.197 24.343L-4.197 24.623L-5.828 24.623L-5.828 24.343Q-5.602 24.343-5.452 24.307Q-5.301 24.271-5.301 24.134L-5.301 22.494Q-5.301 22.159-5.421 21.959Q-5.541 21.759-5.855 21.759Q-6.125 21.759-6.359 21.895Q-6.593 22.032-6.732 22.266Q-6.870 22.500-6.870 22.774L-6.870 24.134Q-6.870 24.271-6.722 24.307Q-6.573 24.343-6.347 24.343L-6.347 24.623L-7.978 24.623L-7.978 24.343Q-7.749 24.343-7.600 24.309Q-7.451 24.274-7.451 24.134L-7.451 22.494Q-7.451 22.159-7.571 21.959Q-7.691 21.759-8.005 21.759Q-8.275 21.759-8.509 21.895Q-8.743 22.032-8.882 22.266Q-9.020 22.500-9.020 22.774L-9.020 24.134Q-9.020 24.271-8.870 24.307Q-8.719 24.343-8.494 24.343L-8.494 24.623M-1.993 24.623L-3.545 24.623L-3.545 24.343Q-3.319 24.343-3.170 24.309Q-3.022 24.274-3.022 24.134L-3.022 22.285Q-3.022 22.097-3.070 22.013Q-3.117 21.930-3.215 21.911Q-3.312 21.892-3.524 21.892L-3.524 21.612L-2.468 21.537L-2.468 24.134Q-2.468 24.274-2.336 24.309Q-2.205 24.343-1.993 24.343L-1.993 24.623M-3.264 20.316Q-3.264 20.145-3.141 20.026Q-3.018 19.906-2.847 19.906Q-2.680 19.906-2.557 20.026Q-2.434 20.145-2.434 20.316Q-2.434 20.491-2.557 20.614Q-2.680 20.737-2.847 20.737Q-3.018 20.737-3.141 20.614Q-3.264 20.491-3.264 20.316M0.335 24.623L-1.299 24.623L-1.299 24.343Q-1.070 24.343-0.921 24.309Q-0.773 24.274-0.773 24.134L-0.773 22.285Q-0.773 22.015-0.880 21.954Q-0.988 21.892-1.299 21.892L-1.299 21.612L-0.239 21.537L-0.239 22.186Q-0.069 21.878 0.236 21.707Q0.540 21.537 0.885 21.537Q1.391 21.537 1.675 21.760Q1.958 21.984 1.958 22.480L1.958 24.134Q1.958 24.271 2.107 24.307Q2.256 24.343 2.481 24.343L2.481 24.623L0.851 24.623L0.851 24.343Q1.080 24.343 1.229 24.309Q1.377 24.274 1.377 24.134L1.377 22.494Q1.377 22.159 1.258 21.959Q1.138 21.759 0.824 21.759Q0.554 21.759 0.319 21.895Q0.085 22.032-0.053 22.266Q-0.192 22.500-0.192 22.774L-0.192 24.134Q-0.192 24.271-0.041 24.307Q0.109 24.343 0.335 24.343L0.335 24.623M3.127 23.895Q3.127 23.563 3.351 23.336Q3.575 23.109 3.919 22.981Q4.262 22.852 4.635 22.800Q5.007 22.747 5.311 22.747L5.311 22.494Q5.311 22.289 5.204 22.109Q5.096 21.930 4.915 21.827Q4.734 21.725 4.525 21.725Q4.118 21.725 3.883 21.817Q3.971 21.854 4.018 21.938Q4.064 22.022 4.064 22.124Q4.064 22.220 4.018 22.299Q3.971 22.377 3.891 22.422Q3.811 22.466 3.722 22.466Q3.572 22.466 3.471 22.369Q3.370 22.271 3.370 22.124Q3.370 21.502 4.525 21.502Q4.737 21.502 4.987 21.566Q5.236 21.629 5.438 21.748Q5.639 21.868 5.766 22.053Q5.892 22.237 5.892 22.480L5.892 24.056Q5.892 24.172 5.954 24.268Q6.015 24.363 6.128 24.363Q6.238 24.363 6.303 24.269Q6.367 24.175 6.367 24.056L6.367 23.608L6.634 23.608L6.634 24.056Q6.634 24.326 6.407 24.491Q6.179 24.657 5.899 24.657Q5.691 24.657 5.554 24.503Q5.417 24.350 5.393 24.134Q5.246 24.401 4.964 24.546Q4.682 24.691 4.358 24.691Q4.081 24.691 3.797 24.616Q3.513 24.541 3.320 24.362Q3.127 24.182 3.127 23.895M3.742 23.895Q3.742 24.069 3.843 24.199Q3.944 24.329 4.100 24.399Q4.255 24.469 4.419 24.469Q4.638 24.469 4.846 24.372Q5.055 24.274 5.183 24.093Q5.311 23.912 5.311 23.686L5.311 22.958Q4.987 22.958 4.621 23.049Q4.255 23.140 3.999 23.352Q3.742 23.563 3.742 23.895M8.719 24.623L7.116 24.623L7.116 24.343Q7.342 24.343 7.490 24.309Q7.639 24.274 7.639 24.134L7.639 20.515Q7.639 20.245 7.531 20.183Q7.424 20.122 7.116 20.122L7.116 19.841L8.193 19.766L8.193 24.134Q8.193 24.271 8.343 24.307Q8.493 24.343 8.719 24.343\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-dasharray:3.0,3.0;stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M106.136-41.42C70.95-71.87 21.232-70.89-10.832-40.856\"\u002F>\u003Cpath stroke=\"none\" d=\"m-12.73-39.08 4.458-1.326-2.56-.451-.284-2.585\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(45.507 -30.573)\">\u003Cpath d=\"M-18.215 24.623L-19.951 24.623L-19.951 24.343Q-19.722 24.343-19.573 24.309Q-19.425 24.274-19.425 24.134L-19.425 22.285Q-19.425 22.015-19.532 21.954Q-19.640 21.892-19.951 21.892L-19.951 21.612L-18.922 21.537L-18.922 22.244Q-18.792 21.936-18.550 21.737Q-18.307 21.537-17.989 21.537Q-17.770 21.537-17.599 21.661Q-17.428 21.786-17.428 21.998Q-17.428 22.135-17.528 22.234Q-17.627 22.333-17.760 22.333Q-17.897 22.333-17.996 22.234Q-18.095 22.135-18.095 21.998Q-18.095 21.858-17.996 21.759Q-18.286 21.759-18.486 21.955Q-18.686 22.152-18.779 22.446Q-18.871 22.740-18.871 23.020L-18.871 24.134Q-18.871 24.343-18.215 24.343L-18.215 24.623M-16.885 23.088Q-16.885 22.767-16.760 22.478Q-16.635 22.189-16.410 21.966Q-16.184 21.742-15.889 21.622Q-15.593 21.502-15.275 21.502Q-14.947 21.502-14.685 21.602Q-14.424 21.701-14.248 21.883Q-14.072 22.066-13.978 22.324Q-13.884 22.582-13.884 22.914Q-13.884 23.006-13.966 23.027L-16.222 23.027L-16.222 23.088Q-16.222 23.676-15.938 24.059Q-15.654 24.442-15.087 24.442Q-14.766 24.442-14.498 24.249Q-14.229 24.056-14.140 23.741Q-14.133 23.700-14.058 23.686L-13.966 23.686Q-13.884 23.710-13.884 23.782Q-13.884 23.789-13.891 23.816Q-14.004 24.213-14.374 24.452Q-14.745 24.691-15.169 24.691Q-15.607 24.691-16.007 24.483Q-16.406 24.274-16.646 23.907Q-16.885 23.540-16.885 23.088M-16.215 22.818L-14.400 22.818Q-14.400 22.541-14.498 22.289Q-14.595 22.036-14.793 21.880Q-14.991 21.725-15.275 21.725Q-15.552 21.725-15.766 21.883Q-15.979 22.042-16.097 22.297Q-16.215 22.552-16.215 22.818M-13.296 24.616L-13.296 23.553Q-13.296 23.529-13.269 23.502Q-13.241 23.475-13.217 23.475L-13.108 23.475Q-13.043 23.475-13.029 23.533Q-12.934 23.967-12.688 24.218Q-12.442 24.469-12.028 24.469Q-11.686 24.469-11.433 24.336Q-11.180 24.203-11.180 23.895Q-11.180 23.738-11.274 23.623Q-11.368 23.509-11.507 23.440Q-11.645 23.372-11.813 23.334L-12.394 23.235Q-12.749 23.167-13.023 22.946Q-13.296 22.726-13.296 22.384Q-13.296 22.135-13.185 21.960Q-13.074 21.786-12.888 21.687Q-12.701 21.588-12.486 21.545Q-12.271 21.502-12.028 21.502Q-11.614 21.502-11.334 21.684L-11.119 21.509Q-11.109 21.506-11.102 21.504Q-11.095 21.502-11.085 21.502L-11.033 21.502Q-11.006 21.502-10.982 21.526Q-10.958 21.550-10.958 21.578L-10.958 22.425Q-10.958 22.446-10.982 22.473Q-11.006 22.500-11.033 22.500L-11.146 22.500Q-11.174 22.500-11.199 22.475Q-11.225 22.449-11.225 22.425Q-11.225 22.189-11.331 22.025Q-11.437 21.861-11.620 21.779Q-11.802 21.697-12.035 21.697Q-12.363 21.697-12.619 21.800Q-12.876 21.902-12.876 22.179Q-12.876 22.374-12.693 22.483Q-12.510 22.593-12.281 22.634L-11.707 22.740Q-11.461 22.788-11.247 22.916Q-11.033 23.044-10.897 23.247Q-10.760 23.451-10.760 23.700Q-10.760 24.213-11.126 24.452Q-11.491 24.691-12.028 24.691Q-12.524 24.691-12.855 24.397L-13.122 24.671Q-13.142 24.691-13.170 24.691L-13.217 24.691Q-13.241 24.691-13.269 24.664Q-13.296 24.637-13.296 24.616M-10.172 23.088Q-10.172 22.767-10.047 22.478Q-9.923 22.189-9.697 21.966Q-9.471 21.742-9.176 21.622Q-8.880 21.502-8.562 21.502Q-8.234 21.502-7.973 21.602Q-7.711 21.701-7.535 21.883Q-7.359 22.066-7.265 22.324Q-7.171 22.582-7.171 22.914Q-7.171 23.006-7.253 23.027L-9.509 23.027L-9.509 23.088Q-9.509 23.676-9.225 24.059Q-8.942 24.442-8.374 24.442Q-8.053 24.442-7.785 24.249Q-7.516 24.056-7.427 23.741Q-7.421 23.700-7.345 23.686L-7.253 23.686Q-7.171 23.710-7.171 23.782Q-7.171 23.789-7.178 23.816Q-7.291 24.213-7.662 24.452Q-8.032 24.691-8.456 24.691Q-8.894 24.691-9.294 24.483Q-9.694 24.274-9.933 23.907Q-10.172 23.540-10.172 23.088M-9.502 22.818L-7.687 22.818Q-7.687 22.541-7.785 22.289Q-7.882 22.036-8.080 21.880Q-8.279 21.725-8.562 21.725Q-8.839 21.725-9.053 21.883Q-9.266 22.042-9.384 22.297Q-9.502 22.552-9.502 22.818M-6.057 23.782L-6.057 21.885L-6.696 21.885L-6.696 21.663Q-6.378 21.663-6.161 21.453Q-5.944 21.243-5.843 20.933Q-5.742 20.624-5.742 20.316L-5.476 20.316L-5.476 21.605L-4.399 21.605L-4.399 21.885L-5.476 21.885L-5.476 23.769Q-5.476 24.045-5.372 24.244Q-5.267 24.442-5.008 24.442Q-4.850 24.442-4.744 24.338Q-4.638 24.233-4.589 24.080Q-4.539 23.926-4.539 23.769L-4.539 23.355L-4.273 23.355L-4.273 23.782Q-4.273 24.008-4.372 24.218Q-4.471 24.428-4.655 24.560Q-4.840 24.691-5.069 24.691Q-5.507 24.691-5.782 24.454Q-6.057 24.216-6.057 23.782\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(45.507 -30.573)\">\u003Cpath d=\"M-0.214 23.782L-0.214 21.885L-0.853 21.885L-0.853 21.663Q-0.535 21.663-0.318 21.453Q-0.101 21.243-0.001 20.933Q0.100 20.624 0.100 20.316L0.367 20.316L0.367 21.605L1.444 21.605L1.444 21.885L0.367 21.885L0.367 23.769Q0.367 24.045 0.471 24.244Q0.575 24.442 0.835 24.442Q0.992 24.442 1.098 24.338Q1.204 24.233 1.254 24.080Q1.303 23.926 1.303 23.769L1.303 23.355L1.570 23.355L1.570 23.782Q1.570 24.008 1.471 24.218Q1.372 24.428 1.187 24.560Q1.003 24.691 0.774 24.691Q0.336 24.691 0.061 24.454Q-0.214 24.216-0.214 23.782M2.339 23.140Q2.339 22.798 2.474 22.499Q2.609 22.200 2.848 21.976Q3.088 21.752 3.405 21.627Q3.723 21.502 4.055 21.502Q4.499 21.502 4.899 21.718Q5.299 21.933 5.533 22.311Q5.767 22.688 5.767 23.140Q5.767 23.481 5.625 23.765Q5.484 24.049 5.239 24.256Q4.995 24.462 4.685 24.577Q4.376 24.691 4.055 24.691Q3.624 24.691 3.223 24.490Q2.821 24.288 2.580 23.936Q2.339 23.584 2.339 23.140M4.055 24.442Q4.656 24.442 4.880 24.064Q5.104 23.686 5.104 23.054Q5.104 22.442 4.870 22.083Q4.636 21.725 4.055 21.725Q3.002 21.725 3.002 23.054Q3.002 23.686 3.228 24.064Q3.453 24.442 4.055 24.442\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(45.507 -30.573)\">\u003Cpath d=\"M9.065 24.616L9.065 23.553Q9.065 23.529 9.093 23.502Q9.120 23.475 9.144 23.475L9.253 23.475Q9.318 23.475 9.332 23.533Q9.428 23.967 9.674 24.218Q9.920 24.469 10.334 24.469Q10.675 24.469 10.928 24.336Q11.181 24.203 11.181 23.895Q11.181 23.738 11.087 23.623Q10.993 23.509 10.855 23.440Q10.716 23.372 10.549 23.334L9.968 23.235Q9.612 23.167 9.339 22.946Q9.065 22.726 9.065 22.384Q9.065 22.135 9.177 21.960Q9.288 21.786 9.474 21.687Q9.660 21.588 9.876 21.545Q10.091 21.502 10.334 21.502Q10.747 21.502 11.027 21.684L11.243 21.509Q11.253 21.506 11.260 21.504Q11.267 21.502 11.277 21.502L11.328 21.502Q11.355 21.502 11.379 21.526Q11.403 21.550 11.403 21.578L11.403 22.425Q11.403 22.446 11.379 22.473Q11.355 22.500 11.328 22.500L11.215 22.500Q11.188 22.500 11.162 22.475Q11.137 22.449 11.137 22.425Q11.137 22.189 11.031 22.025Q10.925 21.861 10.742 21.779Q10.559 21.697 10.327 21.697Q9.999 21.697 9.742 21.800Q9.486 21.902 9.486 22.179Q9.486 22.374 9.669 22.483Q9.852 22.593 10.081 22.634L10.655 22.740Q10.901 22.788 11.115 22.916Q11.328 23.044 11.465 23.247Q11.602 23.451 11.602 23.700Q11.602 24.213 11.236 24.452Q10.870 24.691 10.334 24.691Q9.838 24.691 9.506 24.397L9.240 24.671Q9.219 24.691 9.192 24.691L9.144 24.691Q9.120 24.691 9.093 24.664Q9.065 24.637 9.065 24.616M12.757 23.782L12.757 21.885L12.118 21.885L12.118 21.663Q12.436 21.663 12.653 21.453Q12.870 21.243 12.970 20.933Q13.071 20.624 13.071 20.316L13.338 20.316L13.338 21.605L14.415 21.605L14.415 21.885L13.338 21.885L13.338 23.769Q13.338 24.045 13.442 24.244Q13.546 24.442 13.806 24.442Q13.963 24.442 14.069 24.338Q14.175 24.233 14.225 24.080Q14.274 23.926 14.274 23.769L14.274 23.355L14.541 23.355L14.541 23.782Q14.541 24.008 14.442 24.218Q14.343 24.428 14.158 24.560Q13.974 24.691 13.745 24.691Q13.307 24.691 13.032 24.454Q12.757 24.216 12.757 23.782M15.409 23.895Q15.409 23.563 15.633 23.336Q15.857 23.109 16.200 22.981Q16.544 22.852 16.917 22.800Q17.289 22.747 17.593 22.747L17.593 22.494Q17.593 22.289 17.486 22.109Q17.378 21.930 17.197 21.827Q17.016 21.725 16.807 21.725Q16.400 21.725 16.165 21.817Q16.253 21.854 16.300 21.938Q16.346 22.022 16.346 22.124Q16.346 22.220 16.300 22.299Q16.253 22.377 16.173 22.422Q16.093 22.466 16.004 22.466Q15.854 22.466 15.753 22.369Q15.652 22.271 15.652 22.124Q15.652 21.502 16.807 21.502Q17.019 21.502 17.269 21.566Q17.518 21.629 17.720 21.748Q17.921 21.868 18.048 22.053Q18.174 22.237 18.174 22.480L18.174 24.056Q18.174 24.172 18.236 24.268Q18.297 24.363 18.410 24.363Q18.520 24.363 18.584 24.269Q18.649 24.175 18.649 24.056L18.649 23.608L18.916 23.608L18.916 24.056Q18.916 24.326 18.689 24.491Q18.461 24.657 18.181 24.657Q17.973 24.657 17.836 24.503Q17.699 24.350 17.675 24.134Q17.528 24.401 17.246 24.546Q16.964 24.691 16.640 24.691Q16.363 24.691 16.079 24.616Q15.795 24.541 15.602 24.362Q15.409 24.182 15.409 23.895M16.024 23.895Q16.024 24.069 16.125 24.199Q16.226 24.329 16.382 24.399Q16.537 24.469 16.701 24.469Q16.920 24.469 17.128 24.372Q17.337 24.274 17.465 24.093Q17.593 23.912 17.593 23.686L17.593 22.958Q17.269 22.958 16.903 23.049Q16.537 23.140 16.281 23.352Q16.024 23.563 16.024 23.895M21.083 24.623L19.347 24.623L19.347 24.343Q19.576 24.343 19.724 24.309Q19.873 24.274 19.873 24.134L19.873 22.285Q19.873 22.015 19.765 21.954Q19.658 21.892 19.347 21.892L19.347 21.612L20.376 21.537L20.376 22.244Q20.505 21.936 20.748 21.737Q20.991 21.537 21.309 21.537Q21.527 21.537 21.698 21.661Q21.869 21.786 21.869 21.998Q21.869 22.135 21.770 22.234Q21.671 22.333 21.538 22.333Q21.401 22.333 21.302 22.234Q21.203 22.135 21.203 21.998Q21.203 21.858 21.302 21.759Q21.011 21.759 20.811 21.955Q20.611 22.152 20.519 22.446Q20.427 22.740 20.427 23.020L20.427 24.134Q20.427 24.343 21.083 24.343L21.083 24.623M22.980 23.782L22.980 21.885L22.341 21.885L22.341 21.663Q22.659 21.663 22.876 21.453Q23.093 21.243 23.194 20.933Q23.294 20.624 23.294 20.316L23.561 20.316L23.561 21.605L24.638 21.605L24.638 21.885L23.561 21.885L23.561 23.769Q23.561 24.045 23.665 24.244Q23.770 24.442 24.029 24.442Q24.187 24.442 24.293 24.338Q24.398 24.233 24.448 24.080Q24.498 23.926 24.498 23.769L24.498 23.355L24.764 23.355L24.764 23.782Q24.764 24.008 24.665 24.218Q24.566 24.428 24.381 24.560Q24.197 24.691 23.968 24.691Q23.530 24.691 23.255 24.454Q22.980 24.216 22.980 23.782\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-41.992 1.674)\">\u003Cpath d=\"M-19.965 23.112Q-19.965 22.784-19.830 22.483Q-19.695 22.183-19.459 21.962Q-19.223 21.742-18.919 21.622Q-18.614 21.502-18.290 21.502Q-17.784 21.502-17.435 21.605Q-17.087 21.707-17.087 22.083Q-17.087 22.230-17.184 22.331Q-17.281 22.432-17.428 22.432Q-17.582 22.432-17.681 22.333Q-17.780 22.234-17.780 22.083Q-17.780 21.895-17.640 21.803Q-17.842 21.752-18.283 21.752Q-18.638 21.752-18.867 21.948Q-19.096 22.145-19.197 22.454Q-19.298 22.764-19.298 23.112Q-19.298 23.461-19.172 23.767Q-19.045 24.073-18.790 24.257Q-18.536 24.442-18.180 24.442Q-17.958 24.442-17.774 24.358Q-17.589 24.274-17.454 24.119Q-17.319 23.963-17.261 23.755Q-17.247 23.700-17.193 23.700L-17.080 23.700Q-17.049 23.700-17.027 23.724Q-17.005 23.748-17.005 23.782L-17.005 23.803Q-17.090 24.090-17.278 24.288Q-17.466 24.486-17.731 24.589Q-17.996 24.691-18.290 24.691Q-18.720 24.691-19.108 24.485Q-19.496 24.278-19.730 23.915Q-19.965 23.553-19.965 23.112M-16.458 23.140Q-16.458 22.798-16.323 22.499Q-16.188 22.200-15.948 21.976Q-15.709 21.752-15.391 21.627Q-15.073 21.502-14.742 21.502Q-14.298 21.502-13.898 21.718Q-13.498 21.933-13.264 22.311Q-13.029 22.688-13.029 23.140Q-13.029 23.481-13.171 23.765Q-13.313 24.049-13.558 24.256Q-13.802 24.462-14.111 24.577Q-14.421 24.691-14.742 24.691Q-15.173 24.691-15.574 24.490Q-15.976 24.288-16.217 23.936Q-16.458 23.584-16.458 23.140M-14.742 24.442Q-14.140 24.442-13.916 24.064Q-13.693 23.686-13.693 23.054Q-13.693 22.442-13.927 22.083Q-14.161 21.725-14.742 21.725Q-15.795 21.725-15.795 23.054Q-15.795 23.686-15.569 24.064Q-15.343 24.442-14.742 24.442M-10.753 24.623L-12.387 24.623L-12.387 24.343Q-12.158 24.343-12.009 24.309Q-11.861 24.274-11.861 24.134L-11.861 22.285Q-11.861 22.015-11.968 21.954Q-12.076 21.892-12.387 21.892L-12.387 21.612L-11.327 21.537L-11.327 22.186Q-11.156 21.878-10.852 21.707Q-10.548 21.537-10.203 21.537Q-9.697 21.537-9.413 21.760Q-9.130 21.984-9.130 22.480L-9.130 24.134Q-9.130 24.271-8.981 24.307Q-8.832 24.343-8.607 24.343L-8.607 24.623L-10.237 24.623L-10.237 24.343Q-10.008 24.343-9.859 24.309Q-9.711 24.274-9.711 24.134L-9.711 22.494Q-9.711 22.159-9.830 21.959Q-9.950 21.759-10.264 21.759Q-10.534 21.759-10.769 21.895Q-11.003 22.032-11.141 22.266Q-11.279 22.500-11.279 22.774L-11.279 24.134Q-11.279 24.271-11.129 24.307Q-10.979 24.343-10.753 24.343\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-41.992 1.674)\">\u003Cpath d=\"M-7.695 23.782L-7.695 21.885L-8.334 21.885L-8.334 21.663Q-8.016 21.663-7.799 21.453Q-7.582 21.243-7.482 20.933Q-7.381 20.624-7.381 20.316L-7.114 20.316L-7.114 21.605L-6.037 21.605L-6.037 21.885L-7.114 21.885L-7.114 23.769Q-7.114 24.045-7.010 24.244Q-6.906 24.442-6.646 24.442Q-6.489 24.442-6.383 24.338Q-6.277 24.233-6.227 24.080Q-6.178 23.926-6.178 23.769L-6.178 23.355L-5.911 23.355L-5.911 23.782Q-5.911 24.008-6.010 24.218Q-6.109 24.428-6.294 24.560Q-6.478 24.691-6.707 24.691Q-7.145 24.691-7.420 24.454Q-7.695 24.216-7.695 23.782M-3.484 24.623L-5.036 24.623L-5.036 24.343Q-4.810 24.343-4.662 24.309Q-4.513 24.274-4.513 24.134L-4.513 22.285Q-4.513 22.097-4.561 22.013Q-4.609 21.930-4.706 21.911Q-4.804 21.892-5.015 21.892L-5.015 21.612L-3.959 21.537L-3.959 24.134Q-3.959 24.274-3.828 24.309Q-3.696 24.343-3.484 24.343L-3.484 24.623M-4.756 20.316Q-4.756 20.145-4.633 20.026Q-4.510 19.906-4.339 19.906Q-4.171 19.906-4.048 20.026Q-3.925 20.145-3.925 20.316Q-3.925 20.491-4.048 20.614Q-4.171 20.737-4.339 20.737Q-4.510 20.737-4.633 20.614Q-4.756 20.491-4.756 20.316M-1.157 24.623L-2.790 24.623L-2.790 24.343Q-2.561 24.343-2.413 24.309Q-2.264 24.274-2.264 24.134L-2.264 22.285Q-2.264 22.015-2.372 21.954Q-2.479 21.892-2.790 21.892L-2.790 21.612L-1.731 21.537L-1.731 22.186Q-1.560 21.878-1.256 21.707Q-0.952 21.537-0.606 21.537Q-0.100 21.537 0.183 21.760Q0.467 21.984 0.467 22.480L0.467 24.134Q0.467 24.271 0.616 24.307Q0.764 24.343 0.990 24.343L0.990 24.623L-0.640 24.623L-0.640 24.343Q-0.411 24.343-0.263 24.309Q-0.114 24.274-0.114 24.134L-0.114 22.494Q-0.114 22.159-0.234 21.959Q-0.353 21.759-0.668 21.759Q-0.938 21.759-1.172 21.895Q-1.406 22.032-1.545 22.266Q-1.683 22.500-1.683 22.774L-1.683 24.134Q-1.683 24.271-1.533 24.307Q-1.382 24.343-1.157 24.343\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-41.992 1.674)\">\u003Cpath d=\"M1.944 23.789L1.944 22.285Q1.944 22.015 1.836 21.954Q1.728 21.892 1.417 21.892L1.417 21.612L2.525 21.537L2.525 23.769L2.525 23.789Q2.525 24.069 2.576 24.213Q2.627 24.356 2.769 24.413Q2.911 24.469 3.198 24.469Q3.451 24.469 3.656 24.329Q3.861 24.189 3.977 23.963Q4.094 23.738 4.094 23.488L4.094 22.285Q4.094 22.015 3.986 21.954Q3.878 21.892 3.567 21.892L3.567 21.612L4.675 21.537L4.675 23.950Q4.675 24.141 4.728 24.223Q4.781 24.305 4.881 24.324Q4.982 24.343 5.198 24.343L5.198 24.623L4.121 24.691L4.121 24.127Q4.012 24.309 3.866 24.432Q3.721 24.555 3.535 24.623Q3.348 24.691 3.147 24.691Q1.944 24.691 1.944 23.789M7.402 24.623L5.850 24.623L5.850 24.343Q6.076 24.343 6.225 24.309Q6.373 24.274 6.373 24.134L6.373 22.285Q6.373 22.097 6.326 22.013Q6.278 21.930 6.180 21.911Q6.083 21.892 5.871 21.892L5.871 21.612L6.927 21.537L6.927 24.134Q6.927 24.274 7.059 24.309Q7.190 24.343 7.402 24.343L7.402 24.623M6.131 20.316Q6.131 20.145 6.254 20.026Q6.377 19.906 6.548 19.906Q6.715 19.906 6.838 20.026Q6.961 20.145 6.961 20.316Q6.961 20.491 6.838 20.614Q6.715 20.737 6.548 20.737Q6.377 20.737 6.254 20.614Q6.131 20.491 6.131 20.316M9.730 24.623L8.096 24.623L8.096 24.343Q8.325 24.343 8.474 24.309Q8.622 24.274 8.622 24.134L8.622 22.285Q8.622 22.015 8.515 21.954Q8.407 21.892 8.096 21.892L8.096 21.612L9.156 21.537L9.156 22.186Q9.326 21.878 9.631 21.707Q9.935 21.537 10.280 21.537Q10.786 21.537 11.070 21.760Q11.353 21.984 11.353 22.480L11.353 24.134Q11.353 24.271 11.502 24.307Q11.651 24.343 11.876 24.343L11.876 24.623L10.246 24.623L10.246 24.343Q10.475 24.343 10.624 24.309Q10.772 24.274 10.772 24.134L10.772 22.494Q10.772 22.159 10.653 21.959Q10.533 21.759 10.219 21.759Q9.949 21.759 9.714 21.895Q9.480 22.032 9.342 22.266Q9.203 22.500 9.203 22.774L9.203 24.134Q9.203 24.271 9.354 24.307Q9.504 24.343 9.730 24.343L9.730 24.623M12.423 25.156Q12.423 24.910 12.620 24.726Q12.816 24.541 13.073 24.462Q12.936 24.350 12.864 24.189Q12.792 24.028 12.792 23.847Q12.792 23.526 13.004 23.280Q12.669 22.982 12.669 22.572Q12.669 22.111 13.059 21.824Q13.449 21.537 13.927 21.537Q14.399 21.537 14.734 21.783Q14.908 21.629 15.118 21.547Q15.328 21.465 15.557 21.465Q15.722 21.465 15.843 21.572Q15.964 21.680 15.964 21.844Q15.964 21.940 15.892 22.012Q15.821 22.083 15.728 22.083Q15.629 22.083 15.559 22.010Q15.489 21.936 15.489 21.837Q15.489 21.783 15.503 21.752L15.510 21.738Q15.516 21.718 15.525 21.707Q15.534 21.697 15.537 21.690Q15.181 21.690 14.894 21.913Q15.181 22.206 15.181 22.572Q15.181 22.887 14.997 23.119Q14.812 23.352 14.524 23.480Q14.235 23.608 13.927 23.608Q13.725 23.608 13.534 23.558Q13.343 23.509 13.165 23.399Q13.073 23.526 13.073 23.669Q13.073 23.851 13.201 23.986Q13.329 24.121 13.513 24.121L14.146 24.121Q14.594 24.121 14.963 24.192Q15.332 24.264 15.592 24.493Q15.851 24.722 15.851 25.156Q15.851 25.477 15.556 25.679Q15.260 25.881 14.857 25.970Q14.453 26.059 14.139 26.059Q13.821 26.059 13.418 25.970Q13.014 25.881 12.719 25.679Q12.423 25.477 12.423 25.156M12.878 25.156Q12.878 25.385 13.097 25.534Q13.315 25.683 13.607 25.751Q13.900 25.819 14.139 25.819Q14.303 25.819 14.512 25.783Q14.720 25.748 14.927 25.667Q15.134 25.587 15.265 25.459Q15.397 25.331 15.397 25.156Q15.397 24.804 15.016 24.710Q14.635 24.616 14.132 24.616L13.513 24.616Q13.274 24.616 13.076 24.767Q12.878 24.917 12.878 25.156M13.927 23.369Q14.594 23.369 14.594 22.572Q14.594 21.772 13.927 21.772Q13.257 21.772 13.257 22.572Q13.257 23.369 13.927 23.369M16.846 24.203Q16.846 24.035 16.969 23.912Q17.092 23.789 17.266 23.789Q17.434 23.789 17.557 23.912Q17.680 24.035 17.680 24.203Q17.680 24.377 17.557 24.500Q17.434 24.623 17.266 24.623Q17.092 24.623 16.969 24.500Q16.846 24.377 16.846 24.203M16.846 22.019Q16.846 21.851 16.969 21.728Q17.092 21.605 17.266 21.605Q17.434 21.605 17.557 21.728Q17.680 21.851 17.680 22.019Q17.680 22.193 17.557 22.316Q17.434 22.439 17.266 22.439Q17.092 22.439 16.969 22.316Q16.846 22.193 16.846 22.019\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-10.32 24.623c0-5.5-4.459-9.959-9.959-9.959s-9.958 4.459-9.958 9.959 4.458 9.958 9.958 9.958 9.959-4.458 9.959-9.958Zm-9.959 0\"\u002F>\u003Cg transform=\"translate(-3.801 2.578)\">\u003Cpath d=\"M-19.998 24.615L-19.998 23.393Q-19.998 23.365-19.966 23.334Q-19.935 23.303-19.912 23.303L-19.806 23.303Q-19.736 23.303-19.720 23.365Q-19.658 23.686-19.519 23.926Q-19.381 24.166-19.148 24.307Q-18.916 24.447-18.607 24.447Q-18.369 24.447-18.160 24.387Q-17.951 24.326-17.814 24.178Q-17.677 24.029-17.677 23.783Q-17.677 23.529-17.888 23.363Q-18.099 23.197-18.369 23.143L-18.990 23.029Q-19.396 22.951-19.697 22.695Q-19.998 22.439-19.998 22.064Q-19.998 21.697-19.797 21.475Q-19.595 21.252-19.271 21.154Q-18.947 21.057-18.607 21.057Q-18.142 21.057-17.845 21.264L-17.623 21.080Q-17.599 21.057-17.568 21.057L-17.517 21.057Q-17.486 21.057-17.459 21.084Q-17.431 21.111-17.431 21.143L-17.431 22.127Q-17.431 22.158-17.457 22.187Q-17.482 22.217-17.517 22.217L-17.623 22.217Q-17.658 22.217-17.685 22.189Q-17.713 22.162-17.713 22.127Q-17.713 21.728-17.965 21.508Q-18.216 21.287-18.615 21.287Q-18.970 21.287-19.254 21.410Q-19.537 21.533-19.537 21.838Q-19.537 22.057-19.336 22.189Q-19.134 22.322-18.888 22.365L-18.263 22.478Q-17.834 22.568-17.525 22.865Q-17.216 23.162-17.216 23.576Q-17.216 24.146-17.615 24.424Q-18.013 24.701-18.607 24.701Q-19.158 24.701-19.509 24.365L-19.806 24.678Q-19.830 24.701-19.865 24.701L-19.912 24.701Q-19.935 24.701-19.966 24.670Q-19.998 24.639-19.998 24.615M-14.806 24.791Q-15.509 24.791-15.910 24.391Q-16.310 23.990-16.455 23.381Q-16.599 22.771-16.599 22.072Q-16.599 21.549-16.529 21.086Q-16.459 20.623-16.265 20.211Q-16.072 19.799-15.715 19.551Q-15.357 19.303-14.806 19.303Q-14.256 19.303-13.898 19.551Q-13.541 19.799-13.349 20.209Q-13.158 20.619-13.088 21.088Q-13.017 21.557-13.017 22.072Q-13.017 22.771-13.160 23.379Q-13.302 23.986-13.703 24.389Q-14.103 24.791-14.806 24.791M-14.806 24.533Q-14.334 24.533-14.101 24.098Q-13.869 23.662-13.814 23.123Q-13.759 22.584-13.759 21.943Q-13.759 20.947-13.943 20.254Q-14.127 19.561-14.806 19.561Q-15.174 19.561-15.394 19.799Q-15.615 20.037-15.711 20.394Q-15.806 20.752-15.832 21.123Q-15.857 21.494-15.857 21.943Q-15.857 22.584-15.802 23.123Q-15.748 23.662-15.515 24.098Q-15.283 24.533-14.806 24.533\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M35.204 24.623c0-5.5-4.458-9.959-9.958-9.959s-9.959 4.459-9.959 9.959 4.459 9.958 9.959 9.958 9.958-4.458 9.958-9.958Zm-9.958 0\"\u002F>\u003Cg transform=\"translate(41.723 2.578)\">\u003Cpath d=\"M-19.998 24.615L-19.998 23.393Q-19.998 23.365-19.966 23.334Q-19.935 23.303-19.912 23.303L-19.806 23.303Q-19.736 23.303-19.720 23.365Q-19.658 23.686-19.519 23.926Q-19.381 24.166-19.148 24.307Q-18.916 24.447-18.607 24.447Q-18.369 24.447-18.160 24.387Q-17.951 24.326-17.814 24.178Q-17.677 24.029-17.677 23.783Q-17.677 23.529-17.888 23.363Q-18.099 23.197-18.369 23.143L-18.990 23.029Q-19.396 22.951-19.697 22.695Q-19.998 22.439-19.998 22.064Q-19.998 21.697-19.797 21.475Q-19.595 21.252-19.271 21.154Q-18.947 21.057-18.607 21.057Q-18.142 21.057-17.845 21.264L-17.623 21.080Q-17.599 21.057-17.568 21.057L-17.517 21.057Q-17.486 21.057-17.459 21.084Q-17.431 21.111-17.431 21.143L-17.431 22.127Q-17.431 22.158-17.457 22.187Q-17.482 22.217-17.517 22.217L-17.623 22.217Q-17.658 22.217-17.685 22.189Q-17.713 22.162-17.713 22.127Q-17.713 21.728-17.965 21.508Q-18.216 21.287-18.615 21.287Q-18.970 21.287-19.254 21.410Q-19.537 21.533-19.537 21.838Q-19.537 22.057-19.336 22.189Q-19.134 22.322-18.888 22.365L-18.263 22.478Q-17.834 22.568-17.525 22.865Q-17.216 23.162-17.216 23.576Q-17.216 24.146-17.615 24.424Q-18.013 24.701-18.607 24.701Q-19.158 24.701-19.509 24.365L-19.806 24.678Q-19.830 24.701-19.865 24.701L-19.912 24.701Q-19.935 24.701-19.966 24.670Q-19.998 24.639-19.998 24.615M-13.334 24.623L-16.127 24.623L-16.127 24.326Q-15.064 24.326-15.064 24.064L-15.064 19.896Q-15.494 20.111-16.174 20.111L-16.174 19.814Q-15.154 19.814-14.638 19.303L-14.494 19.303Q-14.420 19.322-14.400 19.400L-14.400 24.064Q-14.400 24.326-13.334 24.326\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M80.728 24.623c0-5.5-4.458-9.959-9.958-9.959s-9.959 4.459-9.959 9.959 4.459 9.958 9.959 9.958 9.958-4.458 9.958-9.958Zm-9.958 0\"\u002F>\u003Cg transform=\"translate(87.247 2.578)\">\u003Cpath d=\"M-19.998 24.615L-19.998 23.393Q-19.998 23.365-19.966 23.334Q-19.935 23.303-19.912 23.303L-19.806 23.303Q-19.736 23.303-19.720 23.365Q-19.658 23.686-19.519 23.926Q-19.381 24.166-19.148 24.307Q-18.916 24.447-18.607 24.447Q-18.369 24.447-18.160 24.387Q-17.951 24.326-17.814 24.178Q-17.677 24.029-17.677 23.783Q-17.677 23.529-17.888 23.363Q-18.099 23.197-18.369 23.143L-18.990 23.029Q-19.396 22.951-19.697 22.695Q-19.998 22.439-19.998 22.064Q-19.998 21.697-19.797 21.475Q-19.595 21.252-19.271 21.154Q-18.947 21.057-18.607 21.057Q-18.142 21.057-17.845 21.264L-17.623 21.080Q-17.599 21.057-17.568 21.057L-17.517 21.057Q-17.486 21.057-17.459 21.084Q-17.431 21.111-17.431 21.143L-17.431 22.127Q-17.431 22.158-17.457 22.187Q-17.482 22.217-17.517 22.217L-17.623 22.217Q-17.658 22.217-17.685 22.189Q-17.713 22.162-17.713 22.127Q-17.713 21.728-17.965 21.508Q-18.216 21.287-18.615 21.287Q-18.970 21.287-19.254 21.410Q-19.537 21.533-19.537 21.838Q-19.537 22.057-19.336 22.189Q-19.134 22.322-18.888 22.365L-18.263 22.478Q-17.834 22.568-17.525 22.865Q-17.216 23.162-17.216 23.576Q-17.216 24.146-17.615 24.424Q-18.013 24.701-18.607 24.701Q-19.158 24.701-19.509 24.365L-19.806 24.678Q-19.830 24.701-19.865 24.701L-19.912 24.701Q-19.935 24.701-19.966 24.670Q-19.998 24.639-19.998 24.615M-13.341 24.623L-16.502 24.623L-16.502 24.416Q-16.502 24.389-16.478 24.357L-15.127 22.959Q-14.748 22.572-14.500 22.283Q-14.252 21.994-14.078 21.637Q-13.904 21.279-13.904 20.889Q-13.904 20.541-14.037 20.248Q-14.170 19.955-14.424 19.777Q-14.677 19.600-15.033 19.600Q-15.392 19.600-15.683 19.795Q-15.974 19.990-16.119 20.318L-16.064 20.318Q-15.881 20.318-15.756 20.439Q-15.631 20.561-15.631 20.752Q-15.631 20.932-15.756 21.061Q-15.881 21.189-16.064 21.189Q-16.244 21.189-16.373 21.061Q-16.502 20.932-16.502 20.752Q-16.502 20.350-16.281 20.014Q-16.060 19.678-15.695 19.490Q-15.330 19.303-14.927 19.303Q-14.447 19.303-14.031 19.490Q-13.615 19.678-13.363 20.039Q-13.111 20.400-13.111 20.889Q-13.111 21.248-13.265 21.551Q-13.420 21.853-13.672 22.113Q-13.924 22.373-14.273 22.658Q-14.623 22.943-14.791 23.096L-15.720 23.936L-15.006 23.936Q-13.631 23.936-13.591 23.896Q-13.521 23.818-13.478 23.633Q-13.435 23.447-13.392 23.158L-13.111 23.158\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-warn)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M126.253 24.623c0-5.5-4.459-9.959-9.959-9.959s-9.958 4.459-9.958 9.959 4.458 9.958 9.958 9.958 9.959-4.458 9.959-9.958Zm-9.959 0\"\u002F>\u003Cg transform=\"translate(134.448 .944)\">\u003Cpath d=\"M-20.041 25.232Q-20.041 24.951-19.830 24.740Q-19.619 24.529-19.334 24.439Q-19.490 24.314-19.568 24.125Q-19.646 23.936-19.646 23.736Q-19.646 23.381-19.416 23.088Q-19.783 22.748-19.783 22.279Q-19.783 21.928-19.580 21.658Q-19.377 21.389-19.056 21.242Q-18.736 21.096-18.392 21.096Q-17.873 21.096-17.502 21.377Q-17.138 21.006-16.591 21.006Q-16.412 21.006-16.285 21.133Q-16.158 21.260-16.158 21.439Q-16.158 21.545-16.236 21.623Q-16.314 21.701-16.424 21.701Q-16.533 21.701-16.609 21.625Q-16.685 21.549-16.685 21.439Q-16.685 21.338-16.646 21.287Q-16.638 21.279-16.634 21.273Q-16.631 21.268-16.631 21.264Q-17.006 21.264-17.326 21.518Q-17.006 21.857-17.006 22.279Q-17.006 22.549-17.123 22.766Q-17.240 22.982-17.445 23.141Q-17.650 23.299-17.892 23.381Q-18.134 23.463-18.392 23.463Q-18.611 23.463-18.824 23.404Q-19.037 23.346-19.232 23.225Q-19.326 23.365-19.326 23.545Q-19.326 23.752-19.189 23.904Q-19.052 24.057-18.845 24.057L-18.150 24.057Q-17.662 24.057-17.250 24.141Q-16.838 24.225-16.558 24.482Q-16.279 24.740-16.279 25.232Q-16.279 25.596-16.599 25.828Q-16.920 26.061-17.361 26.162Q-17.802 26.264-18.158 26.264Q-18.513 26.264-18.957 26.162Q-19.400 26.061-19.720 25.828Q-20.041 25.596-20.041 25.232M-19.537 25.232Q-19.537 25.428-19.392 25.576Q-19.248 25.725-19.035 25.814Q-18.822 25.904-18.582 25.951Q-18.341 25.998-18.158 25.998Q-17.916 25.998-17.586 25.920Q-17.256 25.842-17.019 25.668Q-16.783 25.494-16.783 25.232Q-16.783 24.826-17.193 24.717Q-17.603 24.607-18.166 24.607L-18.845 24.607Q-19.115 24.607-19.326 24.785Q-19.537 24.963-19.537 25.232M-18.392 23.197Q-17.670 23.197-17.670 22.279Q-17.670 21.357-18.392 21.357Q-19.119 21.357-19.119 22.279Q-19.119 23.197-18.392 23.197\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-10.12 24.623h22.607\"\u002F>\u003Cpath stroke=\"none\" d=\"m15.087 24.623-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M35.404 24.623h22.608\"\u002F>\u003Cpath stroke=\"none\" d=\"m60.611 24.623-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M80.928 24.623h22.408\"\u002F>\u003Cpath stroke=\"none\" d=\"m105.936 24.623-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(113.844 -16.664)\">\u003Cpath d=\"M-20.006 25.156Q-20.006 24.910-19.809 24.726Q-19.612 24.541-19.356 24.462Q-19.493 24.350-19.565 24.189Q-19.636 24.028-19.636 23.847Q-19.636 23.526-19.425 23.280Q-19.759 22.982-19.759 22.572Q-19.759 22.111-19.370 21.824Q-18.980 21.537-18.502 21.537Q-18.030 21.537-17.695 21.783Q-17.521 21.629-17.310 21.547Q-17.100 21.465-16.871 21.465Q-16.707 21.465-16.586 21.572Q-16.465 21.680-16.465 21.844Q-16.465 21.940-16.536 22.012Q-16.608 22.083-16.700 22.083Q-16.800 22.083-16.870 22.010Q-16.940 21.936-16.940 21.837Q-16.940 21.783-16.926 21.752L-16.919 21.738Q-16.912 21.718-16.904 21.707Q-16.895 21.697-16.892 21.690Q-17.247 21.690-17.534 21.913Q-17.247 22.206-17.247 22.572Q-17.247 22.887-17.432 23.119Q-17.616 23.352-17.905 23.480Q-18.194 23.608-18.502 23.608Q-18.703 23.608-18.895 23.558Q-19.086 23.509-19.264 23.399Q-19.356 23.526-19.356 23.669Q-19.356 23.851-19.228 23.986Q-19.100 24.121-18.915 24.121L-18.283 24.121Q-17.835 24.121-17.466 24.192Q-17.097 24.264-16.837 24.493Q-16.577 24.722-16.577 25.156Q-16.577 25.477-16.873 25.679Q-17.169 25.881-17.572 25.970Q-17.975 26.059-18.290 26.059Q-18.608 26.059-19.011 25.970Q-19.414 25.881-19.710 25.679Q-20.006 25.477-20.006 25.156M-19.551 25.156Q-19.551 25.385-19.332 25.534Q-19.113 25.683-18.821 25.751Q-18.529 25.819-18.290 25.819Q-18.126 25.819-17.917 25.783Q-17.709 25.748-17.502 25.667Q-17.295 25.587-17.164 25.459Q-17.032 25.331-17.032 25.156Q-17.032 24.804-17.413 24.710Q-17.794 24.616-18.297 24.616L-18.915 24.616Q-19.154 24.616-19.353 24.767Q-19.551 24.917-19.551 25.156M-18.502 23.369Q-17.835 23.369-17.835 22.572Q-17.835 21.772-18.502 21.772Q-19.172 21.772-19.172 22.572Q-19.172 23.369-18.502 23.369M-15.925 23.895Q-15.925 23.563-15.701 23.336Q-15.477 23.109-15.133 22.981Q-14.790 22.852-14.417 22.800Q-14.045 22.747-13.740 22.747L-13.740 22.494Q-13.740 22.289-13.848 22.109Q-13.956 21.930-14.137 21.827Q-14.318 21.725-14.527 21.725Q-14.933 21.725-15.169 21.817Q-15.080 21.854-15.034 21.938Q-14.988 22.022-14.988 22.124Q-14.988 22.220-15.034 22.299Q-15.080 22.377-15.161 22.422Q-15.241 22.466-15.330 22.466Q-15.480 22.466-15.581 22.369Q-15.682 22.271-15.682 22.124Q-15.682 21.502-14.527 21.502Q-14.315 21.502-14.065 21.566Q-13.816 21.629-13.614 21.748Q-13.412 21.868-13.286 22.053Q-13.159 22.237-13.159 22.480L-13.159 24.056Q-13.159 24.172-13.098 24.268Q-13.036 24.363-12.924 24.363Q-12.814 24.363-12.749 24.269Q-12.684 24.175-12.684 24.056L-12.684 23.608L-12.418 23.608L-12.418 24.056Q-12.418 24.326-12.645 24.491Q-12.872 24.657-13.153 24.657Q-13.361 24.657-13.498 24.503Q-13.634 24.350-13.658 24.134Q-13.805 24.401-14.087 24.546Q-14.369 24.691-14.694 24.691Q-14.971 24.691-15.255 24.616Q-15.538 24.541-15.731 24.362Q-15.925 24.182-15.925 23.895M-15.309 23.895Q-15.309 24.069-15.208 24.199Q-15.108 24.329-14.952 24.399Q-14.797 24.469-14.633 24.469Q-14.414 24.469-14.205 24.372Q-13.997 24.274-13.869 24.093Q-13.740 23.912-13.740 23.686L-13.740 22.958Q-14.065 22.958-14.431 23.049Q-14.797 23.140-15.053 23.352Q-15.309 23.563-15.309 23.895M-10.319 24.623L-11.953 24.623L-11.953 24.343Q-11.724 24.343-11.575 24.309Q-11.426 24.274-11.426 24.134L-11.426 22.285Q-11.426 22.015-11.534 21.954Q-11.642 21.892-11.953 21.892L-11.953 21.612L-10.893 21.537L-10.893 22.186Q-10.722 21.878-10.418 21.707Q-10.114 21.537-9.769 21.537Q-9.369 21.537-9.092 21.677Q-8.815 21.817-8.730 22.165Q-8.562 21.872-8.263 21.704Q-7.964 21.537-7.619 21.537Q-7.113 21.537-6.829 21.760Q-6.546 21.984-6.546 22.480L-6.546 24.134Q-6.546 24.271-6.397 24.307Q-6.248 24.343-6.023 24.343L-6.023 24.623L-7.653 24.623L-7.653 24.343Q-7.427 24.343-7.277 24.307Q-7.127 24.271-7.127 24.134L-7.127 22.494Q-7.127 22.159-7.246 21.959Q-7.366 21.759-7.680 21.759Q-7.950 21.759-8.185 21.895Q-8.419 22.032-8.557 22.266Q-8.696 22.500-8.696 22.774L-8.696 24.134Q-8.696 24.271-8.547 24.307Q-8.398 24.343-8.173 24.343L-8.173 24.623L-9.803 24.623L-9.803 24.343Q-9.574 24.343-9.425 24.309Q-9.277 24.274-9.277 24.134L-9.277 22.494Q-9.277 22.159-9.396 21.959Q-9.516 21.759-9.830 21.759Q-10.100 21.759-10.334 21.895Q-10.569 22.032-10.707 22.266Q-10.845 22.500-10.845 22.774L-10.845 24.134Q-10.845 24.271-10.695 24.307Q-10.545 24.343-10.319 24.343L-10.319 24.623M-3.753 24.623L-5.387 24.623L-5.387 24.343Q-5.158 24.343-5.009 24.309Q-4.861 24.274-4.861 24.134L-4.861 22.285Q-4.861 22.015-4.968 21.954Q-5.076 21.892-5.387 21.892L-5.387 21.612L-4.327 21.537L-4.327 22.186Q-4.156 21.878-3.852 21.707Q-3.548 21.537-3.203 21.537Q-2.803 21.537-2.526 21.677Q-2.249 21.817-2.164 22.165Q-1.996 21.872-1.697 21.704Q-1.398 21.537-1.053 21.537Q-0.547 21.537-0.263 21.760Q0.020 21.984 0.020 22.480L0.020 24.134Q0.020 24.271 0.169 24.307Q0.318 24.343 0.543 24.343L0.543 24.623L-1.087 24.623L-1.087 24.343Q-0.862 24.343-0.711 24.307Q-0.561 24.271-0.561 24.134L-0.561 22.494Q-0.561 22.159-0.680 21.959Q-0.800 21.759-1.114 21.759Q-1.384 21.759-1.619 21.895Q-1.853 22.032-1.991 22.266Q-2.130 22.500-2.130 22.774L-2.130 24.134Q-2.130 24.271-1.981 24.307Q-1.832 24.343-1.607 24.343L-1.607 24.623L-3.237 24.623L-3.237 24.343Q-3.008 24.343-2.859 24.309Q-2.711 24.274-2.711 24.134L-2.711 22.494Q-2.711 22.159-2.830 21.959Q-2.950 21.759-3.264 21.759Q-3.534 21.759-3.769 21.895Q-4.003 22.032-4.141 22.266Q-4.279 22.500-4.279 22.774L-4.279 24.134Q-4.279 24.271-4.129 24.307Q-3.979 24.343-3.753 24.343L-3.753 24.623M1.189 23.895Q1.189 23.563 1.413 23.336Q1.637 23.109 1.981 22.981Q2.324 22.852 2.697 22.800Q3.069 22.747 3.373 22.747L3.373 22.494Q3.373 22.289 3.266 22.109Q3.158 21.930 2.977 21.827Q2.796 21.725 2.587 21.725Q2.180 21.725 1.945 21.817Q2.034 21.854 2.080 21.938Q2.126 22.022 2.126 22.124Q2.126 22.220 2.080 22.299Q2.034 22.377 1.953 22.422Q1.873 22.466 1.784 22.466Q1.634 22.466 1.533 22.369Q1.432 22.271 1.432 22.124Q1.432 21.502 2.587 21.502Q2.799 21.502 3.049 21.566Q3.298 21.629 3.500 21.748Q3.701 21.868 3.828 22.053Q3.954 22.237 3.954 22.480L3.954 24.056Q3.954 24.172 4.016 24.268Q4.077 24.363 4.190 24.363Q4.300 24.363 4.365 24.269Q4.429 24.175 4.429 24.056L4.429 23.608L4.696 23.608L4.696 24.056Q4.696 24.326 4.469 24.491Q4.242 24.657 3.961 24.657Q3.753 24.657 3.616 24.503Q3.479 24.350 3.455 24.134Q3.308 24.401 3.026 24.546Q2.744 24.691 2.420 24.691Q2.143 24.691 1.859 24.616Q1.575 24.541 1.382 24.362Q1.189 24.182 1.189 23.895M1.804 23.895Q1.804 24.069 1.905 24.199Q2.006 24.329 2.162 24.399Q2.317 24.469 2.481 24.469Q2.700 24.469 2.909 24.372Q3.117 24.274 3.245 24.093Q3.373 23.912 3.373 23.686L3.373 22.958Q3.049 22.958 2.683 23.049Q2.317 23.140 2.061 23.352Q1.804 23.563 1.804 23.895M7.242 26.373Q6.692 25.973 6.321 25.418Q5.950 24.862 5.769 24.216Q5.588 23.570 5.588 22.873Q5.588 22.360 5.689 21.865Q5.790 21.369 5.995 20.918Q6.200 20.467 6.513 20.075Q6.825 19.684 7.242 19.380Q7.253 19.376 7.260 19.375Q7.266 19.373 7.277 19.373L7.345 19.373Q7.379 19.373 7.401 19.397Q7.424 19.421 7.424 19.458Q7.424 19.503 7.396 19.520Q7.048 19.821 6.795 20.205Q6.542 20.590 6.390 21.031Q6.238 21.472 6.166 21.928Q6.094 22.384 6.094 22.873Q6.094 23.874 6.403 24.761Q6.713 25.648 7.396 26.233Q7.424 26.250 7.424 26.294Q7.424 26.332 7.401 26.356Q7.379 26.380 7.345 26.380L7.277 26.380Q7.270 26.376 7.261 26.375Q7.253 26.373 7.242 26.373M8.193 25.156Q8.193 24.910 8.389 24.726Q8.586 24.541 8.842 24.462Q8.705 24.350 8.634 24.189Q8.562 24.028 8.562 23.847Q8.562 23.526 8.774 23.280Q8.439 22.982 8.439 22.572Q8.439 22.111 8.828 21.824Q9.218 21.537 9.697 21.537Q10.168 21.537 10.503 21.783Q10.678 21.629 10.888 21.547Q11.098 21.465 11.327 21.465Q11.491 21.465 11.612 21.572Q11.734 21.680 11.734 21.844Q11.734 21.940 11.662 22.012Q11.590 22.083 11.498 22.083Q11.399 22.083 11.329 22.010Q11.259 21.936 11.259 21.837Q11.259 21.783 11.272 21.752L11.279 21.738Q11.286 21.718 11.294 21.707Q11.303 21.697 11.306 21.690Q10.951 21.690 10.664 21.913Q10.951 22.206 10.951 22.572Q10.951 22.887 10.766 23.119Q10.582 23.352 10.293 23.480Q10.004 23.608 9.697 23.608Q9.495 23.608 9.304 23.558Q9.112 23.509 8.934 23.399Q8.842 23.526 8.842 23.669Q8.842 23.851 8.970 23.986Q9.098 24.121 9.283 24.121L9.915 24.121Q10.363 24.121 10.732 24.192Q11.101 24.264 11.361 24.493Q11.621 24.722 11.621 25.156Q11.621 25.477 11.325 25.679Q11.030 25.881 10.626 25.970Q10.223 26.059 9.909 26.059Q9.591 26.059 9.187 25.970Q8.784 25.881 8.488 25.679Q8.193 25.477 8.193 25.156M8.647 25.156Q8.647 25.385 8.866 25.534Q9.085 25.683 9.377 25.751Q9.669 25.819 9.909 25.819Q10.073 25.819 10.281 25.783Q10.490 25.748 10.696 25.667Q10.903 25.587 11.035 25.459Q11.166 25.331 11.166 25.156Q11.166 24.804 10.785 24.710Q10.404 24.616 9.902 24.616L9.283 24.616Q9.044 24.616 8.846 24.767Q8.647 24.917 8.647 25.156M9.697 23.369Q10.363 23.369 10.363 22.572Q10.363 21.772 9.697 21.772Q9.027 21.772 9.027 22.572Q9.027 23.369 9.697 23.369M12.537 26.380L12.469 26.380Q12.434 26.380 12.412 26.354Q12.390 26.329 12.390 26.294Q12.390 26.250 12.421 26.233Q12.776 25.929 13.026 25.539Q13.275 25.149 13.427 24.717Q13.579 24.285 13.649 23.816Q13.720 23.348 13.720 22.873Q13.720 22.394 13.649 21.928Q13.579 21.461 13.426 21.026Q13.272 20.590 13.021 20.202Q12.769 19.814 12.421 19.520Q12.390 19.503 12.390 19.458Q12.390 19.424 12.412 19.399Q12.434 19.373 12.469 19.373L12.537 19.373Q12.547 19.373 12.556 19.375Q12.564 19.376 12.575 19.380Q13.118 19.780 13.491 20.333Q13.863 20.887 14.044 21.533Q14.225 22.179 14.225 22.873Q14.225 23.574 14.044 24.221Q13.863 24.869 13.489 25.423Q13.115 25.977 12.575 26.373Q12.564 26.373 12.556 26.375Q12.547 26.376 12.537 26.380M20.504 23.816L15.671 23.816Q15.603 23.806 15.557 23.760Q15.511 23.714 15.511 23.642Q15.511 23.577 15.557 23.531Q15.603 23.485 15.671 23.475L20.504 23.475Q20.573 23.485 20.619 23.531Q20.665 23.577 20.665 23.642Q20.665 23.714 20.619 23.760Q20.573 23.806 20.504 23.816M20.504 22.278L15.671 22.278Q15.603 22.268 15.557 22.222Q15.511 22.176 15.511 22.104Q15.511 21.960 15.671 21.936L20.504 21.936Q20.665 21.960 20.665 22.104Q20.665 22.176 20.619 22.222Q20.573 22.268 20.504 22.278M23.150 24.763Q22.514 24.763 22.150 24.418Q21.786 24.073 21.651 23.548Q21.516 23.023 21.516 22.398Q21.516 21.373 21.871 20.674Q22.227 19.975 23.150 19.975Q24.076 19.975 24.428 20.674Q24.780 21.373 24.780 22.398Q24.780 23.023 24.645 23.548Q24.510 24.073 24.148 24.418Q23.785 24.763 23.150 24.763M23.150 24.538Q23.587 24.538 23.801 24.163Q24.014 23.789 24.064 23.322Q24.114 22.856 24.114 22.278Q24.114 21.725 24.064 21.297Q24.014 20.870 23.803 20.535Q23.591 20.200 23.150 20.200Q22.808 20.200 22.605 20.407Q22.401 20.614 22.314 20.926Q22.227 21.239 22.205 21.555Q22.182 21.872 22.182 22.278Q22.182 22.695 22.205 23.037Q22.227 23.379 22.316 23.727Q22.405 24.076 22.610 24.307Q22.815 24.538 23.150 24.538\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(16.33 19.994)\">\u003Cpath d=\"M-20.006 25.156Q-20.006 24.910-19.809 24.726Q-19.612 24.541-19.356 24.462Q-19.493 24.350-19.565 24.189Q-19.636 24.028-19.636 23.847Q-19.636 23.526-19.425 23.280Q-19.759 22.982-19.759 22.572Q-19.759 22.111-19.370 21.824Q-18.980 21.537-18.502 21.537Q-18.030 21.537-17.695 21.783Q-17.521 21.629-17.310 21.547Q-17.100 21.465-16.871 21.465Q-16.707 21.465-16.586 21.572Q-16.465 21.680-16.465 21.844Q-16.465 21.940-16.536 22.012Q-16.608 22.083-16.700 22.083Q-16.800 22.083-16.870 22.010Q-16.940 21.936-16.940 21.837Q-16.940 21.783-16.926 21.752L-16.919 21.738Q-16.912 21.718-16.904 21.707Q-16.895 21.697-16.892 21.690Q-17.247 21.690-17.534 21.913Q-17.247 22.206-17.247 22.572Q-17.247 22.887-17.432 23.119Q-17.616 23.352-17.905 23.480Q-18.194 23.608-18.502 23.608Q-18.703 23.608-18.895 23.558Q-19.086 23.509-19.264 23.399Q-19.356 23.526-19.356 23.669Q-19.356 23.851-19.228 23.986Q-19.100 24.121-18.915 24.121L-18.283 24.121Q-17.835 24.121-17.466 24.192Q-17.097 24.264-16.837 24.493Q-16.577 24.722-16.577 25.156Q-16.577 25.477-16.873 25.679Q-17.169 25.881-17.572 25.970Q-17.975 26.059-18.290 26.059Q-18.608 26.059-19.011 25.970Q-19.414 25.881-19.710 25.679Q-20.006 25.477-20.006 25.156M-19.551 25.156Q-19.551 25.385-19.332 25.534Q-19.113 25.683-18.821 25.751Q-18.529 25.819-18.290 25.819Q-18.126 25.819-17.917 25.783Q-17.709 25.748-17.502 25.667Q-17.295 25.587-17.164 25.459Q-17.032 25.331-17.032 25.156Q-17.032 24.804-17.413 24.710Q-17.794 24.616-18.297 24.616L-18.915 24.616Q-19.154 24.616-19.353 24.767Q-19.551 24.917-19.551 25.156M-18.502 23.369Q-17.835 23.369-17.835 22.572Q-17.835 21.772-18.502 21.772Q-19.172 21.772-19.172 22.572Q-19.172 23.369-18.502 23.369M-15.925 23.895Q-15.925 23.563-15.701 23.336Q-15.477 23.109-15.133 22.981Q-14.790 22.852-14.417 22.800Q-14.045 22.747-13.740 22.747L-13.740 22.494Q-13.740 22.289-13.848 22.109Q-13.956 21.930-14.137 21.827Q-14.318 21.725-14.527 21.725Q-14.933 21.725-15.169 21.817Q-15.080 21.854-15.034 21.938Q-14.988 22.022-14.988 22.124Q-14.988 22.220-15.034 22.299Q-15.080 22.377-15.161 22.422Q-15.241 22.466-15.330 22.466Q-15.480 22.466-15.581 22.369Q-15.682 22.271-15.682 22.124Q-15.682 21.502-14.527 21.502Q-14.315 21.502-14.065 21.566Q-13.816 21.629-13.614 21.748Q-13.412 21.868-13.286 22.053Q-13.159 22.237-13.159 22.480L-13.159 24.056Q-13.159 24.172-13.098 24.268Q-13.036 24.363-12.924 24.363Q-12.814 24.363-12.749 24.269Q-12.684 24.175-12.684 24.056L-12.684 23.608L-12.418 23.608L-12.418 24.056Q-12.418 24.326-12.645 24.491Q-12.872 24.657-13.153 24.657Q-13.361 24.657-13.498 24.503Q-13.634 24.350-13.658 24.134Q-13.805 24.401-14.087 24.546Q-14.369 24.691-14.694 24.691Q-14.971 24.691-15.255 24.616Q-15.538 24.541-15.731 24.362Q-15.925 24.182-15.925 23.895M-15.309 23.895Q-15.309 24.069-15.208 24.199Q-15.108 24.329-14.952 24.399Q-14.797 24.469-14.633 24.469Q-14.414 24.469-14.205 24.372Q-13.997 24.274-13.869 24.093Q-13.740 23.912-13.740 23.686L-13.740 22.958Q-14.065 22.958-14.431 23.049Q-14.797 23.140-15.053 23.352Q-15.309 23.563-15.309 23.895M-10.319 24.623L-11.953 24.623L-11.953 24.343Q-11.724 24.343-11.575 24.309Q-11.426 24.274-11.426 24.134L-11.426 22.285Q-11.426 22.015-11.534 21.954Q-11.642 21.892-11.953 21.892L-11.953 21.612L-10.893 21.537L-10.893 22.186Q-10.722 21.878-10.418 21.707Q-10.114 21.537-9.769 21.537Q-9.369 21.537-9.092 21.677Q-8.815 21.817-8.730 22.165Q-8.562 21.872-8.263 21.704Q-7.964 21.537-7.619 21.537Q-7.113 21.537-6.829 21.760Q-6.546 21.984-6.546 22.480L-6.546 24.134Q-6.546 24.271-6.397 24.307Q-6.248 24.343-6.023 24.343L-6.023 24.623L-7.653 24.623L-7.653 24.343Q-7.427 24.343-7.277 24.307Q-7.127 24.271-7.127 24.134L-7.127 22.494Q-7.127 22.159-7.246 21.959Q-7.366 21.759-7.680 21.759Q-7.950 21.759-8.185 21.895Q-8.419 22.032-8.557 22.266Q-8.696 22.500-8.696 22.774L-8.696 24.134Q-8.696 24.271-8.547 24.307Q-8.398 24.343-8.173 24.343L-8.173 24.623L-9.803 24.623L-9.803 24.343Q-9.574 24.343-9.425 24.309Q-9.277 24.274-9.277 24.134L-9.277 22.494Q-9.277 22.159-9.396 21.959Q-9.516 21.759-9.830 21.759Q-10.100 21.759-10.334 21.895Q-10.569 22.032-10.707 22.266Q-10.845 22.500-10.845 22.774L-10.845 24.134Q-10.845 24.271-10.695 24.307Q-10.545 24.343-10.319 24.343L-10.319 24.623M-3.753 24.623L-5.387 24.623L-5.387 24.343Q-5.158 24.343-5.009 24.309Q-4.861 24.274-4.861 24.134L-4.861 22.285Q-4.861 22.015-4.968 21.954Q-5.076 21.892-5.387 21.892L-5.387 21.612L-4.327 21.537L-4.327 22.186Q-4.156 21.878-3.852 21.707Q-3.548 21.537-3.203 21.537Q-2.803 21.537-2.526 21.677Q-2.249 21.817-2.164 22.165Q-1.996 21.872-1.697 21.704Q-1.398 21.537-1.053 21.537Q-0.547 21.537-0.263 21.760Q0.020 21.984 0.020 22.480L0.020 24.134Q0.020 24.271 0.169 24.307Q0.318 24.343 0.543 24.343L0.543 24.623L-1.087 24.623L-1.087 24.343Q-0.862 24.343-0.711 24.307Q-0.561 24.271-0.561 24.134L-0.561 22.494Q-0.561 22.159-0.680 21.959Q-0.800 21.759-1.114 21.759Q-1.384 21.759-1.619 21.895Q-1.853 22.032-1.991 22.266Q-2.130 22.500-2.130 22.774L-2.130 24.134Q-2.130 24.271-1.981 24.307Q-1.832 24.343-1.607 24.343L-1.607 24.623L-3.237 24.623L-3.237 24.343Q-3.008 24.343-2.859 24.309Q-2.711 24.274-2.711 24.134L-2.711 22.494Q-2.711 22.159-2.830 21.959Q-2.950 21.759-3.264 21.759Q-3.534 21.759-3.769 21.895Q-4.003 22.032-4.141 22.266Q-4.279 22.500-4.279 22.774L-4.279 24.134Q-4.279 24.271-4.129 24.307Q-3.979 24.343-3.753 24.343L-3.753 24.623M1.189 23.895Q1.189 23.563 1.413 23.336Q1.637 23.109 1.981 22.981Q2.324 22.852 2.697 22.800Q3.069 22.747 3.373 22.747L3.373 22.494Q3.373 22.289 3.266 22.109Q3.158 21.930 2.977 21.827Q2.796 21.725 2.587 21.725Q2.180 21.725 1.945 21.817Q2.034 21.854 2.080 21.938Q2.126 22.022 2.126 22.124Q2.126 22.220 2.080 22.299Q2.034 22.377 1.953 22.422Q1.873 22.466 1.784 22.466Q1.634 22.466 1.533 22.369Q1.432 22.271 1.432 22.124Q1.432 21.502 2.587 21.502Q2.799 21.502 3.049 21.566Q3.298 21.629 3.500 21.748Q3.701 21.868 3.828 22.053Q3.954 22.237 3.954 22.480L3.954 24.056Q3.954 24.172 4.016 24.268Q4.077 24.363 4.190 24.363Q4.300 24.363 4.365 24.269Q4.429 24.175 4.429 24.056L4.429 23.608L4.696 23.608L4.696 24.056Q4.696 24.326 4.469 24.491Q4.242 24.657 3.961 24.657Q3.753 24.657 3.616 24.503Q3.479 24.350 3.455 24.134Q3.308 24.401 3.026 24.546Q2.744 24.691 2.420 24.691Q2.143 24.691 1.859 24.616Q1.575 24.541 1.382 24.362Q1.189 24.182 1.189 23.895M1.804 23.895Q1.804 24.069 1.905 24.199Q2.006 24.329 2.162 24.399Q2.317 24.469 2.481 24.469Q2.700 24.469 2.909 24.372Q3.117 24.274 3.245 24.093Q3.373 23.912 3.373 23.686L3.373 22.958Q3.049 22.958 2.683 23.049Q2.317 23.140 2.061 23.352Q1.804 23.563 1.804 23.895\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(16.33 19.994)\">\u003Cpath d=\"M7.886 23.895Q7.886 23.563 8.109 23.336Q8.333 23.109 8.677 22.981Q9.020 22.852 9.393 22.800Q9.765 22.747 10.070 22.747L10.070 22.494Q10.070 22.289 9.962 22.109Q9.854 21.930 9.673 21.827Q9.492 21.725 9.284 21.725Q8.877 21.725 8.641 21.817Q8.730 21.854 8.776 21.938Q8.822 22.022 8.822 22.124Q8.822 22.220 8.776 22.299Q8.730 22.377 8.649 22.422Q8.569 22.466 8.480 22.466Q8.330 22.466 8.229 22.369Q8.128 22.271 8.128 22.124Q8.128 21.502 9.284 21.502Q9.495 21.502 9.745 21.566Q9.994 21.629 10.196 21.748Q10.398 21.868 10.524 22.053Q10.651 22.237 10.651 22.480L10.651 24.056Q10.651 24.172 10.712 24.268Q10.774 24.363 10.887 24.363Q10.996 24.363 11.061 24.269Q11.126 24.175 11.126 24.056L11.126 23.608L11.392 23.608L11.392 24.056Q11.392 24.326 11.165 24.491Q10.938 24.657 10.658 24.657Q10.449 24.657 10.312 24.503Q10.176 24.350 10.152 24.134Q10.005 24.401 9.723 24.546Q9.441 24.691 9.116 24.691Q8.839 24.691 8.555 24.616Q8.272 24.541 8.079 24.362Q7.886 24.182 7.886 23.895M8.501 23.895Q8.501 24.069 8.602 24.199Q8.702 24.329 8.858 24.399Q9.013 24.469 9.178 24.469Q9.396 24.469 9.605 24.372Q9.813 24.274 9.941 24.093Q10.070 23.912 10.070 23.686L10.070 22.958Q9.745 22.958 9.379 23.049Q9.013 23.140 8.757 23.352Q8.501 23.563 8.501 23.895M13.453 25.980L11.823 25.980L11.823 25.700Q12.052 25.700 12.201 25.665Q12.349 25.631 12.349 25.491L12.349 22.145Q12.349 21.974 12.213 21.933Q12.076 21.892 11.823 21.892L11.823 21.612L12.903 21.537L12.903 21.943Q13.125 21.742 13.412 21.639Q13.700 21.537 14.007 21.537Q14.434 21.537 14.798 21.750Q15.162 21.964 15.376 22.328Q15.590 22.692 15.590 23.112Q15.590 23.557 15.350 23.921Q15.111 24.285 14.718 24.488Q14.325 24.691 13.881 24.691Q13.614 24.691 13.366 24.591Q13.118 24.490 12.930 24.309L12.930 25.491Q12.930 25.628 13.079 25.664Q13.228 25.700 13.453 25.700L13.453 25.980M12.930 22.292L12.930 23.902Q13.064 24.155 13.306 24.312Q13.549 24.469 13.826 24.469Q14.154 24.469 14.407 24.268Q14.660 24.066 14.793 23.748Q14.927 23.430 14.927 23.112Q14.927 22.883 14.862 22.654Q14.797 22.425 14.669 22.227Q14.540 22.029 14.346 21.909Q14.151 21.790 13.918 21.790Q13.624 21.790 13.356 21.919Q13.088 22.049 12.930 22.292M17.869 25.980L16.239 25.980L16.239 25.700Q16.468 25.700 16.617 25.665Q16.765 25.631 16.765 25.491L16.765 22.145Q16.765 21.974 16.629 21.933Q16.492 21.892 16.239 21.892L16.239 21.612L17.319 21.537L17.319 21.943Q17.541 21.742 17.828 21.639Q18.116 21.537 18.423 21.537Q18.850 21.537 19.214 21.750Q19.578 21.964 19.792 22.328Q20.006 22.692 20.006 23.112Q20.006 23.557 19.766 23.921Q19.527 24.285 19.134 24.488Q18.741 24.691 18.297 24.691Q18.030 24.691 17.782 24.591Q17.534 24.490 17.346 24.309L17.346 25.491Q17.346 25.628 17.495 25.664Q17.644 25.700 17.869 25.700L17.869 25.980M17.346 22.292L17.346 23.902Q17.480 24.155 17.722 24.312Q17.965 24.469 18.242 24.469Q18.570 24.469 18.823 24.268Q19.076 24.066 19.209 23.748Q19.343 23.430 19.343 23.112Q19.343 22.883 19.278 22.654Q19.213 22.425 19.085 22.227Q18.956 22.029 18.762 21.909Q18.567 21.790 18.334 21.790Q18.040 21.790 17.772 21.919Q17.504 22.049 17.346 22.292M22.391 24.623L20.655 24.623L20.655 24.343Q20.884 24.343 21.033 24.309Q21.181 24.274 21.181 24.134L21.181 22.285Q21.181 22.015 21.074 21.954Q20.966 21.892 20.655 21.892L20.655 21.612L21.684 21.537L21.684 22.244Q21.814 21.936 22.056 21.737Q22.299 21.537 22.617 21.537Q22.836 21.537 23.007 21.661Q23.178 21.786 23.178 21.998Q23.178 22.135 23.078 22.234Q22.979 22.333 22.846 22.333Q22.709 22.333 22.610 22.234Q22.511 22.135 22.511 21.998Q22.511 21.858 22.610 21.759Q22.320 21.759 22.120 21.955Q21.920 22.152 21.827 22.446Q21.735 22.740 21.735 23.020L21.735 24.134Q21.735 24.343 22.391 24.343L22.391 24.623M23.721 23.140Q23.721 22.798 23.856 22.499Q23.991 22.200 24.230 21.976Q24.470 21.752 24.787 21.627Q25.105 21.502 25.437 21.502Q25.881 21.502 26.281 21.718Q26.681 21.933 26.915 22.311Q27.149 22.688 27.149 23.140Q27.149 23.481 27.007 23.765Q26.866 24.049 26.621 24.256Q26.377 24.462 26.067 24.577Q25.758 24.691 25.437 24.691Q25.006 24.691 24.605 24.490Q24.203 24.288 23.962 23.936Q23.721 23.584 23.721 23.140M25.437 24.442Q26.038 24.442 26.262 24.064Q26.486 23.686 26.486 23.054Q26.486 22.442 26.252 22.083Q26.018 21.725 25.437 21.725Q24.384 21.725 24.384 23.054Q24.384 23.686 24.610 24.064Q24.835 24.442 25.437 24.442\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(16.33 19.994)\">\u003Cpath d=\"M28.725 24.623L27.402 24.623L27.402 24.343Q27.963 24.343 28.342 23.943L29.057 23.146L28.144 22.097Q28.007 21.950 27.859 21.918Q27.710 21.885 27.443 21.885L27.443 21.605L28.944 21.605L28.944 21.885Q28.752 21.885 28.752 22.019Q28.752 22.049 28.783 22.097L29.378 22.781L29.819 22.285Q29.932 22.155 29.932 22.039Q29.932 21.977 29.894 21.931Q29.856 21.885 29.798 21.885L29.798 21.605L31.114 21.605L31.114 21.885Q30.554 21.885 30.174 22.285L29.552 22.986L30.547 24.134Q30.646 24.233 30.747 24.278Q30.848 24.322 30.959 24.332Q31.070 24.343 31.248 24.343L31.248 24.623L29.754 24.623L29.754 24.343Q29.819 24.343 29.879 24.309Q29.938 24.274 29.938 24.209Q29.938 24.162 29.908 24.134L29.231 23.348L28.698 23.943Q28.585 24.073 28.585 24.189Q28.585 24.254 28.626 24.298Q28.667 24.343 28.725 24.343\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(16.33 19.994)\">\u003Cpath d=\"M37.450 24.623L34.920 24.623L34.920 24.343Q35.888 24.343 35.888 24.134L35.888 20.515Q35.495 20.703 34.873 20.703L34.873 20.422Q35.290 20.422 35.654 20.321Q36.018 20.221 36.274 19.975L36.400 19.975Q36.465 19.992 36.482 20.060L36.482 24.134Q36.482 24.343 37.450 24.343\" 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)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M108.597 17.692C73.402-13.928 22.395-13.872-10.8 16.084\"\u002F>\u003Cpath stroke=\"none\" d=\"m-12.73 17.826 4.482-1.243-2.552-.5-.235-2.588\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">State-dependent termination \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0556em;\">γ\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">s\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> unifies the settings. Top: the classic episodic view, with a special terminal state and a reset. Bottom: the same experience as one continuing stream where the goal state has \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0556em;\">γ\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\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> and jumps back to the start; a bootstrapping state has \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6776em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0556em;\">γ\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\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>. Setting \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6582em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0556em;\">γ\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\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> off the goal recovers the episodic case; a constant \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0556em;\">γ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> recovers ordinary discounting.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:421.232px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 315.924 173.079\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" d=\"M-23.808-20.77h204.86\" style=\"stroke-dasharray:3.0,3.0\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(209.815 -55.155)\">\u003Cpath d=\"M-22.687 36.135L-22.954 36.135L-22.954 32.027Q-22.954 31.757-23.061 31.695Q-23.169 31.634-23.480 31.634L-23.480 31.353L-22.400 31.278L-22.400 33.448Q-22.191 33.257-21.906 33.153Q-21.620 33.049-21.323 33.049Q-21.005 33.049-20.708 33.170Q-20.411 33.291-20.188 33.507Q-19.966 33.722-19.840 34.007Q-19.713 34.293-19.713 34.624Q-19.713 35.069-19.953 35.433Q-20.192 35.797-20.585 36Q-20.978 36.203-21.422 36.203Q-21.617 36.203-21.807 36.147Q-21.996 36.091-22.157 35.986Q-22.318 35.882-22.458 35.721L-22.687 36.135M-22.372 33.790L-22.372 35.407Q-22.236 35.667-21.995 35.824Q-21.754 35.981-21.477 35.981Q-21.183 35.981-20.971 35.874Q-20.759 35.766-20.626 35.574Q-20.493 35.383-20.434 35.144Q-20.376 34.905-20.376 34.624Q-20.376 34.265-20.470 33.961Q-20.564 33.657-20.792 33.464Q-21.019 33.271-21.385 33.271Q-21.685 33.271-21.952 33.407Q-22.219 33.544-22.372 33.790M-19.019 35.407Q-19.019 35.075-18.796 34.848Q-18.572 34.621-18.228 34.493Q-17.885 34.364-17.512 34.312Q-17.140 34.259-16.835 34.259L-16.835 34.006Q-16.835 33.801-16.943 33.621Q-17.051 33.442-17.232 33.339Q-17.413 33.237-17.621 33.237Q-18.028 33.237-18.264 33.329Q-18.175 33.366-18.129 33.450Q-18.083 33.534-18.083 33.636Q-18.083 33.732-18.129 33.811Q-18.175 33.889-18.256 33.934Q-18.336 33.978-18.425 33.978Q-18.575 33.978-18.676 33.881Q-18.777 33.783-18.777 33.636Q-18.777 33.014-17.621 33.014Q-17.410 33.014-17.160 33.078Q-16.911 33.141-16.709 33.260Q-16.507 33.380-16.381 33.565Q-16.254 33.749-16.254 33.992L-16.254 35.568Q-16.254 35.684-16.193 35.780Q-16.131 35.875-16.018 35.875Q-15.909 35.875-15.844 35.781Q-15.779 35.687-15.779 35.568L-15.779 35.120L-15.513 35.120L-15.513 35.568Q-15.513 35.838-15.740 36.003Q-15.967 36.169-16.247 36.169Q-16.456 36.169-16.593 36.015Q-16.729 35.862-16.753 35.646Q-16.900 35.913-17.182 36.058Q-17.464 36.203-17.789 36.203Q-18.066 36.203-18.350 36.128Q-18.633 36.053-18.826 35.874Q-19.019 35.694-19.019 35.407M-18.404 35.407Q-18.404 35.581-18.303 35.711Q-18.203 35.841-18.047 35.911Q-17.891 35.981-17.727 35.981Q-17.509 35.981-17.300 35.884Q-17.092 35.786-16.964 35.605Q-16.835 35.424-16.835 35.198L-16.835 34.470Q-17.160 34.470-17.526 34.561Q-17.891 34.652-18.148 34.864Q-18.404 35.075-18.404 35.407M-15.096 36.128L-15.096 35.065Q-15.096 35.041-15.068 35.014Q-15.041 34.987-15.017 34.987L-14.908 34.987Q-14.843 34.987-14.829 35.045Q-14.733 35.479-14.487 35.730Q-14.241 35.981-13.828 35.981Q-13.486 35.981-13.233 35.848Q-12.980 35.715-12.980 35.407Q-12.980 35.250-13.074 35.135Q-13.168 35.021-13.306 34.952Q-13.445 34.884-13.612 34.846L-14.193 34.747Q-14.549 34.679-14.822 34.458Q-15.096 34.238-15.096 33.896Q-15.096 33.647-14.985 33.472Q-14.873 33.298-14.687 33.199Q-14.501 33.100-14.286 33.057Q-14.070 33.014-13.828 33.014Q-13.414 33.014-13.134 33.196L-12.918 33.021Q-12.908 33.018-12.901 33.016Q-12.894 33.014-12.884 33.014L-12.833 33.014Q-12.806 33.014-12.782 33.038Q-12.758 33.062-12.758 33.090L-12.758 33.937Q-12.758 33.958-12.782 33.985Q-12.806 34.012-12.833 34.012L-12.946 34.012Q-12.973 34.012-12.999 33.987Q-13.024 33.961-13.024 33.937Q-13.024 33.701-13.130 33.537Q-13.236 33.373-13.419 33.291Q-13.602 33.209-13.834 33.209Q-14.162 33.209-14.419 33.312Q-14.675 33.414-14.675 33.691Q-14.675 33.886-14.492 33.995Q-14.309 34.105-14.080 34.146L-13.506 34.252Q-13.260 34.300-13.047 34.428Q-12.833 34.556-12.696 34.759Q-12.559 34.963-12.559 35.212Q-12.559 35.725-12.925 35.964Q-13.291 36.203-13.828 36.203Q-14.323 36.203-14.655 35.909L-14.921 36.183Q-14.942 36.203-14.969 36.203L-15.017 36.203Q-15.041 36.203-15.068 36.176Q-15.096 36.149-15.096 36.128M-11.972 34.600Q-11.972 34.279-11.847 33.990Q-11.722 33.701-11.496 33.478Q-11.271 33.254-10.975 33.134Q-10.680 33.014-10.362 33.014Q-10.034 33.014-9.772 33.114Q-9.511 33.213-9.335 33.395Q-9.159 33.578-9.065 33.836Q-8.971 34.094-8.971 34.426Q-8.971 34.518-9.053 34.539L-11.308 34.539L-11.308 34.600Q-11.308 35.188-11.025 35.571Q-10.741 35.954-10.174 35.954Q-9.852 35.954-9.584 35.761Q-9.316 35.568-9.227 35.253Q-9.220 35.212-9.145 35.198L-9.053 35.198Q-8.971 35.222-8.971 35.294Q-8.971 35.301-8.977 35.328Q-9.090 35.725-9.461 35.964Q-9.832 36.203-10.256 36.203Q-10.693 36.203-11.093 35.995Q-11.493 35.786-11.732 35.419Q-11.972 35.052-11.972 34.600M-11.302 34.330L-9.487 34.330Q-9.487 34.053-9.584 33.801Q-9.682 33.548-9.880 33.392Q-10.078 33.237-10.362 33.237Q-10.639 33.237-10.852 33.395Q-11.066 33.554-11.184 33.809Q-11.302 34.064-11.302 34.330M-6.715 36.135L-8.318 36.135L-8.318 35.855Q-8.092 35.855-7.943 35.821Q-7.795 35.786-7.795 35.646L-7.795 32.027Q-7.795 31.757-7.902 31.695Q-8.010 31.634-8.318 31.634L-8.318 31.353L-7.241 31.278L-7.241 35.646Q-7.241 35.783-7.091 35.819Q-6.940 35.855-6.715 35.855L-6.715 36.135M-4.503 36.135L-6.055 36.135L-6.055 35.855Q-5.829 35.855-5.681 35.821Q-5.532 35.786-5.532 35.646L-5.532 33.797Q-5.532 33.609-5.580 33.525Q-5.628 33.442-5.725 33.423Q-5.823 33.404-6.035 33.404L-6.035 33.124L-4.978 33.049L-4.978 35.646Q-4.978 35.786-4.847 35.821Q-4.715 35.855-4.503 35.855L-4.503 36.135M-5.775 31.828Q-5.775 31.657-5.652 31.538Q-5.529 31.418-5.358 31.418Q-5.190 31.418-5.067 31.538Q-4.944 31.657-4.944 31.828Q-4.944 32.003-5.067 32.126Q-5.190 32.249-5.358 32.249Q-5.529 32.249-5.652 32.126Q-5.775 32.003-5.775 31.828M-2.176 36.135L-3.809 36.135L-3.809 35.855Q-3.580 35.855-3.432 35.821Q-3.283 35.786-3.283 35.646L-3.283 33.797Q-3.283 33.527-3.391 33.466Q-3.498 33.404-3.809 33.404L-3.809 33.124L-2.750 33.049L-2.750 33.698Q-2.579 33.390-2.275 33.219Q-1.971 33.049-1.625 33.049Q-1.120 33.049-0.836 33.272Q-0.552 33.496-0.552 33.992L-0.552 35.646Q-0.552 35.783-0.403 35.819Q-0.255 35.855-0.029 35.855L-0.029 36.135L-1.660 36.135L-1.660 35.855Q-1.431 35.855-1.282 35.821Q-1.133 35.786-1.133 35.646L-1.133 34.006Q-1.133 33.671-1.253 33.471Q-1.372 33.271-1.687 33.271Q-1.957 33.271-2.191 33.407Q-2.425 33.544-2.564 33.778Q-2.702 34.012-2.702 34.286L-2.702 35.646Q-2.702 35.783-2.552 35.819Q-2.401 35.855-2.176 35.855L-2.176 36.135M0.518 34.600Q0.518 34.279 0.642 33.990Q0.767 33.701 0.993 33.478Q1.218 33.254 1.514 33.134Q1.810 33.014 2.128 33.014Q2.456 33.014 2.717 33.114Q2.979 33.213 3.155 33.395Q3.331 33.578 3.425 33.836Q3.519 34.094 3.519 34.426Q3.519 34.518 3.437 34.539L1.181 34.539L1.181 34.600Q1.181 35.188 1.464 35.571Q1.748 35.954 2.316 35.954Q2.637 35.954 2.905 35.761Q3.173 35.568 3.262 35.253Q3.269 35.212 3.344 35.198L3.437 35.198Q3.519 35.222 3.519 35.294Q3.519 35.301 3.512 35.328Q3.399 35.725 3.028 35.964Q2.657 36.203 2.234 36.203Q1.796 36.203 1.396 35.995Q0.996 35.786 0.757 35.419Q0.518 35.052 0.518 34.600M1.188 34.330L3.003 34.330Q3.003 34.053 2.905 33.801Q2.808 33.548 2.609 33.392Q2.411 33.237 2.128 33.237Q1.851 33.237 1.637 33.395Q1.423 33.554 1.306 33.809Q1.188 34.064 1.188 34.330\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(209.815 -55.155)\">\u003Cpath d=\"M8.417 36.108L7.289 33.609Q7.217 33.462 7.087 33.430Q6.957 33.397 6.728 33.397L6.728 33.117L8.242 33.117L8.242 33.397Q7.890 33.397 7.890 33.544Q7.890 33.589 7.901 33.609L8.765 35.527L9.545 33.797Q9.579 33.729 9.579 33.650Q9.579 33.537 9.495 33.467Q9.411 33.397 9.292 33.397L9.292 33.117L10.488 33.117L10.488 33.397Q10.269 33.397 10.098 33.500Q9.928 33.602 9.839 33.797L8.803 36.108Q8.755 36.203 8.649 36.203L8.571 36.203Q8.465 36.203 8.417 36.108M13.158 37.885Q12.607 37.485 12.236 36.930Q11.866 36.374 11.684 35.728Q11.503 35.082 11.503 34.385Q11.503 33.872 11.604 33.377Q11.705 32.881 11.910 32.430Q12.115 31.979 12.428 31.587Q12.741 31.196 13.158 30.892Q13.168 30.888 13.175 30.887Q13.181 30.885 13.192 30.885L13.260 30.885Q13.294 30.885 13.316 30.909Q13.339 30.933 13.339 30.970Q13.339 31.015 13.311 31.032Q12.963 31.333 12.710 31.717Q12.457 32.102 12.305 32.543Q12.153 32.984 12.081 33.440Q12.009 33.896 12.009 34.385Q12.009 35.386 12.318 36.273Q12.628 37.160 13.311 37.745Q13.339 37.762 13.339 37.806Q13.339 37.844 13.316 37.868Q13.294 37.892 13.260 37.892L13.192 37.892Q13.185 37.888 13.176 37.887Q13.168 37.885 13.158 37.885M14.323 36.197L14.323 34.624Q14.323 34.597 14.349 34.571Q14.374 34.546 14.402 34.546L14.514 34.546Q14.542 34.546 14.566 34.573Q14.590 34.600 14.590 34.624Q14.590 34.969 14.721 35.233Q14.853 35.496 15.082 35.665Q15.311 35.834 15.613 35.915Q15.916 35.995 16.258 35.995Q16.524 35.995 16.760 35.867Q16.996 35.739 17.141 35.516Q17.286 35.294 17.286 35.028Q17.286 34.805 17.180 34.609Q17.075 34.412 16.893 34.277Q16.712 34.142 16.487 34.091L15.458 33.859Q15.147 33.787 14.887 33.601Q14.627 33.414 14.475 33.143Q14.323 32.871 14.323 32.556Q14.323 32.170 14.537 31.863Q14.750 31.555 15.097 31.384Q15.444 31.213 15.824 31.213Q16.053 31.213 16.282 31.266Q16.511 31.319 16.709 31.427Q16.907 31.534 17.061 31.698L17.355 31.258Q17.379 31.213 17.420 31.213L17.468 31.213Q17.498 31.213 17.521 31.239Q17.543 31.264 17.543 31.292L17.543 32.867Q17.543 32.888 17.519 32.915Q17.495 32.943 17.468 32.943L17.355 32.943Q17.293 32.943 17.280 32.867Q17.239 32.454 17.057 32.134Q16.876 31.815 16.565 31.640Q16.254 31.466 15.824 31.466Q15.574 31.466 15.335 31.577Q15.096 31.688 14.945 31.886Q14.795 32.085 14.795 32.348Q14.795 32.560 14.902 32.741Q15.010 32.922 15.186 33.042Q15.362 33.161 15.571 33.202L16.599 33.431Q16.917 33.503 17.184 33.708Q17.450 33.913 17.603 34.207Q17.755 34.501 17.755 34.833Q17.755 35.226 17.550 35.561Q17.345 35.896 16.999 36.085Q16.654 36.275 16.258 36.275Q15.837 36.275 15.458 36.162Q15.078 36.050 14.808 35.800L14.514 36.234Q14.487 36.275 14.450 36.275L14.402 36.275Q14.374 36.275 14.349 36.250Q14.323 36.224 14.323 36.197M19.091 35.294L19.091 33.397L18.452 33.397L18.452 33.175Q18.770 33.175 18.987 32.965Q19.204 32.755 19.305 32.445Q19.406 32.136 19.406 31.828L19.672 31.828L19.672 33.117L20.749 33.117L20.749 33.397L19.672 33.397L19.672 35.281Q19.672 35.557 19.776 35.756Q19.881 35.954 20.140 35.954Q20.298 35.954 20.404 35.850Q20.510 35.745 20.559 35.592Q20.609 35.438 20.609 35.281L20.609 34.867L20.875 34.867L20.875 35.294Q20.875 35.520 20.776 35.730Q20.677 35.940 20.492 36.072Q20.308 36.203 20.079 36.203Q19.641 36.203 19.366 35.966Q19.091 35.728 19.091 35.294M22.007 37.892L21.938 37.892Q21.904 37.892 21.882 37.866Q21.860 37.841 21.860 37.806Q21.860 37.762 21.890 37.745Q22.246 37.441 22.495 37.051Q22.745 36.661 22.897 36.229Q23.049 35.797 23.119 35.328Q23.189 34.860 23.189 34.385Q23.189 33.906 23.119 33.440Q23.049 32.973 22.895 32.538Q22.742 32.102 22.490 31.714Q22.239 31.326 21.890 31.032Q21.860 31.015 21.860 30.970Q21.860 30.936 21.882 30.911Q21.904 30.885 21.938 30.885L22.007 30.885Q22.017 30.885 22.025 30.887Q22.034 30.888 22.044 30.892Q22.588 31.292 22.960 31.845Q23.333 32.399 23.514 33.045Q23.695 33.691 23.695 34.385Q23.695 35.086 23.514 35.733Q23.333 36.381 22.959 36.935Q22.584 37.489 22.044 37.885Q22.034 37.885 22.025 37.887Q22.017 37.888 22.007 37.892\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-31.35 -54.514)\">\u003Cpath d=\"M-21.812 36.135L-23.446 36.135L-23.446 35.855Q-23.217 35.855-23.068 35.821Q-22.919 35.786-22.919 35.646L-22.919 33.797Q-22.919 33.527-23.027 33.466Q-23.135 33.404-23.446 33.404L-23.446 33.124L-22.386 33.049L-22.386 33.698Q-22.215 33.390-21.911 33.219Q-21.607 33.049-21.262 33.049Q-20.756 33.049-20.472 33.272Q-20.188 33.496-20.188 33.992L-20.188 35.646Q-20.188 35.783-20.040 35.819Q-19.891 35.855-19.665 35.855L-19.665 36.135L-21.296 36.135L-21.296 35.855Q-21.067 35.855-20.918 35.821Q-20.769 35.786-20.769 35.646L-20.769 34.006Q-20.769 33.671-20.889 33.471Q-21.009 33.271-21.323 33.271Q-21.593 33.271-21.827 33.407Q-22.061 33.544-22.200 33.778Q-22.338 34.012-22.338 34.286L-22.338 35.646Q-22.338 35.783-22.188 35.819Q-22.037 35.855-21.812 35.855L-21.812 36.135M-19.119 34.652Q-19.119 34.310-18.984 34.011Q-18.849 33.712-18.609 33.488Q-18.370 33.264-18.052 33.139Q-17.734 33.014-17.403 33.014Q-16.958 33.014-16.558 33.230Q-16.159 33.445-15.924 33.823Q-15.690 34.200-15.690 34.652Q-15.690 34.993-15.832 35.277Q-15.974 35.561-16.218 35.768Q-16.463 35.974-16.772 36.089Q-17.081 36.203-17.403 36.203Q-17.833 36.203-18.235 36.002Q-18.637 35.800-18.878 35.448Q-19.119 35.096-19.119 34.652M-17.403 35.954Q-16.801 35.954-16.577 35.576Q-16.353 35.198-16.353 34.566Q-16.353 33.954-16.588 33.595Q-16.822 33.237-17.403 33.237Q-18.455 33.237-18.455 34.566Q-18.455 35.198-18.230 35.576Q-18.004 35.954-17.403 35.954\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-31.35 -54.514)\">\u003Cpath d=\"M-12.222 33.742Q-12.222 33.216-12.005 32.748Q-11.788 32.280-11.405 31.934Q-11.023 31.589-10.539 31.401Q-10.055 31.213-9.525 31.213Q-9.122 31.213-8.758 31.370Q-8.394 31.528-8.110 31.822L-7.687 31.240Q-7.652 31.213-7.628 31.213L-7.581 31.213Q-7.550 31.213-7.526 31.237Q-7.502 31.261-7.502 31.292L-7.502 33.155Q-7.502 33.178-7.528 33.204Q-7.553 33.230-7.581 33.230L-7.707 33.230Q-7.769 33.230-7.782 33.155Q-7.813 32.840-7.948 32.536Q-8.083 32.232-8.298 31.998Q-8.514 31.763-8.803 31.628Q-9.091 31.493-9.419 31.493Q-10.062 31.493-10.520 31.787Q-10.978 32.081-11.211 32.594Q-11.443 33.107-11.443 33.742Q-11.443 34.214-11.313 34.623Q-11.183 35.031-10.923 35.344Q-10.664 35.656-10.284 35.826Q-9.905 35.995-9.413 35.995Q-9.085 35.995-8.791 35.879Q-8.497 35.762-8.263 35.547Q-8.028 35.332-7.899 35.043Q-7.769 34.754-7.769 34.426Q-7.769 34.399-7.741 34.375Q-7.714 34.351-7.693 34.351L-7.581 34.351Q-7.543 34.351-7.523 34.376Q-7.502 34.402-7.502 34.440Q-7.502 34.836-7.668 35.173Q-7.834 35.510-8.121 35.757Q-8.408 36.005-8.777 36.140Q-9.146 36.275-9.525 36.275Q-10.045 36.275-10.537 36.085Q-11.029 35.896-11.409 35.552Q-11.788 35.209-12.005 34.740Q-12.222 34.272-12.222 33.742M-4.255 36.176L-6.097 31.845Q-6.166 31.702-6.347 31.668Q-6.528 31.634-6.791 31.634L-6.791 31.353L-4.822 31.353L-4.822 31.634Q-5.376 31.634-5.376 31.808Q-5.373 31.818-5.371 31.828Q-5.369 31.839-5.369 31.845L-3.848 35.421L-2.416 32.033Q-2.392 31.996-2.392 31.941Q-2.392 31.781-2.543 31.707Q-2.693 31.634-2.878 31.634L-2.878 31.353L-1.329 31.353L-1.329 31.634Q-1.589 31.634-1.799 31.728Q-2.009 31.822-2.105 32.033L-3.862 36.176Q-3.913 36.275-4.009 36.275L-4.108 36.275Q-4.207 36.275-4.255 36.176M-0.393 35.715Q-0.393 35.547-0.270 35.424Q-0.147 35.301 0.028 35.301Q0.195 35.301 0.318 35.424Q0.441 35.547 0.441 35.715Q0.441 35.889 0.318 36.012Q0.195 36.135 0.028 36.135Q-0.147 36.135-0.270 36.012Q-0.393 35.889-0.393 35.715M-0.393 33.531Q-0.393 33.363-0.270 33.240Q-0.147 33.117 0.028 33.117Q0.195 33.117 0.318 33.240Q0.441 33.363 0.441 33.531Q0.441 33.705 0.318 33.828Q0.195 33.951 0.028 33.951Q-0.147 33.951-0.270 33.828Q-0.393 33.705-0.393 33.531\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"M-3.892-20.77v-34.144\" style=\"stroke-width:.8\"\u002F>\u003Cpath fill=\"var(--tk-warn)\" stroke=\"none\" d=\"M-2.292-54.914a1.6 1.6 0 1 0-3.2 0 1.6 1.6 0 0 0 3.2 0m-1.6 0\"\u002F>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"M24.561-20.77v31.298\" style=\"stroke-width:.8\"\u002F>\u003Cpath fill=\"var(--tk-warn)\" stroke=\"none\" d=\"M26.161 10.528a1.6 1.6 0 1 0-3.2 0 1.6 1.6 0 0 0 3.2 0m-1.6 0\"\u002F>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"M53.014-20.77v-25.608\" style=\"stroke-width:.8\"\u002F>\u003Cpath fill=\"var(--tk-warn)\" stroke=\"none\" d=\"M54.614-46.378a1.6 1.6 0 1 0-3.2 0 1.6 1.6 0 0 0 3.2 0m-1.6 0\"\u002F>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"M81.467-20.77V7.682\" style=\"stroke-width:.8\"\u002F>\u003Cpath fill=\"var(--tk-warn)\" stroke=\"none\" d=\"M83.067 7.682a1.6 1.6 0 1 0-3.2 0 1.6 1.6 0 0 0 3.2 0m-1.6 0\"\u002F>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"M109.92-20.77v-36.99\" style=\"stroke-width:.8\"\u002F>\u003Cpath fill=\"var(--tk-warn)\" stroke=\"none\" d=\"M111.52-57.76a1.6 1.6 0 1 0-3.2 0 1.6 1.6 0 0 0 3.2 0m-1.6 0\"\u002F>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"M138.372-20.77V4.837\" style=\"stroke-width:.8\"\u002F>\u003Cpath fill=\"var(--tk-warn)\" stroke=\"none\" d=\"M139.972 4.837a1.6 1.6 0 1 0-3.2 0 1.6 1.6 0 0 0 3.2 0m-1.6 0\"\u002F>\u003Cpath fill=\"none\" d=\"M-23.808 70.278h204.86\" style=\"stroke-dasharray:3.0,3.0\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(209.815 35.893)\">\u003Cpath d=\"M-22.687 36.135L-22.954 36.135L-22.954 32.027Q-22.954 31.757-23.061 31.695Q-23.169 31.634-23.480 31.634L-23.480 31.353L-22.400 31.278L-22.400 33.448Q-22.191 33.257-21.906 33.153Q-21.620 33.049-21.323 33.049Q-21.005 33.049-20.708 33.170Q-20.411 33.291-20.188 33.507Q-19.966 33.722-19.840 34.007Q-19.713 34.293-19.713 34.624Q-19.713 35.069-19.953 35.433Q-20.192 35.797-20.585 36Q-20.978 36.203-21.422 36.203Q-21.617 36.203-21.807 36.147Q-21.996 36.091-22.157 35.986Q-22.318 35.882-22.458 35.721L-22.687 36.135M-22.372 33.790L-22.372 35.407Q-22.236 35.667-21.995 35.824Q-21.754 35.981-21.477 35.981Q-21.183 35.981-20.971 35.874Q-20.759 35.766-20.626 35.574Q-20.493 35.383-20.434 35.144Q-20.376 34.905-20.376 34.624Q-20.376 34.265-20.470 33.961Q-20.564 33.657-20.792 33.464Q-21.019 33.271-21.385 33.271Q-21.685 33.271-21.952 33.407Q-22.219 33.544-22.372 33.790M-19.019 35.407Q-19.019 35.075-18.796 34.848Q-18.572 34.621-18.228 34.493Q-17.885 34.364-17.512 34.312Q-17.140 34.259-16.835 34.259L-16.835 34.006Q-16.835 33.801-16.943 33.621Q-17.051 33.442-17.232 33.339Q-17.413 33.237-17.621 33.237Q-18.028 33.237-18.264 33.329Q-18.175 33.366-18.129 33.450Q-18.083 33.534-18.083 33.636Q-18.083 33.732-18.129 33.811Q-18.175 33.889-18.256 33.934Q-18.336 33.978-18.425 33.978Q-18.575 33.978-18.676 33.881Q-18.777 33.783-18.777 33.636Q-18.777 33.014-17.621 33.014Q-17.410 33.014-17.160 33.078Q-16.911 33.141-16.709 33.260Q-16.507 33.380-16.381 33.565Q-16.254 33.749-16.254 33.992L-16.254 35.568Q-16.254 35.684-16.193 35.780Q-16.131 35.875-16.018 35.875Q-15.909 35.875-15.844 35.781Q-15.779 35.687-15.779 35.568L-15.779 35.120L-15.513 35.120L-15.513 35.568Q-15.513 35.838-15.740 36.003Q-15.967 36.169-16.247 36.169Q-16.456 36.169-16.593 36.015Q-16.729 35.862-16.753 35.646Q-16.900 35.913-17.182 36.058Q-17.464 36.203-17.789 36.203Q-18.066 36.203-18.350 36.128Q-18.633 36.053-18.826 35.874Q-19.019 35.694-19.019 35.407M-18.404 35.407Q-18.404 35.581-18.303 35.711Q-18.203 35.841-18.047 35.911Q-17.891 35.981-17.727 35.981Q-17.509 35.981-17.300 35.884Q-17.092 35.786-16.964 35.605Q-16.835 35.424-16.835 35.198L-16.835 34.470Q-17.160 34.470-17.526 34.561Q-17.891 34.652-18.148 34.864Q-18.404 35.075-18.404 35.407M-15.096 36.128L-15.096 35.065Q-15.096 35.041-15.068 35.014Q-15.041 34.987-15.017 34.987L-14.908 34.987Q-14.843 34.987-14.829 35.045Q-14.733 35.479-14.487 35.730Q-14.241 35.981-13.828 35.981Q-13.486 35.981-13.233 35.848Q-12.980 35.715-12.980 35.407Q-12.980 35.250-13.074 35.135Q-13.168 35.021-13.306 34.952Q-13.445 34.884-13.612 34.846L-14.193 34.747Q-14.549 34.679-14.822 34.458Q-15.096 34.238-15.096 33.896Q-15.096 33.647-14.985 33.472Q-14.873 33.298-14.687 33.199Q-14.501 33.100-14.286 33.057Q-14.070 33.014-13.828 33.014Q-13.414 33.014-13.134 33.196L-12.918 33.021Q-12.908 33.018-12.901 33.016Q-12.894 33.014-12.884 33.014L-12.833 33.014Q-12.806 33.014-12.782 33.038Q-12.758 33.062-12.758 33.090L-12.758 33.937Q-12.758 33.958-12.782 33.985Q-12.806 34.012-12.833 34.012L-12.946 34.012Q-12.973 34.012-12.999 33.987Q-13.024 33.961-13.024 33.937Q-13.024 33.701-13.130 33.537Q-13.236 33.373-13.419 33.291Q-13.602 33.209-13.834 33.209Q-14.162 33.209-14.419 33.312Q-14.675 33.414-14.675 33.691Q-14.675 33.886-14.492 33.995Q-14.309 34.105-14.080 34.146L-13.506 34.252Q-13.260 34.300-13.047 34.428Q-12.833 34.556-12.696 34.759Q-12.559 34.963-12.559 35.212Q-12.559 35.725-12.925 35.964Q-13.291 36.203-13.828 36.203Q-14.323 36.203-14.655 35.909L-14.921 36.183Q-14.942 36.203-14.969 36.203L-15.017 36.203Q-15.041 36.203-15.068 36.176Q-15.096 36.149-15.096 36.128M-11.972 34.600Q-11.972 34.279-11.847 33.990Q-11.722 33.701-11.496 33.478Q-11.271 33.254-10.975 33.134Q-10.680 33.014-10.362 33.014Q-10.034 33.014-9.772 33.114Q-9.511 33.213-9.335 33.395Q-9.159 33.578-9.065 33.836Q-8.971 34.094-8.971 34.426Q-8.971 34.518-9.053 34.539L-11.308 34.539L-11.308 34.600Q-11.308 35.188-11.025 35.571Q-10.741 35.954-10.174 35.954Q-9.852 35.954-9.584 35.761Q-9.316 35.568-9.227 35.253Q-9.220 35.212-9.145 35.198L-9.053 35.198Q-8.971 35.222-8.971 35.294Q-8.971 35.301-8.977 35.328Q-9.090 35.725-9.461 35.964Q-9.832 36.203-10.256 36.203Q-10.693 36.203-11.093 35.995Q-11.493 35.786-11.732 35.419Q-11.972 35.052-11.972 34.600M-11.302 34.330L-9.487 34.330Q-9.487 34.053-9.584 33.801Q-9.682 33.548-9.880 33.392Q-10.078 33.237-10.362 33.237Q-10.639 33.237-10.852 33.395Q-11.066 33.554-11.184 33.809Q-11.302 34.064-11.302 34.330M-6.715 36.135L-8.318 36.135L-8.318 35.855Q-8.092 35.855-7.943 35.821Q-7.795 35.786-7.795 35.646L-7.795 32.027Q-7.795 31.757-7.902 31.695Q-8.010 31.634-8.318 31.634L-8.318 31.353L-7.241 31.278L-7.241 35.646Q-7.241 35.783-7.091 35.819Q-6.940 35.855-6.715 35.855L-6.715 36.135M-4.503 36.135L-6.055 36.135L-6.055 35.855Q-5.829 35.855-5.681 35.821Q-5.532 35.786-5.532 35.646L-5.532 33.797Q-5.532 33.609-5.580 33.525Q-5.628 33.442-5.725 33.423Q-5.823 33.404-6.035 33.404L-6.035 33.124L-4.978 33.049L-4.978 35.646Q-4.978 35.786-4.847 35.821Q-4.715 35.855-4.503 35.855L-4.503 36.135M-5.775 31.828Q-5.775 31.657-5.652 31.538Q-5.529 31.418-5.358 31.418Q-5.190 31.418-5.067 31.538Q-4.944 31.657-4.944 31.828Q-4.944 32.003-5.067 32.126Q-5.190 32.249-5.358 32.249Q-5.529 32.249-5.652 32.126Q-5.775 32.003-5.775 31.828M-2.176 36.135L-3.809 36.135L-3.809 35.855Q-3.580 35.855-3.432 35.821Q-3.283 35.786-3.283 35.646L-3.283 33.797Q-3.283 33.527-3.391 33.466Q-3.498 33.404-3.809 33.404L-3.809 33.124L-2.750 33.049L-2.750 33.698Q-2.579 33.390-2.275 33.219Q-1.971 33.049-1.625 33.049Q-1.120 33.049-0.836 33.272Q-0.552 33.496-0.552 33.992L-0.552 35.646Q-0.552 35.783-0.403 35.819Q-0.255 35.855-0.029 35.855L-0.029 36.135L-1.660 36.135L-1.660 35.855Q-1.431 35.855-1.282 35.821Q-1.133 35.786-1.133 35.646L-1.133 34.006Q-1.133 33.671-1.253 33.471Q-1.372 33.271-1.687 33.271Q-1.957 33.271-2.191 33.407Q-2.425 33.544-2.564 33.778Q-2.702 34.012-2.702 34.286L-2.702 35.646Q-2.702 35.783-2.552 35.819Q-2.401 35.855-2.176 35.855L-2.176 36.135M0.518 34.600Q0.518 34.279 0.642 33.990Q0.767 33.701 0.993 33.478Q1.218 33.254 1.514 33.134Q1.810 33.014 2.128 33.014Q2.456 33.014 2.717 33.114Q2.979 33.213 3.155 33.395Q3.331 33.578 3.425 33.836Q3.519 34.094 3.519 34.426Q3.519 34.518 3.437 34.539L1.181 34.539L1.181 34.600Q1.181 35.188 1.464 35.571Q1.748 35.954 2.316 35.954Q2.637 35.954 2.905 35.761Q3.173 35.568 3.262 35.253Q3.269 35.212 3.344 35.198L3.437 35.198Q3.519 35.222 3.519 35.294Q3.519 35.301 3.512 35.328Q3.399 35.725 3.028 35.964Q2.657 36.203 2.234 36.203Q1.796 36.203 1.396 35.995Q0.996 35.786 0.757 35.419Q0.518 35.052 0.518 34.600M1.188 34.330L3.003 34.330Q3.003 34.053 2.905 33.801Q2.808 33.548 2.609 33.392Q2.411 33.237 2.128 33.237Q1.851 33.237 1.637 33.395Q1.423 33.554 1.306 33.809Q1.188 34.064 1.188 34.330\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(209.815 35.893)\">\u003Cpath d=\"M8.417 36.108L7.289 33.609Q7.217 33.462 7.087 33.430Q6.957 33.397 6.728 33.397L6.728 33.117L8.242 33.117L8.242 33.397Q7.890 33.397 7.890 33.544Q7.890 33.589 7.901 33.609L8.765 35.527L9.545 33.797Q9.579 33.729 9.579 33.650Q9.579 33.537 9.495 33.467Q9.411 33.397 9.292 33.397L9.292 33.117L10.488 33.117L10.488 33.397Q10.269 33.397 10.098 33.500Q9.928 33.602 9.839 33.797L8.803 36.108Q8.755 36.203 8.649 36.203L8.571 36.203Q8.465 36.203 8.417 36.108M13.158 37.885Q12.607 37.485 12.236 36.930Q11.866 36.374 11.684 35.728Q11.503 35.082 11.503 34.385Q11.503 33.872 11.604 33.377Q11.705 32.881 11.910 32.430Q12.115 31.979 12.428 31.587Q12.741 31.196 13.158 30.892Q13.168 30.888 13.175 30.887Q13.181 30.885 13.192 30.885L13.260 30.885Q13.294 30.885 13.316 30.909Q13.339 30.933 13.339 30.970Q13.339 31.015 13.311 31.032Q12.963 31.333 12.710 31.717Q12.457 32.102 12.305 32.543Q12.153 32.984 12.081 33.440Q12.009 33.896 12.009 34.385Q12.009 35.386 12.318 36.273Q12.628 37.160 13.311 37.745Q13.339 37.762 13.339 37.806Q13.339 37.844 13.316 37.868Q13.294 37.892 13.260 37.892L13.192 37.892Q13.185 37.888 13.176 37.887Q13.168 37.885 13.158 37.885M14.323 36.197L14.323 34.624Q14.323 34.597 14.349 34.571Q14.374 34.546 14.402 34.546L14.514 34.546Q14.542 34.546 14.566 34.573Q14.590 34.600 14.590 34.624Q14.590 34.969 14.721 35.233Q14.853 35.496 15.082 35.665Q15.311 35.834 15.613 35.915Q15.916 35.995 16.258 35.995Q16.524 35.995 16.760 35.867Q16.996 35.739 17.141 35.516Q17.286 35.294 17.286 35.028Q17.286 34.805 17.180 34.609Q17.075 34.412 16.893 34.277Q16.712 34.142 16.487 34.091L15.458 33.859Q15.147 33.787 14.887 33.601Q14.627 33.414 14.475 33.143Q14.323 32.871 14.323 32.556Q14.323 32.170 14.537 31.863Q14.750 31.555 15.097 31.384Q15.444 31.213 15.824 31.213Q16.053 31.213 16.282 31.266Q16.511 31.319 16.709 31.427Q16.907 31.534 17.061 31.698L17.355 31.258Q17.379 31.213 17.420 31.213L17.468 31.213Q17.498 31.213 17.521 31.239Q17.543 31.264 17.543 31.292L17.543 32.867Q17.543 32.888 17.519 32.915Q17.495 32.943 17.468 32.943L17.355 32.943Q17.293 32.943 17.280 32.867Q17.239 32.454 17.057 32.134Q16.876 31.815 16.565 31.640Q16.254 31.466 15.824 31.466Q15.574 31.466 15.335 31.577Q15.096 31.688 14.945 31.886Q14.795 32.085 14.795 32.348Q14.795 32.560 14.902 32.741Q15.010 32.922 15.186 33.042Q15.362 33.161 15.571 33.202L16.599 33.431Q16.917 33.503 17.184 33.708Q17.450 33.913 17.603 34.207Q17.755 34.501 17.755 34.833Q17.755 35.226 17.550 35.561Q17.345 35.896 16.999 36.085Q16.654 36.275 16.258 36.275Q15.837 36.275 15.458 36.162Q15.078 36.050 14.808 35.800L14.514 36.234Q14.487 36.275 14.450 36.275L14.402 36.275Q14.374 36.275 14.349 36.250Q14.323 36.224 14.323 36.197M19.091 35.294L19.091 33.397L18.452 33.397L18.452 33.175Q18.770 33.175 18.987 32.965Q19.204 32.755 19.305 32.445Q19.406 32.136 19.406 31.828L19.672 31.828L19.672 33.117L20.749 33.117L20.749 33.397L19.672 33.397L19.672 35.281Q19.672 35.557 19.776 35.756Q19.881 35.954 20.140 35.954Q20.298 35.954 20.404 35.850Q20.510 35.745 20.559 35.592Q20.609 35.438 20.609 35.281L20.609 34.867L20.875 34.867L20.875 35.294Q20.875 35.520 20.776 35.730Q20.677 35.940 20.492 36.072Q20.308 36.203 20.079 36.203Q19.641 36.203 19.366 35.966Q19.091 35.728 19.091 35.294M22.007 37.892L21.938 37.892Q21.904 37.892 21.882 37.866Q21.860 37.841 21.860 37.806Q21.860 37.762 21.890 37.745Q22.246 37.441 22.495 37.051Q22.745 36.661 22.897 36.229Q23.049 35.797 23.119 35.328Q23.189 34.860 23.189 34.385Q23.189 33.906 23.119 33.440Q23.049 32.973 22.895 32.538Q22.742 32.102 22.490 31.714Q22.239 31.326 21.890 31.032Q21.860 31.015 21.860 30.970Q21.860 30.936 21.882 30.911Q21.904 30.885 21.938 30.885L22.007 30.885Q22.017 30.885 22.025 30.887Q22.034 30.888 22.044 30.892Q22.588 31.292 22.960 31.845Q23.333 32.399 23.514 33.045Q23.695 33.691 23.695 34.385Q23.695 35.086 23.514 35.733Q23.333 36.381 22.959 36.935Q22.584 37.489 22.044 37.885Q22.034 37.885 22.025 37.887Q22.017 37.888 22.007 37.892\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-38.462 36.574)\">\u003Cpath d=\"M-22.106 36.108L-23.087 33.609Q-23.148 33.466-23.266 33.431Q-23.384 33.397-23.600 33.397L-23.600 33.117L-22.120 33.117L-22.120 33.397Q-22.499 33.397-22.499 33.558Q-22.499 33.568-22.485 33.609L-21.771 35.441L-21.098 33.736Q-21.128 33.664-21.128 33.636Q-21.128 33.609-21.156 33.609Q-21.217 33.462-21.335 33.430Q-21.453 33.397-21.665 33.397L-21.665 33.117L-20.267 33.117L-20.267 33.397Q-20.643 33.397-20.643 33.558Q-20.643 33.589-20.636 33.609L-19.881 35.547L-19.194 33.797Q-19.173 33.746-19.173 33.691Q-19.173 33.551-19.286 33.474Q-19.399 33.397-19.539 33.397L-19.539 33.117L-18.319 33.117L-18.319 33.397Q-18.524 33.397-18.679 33.503Q-18.835 33.609-18.907 33.797L-19.812 36.108Q-19.847 36.203-19.959 36.203L-20.028 36.203Q-20.137 36.203-20.175 36.108L-20.957 34.105L-21.744 36.108Q-21.778 36.203-21.891 36.203L-21.959 36.203Q-22.068 36.203-22.106 36.108M-16.172 36.135L-17.724 36.135L-17.724 35.855Q-17.498 35.855-17.350 35.821Q-17.201 35.786-17.201 35.646L-17.201 33.797Q-17.201 33.609-17.249 33.525Q-17.297 33.442-17.394 33.423Q-17.492 33.404-17.704 33.404L-17.704 33.124L-16.647 33.049L-16.647 35.646Q-16.647 35.786-16.516 35.821Q-16.384 35.855-16.172 35.855L-16.172 36.135M-17.444 31.828Q-17.444 31.657-17.321 31.538Q-17.198 31.418-17.027 31.418Q-16.859 31.418-16.736 31.538Q-16.613 31.657-16.613 31.828Q-16.613 32.003-16.736 32.126Q-16.859 32.249-17.027 32.249Q-17.198 32.249-17.321 32.126Q-17.444 32.003-17.444 31.828M-15 35.294L-15 33.397L-15.639 33.397L-15.639 33.175Q-15.321 33.175-15.104 32.965Q-14.887 32.755-14.786 32.445Q-14.685 32.136-14.685 31.828L-14.419 31.828L-14.419 33.117L-13.342 33.117L-13.342 33.397L-14.419 33.397L-14.419 35.281Q-14.419 35.557-14.315 35.756Q-14.210 35.954-13.951 35.954Q-13.793 35.954-13.687 35.850Q-13.581 35.745-13.532 35.592Q-13.482 35.438-13.482 35.281L-13.482 34.867L-13.216 34.867L-13.216 35.294Q-13.216 35.520-13.315 35.730Q-13.414 35.940-13.599 36.072Q-13.783 36.203-14.012 36.203Q-14.450 36.203-14.725 35.966Q-15 35.728-15 35.294M-10.724 36.135L-12.358 36.135L-12.358 35.855Q-12.129 35.855-11.980 35.821Q-11.831 35.786-11.831 35.646L-11.831 32.027Q-11.831 31.757-11.939 31.695Q-12.047 31.634-12.358 31.634L-12.358 31.353L-11.278 31.278L-11.278 33.664Q-11.172 33.479-10.994 33.337Q-10.816 33.196-10.608 33.122Q-10.399 33.049-10.174 33.049Q-9.668 33.049-9.384 33.272Q-9.100 33.496-9.100 33.992L-9.100 35.646Q-9.100 35.783-8.952 35.819Q-8.803 35.855-8.578 35.855L-8.578 36.135L-10.208 36.135L-10.208 35.855Q-9.979 35.855-9.830 35.821Q-9.682 35.786-9.682 35.646L-9.682 34.006Q-9.682 33.671-9.801 33.471Q-9.921 33.271-10.235 33.271Q-10.505 33.271-10.739 33.407Q-10.974 33.544-11.112 33.778Q-11.250 34.012-11.250 34.286L-11.250 35.646Q-11.250 35.783-11.100 35.819Q-10.950 35.855-10.724 35.855\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-38.462 36.574)\">\u003Cpath d=\"M-5.111 33.742Q-5.111 33.216-4.894 32.748Q-4.677 32.280-4.294 31.934Q-3.912 31.589-3.428 31.401Q-2.944 31.213-2.414 31.213Q-2.011 31.213-1.647 31.370Q-1.283 31.528-0.999 31.822L-0.576 31.240Q-0.541 31.213-0.517 31.213L-0.470 31.213Q-0.439 31.213-0.415 31.237Q-0.391 31.261-0.391 31.292L-0.391 33.155Q-0.391 33.178-0.417 33.204Q-0.442 33.230-0.470 33.230L-0.596 33.230Q-0.658 33.230-0.671 33.155Q-0.702 32.840-0.837 32.536Q-0.972 32.232-1.187 31.998Q-1.403 31.763-1.692 31.628Q-1.980 31.493-2.308 31.493Q-2.951 31.493-3.409 31.787Q-3.867 32.081-4.100 32.594Q-4.332 33.107-4.332 33.742Q-4.332 34.214-4.202 34.623Q-4.072 35.031-3.812 35.344Q-3.553 35.656-3.173 35.826Q-2.794 35.995-2.302 35.995Q-1.974 35.995-1.680 35.879Q-1.386 35.762-1.152 35.547Q-0.917 35.332-0.787 35.043Q-0.658 34.754-0.658 34.426Q-0.658 34.399-0.630 34.375Q-0.603 34.351-0.582 34.351L-0.470 34.351Q-0.432 34.351-0.412 34.376Q-0.391 34.402-0.391 34.440Q-0.391 34.836-0.557 35.173Q-0.723 35.510-1.010 35.757Q-1.297 36.005-1.666 36.140Q-2.035 36.275-2.414 36.275Q-2.934 36.275-3.426 36.085Q-3.918 35.896-4.298 35.552Q-4.677 35.209-4.894 34.740Q-5.111 34.272-5.111 33.742M2.856 36.176L1.014 31.845Q0.945 31.702 0.764 31.668Q0.583 31.634 0.320 31.634L0.320 31.353L2.289 31.353L2.289 31.634Q1.735 31.634 1.735 31.808Q1.738 31.818 1.740 31.828Q1.742 31.839 1.742 31.845L3.263 35.421L4.695 32.033Q4.719 31.996 4.719 31.941Q4.719 31.781 4.568 31.707Q4.418 31.634 4.233 31.634L4.233 31.353L5.782 31.353L5.782 31.634Q5.522 31.634 5.312 31.728Q5.102 31.822 5.006 32.033L3.249 36.176Q3.198 36.275 3.102 36.275L3.003 36.275Q2.904 36.275 2.856 36.176M6.718 35.715Q6.718 35.547 6.841 35.424Q6.964 35.301 7.139 35.301Q7.306 35.301 7.429 35.424Q7.552 35.547 7.552 35.715Q7.552 35.889 7.429 36.012Q7.306 36.135 7.139 36.135Q6.964 36.135 6.841 36.012Q6.718 35.889 6.718 35.715M6.718 33.531Q6.718 33.363 6.841 33.240Q6.964 33.117 7.139 33.117Q7.306 33.117 7.429 33.240Q7.552 33.363 7.552 33.531Q7.552 33.705 7.429 33.828Q7.306 33.951 7.139 33.951Q6.964 33.951 6.841 33.828Q6.718 33.705 6.718 33.531\" 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-3.892 70.278V57.475\" style=\"stroke-width:.8\"\u002F>\u003Cpath fill=\"var(--tk-accent)\" stroke=\"none\" d=\"M-2.292 57.475a1.6 1.6 0 1 0-3.2 0 1.6 1.6 0 0 0 3.2 0m-1.6 0\"\u002F>\u003Cpath fill=\"none\" stroke=\"var(--tk-accent)\" d=\"M24.561 70.278v8.536\" style=\"stroke-width:.8\"\u002F>\u003Cpath fill=\"var(--tk-accent)\" stroke=\"none\" d=\"M26.161 78.814a1.6 1.6 0 1 0-3.2 0 1.6 1.6 0 0 0 3.2 0m-1.6 0\"\u002F>\u003Cpath fill=\"none\" stroke=\"var(--tk-accent)\" d=\"M53.014 70.278V60.32\" style=\"stroke-width:.8\"\u002F>\u003Cpath fill=\"var(--tk-accent)\" stroke=\"none\" d=\"M54.614 60.32a1.6 1.6 0 1 0-3.2 0 1.6 1.6 0 0 0 3.2 0m-1.6 0\"\u002F>\u003Cpath fill=\"none\" stroke=\"var(--tk-accent)\" d=\"M81.467 70.278v7.113\" style=\"stroke-width:.8\"\u002F>\u003Cpath fill=\"var(--tk-accent)\" stroke=\"none\" d=\"M83.067 77.391a1.6 1.6 0 1 0-3.2 0 1.6 1.6 0 0 0 3.2 0m-1.6 0\"\u002F>\u003Cpath fill=\"none\" stroke=\"var(--tk-accent)\" d=\"M109.92 70.278V56.052\" style=\"stroke-width:.8\"\u002F>\u003Cpath fill=\"var(--tk-accent)\" stroke=\"none\" d=\"M111.52 56.052a1.6 1.6 0 1 0-3.2 0 1.6 1.6 0 0 0 3.2 0m-1.6 0\"\u002F>\u003Cpath fill=\"none\" stroke=\"var(--tk-accent)\" d=\"M138.372 70.278v5.69\" style=\"stroke-width:.8\"\u002F>\u003Cpath fill=\"var(--tk-accent)\" stroke=\"none\" d=\"M139.972 75.969a1.6 1.6 0 1 0-3.2 0 1.6 1.6 0 0 0 3.2 0m-1.6 0\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(79.211 -100.21)\">\u003Cpath d=\"M-21.812 36.135L-23.446 36.135L-23.446 35.855Q-23.217 35.855-23.068 35.821Q-22.919 35.786-22.919 35.646L-22.919 32.027Q-22.919 31.757-23.027 31.695Q-23.135 31.634-23.446 31.634L-23.446 31.353L-22.366 31.278L-22.366 33.664Q-22.260 33.479-22.082 33.337Q-21.904 33.196-21.696 33.122Q-21.487 33.049-21.262 33.049Q-20.756 33.049-20.472 33.272Q-20.188 33.496-20.188 33.992L-20.188 35.646Q-20.188 35.783-20.040 35.819Q-19.891 35.855-19.665 35.855L-19.665 36.135L-21.296 36.135L-21.296 35.855Q-21.067 35.855-20.918 35.821Q-20.769 35.786-20.769 35.646L-20.769 34.006Q-20.769 33.671-20.889 33.471Q-21.009 33.271-21.323 33.271Q-21.593 33.271-21.827 33.407Q-22.061 33.544-22.200 33.778Q-22.338 34.012-22.338 34.286L-22.338 35.646Q-22.338 35.783-22.188 35.819Q-22.037 35.855-21.812 35.855L-21.812 36.135M-17.461 36.135L-19.013 36.135L-19.013 35.855Q-18.787 35.855-18.638 35.821Q-18.490 35.786-18.490 35.646L-18.490 33.797Q-18.490 33.609-18.537 33.525Q-18.585 33.442-18.683 33.423Q-18.780 33.404-18.992 33.404L-18.992 33.124L-17.936 33.049L-17.936 35.646Q-17.936 35.786-17.804 35.821Q-17.673 35.855-17.461 35.855L-17.461 36.135M-18.732 31.828Q-18.732 31.657-18.609 31.538Q-18.486 31.418-18.315 31.418Q-18.148 31.418-18.025 31.538Q-17.902 31.657-17.902 31.828Q-17.902 32.003-18.025 32.126Q-18.148 32.249-18.315 32.249Q-18.486 32.249-18.609 32.126Q-18.732 32.003-18.732 31.828M-16.856 36.668Q-16.856 36.422-16.659 36.238Q-16.463 36.053-16.206 35.974Q-16.343 35.862-16.415 35.701Q-16.487 35.540-16.487 35.359Q-16.487 35.038-16.275 34.792Q-16.610 34.494-16.610 34.084Q-16.610 33.623-16.220 33.336Q-15.830 33.049-15.352 33.049Q-14.880 33.049-14.545 33.295Q-14.371 33.141-14.161 33.059Q-13.951 32.977-13.722 32.977Q-13.558 32.977-13.436 33.084Q-13.315 33.192-13.315 33.356Q-13.315 33.452-13.387 33.524Q-13.458 33.595-13.551 33.595Q-13.650 33.595-13.720 33.522Q-13.790 33.448-13.790 33.349Q-13.790 33.295-13.776 33.264L-13.769 33.250Q-13.763 33.230-13.754 33.219Q-13.745 33.209-13.742 33.202Q-14.098 33.202-14.385 33.425Q-14.098 33.718-14.098 34.084Q-14.098 34.399-14.282 34.631Q-14.467 34.864-14.756 34.992Q-15.044 35.120-15.352 35.120Q-15.554 35.120-15.745 35.070Q-15.936 35.021-16.114 34.911Q-16.206 35.038-16.206 35.181Q-16.206 35.363-16.078 35.498Q-15.950 35.633-15.766 35.633L-15.133 35.633Q-14.685 35.633-14.316 35.704Q-13.947 35.776-13.687 36.005Q-13.428 36.234-13.428 36.668Q-13.428 36.989-13.723 37.191Q-14.019 37.393-14.422 37.482Q-14.826 37.571-15.140 37.571Q-15.458 37.571-15.861 37.482Q-16.265 37.393-16.560 37.191Q-16.856 36.989-16.856 36.668M-16.401 36.668Q-16.401 36.897-16.183 37.046Q-15.964 37.195-15.672 37.263Q-15.379 37.331-15.140 37.331Q-14.976 37.331-14.767 37.295Q-14.559 37.260-14.352 37.179Q-14.145 37.099-14.014 36.971Q-13.882 36.843-13.882 36.668Q-13.882 36.316-14.263 36.222Q-14.644 36.128-15.147 36.128L-15.766 36.128Q-16.005 36.128-16.203 36.279Q-16.401 36.429-16.401 36.668M-15.352 34.881Q-14.685 34.881-14.685 34.084Q-14.685 33.284-15.352 33.284Q-16.022 33.284-16.022 34.084Q-16.022 34.881-15.352 34.881M-11.151 36.135L-12.785 36.135L-12.785 35.855Q-12.556 35.855-12.407 35.821Q-12.259 35.786-12.259 35.646L-12.259 32.027Q-12.259 31.757-12.366 31.695Q-12.474 31.634-12.785 31.634L-12.785 31.353L-11.705 31.278L-11.705 33.664Q-11.599 33.479-11.421 33.337Q-11.244 33.196-11.035 33.122Q-10.827 33.049-10.601 33.049Q-10.095 33.049-9.811 33.272Q-9.528 33.496-9.528 33.992L-9.528 35.646Q-9.528 35.783-9.379 35.819Q-9.230 35.855-9.005 35.855L-9.005 36.135L-10.635 36.135L-10.635 35.855Q-10.406 35.855-10.257 35.821Q-10.109 35.786-10.109 35.646L-10.109 34.006Q-10.109 33.671-10.228 33.471Q-10.348 33.271-10.662 33.271Q-10.933 33.271-11.167 33.407Q-11.401 33.544-11.539 33.778Q-11.678 34.012-11.678 34.286L-11.678 35.646Q-11.678 35.783-11.527 35.819Q-11.377 35.855-11.151 35.855\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(79.211 -100.21)\">\u003Cpath d=\"M-4.127 36.108L-5.255 33.609Q-5.327 33.462-5.457 33.430Q-5.587 33.397-5.816 33.397L-5.816 33.117L-4.302 33.117L-4.302 33.397Q-4.654 33.397-4.654 33.544Q-4.654 33.589-4.643 33.609L-3.779 35.527L-2.999 33.797Q-2.965 33.729-2.965 33.650Q-2.965 33.537-3.049 33.467Q-3.133 33.397-3.252 33.397L-3.252 33.117L-2.056 33.117L-2.056 33.397Q-2.275 33.397-2.446 33.500Q-2.616 33.602-2.705 33.797L-3.741 36.108Q-3.789 36.203-3.895 36.203L-3.973 36.203Q-4.079 36.203-4.127 36.108\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(79.211 -100.21)\">\u003Cpath d=\"M-1.887 35.407Q-1.887 35.075-1.664 34.848Q-1.440 34.621-1.096 34.493Q-0.753 34.364-0.380 34.312Q-0.008 34.259 0.297 34.259L0.297 34.006Q0.297 33.801 0.189 33.621Q0.081 33.442-0.100 33.339Q-0.281 33.237-0.489 33.237Q-0.896 33.237-1.132 33.329Q-1.043 33.366-0.997 33.450Q-0.951 33.534-0.951 33.636Q-0.951 33.732-0.997 33.811Q-1.043 33.889-1.124 33.934Q-1.204 33.978-1.293 33.978Q-1.443 33.978-1.544 33.881Q-1.645 33.783-1.645 33.636Q-1.645 33.014-0.489 33.014Q-0.278 33.014-0.028 33.078Q0.221 33.141 0.423 33.260Q0.625 33.380 0.751 33.565Q0.878 33.749 0.878 33.992L0.878 35.568Q0.878 35.684 0.939 35.780Q1.001 35.875 1.114 35.875Q1.223 35.875 1.288 35.781Q1.353 35.687 1.353 35.568L1.353 35.120L1.619 35.120L1.619 35.568Q1.619 35.838 1.392 36.003Q1.165 36.169 0.885 36.169Q0.676 36.169 0.539 36.015Q0.403 35.862 0.379 35.646Q0.232 35.913-0.050 36.058Q-0.332 36.203-0.657 36.203Q-0.934 36.203-1.218 36.128Q-1.501 36.053-1.694 35.874Q-1.887 35.694-1.887 35.407M-1.272 35.407Q-1.272 35.581-1.171 35.711Q-1.071 35.841-0.915 35.911Q-0.760 35.981-0.595 35.981Q-0.377 35.981-0.168 35.884Q0.040 35.786 0.168 35.605Q0.297 35.424 0.297 35.198L0.297 34.470Q-0.028 34.470-0.394 34.561Q-0.760 34.652-1.016 34.864Q-1.272 35.075-1.272 35.407M3.786 36.135L2.050 36.135L2.050 35.855Q2.279 35.855 2.428 35.821Q2.576 35.786 2.576 35.646L2.576 33.797Q2.576 33.527 2.469 33.466Q2.361 33.404 2.050 33.404L2.050 33.124L3.079 33.049L3.079 33.756Q3.209 33.448 3.451 33.249Q3.694 33.049 4.012 33.049Q4.231 33.049 4.402 33.173Q4.573 33.298 4.573 33.510Q4.573 33.647 4.473 33.746Q4.374 33.845 4.241 33.845Q4.104 33.845 4.005 33.746Q3.906 33.647 3.906 33.510Q3.906 33.370 4.005 33.271Q3.715 33.271 3.515 33.467Q3.315 33.664 3.222 33.958Q3.130 34.252 3.130 34.532L3.130 35.646Q3.130 35.855 3.786 35.855L3.786 36.135M6.774 36.135L5.222 36.135L5.222 35.855Q5.448 35.855 5.596 35.821Q5.745 35.786 5.745 35.646L5.745 33.797Q5.745 33.609 5.697 33.525Q5.649 33.442 5.552 33.423Q5.454 33.404 5.242 33.404L5.242 33.124L6.299 33.049L6.299 35.646Q6.299 35.786 6.430 35.821Q6.562 35.855 6.774 35.855L6.774 36.135M5.502 31.828Q5.502 31.657 5.625 31.538Q5.748 31.418 5.919 31.418Q6.087 31.418 6.210 31.538Q6.333 31.657 6.333 31.828Q6.333 32.003 6.210 32.126Q6.087 32.249 5.919 32.249Q5.748 32.249 5.625 32.126Q5.502 32.003 5.502 31.828M7.478 35.407Q7.478 35.075 7.702 34.848Q7.926 34.621 8.269 34.493Q8.613 34.364 8.985 34.312Q9.358 34.259 9.662 34.259L9.662 34.006Q9.662 33.801 9.554 33.621Q9.447 33.442 9.265 33.339Q9.084 33.237 8.876 33.237Q8.469 33.237 8.233 33.329Q8.322 33.366 8.368 33.450Q8.414 33.534 8.414 33.636Q8.414 33.732 8.368 33.811Q8.322 33.889 8.242 33.934Q8.161 33.978 8.073 33.978Q7.922 33.978 7.821 33.881Q7.720 33.783 7.720 33.636Q7.720 33.014 8.876 33.014Q9.088 33.014 9.337 33.078Q9.587 33.141 9.788 33.260Q9.990 33.380 10.116 33.565Q10.243 33.749 10.243 33.992L10.243 35.568Q10.243 35.684 10.304 35.780Q10.366 35.875 10.479 35.875Q10.588 35.875 10.653 35.781Q10.718 35.687 10.718 35.568L10.718 35.120L10.985 35.120L10.985 35.568Q10.985 35.838 10.757 36.003Q10.530 36.169 10.250 36.169Q10.041 36.169 9.905 36.015Q9.768 35.862 9.744 35.646Q9.597 35.913 9.315 36.058Q9.033 36.203 8.708 36.203Q8.431 36.203 8.148 36.128Q7.864 36.053 7.671 35.874Q7.478 35.694 7.478 35.407M8.093 35.407Q8.093 35.581 8.194 35.711Q8.295 35.841 8.450 35.911Q8.606 35.981 8.770 35.981Q8.989 35.981 9.197 35.884Q9.406 35.786 9.534 35.605Q9.662 35.424 9.662 35.198L9.662 34.470Q9.337 34.470 8.971 34.561Q8.606 34.652 8.349 34.864Q8.093 35.075 8.093 35.407M13.083 36.135L11.449 36.135L11.449 35.855Q11.678 35.855 11.827 35.821Q11.976 35.786 11.976 35.646L11.976 33.797Q11.976 33.527 11.868 33.466Q11.761 33.404 11.449 33.404L11.449 33.124L12.509 33.049L12.509 33.698Q12.680 33.390 12.984 33.219Q13.288 33.049 13.634 33.049Q14.139 33.049 14.423 33.272Q14.707 33.496 14.707 33.992L14.707 35.646Q14.707 35.783 14.855 35.819Q15.004 35.855 15.230 35.855L15.230 36.135L13.599 36.135L13.599 35.855Q13.828 35.855 13.977 35.821Q14.126 35.786 14.126 35.646L14.126 34.006Q14.126 33.671 14.006 33.471Q13.886 33.271 13.572 33.271Q13.302 33.271 13.068 33.407Q12.834 33.544 12.695 33.778Q12.557 34.012 12.557 34.286L12.557 35.646Q12.557 35.783 12.707 35.819Q12.858 35.855 13.083 35.855L13.083 36.135M15.818 34.624Q15.818 34.296 15.953 33.995Q16.088 33.695 16.323 33.474Q16.559 33.254 16.864 33.134Q17.168 33.014 17.492 33.014Q17.998 33.014 18.347 33.117Q18.696 33.219 18.696 33.595Q18.696 33.742 18.598 33.843Q18.501 33.944 18.354 33.944Q18.200 33.944 18.101 33.845Q18.002 33.746 18.002 33.595Q18.002 33.407 18.142 33.315Q17.940 33.264 17.499 33.264Q17.144 33.264 16.915 33.460Q16.686 33.657 16.585 33.966Q16.484 34.276 16.484 34.624Q16.484 34.973 16.611 35.279Q16.737 35.585 16.992 35.769Q17.246 35.954 17.602 35.954Q17.824 35.954 18.009 35.870Q18.193 35.786 18.328 35.631Q18.463 35.475 18.521 35.267Q18.535 35.212 18.590 35.212L18.702 35.212Q18.733 35.212 18.755 35.236Q18.778 35.260 18.778 35.294L18.778 35.315Q18.692 35.602 18.504 35.800Q18.316 35.998 18.051 36.101Q17.786 36.203 17.492 36.203Q17.062 36.203 16.674 35.997Q16.286 35.790 16.052 35.427Q15.818 35.065 15.818 34.624M19.324 34.600Q19.324 34.279 19.449 33.990Q19.574 33.701 19.800 33.478Q20.025 33.254 20.321 33.134Q20.616 33.014 20.934 33.014Q21.262 33.014 21.524 33.114Q21.785 33.213 21.961 33.395Q22.137 33.578 22.231 33.836Q22.325 34.094 22.325 34.426Q22.325 34.518 22.243 34.539L19.988 34.539L19.988 34.600Q19.988 35.188 20.271 35.571Q20.555 35.954 21.122 35.954Q21.444 35.954 21.712 35.761Q21.980 35.568 22.069 35.253Q22.076 35.212 22.151 35.198L22.243 35.198Q22.325 35.222 22.325 35.294Q22.325 35.301 22.319 35.328Q22.206 35.725 21.835 35.964Q21.464 36.203 21.040 36.203Q20.603 36.203 20.203 35.995Q19.803 35.786 19.564 35.419Q19.324 35.052 19.324 34.600M19.994 34.330L21.809 34.330Q21.809 34.053 21.712 33.801Q21.615 33.548 21.416 33.392Q21.218 33.237 20.934 33.237Q20.657 33.237 20.444 33.395Q20.230 33.554 20.112 33.809Q19.994 34.064 19.994 34.330\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(73.232 55.34)\">\u003Cpath d=\"M-21.904 36.108L-23.032 33.609Q-23.104 33.462-23.234 33.430Q-23.364 33.397-23.593 33.397L-23.593 33.117L-22.079 33.117L-22.079 33.397Q-22.431 33.397-22.431 33.544Q-22.431 33.589-22.420 33.609L-21.556 35.527L-20.776 33.797Q-20.742 33.729-20.742 33.650Q-20.742 33.537-20.826 33.467Q-20.910 33.397-21.029 33.397L-21.029 33.117L-19.833 33.117L-19.833 33.397Q-20.052 33.397-20.223 33.500Q-20.393 33.602-20.482 33.797L-21.518 36.108Q-21.566 36.203-21.672 36.203L-21.750 36.203Q-21.856 36.203-21.904 36.108\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(73.232 55.34)\">\u003Cpath d=\"M-19.665 35.407Q-19.665 35.075-19.442 34.848Q-19.218 34.621-18.874 34.493Q-18.531 34.364-18.158 34.312Q-17.786 34.259-17.481 34.259L-17.481 34.006Q-17.481 33.801-17.589 33.621Q-17.697 33.442-17.878 33.339Q-18.059 33.237-18.267 33.237Q-18.674 33.237-18.910 33.329Q-18.821 33.366-18.775 33.450Q-18.729 33.534-18.729 33.636Q-18.729 33.732-18.775 33.811Q-18.821 33.889-18.902 33.934Q-18.982 33.978-19.071 33.978Q-19.221 33.978-19.322 33.881Q-19.423 33.783-19.423 33.636Q-19.423 33.014-18.267 33.014Q-18.056 33.014-17.806 33.078Q-17.557 33.141-17.355 33.260Q-17.153 33.380-17.027 33.565Q-16.900 33.749-16.900 33.992L-16.900 35.568Q-16.900 35.684-16.839 35.780Q-16.777 35.875-16.664 35.875Q-16.555 35.875-16.490 35.781Q-16.425 35.687-16.425 35.568L-16.425 35.120L-16.159 35.120L-16.159 35.568Q-16.159 35.838-16.386 36.003Q-16.613 36.169-16.893 36.169Q-17.102 36.169-17.239 36.015Q-17.375 35.862-17.399 35.646Q-17.546 35.913-17.828 36.058Q-18.110 36.203-18.435 36.203Q-18.712 36.203-18.996 36.128Q-19.279 36.053-19.472 35.874Q-19.665 35.694-19.665 35.407M-19.050 35.407Q-19.050 35.581-18.949 35.711Q-18.849 35.841-18.693 35.911Q-18.538 35.981-18.373 35.981Q-18.155 35.981-17.946 35.884Q-17.738 35.786-17.610 35.605Q-17.481 35.424-17.481 35.198L-17.481 34.470Q-17.806 34.470-18.172 34.561Q-18.538 34.652-18.794 34.864Q-19.050 35.075-19.050 35.407M-13.992 36.135L-15.728 36.135L-15.728 35.855Q-15.499 35.855-15.350 35.821Q-15.202 35.786-15.202 35.646L-15.202 33.797Q-15.202 33.527-15.309 33.466Q-15.417 33.404-15.728 33.404L-15.728 33.124L-14.699 33.049L-14.699 33.756Q-14.569 33.448-14.327 33.249Q-14.084 33.049-13.766 33.049Q-13.547 33.049-13.376 33.173Q-13.205 33.298-13.205 33.510Q-13.205 33.647-13.305 33.746Q-13.404 33.845-13.537 33.845Q-13.674 33.845-13.773 33.746Q-13.872 33.647-13.872 33.510Q-13.872 33.370-13.773 33.271Q-14.063 33.271-14.263 33.467Q-14.463 33.664-14.556 33.958Q-14.648 34.252-14.648 34.532L-14.648 35.646Q-14.648 35.855-13.992 35.855L-13.992 36.135M-11.004 36.135L-12.556 36.135L-12.556 35.855Q-12.330 35.855-12.182 35.821Q-12.033 35.786-12.033 35.646L-12.033 33.797Q-12.033 33.609-12.081 33.525Q-12.129 33.442-12.226 33.423Q-12.324 33.404-12.536 33.404L-12.536 33.124L-11.479 33.049L-11.479 35.646Q-11.479 35.786-11.348 35.821Q-11.216 35.855-11.004 35.855L-11.004 36.135M-12.276 31.828Q-12.276 31.657-12.153 31.538Q-12.030 31.418-11.859 31.418Q-11.691 31.418-11.568 31.538Q-11.445 31.657-11.445 31.828Q-11.445 32.003-11.568 32.126Q-11.691 32.249-11.859 32.249Q-12.030 32.249-12.153 32.126Q-12.276 32.003-12.276 31.828M-10.300 35.407Q-10.300 35.075-10.076 34.848Q-9.852 34.621-9.509 34.493Q-9.165 34.364-8.793 34.312Q-8.420 34.259-8.116 34.259L-8.116 34.006Q-8.116 33.801-8.224 33.621Q-8.331 33.442-8.513 33.339Q-8.694 33.237-8.902 33.237Q-9.309 33.237-9.545 33.329Q-9.456 33.366-9.410 33.450Q-9.364 33.534-9.364 33.636Q-9.364 33.732-9.410 33.811Q-9.456 33.889-9.536 33.934Q-9.617 33.978-9.705 33.978Q-9.856 33.978-9.957 33.881Q-10.058 33.783-10.058 33.636Q-10.058 33.014-8.902 33.014Q-8.690 33.014-8.441 33.078Q-8.191 33.141-7.990 33.260Q-7.788 33.380-7.662 33.565Q-7.535 33.749-7.535 33.992L-7.535 35.568Q-7.535 35.684-7.474 35.780Q-7.412 35.875-7.299 35.875Q-7.190 35.875-7.125 35.781Q-7.060 35.687-7.060 35.568L-7.060 35.120L-6.793 35.120L-6.793 35.568Q-6.793 35.838-7.021 36.003Q-7.248 36.169-7.528 36.169Q-7.737 36.169-7.873 36.015Q-8.010 35.862-8.034 35.646Q-8.181 35.913-8.463 36.058Q-8.745 36.203-9.070 36.203Q-9.347 36.203-9.630 36.128Q-9.914 36.053-10.107 35.874Q-10.300 35.694-10.300 35.407M-9.685 35.407Q-9.685 35.581-9.584 35.711Q-9.483 35.841-9.328 35.911Q-9.172 35.981-9.008 35.981Q-8.789 35.981-8.581 35.884Q-8.372 35.786-8.244 35.605Q-8.116 35.424-8.116 35.198L-8.116 34.470Q-8.441 34.470-8.807 34.561Q-9.172 34.652-9.429 34.864Q-9.685 35.075-9.685 35.407M-4.695 36.135L-6.329 36.135L-6.329 35.855Q-6.100 35.855-5.951 35.821Q-5.802 35.786-5.802 35.646L-5.802 33.797Q-5.802 33.527-5.910 33.466Q-6.017 33.404-6.329 33.404L-6.329 33.124L-5.269 33.049L-5.269 33.698Q-5.098 33.390-4.794 33.219Q-4.490 33.049-4.144 33.049Q-3.639 33.049-3.355 33.272Q-3.071 33.496-3.071 33.992L-3.071 35.646Q-3.071 35.783-2.923 35.819Q-2.774 35.855-2.548 35.855L-2.548 36.135L-4.179 36.135L-4.179 35.855Q-3.950 35.855-3.801 35.821Q-3.652 35.786-3.652 35.646L-3.652 34.006Q-3.652 33.671-3.772 33.471Q-3.892 33.271-4.206 33.271Q-4.476 33.271-4.710 33.407Q-4.944 33.544-5.083 33.778Q-5.221 34.012-5.221 34.286L-5.221 35.646Q-5.221 35.783-5.071 35.819Q-4.920 35.855-4.695 35.855L-4.695 36.135M-1.960 34.624Q-1.960 34.296-1.825 33.995Q-1.690 33.695-1.455 33.474Q-1.219 33.254-0.914 33.134Q-0.610 33.014-0.286 33.014Q0.220 33.014 0.569 33.117Q0.918 33.219 0.918 33.595Q0.918 33.742 0.820 33.843Q0.723 33.944 0.576 33.944Q0.422 33.944 0.323 33.845Q0.224 33.746 0.224 33.595Q0.224 33.407 0.364 33.315Q0.162 33.264-0.279 33.264Q-0.634 33.264-0.863 33.460Q-1.092 33.657-1.193 33.966Q-1.294 34.276-1.294 34.624Q-1.294 34.973-1.167 35.279Q-1.041 35.585-0.786 35.769Q-0.532 35.954-0.176 35.954Q0.046 35.954 0.231 35.870Q0.415 35.786 0.550 35.631Q0.685 35.475 0.743 35.267Q0.757 35.212 0.812 35.212L0.924 35.212Q0.955 35.212 0.977 35.236Q1 35.260 1 35.294L1 35.315Q0.914 35.602 0.726 35.800Q0.538 35.998 0.273 36.101Q0.008 36.203-0.286 36.203Q-0.716 36.203-1.104 35.997Q-1.492 35.790-1.726 35.427Q-1.960 35.065-1.960 34.624M1.546 34.600Q1.546 34.279 1.671 33.990Q1.796 33.701 2.022 33.478Q2.247 33.254 2.543 33.134Q2.838 33.014 3.156 33.014Q3.484 33.014 3.746 33.114Q4.007 33.213 4.183 33.395Q4.359 33.578 4.453 33.836Q4.547 34.094 4.547 34.426Q4.547 34.518 4.465 34.539L2.210 34.539L2.210 34.600Q2.210 35.188 2.493 35.571Q2.777 35.954 3.344 35.954Q3.666 35.954 3.934 35.761Q4.202 35.568 4.291 35.253Q4.298 35.212 4.373 35.198L4.465 35.198Q4.547 35.222 4.547 35.294Q4.547 35.301 4.541 35.328Q4.428 35.725 4.057 35.964Q3.686 36.203 3.262 36.203Q2.825 36.203 2.425 35.995Q2.025 35.786 1.786 35.419Q1.546 35.052 1.546 34.600M2.216 34.330L4.031 34.330Q4.031 34.053 3.934 33.801Q3.837 33.548 3.638 33.392Q3.440 33.237 3.156 33.237Q2.879 33.237 2.666 33.395Q2.452 33.554 2.334 33.809Q2.216 34.064 2.216 34.330\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(73.232 55.34)\">\u003Cpath d=\"M9.610 36.135L7.874 36.135L7.874 35.855Q8.103 35.855 8.252 35.821Q8.400 35.786 8.400 35.646L8.400 33.797Q8.400 33.527 8.293 33.466Q8.185 33.404 7.874 33.404L7.874 33.124L8.903 33.049L8.903 33.756Q9.033 33.448 9.275 33.249Q9.518 33.049 9.836 33.049Q10.055 33.049 10.226 33.173Q10.397 33.298 10.397 33.510Q10.397 33.647 10.297 33.746Q10.198 33.845 10.065 33.845Q9.928 33.845 9.829 33.746Q9.730 33.647 9.730 33.510Q9.730 33.370 9.829 33.271Q9.539 33.271 9.339 33.467Q9.139 33.664 9.046 33.958Q8.954 34.252 8.954 34.532L8.954 35.646Q8.954 35.855 9.610 35.855L9.610 36.135M10.940 34.600Q10.940 34.279 11.065 33.990Q11.190 33.701 11.415 33.478Q11.641 33.254 11.936 33.134Q12.232 33.014 12.550 33.014Q12.878 33.014 13.140 33.114Q13.401 33.213 13.577 33.395Q13.753 33.578 13.847 33.836Q13.941 34.094 13.941 34.426Q13.941 34.518 13.859 34.539L11.603 34.539L11.603 34.600Q11.603 35.188 11.887 35.571Q12.171 35.954 12.738 35.954Q13.059 35.954 13.327 35.761Q13.596 35.568 13.685 35.253Q13.692 35.212 13.767 35.198L13.859 35.198Q13.941 35.222 13.941 35.294Q13.941 35.301 13.934 35.328Q13.821 35.725 13.451 35.964Q13.080 36.203 12.656 36.203Q12.218 36.203 11.818 35.995Q11.419 35.786 11.179 35.419Q10.940 35.052 10.940 34.600M11.610 34.330L13.425 34.330Q13.425 34.053 13.327 33.801Q13.230 33.548 13.032 33.392Q12.834 33.237 12.550 33.237Q12.273 33.237 12.059 33.395Q11.846 33.554 11.728 33.809Q11.610 34.064 11.610 34.330M14.529 34.624Q14.529 34.286 14.669 33.995Q14.809 33.705 15.054 33.491Q15.298 33.278 15.602 33.163Q15.906 33.049 16.231 33.049Q16.501 33.049 16.764 33.148Q17.027 33.247 17.219 33.425L17.219 32.027Q17.219 31.757 17.111 31.695Q17.004 31.634 16.692 31.634L16.692 31.353L17.769 31.278L17.769 35.462Q17.769 35.650 17.824 35.733Q17.879 35.817 17.979 35.836Q18.080 35.855 18.296 35.855L18.296 36.135L17.188 36.203L17.188 35.786Q16.771 36.203 16.146 36.203Q15.715 36.203 15.342 35.991Q14.970 35.780 14.749 35.419Q14.529 35.058 14.529 34.624M16.204 35.981Q16.412 35.981 16.598 35.909Q16.785 35.838 16.939 35.701Q17.092 35.564 17.188 35.386L17.188 33.777Q17.103 33.630 16.957 33.510Q16.812 33.390 16.643 33.331Q16.474 33.271 16.293 33.271Q15.732 33.271 15.464 33.660Q15.195 34.050 15.195 34.631Q15.195 35.202 15.430 35.592Q15.664 35.981 16.204 35.981M19.519 35.301L19.519 33.797Q19.519 33.527 19.411 33.466Q19.304 33.404 18.993 33.404L18.993 33.124L20.100 33.049L20.100 35.281L20.100 35.301Q20.100 35.581 20.151 35.725Q20.203 35.868 20.345 35.925Q20.486 35.981 20.774 35.981Q21.026 35.981 21.232 35.841Q21.437 35.701 21.553 35.475Q21.669 35.250 21.669 35L21.669 33.797Q21.669 33.527 21.561 33.466Q21.454 33.404 21.143 33.404L21.143 33.124L22.250 33.049L22.250 35.462Q22.250 35.653 22.303 35.735Q22.356 35.817 22.457 35.836Q22.558 35.855 22.773 35.855L22.773 36.135L21.696 36.203L21.696 35.639Q21.587 35.821 21.442 35.944Q21.296 36.067 21.110 36.135Q20.924 36.203 20.722 36.203Q19.519 36.203 19.519 35.301M23.361 34.624Q23.361 34.296 23.496 33.995Q23.631 33.695 23.867 33.474Q24.103 33.254 24.407 33.134Q24.711 33.014 25.036 33.014Q25.542 33.014 25.890 33.117Q26.239 33.219 26.239 33.595Q26.239 33.742 26.141 33.843Q26.044 33.944 25.897 33.944Q25.743 33.944 25.644 33.845Q25.545 33.746 25.545 33.595Q25.545 33.407 25.685 33.315Q25.483 33.264 25.043 33.264Q24.687 33.264 24.458 33.460Q24.229 33.657 24.128 33.966Q24.027 34.276 24.027 34.624Q24.027 34.973 24.154 35.279Q24.280 35.585 24.535 35.769Q24.790 35.954 25.145 35.954Q25.367 35.954 25.552 35.870Q25.736 35.786 25.871 35.631Q26.006 35.475 26.065 35.267Q26.078 35.212 26.133 35.212L26.246 35.212Q26.276 35.212 26.299 35.236Q26.321 35.260 26.321 35.294L26.321 35.315Q26.235 35.602 26.047 35.800Q25.859 35.998 25.595 36.101Q25.330 36.203 25.036 36.203Q24.605 36.203 24.217 35.997Q23.829 35.790 23.595 35.427Q23.361 35.065 23.361 34.624M26.868 34.600Q26.868 34.279 26.993 33.990Q27.117 33.701 27.343 33.478Q27.568 33.254 27.864 33.134Q28.160 33.014 28.478 33.014Q28.806 33.014 29.067 33.114Q29.329 33.213 29.505 33.395Q29.681 33.578 29.775 33.836Q29.869 34.094 29.869 34.426Q29.869 34.518 29.787 34.539L27.531 34.539L27.531 34.600Q27.531 35.188 27.815 35.571Q28.098 35.954 28.666 35.954Q28.987 35.954 29.255 35.761Q29.524 35.568 29.612 35.253Q29.619 35.212 29.694 35.198L29.787 35.198Q29.869 35.222 29.869 35.294Q29.869 35.301 29.862 35.328Q29.749 35.725 29.378 35.964Q29.007 36.203 28.584 36.203Q28.146 36.203 27.746 35.995Q27.346 35.786 27.107 35.419Q26.868 35.052 26.868 34.600M27.538 34.330L29.353 34.330Q29.353 34.053 29.255 33.801Q29.158 33.548 28.960 33.392Q28.761 33.237 28.478 33.237Q28.201 33.237 27.987 33.395Q27.774 33.554 27.656 33.809Q27.538 34.064 27.538 34.330M30.457 34.624Q30.457 34.286 30.597 33.995Q30.737 33.705 30.981 33.491Q31.226 33.278 31.530 33.163Q31.834 33.049 32.159 33.049Q32.429 33.049 32.692 33.148Q32.955 33.247 33.147 33.425L33.147 32.027Q33.147 31.757 33.039 31.695Q32.931 31.634 32.620 31.634L32.620 31.353L33.697 31.278L33.697 35.462Q33.697 35.650 33.752 35.733Q33.806 35.817 33.907 35.836Q34.008 35.855 34.223 35.855L34.223 36.135L33.116 36.203L33.116 35.786Q32.699 36.203 32.073 36.203Q31.643 36.203 31.270 35.991Q30.898 35.780 30.677 35.419Q30.457 35.058 30.457 34.624M32.131 35.981Q32.340 35.981 32.526 35.909Q32.713 35.838 32.866 35.701Q33.020 35.564 33.116 35.386L33.116 33.777Q33.030 33.630 32.885 33.510Q32.740 33.390 32.571 33.331Q32.401 33.271 32.220 33.271Q31.660 33.271 31.391 33.660Q31.123 34.050 31.123 34.631Q31.123 35.202 31.357 35.592Q31.591 35.981 32.131 35.981\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">The control variate at work. Without it (top), a large importance ratio \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\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:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> scales the whole target and the estimate whipsaws far from the baseline \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6944em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.2222em;\">\u003Cspan class=\"mord\">^\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">S\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>. With the correction \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\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:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mord accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6944em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">v\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.2222em;\">\u003Cspan class=\"mord\">^\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">S\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> (bottom), the same ratios produce a target that stays near the baseline; the correction has zero mean, so no bias is added.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:468.222px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 351.166 89.113\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" d=\"M-68.737-43.417H-6.14V-71.87h-62.596Z\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-17.67 5.278)\">\u003Cpath d=\"M-35.481-64.287L-37.111-64.287L-37.111-64.567Q-36.882-64.567-36.733-64.602Q-36.585-64.636-36.585-64.776L-36.585-68.122Q-36.585-68.293-36.721-68.334Q-36.858-68.375-37.111-68.375L-37.111-68.655L-36.031-68.730L-36.031-68.324Q-35.809-68.525-35.522-68.628Q-35.234-68.730-34.927-68.730Q-34.500-68.730-34.136-68.517Q-33.772-68.303-33.558-67.939Q-33.344-67.575-33.344-67.155Q-33.344-66.710-33.584-66.346Q-33.823-65.982-34.216-65.779Q-34.609-65.576-35.053-65.576Q-35.320-65.576-35.568-65.676Q-35.815-65.777-36.003-65.958L-36.003-64.776Q-36.003-64.639-35.855-64.603Q-35.706-64.567-35.481-64.567L-35.481-64.287M-36.003-67.975L-36.003-66.365Q-35.870-66.112-35.627-65.955Q-35.385-65.798-35.108-65.798Q-34.780-65.798-34.527-65.999Q-34.274-66.201-34.141-66.519Q-34.007-66.837-34.007-67.155Q-34.007-67.384-34.072-67.613Q-34.137-67.842-34.265-68.040Q-34.394-68.238-34.588-68.358Q-34.783-68.477-35.016-68.477Q-35.310-68.477-35.578-68.348Q-35.846-68.218-36.003-67.975M-30.959-65.644L-32.695-65.644L-32.695-65.924Q-32.466-65.924-32.317-65.958Q-32.168-65.993-32.168-66.133L-32.168-67.982Q-32.168-68.252-32.276-68.313Q-32.384-68.375-32.695-68.375L-32.695-68.655L-31.666-68.730L-31.666-68.023Q-31.536-68.331-31.293-68.530Q-31.051-68.730-30.733-68.730Q-30.514-68.730-30.343-68.606Q-30.172-68.481-30.172-68.269Q-30.172-68.132-30.272-68.033Q-30.371-67.934-30.504-67.934Q-30.641-67.934-30.740-68.033Q-30.839-68.132-30.839-68.269Q-30.839-68.409-30.740-68.508Q-31.030-68.508-31.230-68.312Q-31.430-68.115-31.522-67.821Q-31.615-67.527-31.615-67.247L-31.615-66.133Q-31.615-65.924-30.959-65.924L-30.959-65.644M-29.629-67.179Q-29.629-67.500-29.504-67.789Q-29.379-68.078-29.154-68.301Q-28.928-68.525-28.633-68.645Q-28.337-68.765-28.019-68.765Q-27.691-68.765-27.429-68.665Q-27.168-68.566-26.992-68.384Q-26.816-68.201-26.722-67.943Q-26.628-67.685-26.628-67.353Q-26.628-67.261-26.710-67.240L-28.966-67.240L-28.966-67.179Q-28.966-66.591-28.682-66.208Q-28.398-65.825-27.831-65.825Q-27.510-65.825-27.241-66.018Q-26.973-66.211-26.884-66.526Q-26.877-66.567-26.802-66.581L-26.710-66.581Q-26.628-66.557-26.628-66.485Q-26.628-66.478-26.635-66.451Q-26.748-66.054-27.118-65.815Q-27.489-65.576-27.913-65.576Q-28.351-65.576-28.751-65.784Q-29.150-65.993-29.390-66.360Q-29.629-66.727-29.629-67.179M-28.959-67.449L-27.144-67.449Q-27.144-67.726-27.241-67.978Q-27.339-68.231-27.537-68.387Q-27.735-68.542-28.019-68.542Q-28.296-68.542-28.510-68.384Q-28.723-68.225-28.841-67.970Q-28.959-67.715-28.959-67.449M-24.451-65.671L-25.579-68.170Q-25.650-68.317-25.780-68.349Q-25.910-68.382-26.139-68.382L-26.139-68.662L-24.625-68.662L-24.625-68.382Q-24.977-68.382-24.977-68.235Q-24.977-68.190-24.967-68.170L-24.102-66.252L-23.323-67.982Q-23.289-68.050-23.289-68.129Q-23.289-68.242-23.372-68.312Q-23.456-68.382-23.576-68.382L-23.576-68.662L-22.379-68.662L-22.379-68.382Q-22.598-68.382-22.769-68.279Q-22.940-68.177-23.029-67.982L-24.064-65.671Q-24.112-65.576-24.218-65.576L-24.297-65.576Q-24.403-65.576-24.451-65.671\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-17.67 5.278)\">\u003Cpath d=\"M-18.605-66.485L-18.605-68.382L-19.244-68.382L-19.244-68.604Q-18.926-68.604-18.709-68.814Q-18.492-69.024-18.392-69.334Q-18.291-69.643-18.291-69.951L-18.024-69.951L-18.024-68.662L-16.947-68.662L-16.947-68.382L-18.024-68.382L-18.024-66.498Q-18.024-66.222-17.920-66.023Q-17.816-65.825-17.556-65.825Q-17.399-65.825-17.293-65.929Q-17.187-66.034-17.137-66.187Q-17.088-66.341-17.088-66.498L-17.088-66.912L-16.821-66.912L-16.821-66.485Q-16.821-66.259-16.920-66.049Q-17.019-65.839-17.204-65.707Q-17.388-65.576-17.617-65.576Q-18.055-65.576-18.330-65.813Q-18.605-66.051-18.605-66.485M-14.261-65.644L-15.997-65.644L-15.997-65.924Q-15.768-65.924-15.620-65.958Q-15.471-65.993-15.471-66.133L-15.471-67.982Q-15.471-68.252-15.579-68.313Q-15.686-68.375-15.997-68.375L-15.997-68.655L-14.968-68.730L-14.968-68.023Q-14.839-68.331-14.596-68.530Q-14.353-68.730-14.035-68.730Q-13.817-68.730-13.646-68.606Q-13.475-68.481-13.475-68.269Q-13.475-68.132-13.574-68.033Q-13.673-67.934-13.806-67.934Q-13.943-67.934-14.042-68.033Q-14.141-68.132-14.141-68.269Q-14.141-68.409-14.042-68.508Q-14.333-68.508-14.533-68.312Q-14.733-68.115-14.825-67.821Q-14.917-67.527-14.917-67.247L-14.917-66.133Q-14.917-65.924-14.261-65.924L-14.261-65.644M-12.832-66.372Q-12.832-66.704-12.608-66.931Q-12.384-67.158-12.041-67.286Q-11.697-67.415-11.325-67.467Q-10.952-67.520-10.648-67.520L-10.648-67.773Q-10.648-67.978-10.756-68.158Q-10.863-68.337-11.045-68.440Q-11.226-68.542-11.434-68.542Q-11.841-68.542-12.077-68.450Q-11.988-68.413-11.942-68.329Q-11.896-68.245-11.896-68.143Q-11.896-68.047-11.942-67.968Q-11.988-67.890-12.068-67.845Q-12.149-67.801-12.238-67.801Q-12.388-67.801-12.489-67.898Q-12.590-67.996-12.590-68.143Q-12.590-68.765-11.434-68.765Q-11.222-68.765-10.973-68.701Q-10.723-68.638-10.522-68.519Q-10.320-68.399-10.194-68.214Q-10.067-68.030-10.067-67.787L-10.067-66.211Q-10.067-66.095-10.006-65.999Q-9.944-65.904-9.831-65.904Q-9.722-65.904-9.657-65.998Q-9.592-66.092-9.592-66.211L-9.592-66.659L-9.325-66.659L-9.325-66.211Q-9.325-65.941-9.553-65.776Q-9.780-65.610-10.060-65.610Q-10.269-65.610-10.405-65.764Q-10.542-65.917-10.566-66.133Q-10.713-65.866-10.995-65.721Q-11.277-65.576-11.602-65.576Q-11.879-65.576-12.162-65.651Q-12.446-65.726-12.639-65.905Q-12.832-66.085-12.832-66.372M-12.217-66.372Q-12.217-66.198-12.116-66.068Q-12.015-65.938-11.860-65.868Q-11.704-65.798-11.540-65.798Q-11.321-65.798-11.113-65.895Q-10.904-65.993-10.776-66.174Q-10.648-66.355-10.648-66.581L-10.648-67.309Q-10.973-67.309-11.339-67.218Q-11.704-67.127-11.961-66.915Q-12.217-66.704-12.217-66.372M-8.908-67.155Q-8.908-67.483-8.773-67.784Q-8.638-68.084-8.403-68.305Q-8.167-68.525-7.863-68.645Q-7.558-68.765-7.234-68.765Q-6.728-68.765-6.379-68.662Q-6.030-68.560-6.030-68.184Q-6.030-68.037-6.128-67.936Q-6.225-67.835-6.372-67.835Q-6.526-67.835-6.625-67.934Q-6.724-68.033-6.724-68.184Q-6.724-68.372-6.584-68.464Q-6.786-68.515-7.227-68.515Q-7.582-68.515-7.811-68.319Q-8.040-68.122-8.141-67.813Q-8.242-67.503-8.242-67.155Q-8.242-66.806-8.115-66.500Q-7.989-66.194-7.734-66.010Q-7.480-65.825-7.124-65.825Q-6.902-65.825-6.717-65.909Q-6.533-65.993-6.398-66.148Q-6.263-66.304-6.205-66.512Q-6.191-66.567-6.136-66.567L-6.024-66.567Q-5.993-66.567-5.971-66.543Q-5.948-66.519-5.948-66.485L-5.948-66.464Q-6.034-66.177-6.222-65.979Q-6.410-65.781-6.675-65.678Q-6.940-65.576-7.234-65.576Q-7.664-65.576-8.052-65.782Q-8.440-65.989-8.674-66.352Q-8.908-66.714-8.908-67.155M-5.402-67.179Q-5.402-67.500-5.277-67.789Q-5.152-68.078-4.926-68.301Q-4.701-68.525-4.405-68.645Q-4.110-68.765-3.792-68.765Q-3.464-68.765-3.202-68.665Q-2.941-68.566-2.765-68.384Q-2.589-68.201-2.495-67.943Q-2.401-67.685-2.401-67.353Q-2.401-67.261-2.483-67.240L-4.738-67.240L-4.738-67.179Q-4.738-66.591-4.455-66.208Q-4.171-65.825-3.604-65.825Q-3.282-65.825-3.014-66.018Q-2.746-66.211-2.657-66.526Q-2.650-66.567-2.575-66.581L-2.483-66.581Q-2.401-66.557-2.401-66.485Q-2.401-66.478-2.407-66.451Q-2.520-66.054-2.891-65.815Q-3.262-65.576-3.686-65.576Q-4.123-65.576-4.523-65.784Q-4.923-65.993-5.162-66.360Q-5.402-66.727-5.402-67.179M-4.732-67.449L-2.917-67.449Q-2.917-67.726-3.014-67.978Q-3.112-68.231-3.310-68.387Q-3.508-68.542-3.792-68.542Q-4.069-68.542-4.282-68.384Q-4.496-68.225-4.614-67.970Q-4.732-67.715-4.732-67.449\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-17.67 5.278)\">\u003Cpath d=\"M-26.529-57.644L-29.201-57.644Q-29.246-57.644-29.273-57.671Q-29.301-57.699-29.301-57.743L-29.301-57.811Q-29.301-57.852-29.273-57.883L-27.164-60.436L-27.803-60.436Q-28.227-60.436-28.460-60.370Q-28.692-60.303-28.812-60.093Q-28.931-59.883-28.931-59.462L-29.195-59.462L-29.113-60.662L-26.522-60.662Q-26.481-60.662-26.452-60.635Q-26.423-60.607-26.423-60.563L-26.423-60.515Q-26.423-60.471-26.447-60.443L-28.552-57.897L-27.872-57.897Q-27.544-57.897-27.327-57.933Q-27.110-57.969-26.942-58.119Q-26.802-58.256-26.749-58.480Q-26.696-58.704-26.669-59.032L-26.402-59.032L-26.529-57.644M-23.582-55.894Q-24.133-56.294-24.503-56.849Q-24.874-57.405-25.055-58.051Q-25.237-58.697-25.237-59.394Q-25.237-59.907-25.136-60.402Q-25.035-60.898-24.830-61.349Q-24.625-61.800-24.312-62.192Q-23.999-62.583-23.582-62.887Q-23.572-62.891-23.565-62.892Q-23.558-62.894-23.548-62.894L-23.480-62.894Q-23.446-62.894-23.423-62.870Q-23.401-62.846-23.401-62.809Q-23.401-62.764-23.428-62.747Q-23.777-62.446-24.030-62.062Q-24.283-61.677-24.435-61.236Q-24.587-60.795-24.659-60.339Q-24.731-59.883-24.731-59.394Q-24.731-58.393-24.421-57.506Q-24.112-56.619-23.428-56.034Q-23.401-56.017-23.401-55.973Q-23.401-55.935-23.423-55.911Q-23.446-55.887-23.480-55.887L-23.548-55.887Q-23.555-55.891-23.564-55.892Q-23.572-55.894-23.582-55.894M-22.065-58.485L-22.065-60.382L-22.704-60.382L-22.704-60.604Q-22.386-60.604-22.169-60.814Q-21.952-61.024-21.851-61.334Q-21.750-61.643-21.750-61.951L-21.484-61.951L-21.484-60.662L-20.407-60.662L-20.407-60.382L-21.484-60.382L-21.484-58.498Q-21.484-58.222-21.379-58.023Q-21.275-57.825-21.015-57.825Q-20.858-57.825-20.752-57.929Q-20.646-58.034-20.597-58.187Q-20.547-58.341-20.547-58.498L-20.547-58.912L-20.281-58.912L-20.281-58.485Q-20.281-58.259-20.380-58.049Q-20.479-57.839-20.663-57.707Q-20.848-57.576-21.077-57.576Q-21.514-57.576-21.790-57.813Q-22.065-58.051-22.065-58.485M-17.587-58.898L-19.645-58.898L-19.645-59.401L-17.587-59.401L-17.587-58.898M-13.773-57.644L-16.302-57.644L-16.302-57.924Q-15.335-57.924-15.335-58.133L-15.335-61.752Q-15.728-61.564-16.350-61.564L-16.350-61.845Q-15.933-61.845-15.569-61.946Q-15.205-62.046-14.949-62.292L-14.822-62.292Q-14.757-62.275-14.740-62.207L-14.740-58.133Q-14.740-57.924-13.773-57.924L-13.773-57.644M-12.481-55.887L-12.549-55.887Q-12.583-55.887-12.605-55.913Q-12.628-55.938-12.628-55.973Q-12.628-56.017-12.597-56.034Q-12.241-56.338-11.992-56.728Q-11.742-57.118-11.590-57.550Q-11.438-57.982-11.368-58.451Q-11.298-58.919-11.298-59.394Q-11.298-59.873-11.368-60.339Q-11.438-60.806-11.592-61.241Q-11.746-61.677-11.997-62.065Q-12.248-62.453-12.597-62.747Q-12.628-62.764-12.628-62.809Q-12.628-62.843-12.605-62.868Q-12.583-62.894-12.549-62.894L-12.481-62.894Q-12.470-62.894-12.462-62.892Q-12.453-62.891-12.443-62.887Q-11.900-62.487-11.527-61.934Q-11.155-61.380-10.973-60.734Q-10.792-60.088-10.792-59.394Q-10.792-58.693-10.973-58.046Q-11.155-57.398-11.529-56.844Q-11.903-56.290-12.443-55.894Q-12.453-55.894-12.462-55.892Q-12.470-55.891-12.481-55.887\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M14.187-43.417h78.846V-71.87H14.187Z\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(54.958 5.556)\">\u003Cpath d=\"M-16.333-67.155Q-16.333-67.493-16.192-67.784Q-16.052-68.074-15.808-68.288Q-15.564-68.501-15.259-68.616Q-14.955-68.730-14.630-68.730Q-14.360-68.730-14.097-68.631Q-13.834-68.532-13.643-68.354L-13.643-69.752Q-13.643-70.022-13.750-70.084Q-13.858-70.145-14.169-70.145L-14.169-70.426L-13.092-70.501L-13.092-66.317Q-13.092-66.129-13.038-66.046Q-12.983-65.962-12.882-65.943Q-12.781-65.924-12.566-65.924L-12.566-65.644L-13.673-65.576L-13.673-65.993Q-14.090-65.576-14.716-65.576Q-15.147-65.576-15.519-65.788Q-15.892-65.999-16.112-66.360Q-16.333-66.721-16.333-67.155M-14.658-65.798Q-14.449-65.798-14.263-65.870Q-14.077-65.941-13.923-66.078Q-13.769-66.215-13.673-66.393L-13.673-68.002Q-13.759-68.149-13.904-68.269Q-14.049-68.389-14.219-68.448Q-14.388-68.508-14.569-68.508Q-15.129-68.508-15.398-68.119Q-15.666-67.729-15.666-67.148Q-15.666-66.577-15.432-66.187Q-15.198-65.798-14.658-65.798M-11.958-67.179Q-11.958-67.500-11.833-67.789Q-11.708-68.078-11.482-68.301Q-11.257-68.525-10.961-68.645Q-10.666-68.765-10.348-68.765Q-10.020-68.765-9.758-68.665Q-9.497-68.566-9.321-68.384Q-9.145-68.201-9.051-67.943Q-8.957-67.685-8.957-67.353Q-8.957-67.261-9.039-67.240L-11.294-67.240L-11.294-67.179Q-11.294-66.591-11.011-66.208Q-10.727-65.825-10.160-65.825Q-9.838-65.825-9.570-66.018Q-9.302-66.211-9.213-66.526Q-9.206-66.567-9.131-66.581L-9.039-66.581Q-8.957-66.557-8.957-66.485Q-8.957-66.478-8.963-66.451Q-9.076-66.054-9.447-65.815Q-9.818-65.576-10.242-65.576Q-10.679-65.576-11.079-65.784Q-11.479-65.993-11.718-66.360Q-11.958-66.727-11.958-67.179M-11.288-67.449L-9.473-67.449Q-9.473-67.726-9.570-67.978Q-9.668-68.231-9.866-68.387Q-10.064-68.542-10.348-68.542Q-10.625-68.542-10.838-68.384Q-11.052-68.225-11.170-67.970Q-11.288-67.715-11.288-67.449M-8.369-67.155Q-8.369-67.483-8.234-67.784Q-8.099-68.084-7.863-68.305Q-7.627-68.525-7.323-68.645Q-7.019-68.765-6.694-68.765Q-6.188-68.765-5.839-68.662Q-5.491-68.560-5.491-68.184Q-5.491-68.037-5.588-67.936Q-5.686-67.835-5.833-67.835Q-5.986-67.835-6.085-67.934Q-6.185-68.033-6.185-68.184Q-6.185-68.372-6.044-68.464Q-6.246-68.515-6.687-68.515Q-7.043-68.515-7.272-68.319Q-7.501-68.122-7.601-67.813Q-7.702-67.503-7.702-67.155Q-7.702-66.806-7.576-66.500Q-7.449-66.194-7.195-66.010Q-6.940-65.825-6.584-65.825Q-6.362-65.825-6.178-65.909Q-5.993-65.993-5.858-66.148Q-5.723-66.304-5.665-66.512Q-5.651-66.567-5.597-66.567L-5.484-66.567Q-5.453-66.567-5.431-66.543Q-5.409-66.519-5.409-66.485L-5.409-66.464Q-5.494-66.177-5.682-65.979Q-5.870-65.781-6.135-65.678Q-6.400-65.576-6.694-65.576Q-7.125-65.576-7.512-65.782Q-7.900-65.989-8.135-66.352Q-8.369-66.714-8.369-67.155M-4.763-66.372Q-4.763-66.704-4.539-66.931Q-4.315-67.158-3.971-67.286Q-3.628-67.415-3.255-67.467Q-2.883-67.520-2.579-67.520L-2.579-67.773Q-2.579-67.978-2.686-68.158Q-2.794-68.337-2.975-68.440Q-3.156-68.542-3.365-68.542Q-3.772-68.542-4.007-68.450Q-3.918-68.413-3.872-68.329Q-3.826-68.245-3.826-68.143Q-3.826-68.047-3.872-67.968Q-3.918-67.890-3.999-67.845Q-4.079-67.801-4.168-67.801Q-4.318-67.801-4.419-67.898Q-4.520-67.996-4.520-68.143Q-4.520-68.765-3.365-68.765Q-3.153-68.765-2.903-68.701Q-2.654-68.638-2.452-68.519Q-2.251-68.399-2.124-68.214Q-1.998-68.030-1.998-67.787L-1.998-66.211Q-1.998-66.095-1.936-65.999Q-1.875-65.904-1.762-65.904Q-1.652-65.904-1.587-65.998Q-1.522-66.092-1.522-66.211L-1.522-66.659L-1.256-66.659L-1.256-66.211Q-1.256-65.941-1.483-65.776Q-1.710-65.610-1.991-65.610Q-2.199-65.610-2.336-65.764Q-2.473-65.917-2.497-66.133Q-2.644-65.866-2.926-65.721Q-3.208-65.576-3.532-65.576Q-3.809-65.576-4.093-65.651Q-4.376-65.726-4.570-65.905Q-4.763-66.085-4.763-66.372M-4.147-66.372Q-4.147-66.198-4.047-66.068Q-3.946-65.938-3.790-65.868Q-3.635-65.798-3.471-65.798Q-3.252-65.798-3.043-65.895Q-2.835-65.993-2.707-66.174Q-2.579-66.355-2.579-66.581L-2.579-67.309Q-2.903-67.309-3.269-67.218Q-3.635-67.127-3.891-66.915Q-4.147-66.704-4.147-66.372\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(54.958 5.556)\">\u003Cpath d=\"M-0.699-64.509Q-0.569-64.441-0.432-64.441Q-0.261-64.441-0.111-64.530Q0.040-64.619 0.151-64.764Q0.262-64.909 0.340-65.077L0.604-65.644L-0.565-68.170Q-0.640-68.317-0.770-68.349Q-0.900-68.382-1.133-68.382L-1.133-68.662L0.388-68.662L0.388-68.382Q0.040-68.382 0.040-68.235Q0.043-68.214 0.045-68.197Q0.047-68.180 0.047-68.170L0.904-66.311L1.677-67.982Q1.711-68.050 1.711-68.129Q1.711-68.242 1.627-68.312Q1.544-68.382 1.431-68.382L1.431-68.662L2.627-68.662L2.627-68.382Q2.408-68.382 2.236-68.278Q2.063-68.173 1.971-67.982L0.634-65.077Q0.464-64.707 0.194-64.461Q-0.077-64.215-0.432-64.215Q-0.702-64.215-0.921-64.381Q-1.140-64.547-1.140-64.810Q-1.140-64.947-1.047-65.036Q-0.955-65.124-0.815-65.124Q-0.678-65.124-0.589-65.036Q-0.500-64.947-0.500-64.810Q-0.500-64.707-0.553-64.629Q-0.606-64.550-0.699-64.509\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(54.958 5.556)\">\u003Cpath d=\"M6.669-65.644L6.402-65.644L6.402-69.752Q6.402-70.022 6.295-70.084Q6.187-70.145 5.876-70.145L5.876-70.426L6.956-70.501L6.956-68.331Q7.165-68.522 7.450-68.626Q7.736-68.730 8.033-68.730Q8.351-68.730 8.648-68.609Q8.945-68.488 9.168-68.272Q9.390-68.057 9.516-67.772Q9.643-67.486 9.643-67.155Q9.643-66.710 9.403-66.346Q9.164-65.982 8.771-65.779Q8.378-65.576 7.934-65.576Q7.739-65.576 7.549-65.632Q7.360-65.688 7.199-65.793Q7.038-65.897 6.898-66.058L6.669-65.644M6.984-67.989L6.984-66.372Q7.120-66.112 7.361-65.955Q7.602-65.798 7.879-65.798Q8.173-65.798 8.385-65.905Q8.597-66.013 8.730-66.205Q8.863-66.396 8.922-66.635Q8.980-66.874 8.980-67.155Q8.980-67.514 8.886-67.818Q8.792-68.122 8.564-68.315Q8.337-68.508 7.971-68.508Q7.671-68.508 7.404-68.372Q7.137-68.235 6.984-67.989\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(54.958 5.556)\">\u003Cpath d=\"M10.398-64.509Q10.528-64.441 10.665-64.441Q10.836-64.441 10.986-64.530Q11.137-64.619 11.248-64.764Q11.359-64.909 11.437-65.077L11.701-65.644L10.532-68.170Q10.457-68.317 10.327-68.349Q10.197-68.382 9.964-68.382L9.964-68.662L11.485-68.662L11.485-68.382Q11.137-68.382 11.137-68.235Q11.140-68.214 11.142-68.197Q11.144-68.180 11.144-68.170L12.001-66.311L12.774-67.982Q12.808-68.050 12.808-68.129Q12.808-68.242 12.724-68.312Q12.641-68.382 12.528-68.382L12.528-68.662L13.724-68.662L13.724-68.382Q13.505-68.382 13.333-68.278Q13.160-68.173 13.068-67.982L11.731-65.077Q11.561-64.707 11.291-64.461Q11.020-64.215 10.665-64.215Q10.395-64.215 10.176-64.381Q9.957-64.547 9.957-64.810Q9.957-64.947 10.050-65.036Q10.142-65.124 10.282-65.124Q10.419-65.124 10.508-65.036Q10.597-64.947 10.597-64.810Q10.597-64.707 10.544-64.629Q10.491-64.550 10.398-64.509\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(54.958 5.556)\">\u003Cpath d=\"M-37.166-57.111Q-37.166-57.357-36.969-57.541Q-36.772-57.726-36.516-57.805Q-36.653-57.917-36.725-58.078Q-36.796-58.239-36.796-58.420Q-36.796-58.741-36.585-58.987Q-36.919-59.285-36.919-59.695Q-36.919-60.156-36.530-60.443Q-36.140-60.730-35.662-60.730Q-35.190-60.730-34.855-60.484Q-34.681-60.638-34.470-60.720Q-34.260-60.802-34.031-60.802Q-33.867-60.802-33.746-60.695Q-33.625-60.587-33.625-60.423Q-33.625-60.327-33.696-60.255Q-33.768-60.184-33.860-60.184Q-33.960-60.184-34.030-60.257Q-34.100-60.331-34.100-60.430Q-34.100-60.484-34.086-60.515L-34.079-60.529Q-34.072-60.549-34.064-60.560Q-34.055-60.570-34.052-60.577Q-34.407-60.577-34.694-60.354Q-34.407-60.061-34.407-59.695Q-34.407-59.380-34.592-59.148Q-34.776-58.915-35.065-58.787Q-35.354-58.659-35.662-58.659Q-35.863-58.659-36.055-58.709Q-36.246-58.758-36.424-58.868Q-36.516-58.741-36.516-58.598Q-36.516-58.416-36.388-58.281Q-36.260-58.146-36.075-58.146L-35.443-58.146Q-34.995-58.146-34.626-58.075Q-34.257-58.003-33.997-57.774Q-33.737-57.545-33.737-57.111Q-33.737-56.790-34.033-56.588Q-34.329-56.386-34.732-56.297Q-35.135-56.208-35.450-56.208Q-35.768-56.208-36.171-56.297Q-36.574-56.386-36.870-56.588Q-37.166-56.790-37.166-57.111M-36.711-57.111Q-36.711-56.882-36.492-56.733Q-36.273-56.584-35.981-56.516Q-35.689-56.448-35.450-56.448Q-35.286-56.448-35.077-56.484Q-34.869-56.519-34.662-56.600Q-34.455-56.680-34.324-56.808Q-34.192-56.936-34.192-57.111Q-34.192-57.463-34.573-57.557Q-34.954-57.651-35.457-57.651L-36.075-57.651Q-36.314-57.651-36.513-57.500Q-36.711-57.350-36.711-57.111M-35.662-58.898Q-34.995-58.898-34.995-59.695Q-34.995-60.495-35.662-60.495Q-36.332-60.495-36.332-59.695Q-36.332-58.898-35.662-58.898M-33.085-58.372Q-33.085-58.704-32.861-58.931Q-32.637-59.158-32.293-59.286Q-31.950-59.415-31.577-59.467Q-31.205-59.520-30.900-59.520L-30.900-59.773Q-30.900-59.978-31.008-60.158Q-31.116-60.337-31.297-60.440Q-31.478-60.542-31.687-60.542Q-32.093-60.542-32.329-60.450Q-32.240-60.413-32.194-60.329Q-32.148-60.245-32.148-60.143Q-32.148-60.047-32.194-59.968Q-32.240-59.890-32.321-59.845Q-32.401-59.801-32.490-59.801Q-32.640-59.801-32.741-59.898Q-32.842-59.996-32.842-60.143Q-32.842-60.765-31.687-60.765Q-31.475-60.765-31.225-60.701Q-30.976-60.638-30.774-60.519Q-30.572-60.399-30.446-60.214Q-30.319-60.030-30.319-59.787L-30.319-58.211Q-30.319-58.095-30.258-57.999Q-30.196-57.904-30.084-57.904Q-29.974-57.904-29.909-57.998Q-29.844-58.092-29.844-58.211L-29.844-58.659L-29.578-58.659L-29.578-58.211Q-29.578-57.941-29.805-57.776Q-30.032-57.610-30.313-57.610Q-30.521-57.610-30.658-57.764Q-30.794-57.917-30.818-58.133Q-30.965-57.866-31.247-57.721Q-31.529-57.576-31.854-57.576Q-32.131-57.576-32.415-57.651Q-32.698-57.726-32.891-57.905Q-33.085-58.085-33.085-58.372M-32.469-58.372Q-32.469-58.198-32.368-58.068Q-32.268-57.938-32.112-57.868Q-31.957-57.798-31.793-57.798Q-31.574-57.798-31.365-57.895Q-31.157-57.993-31.029-58.174Q-30.900-58.355-30.900-58.581L-30.900-59.309Q-31.225-59.309-31.591-59.218Q-31.957-59.127-32.213-58.915Q-32.469-58.704-32.469-58.372M-27.479-57.644L-29.113-57.644L-29.113-57.924Q-28.884-57.924-28.735-57.958Q-28.586-57.993-28.586-58.133L-28.586-59.982Q-28.586-60.252-28.694-60.313Q-28.802-60.375-29.113-60.375L-29.113-60.655L-28.053-60.730L-28.053-60.081Q-27.882-60.389-27.578-60.560Q-27.274-60.730-26.929-60.730Q-26.529-60.730-26.252-60.590Q-25.975-60.450-25.890-60.102Q-25.722-60.395-25.423-60.563Q-25.124-60.730-24.779-60.730Q-24.273-60.730-23.989-60.507Q-23.706-60.283-23.706-59.787L-23.706-58.133Q-23.706-57.996-23.557-57.960Q-23.408-57.924-23.183-57.924L-23.183-57.644L-24.813-57.644L-24.813-57.924Q-24.587-57.924-24.437-57.960Q-24.287-57.996-24.287-58.133L-24.287-59.773Q-24.287-60.108-24.406-60.308Q-24.526-60.508-24.840-60.508Q-25.110-60.508-25.345-60.372Q-25.579-60.235-25.717-60.001Q-25.856-59.767-25.856-59.493L-25.856-58.133Q-25.856-57.996-25.707-57.960Q-25.558-57.924-25.333-57.924L-25.333-57.644L-26.963-57.644L-26.963-57.924Q-26.734-57.924-26.585-57.958Q-26.437-57.993-26.437-58.133L-26.437-59.773Q-26.437-60.108-26.556-60.308Q-26.676-60.508-26.990-60.508Q-27.260-60.508-27.494-60.372Q-27.729-60.235-27.867-60.001Q-28.005-59.767-28.005-59.493L-28.005-58.133Q-28.005-57.996-27.855-57.960Q-27.705-57.924-27.479-57.924L-27.479-57.644M-20.913-57.644L-22.547-57.644L-22.547-57.924Q-22.318-57.924-22.169-57.958Q-22.021-57.993-22.021-58.133L-22.021-59.982Q-22.021-60.252-22.128-60.313Q-22.236-60.375-22.547-60.375L-22.547-60.655L-21.487-60.730L-21.487-60.081Q-21.316-60.389-21.012-60.560Q-20.708-60.730-20.363-60.730Q-19.963-60.730-19.686-60.590Q-19.409-60.450-19.324-60.102Q-19.156-60.395-18.857-60.563Q-18.558-60.730-18.213-60.730Q-17.707-60.730-17.423-60.507Q-17.140-60.283-17.140-59.787L-17.140-58.133Q-17.140-57.996-16.991-57.960Q-16.842-57.924-16.617-57.924L-16.617-57.644L-18.247-57.644L-18.247-57.924Q-18.022-57.924-17.871-57.960Q-17.721-57.996-17.721-58.133L-17.721-59.773Q-17.721-60.108-17.840-60.308Q-17.960-60.508-18.274-60.508Q-18.544-60.508-18.779-60.372Q-19.013-60.235-19.151-60.001Q-19.290-59.767-19.290-59.493L-19.290-58.133Q-19.290-57.996-19.141-57.960Q-18.992-57.924-18.767-57.924L-18.767-57.644L-20.397-57.644L-20.397-57.924Q-20.168-57.924-20.019-57.958Q-19.871-57.993-19.871-58.133L-19.871-59.773Q-19.871-60.108-19.990-60.308Q-20.110-60.508-20.424-60.508Q-20.694-60.508-20.929-60.372Q-21.163-60.235-21.301-60.001Q-21.439-59.767-21.439-59.493L-21.439-58.133Q-21.439-57.996-21.289-57.960Q-21.139-57.924-20.913-57.924L-20.913-57.644M-15.971-58.372Q-15.971-58.704-15.747-58.931Q-15.523-59.158-15.179-59.286Q-14.836-59.415-14.463-59.467Q-14.091-59.520-13.787-59.520L-13.787-59.773Q-13.787-59.978-13.894-60.158Q-14.002-60.337-14.183-60.440Q-14.364-60.542-14.573-60.542Q-14.980-60.542-15.215-60.450Q-15.127-60.413-15.080-60.329Q-15.034-60.245-15.034-60.143Q-15.034-60.047-15.080-59.968Q-15.127-59.890-15.207-59.845Q-15.287-59.801-15.376-59.801Q-15.526-59.801-15.627-59.898Q-15.728-59.996-15.728-60.143Q-15.728-60.765-14.573-60.765Q-14.361-60.765-14.111-60.701Q-13.862-60.638-13.660-60.519Q-13.459-60.399-13.332-60.214Q-13.206-60.030-13.206-59.787L-13.206-58.211Q-13.206-58.095-13.144-57.999Q-13.083-57.904-12.970-57.904Q-12.860-57.904-12.795-57.998Q-12.731-58.092-12.731-58.211L-12.731-58.659L-12.464-58.659L-12.464-58.211Q-12.464-57.941-12.691-57.776Q-12.918-57.610-13.199-57.610Q-13.407-57.610-13.544-57.764Q-13.681-57.917-13.705-58.133Q-13.852-57.866-14.134-57.721Q-14.416-57.576-14.740-57.576Q-15.017-57.576-15.301-57.651Q-15.585-57.726-15.778-57.905Q-15.971-58.085-15.971-58.372M-15.356-58.372Q-15.356-58.198-15.255-58.068Q-15.154-57.938-14.998-57.868Q-14.843-57.798-14.679-57.798Q-14.460-57.798-14.252-57.895Q-14.043-57.993-13.915-58.174Q-13.787-58.355-13.787-58.581L-13.787-59.309Q-14.111-59.309-14.477-59.218Q-14.843-59.127-15.099-58.915Q-15.356-58.704-15.356-58.372M-9.918-55.894Q-10.468-56.294-10.839-56.849Q-11.210-57.405-11.391-58.051Q-11.572-58.697-11.572-59.394Q-11.572-59.907-11.471-60.402Q-11.370-60.898-11.165-61.349Q-10.960-61.800-10.647-62.192Q-10.335-62.583-9.918-62.887Q-9.907-62.891-9.900-62.892Q-9.894-62.894-9.883-62.894L-9.815-62.894Q-9.781-62.894-9.759-62.870Q-9.736-62.846-9.736-62.809Q-9.736-62.764-9.764-62.747Q-10.112-62.446-10.365-62.062Q-10.618-61.677-10.770-61.236Q-10.922-60.795-10.994-60.339Q-11.066-59.883-11.066-59.394Q-11.066-58.393-10.757-57.506Q-10.447-56.619-9.764-56.034Q-9.736-56.017-9.736-55.973Q-9.736-55.935-9.759-55.911Q-9.781-55.887-9.815-55.887L-9.883-55.887Q-9.890-55.891-9.899-55.892Q-9.907-55.894-9.918-55.894M-8.400-58.485L-8.400-60.382L-9.039-60.382L-9.039-60.604Q-8.721-60.604-8.504-60.814Q-8.287-61.024-8.186-61.334Q-8.085-61.643-8.085-61.951L-7.819-61.951L-7.819-60.662L-6.742-60.662L-6.742-60.382L-7.819-60.382L-7.819-58.498Q-7.819-58.222-7.715-58.023Q-7.610-57.825-7.351-57.825Q-7.193-57.825-7.087-57.929Q-6.981-58.034-6.932-58.187Q-6.882-58.341-6.882-58.498L-6.882-58.912L-6.616-58.912L-6.616-58.485Q-6.616-58.259-6.715-58.049Q-6.814-57.839-6.999-57.707Q-7.183-57.576-7.412-57.576Q-7.850-57.576-8.125-57.813Q-8.400-58.051-8.400-58.485M-5.484-55.887L-5.553-55.887Q-5.587-55.887-5.609-55.913Q-5.631-55.938-5.631-55.973Q-5.631-56.017-5.601-56.034Q-5.245-56.338-4.996-56.728Q-4.746-57.118-4.594-57.550Q-4.442-57.982-4.372-58.451Q-4.302-58.919-4.302-59.394Q-4.302-59.873-4.372-60.339Q-4.442-60.806-4.596-61.241Q-4.750-61.677-5.001-62.065Q-5.252-62.453-5.601-62.747Q-5.631-62.764-5.631-62.809Q-5.631-62.843-5.609-62.868Q-5.587-62.894-5.553-62.894L-5.484-62.894Q-5.474-62.894-5.466-62.892Q-5.457-62.891-5.447-62.887Q-4.903-62.487-4.531-61.934Q-4.158-61.380-3.977-60.734Q-3.796-60.088-3.796-59.394Q-3.796-58.693-3.977-58.046Q-4.158-57.398-4.533-56.844Q-4.907-56.290-5.447-55.894Q-5.457-55.894-5.466-55.892Q-5.474-55.891-5.484-55.887\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(54.958 5.556)\">\u003Cpath d=\"M1.710-57.644L0.107-57.644L0.107-57.924Q0.333-57.924 0.482-57.958Q0.630-57.993 0.630-58.133L0.630-61.752Q0.630-62.022 0.523-62.084Q0.415-62.145 0.107-62.145L0.107-62.426L1.184-62.501L1.184-58.133Q1.184-57.996 1.334-57.960Q1.485-57.924 1.710-57.924L1.710-57.644M2.363-58.372Q2.363-58.704 2.587-58.931Q2.811-59.158 3.155-59.286Q3.498-59.415 3.871-59.467Q4.243-59.520 4.547-59.520L4.547-59.773Q4.547-59.978 4.440-60.158Q4.332-60.337 4.151-60.440Q3.970-60.542 3.761-60.542Q3.354-60.542 3.119-60.450Q3.207-60.413 3.254-60.329Q3.300-60.245 3.300-60.143Q3.300-60.047 3.254-59.968Q3.207-59.890 3.127-59.845Q3.047-59.801 2.958-59.801Q2.808-59.801 2.707-59.898Q2.606-59.996 2.606-60.143Q2.606-60.765 3.761-60.765Q3.973-60.765 4.223-60.701Q4.472-60.638 4.674-60.519Q4.875-60.399 5.002-60.214Q5.128-60.030 5.128-59.787L5.128-58.211Q5.128-58.095 5.190-57.999Q5.251-57.904 5.364-57.904Q5.474-57.904 5.539-57.998Q5.603-58.092 5.603-58.211L5.603-58.659L5.870-58.659L5.870-58.211Q5.870-57.941 5.643-57.776Q5.415-57.610 5.135-57.610Q4.927-57.610 4.790-57.764Q4.653-57.917 4.629-58.133Q4.482-57.866 4.200-57.721Q3.918-57.576 3.594-57.576Q3.317-57.576 3.033-57.651Q2.749-57.726 2.556-57.905Q2.363-58.085 2.363-58.372M2.978-58.372Q2.978-58.198 3.079-58.068Q3.180-57.938 3.336-57.868Q3.491-57.798 3.655-57.798Q3.874-57.798 4.082-57.895Q4.291-57.993 4.419-58.174Q4.547-58.355 4.547-58.581L4.547-59.309Q4.223-59.309 3.857-59.218Q3.491-59.127 3.235-58.915Q2.978-58.704 2.978-58.372M7.969-57.644L6.335-57.644L6.335-57.924Q6.564-57.924 6.713-57.958Q6.861-57.993 6.861-58.133L6.861-59.982Q6.861-60.252 6.754-60.313Q6.646-60.375 6.335-60.375L6.335-60.655L7.395-60.730L7.395-60.081Q7.565-60.389 7.870-60.560Q8.174-60.730 8.519-60.730Q8.919-60.730 9.196-60.590Q9.473-60.450 9.558-60.102Q9.726-60.395 10.025-60.563Q10.324-60.730 10.669-60.730Q11.175-60.730 11.458-60.507Q11.742-60.283 11.742-59.787L11.742-58.133Q11.742-57.996 11.891-57.960Q12.040-57.924 12.265-57.924L12.265-57.644L10.635-57.644L10.635-57.924Q10.860-57.924 11.011-57.960Q11.161-57.996 11.161-58.133L11.161-59.773Q11.161-60.108 11.041-60.308Q10.922-60.508 10.607-60.508Q10.337-60.508 10.103-60.372Q9.869-60.235 9.731-60.001Q9.592-59.767 9.592-59.493L9.592-58.133Q9.592-57.996 9.741-57.960Q9.890-57.924 10.115-57.924L10.115-57.644L8.485-57.644L8.485-57.924Q8.714-57.924 8.863-57.958Q9.011-57.993 9.011-58.133L9.011-59.773Q9.011-60.108 8.892-60.308Q8.772-60.508 8.457-60.508Q8.187-60.508 7.953-60.372Q7.719-60.235 7.581-60.001Q7.442-59.767 7.442-59.493L7.442-58.133Q7.442-57.996 7.593-57.960Q7.743-57.924 7.969-57.924\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(54.958 5.556)\">\u003Cpath d=\"M13.453-57.644L13.186-57.644L13.186-61.752Q13.186-62.022 13.079-62.084Q12.971-62.145 12.660-62.145L12.660-62.426L13.740-62.501L13.740-60.331Q13.949-60.522 14.234-60.626Q14.520-60.730 14.817-60.730Q15.135-60.730 15.432-60.609Q15.729-60.488 15.952-60.272Q16.174-60.057 16.300-59.772Q16.427-59.486 16.427-59.155Q16.427-58.710 16.187-58.346Q15.948-57.982 15.555-57.779Q15.162-57.576 14.718-57.576Q14.523-57.576 14.333-57.632Q14.144-57.688 13.983-57.793Q13.822-57.897 13.682-58.058L13.453-57.644M13.768-59.989L13.768-58.372Q13.904-58.112 14.145-57.955Q14.386-57.798 14.663-57.798Q14.957-57.798 15.169-57.905Q15.381-58.013 15.514-58.205Q15.647-58.396 15.706-58.635Q15.764-58.874 15.764-59.155Q15.764-59.514 15.670-59.818Q15.576-60.122 15.348-60.315Q15.121-60.508 14.755-60.508Q14.455-60.508 14.188-60.372Q13.921-60.235 13.768-59.989\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(54.958 5.556)\">\u003Cpath d=\"M17.278-59.155Q17.278-59.493 17.419-59.784Q17.559-60.074 17.803-60.288Q18.047-60.501 18.352-60.616Q18.656-60.730 18.981-60.730Q19.251-60.730 19.514-60.631Q19.777-60.532 19.968-60.354L19.968-61.752Q19.968-62.022 19.861-62.084Q19.753-62.145 19.442-62.145L19.442-62.426L20.519-62.501L20.519-58.317Q20.519-58.129 20.573-58.046Q20.628-57.962 20.729-57.943Q20.830-57.924 21.045-57.924L21.045-57.644L19.938-57.576L19.938-57.993Q19.521-57.576 18.895-57.576Q18.464-57.576 18.092-57.788Q17.719-57.999 17.499-58.360Q17.278-58.721 17.278-59.155M18.953-57.798Q19.162-57.798 19.348-57.870Q19.534-57.941 19.688-58.078Q19.842-58.215 19.938-58.393L19.938-60.002Q19.852-60.149 19.707-60.269Q19.562-60.389 19.392-60.448Q19.223-60.508 19.042-60.508Q18.482-60.508 18.213-60.119Q17.945-59.729 17.945-59.148Q17.945-58.577 18.179-58.187Q18.413-57.798 18.953-57.798M21.753-58.372Q21.753-58.704 21.976-58.931Q22.200-59.158 22.544-59.286Q22.887-59.415 23.260-59.467Q23.632-59.520 23.937-59.520L23.937-59.773Q23.937-59.978 23.829-60.158Q23.721-60.337 23.540-60.440Q23.359-60.542 23.151-60.542Q22.744-60.542 22.508-60.450Q22.597-60.413 22.643-60.329Q22.689-60.245 22.689-60.143Q22.689-60.047 22.643-59.968Q22.597-59.890 22.516-59.845Q22.436-59.801 22.347-59.801Q22.197-59.801 22.096-59.898Q21.995-59.996 21.995-60.143Q21.995-60.765 23.151-60.765Q23.362-60.765 23.612-60.701Q23.861-60.638 24.063-60.519Q24.265-60.399 24.391-60.214Q24.518-60.030 24.518-59.787L24.518-58.211Q24.518-58.095 24.579-57.999Q24.641-57.904 24.754-57.904Q24.863-57.904 24.928-57.998Q24.993-58.092 24.993-58.211L24.993-58.659L25.259-58.659L25.259-58.211Q25.259-57.941 25.032-57.776Q24.805-57.610 24.525-57.610Q24.316-57.610 24.179-57.764Q24.043-57.917 24.019-58.133Q23.872-57.866 23.590-57.721Q23.308-57.576 22.983-57.576Q22.706-57.576 22.422-57.651Q22.139-57.726 21.946-57.905Q21.753-58.085 21.753-58.372M22.368-58.372Q22.368-58.198 22.469-58.068Q22.569-57.938 22.725-57.868Q22.881-57.798 23.045-57.798Q23.263-57.798 23.472-57.895Q23.680-57.993 23.808-58.174Q23.937-58.355 23.937-58.581L23.937-59.309Q23.612-59.309 23.246-59.218Q22.881-59.127 22.624-58.915Q22.368-58.704 22.368-58.372M27.806-55.894Q27.256-56.294 26.885-56.849Q26.514-57.405 26.333-58.051Q26.151-58.697 26.151-59.394Q26.151-59.907 26.252-60.402Q26.353-60.898 26.558-61.349Q26.763-61.800 27.076-62.192Q27.389-62.583 27.806-62.887Q27.816-62.891 27.823-62.892Q27.830-62.894 27.840-62.894L27.908-62.894Q27.943-62.894 27.965-62.870Q27.987-62.846 27.987-62.809Q27.987-62.764 27.960-62.747Q27.611-62.446 27.358-62.062Q27.105-61.677 26.953-61.236Q26.801-60.795 26.729-60.339Q26.657-59.883 26.657-59.394Q26.657-58.393 26.967-57.506Q27.276-56.619 27.960-56.034Q27.987-56.017 27.987-55.973Q27.987-55.935 27.965-55.911Q27.943-55.887 27.908-55.887L27.840-55.887Q27.833-55.891 27.825-55.892Q27.816-55.894 27.806-55.894M29.323-58.485L29.323-60.382L28.684-60.382L28.684-60.604Q29.002-60.604 29.219-60.814Q29.436-61.024 29.537-61.334Q29.638-61.643 29.638-61.951L29.904-61.951L29.904-60.662L30.981-60.662L30.981-60.382L29.904-60.382L29.904-58.498Q29.904-58.222 30.009-58.023Q30.113-57.825 30.373-57.825Q30.530-57.825 30.636-57.929Q30.742-58.034 30.791-58.187Q30.841-58.341 30.841-58.498L30.841-58.912L31.108-58.912L31.108-58.485Q31.108-58.259 31.008-58.049Q30.909-57.839 30.725-57.707Q30.540-57.576 30.311-57.576Q29.874-57.576 29.599-57.813Q29.323-58.051 29.323-58.485M32.239-55.887L32.171-55.887Q32.136-55.887 32.114-55.913Q32.092-55.938 32.092-55.973Q32.092-56.017 32.123-56.034Q32.478-56.338 32.728-56.728Q32.977-57.118 33.129-57.550Q33.281-57.982 33.351-58.451Q33.422-58.919 33.422-59.394Q33.422-59.873 33.351-60.339Q33.281-60.806 33.128-61.241Q32.974-61.677 32.723-62.065Q32.471-62.453 32.123-62.747Q32.092-62.764 32.092-62.809Q32.092-62.843 32.114-62.868Q32.136-62.894 32.171-62.894L32.239-62.894Q32.249-62.894 32.258-62.892Q32.266-62.891 32.276-62.887Q32.820-62.487 33.193-61.934Q33.565-61.380 33.746-60.734Q33.927-60.088 33.927-59.394Q33.927-58.693 33.746-58.046Q33.565-57.398 33.191-56.844Q32.817-56.290 32.276-55.894Q32.266-55.894 32.258-55.892Q32.249-55.891 32.239-55.887\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M22.312 10.643h62.596V-17.81H22.312Z\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(68.962 59.616)\">\u003Cpath d=\"M-37.066-66.372Q-37.066-66.704-36.843-66.931Q-36.619-67.158-36.275-67.286Q-35.932-67.415-35.559-67.467Q-35.187-67.520-34.882-67.520L-34.882-67.773Q-34.882-67.978-34.990-68.158Q-35.098-68.337-35.279-68.440Q-35.460-68.542-35.668-68.542Q-36.075-68.542-36.311-68.450Q-36.222-68.413-36.176-68.329Q-36.130-68.245-36.130-68.143Q-36.130-68.047-36.176-67.968Q-36.222-67.890-36.303-67.845Q-36.383-67.801-36.472-67.801Q-36.622-67.801-36.723-67.898Q-36.824-67.996-36.824-68.143Q-36.824-68.765-35.668-68.765Q-35.457-68.765-35.207-68.701Q-34.958-68.638-34.756-68.519Q-34.554-68.399-34.428-68.214Q-34.301-68.030-34.301-67.787L-34.301-66.211Q-34.301-66.095-34.240-65.999Q-34.178-65.904-34.065-65.904Q-33.956-65.904-33.891-65.998Q-33.826-66.092-33.826-66.211L-33.826-66.659L-33.560-66.659L-33.560-66.211Q-33.560-65.941-33.787-65.776Q-34.014-65.610-34.294-65.610Q-34.503-65.610-34.640-65.764Q-34.776-65.917-34.800-66.133Q-34.947-65.866-35.229-65.721Q-35.511-65.576-35.836-65.576Q-36.113-65.576-36.397-65.651Q-36.680-65.726-36.873-65.905Q-37.066-66.085-37.066-66.372M-36.451-66.372Q-36.451-66.198-36.350-66.068Q-36.250-65.938-36.094-65.868Q-35.939-65.798-35.774-65.798Q-35.556-65.798-35.347-65.895Q-35.139-65.993-35.011-66.174Q-34.882-66.355-34.882-66.581L-34.882-67.309Q-35.207-67.309-35.573-67.218Q-35.939-67.127-36.195-66.915Q-36.451-66.704-36.451-66.372M-33.143-67.155Q-33.143-67.493-33.002-67.784Q-32.862-68.074-32.618-68.288Q-32.374-68.501-32.069-68.616Q-31.765-68.730-31.440-68.730Q-31.170-68.730-30.907-68.631Q-30.644-68.532-30.453-68.354L-30.453-69.752Q-30.453-70.022-30.560-70.084Q-30.668-70.145-30.979-70.145L-30.979-70.426L-29.902-70.501L-29.902-66.317Q-29.902-66.129-29.848-66.046Q-29.793-65.962-29.692-65.943Q-29.591-65.924-29.376-65.924L-29.376-65.644L-30.483-65.576L-30.483-65.993Q-30.900-65.576-31.526-65.576Q-31.957-65.576-32.329-65.788Q-32.702-65.999-32.922-66.360Q-33.143-66.721-33.143-67.155M-31.468-65.798Q-31.259-65.798-31.073-65.870Q-30.887-65.941-30.733-66.078Q-30.579-66.215-30.483-66.393L-30.483-68.002Q-30.569-68.149-30.714-68.269Q-30.859-68.389-31.029-68.448Q-31.198-68.508-31.379-68.508Q-31.939-68.508-32.208-68.119Q-32.476-67.729-32.476-67.148Q-32.476-66.577-32.242-66.187Q-32.008-65.798-31.468-65.798M-28.727-67.155Q-28.727-67.493-28.586-67.784Q-28.446-68.074-28.202-68.288Q-27.958-68.501-27.653-68.616Q-27.349-68.730-27.024-68.730Q-26.754-68.730-26.491-68.631Q-26.228-68.532-26.037-68.354L-26.037-69.752Q-26.037-70.022-26.144-70.084Q-26.252-70.145-26.563-70.145L-26.563-70.426L-25.486-70.501L-25.486-66.317Q-25.486-66.129-25.432-66.046Q-25.377-65.962-25.276-65.943Q-25.175-65.924-24.960-65.924L-24.960-65.644L-26.067-65.576L-26.067-65.993Q-26.484-65.576-27.110-65.576Q-27.541-65.576-27.913-65.788Q-28.286-65.999-28.506-66.360Q-28.727-66.721-28.727-67.155M-27.052-65.798Q-26.843-65.798-26.657-65.870Q-26.471-65.941-26.317-66.078Q-26.163-66.215-26.067-66.393L-26.067-68.002Q-26.153-68.149-26.298-68.269Q-26.443-68.389-26.613-68.448Q-26.782-68.508-26.963-68.508Q-27.523-68.508-27.792-68.119Q-28.060-67.729-28.060-67.148Q-28.060-66.577-27.826-66.187Q-27.592-65.798-27.052-65.798\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.962 59.616)\">\u003Cpath d=\"M-21.652-65.111Q-21.652-65.357-21.455-65.541Q-21.258-65.726-21.002-65.805Q-21.139-65.917-21.211-66.078Q-21.282-66.239-21.282-66.420Q-21.282-66.741-21.071-66.987Q-21.405-67.285-21.405-67.695Q-21.405-68.156-21.016-68.443Q-20.626-68.730-20.148-68.730Q-19.676-68.730-19.341-68.484Q-19.167-68.638-18.956-68.720Q-18.746-68.802-18.517-68.802Q-18.353-68.802-18.232-68.695Q-18.111-68.587-18.111-68.423Q-18.111-68.327-18.182-68.255Q-18.254-68.184-18.346-68.184Q-18.446-68.184-18.516-68.257Q-18.586-68.331-18.586-68.430Q-18.586-68.484-18.572-68.515L-18.565-68.529Q-18.558-68.549-18.550-68.560Q-18.541-68.570-18.538-68.577Q-18.893-68.577-19.180-68.354Q-18.893-68.061-18.893-67.695Q-18.893-67.380-19.078-67.148Q-19.262-66.915-19.551-66.787Q-19.840-66.659-20.148-66.659Q-20.349-66.659-20.541-66.709Q-20.732-66.758-20.910-66.868Q-21.002-66.741-21.002-66.598Q-21.002-66.416-20.874-66.281Q-20.746-66.146-20.561-66.146L-19.929-66.146Q-19.481-66.146-19.112-66.075Q-18.743-66.003-18.483-65.774Q-18.223-65.545-18.223-65.111Q-18.223-64.790-18.519-64.588Q-18.815-64.386-19.218-64.297Q-19.621-64.208-19.936-64.208Q-20.254-64.208-20.657-64.297Q-21.060-64.386-21.356-64.588Q-21.652-64.790-21.652-65.111M-21.197-65.111Q-21.197-64.882-20.978-64.733Q-20.759-64.584-20.467-64.516Q-20.175-64.448-19.936-64.448Q-19.772-64.448-19.563-64.484Q-19.355-64.519-19.148-64.600Q-18.941-64.680-18.810-64.808Q-18.678-64.936-18.678-65.111Q-18.678-65.463-19.059-65.557Q-19.440-65.651-19.943-65.651L-20.561-65.651Q-20.800-65.651-20.999-65.500Q-21.197-65.350-21.197-65.111M-20.148-66.898Q-19.481-66.898-19.481-67.695Q-19.481-68.495-20.148-68.495Q-20.818-68.495-20.818-67.695Q-20.818-66.898-20.148-66.898M-15.879-65.644L-17.615-65.644L-17.615-65.924Q-17.386-65.924-17.237-65.958Q-17.089-65.993-17.089-66.133L-17.089-67.982Q-17.089-68.252-17.196-68.313Q-17.304-68.375-17.615-68.375L-17.615-68.655L-16.586-68.730L-16.586-68.023Q-16.456-68.331-16.214-68.530Q-15.971-68.730-15.653-68.730Q-15.434-68.730-15.263-68.606Q-15.092-68.481-15.092-68.269Q-15.092-68.132-15.192-68.033Q-15.291-67.934-15.424-67.934Q-15.561-67.934-15.660-68.033Q-15.759-68.132-15.759-68.269Q-15.759-68.409-15.660-68.508Q-15.950-68.508-16.150-68.312Q-16.350-68.115-16.443-67.821Q-16.535-67.527-16.535-67.247L-16.535-66.133Q-16.535-65.924-15.879-65.924L-15.879-65.644M-14.450-66.372Q-14.450-66.704-14.226-66.931Q-14.002-67.158-13.659-67.286Q-13.315-67.415-12.943-67.467Q-12.570-67.520-12.266-67.520L-12.266-67.773Q-12.266-67.978-12.373-68.158Q-12.481-68.337-12.662-68.440Q-12.843-68.542-13.052-68.542Q-13.459-68.542-13.695-68.450Q-13.606-68.413-13.560-68.329Q-13.513-68.245-13.513-68.143Q-13.513-68.047-13.560-67.968Q-13.606-67.890-13.686-67.845Q-13.766-67.801-13.855-67.801Q-14.006-67.801-14.106-67.898Q-14.207-67.996-14.207-68.143Q-14.207-68.765-13.052-68.765Q-12.840-68.765-12.591-68.701Q-12.341-68.638-12.139-68.519Q-11.938-68.399-11.811-68.214Q-11.685-68.030-11.685-67.787L-11.685-66.211Q-11.685-66.095-11.623-65.999Q-11.562-65.904-11.449-65.904Q-11.340-65.904-11.275-65.998Q-11.210-66.092-11.210-66.211L-11.210-66.659L-10.943-66.659L-10.943-66.211Q-10.943-65.941-11.170-65.776Q-11.398-65.610-11.678-65.610Q-11.886-65.610-12.023-65.764Q-12.160-65.917-12.184-66.133Q-12.331-65.866-12.613-65.721Q-12.895-65.576-13.219-65.576Q-13.496-65.576-13.780-65.651Q-14.064-65.726-14.257-65.905Q-14.450-66.085-14.450-66.372M-13.835-66.372Q-13.835-66.198-13.734-66.068Q-13.633-65.938-13.477-65.868Q-13.322-65.798-13.158-65.798Q-12.939-65.798-12.731-65.895Q-12.522-65.993-12.394-66.174Q-12.266-66.355-12.266-66.581L-12.266-67.309Q-12.591-67.309-12.956-67.218Q-13.322-67.127-13.578-66.915Q-13.835-66.704-13.835-66.372M-10.526-67.155Q-10.526-67.493-10.386-67.784Q-10.246-68.074-10.001-68.288Q-9.757-68.501-9.453-68.616Q-9.149-68.730-8.824-68.730Q-8.554-68.730-8.291-68.631Q-8.028-68.532-7.836-68.354L-7.836-69.752Q-7.836-70.022-7.944-70.084Q-8.051-70.145-8.363-70.145L-8.363-70.426L-7.286-70.501L-7.286-66.317Q-7.286-66.129-7.231-66.046Q-7.176-65.962-7.076-65.943Q-6.975-65.924-6.759-65.924L-6.759-65.644L-7.867-65.576L-7.867-65.993Q-8.284-65.576-8.909-65.576Q-9.340-65.576-9.713-65.788Q-10.085-65.999-10.306-66.360Q-10.526-66.721-10.526-67.155M-8.851-65.798Q-8.643-65.798-8.456-65.870Q-8.270-65.941-8.116-66.078Q-7.963-66.215-7.867-66.393L-7.867-68.002Q-7.952-68.149-8.098-68.269Q-8.243-68.389-8.412-68.448Q-8.581-68.508-8.762-68.508Q-9.323-68.508-9.591-68.119Q-9.860-67.729-9.860-67.148Q-9.860-66.577-9.625-66.187Q-9.391-65.798-8.851-65.798M-4.493-65.644L-6.045-65.644L-6.045-65.924Q-5.820-65.924-5.671-65.958Q-5.522-65.993-5.522-66.133L-5.522-67.982Q-5.522-68.170-5.570-68.254Q-5.618-68.337-5.715-68.356Q-5.813-68.375-6.025-68.375L-6.025-68.655L-4.968-68.730L-4.968-66.133Q-4.968-65.993-4.837-65.958Q-4.705-65.924-4.493-65.924L-4.493-65.644M-5.765-69.951Q-5.765-70.122-5.642-70.241Q-5.519-70.361-5.348-70.361Q-5.180-70.361-5.057-70.241Q-4.934-70.122-4.934-69.951Q-4.934-69.776-5.057-69.653Q-5.180-69.530-5.348-69.530Q-5.519-69.530-5.642-69.653Q-5.765-69.776-5.765-69.951M-3.888-67.179Q-3.888-67.500-3.764-67.789Q-3.639-68.078-3.413-68.301Q-3.188-68.525-2.892-68.645Q-2.596-68.765-2.279-68.765Q-1.950-68.765-1.689-68.665Q-1.427-68.566-1.251-68.384Q-1.075-68.201-0.981-67.943Q-0.887-67.685-0.887-67.353Q-0.887-67.261-0.969-67.240L-3.225-67.240L-3.225-67.179Q-3.225-66.591-2.942-66.208Q-2.658-65.825-2.091-65.825Q-1.769-65.825-1.501-66.018Q-1.233-66.211-1.144-66.526Q-1.137-66.567-1.062-66.581L-0.969-66.581Q-0.887-66.557-0.887-66.485Q-0.887-66.478-0.894-66.451Q-1.007-66.054-1.378-65.815Q-1.749-65.576-2.173-65.576Q-2.610-65.576-3.010-65.784Q-3.410-65.993-3.649-66.360Q-3.888-66.727-3.888-67.179M-3.218-67.449L-1.404-67.449Q-1.404-67.726-1.501-67.978Q-1.598-68.231-1.797-68.387Q-1.995-68.542-2.279-68.542Q-2.555-68.542-2.769-68.384Q-2.983-68.225-3.101-67.970Q-3.218-67.715-3.218-67.449M1.382-65.644L-0.252-65.644L-0.252-65.924Q-0.023-65.924 0.126-65.958Q0.275-65.993 0.275-66.133L0.275-67.982Q0.275-68.252 0.167-68.313Q0.059-68.375-0.252-68.375L-0.252-68.655L0.808-68.730L0.808-68.081Q0.979-68.389 1.283-68.560Q1.587-68.730 1.932-68.730Q2.438-68.730 2.722-68.507Q3.006-68.283 3.006-67.787L3.006-66.133Q3.006-65.996 3.154-65.960Q3.303-65.924 3.529-65.924L3.529-65.644L1.898-65.644L1.898-65.924Q2.127-65.924 2.276-65.958Q2.425-65.993 2.425-66.133L2.425-67.773Q2.425-68.108 2.305-68.308Q2.185-68.508 1.871-68.508Q1.601-68.508 1.367-68.372Q1.133-68.235 0.994-68.001Q0.856-67.767 0.856-67.493L0.856-66.133Q0.856-65.996 1.006-65.960Q1.157-65.924 1.382-65.924\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.962 59.616)\">\u003Cpath d=\"M4.451-66.485L4.451-68.382L3.812-68.382L3.812-68.604Q4.130-68.604 4.347-68.814Q4.564-69.024 4.664-69.334Q4.765-69.643 4.765-69.951L5.032-69.951L5.032-68.662L6.109-68.662L6.109-68.382L5.032-68.382L5.032-66.498Q5.032-66.222 5.136-66.023Q5.240-65.825 5.500-65.825Q5.657-65.825 5.763-65.929Q5.869-66.034 5.919-66.187Q5.968-66.341 5.968-66.498L5.968-66.912L6.235-66.912L6.235-66.485Q6.235-66.259 6.136-66.049Q6.037-65.839 5.852-65.707Q5.668-65.576 5.439-65.576Q5.001-65.576 4.726-65.813Q4.451-66.051 4.451-66.485\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.962 59.616)\">\u003Cpath d=\"M-33.180-57.111Q-33.180-57.357-32.983-57.541Q-32.786-57.726-32.530-57.805Q-32.667-57.917-32.739-58.078Q-32.810-58.239-32.810-58.420Q-32.810-58.741-32.599-58.987Q-32.933-59.285-32.933-59.695Q-32.933-60.156-32.544-60.443Q-32.154-60.730-31.676-60.730Q-31.204-60.730-30.869-60.484Q-30.695-60.638-30.484-60.720Q-30.274-60.802-30.045-60.802Q-29.881-60.802-29.760-60.695Q-29.639-60.587-29.639-60.423Q-29.639-60.327-29.710-60.255Q-29.782-60.184-29.874-60.184Q-29.974-60.184-30.044-60.257Q-30.114-60.331-30.114-60.430Q-30.114-60.484-30.100-60.515L-30.093-60.529Q-30.086-60.549-30.078-60.560Q-30.069-60.570-30.066-60.577Q-30.421-60.577-30.708-60.354Q-30.421-60.061-30.421-59.695Q-30.421-59.380-30.606-59.148Q-30.790-58.915-31.079-58.787Q-31.368-58.659-31.676-58.659Q-31.877-58.659-32.069-58.709Q-32.260-58.758-32.438-58.868Q-32.530-58.741-32.530-58.598Q-32.530-58.416-32.402-58.281Q-32.274-58.146-32.089-58.146L-31.457-58.146Q-31.009-58.146-30.640-58.075Q-30.271-58.003-30.011-57.774Q-29.751-57.545-29.751-57.111Q-29.751-56.790-30.047-56.588Q-30.343-56.386-30.746-56.297Q-31.149-56.208-31.464-56.208Q-31.782-56.208-32.185-56.297Q-32.588-56.386-32.884-56.588Q-33.180-56.790-33.180-57.111M-32.725-57.111Q-32.725-56.882-32.506-56.733Q-32.287-56.584-31.995-56.516Q-31.703-56.448-31.464-56.448Q-31.300-56.448-31.091-56.484Q-30.883-56.519-30.676-56.600Q-30.469-56.680-30.338-56.808Q-30.206-56.936-30.206-57.111Q-30.206-57.463-30.587-57.557Q-30.968-57.651-31.471-57.651L-32.089-57.651Q-32.328-57.651-32.527-57.500Q-32.725-57.350-32.725-57.111M-31.676-58.898Q-31.009-58.898-31.009-59.695Q-31.009-60.495-31.676-60.495Q-32.346-60.495-32.346-59.695Q-32.346-58.898-31.676-58.898M-27.407-57.644L-29.143-57.644L-29.143-57.924Q-28.914-57.924-28.765-57.958Q-28.617-57.993-28.617-58.133L-28.617-59.982Q-28.617-60.252-28.724-60.313Q-28.832-60.375-29.143-60.375L-29.143-60.655L-28.114-60.730L-28.114-60.023Q-27.984-60.331-27.742-60.530Q-27.499-60.730-27.181-60.730Q-26.962-60.730-26.791-60.606Q-26.620-60.481-26.620-60.269Q-26.620-60.132-26.720-60.033Q-26.819-59.934-26.952-59.934Q-27.089-59.934-27.188-60.033Q-27.287-60.132-27.287-60.269Q-27.287-60.409-27.188-60.508Q-27.478-60.508-27.678-60.312Q-27.878-60.115-27.971-59.821Q-28.063-59.527-28.063-59.247L-28.063-58.133Q-28.063-57.924-27.407-57.924L-27.407-57.644M-25.978-58.372Q-25.978-58.704-25.754-58.931Q-25.530-59.158-25.187-59.286Q-24.843-59.415-24.471-59.467Q-24.098-59.520-23.794-59.520L-23.794-59.773Q-23.794-59.978-23.901-60.158Q-24.009-60.337-24.190-60.440Q-24.371-60.542-24.580-60.542Q-24.987-60.542-25.223-60.450Q-25.134-60.413-25.088-60.329Q-25.041-60.245-25.041-60.143Q-25.041-60.047-25.088-59.968Q-25.134-59.890-25.214-59.845Q-25.294-59.801-25.383-59.801Q-25.534-59.801-25.634-59.898Q-25.735-59.996-25.735-60.143Q-25.735-60.765-24.580-60.765Q-24.368-60.765-24.119-60.701Q-23.869-60.638-23.667-60.519Q-23.466-60.399-23.339-60.214Q-23.213-60.030-23.213-59.787L-23.213-58.211Q-23.213-58.095-23.151-57.999Q-23.090-57.904-22.977-57.904Q-22.868-57.904-22.803-57.998Q-22.738-58.092-22.738-58.211L-22.738-58.659L-22.471-58.659L-22.471-58.211Q-22.471-57.941-22.698-57.776Q-22.926-57.610-23.206-57.610Q-23.414-57.610-23.551-57.764Q-23.688-57.917-23.712-58.133Q-23.859-57.866-24.141-57.721Q-24.423-57.576-24.747-57.576Q-25.024-57.576-25.308-57.651Q-25.592-57.726-25.785-57.905Q-25.978-58.085-25.978-58.372M-25.363-58.372Q-25.363-58.198-25.262-58.068Q-25.161-57.938-25.005-57.868Q-24.850-57.798-24.686-57.798Q-24.467-57.798-24.259-57.895Q-24.050-57.993-23.922-58.174Q-23.794-58.355-23.794-58.581L-23.794-59.309Q-24.119-59.309-24.484-59.218Q-24.850-59.127-25.106-58.915Q-25.363-58.704-25.363-58.372M-22.054-59.155Q-22.054-59.493-21.914-59.784Q-21.774-60.074-21.529-60.288Q-21.285-60.501-20.981-60.616Q-20.677-60.730-20.352-60.730Q-20.082-60.730-19.819-60.631Q-19.556-60.532-19.364-60.354L-19.364-61.752Q-19.364-62.022-19.472-62.084Q-19.579-62.145-19.891-62.145L-19.891-62.426L-18.814-62.501L-18.814-58.317Q-18.814-58.129-18.759-58.046Q-18.704-57.962-18.604-57.943Q-18.503-57.924-18.287-57.924L-18.287-57.644L-19.395-57.576L-19.395-57.993Q-19.812-57.576-20.437-57.576Q-20.868-57.576-21.241-57.788Q-21.613-57.999-21.834-58.360Q-22.054-58.721-22.054-59.155M-20.379-57.798Q-20.171-57.798-19.984-57.870Q-19.798-57.941-19.644-58.078Q-19.491-58.215-19.395-58.393L-19.395-60.002Q-19.480-60.149-19.626-60.269Q-19.771-60.389-19.940-60.448Q-20.109-60.508-20.290-60.508Q-20.851-60.508-21.119-60.119Q-21.388-59.729-21.388-59.148Q-21.388-58.577-21.153-58.187Q-20.919-57.798-20.379-57.798\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.962 59.616)\">\u003Cpath d=\"M-13.340-57.671L-14.468-60.170Q-14.540-60.317-14.670-60.349Q-14.800-60.382-15.029-60.382L-15.029-60.662L-13.515-60.662L-13.515-60.382Q-13.867-60.382-13.867-60.235Q-13.867-60.190-13.856-60.170L-12.992-58.252L-12.212-59.982Q-12.178-60.050-12.178-60.129Q-12.178-60.242-12.262-60.312Q-12.346-60.382-12.465-60.382L-12.465-60.662L-11.269-60.662L-11.269-60.382Q-11.488-60.382-11.659-60.279Q-11.829-60.177-11.918-59.982L-12.954-57.671Q-13.002-57.576-13.108-57.576L-13.186-57.576Q-13.292-57.576-13.340-57.671M-8.599-55.894Q-9.150-56.294-9.521-56.849Q-9.891-57.405-10.073-58.051Q-10.254-58.697-10.254-59.394Q-10.254-59.907-10.153-60.402Q-10.052-60.898-9.847-61.349Q-9.642-61.800-9.329-62.192Q-9.016-62.583-8.599-62.887Q-8.589-62.891-8.582-62.892Q-8.576-62.894-8.565-62.894L-8.497-62.894Q-8.463-62.894-8.441-62.870Q-8.418-62.846-8.418-62.809Q-8.418-62.764-8.446-62.747Q-8.794-62.446-9.047-62.062Q-9.300-61.677-9.452-61.236Q-9.604-60.795-9.676-60.339Q-9.748-59.883-9.748-59.394Q-9.748-58.393-9.439-57.506Q-9.129-56.619-8.446-56.034Q-8.418-56.017-8.418-55.973Q-8.418-55.935-8.441-55.911Q-8.463-55.887-8.497-55.887L-8.565-55.887Q-8.572-55.891-8.581-55.892Q-8.589-55.894-8.599-55.894M-7.434-57.582L-7.434-59.155Q-7.434-59.182-7.408-59.208Q-7.383-59.233-7.355-59.233L-7.243-59.233Q-7.215-59.233-7.191-59.206Q-7.167-59.179-7.167-59.155Q-7.167-58.810-7.036-58.546Q-6.904-58.283-6.675-58.114Q-6.446-57.945-6.144-57.864Q-5.841-57.784-5.499-57.784Q-5.233-57.784-4.997-57.912Q-4.761-58.040-4.616-58.263Q-4.471-58.485-4.471-58.751Q-4.471-58.974-4.577-59.170Q-4.682-59.367-4.864-59.502Q-5.045-59.637-5.270-59.688L-6.299-59.920Q-6.610-59.992-6.870-60.178Q-7.130-60.365-7.282-60.636Q-7.434-60.908-7.434-61.223Q-7.434-61.609-7.220-61.916Q-7.007-62.224-6.660-62.395Q-6.313-62.566-5.933-62.566Q-5.704-62.566-5.475-62.513Q-5.246-62.460-5.048-62.352Q-4.850-62.245-4.696-62.081L-4.402-62.521Q-4.378-62.566-4.337-62.566L-4.289-62.566Q-4.259-62.566-4.236-62.540Q-4.214-62.515-4.214-62.487L-4.214-60.912Q-4.214-60.891-4.238-60.864Q-4.262-60.836-4.289-60.836L-4.402-60.836Q-4.464-60.836-4.477-60.912Q-4.518-61.325-4.700-61.645Q-4.881-61.964-5.192-62.139Q-5.503-62.313-5.933-62.313Q-6.183-62.313-6.422-62.202Q-6.661-62.091-6.812-61.893Q-6.962-61.694-6.962-61.431Q-6.962-61.219-6.855-61.038Q-6.747-60.857-6.571-60.737Q-6.395-60.618-6.186-60.577L-5.158-60.348Q-4.840-60.276-4.573-60.071Q-4.306-59.866-4.154-59.572Q-4.002-59.278-4.002-58.946Q-4.002-58.553-4.207-58.218Q-4.412-57.883-4.758-57.694Q-5.103-57.504-5.499-57.504Q-5.920-57.504-6.299-57.617Q-6.679-57.729-6.949-57.979L-7.243-57.545Q-7.270-57.504-7.307-57.504L-7.355-57.504Q-7.383-57.504-7.408-57.529Q-7.434-57.555-7.434-57.582M-2.666-58.485L-2.666-60.382L-3.305-60.382L-3.305-60.604Q-2.987-60.604-2.770-60.814Q-2.553-61.024-2.452-61.334Q-2.351-61.643-2.351-61.951L-2.085-61.951L-2.085-60.662L-1.008-60.662L-1.008-60.382L-2.085-60.382L-2.085-58.498Q-2.085-58.222-1.981-58.023Q-1.876-57.825-1.617-57.825Q-1.459-57.825-1.353-57.929Q-1.247-58.034-1.198-58.187Q-1.148-58.341-1.148-58.498L-1.148-58.912L-0.882-58.912L-0.882-58.485Q-0.882-58.259-0.981-58.049Q-1.080-57.839-1.265-57.707Q-1.449-57.576-1.678-57.576Q-2.116-57.576-2.391-57.813Q-2.666-58.051-2.666-58.485M0.250-55.887L0.181-55.887Q0.147-55.887 0.125-55.913Q0.103-55.938 0.103-55.973Q0.103-56.017 0.133-56.034Q0.489-56.338 0.738-56.728Q0.988-57.118 1.140-57.550Q1.292-57.982 1.362-58.451Q1.432-58.919 1.432-59.394Q1.432-59.873 1.362-60.339Q1.292-60.806 1.138-61.241Q0.985-61.677 0.733-62.065Q0.482-62.453 0.133-62.747Q0.103-62.764 0.103-62.809Q0.103-62.843 0.125-62.868Q0.147-62.894 0.181-62.894L0.250-62.894Q0.260-62.894 0.268-62.892Q0.277-62.891 0.287-62.887Q0.831-62.487 1.203-61.934Q1.576-61.380 1.757-60.734Q1.938-60.088 1.938-59.394Q1.938-58.693 1.757-58.046Q1.576-57.398 1.202-56.844Q0.827-56.290 0.287-55.894Q0.277-55.894 0.268-55.892Q0.260-55.891 0.250-55.887\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M119.052-43.417h62.596V-71.87h-62.596Z\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(173.975 5.556)\">\u003Cpath d=\"M-37.125-65.651L-37.125-66.714Q-37.125-66.738-37.097-66.765Q-37.070-66.792-37.046-66.792L-36.937-66.792Q-36.872-66.792-36.858-66.734Q-36.762-66.300-36.516-66.049Q-36.270-65.798-35.856-65.798Q-35.515-65.798-35.262-65.931Q-35.009-66.064-35.009-66.372Q-35.009-66.529-35.103-66.644Q-35.197-66.758-35.335-66.827Q-35.474-66.895-35.641-66.933L-36.222-67.032Q-36.578-67.100-36.851-67.321Q-37.125-67.541-37.125-67.883Q-37.125-68.132-37.013-68.307Q-36.902-68.481-36.716-68.580Q-36.530-68.679-36.314-68.722Q-36.099-68.765-35.856-68.765Q-35.443-68.765-35.163-68.583L-34.947-68.758Q-34.937-68.761-34.930-68.763Q-34.923-68.765-34.913-68.765L-34.862-68.765Q-34.835-68.765-34.811-68.741Q-34.787-68.717-34.787-68.689L-34.787-67.842Q-34.787-67.821-34.811-67.794Q-34.835-67.767-34.862-67.767L-34.975-67.767Q-35.002-67.767-35.028-67.792Q-35.053-67.818-35.053-67.842Q-35.053-68.078-35.159-68.242Q-35.265-68.406-35.448-68.488Q-35.631-68.570-35.863-68.570Q-36.191-68.570-36.448-68.467Q-36.704-68.365-36.704-68.088Q-36.704-67.893-36.521-67.784Q-36.338-67.674-36.109-67.633L-35.535-67.527Q-35.289-67.479-35.075-67.351Q-34.862-67.223-34.725-67.020Q-34.588-66.816-34.588-66.567Q-34.588-66.054-34.954-65.815Q-35.320-65.576-35.856-65.576Q-36.352-65.576-36.684-65.870L-36.950-65.596Q-36.971-65.576-36.998-65.576L-37.046-65.576Q-37.070-65.576-37.097-65.603Q-37.125-65.630-37.125-65.651M-33.960-67.155Q-33.960-67.483-33.824-67.784Q-33.689-68.084-33.454-68.305Q-33.218-68.525-32.914-68.645Q-32.609-68.765-32.285-68.765Q-31.779-68.765-31.430-68.662Q-31.082-68.560-31.082-68.184Q-31.082-68.037-31.179-67.936Q-31.276-67.835-31.423-67.835Q-31.577-67.835-31.676-67.934Q-31.775-68.033-31.775-68.184Q-31.775-68.372-31.635-68.464Q-31.837-68.515-32.278-68.515Q-32.633-68.515-32.862-68.319Q-33.091-68.122-33.192-67.813Q-33.293-67.503-33.293-67.155Q-33.293-66.806-33.167-66.500Q-33.040-66.194-32.785-66.010Q-32.531-65.825-32.175-65.825Q-31.953-65.825-31.769-65.909Q-31.584-65.993-31.449-66.148Q-31.314-66.304-31.256-66.512Q-31.242-66.567-31.188-66.567L-31.075-66.567Q-31.044-66.567-31.022-66.543Q-31-66.519-31-66.485L-31-66.464Q-31.085-66.177-31.273-65.979Q-31.461-65.781-31.726-65.678Q-31.991-65.576-32.285-65.576Q-32.715-65.576-33.103-65.782Q-33.491-65.989-33.725-66.352Q-33.960-66.714-33.960-67.155M-30.354-66.372Q-30.354-66.704-30.130-66.931Q-29.906-67.158-29.562-67.286Q-29.219-67.415-28.846-67.467Q-28.474-67.520-28.169-67.520L-28.169-67.773Q-28.169-67.978-28.277-68.158Q-28.385-68.337-28.566-68.440Q-28.747-68.542-28.956-68.542Q-29.362-68.542-29.598-68.450Q-29.509-68.413-29.463-68.329Q-29.417-68.245-29.417-68.143Q-29.417-68.047-29.463-67.968Q-29.509-67.890-29.590-67.845Q-29.670-67.801-29.759-67.801Q-29.909-67.801-30.010-67.898Q-30.111-67.996-30.111-68.143Q-30.111-68.765-28.956-68.765Q-28.744-68.765-28.494-68.701Q-28.245-68.638-28.043-68.519Q-27.841-68.399-27.715-68.214Q-27.588-68.030-27.588-67.787L-27.588-66.211Q-27.588-66.095-27.527-65.999Q-27.465-65.904-27.353-65.904Q-27.243-65.904-27.178-65.998Q-27.113-66.092-27.113-66.211L-27.113-66.659L-26.847-66.659L-26.847-66.211Q-26.847-65.941-27.074-65.776Q-27.301-65.610-27.582-65.610Q-27.790-65.610-27.927-65.764Q-28.064-65.917-28.087-66.133Q-28.234-65.866-28.516-65.721Q-28.798-65.576-29.123-65.576Q-29.400-65.576-29.684-65.651Q-29.967-65.726-30.160-65.905Q-30.354-66.085-30.354-66.372M-29.738-66.372Q-29.738-66.198-29.637-66.068Q-29.537-65.938-29.381-65.868Q-29.226-65.798-29.062-65.798Q-28.843-65.798-28.634-65.895Q-28.426-65.993-28.298-66.174Q-28.169-66.355-28.169-66.581L-28.169-67.309Q-28.494-67.309-28.860-67.218Q-29.226-67.127-29.482-66.915Q-29.738-66.704-29.738-66.372M-24.762-65.644L-26.365-65.644L-26.365-65.924Q-26.139-65.924-25.991-65.958Q-25.842-65.993-25.842-66.133L-25.842-69.752Q-25.842-70.022-25.949-70.084Q-26.057-70.145-26.365-70.145L-26.365-70.426L-25.288-70.501L-25.288-66.133Q-25.288-65.996-25.138-65.960Q-24.987-65.924-24.762-65.924L-24.762-65.644M-24.208-67.179Q-24.208-67.500-24.083-67.789Q-23.959-68.078-23.733-68.301Q-23.507-68.525-23.212-68.645Q-22.916-68.765-22.598-68.765Q-22.270-68.765-22.009-68.665Q-21.747-68.566-21.571-68.384Q-21.395-68.201-21.301-67.943Q-21.207-67.685-21.207-67.353Q-21.207-67.261-21.289-67.240L-23.545-67.240L-23.545-67.179Q-23.545-66.591-23.261-66.208Q-22.978-65.825-22.410-65.825Q-22.089-65.825-21.821-66.018Q-21.552-66.211-21.463-66.526Q-21.457-66.567-21.381-66.581L-21.289-66.581Q-21.207-66.557-21.207-66.485Q-21.207-66.478-21.214-66.451Q-21.327-66.054-21.698-65.815Q-22.068-65.576-22.492-65.576Q-22.930-65.576-23.330-65.784Q-23.730-65.993-23.969-66.360Q-24.208-66.727-24.208-67.179M-23.538-67.449L-21.723-67.449Q-21.723-67.726-21.821-67.978Q-21.918-68.231-22.116-68.387Q-22.314-68.542-22.598-68.542Q-22.875-68.542-23.089-68.384Q-23.302-68.225-23.420-67.970Q-23.538-67.715-23.538-67.449\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(173.975 5.556)\">\u003Cpath d=\"M-17.094-65.644L-17.361-65.644L-17.361-69.752Q-17.361-70.022-17.468-70.084Q-17.576-70.145-17.887-70.145L-17.887-70.426L-16.807-70.501L-16.807-68.331Q-16.598-68.522-16.313-68.626Q-16.027-68.730-15.730-68.730Q-15.412-68.730-15.115-68.609Q-14.818-68.488-14.595-68.272Q-14.373-68.057-14.247-67.772Q-14.120-67.486-14.120-67.155Q-14.120-66.710-14.360-66.346Q-14.599-65.982-14.992-65.779Q-15.385-65.576-15.829-65.576Q-16.024-65.576-16.214-65.632Q-16.403-65.688-16.564-65.793Q-16.725-65.897-16.865-66.058L-17.094-65.644M-16.779-67.989L-16.779-66.372Q-16.643-66.112-16.402-65.955Q-16.161-65.798-15.884-65.798Q-15.590-65.798-15.378-65.905Q-15.166-66.013-15.033-66.205Q-14.900-66.396-14.841-66.635Q-14.783-66.874-14.783-67.155Q-14.783-67.514-14.877-67.818Q-14.971-68.122-15.199-68.315Q-15.426-68.508-15.792-68.508Q-16.092-68.508-16.359-68.372Q-16.626-68.235-16.779-67.989\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(173.975 5.556)\">\u003Cpath d=\"M-13.365-64.509Q-13.235-64.441-13.098-64.441Q-12.927-64.441-12.777-64.530Q-12.626-64.619-12.515-64.764Q-12.404-64.909-12.326-65.077L-12.062-65.644L-13.231-68.170Q-13.306-68.317-13.436-68.349Q-13.566-68.382-13.799-68.382L-13.799-68.662L-12.278-68.662L-12.278-68.382Q-12.626-68.382-12.626-68.235Q-12.623-68.214-12.621-68.197Q-12.619-68.180-12.619-68.170L-11.762-66.311L-10.989-67.982Q-10.955-68.050-10.955-68.129Q-10.955-68.242-11.039-68.312Q-11.122-68.382-11.235-68.382L-11.235-68.662L-10.039-68.662L-10.039-68.382Q-10.258-68.382-10.430-68.278Q-10.603-68.173-10.695-67.982L-12.032-65.077Q-12.202-64.707-12.472-64.461Q-12.743-64.215-13.098-64.215Q-13.368-64.215-13.587-64.381Q-13.806-64.547-13.806-64.810Q-13.806-64.947-13.713-65.036Q-13.621-65.124-13.481-65.124Q-13.344-65.124-13.255-65.036Q-13.166-64.947-13.166-64.810Q-13.166-64.707-13.219-64.629Q-13.272-64.550-13.365-64.509\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(173.975 5.556)\">\u003Cpath d=\"M-32.013-57.644L-33.749-57.644L-33.749-57.924Q-33.520-57.924-33.371-57.958Q-33.223-57.993-33.223-58.133L-33.223-59.982Q-33.223-60.252-33.330-60.313Q-33.438-60.375-33.749-60.375L-33.749-60.655L-32.720-60.730L-32.720-60.023Q-32.590-60.331-32.348-60.530Q-32.105-60.730-31.787-60.730Q-31.568-60.730-31.397-60.606Q-31.226-60.481-31.226-60.269Q-31.226-60.132-31.326-60.033Q-31.425-59.934-31.558-59.934Q-31.695-59.934-31.794-60.033Q-31.893-60.132-31.893-60.269Q-31.893-60.409-31.794-60.508Q-32.084-60.508-32.284-60.312Q-32.484-60.115-32.577-59.821Q-32.669-59.527-32.669-59.247L-32.669-58.133Q-32.669-57.924-32.013-57.924L-32.013-57.644M-28.960-57.644L-30.594-57.644L-30.594-57.924Q-30.365-57.924-30.216-57.958Q-30.068-57.993-30.068-58.133L-30.068-61.752Q-30.068-62.022-30.175-62.084Q-30.283-62.145-30.594-62.145L-30.594-62.426L-29.514-62.501L-29.514-60.115Q-29.408-60.300-29.230-60.442Q-29.053-60.583-28.844-60.657Q-28.636-60.730-28.410-60.730Q-27.904-60.730-27.620-60.507Q-27.337-60.283-27.337-59.787L-27.337-58.133Q-27.337-57.996-27.188-57.960Q-27.039-57.924-26.814-57.924L-26.814-57.644L-28.444-57.644L-28.444-57.924Q-28.215-57.924-28.067-57.958Q-27.918-57.993-27.918-58.133L-27.918-59.773Q-27.918-60.108-28.037-60.308Q-28.157-60.508-28.472-60.508Q-28.742-60.508-28.976-60.372Q-29.210-60.235-29.348-60.001Q-29.487-59.767-29.487-59.493L-29.487-58.133Q-29.487-57.996-29.336-57.960Q-29.186-57.924-28.960-57.924L-28.960-57.644M-26.267-59.127Q-26.267-59.469-26.132-59.768Q-25.997-60.067-25.758-60.291Q-25.518-60.515-25.201-60.640Q-24.883-60.765-24.551-60.765Q-24.107-60.765-23.707-60.549Q-23.307-60.334-23.073-59.956Q-22.839-59.579-22.839-59.127Q-22.839-58.786-22.981-58.502Q-23.122-58.218-23.367-58.011Q-23.611-57.805-23.921-57.690Q-24.230-57.576-24.551-57.576Q-24.982-57.576-25.383-57.777Q-25.785-57.979-26.026-58.331Q-26.267-58.683-26.267-59.127M-24.551-57.825Q-23.950-57.825-23.726-58.203Q-23.502-58.581-23.502-59.213Q-23.502-59.825-23.736-60.184Q-23.970-60.542-24.551-60.542Q-25.604-60.542-25.604-59.213Q-25.604-58.581-25.378-58.203Q-25.153-57.825-24.551-57.825M-20.115-55.894Q-20.665-56.294-21.036-56.849Q-21.407-57.405-21.588-58.051Q-21.769-58.697-21.769-59.394Q-21.769-59.907-21.668-60.402Q-21.567-60.898-21.362-61.349Q-21.157-61.800-20.844-62.192Q-20.532-62.583-20.115-62.887Q-20.104-62.891-20.098-62.892Q-20.091-62.894-20.080-62.894L-20.012-62.894Q-19.978-62.894-19.956-62.870Q-19.933-62.846-19.933-62.809Q-19.933-62.764-19.961-62.747Q-20.309-62.446-20.562-62.062Q-20.815-61.677-20.967-61.236Q-21.119-60.795-21.191-60.339Q-21.263-59.883-21.263-59.394Q-21.263-58.393-20.954-57.506Q-20.644-56.619-19.961-56.034Q-19.933-56.017-19.933-55.973Q-19.933-55.935-19.956-55.911Q-19.978-55.887-20.012-55.887L-20.080-55.887Q-20.087-55.891-20.096-55.892Q-20.104-55.894-20.115-55.894M-18.597-58.485L-18.597-60.382L-19.236-60.382L-19.236-60.604Q-18.918-60.604-18.701-60.814Q-18.484-61.024-18.383-61.334Q-18.283-61.643-18.283-61.951L-18.016-61.951L-18.016-60.662L-16.939-60.662L-16.939-60.382L-18.016-60.382L-18.016-58.498Q-18.016-58.222-17.912-58.023Q-17.807-57.825-17.548-57.825Q-17.390-57.825-17.285-57.929Q-17.179-58.034-17.129-58.187Q-17.079-58.341-17.079-58.498L-17.079-58.912L-16.813-58.912L-16.813-58.485Q-16.813-58.259-16.912-58.049Q-17.011-57.839-17.196-57.707Q-17.380-57.576-17.609-57.576Q-18.047-57.576-18.322-57.813Q-18.597-58.051-18.597-58.485M-15.681-55.887L-15.750-55.887Q-15.784-55.887-15.806-55.913Q-15.828-55.938-15.828-55.973Q-15.828-56.017-15.798-56.034Q-15.442-56.338-15.193-56.728Q-14.943-57.118-14.791-57.550Q-14.639-57.982-14.569-58.451Q-14.499-58.919-14.499-59.394Q-14.499-59.873-14.569-60.339Q-14.639-60.806-14.793-61.241Q-14.947-61.677-15.198-62.065Q-15.449-62.453-15.798-62.747Q-15.828-62.764-15.828-62.809Q-15.828-62.843-15.806-62.868Q-15.784-62.894-15.750-62.894L-15.681-62.894Q-15.671-62.894-15.663-62.892Q-15.654-62.891-15.644-62.887Q-15.100-62.487-14.728-61.934Q-14.355-61.380-14.174-60.734Q-13.993-60.088-13.993-59.394Q-13.993-58.693-14.174-58.046Q-14.355-57.398-14.730-56.844Q-15.104-56.290-15.644-55.894Q-15.654-55.894-15.663-55.892Q-15.671-55.891-15.681-55.887\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M210.1-43.417h62.596V-71.87H210.1Z\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(261.976 5.278)\">\u003Cpath d=\"M-35.443-65.644L-37.077-65.644L-37.077-65.924Q-36.848-65.924-36.699-65.958Q-36.550-65.993-36.550-66.133L-36.550-67.982Q-36.550-68.252-36.658-68.313Q-36.766-68.375-37.077-68.375L-37.077-68.655L-36.017-68.730L-36.017-68.081Q-35.846-68.389-35.542-68.560Q-35.238-68.730-34.893-68.730Q-34.387-68.730-34.103-68.507Q-33.819-68.283-33.819-67.787L-33.819-66.133Q-33.819-65.996-33.671-65.960Q-33.522-65.924-33.296-65.924L-33.296-65.644L-34.927-65.644L-34.927-65.924Q-34.698-65.924-34.549-65.958Q-34.400-65.993-34.400-66.133L-34.400-67.773Q-34.400-68.108-34.520-68.308Q-34.640-68.508-34.954-68.508Q-35.224-68.508-35.458-68.372Q-35.692-68.235-35.831-68.001Q-35.969-67.767-35.969-67.493L-35.969-66.133Q-35.969-65.996-35.819-65.960Q-35.668-65.924-35.443-65.924L-35.443-65.644M-32.750-67.179Q-32.750-67.500-32.625-67.789Q-32.500-68.078-32.274-68.301Q-32.049-68.525-31.753-68.645Q-31.458-68.765-31.140-68.765Q-30.812-68.765-30.550-68.665Q-30.289-68.566-30.113-68.384Q-29.937-68.201-29.843-67.943Q-29.749-67.685-29.749-67.353Q-29.749-67.261-29.831-67.240L-32.086-67.240L-32.086-67.179Q-32.086-66.591-31.803-66.208Q-31.519-65.825-30.952-65.825Q-30.630-65.825-30.362-66.018Q-30.094-66.211-30.005-66.526Q-29.998-66.567-29.923-66.581L-29.831-66.581Q-29.749-66.557-29.749-66.485Q-29.749-66.478-29.755-66.451Q-29.868-66.054-30.239-65.815Q-30.610-65.576-31.034-65.576Q-31.471-65.576-31.871-65.784Q-32.271-65.993-32.510-66.360Q-32.750-66.727-32.750-67.179M-32.080-67.449L-30.265-67.449Q-30.265-67.726-30.362-67.978Q-30.460-68.231-30.658-68.387Q-30.856-68.542-31.140-68.542Q-31.417-68.542-31.630-68.384Q-31.844-68.225-31.962-67.970Q-32.080-67.715-32.080-67.449M-27.773-65.671L-28.754-68.170Q-28.815-68.313-28.933-68.348Q-29.051-68.382-29.267-68.382L-29.267-68.662L-27.787-68.662L-27.787-68.382Q-28.166-68.382-28.166-68.221Q-28.166-68.211-28.152-68.170L-27.438-66.338L-26.765-68.043Q-26.795-68.115-26.795-68.143Q-26.795-68.170-26.823-68.170Q-26.884-68.317-27.002-68.349Q-27.120-68.382-27.332-68.382L-27.332-68.662L-25.934-68.662L-25.934-68.382Q-26.310-68.382-26.310-68.221Q-26.310-68.190-26.303-68.170L-25.548-66.232L-24.861-67.982Q-24.840-68.033-24.840-68.088Q-24.840-68.228-24.953-68.305Q-25.066-68.382-25.206-68.382L-25.206-68.662L-23.986-68.662L-23.986-68.382Q-24.191-68.382-24.346-68.276Q-24.502-68.170-24.574-67.982L-25.480-65.671Q-25.514-65.576-25.627-65.576L-25.695-65.576Q-25.804-65.576-25.842-65.671L-26.625-67.674L-27.411-65.671Q-27.445-65.576-27.558-65.576L-27.626-65.576Q-27.735-65.576-27.773-65.671\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(261.976 5.278)\">\u003Cpath d=\"M-20.223-66.485L-20.223-68.382L-20.862-68.382L-20.862-68.604Q-20.544-68.604-20.327-68.814Q-20.110-69.024-20.010-69.334Q-19.909-69.643-19.909-69.951L-19.642-69.951L-19.642-68.662L-18.565-68.662L-18.565-68.382L-19.642-68.382L-19.642-66.498Q-19.642-66.222-19.538-66.023Q-19.434-65.825-19.174-65.825Q-19.017-65.825-18.911-65.929Q-18.805-66.034-18.755-66.187Q-18.706-66.341-18.706-66.498L-18.706-66.912L-18.439-66.912L-18.439-66.485Q-18.439-66.259-18.538-66.049Q-18.637-65.839-18.822-65.707Q-19.006-65.576-19.235-65.576Q-19.673-65.576-19.948-65.813Q-20.223-66.051-20.223-66.485M-15.879-65.644L-17.615-65.644L-17.615-65.924Q-17.386-65.924-17.238-65.958Q-17.089-65.993-17.089-66.133L-17.089-67.982Q-17.089-68.252-17.197-68.313Q-17.304-68.375-17.615-68.375L-17.615-68.655L-16.586-68.730L-16.586-68.023Q-16.457-68.331-16.214-68.530Q-15.971-68.730-15.653-68.730Q-15.435-68.730-15.264-68.606Q-15.093-68.481-15.093-68.269Q-15.093-68.132-15.192-68.033Q-15.291-67.934-15.424-67.934Q-15.561-67.934-15.660-68.033Q-15.759-68.132-15.759-68.269Q-15.759-68.409-15.660-68.508Q-15.951-68.508-16.151-68.312Q-16.351-68.115-16.443-67.821Q-16.535-67.527-16.535-67.247L-16.535-66.133Q-16.535-65.924-15.879-65.924L-15.879-65.644M-14.450-66.372Q-14.450-66.704-14.226-66.931Q-14.002-67.158-13.659-67.286Q-13.315-67.415-12.943-67.467Q-12.570-67.520-12.266-67.520L-12.266-67.773Q-12.266-67.978-12.374-68.158Q-12.481-68.337-12.663-68.440Q-12.844-68.542-13.052-68.542Q-13.459-68.542-13.695-68.450Q-13.606-68.413-13.560-68.329Q-13.514-68.245-13.514-68.143Q-13.514-68.047-13.560-67.968Q-13.606-67.890-13.686-67.845Q-13.767-67.801-13.856-67.801Q-14.006-67.801-14.107-67.898Q-14.208-67.996-14.208-68.143Q-14.208-68.765-13.052-68.765Q-12.840-68.765-12.591-68.701Q-12.341-68.638-12.140-68.519Q-11.938-68.399-11.812-68.214Q-11.685-68.030-11.685-67.787L-11.685-66.211Q-11.685-66.095-11.624-65.999Q-11.562-65.904-11.449-65.904Q-11.340-65.904-11.275-65.998Q-11.210-66.092-11.210-66.211L-11.210-66.659L-10.943-66.659L-10.943-66.211Q-10.943-65.941-11.171-65.776Q-11.398-65.610-11.678-65.610Q-11.887-65.610-12.023-65.764Q-12.160-65.917-12.184-66.133Q-12.331-65.866-12.613-65.721Q-12.895-65.576-13.220-65.576Q-13.497-65.576-13.780-65.651Q-14.064-65.726-14.257-65.905Q-14.450-66.085-14.450-66.372M-13.835-66.372Q-13.835-66.198-13.734-66.068Q-13.633-65.938-13.478-65.868Q-13.322-65.798-13.158-65.798Q-12.939-65.798-12.731-65.895Q-12.522-65.993-12.394-66.174Q-12.266-66.355-12.266-66.581L-12.266-67.309Q-12.591-67.309-12.957-67.218Q-13.322-67.127-13.579-66.915Q-13.835-66.704-13.835-66.372M-10.526-67.155Q-10.526-67.483-10.391-67.784Q-10.256-68.084-10.021-68.305Q-9.785-68.525-9.481-68.645Q-9.176-68.765-8.852-68.765Q-8.346-68.765-7.997-68.662Q-7.648-68.560-7.648-68.184Q-7.648-68.037-7.746-67.936Q-7.843-67.835-7.990-67.835Q-8.144-67.835-8.243-67.934Q-8.342-68.033-8.342-68.184Q-8.342-68.372-8.202-68.464Q-8.404-68.515-8.845-68.515Q-9.200-68.515-9.429-68.319Q-9.658-68.122-9.759-67.813Q-9.860-67.503-9.860-67.155Q-9.860-66.806-9.733-66.500Q-9.607-66.194-9.352-66.010Q-9.098-65.825-8.742-65.825Q-8.520-65.825-8.335-65.909Q-8.151-65.993-8.016-66.148Q-7.881-66.304-7.823-66.512Q-7.809-66.567-7.754-66.567L-7.642-66.567Q-7.611-66.567-7.589-66.543Q-7.566-66.519-7.566-66.485L-7.566-66.464Q-7.652-66.177-7.840-65.979Q-8.028-65.781-8.293-65.678Q-8.558-65.576-8.852-65.576Q-9.282-65.576-9.670-65.782Q-10.058-65.989-10.292-66.352Q-10.526-66.714-10.526-67.155M-7.020-67.179Q-7.020-67.500-6.895-67.789Q-6.770-68.078-6.544-68.301Q-6.319-68.525-6.023-68.645Q-5.728-68.765-5.410-68.765Q-5.082-68.765-4.820-68.665Q-4.559-68.566-4.383-68.384Q-4.207-68.201-4.113-67.943Q-4.019-67.685-4.019-67.353Q-4.019-67.261-4.101-67.240L-6.356-67.240L-6.356-67.179Q-6.356-66.591-6.073-66.208Q-5.789-65.825-5.222-65.825Q-4.900-65.825-4.632-66.018Q-4.364-66.211-4.275-66.526Q-4.268-66.567-4.193-66.581L-4.101-66.581Q-4.019-66.557-4.019-66.485Q-4.019-66.478-4.025-66.451Q-4.138-66.054-4.509-65.815Q-4.880-65.576-5.304-65.576Q-5.741-65.576-6.141-65.784Q-6.541-65.993-6.780-66.360Q-7.020-66.727-7.020-67.179M-6.350-67.449L-4.535-67.449Q-4.535-67.726-4.632-67.978Q-4.730-68.231-4.928-68.387Q-5.126-68.542-5.410-68.542Q-5.687-68.542-5.900-68.384Q-6.114-68.225-6.232-67.970Q-6.350-67.715-6.350-67.449\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(261.976 5.278)\">\u003Cpath d=\"M-23.998-57.644L-26.670-57.644Q-26.715-57.644-26.742-57.671Q-26.770-57.699-26.770-57.743L-26.770-57.811Q-26.770-57.852-26.742-57.883L-24.633-60.436L-25.272-60.436Q-25.696-60.436-25.929-60.370Q-26.161-60.303-26.281-60.093Q-26.400-59.883-26.400-59.462L-26.664-59.462L-26.582-60.662L-23.991-60.662Q-23.950-60.662-23.921-60.635Q-23.892-60.607-23.892-60.563L-23.892-60.515Q-23.892-60.471-23.916-60.443L-26.021-57.897L-25.341-57.897Q-25.013-57.897-24.796-57.933Q-24.579-57.969-24.411-58.119Q-24.271-58.256-24.218-58.480Q-24.165-58.704-24.138-59.032L-23.871-59.032L-23.998-57.644M-21.051-55.894Q-21.602-56.294-21.972-56.849Q-22.343-57.405-22.524-58.051Q-22.706-58.697-22.706-59.394Q-22.706-59.907-22.605-60.402Q-22.504-60.898-22.299-61.349Q-22.094-61.800-21.781-62.192Q-21.468-62.583-21.051-62.887Q-21.041-62.891-21.034-62.892Q-21.027-62.894-21.017-62.894L-20.949-62.894Q-20.915-62.894-20.892-62.870Q-20.870-62.846-20.870-62.809Q-20.870-62.764-20.897-62.747Q-21.246-62.446-21.499-62.062Q-21.752-61.677-21.904-61.236Q-22.056-60.795-22.128-60.339Q-22.200-59.883-22.200-59.394Q-22.200-58.393-21.890-57.506Q-21.581-56.619-20.897-56.034Q-20.870-56.017-20.870-55.973Q-20.870-55.935-20.892-55.911Q-20.915-55.887-20.949-55.887L-21.017-55.887Q-21.024-55.891-21.033-55.892Q-21.041-55.894-21.051-55.894M-19.534-58.485L-19.534-60.382L-20.173-60.382L-20.173-60.604Q-19.855-60.604-19.638-60.814Q-19.421-61.024-19.320-61.334Q-19.219-61.643-19.219-61.951L-18.953-61.951L-18.953-60.662L-17.876-60.662L-17.876-60.382L-18.953-60.382L-18.953-58.498Q-18.953-58.222-18.848-58.023Q-18.744-57.825-18.484-57.825Q-18.327-57.825-18.221-57.929Q-18.115-58.034-18.066-58.187Q-18.016-58.341-18.016-58.498L-18.016-58.912L-17.750-58.912L-17.750-58.485Q-17.750-58.259-17.849-58.049Q-17.948-57.839-18.132-57.707Q-18.317-57.576-18.546-57.576Q-18.983-57.576-19.259-57.813Q-19.534-58.051-19.534-58.485M-16.618-55.887L-16.687-55.887Q-16.721-55.887-16.743-55.913Q-16.765-55.938-16.765-55.973Q-16.765-56.017-16.734-56.034Q-16.379-56.338-16.129-56.728Q-15.880-57.118-15.728-57.550Q-15.576-57.982-15.506-58.451Q-15.436-58.919-15.436-59.394Q-15.436-59.873-15.506-60.339Q-15.576-60.806-15.730-61.241Q-15.883-61.677-16.135-62.065Q-16.386-62.453-16.734-62.747Q-16.765-62.764-16.765-62.809Q-16.765-62.843-16.743-62.868Q-16.721-62.894-16.687-62.894L-16.618-62.894Q-16.608-62.894-16.599-62.892Q-16.591-62.891-16.581-62.887Q-16.037-62.487-15.665-61.934Q-15.292-61.380-15.111-60.734Q-14.930-60.088-14.930-59.394Q-14.930-58.693-15.111-58.046Q-15.292-57.398-15.666-56.844Q-16.041-56.290-16.581-55.894Q-16.591-55.894-16.599-55.892Q-16.608-55.891-16.618-55.887\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-5.94-57.644h17.327\"\u002F>\u003Cpath stroke=\"none\" d=\"m13.987-57.644-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M53.61-18.01v-22.607\"\u002F>\u003Cpath stroke=\"none\" d=\"m53.61-43.217-2.08 4.16 2.08-1.56 2.08 1.56\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M93.233-57.644h22.819\"\u002F>\u003Cpath stroke=\"none\" d=\"m118.652-57.644-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M182.048-57.644H207.3\"\u002F>\u003Cpath stroke=\"none\" d=\"m209.9-57.644-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(166.024 30.123)\">\u003Cpath d=\"M-35.375-57.644L-37.111-57.644L-37.111-57.924Q-36.882-57.924-36.733-57.958Q-36.585-57.993-36.585-58.133L-36.585-59.982Q-36.585-60.252-36.692-60.313Q-36.800-60.375-37.111-60.375L-37.111-60.655L-36.082-60.730L-36.082-60.023Q-35.952-60.331-35.710-60.530Q-35.467-60.730-35.149-60.730Q-34.930-60.730-34.759-60.606Q-34.588-60.481-34.588-60.269Q-34.588-60.132-34.688-60.033Q-34.787-59.934-34.920-59.934Q-35.057-59.934-35.156-60.033Q-35.255-60.132-35.255-60.269Q-35.255-60.409-35.156-60.508Q-35.446-60.508-35.646-60.312Q-35.846-60.115-35.939-59.821Q-36.031-59.527-36.031-59.247L-36.031-58.133Q-36.031-57.924-35.375-57.924L-35.375-57.644M-32.322-57.644L-33.956-57.644L-33.956-57.924Q-33.727-57.924-33.578-57.958Q-33.430-57.993-33.430-58.133L-33.430-61.752Q-33.430-62.022-33.537-62.084Q-33.645-62.145-33.956-62.145L-33.956-62.426L-32.876-62.501L-32.876-60.115Q-32.770-60.300-32.592-60.442Q-32.415-60.583-32.206-60.657Q-31.998-60.730-31.772-60.730Q-31.266-60.730-30.982-60.507Q-30.699-60.283-30.699-59.787L-30.699-58.133Q-30.699-57.996-30.550-57.960Q-30.401-57.924-30.176-57.924L-30.176-57.644L-31.806-57.644L-31.806-57.924Q-31.577-57.924-31.429-57.958Q-31.280-57.993-31.280-58.133L-31.280-59.773Q-31.280-60.108-31.399-60.308Q-31.519-60.508-31.834-60.508Q-32.104-60.508-32.338-60.372Q-32.572-60.235-32.710-60.001Q-32.849-59.767-32.849-59.493L-32.849-58.133Q-32.849-57.996-32.698-57.960Q-32.548-57.924-32.322-57.924L-32.322-57.644M-29.629-59.127Q-29.629-59.469-29.494-59.768Q-29.359-60.067-29.120-60.291Q-28.880-60.515-28.563-60.640Q-28.245-60.765-27.913-60.765Q-27.469-60.765-27.069-60.549Q-26.669-60.334-26.435-59.956Q-26.201-59.579-26.201-59.127Q-26.201-58.786-26.343-58.502Q-26.484-58.218-26.729-58.011Q-26.973-57.805-27.283-57.690Q-27.592-57.576-27.913-57.576Q-28.344-57.576-28.745-57.777Q-29.147-57.979-29.388-58.331Q-29.629-58.683-29.629-59.127M-27.913-57.825Q-27.312-57.825-27.088-58.203Q-26.864-58.581-26.864-59.213Q-26.864-59.825-27.098-60.184Q-27.332-60.542-27.913-60.542Q-28.966-60.542-28.966-59.213Q-28.966-58.581-28.740-58.203Q-28.515-57.825-27.913-57.825\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(166.024 30.123)\">\u003Cpath d=\"M-17.734-58.451L-22.567-58.451Q-22.635-58.461-22.681-58.507Q-22.727-58.553-22.727-58.625Q-22.727-58.690-22.681-58.736Q-22.635-58.782-22.567-58.792L-17.734-58.792Q-17.665-58.782-17.619-58.736Q-17.573-58.690-17.573-58.625Q-17.573-58.553-17.619-58.507Q-17.665-58.461-17.734-58.451M-17.734-59.989L-22.567-59.989Q-22.635-59.999-22.681-60.045Q-22.727-60.091-22.727-60.163Q-22.727-60.307-22.567-60.331L-17.734-60.331Q-17.573-60.307-17.573-60.163Q-17.573-60.091-17.619-60.045Q-17.665-59.999-17.734-59.989\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(166.024 30.123)\">\u003Cpath d=\"M-12.425-56.287L-14.055-56.287L-14.055-56.567Q-13.826-56.567-13.677-56.602Q-13.529-56.636-13.529-56.776L-13.529-60.122Q-13.529-60.293-13.665-60.334Q-13.802-60.375-14.055-60.375L-14.055-60.655L-12.975-60.730L-12.975-60.324Q-12.753-60.525-12.466-60.628Q-12.178-60.730-11.871-60.730Q-11.444-60.730-11.080-60.517Q-10.716-60.303-10.502-59.939Q-10.288-59.575-10.288-59.155Q-10.288-58.710-10.528-58.346Q-10.767-57.982-11.160-57.779Q-11.553-57.576-11.997-57.576Q-12.264-57.576-12.512-57.676Q-12.759-57.777-12.947-57.958L-12.947-56.776Q-12.947-56.639-12.799-56.603Q-12.650-56.567-12.425-56.567L-12.425-56.287M-12.947-59.975L-12.947-58.365Q-12.814-58.112-12.571-57.955Q-12.329-57.798-12.052-57.798Q-11.724-57.798-11.471-57.999Q-11.218-58.201-11.085-58.519Q-10.951-58.837-10.951-59.155Q-10.951-59.384-11.016-59.613Q-11.081-59.842-11.209-60.040Q-11.338-60.238-11.532-60.358Q-11.727-60.477-11.960-60.477Q-12.254-60.477-12.522-60.348Q-12.790-60.218-12.947-59.975M-8.036-57.644L-9.588-57.644L-9.588-57.924Q-9.362-57.924-9.213-57.958Q-9.065-57.993-9.065-58.133L-9.065-59.982Q-9.065-60.170-9.112-60.254Q-9.160-60.337-9.258-60.356Q-9.355-60.375-9.567-60.375L-9.567-60.655L-8.511-60.730L-8.511-58.133Q-8.511-57.993-8.379-57.958Q-8.248-57.924-8.036-57.924L-8.036-57.644M-9.307-61.951Q-9.307-62.122-9.184-62.241Q-9.061-62.361-8.890-62.361Q-8.723-62.361-8.600-62.241Q-8.477-62.122-8.477-61.951Q-8.477-61.776-8.600-61.653Q-8.723-61.530-8.890-61.530Q-9.061-61.530-9.184-61.653Q-9.307-61.776-9.307-61.951\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(166.024 30.123)\">\u003Cpath d=\"M-4.519-56.061Q-4.519-56.079-4.506-56.126L-1.850-62.788Q-1.795-62.894-1.689-62.894Q-1.624-62.894-1.573-62.843Q-1.522-62.791-1.522-62.727Q-1.522-62.703-1.523-62.691Q-1.525-62.679-1.529-62.662L-4.181-56Q-4.253-55.894-4.341-55.894Q-4.410-55.894-4.465-55.945Q-4.519-55.997-4.519-56.061\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(166.024 30.123)\">\u003Cpath d=\"M2.794-57.644L2.527-57.644L2.527-61.752Q2.527-62.022 2.420-62.084Q2.312-62.145 2.001-62.145L2.001-62.426L3.081-62.501L3.081-60.331Q3.290-60.522 3.575-60.626Q3.861-60.730 4.158-60.730Q4.476-60.730 4.773-60.609Q5.070-60.488 5.293-60.272Q5.515-60.057 5.641-59.772Q5.768-59.486 5.768-59.155Q5.768-58.710 5.528-58.346Q5.289-57.982 4.896-57.779Q4.503-57.576 4.059-57.576Q3.864-57.576 3.674-57.632Q3.485-57.688 3.324-57.793Q3.163-57.897 3.023-58.058L2.794-57.644M3.109-59.989L3.109-58.372Q3.245-58.112 3.486-57.955Q3.727-57.798 4.004-57.798Q4.298-57.798 4.510-57.905Q4.722-58.013 4.855-58.205Q4.988-58.396 5.047-58.635Q5.105-58.874 5.105-59.155Q5.105-59.514 5.011-59.818Q4.917-60.122 4.689-60.315Q4.462-60.508 4.096-60.508Q3.796-60.508 3.529-60.372Q3.262-60.235 3.109-59.989\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">Where the importance ratio enters the trace. The trace at \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:0.0576em;\">S\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> decays the previous trace by \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\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0556em;\">γ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>, adds the new gradient, and scales the whole thing by \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\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:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">S\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mord\">\u002F\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">S\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>. When \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>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> agrees with \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>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>, \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\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:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\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\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> and this is the on-policy accumulating trace; a target action that \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>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> rarely takes gives \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.7335em;vertical-align:-0.1944em;\">\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:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\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\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> and amplifies the trace.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:415.364px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 311.523 131.048\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg stroke=\"none\" font-family=\"cmr9\" font-size=\"9\">\u003Cg transform=\"translate(59.084 -111.058)\">\u003Cpath d=\"M-39.943 48.959L-41.806 43.334Q-41.881 43.141-42.078 43.090Q-42.276 43.039-42.606 43.039L-42.606 42.723L-40.193 42.723L-40.193 43.039Q-40.949 43.039-40.949 43.290Q-40.949 43.316-40.940 43.325L-39.472 47.772L-38.176 43.883L-38.365 43.334Q-38.440 43.145-38.637 43.092Q-38.835 43.039-39.169 43.039L-39.169 42.723L-36.756 42.723L-36.756 43.039Q-37.512 43.039-37.512 43.290Q-37.512 43.312-37.495 43.356L-36.027 47.772L-34.643 43.589Q-34.634 43.562-34.634 43.496Q-34.634 43.259-34.878 43.149Q-35.122 43.039-35.399 43.039L-35.399 42.723L-33.452 42.723L-33.452 43.039Q-33.773 43.039-34.003 43.171Q-34.234 43.303-34.335 43.589L-36.115 48.959Q-36.168 49.069-36.277 49.069L-36.343 49.069Q-36.462 49.069-36.493 48.959L-38.027 44.345L-39.565 48.959Q-39.595 49.069-39.718 49.069L-39.780 49.069Q-39.899 49.069-39.943 48.959\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(59.084 -111.058)\">\u003Cpath d=\"M-33.674 47.961Q-33.674 47.421-33.241 47.087Q-32.808 46.753-32.202 46.614Q-31.595 46.476-31.064 46.476L-31.064 46.142Q-31.064 45.883-31.182 45.639Q-31.301 45.395-31.510 45.248Q-31.718 45.100-31.991 45.100Q-32.553 45.100-32.865 45.298Q-32.716 45.325-32.624 45.443Q-32.531 45.562-32.531 45.720Q-32.531 45.896-32.657 46.026Q-32.782 46.155-32.962 46.155Q-33.151 46.155-33.278 46.028Q-33.406 45.900-33.406 45.720Q-33.406 45.250-32.966 45.043Q-32.527 44.837-31.991 44.837Q-31.701 44.837-31.413 44.925Q-31.125 45.013-30.892 45.173Q-30.659 45.333-30.510 45.573Q-30.360 45.812-30.360 46.107L-30.360 48.142Q-30.360 48.295-30.286 48.434Q-30.211 48.572-30.066 48.572Q-29.912 48.572-29.840 48.436Q-29.767 48.300-29.767 48.142L-29.767 47.566L-29.481 47.566L-29.481 48.142Q-29.481 48.475-29.730 48.700Q-29.978 48.924-30.308 48.924Q-30.567 48.924-30.749 48.730Q-30.932 48.537-30.976 48.260Q-31.143 48.581-31.477 48.777Q-31.811 48.972-32.180 48.972Q-32.733 48.972-33.204 48.724Q-33.674 48.475-33.674 47.961M-32.918 47.961Q-32.918 48.273-32.676 48.491Q-32.435 48.708-32.118 48.708Q-31.683 48.708-31.373 48.399Q-31.064 48.089-31.064 47.658L-31.064 46.731Q-31.481 46.731-31.907 46.858Q-32.334 46.986-32.626 47.263Q-32.918 47.539-32.918 47.961M-28.488 47.799L-28.488 45.307L-29.253 45.307L-29.253 45.048Q-28.849 45.048-28.583 44.782Q-28.317 44.516-28.196 44.116Q-28.075 43.716-28.075 43.334L-27.785 43.334L-27.785 44.991L-26.498 44.991L-26.498 45.307L-27.785 45.307L-27.785 47.764Q-27.785 48.133-27.660 48.407Q-27.535 48.682-27.210 48.682Q-26.911 48.682-26.772 48.388Q-26.634 48.093-26.634 47.764L-26.634 47.241L-26.348 47.241L-26.348 47.799Q-26.348 48.076-26.458 48.348Q-26.568 48.621-26.781 48.796Q-26.994 48.972-27.275 48.972Q-27.636 48.972-27.908 48.834Q-28.181 48.695-28.335 48.432Q-28.488 48.168-28.488 47.799M-23.549 48.871L-25.584 48.871L-25.584 48.555Q-25.272 48.555-25.080 48.502Q-24.889 48.449-24.889 48.260L-24.889 43.545Q-24.889 43.303-24.960 43.195Q-25.030 43.088-25.164 43.064Q-25.298 43.039-25.584 43.039L-25.584 42.723L-24.212 42.626L-24.212 46.871L-23.008 45.856Q-22.802 45.685-22.802 45.505Q-22.802 45.408-22.870 45.358Q-22.938 45.307-23.035 45.307L-23.035 44.991L-21.325 44.991L-21.325 45.307Q-21.923 45.307-22.569 45.856L-23.215 46.406L-22.055 48.005Q-21.879 48.243-21.776 48.348Q-21.672 48.454-21.521 48.504Q-21.369 48.555-21.119 48.555L-21.119 48.871L-22.947 48.871L-22.947 48.555Q-22.648 48.555-22.648 48.366Q-22.648 48.247-22.819 48.005L-23.694 46.810L-24.243 47.280L-24.243 48.260Q-24.243 48.449-24.050 48.502Q-23.856 48.555-23.549 48.555L-23.549 48.871M-18.662 48.871L-20.648 48.871L-20.648 48.555Q-20.341 48.555-20.150 48.502Q-19.959 48.449-19.959 48.260L-19.959 45.812Q-19.959 45.566-20.024 45.461Q-20.090 45.355-20.216 45.331Q-20.341 45.307-20.613 45.307L-20.613 44.991L-19.282 44.894L-19.282 48.260Q-19.282 48.454-19.117 48.504Q-18.952 48.555-18.662 48.555L-18.662 48.871M-20.262 43.347Q-20.262 43.141-20.112 42.991Q-19.963 42.842-19.761 42.842Q-19.629 42.842-19.512 42.912Q-19.396 42.982-19.326 43.099Q-19.255 43.215-19.255 43.347Q-19.255 43.549-19.405 43.699Q-19.554 43.848-19.761 43.848Q-19.963 43.848-20.112 43.699Q-20.262 43.549-20.262 43.347M-16.003 48.871L-18.091 48.871L-18.091 48.555Q-17.783 48.555-17.592 48.502Q-17.401 48.449-17.401 48.260L-17.401 45.812Q-17.401 45.571-17.471 45.463Q-17.542 45.355-17.676 45.331Q-17.810 45.307-18.091 45.307L-18.091 44.991L-16.751 44.894L-16.751 45.729Q-16.553 45.347-16.199 45.120Q-15.845 44.894-15.419 44.894Q-14.140 44.894-14.140 46.107L-14.140 48.260Q-14.140 48.449-13.949 48.502Q-13.758 48.555-13.450 48.555L-13.450 48.871L-15.538 48.871L-15.538 48.555Q-15.226 48.555-15.034 48.502Q-14.843 48.449-14.843 48.260L-14.843 46.142Q-14.843 45.883-14.887 45.661Q-14.931 45.439-15.076 45.296Q-15.221 45.153-15.480 45.153Q-15.823 45.153-16.105 45.342Q-16.386 45.531-16.542 45.843Q-16.698 46.155-16.698 46.502L-16.698 48.260Q-16.698 48.449-16.504 48.502Q-16.311 48.555-16.003 48.555L-16.003 48.871M-12.945 48.889L-12.945 47.447Q-12.945 47.416-12.916 47.392Q-12.888 47.368-12.857 47.368L-12.747 47.368Q-12.712 47.368-12.690 47.390Q-12.668 47.412-12.659 47.447Q-12.400 48.708-11.433 48.708Q-11.007 48.708-10.712 48.524Q-10.418 48.339-10.418 47.935Q-10.418 47.641-10.649 47.445Q-10.879 47.249-11.191 47.188L-11.793 47.069Q-12.259 46.981-12.602 46.698Q-12.945 46.414-12.945 45.975Q-12.945 45.382-12.508 45.109Q-12.070 44.837-11.433 44.837Q-10.954 44.837-10.607 45.083L-10.356 44.859Q-10.299 44.837-10.299 44.837L-10.247 44.837Q-10.220 44.837-10.187 44.863Q-10.154 44.890-10.154 44.920L-10.154 46.080Q-10.154 46.111-10.189 46.138Q-10.225 46.164-10.247 46.164L-10.356 46.164Q-10.378 46.164-10.411 46.135Q-10.444 46.107-10.444 46.080Q-10.444 45.615-10.710 45.344Q-10.976 45.074-11.442 45.074Q-11.846 45.074-12.149 45.219Q-12.453 45.364-12.453 45.720Q-12.453 45.966-12.235 46.120Q-12.018 46.274-11.741 46.331L-11.112 46.458Q-10.796 46.520-10.526 46.687Q-10.255 46.854-10.088 47.120Q-9.921 47.386-9.921 47.702Q-9.921 48.344-10.348 48.658Q-10.774 48.972-11.433 48.972Q-11.706 48.972-11.971 48.878Q-12.237 48.783-12.418 48.594L-12.738 48.933Q-12.756 48.972-12.804 48.972L-12.857 48.972Q-12.879 48.972-12.912 48.944Q-12.945 48.915-12.945 48.889M-8.709 45.237Q-8.709 45.184-8.673 45.162Q-8.331 44.863-8.146 44.448Q-7.961 44.033-7.961 43.589L-7.961 43.488Q-8.106 43.633-8.331 43.633Q-8.476 43.633-8.590 43.567Q-8.704 43.501-8.770 43.389Q-8.836 43.277-8.836 43.127Q-8.836 42.908-8.695 42.767Q-8.555 42.626-8.331 42.626Q-8.005 42.626-7.854 42.925Q-7.702 43.224-7.702 43.589Q-7.702 43.927-7.797 44.246Q-7.891 44.564-8.074 44.850Q-8.256 45.136-8.502 45.351Q-8.528 45.377-8.564 45.377Q-8.621 45.377-8.665 45.333Q-8.709 45.289-8.709 45.237M-6.735 48.889L-6.735 47.447Q-6.735 47.416-6.707 47.392Q-6.678 47.368-6.647 47.368L-6.538 47.368Q-6.502 47.368-6.480 47.390Q-6.459 47.412-6.450 47.447Q-6.190 48.708-5.224 48.708Q-4.797 48.708-4.503 48.524Q-4.209 48.339-4.209 47.935Q-4.209 47.641-4.439 47.445Q-4.670 47.249-4.982 47.188L-5.584 47.069Q-6.050 46.981-6.393 46.698Q-6.735 46.414-6.735 45.975Q-6.735 45.382-6.298 45.109Q-5.861 44.837-5.224 44.837Q-4.745 44.837-4.397 45.083L-4.147 44.859Q-4.090 44.837-4.090 44.837L-4.037 44.837Q-4.011 44.837-3.978 44.863Q-3.945 44.890-3.945 44.920L-3.945 46.080Q-3.945 46.111-3.980 46.138Q-4.015 46.164-4.037 46.164L-4.147 46.164Q-4.169 46.164-4.202 46.135Q-4.235 46.107-4.235 46.080Q-4.235 45.615-4.501 45.344Q-4.767 45.074-5.232 45.074Q-5.637 45.074-5.940 45.219Q-6.243 45.364-6.243 45.720Q-6.243 45.966-6.026 46.120Q-5.808 46.274-5.531 46.331L-4.903 46.458Q-4.586 46.520-4.316 46.687Q-4.046 46.854-3.879 47.120Q-3.712 47.386-3.712 47.702Q-3.712 48.344-4.138 48.658Q-4.564 48.972-5.224 48.972Q-5.496 48.972-5.762 48.878Q-6.028 48.783-6.208 48.594L-6.529 48.933Q-6.546 48.972-6.595 48.972L-6.647 48.972Q-6.669 48.972-6.702 48.944Q-6.735 48.915-6.735 48.889\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(59.084 -111.058)\">\u003Cpath d=\"M3.310 49.069Q2.664 49.069 2.104 48.809Q1.544 48.550 1.122 48.093Q0.700 47.636 0.467 47.045Q0.234 46.454 0.234 45.830Q0.234 45.197 0.465 44.591Q0.696 43.984 1.113 43.523Q1.531 43.061 2.095 42.793Q2.660 42.525 3.310 42.525Q3.965 42.525 4.534 42.793Q5.103 43.061 5.519 43.527Q5.934 43.993 6.162 44.589Q6.391 45.184 6.391 45.830Q6.391 46.291 6.266 46.737Q6.140 47.184 5.892 47.590Q5.644 47.996 5.292 48.306Q4.941 48.616 4.519 48.809Q4.673 49.187 4.870 49.425Q5.068 49.662 5.389 49.662Q5.604 49.662 5.798 49.557Q5.991 49.451 6.105 49.262Q6.220 49.073 6.220 48.862Q6.220 48.823 6.255 48.788Q6.290 48.752 6.334 48.752Q6.382 48.752 6.413 48.790Q6.444 48.827 6.444 48.880Q6.444 49.539 6.178 50.077Q5.912 50.616 5.327 50.616Q4.813 50.616 4.589 50.257Q4.365 49.899 4.246 49.293L4.176 48.933Q3.763 49.069 3.310 49.069M2.796 48.287Q2.796 48.502 2.943 48.642Q3.091 48.783 3.310 48.783Q3.719 48.783 4.101 48.603Q4.005 48.216 3.831 47.994Q3.657 47.772 3.310 47.772Q3.095 47.772 2.946 47.926Q2.796 48.080 2.796 48.287M2.537 48.287Q2.537 48.084 2.642 47.904Q2.748 47.724 2.928 47.619Q3.108 47.513 3.310 47.513Q3.706 47.513 3.945 47.768Q4.185 48.023 4.365 48.449Q4.761 48.177 5.009 47.750Q5.257 47.324 5.360 46.843Q5.464 46.362 5.464 45.830Q5.464 44.494 4.888 43.677Q4.611 43.290 4.194 43.053Q3.776 42.815 3.310 42.815Q2.849 42.815 2.431 43.053Q2.014 43.290 1.737 43.677Q1.161 44.485 1.161 45.830Q1.161 46.770 1.509 47.548Q1.856 48.326 2.629 48.647Q2.537 48.493 2.537 48.287M9.762 51.112Q9.256 50.725 8.887 50.220Q8.518 49.715 8.274 49.115Q8.030 48.515 7.916 47.898Q7.802 47.280 7.802 46.621Q7.802 45.962 7.916 45.347Q8.030 44.731 8.270 44.138Q8.509 43.545 8.883 43.035Q9.256 42.525 9.762 42.139Q9.797 42.121 9.819 42.121L9.898 42.121Q9.986 42.121 9.986 42.222Q9.986 42.257 9.950 42.292Q9.388 42.815 9.043 43.521Q8.698 44.226 8.551 45.008Q8.404 45.790 8.404 46.621Q8.404 47.245 8.483 47.829Q8.562 48.414 8.740 48.981Q8.918 49.548 9.217 50.049Q9.515 50.550 9.950 50.958Q9.986 50.994 9.986 51.033Q9.986 51.130 9.898 51.130L9.819 51.130Q9.797 51.130 9.762 51.112M12.868 48.871L10.807 48.871L10.807 48.555Q11.115 48.555 11.306 48.502Q11.497 48.449 11.497 48.260L11.497 43.545Q11.497 43.303 11.427 43.195Q11.357 43.088 11.223 43.064Q11.089 43.039 10.807 43.039L10.807 42.723L12.174 42.626L12.174 48.260Q12.174 48.449 12.367 48.502Q12.561 48.555 12.868 48.555L12.868 48.871M13.435 47.961Q13.435 47.421 13.868 47.087Q14.301 46.753 14.907 46.614Q15.514 46.476 16.046 46.476L16.046 46.142Q16.046 45.883 15.927 45.639Q15.808 45.395 15.600 45.248Q15.391 45.100 15.118 45.100Q14.556 45.100 14.244 45.298Q14.393 45.325 14.486 45.443Q14.578 45.562 14.578 45.720Q14.578 45.896 14.453 46.026Q14.327 46.155 14.147 46.155Q13.958 46.155 13.831 46.028Q13.703 45.900 13.703 45.720Q13.703 45.250 14.143 45.043Q14.582 44.837 15.118 44.837Q15.408 44.837 15.696 44.925Q15.984 45.013 16.217 45.173Q16.450 45.333 16.599 45.573Q16.749 45.812 16.749 46.107L16.749 48.142Q16.749 48.295 16.824 48.434Q16.898 48.572 17.043 48.572Q17.197 48.572 17.270 48.436Q17.342 48.300 17.342 48.142L17.342 47.566L17.628 47.566L17.628 48.142Q17.628 48.475 17.379 48.700Q17.131 48.924 16.802 48.924Q16.542 48.924 16.360 48.730Q16.178 48.537 16.134 48.260Q15.967 48.581 15.633 48.777Q15.299 48.972 14.929 48.972Q14.376 48.972 13.906 48.724Q13.435 48.475 13.435 47.961M14.191 47.961Q14.191 48.273 14.433 48.491Q14.675 48.708 14.991 48.708Q15.426 48.708 15.736 48.399Q16.046 48.089 16.046 47.658L16.046 46.731Q15.628 46.731 15.202 46.858Q14.776 46.986 14.483 47.263Q14.191 47.539 14.191 47.961M20.071 48.871L17.984 48.871L17.984 48.555Q18.291 48.555 18.482 48.502Q18.674 48.449 18.674 48.260L18.674 45.812Q18.674 45.571 18.603 45.463Q18.533 45.355 18.399 45.331Q18.265 45.307 17.984 45.307L17.984 44.991L19.324 44.894L19.324 45.729Q19.522 45.351 19.882 45.122Q20.242 44.894 20.664 44.894Q21.710 44.894 21.895 45.703Q22.097 45.333 22.455 45.114Q22.813 44.894 23.231 44.894Q23.855 44.894 24.180 45.188Q24.505 45.483 24.505 46.107L24.505 48.260Q24.505 48.449 24.699 48.502Q24.892 48.555 25.199 48.555L25.199 48.871L23.112 48.871L23.112 48.555Q23.420 48.555 23.613 48.502Q23.806 48.449 23.806 48.260L23.806 46.142Q23.806 45.711 23.679 45.432Q23.552 45.153 23.165 45.153Q22.822 45.153 22.539 45.342Q22.255 45.531 22.099 45.843Q21.943 46.155 21.943 46.502L21.943 48.260Q21.943 48.449 22.134 48.502Q22.325 48.555 22.633 48.555L22.633 48.871L20.546 48.871L20.546 48.555Q20.858 48.555 21.049 48.502Q21.240 48.449 21.240 48.260L21.240 46.142Q21.240 45.883 21.196 45.661Q21.152 45.439 21.007 45.296Q20.862 45.153 20.603 45.153Q20.067 45.153 19.722 45.560Q19.377 45.966 19.377 46.502L19.377 48.260Q19.377 48.449 19.570 48.502Q19.763 48.555 20.071 48.555\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(59.084 -111.058)\">\u003Cpath d=\"M26.392 48.871L26.102 48.871L26.102 43.545Q26.102 43.303 26.032 43.195Q25.961 43.088 25.827 43.064Q25.693 43.039 25.407 43.039L25.407 42.723L26.779 42.626L26.779 45.426Q27.025 45.175 27.359 45.035Q27.693 44.894 28.044 44.894Q28.444 44.894 28.807 45.057Q29.169 45.219 29.426 45.498Q29.683 45.777 29.833 46.155Q29.982 46.533 29.982 46.929Q29.982 47.491 29.705 47.957Q29.428 48.423 28.954 48.697Q28.479 48.972 27.930 48.972Q27.565 48.972 27.244 48.803Q26.924 48.634 26.704 48.339L26.392 48.871M26.805 45.839L26.805 47.970Q26.963 48.304 27.247 48.506Q27.530 48.708 27.873 48.708Q28.563 48.708 28.866 48.188Q29.169 47.667 29.169 46.929Q29.169 46.204 28.903 45.678Q28.637 45.153 27.974 45.153Q27.614 45.153 27.299 45.338Q26.985 45.522 26.805 45.839\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(59.084 -111.058)\">\u003Cpath d=\"M32.803 48.972Q32.258 48.972 31.814 48.689Q31.370 48.405 31.116 47.933Q30.861 47.460 30.861 46.929Q30.861 46.375 31.137 45.909Q31.414 45.443 31.887 45.169Q32.359 44.894 32.904 44.894Q33.238 44.894 33.541 45.026Q33.845 45.158 34.064 45.395L34.064 43.545Q34.064 43.303 33.994 43.195Q33.924 43.088 33.790 43.064Q33.656 43.039 33.370 43.039L33.370 42.723L34.737 42.626L34.737 48.054Q34.737 48.291 34.807 48.399Q34.877 48.506 35.013 48.530Q35.150 48.555 35.431 48.555L35.431 48.871L34.038 48.972L34.038 48.423Q33.787 48.686 33.469 48.829Q33.150 48.972 32.803 48.972M32.865 48.708Q33.234 48.708 33.543 48.497Q33.853 48.287 34.038 47.944L34.038 45.812Q33.866 45.509 33.585 45.331Q33.304 45.153 32.966 45.153Q32.276 45.153 31.972 45.674Q31.669 46.195 31.669 46.937Q31.669 47.658 31.939 48.183Q32.210 48.708 32.865 48.708M36.059 47.961Q36.059 47.421 36.492 47.087Q36.925 46.753 37.532 46.614Q38.138 46.476 38.670 46.476L38.670 46.142Q38.670 45.883 38.551 45.639Q38.432 45.395 38.224 45.248Q38.015 45.100 37.742 45.100Q37.180 45.100 36.868 45.298Q37.017 45.325 37.110 45.443Q37.202 45.562 37.202 45.720Q37.202 45.896 37.077 46.026Q36.951 46.155 36.771 46.155Q36.582 46.155 36.455 46.028Q36.327 45.900 36.327 45.720Q36.327 45.250 36.767 45.043Q37.206 44.837 37.742 44.837Q38.032 44.837 38.320 44.925Q38.608 45.013 38.841 45.173Q39.074 45.333 39.223 45.573Q39.373 45.812 39.373 46.107L39.373 48.142Q39.373 48.295 39.448 48.434Q39.522 48.572 39.667 48.572Q39.821 48.572 39.894 48.436Q39.966 48.300 39.966 48.142L39.966 47.566L40.252 47.566L40.252 48.142Q40.252 48.475 40.003 48.700Q39.755 48.924 39.426 48.924Q39.166 48.924 38.984 48.730Q38.802 48.537 38.758 48.260Q38.591 48.581 38.257 48.777Q37.923 48.972 37.553 48.972Q37 48.972 36.530 48.724Q36.059 48.475 36.059 47.961M36.815 47.961Q36.815 48.273 37.057 48.491Q37.299 48.708 37.615 48.708Q38.050 48.708 38.360 48.399Q38.670 48.089 38.670 47.658L38.670 46.731Q38.252 46.731 37.826 46.858Q37.400 46.986 37.107 47.263Q36.815 47.539 36.815 47.961M40.994 51.130L40.911 51.130Q40.823 51.130 40.823 51.033Q40.823 50.994 40.858 50.958Q41.693 50.185 42.053 49.051Q42.414 47.917 42.414 46.621Q42.414 46.001 42.335 45.410Q42.256 44.819 42.075 44.261Q41.895 43.703 41.594 43.195Q41.293 42.688 40.858 42.292Q40.823 42.257 40.823 42.222Q40.823 42.121 40.911 42.121L40.994 42.121Q41.012 42.121 41.047 42.139Q41.553 42.521 41.926 43.035Q42.300 43.549 42.537 44.127Q42.774 44.705 42.891 45.333Q43.007 45.962 43.007 46.621Q43.007 47.280 42.891 47.911Q42.774 48.541 42.535 49.126Q42.295 49.710 41.924 50.220Q41.553 50.730 41.047 51.112Q41.012 51.130 40.994 51.130\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath d=\"M-40.501-45.023a2.276 2.276 0 1 0-4.552 0 2.276 2.276 0 0 0 4.552 0Zm-2.276 0\"\u002F>\u003Cpath fill=\"none\" d=\"M-38.225-27.951a4.552 4.552 0 1 0-9.105 0 4.552 4.552 0 0 0 9.105 0ZM-42.777-42.547v9.843\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-16.16 -64.16)\">\u003Cpath d=\"M-39.451 48.871L-41.981 48.871L-41.981 48.591Q-41.013 48.591-41.013 48.382L-41.013 44.763Q-41.406 44.951-42.028 44.951L-42.028 44.670Q-41.611 44.670-41.247 44.569Q-40.883 44.469-40.627 44.223L-40.501 44.223Q-40.436 44.240-40.419 44.308L-40.419 48.382Q-40.419 48.591-39.451 48.591L-39.451 48.871M-36.597 47.617L-38.655 47.617L-38.655 47.114L-36.597 47.114L-36.597 47.617M-34.126 48.871L-35.729 48.871L-35.729 48.591Q-35.504 48.591-35.355 48.557Q-35.206 48.522-35.206 48.382L-35.206 44.763Q-35.206 44.493-35.314 44.431Q-35.422 44.370-35.729 44.370L-35.729 44.089L-34.652 44.014L-34.652 48.382Q-34.652 48.519-34.502 48.555Q-34.352 48.591-34.126 48.591L-34.126 48.871M-33.473 48.143Q-33.473 47.811-33.249 47.584Q-33.026 47.357-32.682 47.229Q-32.339 47.100-31.966 47.048Q-31.593 46.995-31.289 46.995L-31.289 46.742Q-31.289 46.537-31.397 46.357Q-31.505 46.178-31.686 46.075Q-31.867 45.973-32.075 45.973Q-32.482 45.973-32.718 46.065Q-32.629 46.102-32.583 46.186Q-32.537 46.270-32.537 46.372Q-32.537 46.468-32.583 46.547Q-32.629 46.625-32.709 46.670Q-32.790 46.714-32.879 46.714Q-33.029 46.714-33.130 46.617Q-33.231 46.519-33.231 46.372Q-33.231 45.750-32.075 45.750Q-31.863 45.750-31.614 45.814Q-31.364 45.877-31.163 45.996Q-30.961 46.116-30.835 46.301Q-30.708 46.485-30.708 46.728L-30.708 48.304Q-30.708 48.420-30.647 48.516Q-30.585 48.611-30.472 48.611Q-30.363 48.611-30.298 48.517Q-30.233 48.423-30.233 48.304L-30.233 47.856L-29.966 47.856L-29.966 48.304Q-29.966 48.574-30.194 48.739Q-30.421 48.905-30.701 48.905Q-30.910 48.905-31.047 48.751Q-31.183 48.598-31.207 48.382Q-31.354 48.649-31.636 48.794Q-31.918 48.939-32.243 48.939Q-32.520 48.939-32.803 48.864Q-33.087 48.789-33.280 48.610Q-33.473 48.430-33.473 48.143M-32.858 48.143Q-32.858 48.317-32.757 48.447Q-32.656 48.577-32.501 48.647Q-32.345 48.717-32.181 48.717Q-31.963 48.717-31.754 48.620Q-31.546 48.522-31.417 48.341Q-31.289 48.160-31.289 47.934L-31.289 47.206Q-31.614 47.206-31.980 47.297Q-32.345 47.388-32.602 47.600Q-32.858 47.811-32.858 48.143M-27.868 48.871L-29.502 48.871L-29.502 48.591Q-29.273 48.591-29.124 48.557Q-28.975 48.522-28.975 48.382L-28.975 46.533Q-28.975 46.263-29.083 46.202Q-29.191 46.140-29.502 46.140L-29.502 45.860L-28.442 45.785L-28.442 46.434Q-28.271 46.126-27.967 45.955Q-27.663 45.785-27.318 45.785Q-26.918 45.785-26.641 45.925Q-26.364 46.065-26.278 46.413Q-26.111 46.120-25.812 45.952Q-25.513 45.785-25.168 45.785Q-24.662 45.785-24.378 46.008Q-24.094 46.232-24.094 46.728L-24.094 48.382Q-24.094 48.519-23.946 48.555Q-23.797 48.591-23.571 48.591L-23.571 48.871L-25.202 48.871L-25.202 48.591Q-24.976 48.591-24.826 48.555Q-24.675 48.519-24.675 48.382L-24.675 46.742Q-24.675 46.407-24.795 46.207Q-24.915 46.007-25.229 46.007Q-25.499 46.007-25.733 46.143Q-25.967 46.280-26.106 46.514Q-26.244 46.748-26.244 47.022L-26.244 48.382Q-26.244 48.519-26.096 48.555Q-25.947 48.591-25.721 48.591L-25.721 48.871L-27.352 48.871L-27.352 48.591Q-27.123 48.591-26.974 48.557Q-26.825 48.522-26.825 48.382L-26.825 46.742Q-26.825 46.407-26.945 46.207Q-27.065 46.007-27.379 46.007Q-27.649 46.007-27.883 46.143Q-28.117 46.280-28.256 46.514Q-28.394 46.748-28.394 47.022L-28.394 48.382Q-28.394 48.519-28.244 48.555Q-28.093 48.591-27.868 48.591\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-16.16 -64.16)\">\u003Cpath d=\"M-22.371 48.871L-22.638 48.871L-22.638 44.763Q-22.638 44.493-22.745 44.431Q-22.853 44.370-23.164 44.370L-23.164 44.089L-22.084 44.014L-22.084 46.184Q-21.875 45.993-21.590 45.889Q-21.305 45.785-21.007 45.785Q-20.689 45.785-20.392 45.906Q-20.095 46.027-19.872 46.243Q-19.650 46.458-19.524 46.743Q-19.397 47.029-19.397 47.360Q-19.397 47.805-19.637 48.169Q-19.876 48.533-20.269 48.736Q-20.662 48.939-21.106 48.939Q-21.301 48.939-21.491 48.883Q-21.680 48.827-21.841 48.722Q-22.002 48.618-22.142 48.457L-22.371 48.871M-22.056 46.526L-22.056 48.143Q-21.920 48.403-21.679 48.560Q-21.438 48.717-21.161 48.717Q-20.867 48.717-20.655 48.610Q-20.443 48.502-20.310 48.310Q-20.177 48.119-20.118 47.880Q-20.060 47.641-20.060 47.360Q-20.060 47.001-20.154 46.697Q-20.248 46.393-20.476 46.200Q-20.703 46.007-21.069 46.007Q-21.369 46.007-21.636 46.143Q-21.903 46.280-22.056 46.526\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-16.16 -64.16)\">\u003Cpath d=\"M-18.546 47.360Q-18.546 47.022-18.405 46.731Q-18.265 46.441-18.021 46.227Q-17.777 46.014-17.472 45.899Q-17.168 45.785-16.843 45.785Q-16.573 45.785-16.310 45.884Q-16.047 45.983-15.856 46.161L-15.856 44.763Q-15.856 44.493-15.963 44.431Q-16.071 44.370-16.382 44.370L-16.382 44.089L-15.305 44.014L-15.305 48.198Q-15.305 48.386-15.251 48.469Q-15.196 48.553-15.095 48.572Q-14.994 48.591-14.779 48.591L-14.779 48.871L-15.886 48.939L-15.886 48.522Q-16.303 48.939-16.929 48.939Q-17.360 48.939-17.732 48.727Q-18.105 48.516-18.325 48.155Q-18.546 47.794-18.546 47.360M-16.871 48.717Q-16.662 48.717-16.476 48.645Q-16.290 48.574-16.136 48.437Q-15.982 48.300-15.886 48.122L-15.886 46.513Q-15.972 46.366-16.117 46.246Q-16.262 46.126-16.432 46.067Q-16.601 46.007-16.782 46.007Q-17.342 46.007-17.611 46.396Q-17.879 46.786-17.879 47.367Q-17.879 47.938-17.645 48.328Q-17.411 48.717-16.871 48.717M-14.071 48.143Q-14.071 47.811-13.848 47.584Q-13.624 47.357-13.280 47.229Q-12.937 47.100-12.564 47.048Q-12.192 46.995-11.887 46.995L-11.887 46.742Q-11.887 46.537-11.995 46.357Q-12.103 46.178-12.284 46.075Q-12.465 45.973-12.673 45.973Q-13.080 45.973-13.316 46.065Q-13.227 46.102-13.181 46.186Q-13.135 46.270-13.135 46.372Q-13.135 46.468-13.181 46.547Q-13.227 46.625-13.308 46.670Q-13.388 46.714-13.477 46.714Q-13.627 46.714-13.728 46.617Q-13.829 46.519-13.829 46.372Q-13.829 45.750-12.673 45.750Q-12.462 45.750-12.212 45.814Q-11.963 45.877-11.761 45.996Q-11.559 46.116-11.433 46.301Q-11.306 46.485-11.306 46.728L-11.306 48.304Q-11.306 48.420-11.245 48.516Q-11.183 48.611-11.070 48.611Q-10.961 48.611-10.896 48.517Q-10.831 48.423-10.831 48.304L-10.831 47.856L-10.565 47.856L-10.565 48.304Q-10.565 48.574-10.792 48.739Q-11.019 48.905-11.299 48.905Q-11.508 48.905-11.645 48.751Q-11.781 48.598-11.805 48.382Q-11.952 48.649-12.234 48.794Q-12.516 48.939-12.841 48.939Q-13.118 48.939-13.402 48.864Q-13.685 48.789-13.878 48.610Q-14.071 48.430-14.071 48.143M-13.456 48.143Q-13.456 48.317-13.355 48.447Q-13.255 48.577-13.099 48.647Q-12.943 48.717-12.779 48.717Q-12.561 48.717-12.352 48.620Q-12.144 48.522-12.016 48.341Q-11.887 48.160-11.887 47.934L-11.887 47.206Q-12.212 47.206-12.578 47.297Q-12.943 47.388-13.200 47.600Q-13.456 47.811-13.456 48.143\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath d=\"M-6.358-45.023a2.276 2.276 0 1 0-4.552 0 2.276 2.276 0 0 0 4.552 0Zm-2.276 0\"\u002F>\u003Cpath fill=\"none\" d=\"M-4.082-27.951a4.552 4.552 0 1 0-9.104 0 4.552 4.552 0 0 0 9.104 0Zm-4.552 0\"\u002F>\u003Cpath d=\"M-6.358-13.725a2.276 2.276 0 1 0-4.552 0 2.276 2.276 0 0 0 4.552 0Zm-2.276 0\"\u002F>\u003Cpath fill=\"none\" d=\"M-4.082 3.346a4.552 4.552 0 1 0-9.104 0 4.552 4.552 0 0 0 9.104 0ZM-8.634-42.547v9.843m0 9.505v6.998m0 4.952v9.843\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(2.039 -32.473)\">\u003Cpath d=\"M-40.333 50.621Q-40.883 50.221-41.254 49.666Q-41.625 49.110-41.806 48.464Q-41.987 47.818-41.987 47.121Q-41.987 46.608-41.887 46.113Q-41.786 45.617-41.581 45.166Q-41.376 44.715-41.063 44.323Q-40.750 43.932-40.333 43.628Q-40.323 43.624-40.316 43.623Q-40.309 43.621-40.299 43.621L-40.231 43.621Q-40.196 43.621-40.174 43.645Q-40.152 43.669-40.152 43.706Q-40.152 43.751-40.179 43.768Q-40.528 44.069-40.781 44.453Q-41.034 44.838-41.186 45.279Q-41.338 45.720-41.410 46.176Q-41.482 46.632-41.482 47.121Q-41.482 48.122-41.172 49.009Q-40.863 49.896-40.179 50.481Q-40.152 50.498-40.152 50.542Q-40.152 50.580-40.174 50.604Q-40.196 50.628-40.231 50.628L-40.299 50.628Q-40.306 50.624-40.314 50.623Q-40.323 50.621-40.333 50.621M-36.331 48.871L-38.860 48.871L-38.860 48.591Q-37.893 48.591-37.893 48.382L-37.893 44.763Q-38.286 44.951-38.908 44.951L-38.908 44.670Q-38.491 44.670-38.127 44.569Q-37.763 44.469-37.506 44.223L-37.380 44.223Q-37.315 44.240-37.298 44.308L-37.298 48.382Q-37.298 48.591-36.331 48.591L-36.331 48.871M-33.477 47.617L-35.534 47.617L-35.534 47.114L-33.477 47.114L-33.477 47.617M-31.006 48.871L-32.609 48.871L-32.609 48.591Q-32.383 48.591-32.234 48.557Q-32.086 48.522-32.086 48.382L-32.086 44.763Q-32.086 44.493-32.193 44.431Q-32.301 44.370-32.609 44.370L-32.609 44.089L-31.532 44.014L-31.532 48.382Q-31.532 48.519-31.381 48.555Q-31.231 48.591-31.006 48.591L-31.006 48.871M-30.353 48.143Q-30.353 47.811-30.129 47.584Q-29.905 47.357-29.561 47.229Q-29.218 47.100-28.845 47.048Q-28.473 46.995-28.169 46.995L-28.169 46.742Q-28.169 46.537-28.276 46.357Q-28.384 46.178-28.565 46.075Q-28.746 45.973-28.955 45.973Q-29.361 45.973-29.597 46.065Q-29.508 46.102-29.462 46.186Q-29.416 46.270-29.416 46.372Q-29.416 46.468-29.462 46.547Q-29.508 46.625-29.589 46.670Q-29.669 46.714-29.758 46.714Q-29.908 46.714-30.009 46.617Q-30.110 46.519-30.110 46.372Q-30.110 45.750-28.955 45.750Q-28.743 45.750-28.493 45.814Q-28.244 45.877-28.042 45.996Q-27.840 46.116-27.714 46.301Q-27.588 46.485-27.588 46.728L-27.588 48.304Q-27.588 48.420-27.526 48.516Q-27.465 48.611-27.352 48.611Q-27.242 48.611-27.177 48.517Q-27.112 48.423-27.112 48.304L-27.112 47.856L-26.846 47.856L-26.846 48.304Q-26.846 48.574-27.073 48.739Q-27.300 48.905-27.581 48.905Q-27.789 48.905-27.926 48.751Q-28.063 48.598-28.087 48.382Q-28.234 48.649-28.516 48.794Q-28.798 48.939-29.122 48.939Q-29.399 48.939-29.683 48.864Q-29.966 48.789-30.160 48.610Q-30.353 48.430-30.353 48.143M-29.737 48.143Q-29.737 48.317-29.637 48.447Q-29.536 48.577-29.380 48.647Q-29.225 48.717-29.061 48.717Q-28.842 48.717-28.633 48.620Q-28.425 48.522-28.297 48.341Q-28.169 48.160-28.169 47.934L-28.169 47.206Q-28.493 47.206-28.859 47.297Q-29.225 47.388-29.481 47.600Q-29.737 47.811-29.737 48.143M-24.747 48.871L-26.381 48.871L-26.381 48.591Q-26.152 48.591-26.003 48.557Q-25.855 48.522-25.855 48.382L-25.855 46.533Q-25.855 46.263-25.962 46.202Q-26.070 46.140-26.381 46.140L-26.381 45.860L-25.321 45.785L-25.321 46.434Q-25.151 46.126-24.846 45.955Q-24.542 45.785-24.197 45.785Q-23.797 45.785-23.520 45.925Q-23.243 46.065-23.158 46.413Q-22.990 46.120-22.691 45.952Q-22.392 45.785-22.047 45.785Q-21.541 45.785-21.257 46.008Q-20.974 46.232-20.974 46.728L-20.974 48.382Q-20.974 48.519-20.825 48.555Q-20.676 48.591-20.451 48.591L-20.451 48.871L-22.081 48.871L-22.081 48.591Q-21.856 48.591-21.705 48.555Q-21.555 48.519-21.555 48.382L-21.555 46.742Q-21.555 46.407-21.674 46.207Q-21.794 46.007-22.109 46.007Q-22.379 46.007-22.613 46.143Q-22.847 46.280-22.985 46.514Q-23.124 46.748-23.124 47.022L-23.124 48.382Q-23.124 48.519-22.975 48.555Q-22.826 48.591-22.601 48.591L-22.601 48.871L-24.231 48.871L-24.231 48.591Q-24.002 48.591-23.853 48.557Q-23.705 48.522-23.705 48.382L-23.705 46.742Q-23.705 46.407-23.824 46.207Q-23.944 46.007-24.258 46.007Q-24.528 46.007-24.763 46.143Q-24.997 46.280-25.135 46.514Q-25.274 46.748-25.274 47.022L-25.274 48.382Q-25.274 48.519-25.123 48.555Q-24.973 48.591-24.747 48.591\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(2.039 -32.473)\">\u003Cpath d=\"M-19.246 48.871L-19.513 48.871L-19.513 44.763Q-19.513 44.493-19.620 44.431Q-19.728 44.370-20.039 44.370L-20.039 44.089L-18.959 44.014L-18.959 46.184Q-18.750 45.993-18.465 45.889Q-18.180 45.785-17.882 45.785Q-17.564 45.785-17.267 45.906Q-16.970 46.027-16.747 46.243Q-16.525 46.458-16.399 46.743Q-16.272 47.029-16.272 47.360Q-16.272 47.805-16.512 48.169Q-16.751 48.533-17.144 48.736Q-17.537 48.939-17.981 48.939Q-18.176 48.939-18.366 48.883Q-18.555 48.827-18.716 48.722Q-18.877 48.618-19.017 48.457L-19.246 48.871M-18.931 46.526L-18.931 48.143Q-18.795 48.403-18.554 48.560Q-18.313 48.717-18.036 48.717Q-17.742 48.717-17.530 48.610Q-17.318 48.502-17.185 48.310Q-17.052 48.119-16.993 47.880Q-16.935 47.641-16.935 47.360Q-16.935 47.001-17.029 46.697Q-17.123 46.393-17.351 46.200Q-17.578 46.007-17.944 46.007Q-18.244 46.007-18.511 46.143Q-18.778 46.280-18.931 46.526\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(2.039 -32.473)\">\u003Cpath d=\"M-15.421 47.360Q-15.421 47.022-15.280 46.731Q-15.140 46.441-14.896 46.227Q-14.652 46.014-14.347 45.899Q-14.043 45.785-13.718 45.785Q-13.448 45.785-13.185 45.884Q-12.922 45.983-12.731 46.161L-12.731 44.763Q-12.731 44.493-12.838 44.431Q-12.946 44.370-13.257 44.370L-13.257 44.089L-12.180 44.014L-12.180 48.198Q-12.180 48.386-12.126 48.469Q-12.071 48.553-11.970 48.572Q-11.869 48.591-11.654 48.591L-11.654 48.871L-12.761 48.939L-12.761 48.522Q-13.178 48.939-13.804 48.939Q-14.235 48.939-14.607 48.727Q-14.980 48.516-15.200 48.155Q-15.421 47.794-15.421 47.360M-13.746 48.717Q-13.537 48.717-13.351 48.645Q-13.165 48.574-13.011 48.437Q-12.857 48.300-12.761 48.122L-12.761 46.513Q-12.847 46.366-12.992 46.246Q-13.137 46.126-13.307 46.067Q-13.476 46.007-13.657 46.007Q-14.217 46.007-14.486 46.396Q-14.754 46.786-14.754 47.367Q-14.754 47.938-14.520 48.328Q-14.286 48.717-13.746 48.717M-10.946 48.143Q-10.946 47.811-10.723 47.584Q-10.499 47.357-10.155 47.229Q-9.812 47.100-9.439 47.048Q-9.067 46.995-8.762 46.995L-8.762 46.742Q-8.762 46.537-8.870 46.357Q-8.978 46.178-9.159 46.075Q-9.340 45.973-9.548 45.973Q-9.955 45.973-10.191 46.065Q-10.102 46.102-10.056 46.186Q-10.010 46.270-10.010 46.372Q-10.010 46.468-10.056 46.547Q-10.102 46.625-10.183 46.670Q-10.263 46.714-10.352 46.714Q-10.502 46.714-10.603 46.617Q-10.704 46.519-10.704 46.372Q-10.704 45.750-9.548 45.750Q-9.337 45.750-9.087 45.814Q-8.838 45.877-8.636 45.996Q-8.434 46.116-8.308 46.301Q-8.181 46.485-8.181 46.728L-8.181 48.304Q-8.181 48.420-8.120 48.516Q-8.058 48.611-7.945 48.611Q-7.836 48.611-7.771 48.517Q-7.706 48.423-7.706 48.304L-7.706 47.856L-7.440 47.856L-7.440 48.304Q-7.440 48.574-7.667 48.739Q-7.894 48.905-8.174 48.905Q-8.383 48.905-8.520 48.751Q-8.656 48.598-8.680 48.382Q-8.827 48.649-9.109 48.794Q-9.391 48.939-9.716 48.939Q-9.993 48.939-10.277 48.864Q-10.560 48.789-10.753 48.610Q-10.946 48.430-10.946 48.143M-10.331 48.143Q-10.331 48.317-10.230 48.447Q-10.130 48.577-9.974 48.647Q-9.818 48.717-9.654 48.717Q-9.436 48.717-9.227 48.620Q-9.019 48.522-8.891 48.341Q-8.762 48.160-8.762 47.934L-8.762 47.206Q-9.087 47.206-9.453 47.297Q-9.818 47.388-10.075 47.600Q-10.331 47.811-10.331 48.143M-6.701 50.628L-6.770 50.628Q-6.804 50.628-6.826 50.602Q-6.848 50.577-6.848 50.542Q-6.848 50.498-6.818 50.481Q-6.462 50.177-6.213 49.787Q-5.963 49.397-5.811 48.965Q-5.659 48.533-5.589 48.064Q-5.519 47.596-5.519 47.121Q-5.519 46.642-5.589 46.176Q-5.659 45.709-5.813 45.274Q-5.966 44.838-6.218 44.450Q-6.469 44.062-6.818 43.768Q-6.848 43.751-6.848 43.706Q-6.848 43.672-6.826 43.647Q-6.804 43.621-6.770 43.621L-6.701 43.621Q-6.691 43.621-6.683 43.623Q-6.674 43.624-6.664 43.628Q-6.120 44.028-5.748 44.581Q-5.375 45.135-5.194 45.781Q-5.013 46.427-5.013 47.121Q-5.013 47.822-5.194 48.469Q-5.375 49.117-5.749 49.671Q-6.124 50.225-6.664 50.621Q-6.674 50.621-6.683 50.623Q-6.691 50.624-6.701 50.628M-2.234 48.871L-3.837 48.871L-3.837 48.591Q-3.611 48.591-3.463 48.557Q-3.314 48.522-3.314 48.382L-3.314 44.763Q-3.314 44.493-3.422 44.431Q-3.529 44.370-3.837 44.370L-3.837 44.089L-2.760 44.014L-2.760 48.382Q-2.760 48.519-2.610 48.555Q-2.460 48.591-2.234 48.591L-2.234 48.871M-1.581 48.143Q-1.581 47.811-1.357 47.584Q-1.133 47.357-0.790 47.229Q-0.446 47.100-0.074 47.048Q0.299 46.995 0.603 46.995L0.603 46.742Q0.603 46.537 0.495 46.357Q0.388 46.178 0.206 46.075Q0.025 45.973-0.183 45.973Q-0.590 45.973-0.826 46.065Q-0.737 46.102-0.691 46.186Q-0.645 46.270-0.645 46.372Q-0.645 46.468-0.691 46.547Q-0.737 46.625-0.817 46.670Q-0.898 46.714-0.986 46.714Q-1.137 46.714-1.238 46.617Q-1.339 46.519-1.339 46.372Q-1.339 45.750-0.183 45.750Q0.029 45.750 0.278 45.814Q0.528 45.877 0.729 45.996Q0.931 46.116 1.057 46.301Q1.184 46.485 1.184 46.728L1.184 48.304Q1.184 48.420 1.245 48.516Q1.307 48.611 1.420 48.611Q1.529 48.611 1.594 48.517Q1.659 48.423 1.659 48.304L1.659 47.856L1.926 47.856L1.926 48.304Q1.926 48.574 1.698 48.739Q1.471 48.905 1.191 48.905Q0.982 48.905 0.846 48.751Q0.709 48.598 0.685 48.382Q0.538 48.649 0.256 48.794Q-0.026 48.939-0.351 48.939Q-0.628 48.939-0.911 48.864Q-1.195 48.789-1.388 48.610Q-1.581 48.430-1.581 48.143M-0.966 48.143Q-0.966 48.317-0.865 48.447Q-0.764 48.577-0.609 48.647Q-0.453 48.717-0.289 48.717Q-0.070 48.717 0.138 48.620Q0.347 48.522 0.475 48.341Q0.603 48.160 0.603 47.934L0.603 47.206Q0.278 47.206-0.088 47.297Q-0.453 47.388-0.710 47.600Q-0.966 47.811-0.966 48.143M4.024 48.871L2.390 48.871L2.390 48.591Q2.619 48.591 2.768 48.557Q2.917 48.522 2.917 48.382L2.917 46.533Q2.917 46.263 2.809 46.202Q2.702 46.140 2.390 46.140L2.390 45.860L3.450 45.785L3.450 46.434Q3.621 46.126 3.925 45.955Q4.229 45.785 4.575 45.785Q4.974 45.785 5.251 45.925Q5.528 46.065 5.614 46.413Q5.781 46.120 6.080 45.952Q6.379 45.785 6.724 45.785Q7.230 45.785 7.514 46.008Q7.798 46.232 7.798 46.728L7.798 48.382Q7.798 48.519 7.946 48.555Q8.095 48.591 8.321 48.591L8.321 48.871L6.690 48.871L6.690 48.591Q6.916 48.591 7.066 48.555Q7.217 48.519 7.217 48.382L7.217 46.742Q7.217 46.407 7.097 46.207Q6.977 46.007 6.663 46.007Q6.393 46.007 6.159 46.143Q5.925 46.280 5.786 46.514Q5.648 46.748 5.648 47.022L5.648 48.382Q5.648 48.519 5.796 48.555Q5.945 48.591 6.171 48.591L6.171 48.871L4.540 48.871L4.540 48.591Q4.769 48.591 4.918 48.557Q5.067 48.522 5.067 48.382L5.067 46.742Q5.067 46.407 4.947 46.207Q4.828 46.007 4.513 46.007Q4.243 46.007 4.009 46.143Q3.775 46.280 3.636 46.514Q3.498 46.748 3.498 47.022L3.498 48.382Q3.498 48.519 3.648 48.555Q3.799 48.591 4.024 48.591\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(2.039 -32.473)\">\u003Cpath d=\"M9.518 48.871L9.251 48.871L9.251 44.763Q9.251 44.493 9.144 44.431Q9.036 44.370 8.725 44.370L8.725 44.089L9.805 44.014L9.805 46.184Q10.014 45.993 10.299 45.889Q10.585 45.785 10.882 45.785Q11.200 45.785 11.497 45.906Q11.794 46.027 12.017 46.243Q12.239 46.458 12.365 46.743Q12.492 47.029 12.492 47.360Q12.492 47.805 12.252 48.169Q12.013 48.533 11.620 48.736Q11.227 48.939 10.783 48.939Q10.588 48.939 10.398 48.883Q10.209 48.827 10.048 48.722Q9.887 48.618 9.747 48.457L9.518 48.871M9.833 46.526L9.833 48.143Q9.969 48.403 10.210 48.560Q10.451 48.717 10.728 48.717Q11.022 48.717 11.234 48.610Q11.446 48.502 11.579 48.310Q11.712 48.119 11.771 47.880Q11.829 47.641 11.829 47.360Q11.829 47.001 11.735 46.697Q11.641 46.393 11.413 46.200Q11.186 46.007 10.820 46.007Q10.520 46.007 10.253 46.143Q9.986 46.280 9.833 46.526\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(2.039 -32.473)\">\u003Cpath d=\"M13.343 47.360Q13.343 47.022 13.484 46.731Q13.624 46.441 13.868 46.227Q14.112 46.014 14.417 45.899Q14.721 45.785 15.046 45.785Q15.316 45.785 15.579 45.884Q15.842 45.983 16.033 46.161L16.033 44.763Q16.033 44.493 15.926 44.431Q15.818 44.370 15.507 44.370L15.507 44.089L16.584 44.014L16.584 48.198Q16.584 48.386 16.638 48.469Q16.693 48.553 16.794 48.572Q16.895 48.591 17.110 48.591L17.110 48.871L16.003 48.939L16.003 48.522Q15.586 48.939 14.960 48.939Q14.529 48.939 14.157 48.727Q13.784 48.516 13.564 48.155Q13.343 47.794 13.343 47.360M15.018 48.717Q15.227 48.717 15.413 48.645Q15.599 48.574 15.753 48.437Q15.907 48.300 16.003 48.122L16.003 46.513Q15.917 46.366 15.772 46.246Q15.627 46.126 15.457 46.067Q15.288 46.007 15.107 46.007Q14.547 46.007 14.278 46.396Q14.010 46.786 14.010 47.367Q14.010 47.938 14.244 48.328Q14.478 48.717 15.018 48.717M17.818 48.143Q17.818 47.811 18.041 47.584Q18.265 47.357 18.609 47.229Q18.952 47.100 19.325 47.048Q19.697 46.995 20.002 46.995L20.002 46.742Q20.002 46.537 19.894 46.357Q19.786 46.178 19.605 46.075Q19.424 45.973 19.216 45.973Q18.809 45.973 18.573 46.065Q18.662 46.102 18.708 46.186Q18.754 46.270 18.754 46.372Q18.754 46.468 18.708 46.547Q18.662 46.625 18.581 46.670Q18.501 46.714 18.412 46.714Q18.262 46.714 18.161 46.617Q18.060 46.519 18.060 46.372Q18.060 45.750 19.216 45.750Q19.427 45.750 19.677 45.814Q19.926 45.877 20.128 45.996Q20.330 46.116 20.456 46.301Q20.583 46.485 20.583 46.728L20.583 48.304Q20.583 48.420 20.644 48.516Q20.706 48.611 20.819 48.611Q20.928 48.611 20.993 48.517Q21.058 48.423 21.058 48.304L21.058 47.856L21.324 47.856L21.324 48.304Q21.324 48.574 21.097 48.739Q20.870 48.905 20.590 48.905Q20.381 48.905 20.244 48.751Q20.108 48.598 20.084 48.382Q19.937 48.649 19.655 48.794Q19.373 48.939 19.048 48.939Q18.771 48.939 18.487 48.864Q18.204 48.789 18.011 48.610Q17.818 48.430 17.818 48.143M18.433 48.143Q18.433 48.317 18.534 48.447Q18.634 48.577 18.790 48.647Q18.946 48.717 19.110 48.717Q19.328 48.717 19.537 48.620Q19.745 48.522 19.873 48.341Q20.002 48.160 20.002 47.934L20.002 47.206Q19.677 47.206 19.311 47.297Q18.946 47.388 18.689 47.600Q18.433 47.811 18.433 48.143\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath d=\"M27.785-45.023a2.276 2.276 0 1 0-4.552 0 2.276 2.276 0 0 0 4.552 0Zm-2.276 0\"\u002F>\u003Cpath fill=\"none\" d=\"M30.062-27.951a4.552 4.552 0 1 0-9.105 0 4.552 4.552 0 0 0 9.105 0Zm-4.553 0\"\u002F>\u003Cpath d=\"M27.785-13.725a2.276 2.276 0 1 0-4.552 0 2.276 2.276 0 0 0 4.552 0Zm-2.276 0\"\u002F>\u003Cpath fill=\"none\" d=\"M30.062 3.346a4.552 4.552 0 1 0-9.105 0 4.552 4.552 0 0 0 9.105 0Zm-4.553 0\"\u002F>\u003Cpath d=\"M27.785 17.573a2.276 2.276 0 1 0-4.552 0 2.276 2.276 0 0 0 4.552 0Zm-2.276 0\"\u002F>\u003Cpath fill=\"none\" d=\"M30.062 34.645a4.552 4.552 0 1 0-9.105 0 4.552 4.552 0 0 0 9.105 0ZM25.51-42.547v9.843m0 9.505v6.998m0 4.952v9.843m0 9.505v6.998m0 4.952v9.843\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(22.015 -1.176)\">\u003Cpath d=\"M-40.333 50.621Q-40.883 50.221-41.254 49.666Q-41.625 49.110-41.806 48.464Q-41.987 47.818-41.987 47.121Q-41.987 46.608-41.887 46.113Q-41.786 45.617-41.581 45.166Q-41.376 44.715-41.063 44.323Q-40.750 43.932-40.333 43.628Q-40.323 43.624-40.316 43.623Q-40.309 43.621-40.299 43.621L-40.231 43.621Q-40.196 43.621-40.174 43.645Q-40.152 43.669-40.152 43.706Q-40.152 43.751-40.179 43.768Q-40.528 44.069-40.781 44.453Q-41.034 44.838-41.186 45.279Q-41.338 45.720-41.410 46.176Q-41.482 46.632-41.482 47.121Q-41.482 48.122-41.172 49.009Q-40.863 49.896-40.179 50.481Q-40.152 50.498-40.152 50.542Q-40.152 50.580-40.174 50.604Q-40.196 50.628-40.231 50.628L-40.299 50.628Q-40.306 50.624-40.314 50.623Q-40.323 50.621-40.333 50.621M-36.331 48.871L-38.860 48.871L-38.860 48.591Q-37.893 48.591-37.893 48.382L-37.893 44.763Q-38.286 44.951-38.908 44.951L-38.908 44.670Q-38.491 44.670-38.127 44.569Q-37.763 44.469-37.506 44.223L-37.380 44.223Q-37.315 44.240-37.298 44.308L-37.298 48.382Q-37.298 48.591-36.331 48.591L-36.331 48.871M-33.477 47.617L-35.534 47.617L-35.534 47.114L-33.477 47.114L-33.477 47.617M-31.006 48.871L-32.609 48.871L-32.609 48.591Q-32.383 48.591-32.234 48.557Q-32.086 48.522-32.086 48.382L-32.086 44.763Q-32.086 44.493-32.193 44.431Q-32.301 44.370-32.609 44.370L-32.609 44.089L-31.532 44.014L-31.532 48.382Q-31.532 48.519-31.381 48.555Q-31.231 48.591-31.006 48.591L-31.006 48.871M-30.353 48.143Q-30.353 47.811-30.129 47.584Q-29.905 47.357-29.561 47.229Q-29.218 47.100-28.845 47.048Q-28.473 46.995-28.169 46.995L-28.169 46.742Q-28.169 46.537-28.276 46.357Q-28.384 46.178-28.565 46.075Q-28.746 45.973-28.955 45.973Q-29.361 45.973-29.597 46.065Q-29.508 46.102-29.462 46.186Q-29.416 46.270-29.416 46.372Q-29.416 46.468-29.462 46.547Q-29.508 46.625-29.589 46.670Q-29.669 46.714-29.758 46.714Q-29.908 46.714-30.009 46.617Q-30.110 46.519-30.110 46.372Q-30.110 45.750-28.955 45.750Q-28.743 45.750-28.493 45.814Q-28.244 45.877-28.042 45.996Q-27.840 46.116-27.714 46.301Q-27.588 46.485-27.588 46.728L-27.588 48.304Q-27.588 48.420-27.526 48.516Q-27.465 48.611-27.352 48.611Q-27.242 48.611-27.177 48.517Q-27.112 48.423-27.112 48.304L-27.112 47.856L-26.846 47.856L-26.846 48.304Q-26.846 48.574-27.073 48.739Q-27.300 48.905-27.581 48.905Q-27.789 48.905-27.926 48.751Q-28.063 48.598-28.087 48.382Q-28.234 48.649-28.516 48.794Q-28.798 48.939-29.122 48.939Q-29.399 48.939-29.683 48.864Q-29.966 48.789-30.160 48.610Q-30.353 48.430-30.353 48.143M-29.737 48.143Q-29.737 48.317-29.637 48.447Q-29.536 48.577-29.380 48.647Q-29.225 48.717-29.061 48.717Q-28.842 48.717-28.633 48.620Q-28.425 48.522-28.297 48.341Q-28.169 48.160-28.169 47.934L-28.169 47.206Q-28.493 47.206-28.859 47.297Q-29.225 47.388-29.481 47.600Q-29.737 47.811-29.737 48.143M-24.747 48.871L-26.381 48.871L-26.381 48.591Q-26.152 48.591-26.003 48.557Q-25.855 48.522-25.855 48.382L-25.855 46.533Q-25.855 46.263-25.962 46.202Q-26.070 46.140-26.381 46.140L-26.381 45.860L-25.321 45.785L-25.321 46.434Q-25.151 46.126-24.846 45.955Q-24.542 45.785-24.197 45.785Q-23.797 45.785-23.520 45.925Q-23.243 46.065-23.158 46.413Q-22.990 46.120-22.691 45.952Q-22.392 45.785-22.047 45.785Q-21.541 45.785-21.257 46.008Q-20.974 46.232-20.974 46.728L-20.974 48.382Q-20.974 48.519-20.825 48.555Q-20.676 48.591-20.451 48.591L-20.451 48.871L-22.081 48.871L-22.081 48.591Q-21.856 48.591-21.705 48.555Q-21.555 48.519-21.555 48.382L-21.555 46.742Q-21.555 46.407-21.674 46.207Q-21.794 46.007-22.109 46.007Q-22.379 46.007-22.613 46.143Q-22.847 46.280-22.985 46.514Q-23.124 46.748-23.124 47.022L-23.124 48.382Q-23.124 48.519-22.975 48.555Q-22.826 48.591-22.601 48.591L-22.601 48.871L-24.231 48.871L-24.231 48.591Q-24.002 48.591-23.853 48.557Q-23.705 48.522-23.705 48.382L-23.705 46.742Q-23.705 46.407-23.824 46.207Q-23.944 46.007-24.258 46.007Q-24.528 46.007-24.763 46.143Q-24.997 46.280-25.135 46.514Q-25.274 46.748-25.274 47.022L-25.274 48.382Q-25.274 48.519-25.123 48.555Q-24.973 48.591-24.747 48.591\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(22.015 -1.176)\">\u003Cpath d=\"M-19.246 48.871L-19.513 48.871L-19.513 44.763Q-19.513 44.493-19.620 44.431Q-19.728 44.370-20.039 44.370L-20.039 44.089L-18.959 44.014L-18.959 46.184Q-18.750 45.993-18.465 45.889Q-18.180 45.785-17.882 45.785Q-17.564 45.785-17.267 45.906Q-16.970 46.027-16.747 46.243Q-16.525 46.458-16.399 46.743Q-16.272 47.029-16.272 47.360Q-16.272 47.805-16.512 48.169Q-16.751 48.533-17.144 48.736Q-17.537 48.939-17.981 48.939Q-18.176 48.939-18.366 48.883Q-18.555 48.827-18.716 48.722Q-18.877 48.618-19.017 48.457L-19.246 48.871M-18.931 46.526L-18.931 48.143Q-18.795 48.403-18.554 48.560Q-18.313 48.717-18.036 48.717Q-17.742 48.717-17.530 48.610Q-17.318 48.502-17.185 48.310Q-17.052 48.119-16.993 47.880Q-16.935 47.641-16.935 47.360Q-16.935 47.001-17.029 46.697Q-17.123 46.393-17.351 46.200Q-17.578 46.007-17.944 46.007Q-18.244 46.007-18.511 46.143Q-18.778 46.280-18.931 46.526\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(22.015 -1.176)\">\u003Cpath d=\"M-15.421 47.360Q-15.421 47.022-15.280 46.731Q-15.140 46.441-14.896 46.227Q-14.652 46.014-14.347 45.899Q-14.043 45.785-13.718 45.785Q-13.448 45.785-13.185 45.884Q-12.922 45.983-12.731 46.161L-12.731 44.763Q-12.731 44.493-12.838 44.431Q-12.946 44.370-13.257 44.370L-13.257 44.089L-12.180 44.014L-12.180 48.198Q-12.180 48.386-12.126 48.469Q-12.071 48.553-11.970 48.572Q-11.869 48.591-11.654 48.591L-11.654 48.871L-12.761 48.939L-12.761 48.522Q-13.178 48.939-13.804 48.939Q-14.235 48.939-14.607 48.727Q-14.980 48.516-15.200 48.155Q-15.421 47.794-15.421 47.360M-13.746 48.717Q-13.537 48.717-13.351 48.645Q-13.165 48.574-13.011 48.437Q-12.857 48.300-12.761 48.122L-12.761 46.513Q-12.847 46.366-12.992 46.246Q-13.137 46.126-13.307 46.067Q-13.476 46.007-13.657 46.007Q-14.217 46.007-14.486 46.396Q-14.754 46.786-14.754 47.367Q-14.754 47.938-14.520 48.328Q-14.286 48.717-13.746 48.717M-10.946 48.143Q-10.946 47.811-10.723 47.584Q-10.499 47.357-10.155 47.229Q-9.812 47.100-9.439 47.048Q-9.067 46.995-8.762 46.995L-8.762 46.742Q-8.762 46.537-8.870 46.357Q-8.978 46.178-9.159 46.075Q-9.340 45.973-9.548 45.973Q-9.955 45.973-10.191 46.065Q-10.102 46.102-10.056 46.186Q-10.010 46.270-10.010 46.372Q-10.010 46.468-10.056 46.547Q-10.102 46.625-10.183 46.670Q-10.263 46.714-10.352 46.714Q-10.502 46.714-10.603 46.617Q-10.704 46.519-10.704 46.372Q-10.704 45.750-9.548 45.750Q-9.337 45.750-9.087 45.814Q-8.838 45.877-8.636 45.996Q-8.434 46.116-8.308 46.301Q-8.181 46.485-8.181 46.728L-8.181 48.304Q-8.181 48.420-8.120 48.516Q-8.058 48.611-7.945 48.611Q-7.836 48.611-7.771 48.517Q-7.706 48.423-7.706 48.304L-7.706 47.856L-7.440 47.856L-7.440 48.304Q-7.440 48.574-7.667 48.739Q-7.894 48.905-8.174 48.905Q-8.383 48.905-8.520 48.751Q-8.656 48.598-8.680 48.382Q-8.827 48.649-9.109 48.794Q-9.391 48.939-9.716 48.939Q-9.993 48.939-10.277 48.864Q-10.560 48.789-10.753 48.610Q-10.946 48.430-10.946 48.143M-10.331 48.143Q-10.331 48.317-10.230 48.447Q-10.130 48.577-9.974 48.647Q-9.818 48.717-9.654 48.717Q-9.436 48.717-9.227 48.620Q-9.019 48.522-8.891 48.341Q-8.762 48.160-8.762 47.934L-8.762 47.206Q-9.087 47.206-9.453 47.297Q-9.818 47.388-10.075 47.600Q-10.331 47.811-10.331 48.143M-6.701 50.628L-6.770 50.628Q-6.804 50.628-6.826 50.602Q-6.848 50.577-6.848 50.542Q-6.848 50.498-6.818 50.481Q-6.462 50.177-6.213 49.787Q-5.963 49.397-5.811 48.965Q-5.659 48.533-5.589 48.064Q-5.519 47.596-5.519 47.121Q-5.519 46.642-5.589 46.176Q-5.659 45.709-5.813 45.274Q-5.966 44.838-6.218 44.450Q-6.469 44.062-6.818 43.768Q-6.848 43.751-6.848 43.706Q-6.848 43.672-6.826 43.647Q-6.804 43.621-6.770 43.621L-6.701 43.621Q-6.691 43.621-6.683 43.623Q-6.674 43.624-6.664 43.628Q-6.120 44.028-5.748 44.581Q-5.375 45.135-5.194 45.781Q-5.013 46.427-5.013 47.121Q-5.013 47.822-5.194 48.469Q-5.375 49.117-5.749 49.671Q-6.124 50.225-6.664 50.621Q-6.674 50.621-6.683 50.623Q-6.691 50.624-6.701 50.628M-2.234 48.871L-3.837 48.871L-3.837 48.591Q-3.611 48.591-3.463 48.557Q-3.314 48.522-3.314 48.382L-3.314 44.763Q-3.314 44.493-3.422 44.431Q-3.529 44.370-3.837 44.370L-3.837 44.089L-2.760 44.014L-2.760 48.382Q-2.760 48.519-2.610 48.555Q-2.460 48.591-2.234 48.591L-2.234 48.871M-1.581 48.143Q-1.581 47.811-1.357 47.584Q-1.133 47.357-0.790 47.229Q-0.446 47.100-0.074 47.048Q0.299 46.995 0.603 46.995L0.603 46.742Q0.603 46.537 0.495 46.357Q0.388 46.178 0.206 46.075Q0.025 45.973-0.183 45.973Q-0.590 45.973-0.826 46.065Q-0.737 46.102-0.691 46.186Q-0.645 46.270-0.645 46.372Q-0.645 46.468-0.691 46.547Q-0.737 46.625-0.817 46.670Q-0.898 46.714-0.986 46.714Q-1.137 46.714-1.238 46.617Q-1.339 46.519-1.339 46.372Q-1.339 45.750-0.183 45.750Q0.029 45.750 0.278 45.814Q0.528 45.877 0.729 45.996Q0.931 46.116 1.057 46.301Q1.184 46.485 1.184 46.728L1.184 48.304Q1.184 48.420 1.245 48.516Q1.307 48.611 1.420 48.611Q1.529 48.611 1.594 48.517Q1.659 48.423 1.659 48.304L1.659 47.856L1.926 47.856L1.926 48.304Q1.926 48.574 1.698 48.739Q1.471 48.905 1.191 48.905Q0.982 48.905 0.846 48.751Q0.709 48.598 0.685 48.382Q0.538 48.649 0.256 48.794Q-0.026 48.939-0.351 48.939Q-0.628 48.939-0.911 48.864Q-1.195 48.789-1.388 48.610Q-1.581 48.430-1.581 48.143M-0.966 48.143Q-0.966 48.317-0.865 48.447Q-0.764 48.577-0.609 48.647Q-0.453 48.717-0.289 48.717Q-0.070 48.717 0.138 48.620Q0.347 48.522 0.475 48.341Q0.603 48.160 0.603 47.934L0.603 47.206Q0.278 47.206-0.088 47.297Q-0.453 47.388-0.710 47.600Q-0.966 47.811-0.966 48.143M4.024 48.871L2.390 48.871L2.390 48.591Q2.619 48.591 2.768 48.557Q2.917 48.522 2.917 48.382L2.917 46.533Q2.917 46.263 2.809 46.202Q2.702 46.140 2.390 46.140L2.390 45.860L3.450 45.785L3.450 46.434Q3.621 46.126 3.925 45.955Q4.229 45.785 4.575 45.785Q4.974 45.785 5.251 45.925Q5.528 46.065 5.614 46.413Q5.781 46.120 6.080 45.952Q6.379 45.785 6.724 45.785Q7.230 45.785 7.514 46.008Q7.798 46.232 7.798 46.728L7.798 48.382Q7.798 48.519 7.946 48.555Q8.095 48.591 8.321 48.591L8.321 48.871L6.690 48.871L6.690 48.591Q6.916 48.591 7.066 48.555Q7.217 48.519 7.217 48.382L7.217 46.742Q7.217 46.407 7.097 46.207Q6.977 46.007 6.663 46.007Q6.393 46.007 6.159 46.143Q5.925 46.280 5.786 46.514Q5.648 46.748 5.648 47.022L5.648 48.382Q5.648 48.519 5.796 48.555Q5.945 48.591 6.171 48.591L6.171 48.871L4.540 48.871L4.540 48.591Q4.769 48.591 4.918 48.557Q5.067 48.522 5.067 48.382L5.067 46.742Q5.067 46.407 4.947 46.207Q4.828 46.007 4.513 46.007Q4.243 46.007 4.009 46.143Q3.775 46.280 3.636 46.514Q3.498 46.748 3.498 47.022L3.498 48.382Q3.498 48.519 3.648 48.555Q3.799 48.591 4.024 48.591\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(22.015 -1.176)\">\u003Cpath d=\"M9.518 48.871L9.251 48.871L9.251 44.763Q9.251 44.493 9.144 44.431Q9.036 44.370 8.725 44.370L8.725 44.089L9.805 44.014L9.805 46.184Q10.014 45.993 10.299 45.889Q10.585 45.785 10.882 45.785Q11.200 45.785 11.497 45.906Q11.794 46.027 12.017 46.243Q12.239 46.458 12.365 46.743Q12.492 47.029 12.492 47.360Q12.492 47.805 12.252 48.169Q12.013 48.533 11.620 48.736Q11.227 48.939 10.783 48.939Q10.588 48.939 10.398 48.883Q10.209 48.827 10.048 48.722Q9.887 48.618 9.747 48.457L9.518 48.871M9.833 46.526L9.833 48.143Q9.969 48.403 10.210 48.560Q10.451 48.717 10.728 48.717Q11.022 48.717 11.234 48.610Q11.446 48.502 11.579 48.310Q11.712 48.119 11.771 47.880Q11.829 47.641 11.829 47.360Q11.829 47.001 11.735 46.697Q11.641 46.393 11.413 46.200Q11.186 46.007 10.820 46.007Q10.520 46.007 10.253 46.143Q9.986 46.280 9.833 46.526\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(22.015 -1.176)\">\u003Cpath d=\"M13.343 47.360Q13.343 47.022 13.484 46.731Q13.624 46.441 13.868 46.227Q14.112 46.014 14.417 45.899Q14.721 45.785 15.046 45.785Q15.316 45.785 15.579 45.884Q15.842 45.983 16.033 46.161L16.033 44.763Q16.033 44.493 15.926 44.431Q15.818 44.370 15.507 44.370L15.507 44.089L16.584 44.014L16.584 48.198Q16.584 48.386 16.638 48.469Q16.693 48.553 16.794 48.572Q16.895 48.591 17.110 48.591L17.110 48.871L16.003 48.939L16.003 48.522Q15.586 48.939 14.960 48.939Q14.529 48.939 14.157 48.727Q13.784 48.516 13.564 48.155Q13.343 47.794 13.343 47.360M15.018 48.717Q15.227 48.717 15.413 48.645Q15.599 48.574 15.753 48.437Q15.907 48.300 16.003 48.122L16.003 46.513Q15.917 46.366 15.772 46.246Q15.627 46.126 15.457 46.067Q15.288 46.007 15.107 46.007Q14.547 46.007 14.278 46.396Q14.010 46.786 14.010 47.367Q14.010 47.938 14.244 48.328Q14.478 48.717 15.018 48.717M17.818 48.143Q17.818 47.811 18.041 47.584Q18.265 47.357 18.609 47.229Q18.952 47.100 19.325 47.048Q19.697 46.995 20.002 46.995L20.002 46.742Q20.002 46.537 19.894 46.357Q19.786 46.178 19.605 46.075Q19.424 45.973 19.216 45.973Q18.809 45.973 18.573 46.065Q18.662 46.102 18.708 46.186Q18.754 46.270 18.754 46.372Q18.754 46.468 18.708 46.547Q18.662 46.625 18.581 46.670Q18.501 46.714 18.412 46.714Q18.262 46.714 18.161 46.617Q18.060 46.519 18.060 46.372Q18.060 45.750 19.216 45.750Q19.427 45.750 19.677 45.814Q19.926 45.877 20.128 45.996Q20.330 46.116 20.456 46.301Q20.583 46.485 20.583 46.728L20.583 48.304Q20.583 48.420 20.644 48.516Q20.706 48.611 20.819 48.611Q20.928 48.611 20.993 48.517Q21.058 48.423 21.058 48.304L21.058 47.856L21.324 47.856L21.324 48.304Q21.324 48.574 21.097 48.739Q20.870 48.905 20.590 48.905Q20.381 48.905 20.244 48.751Q20.108 48.598 20.084 48.382Q19.937 48.649 19.655 48.794Q19.373 48.939 19.048 48.939Q18.771 48.939 18.487 48.864Q18.204 48.789 18.011 48.610Q17.818 48.430 17.818 48.143M18.433 48.143Q18.433 48.317 18.534 48.447Q18.634 48.577 18.790 48.647Q18.946 48.717 19.110 48.717Q19.328 48.717 19.537 48.620Q19.745 48.522 19.873 48.341Q20.002 48.160 20.002 47.934L20.002 47.206Q19.677 47.206 19.311 47.297Q18.946 47.388 18.689 47.600Q18.433 47.811 18.433 48.143\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(22.015 -1.176)\">\u003Cpath d=\"M26.108 48.871L24.505 48.871L24.505 48.591Q24.731 48.591 24.880 48.557Q25.028 48.522 25.028 48.382L25.028 44.763Q25.028 44.493 24.921 44.431Q24.813 44.370 24.505 44.370L24.505 44.089L25.582 44.014L25.582 48.382Q25.582 48.519 25.732 48.555Q25.883 48.591 26.108 48.591L26.108 48.871M26.761 48.143Q26.761 47.811 26.985 47.584Q27.209 47.357 27.553 47.229Q27.896 47.100 28.269 47.048Q28.641 46.995 28.945 46.995L28.945 46.742Q28.945 46.537 28.838 46.357Q28.730 46.178 28.549 46.075Q28.368 45.973 28.159 45.973Q27.752 45.973 27.517 46.065Q27.605 46.102 27.652 46.186Q27.698 46.270 27.698 46.372Q27.698 46.468 27.652 46.547Q27.605 46.625 27.525 46.670Q27.445 46.714 27.356 46.714Q27.206 46.714 27.105 46.617Q27.004 46.519 27.004 46.372Q27.004 45.750 28.159 45.750Q28.371 45.750 28.621 45.814Q28.870 45.877 29.072 45.996Q29.273 46.116 29.400 46.301Q29.526 46.485 29.526 46.728L29.526 48.304Q29.526 48.420 29.588 48.516Q29.649 48.611 29.762 48.611Q29.872 48.611 29.937 48.517Q30.001 48.423 30.001 48.304L30.001 47.856L30.268 47.856L30.268 48.304Q30.268 48.574 30.041 48.739Q29.814 48.905 29.533 48.905Q29.325 48.905 29.188 48.751Q29.051 48.598 29.027 48.382Q28.880 48.649 28.598 48.794Q28.316 48.939 27.992 48.939Q27.715 48.939 27.431 48.864Q27.147 48.789 26.954 48.610Q26.761 48.430 26.761 48.143M27.376 48.143Q27.376 48.317 27.477 48.447Q27.578 48.577 27.734 48.647Q27.889 48.717 28.053 48.717Q28.272 48.717 28.480 48.620Q28.689 48.522 28.817 48.341Q28.945 48.160 28.945 47.934L28.945 47.206Q28.621 47.206 28.255 47.297Q27.889 47.388 27.633 47.600Q27.376 47.811 27.376 48.143M32.367 48.871L30.733 48.871L30.733 48.591Q30.962 48.591 31.111 48.557Q31.259 48.522 31.259 48.382L31.259 46.533Q31.259 46.263 31.152 46.202Q31.044 46.140 30.733 46.140L30.733 45.860L31.793 45.785L31.793 46.434Q31.963 46.126 32.268 45.955Q32.572 45.785 32.917 45.785Q33.317 45.785 33.594 45.925Q33.871 46.065 33.956 46.413Q34.124 46.120 34.423 45.952Q34.722 45.785 35.067 45.785Q35.573 45.785 35.856 46.008Q36.140 46.232 36.140 46.728L36.140 48.382Q36.140 48.519 36.289 48.555Q36.438 48.591 36.663 48.591L36.663 48.871L35.033 48.871L35.033 48.591Q35.258 48.591 35.409 48.555Q35.559 48.519 35.559 48.382L35.559 46.742Q35.559 46.407 35.439 46.207Q35.320 46.007 35.005 46.007Q34.735 46.007 34.501 46.143Q34.267 46.280 34.129 46.514Q33.990 46.748 33.990 47.022L33.990 48.382Q33.990 48.519 34.139 48.555Q34.288 48.591 34.513 48.591L34.513 48.871L32.883 48.871L32.883 48.591Q33.112 48.591 33.261 48.557Q33.409 48.522 33.409 48.382L33.409 46.742Q33.409 46.407 33.290 46.207Q33.170 46.007 32.855 46.007Q32.585 46.007 32.351 46.143Q32.117 46.280 31.979 46.514Q31.840 46.748 31.840 47.022L31.840 48.382Q31.840 48.519 31.991 48.555Q32.141 48.591 32.367 48.591\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(22.015 -1.176)\">\u003Cpath d=\"M37.851 48.871L37.584 48.871L37.584 44.763Q37.584 44.493 37.477 44.431Q37.369 44.370 37.058 44.370L37.058 44.089L38.138 44.014L38.138 46.184Q38.347 45.993 38.632 45.889Q38.917 45.785 39.215 45.785Q39.533 45.785 39.830 45.906Q40.127 46.027 40.350 46.243Q40.572 46.458 40.698 46.743Q40.825 47.029 40.825 47.360Q40.825 47.805 40.585 48.169Q40.346 48.533 39.953 48.736Q39.560 48.939 39.116 48.939Q38.921 48.939 38.731 48.883Q38.542 48.827 38.381 48.722Q38.220 48.618 38.080 48.457L37.851 48.871M38.166 46.526L38.166 48.143Q38.302 48.403 38.543 48.560Q38.784 48.717 39.061 48.717Q39.355 48.717 39.567 48.610Q39.779 48.502 39.912 48.310Q40.045 48.119 40.104 47.880Q40.162 47.641 40.162 47.360Q40.162 47.001 40.068 46.697Q39.974 46.393 39.746 46.200Q39.519 46.007 39.153 46.007Q38.853 46.007 38.586 46.143Q38.319 46.280 38.166 46.526\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(22.015 -1.176)\">\u003Cpath d=\"M41.676 47.360Q41.676 47.022 41.817 46.731Q41.957 46.441 42.201 46.227Q42.445 46.014 42.750 45.899Q43.054 45.785 43.379 45.785Q43.649 45.785 43.912 45.884Q44.175 45.983 44.366 46.161L44.366 44.763Q44.366 44.493 44.259 44.431Q44.151 44.370 43.840 44.370L43.840 44.089L44.917 44.014L44.917 48.198Q44.917 48.386 44.971 48.469Q45.026 48.553 45.127 48.572Q45.228 48.591 45.443 48.591L45.443 48.871L44.336 48.939L44.336 48.522Q43.919 48.939 43.293 48.939Q42.862 48.939 42.490 48.727Q42.117 48.516 41.897 48.155Q41.676 47.794 41.676 47.360M43.351 48.717Q43.560 48.717 43.746 48.645Q43.932 48.574 44.086 48.437Q44.240 48.300 44.336 48.122L44.336 46.513Q44.250 46.366 44.105 46.246Q43.960 46.126 43.790 46.067Q43.621 46.007 43.440 46.007Q42.880 46.007 42.611 46.396Q42.343 46.786 42.343 47.367Q42.343 47.938 42.577 48.328Q42.811 48.717 43.351 48.717M46.151 48.143Q46.151 47.811 46.374 47.584Q46.598 47.357 46.942 47.229Q47.285 47.100 47.658 47.048Q48.030 46.995 48.335 46.995L48.335 46.742Q48.335 46.537 48.227 46.357Q48.119 46.178 47.938 46.075Q47.757 45.973 47.549 45.973Q47.142 45.973 46.906 46.065Q46.995 46.102 47.041 46.186Q47.087 46.270 47.087 46.372Q47.087 46.468 47.041 46.547Q46.995 46.625 46.914 46.670Q46.834 46.714 46.745 46.714Q46.595 46.714 46.494 46.617Q46.393 46.519 46.393 46.372Q46.393 45.750 47.549 45.750Q47.760 45.750 48.010 45.814Q48.259 45.877 48.461 45.996Q48.663 46.116 48.789 46.301Q48.916 46.485 48.916 46.728L48.916 48.304Q48.916 48.420 48.977 48.516Q49.039 48.611 49.152 48.611Q49.261 48.611 49.326 48.517Q49.391 48.423 49.391 48.304L49.391 47.856L49.657 47.856L49.657 48.304Q49.657 48.574 49.430 48.739Q49.203 48.905 48.923 48.905Q48.714 48.905 48.577 48.751Q48.441 48.598 48.417 48.382Q48.270 48.649 47.988 48.794Q47.706 48.939 47.381 48.939Q47.104 48.939 46.820 48.864Q46.537 48.789 46.344 48.610Q46.151 48.430 46.151 48.143M46.766 48.143Q46.766 48.317 46.867 48.447Q46.967 48.577 47.123 48.647Q47.279 48.717 47.443 48.717Q47.661 48.717 47.870 48.620Q48.078 48.522 48.206 48.341Q48.335 48.160 48.335 47.934L48.335 47.206Q48.010 47.206 47.644 47.297Q47.279 47.388 47.022 47.600Q46.766 47.811 46.766 48.143\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg transform=\"translate(96.189 -56.503)\">\u003Cpath d=\"M-42.063 48.451Q-42.063 48.283-41.940 48.160Q-41.817 48.037-41.642 48.037Q-41.475 48.037-41.352 48.160Q-41.229 48.283-41.229 48.451Q-41.229 48.625-41.352 48.748Q-41.475 48.871-41.642 48.871Q-41.817 48.871-41.940 48.748Q-42.063 48.625-42.063 48.451M-39.800 48.451Q-39.800 48.283-39.677 48.160Q-39.554 48.037-39.380 48.037Q-39.212 48.037-39.089 48.160Q-38.966 48.283-38.966 48.451Q-38.966 48.625-39.089 48.748Q-39.212 48.871-39.380 48.871Q-39.554 48.871-39.677 48.748Q-39.800 48.625-39.800 48.451M-37.537 48.451Q-37.537 48.283-37.414 48.160Q-37.291 48.037-37.117 48.037Q-36.949 48.037-36.826 48.160Q-36.703 48.283-36.703 48.451Q-36.703 48.625-36.826 48.748Q-36.949 48.871-37.117 48.871Q-37.291 48.871-37.414 48.748Q-37.537 48.625-37.537 48.451\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(121.036 -65.211)\">\u003Cpath d=\"M-39.187 49.069Q-39.833 49.069-40.393 48.809Q-40.953 48.550-41.375 48.093Q-41.797 47.636-42.030 47.045Q-42.263 46.454-42.263 45.830Q-42.263 45.197-42.032 44.591Q-41.801 43.984-41.384 43.523Q-40.966 43.061-40.402 42.793Q-39.837 42.525-39.187 42.525Q-38.532 42.525-37.963 42.793Q-37.394 43.061-36.978 43.527Q-36.563 43.993-36.335 44.589Q-36.106 45.184-36.106 45.830Q-36.106 46.454-36.335 47.045Q-36.563 47.636-36.987 48.093Q-37.411 48.550-37.974 48.809Q-38.536 49.069-39.187 49.069M-40.742 47.882Q-40.562 48.146-40.320 48.341Q-40.079 48.537-39.789 48.649Q-39.499 48.761-39.187 48.761Q-38.875 48.761-38.582 48.649Q-38.290 48.537-38.046 48.339Q-37.802 48.142-37.631 47.882Q-37.424 47.592-37.297 47.230Q-37.170 46.867-37.117 46.480Q-37.064 46.094-37.064 45.676Q-37.064 44.393-37.631 43.642Q-37.807 43.391-38.053 43.204Q-38.299 43.017-38.589 42.916Q-38.879 42.815-39.187 42.815Q-39.639 42.815-40.052 43.035Q-40.465 43.255-40.742 43.642Q-41.041 44.041-41.175 44.564Q-41.309 45.087-41.309 45.676Q-41.309 46.094-41.256 46.480Q-41.204 46.867-41.076 47.230Q-40.949 47.592-40.742 47.882M-32.590 48.871L-35.245 48.871L-35.245 48.555Q-34.326 48.555-34.326 48.260L-34.326 43.334Q-34.326 43.039-35.245 43.039L-35.245 42.723L-32.384 42.723Q-32.010 42.723-31.586 42.822Q-31.162 42.921-30.789 43.121Q-30.415 43.321-30.180 43.631Q-29.945 43.940-29.945 44.353Q-29.945 44.723-30.171 45.024Q-30.398 45.325-30.749 45.522Q-31.101 45.720-31.465 45.812Q-30.547 46.138-30.393 46.973L-30.288 47.764Q-30.244 48.098-30.200 48.291Q-30.156 48.484-30.026 48.647Q-29.897 48.809-29.664 48.809Q-29.391 48.809-29.248 48.559Q-29.106 48.309-29.106 48.005Q-29.106 47.966-29.073 47.937Q-29.040 47.909-29 47.909L-28.917 47.909Q-28.868 47.909-28.844 47.942Q-28.820 47.975-28.820 48.023Q-28.820 48.260-28.921 48.508Q-29.022 48.757-29.218 48.913Q-29.413 49.069-29.664 49.069Q-30.323 49.069-30.782 48.744Q-31.241 48.418-31.241 47.781L-31.241 46.990Q-31.241 46.691-31.404 46.445Q-31.567 46.199-31.830 46.063Q-32.094 45.927-32.393 45.927L-33.509 45.927L-33.509 48.260Q-33.509 48.555-32.590 48.555L-32.590 48.871M-33.509 43.334L-33.509 45.667L-32.503 45.667Q-31.751 45.667-31.325 45.369Q-30.899 45.070-30.899 44.353Q-30.899 43.620-31.318 43.329Q-31.738 43.039-32.503 43.039L-33.048 43.039Q-33.298 43.039-33.403 43.088Q-33.509 43.136-33.509 43.334\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cpath d=\"M113.144-45.023a2.276 2.276 0 1 0-4.553 0 2.276 2.276 0 0 0 4.553 0Zm-2.277 0\"\u002F>\u003Cpath fill=\"none\" d=\"M115.42-27.951a4.552 4.552 0 1 0-9.105 0 4.552 4.552 0 0 0 9.105 0Zm-4.553 0\"\u002F>\u003Cpath d=\"M113.144-13.725a2.276 2.276 0 1 0-4.553 0 2.276 2.276 0 0 0 4.553 0Zm-2.277 0\"\u002F>\u003Cpath fill=\"none\" d=\"M115.42 3.346a4.552 4.552 0 1 0-9.105 0 4.552 4.552 0 0 0 9.105 0Zm-4.553 0\"\u002F>\u003Cg fill=\"var(--tk-soft-neutral)\">\u003Cpath d=\"M105.177 28.954h11.38v-11.38h-11.38Z\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M110.867-42.547v9.843m0 9.505v6.998m0 4.952v9.843m0 9.505v9.274\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(71.437 -23.177)\">\u003Cpath d=\"M-40.795 48.871L-42.398 48.871L-42.398 48.591Q-42.172 48.591-42.023 48.557Q-41.875 48.522-41.875 48.382L-41.875 44.763Q-41.875 44.493-41.982 44.431Q-42.090 44.370-42.398 44.370L-42.398 44.089L-41.321 44.014L-41.321 48.382Q-41.321 48.519-41.171 48.555Q-41.020 48.591-40.795 48.591L-40.795 48.871M-40.142 48.143Q-40.142 47.811-39.918 47.584Q-39.694 47.357-39.350 47.229Q-39.007 47.100-38.634 47.048Q-38.262 46.995-37.958 46.995L-37.958 46.742Q-37.958 46.537-38.065 46.357Q-38.173 46.178-38.354 46.075Q-38.535 45.973-38.744 45.973Q-39.151 45.973-39.386 46.065Q-39.298 46.102-39.251 46.186Q-39.205 46.270-39.205 46.372Q-39.205 46.468-39.251 46.547Q-39.298 46.625-39.378 46.670Q-39.458 46.714-39.547 46.714Q-39.697 46.714-39.798 46.617Q-39.899 46.519-39.899 46.372Q-39.899 45.750-38.744 45.750Q-38.532 45.750-38.282 45.814Q-38.033 45.877-37.831 45.996Q-37.630 46.116-37.503 46.301Q-37.377 46.485-37.377 46.728L-37.377 48.304Q-37.377 48.420-37.315 48.516Q-37.254 48.611-37.141 48.611Q-37.031 48.611-36.966 48.517Q-36.902 48.423-36.902 48.304L-36.902 47.856L-36.635 47.856L-36.635 48.304Q-36.635 48.574-36.862 48.739Q-37.090 48.905-37.370 48.905Q-37.578 48.905-37.715 48.751Q-37.852 48.598-37.876 48.382Q-38.023 48.649-38.305 48.794Q-38.587 48.939-38.911 48.939Q-39.188 48.939-39.472 48.864Q-39.756 48.789-39.949 48.610Q-40.142 48.430-40.142 48.143M-39.527 48.143Q-39.527 48.317-39.426 48.447Q-39.325 48.577-39.169 48.647Q-39.014 48.717-38.850 48.717Q-38.631 48.717-38.423 48.620Q-38.214 48.522-38.086 48.341Q-37.958 48.160-37.958 47.934L-37.958 47.206Q-38.282 47.206-38.648 47.297Q-39.014 47.388-39.270 47.600Q-39.527 47.811-39.527 48.143M-34.536 48.871L-36.170 48.871L-36.170 48.591Q-35.941 48.591-35.792 48.557Q-35.644 48.522-35.644 48.382L-35.644 46.533Q-35.644 46.263-35.751 46.202Q-35.859 46.140-36.170 46.140L-36.170 45.860L-35.110 45.785L-35.110 46.434Q-34.940 46.126-34.635 45.955Q-34.331 45.785-33.986 45.785Q-33.586 45.785-33.309 45.925Q-33.032 46.065-32.947 46.413Q-32.779 46.120-32.480 45.952Q-32.181 45.785-31.836 45.785Q-31.330 45.785-31.047 46.008Q-30.763 46.232-30.763 46.728L-30.763 48.382Q-30.763 48.519-30.614 48.555Q-30.465 48.591-30.240 48.591L-30.240 48.871L-31.870 48.871L-31.870 48.591Q-31.645 48.591-31.494 48.555Q-31.344 48.519-31.344 48.382L-31.344 46.742Q-31.344 46.407-31.464 46.207Q-31.583 46.007-31.898 46.007Q-32.168 46.007-32.402 46.143Q-32.636 46.280-32.774 46.514Q-32.913 46.748-32.913 47.022L-32.913 48.382Q-32.913 48.519-32.764 48.555Q-32.615 48.591-32.390 48.591L-32.390 48.871L-34.020 48.871L-34.020 48.591Q-33.791 48.591-33.642 48.557Q-33.494 48.522-33.494 48.382L-33.494 46.742Q-33.494 46.407-33.613 46.207Q-33.733 46.007-34.048 46.007Q-34.318 46.007-34.552 46.143Q-34.786 46.280-34.924 46.514Q-35.063 46.748-35.063 47.022L-35.063 48.382Q-35.063 48.519-34.912 48.555Q-34.762 48.591-34.536 48.591\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(71.437 -23.177)\">\u003Cpath d=\"M-29.052 48.871L-29.319 48.871L-29.319 44.763Q-29.319 44.493-29.426 44.431Q-29.534 44.370-29.845 44.370L-29.845 44.089L-28.765 44.014L-28.765 46.184Q-28.556 45.993-28.271 45.889Q-27.985 45.785-27.688 45.785Q-27.370 45.785-27.073 45.906Q-26.776 46.027-26.553 46.243Q-26.331 46.458-26.205 46.743Q-26.078 47.029-26.078 47.360Q-26.078 47.805-26.318 48.169Q-26.557 48.533-26.950 48.736Q-27.343 48.939-27.787 48.939Q-27.982 48.939-28.172 48.883Q-28.361 48.827-28.522 48.722Q-28.683 48.618-28.823 48.457L-29.052 48.871M-28.737 46.526L-28.737 48.143Q-28.601 48.403-28.360 48.560Q-28.119 48.717-27.842 48.717Q-27.548 48.717-27.336 48.610Q-27.124 48.502-26.991 48.310Q-26.858 48.119-26.799 47.880Q-26.741 47.641-26.741 47.360Q-26.741 47.001-26.835 46.697Q-26.929 46.393-27.157 46.200Q-27.384 46.007-27.750 46.007Q-28.050 46.007-28.317 46.143Q-28.584 46.280-28.737 46.526\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(71.437 -23.177)\">\u003Cpath d=\"M-25.227 47.360Q-25.227 47.022-25.086 46.731Q-24.946 46.441-24.702 46.227Q-24.458 46.014-24.153 45.899Q-23.849 45.785-23.524 45.785Q-23.254 45.785-22.991 45.884Q-22.728 45.983-22.537 46.161L-22.537 44.763Q-22.537 44.493-22.644 44.431Q-22.752 44.370-23.063 44.370L-23.063 44.089L-21.986 44.014L-21.986 48.198Q-21.986 48.386-21.932 48.469Q-21.877 48.553-21.776 48.572Q-21.675 48.591-21.460 48.591L-21.460 48.871L-22.567 48.939L-22.567 48.522Q-22.984 48.939-23.610 48.939Q-24.041 48.939-24.413 48.727Q-24.786 48.516-25.006 48.155Q-25.227 47.794-25.227 47.360M-23.552 48.717Q-23.343 48.717-23.157 48.645Q-22.971 48.574-22.817 48.437Q-22.663 48.300-22.567 48.122L-22.567 46.513Q-22.653 46.366-22.798 46.246Q-22.943 46.126-23.113 46.067Q-23.282 46.007-23.463 46.007Q-24.023 46.007-24.292 46.396Q-24.560 46.786-24.560 47.367Q-24.560 47.938-24.326 48.328Q-24.092 48.717-23.552 48.717M-20.752 48.143Q-20.752 47.811-20.529 47.584Q-20.305 47.357-19.961 47.229Q-19.618 47.100-19.245 47.048Q-18.873 46.995-18.568 46.995L-18.568 46.742Q-18.568 46.537-18.676 46.357Q-18.784 46.178-18.965 46.075Q-19.146 45.973-19.354 45.973Q-19.761 45.973-19.997 46.065Q-19.908 46.102-19.862 46.186Q-19.816 46.270-19.816 46.372Q-19.816 46.468-19.862 46.547Q-19.908 46.625-19.989 46.670Q-20.069 46.714-20.158 46.714Q-20.308 46.714-20.409 46.617Q-20.510 46.519-20.510 46.372Q-20.510 45.750-19.354 45.750Q-19.143 45.750-18.893 45.814Q-18.644 45.877-18.442 45.996Q-18.240 46.116-18.114 46.301Q-17.987 46.485-17.987 46.728L-17.987 48.304Q-17.987 48.420-17.926 48.516Q-17.864 48.611-17.751 48.611Q-17.642 48.611-17.577 48.517Q-17.512 48.423-17.512 48.304L-17.512 47.856L-17.246 47.856L-17.246 48.304Q-17.246 48.574-17.473 48.739Q-17.700 48.905-17.980 48.905Q-18.189 48.905-18.326 48.751Q-18.462 48.598-18.486 48.382Q-18.633 48.649-18.915 48.794Q-19.197 48.939-19.522 48.939Q-19.799 48.939-20.083 48.864Q-20.366 48.789-20.559 48.610Q-20.752 48.430-20.752 48.143M-20.137 48.143Q-20.137 48.317-20.036 48.447Q-19.936 48.577-19.780 48.647Q-19.624 48.717-19.460 48.717Q-19.242 48.717-19.033 48.620Q-18.825 48.522-18.697 48.341Q-18.568 48.160-18.568 47.934L-18.568 47.206Q-18.893 47.206-19.259 47.297Q-19.624 47.388-19.881 47.600Q-20.137 47.811-20.137 48.143\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(71.437 -23.177)\">\u003Cpath d=\"M-13.603 48.030L-13.603 46.133L-14.242 46.133L-14.242 45.911Q-13.924 45.911-13.707 45.701Q-13.490 45.491-13.390 45.181Q-13.289 44.872-13.289 44.564L-13.022 44.564L-13.022 45.853L-11.945 45.853L-11.945 46.133L-13.022 46.133L-13.022 48.017Q-13.022 48.293-12.918 48.492Q-12.814 48.690-12.554 48.690Q-12.397 48.690-12.291 48.586Q-12.185 48.481-12.135 48.328Q-12.086 48.174-12.086 48.017L-12.086 47.603L-11.819 47.603L-11.819 48.030Q-11.819 48.256-11.918 48.466Q-12.017 48.676-12.202 48.808Q-12.386 48.939-12.615 48.939Q-13.053 48.939-13.328 48.702Q-13.603 48.464-13.603 48.030M-11.050 47.388Q-11.050 47.046-10.915 46.747Q-10.780 46.448-10.541 46.224Q-10.301 46-9.984 45.875Q-9.666 45.750-9.334 45.750Q-8.890 45.750-8.490 45.966Q-8.090 46.181-7.856 46.559Q-7.622 46.936-7.622 47.388Q-7.622 47.729-7.764 48.013Q-7.905 48.297-8.150 48.504Q-8.394 48.710-8.704 48.825Q-9.013 48.939-9.334 48.939Q-9.765 48.939-10.166 48.738Q-10.568 48.536-10.809 48.184Q-11.050 47.832-11.050 47.388M-9.334 48.690Q-8.733 48.690-8.509 48.312Q-8.285 47.934-8.285 47.302Q-8.285 46.690-8.519 46.331Q-8.753 45.973-9.334 45.973Q-10.387 45.973-10.387 47.302Q-10.387 47.934-10.161 48.312Q-9.936 48.690-9.334 48.690\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(71.437 -23.177)\">\u003Cpath d=\"M-3.797 48.030L-3.797 46.133L-4.436 46.133L-4.436 45.911Q-4.118 45.911-3.901 45.701Q-3.684 45.491-3.584 45.181Q-3.483 44.872-3.483 44.564L-3.216 44.564L-3.216 45.853L-2.139 45.853L-2.139 46.133L-3.216 46.133L-3.216 48.017Q-3.216 48.293-3.112 48.492Q-3.008 48.690-2.748 48.690Q-2.591 48.690-2.485 48.586Q-2.379 48.481-2.329 48.328Q-2.280 48.174-2.280 48.017L-2.280 47.603L-2.013 47.603L-2.013 48.030Q-2.013 48.256-2.112 48.466Q-2.211 48.676-2.396 48.808Q-2.580 48.939-2.809 48.939Q-3.247 48.939-3.522 48.702Q-3.797 48.464-3.797 48.030M0.479 48.871L-1.155 48.871L-1.155 48.591Q-0.926 48.591-0.777 48.557Q-0.629 48.522-0.629 48.382L-0.629 44.763Q-0.629 44.493-0.736 44.431Q-0.844 44.370-1.155 44.370L-1.155 44.089L-0.075 44.014L-0.075 46.400Q0.031 46.215 0.209 46.073Q0.386 45.932 0.595 45.858Q0.803 45.785 1.029 45.785Q1.535 45.785 1.819 46.008Q2.102 46.232 2.102 46.728L2.102 48.382Q2.102 48.519 2.251 48.555Q2.400 48.591 2.625 48.591L2.625 48.871L0.995 48.871L0.995 48.591Q1.224 48.591 1.372 48.557Q1.521 48.522 1.521 48.382L1.521 46.742Q1.521 46.407 1.402 46.207Q1.282 46.007 0.967 46.007Q0.697 46.007 0.463 46.143Q0.229 46.280 0.091 46.514Q-0.048 46.748-0.048 47.022L-0.048 48.382Q-0.048 48.519 0.103 48.555Q0.253 48.591 0.479 48.591L0.479 48.871M3.172 47.336Q3.172 47.015 3.297 46.726Q3.422 46.437 3.647 46.214Q3.873 45.990 4.168 45.870Q4.464 45.750 4.782 45.750Q5.110 45.750 5.372 45.850Q5.633 45.949 5.809 46.131Q5.985 46.314 6.079 46.572Q6.173 46.830 6.173 47.162Q6.173 47.254 6.091 47.275L3.835 47.275L3.835 47.336Q3.835 47.924 4.119 48.307Q4.403 48.690 4.970 48.690Q5.291 48.690 5.560 48.497Q5.828 48.304 5.917 47.989Q5.924 47.948 5.999 47.934L6.091 47.934Q6.173 47.958 6.173 48.030Q6.173 48.037 6.166 48.064Q6.053 48.461 5.683 48.700Q5.312 48.939 4.888 48.939Q4.450 48.939 4.050 48.731Q3.651 48.522 3.411 48.155Q3.172 47.788 3.172 47.336M3.842 47.066L5.657 47.066Q5.657 46.789 5.560 46.537Q5.462 46.284 5.264 46.128Q5.066 45.973 4.782 45.973Q4.505 45.973 4.291 46.131Q4.078 46.290 3.960 46.545Q3.842 46.800 3.842 47.066\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(71.437 -23.177)\">\u003Cpath d=\"M13.375 48.871L10.637 48.871L10.637 48.591Q10.986 48.591 11.323 48.555Q11.659 48.519 11.659 48.382L11.659 44.581Q11.659 44.438 11.571 44.404Q11.482 44.370 11.297 44.370L10.938 44.370Q10.637 44.370 10.422 44.417Q10.207 44.465 10.050 44.622Q9.913 44.756 9.853 45.034Q9.793 45.313 9.756 45.726L9.489 45.726L9.636 44.089L14.370 44.089L14.517 45.726L14.250 45.726Q14.213 45.313 14.156 45.036Q14.100 44.759 13.956 44.622Q13.796 44.462 13.584 44.416Q13.372 44.370 13.068 44.370L12.716 44.370Q12.531 44.370 12.442 44.404Q12.353 44.438 12.353 44.581L12.353 48.382Q12.353 48.519 12.690 48.555Q13.027 48.591 13.375 48.591L13.375 48.871M17.056 47.617L14.999 47.617L14.999 47.114L17.056 47.114L17.056 47.617M18.386 48.030L18.386 46.133L17.747 46.133L17.747 45.911Q18.065 45.911 18.282 45.701Q18.499 45.491 18.600 45.181Q18.700 44.872 18.700 44.564L18.967 44.564L18.967 45.853L20.044 45.853L20.044 46.133L18.967 46.133L18.967 48.017Q18.967 48.293 19.071 48.492Q19.175 48.690 19.435 48.690Q19.592 48.690 19.698 48.586Q19.804 48.481 19.854 48.328Q19.904 48.174 19.904 48.017L19.904 47.603L20.170 47.603L20.170 48.030Q20.170 48.256 20.071 48.466Q19.972 48.676 19.787 48.808Q19.603 48.939 19.374 48.939Q18.936 48.939 18.661 48.702Q18.386 48.464 18.386 48.030M22.863 47.617L20.806 47.617L20.806 47.114L22.863 47.114L22.863 47.617M26.678 48.871L24.149 48.871L24.149 48.591Q25.116 48.591 25.116 48.382L25.116 44.763Q24.723 44.951 24.101 44.951L24.101 44.670Q24.518 44.670 24.882 44.569Q25.246 44.469 25.502 44.223L25.629 44.223Q25.694 44.240 25.711 44.308L25.711 48.382Q25.711 48.591 26.678 48.591\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath d=\"M158.668-45.023a2.276 2.276 0 1 0-4.552 0 2.276 2.276 0 0 0 4.552 0Zm-2.276 0\"\u002F>\u003Cpath fill=\"none\" d=\"M160.944-27.951a4.552 4.552 0 1 0-9.104 0 4.552 4.552 0 0 0 9.104 0Zm-4.552 0\"\u002F>\u003Cpath d=\"M158.668-13.725a2.276 2.276 0 1 0-4.552 0 2.276 2.276 0 0 0 4.552 0Zm-2.276 0\"\u002F>\u003Cpath fill=\"none\" d=\"M160.944 3.346a4.552 4.552 0 1 0-9.104 0 4.552 4.552 0 0 0 9.104 0Zm-4.552 0\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(206.97 -92.183)\">\u003Cpath d=\"M-42.288 48.933L-42.288 47.360Q-42.288 47.333-42.263 47.307Q-42.237 47.282-42.210 47.282L-42.097 47.282Q-42.069 47.282-42.046 47.309Q-42.022 47.336-42.022 47.360Q-42.022 47.705-41.890 47.969Q-41.758 48.232-41.529 48.401Q-41.300 48.570-40.998 48.651Q-40.695 48.731-40.354 48.731Q-40.087 48.731-39.851 48.603Q-39.615 48.475-39.470 48.252Q-39.325 48.030-39.325 47.764Q-39.325 47.541-39.431 47.345Q-39.537 47.148-39.718 47.013Q-39.899 46.878-40.125 46.827L-41.153 46.595Q-41.465 46.523-41.724 46.337Q-41.984 46.150-42.136 45.879Q-42.288 45.607-42.288 45.292Q-42.288 44.906-42.075 44.599Q-41.861 44.291-41.514 44.120Q-41.167 43.949-40.788 43.949Q-40.559 43.949-40.330 44.002Q-40.101 44.055-39.902 44.163Q-39.704 44.270-39.550 44.434L-39.256 43.994Q-39.233 43.949-39.192 43.949L-39.144 43.949Q-39.113 43.949-39.091 43.975Q-39.069 44-39.069 44.028L-39.069 45.603Q-39.069 45.624-39.092 45.651Q-39.116 45.679-39.144 45.679L-39.256 45.679Q-39.318 45.679-39.332 45.603Q-39.373 45.190-39.554 44.870Q-39.735 44.551-40.046 44.376Q-40.357 44.202-40.788 44.202Q-41.037 44.202-41.277 44.313Q-41.516 44.424-41.666 44.622Q-41.817 44.821-41.817 45.084Q-41.817 45.296-41.709 45.477Q-41.601 45.658-41.425 45.778Q-41.249 45.897-41.041 45.938L-40.012 46.167Q-39.694 46.239-39.427 46.444Q-39.161 46.649-39.009 46.943Q-38.857 47.237-38.857 47.569Q-38.857 47.962-39.062 48.297Q-39.267 48.632-39.612 48.821Q-39.957 49.011-40.354 49.011Q-40.774 49.011-41.153 48.898Q-41.533 48.786-41.803 48.536L-42.097 48.970Q-42.124 49.011-42.162 49.011L-42.210 49.011Q-42.237 49.011-42.263 48.986Q-42.288 48.960-42.288 48.933M-37.520 48.030L-37.520 46.133L-38.159 46.133L-38.159 45.911Q-37.841 45.911-37.624 45.701Q-37.407 45.491-37.307 45.181Q-37.206 44.872-37.206 44.564L-36.939 44.564L-36.939 45.853L-35.862 45.853L-35.862 46.133L-36.939 46.133L-36.939 48.017Q-36.939 48.293-36.835 48.492Q-36.731 48.690-36.471 48.690Q-36.314 48.690-36.208 48.586Q-36.102 48.481-36.052 48.328Q-36.003 48.174-36.003 48.017L-36.003 47.603L-35.736 47.603L-35.736 48.030Q-35.736 48.256-35.835 48.466Q-35.934 48.676-36.119 48.808Q-36.303 48.939-36.532 48.939Q-36.970 48.939-37.245 48.702Q-37.520 48.464-37.520 48.030M-34.427 50.101Q-34.427 50.067-34.400 50.040Q-34.130 49.811-33.981 49.488Q-33.832 49.165-33.832 48.809L-33.832 48.772Q-33.942 48.871-34.106 48.871Q-34.287 48.871-34.406 48.751Q-34.526 48.632-34.526 48.451Q-34.526 48.276-34.406 48.157Q-34.287 48.037-34.106 48.037Q-33.849 48.037-33.730 48.276Q-33.610 48.516-33.610 48.809Q-33.610 49.209-33.779 49.580Q-33.948 49.951-34.246 50.207Q-34.277 50.228-34.304 50.228Q-34.345 50.228-34.386 50.187Q-34.427 50.146-34.427 50.101\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(206.97 -92.183)\">\u003Cpath d=\"M-28.373 48.871L-29.963 48.871L-29.963 48.591Q-29.320 48.591-29.163 48.191L-27.519 43.976Q-27.485 43.881-27.372 43.881L-27.290 43.881Q-27.180 43.881-27.139 43.976L-25.420 48.382Q-25.352 48.522-25.162 48.557Q-24.972 48.591-24.699 48.591L-24.699 48.871L-26.698 48.871L-26.698 48.591Q-26.134 48.591-26.134 48.416Q-26.134 48.399-26.136 48.392Q-26.138 48.386-26.141 48.382L-26.562 47.316L-28.520 47.316L-28.862 48.191Q-28.876 48.191-28.876 48.269Q-28.876 48.430-28.713 48.510Q-28.551 48.591-28.373 48.591L-28.373 48.871M-27.539 44.797L-28.407 47.036L-26.664 47.036\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(206.97 -92.183)\">\u003Cpath d=\"M-23.749 48.030L-23.749 46.133L-24.388 46.133L-24.388 45.911Q-24.070 45.911-23.853 45.701Q-23.636 45.491-23.536 45.181Q-23.435 44.872-23.435 44.564L-23.168 44.564L-23.168 45.853L-22.091 45.853L-22.091 46.133L-23.168 46.133L-23.168 48.017Q-23.168 48.293-23.064 48.492Q-22.960 48.690-22.700 48.690Q-22.543 48.690-22.437 48.586Q-22.331 48.481-22.281 48.328Q-22.232 48.174-22.232 48.017L-22.232 47.603L-21.965 47.603L-21.965 48.030Q-21.965 48.256-22.064 48.466Q-22.163 48.676-22.348 48.808Q-22.532 48.939-22.761 48.939Q-23.199 48.939-23.474 48.702Q-23.749 48.464-23.749 48.030\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg transform=\"translate(206.97 -74.847)\">\u003Cpath d=\"M-42.288 48.933L-42.288 47.360Q-42.288 47.333-42.263 47.307Q-42.237 47.282-42.210 47.282L-42.097 47.282Q-42.069 47.282-42.046 47.309Q-42.022 47.336-42.022 47.360Q-42.022 47.705-41.890 47.969Q-41.758 48.232-41.529 48.401Q-41.300 48.570-40.998 48.651Q-40.695 48.731-40.354 48.731Q-40.087 48.731-39.851 48.603Q-39.615 48.475-39.470 48.252Q-39.325 48.030-39.325 47.764Q-39.325 47.541-39.431 47.345Q-39.537 47.148-39.718 47.013Q-39.899 46.878-40.125 46.827L-41.153 46.595Q-41.465 46.523-41.724 46.337Q-41.984 46.150-42.136 45.879Q-42.288 45.607-42.288 45.292Q-42.288 44.906-42.075 44.599Q-41.861 44.291-41.514 44.120Q-41.167 43.949-40.788 43.949Q-40.559 43.949-40.330 44.002Q-40.101 44.055-39.902 44.163Q-39.704 44.270-39.550 44.434L-39.256 43.994Q-39.233 43.949-39.192 43.949L-39.144 43.949Q-39.113 43.949-39.091 43.975Q-39.069 44-39.069 44.028L-39.069 45.603Q-39.069 45.624-39.092 45.651Q-39.116 45.679-39.144 45.679L-39.256 45.679Q-39.318 45.679-39.332 45.603Q-39.373 45.190-39.554 44.870Q-39.735 44.551-40.046 44.376Q-40.357 44.202-40.788 44.202Q-41.037 44.202-41.277 44.313Q-41.516 44.424-41.666 44.622Q-41.817 44.821-41.817 45.084Q-41.817 45.296-41.709 45.477Q-41.601 45.658-41.425 45.778Q-41.249 45.897-41.041 45.938L-40.012 46.167Q-39.694 46.239-39.427 46.444Q-39.161 46.649-39.009 46.943Q-38.857 47.237-38.857 47.569Q-38.857 47.962-39.062 48.297Q-39.267 48.632-39.612 48.821Q-39.957 49.011-40.354 49.011Q-40.774 49.011-41.153 48.898Q-41.533 48.786-41.803 48.536L-42.097 48.970Q-42.124 49.011-42.162 49.011L-42.210 49.011Q-42.237 49.011-42.263 48.986Q-42.288 48.960-42.288 48.933M-37.520 48.030L-37.520 46.133L-38.159 46.133L-38.159 45.911Q-37.841 45.911-37.624 45.701Q-37.407 45.491-37.307 45.181Q-37.206 44.872-37.206 44.564L-36.939 44.564L-36.939 45.853L-35.862 45.853L-35.862 46.133L-36.939 46.133L-36.939 48.017Q-36.939 48.293-36.835 48.492Q-36.731 48.690-36.471 48.690Q-36.314 48.690-36.208 48.586Q-36.102 48.481-36.052 48.328Q-36.003 48.174-36.003 48.017L-36.003 47.603L-35.736 47.603L-35.736 48.030Q-35.736 48.256-35.835 48.466Q-35.934 48.676-36.119 48.808Q-36.303 48.939-36.532 48.939Q-36.970 48.939-37.245 48.702Q-37.520 48.464-37.520 48.030M-32.342 49.551L-32.342 47.295L-34.591 47.295Q-34.659 47.285-34.705 47.239Q-34.752 47.193-34.752 47.121Q-34.752 46.977-34.591 46.954L-32.342 46.954L-32.342 44.698Q-32.332 44.629-32.286 44.583Q-32.239 44.537-32.168 44.537Q-32.024 44.537-32 44.698L-32 46.954L-29.758 46.954Q-29.597 46.977-29.597 47.121Q-29.597 47.193-29.643 47.239Q-29.690 47.285-29.758 47.295L-32 47.295L-32 49.551Q-32.024 49.712-32.168 49.712Q-32.239 49.712-32.286 49.666Q-32.332 49.620-32.342 49.551M-25.776 48.871L-28.305 48.871L-28.305 48.591Q-27.338 48.591-27.338 48.382L-27.338 44.763Q-27.731 44.951-28.353 44.951L-28.353 44.670Q-27.936 44.670-27.572 44.569Q-27.208 44.469-26.952 44.223L-26.825 44.223Q-26.760 44.240-26.743 44.308L-26.743 48.382Q-26.743 48.591-25.776 48.591\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(206.97 -43.55)\">\u003Cpath d=\"M-42.288 48.933L-42.288 47.360Q-42.288 47.333-42.263 47.307Q-42.237 47.282-42.210 47.282L-42.097 47.282Q-42.069 47.282-42.046 47.309Q-42.022 47.336-42.022 47.360Q-42.022 47.705-41.890 47.969Q-41.758 48.232-41.529 48.401Q-41.300 48.570-40.998 48.651Q-40.695 48.731-40.354 48.731Q-40.087 48.731-39.851 48.603Q-39.615 48.475-39.470 48.252Q-39.325 48.030-39.325 47.764Q-39.325 47.541-39.431 47.345Q-39.537 47.148-39.718 47.013Q-39.899 46.878-40.125 46.827L-41.153 46.595Q-41.465 46.523-41.724 46.337Q-41.984 46.150-42.136 45.879Q-42.288 45.607-42.288 45.292Q-42.288 44.906-42.075 44.599Q-41.861 44.291-41.514 44.120Q-41.167 43.949-40.788 43.949Q-40.559 43.949-40.330 44.002Q-40.101 44.055-39.902 44.163Q-39.704 44.270-39.550 44.434L-39.256 43.994Q-39.233 43.949-39.192 43.949L-39.144 43.949Q-39.113 43.949-39.091 43.975Q-39.069 44-39.069 44.028L-39.069 45.603Q-39.069 45.624-39.092 45.651Q-39.116 45.679-39.144 45.679L-39.256 45.679Q-39.318 45.679-39.332 45.603Q-39.373 45.190-39.554 44.870Q-39.735 44.551-40.046 44.376Q-40.357 44.202-40.788 44.202Q-41.037 44.202-41.277 44.313Q-41.516 44.424-41.666 44.622Q-41.817 44.821-41.817 45.084Q-41.817 45.296-41.709 45.477Q-41.601 45.658-41.425 45.778Q-41.249 45.897-41.041 45.938L-40.012 46.167Q-39.694 46.239-39.427 46.444Q-39.161 46.649-39.009 46.943Q-38.857 47.237-38.857 47.569Q-38.857 47.962-39.062 48.297Q-39.267 48.632-39.612 48.821Q-39.957 49.011-40.354 49.011Q-40.774 49.011-41.153 48.898Q-41.533 48.786-41.803 48.536L-42.097 48.970Q-42.124 49.011-42.162 49.011L-42.210 49.011Q-42.237 49.011-42.263 48.986Q-42.288 48.960-42.288 48.933M-37.520 48.030L-37.520 46.133L-38.159 46.133L-38.159 45.911Q-37.841 45.911-37.624 45.701Q-37.407 45.491-37.307 45.181Q-37.206 44.872-37.206 44.564L-36.939 44.564L-36.939 45.853L-35.862 45.853L-35.862 46.133L-36.939 46.133L-36.939 48.017Q-36.939 48.293-36.835 48.492Q-36.731 48.690-36.471 48.690Q-36.314 48.690-36.208 48.586Q-36.102 48.481-36.052 48.328Q-36.003 48.174-36.003 48.017L-36.003 47.603L-35.736 47.603L-35.736 48.030Q-35.736 48.256-35.835 48.466Q-35.934 48.676-36.119 48.808Q-36.303 48.939-36.532 48.939Q-36.970 48.939-37.245 48.702Q-37.520 48.464-37.520 48.030M-32.342 49.551L-32.342 47.295L-34.591 47.295Q-34.659 47.285-34.705 47.239Q-34.752 47.193-34.752 47.121Q-34.752 46.977-34.591 46.954L-32.342 46.954L-32.342 44.698Q-32.332 44.629-32.286 44.583Q-32.239 44.537-32.168 44.537Q-32.024 44.537-32 44.698L-32 46.954L-29.758 46.954Q-29.597 46.977-29.597 47.121Q-29.597 47.193-29.643 47.239Q-29.690 47.285-29.758 47.295L-32 47.295L-32 49.551Q-32.024 49.712-32.168 49.712Q-32.239 49.712-32.286 49.666Q-32.332 49.620-32.342 49.551M-25.776 48.871L-28.661 48.871L-28.661 48.669Q-28.661 48.639-28.633 48.611L-27.386 47.394Q-27.314 47.319-27.271 47.277Q-27.229 47.234-27.150 47.155Q-26.736 46.742-26.506 46.384Q-26.275 46.027-26.275 45.603Q-26.275 45.371-26.354 45.168Q-26.432 44.964-26.574 44.814Q-26.716 44.663-26.911 44.583Q-27.106 44.503-27.338 44.503Q-27.649 44.503-27.907 44.662Q-28.165 44.821-28.295 45.098L-28.275 45.098Q-28.107 45.098-27.999 45.209Q-27.892 45.320-27.892 45.484Q-27.892 45.641-28.001 45.754Q-28.110 45.867-28.275 45.867Q-28.435 45.867-28.548 45.754Q-28.661 45.641-28.661 45.484Q-28.661 45.108-28.452 44.821Q-28.244 44.534-27.909 44.378Q-27.574 44.223-27.218 44.223Q-26.795 44.223-26.415 44.381Q-26.036 44.540-25.802 44.857Q-25.568 45.173-25.568 45.603Q-25.568 45.914-25.708 46.183Q-25.848 46.451-26.053 46.656Q-26.258 46.861-26.620 47.143Q-26.983 47.425-27.092 47.521L-27.946 48.249L-27.304 48.249Q-27.041 48.249-26.752 48.247Q-26.463 48.246-26.244 48.237Q-26.026 48.228-26.008 48.211Q-25.947 48.146-25.909 47.979Q-25.872 47.811-25.834 47.569L-25.568 47.569\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M156.392-42.547v9.843m0 9.505v6.998m0 4.952v9.843\"\u002F>\u003Cpath d=\"M150.132 20.418a2.276 2.276 0 1 0-4.552 0 2.276 2.276 0 0 0 4.552 0ZM158.668 20.418a2.276 2.276 0 1 0-4.552 0 2.276 2.276 0 0 0 4.552 0Zm-2.276 0\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cpath fill=\"var(--tk-warn)\" d=\"M167.204 20.418a2.276 2.276 0 1 0-4.552 0 2.276 2.276 0 0 0 4.552 0Zm-2.276 0\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"m154.267 7.597-5.304 10.607M156.392 8.099v9.843\"\u002F>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"m158.517 7.597 5.304 10.607\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(216.929 -26.703)\">\u003Cpath d=\"M-40.665 48.871L-42.398 48.871L-42.398 48.591Q-42.172 48.591-42.023 48.557Q-41.875 48.522-41.875 48.382L-41.875 46.133L-42.463 46.133L-42.463 45.853L-41.875 45.853L-41.875 45.036Q-41.875 44.718-41.697 44.470Q-41.519 44.223-41.229 44.082Q-40.938 43.942-40.627 43.942Q-40.371 43.942-40.167 44.084Q-39.964 44.226-39.964 44.469Q-39.964 44.605-40.063 44.704Q-40.162 44.804-40.299 44.804Q-40.436 44.804-40.535 44.704Q-40.634 44.605-40.634 44.469Q-40.634 44.288-40.494 44.195Q-40.572 44.168-40.672 44.168Q-40.880 44.168-41.034 44.301Q-41.188 44.434-41.268 44.638Q-41.348 44.841-41.348 45.050L-41.348 45.853L-40.460 45.853L-40.460 46.133L-41.321 46.133L-41.321 48.382Q-41.321 48.591-40.665 48.591\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(216.929 -26.703)\">\u003Cpath d=\"M-37.808 48.871L-39.360 48.871L-39.360 48.591Q-39.134 48.591-38.985 48.557Q-38.837 48.522-38.837 48.382L-38.837 46.533Q-38.837 46.345-38.885 46.261Q-38.932 46.178-39.030 46.159Q-39.127 46.140-39.339 46.140L-39.339 45.860L-38.283 45.785L-38.283 48.382Q-38.283 48.522-38.151 48.557Q-38.020 48.591-37.808 48.591L-37.808 48.871M-39.079 44.564Q-39.079 44.393-38.956 44.274Q-38.833 44.154-38.662 44.154Q-38.495 44.154-38.372 44.274Q-38.249 44.393-38.249 44.564Q-38.249 44.739-38.372 44.862Q-38.495 44.985-38.662 44.985Q-38.833 44.985-38.956 44.862Q-39.079 44.739-39.079 44.564M-35.412 48.871L-37.148 48.871L-37.148 48.591Q-36.919 48.591-36.770 48.557Q-36.622 48.522-36.622 48.382L-36.622 46.533Q-36.622 46.263-36.729 46.202Q-36.837 46.140-37.148 46.140L-37.148 45.860L-36.119 45.785L-36.119 46.492Q-35.989 46.184-35.747 45.985Q-35.504 45.785-35.186 45.785Q-34.968 45.785-34.797 45.909Q-34.626 46.034-34.626 46.246Q-34.626 46.383-34.725 46.482Q-34.824 46.581-34.957 46.581Q-35.094 46.581-35.193 46.482Q-35.292 46.383-35.292 46.246Q-35.292 46.106-35.193 46.007Q-35.484 46.007-35.684 46.203Q-35.884 46.400-35.976 46.694Q-36.068 46.988-36.068 47.268L-36.068 48.382Q-36.068 48.591-35.412 48.591L-35.412 48.871M-34.041 48.864L-34.041 47.801Q-34.041 47.777-34.014 47.750Q-33.987 47.723-33.963 47.723L-33.853 47.723Q-33.788 47.723-33.775 47.781Q-33.679 48.215-33.433 48.466Q-33.187 48.717-32.773 48.717Q-32.431 48.717-32.178 48.584Q-31.926 48.451-31.926 48.143Q-31.926 47.986-32.020 47.871Q-32.114 47.757-32.252 47.688Q-32.390 47.620-32.558 47.582L-33.139 47.483Q-33.494 47.415-33.768 47.194Q-34.041 46.974-34.041 46.632Q-34.041 46.383-33.930 46.208Q-33.819 46.034-33.633 45.935Q-33.447 45.836-33.231 45.793Q-33.016 45.750-32.773 45.750Q-32.360 45.750-32.079 45.932L-31.864 45.757Q-31.854 45.754-31.847 45.752Q-31.840 45.750-31.830 45.750L-31.779 45.750Q-31.751 45.750-31.727 45.774Q-31.703 45.798-31.703 45.826L-31.703 46.673Q-31.703 46.694-31.727 46.721Q-31.751 46.748-31.779 46.748L-31.891 46.748Q-31.919 46.748-31.944 46.723Q-31.970 46.697-31.970 46.673Q-31.970 46.437-32.076 46.273Q-32.182 46.109-32.365 46.027Q-32.548 45.945-32.780 45.945Q-33.108 45.945-33.364 46.048Q-33.621 46.150-33.621 46.427Q-33.621 46.622-33.438 46.731Q-33.255 46.841-33.026 46.882L-32.452 46.988Q-32.206 47.036-31.992 47.164Q-31.779 47.292-31.642 47.495Q-31.505 47.699-31.505 47.948Q-31.505 48.461-31.871 48.700Q-32.237 48.939-32.773 48.939Q-33.269 48.939-33.600 48.645L-33.867 48.919Q-33.887 48.939-33.915 48.939L-33.963 48.939Q-33.987 48.939-34.014 48.912Q-34.041 48.885-34.041 48.864M-30.350 48.030L-30.350 46.133L-30.989 46.133L-30.989 45.911Q-30.671 45.911-30.454 45.701Q-30.237 45.491-30.136 45.181Q-30.035 44.872-30.035 44.564L-29.769 44.564L-29.769 45.853L-28.692 45.853L-28.692 46.133L-29.769 46.133L-29.769 48.017Q-29.769 48.293-29.665 48.492Q-29.560 48.690-29.301 48.690Q-29.143 48.690-29.037 48.586Q-28.931 48.481-28.882 48.328Q-28.832 48.174-28.832 48.017L-28.832 47.603L-28.566 47.603L-28.566 48.030Q-28.566 48.256-28.665 48.466Q-28.764 48.676-28.948 48.808Q-29.133 48.939-29.362 48.939Q-29.800 48.939-30.075 48.702Q-30.350 48.464-30.350 48.030\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(216.929 -26.703)\">\u003Cpath d=\"M-23.367 48.871L-25.001 48.871L-25.001 48.591Q-24.772 48.591-24.623 48.557Q-24.474 48.522-24.474 48.382L-24.474 46.533Q-24.474 46.263-24.582 46.202Q-24.690 46.140-25.001 46.140L-25.001 45.860L-23.941 45.785L-23.941 46.434Q-23.770 46.126-23.466 45.955Q-23.162 45.785-22.817 45.785Q-22.311 45.785-22.027 46.008Q-21.743 46.232-21.743 46.728L-21.743 48.382Q-21.743 48.519-21.595 48.555Q-21.446 48.591-21.220 48.591L-21.220 48.871L-22.851 48.871L-22.851 48.591Q-22.622 48.591-22.473 48.557Q-22.324 48.522-22.324 48.382L-22.324 46.742Q-22.324 46.407-22.444 46.207Q-22.564 46.007-22.878 46.007Q-23.148 46.007-23.382 46.143Q-23.616 46.280-23.755 46.514Q-23.893 46.748-23.893 47.022L-23.893 48.382Q-23.893 48.519-23.743 48.555Q-23.592 48.591-23.367 48.591L-23.367 48.871M-20.674 47.388Q-20.674 47.046-20.539 46.747Q-20.404 46.448-20.164 46.224Q-19.925 46-19.607 45.875Q-19.289 45.750-18.958 45.750Q-18.513 45.750-18.113 45.966Q-17.714 46.181-17.479 46.559Q-17.245 46.936-17.245 47.388Q-17.245 47.729-17.387 48.013Q-17.529 48.297-17.773 48.504Q-18.018 48.710-18.327 48.825Q-18.636 48.939-18.958 48.939Q-19.388 48.939-19.790 48.738Q-20.192 48.536-20.433 48.184Q-20.674 47.832-20.674 47.388M-18.958 48.690Q-18.356 48.690-18.132 48.312Q-17.908 47.934-17.908 47.302Q-17.908 46.690-18.143 46.331Q-18.377 45.973-18.958 45.973Q-20.010 45.973-20.010 47.302Q-20.010 47.934-19.785 48.312Q-19.559 48.690-18.958 48.690M-14.969 48.871L-16.603 48.871L-16.603 48.591Q-16.374 48.591-16.225 48.557Q-16.076 48.522-16.076 48.382L-16.076 46.533Q-16.076 46.263-16.184 46.202Q-16.292 46.140-16.603 46.140L-16.603 45.860L-15.543 45.785L-15.543 46.434Q-15.372 46.126-15.068 45.955Q-14.764 45.785-14.419 45.785Q-13.913 45.785-13.629 46.008Q-13.345 46.232-13.345 46.728L-13.345 48.382Q-13.345 48.519-13.197 48.555Q-13.048 48.591-12.822 48.591L-12.822 48.871L-14.453 48.871L-14.453 48.591Q-14.224 48.591-14.075 48.557Q-13.926 48.522-13.926 48.382L-13.926 46.742Q-13.926 46.407-14.046 46.207Q-14.166 46.007-14.480 46.007Q-14.750 46.007-14.984 46.143Q-15.218 46.280-15.357 46.514Q-15.495 46.748-15.495 47.022L-15.495 48.382Q-15.495 48.519-15.345 48.555Q-15.195 48.591-14.969 48.591L-14.969 48.871M-10.351 47.617L-12.409 47.617L-12.409 47.114L-10.351 47.114L-10.351 47.617M-9.589 49.404Q-9.589 49.158-9.393 48.974Q-9.196 48.789-8.940 48.710Q-9.076 48.598-9.148 48.437Q-9.220 48.276-9.220 48.095Q-9.220 47.774-9.008 47.528Q-9.343 47.230-9.343 46.820Q-9.343 46.359-8.953 46.072Q-8.564 45.785-8.085 45.785Q-7.613 45.785-7.279 46.031Q-7.104 45.877-6.894 45.795Q-6.684 45.713-6.455 45.713Q-6.291 45.713-6.169 45.820Q-6.048 45.928-6.048 46.092Q-6.048 46.188-6.120 46.260Q-6.192 46.331-6.284 46.331Q-6.383 46.331-6.453 46.258Q-6.523 46.184-6.523 46.085Q-6.523 46.031-6.509 46L-6.503 45.986Q-6.496 45.966-6.487 45.955Q-6.479 45.945-6.475 45.938Q-6.831 45.938-7.118 46.161Q-6.831 46.454-6.831 46.820Q-6.831 47.135-7.015 47.367Q-7.200 47.600-7.489 47.728Q-7.778 47.856-8.085 47.856Q-8.287 47.856-8.478 47.806Q-8.670 47.757-8.847 47.647Q-8.940 47.774-8.940 47.917Q-8.940 48.099-8.811 48.234Q-8.683 48.369-8.499 48.369L-7.866 48.369Q-7.419 48.369-7.050 48.440Q-6.680 48.512-6.421 48.741Q-6.161 48.970-6.161 49.404Q-6.161 49.725-6.457 49.927Q-6.752 50.129-7.155 50.218Q-7.559 50.307-7.873 50.307Q-8.191 50.307-8.594 50.218Q-8.998 50.129-9.293 49.927Q-9.589 49.725-9.589 49.404M-9.134 49.404Q-9.134 49.633-8.916 49.782Q-8.697 49.931-8.405 49.999Q-8.113 50.067-7.873 50.067Q-7.709 50.067-7.501 50.031Q-7.292 49.996-7.085 49.915Q-6.879 49.835-6.747 49.707Q-6.615 49.579-6.615 49.404Q-6.615 49.052-6.997 48.958Q-7.378 48.864-7.880 48.864L-8.499 48.864Q-8.738 48.864-8.936 49.015Q-9.134 49.165-9.134 49.404M-8.085 47.617Q-7.419 47.617-7.419 46.820Q-7.419 46.020-8.085 46.020Q-8.755 46.020-8.755 46.820Q-8.755 47.617-8.085 47.617M-3.816 48.871L-5.552 48.871L-5.552 48.591Q-5.323 48.591-5.175 48.557Q-5.026 48.522-5.026 48.382L-5.026 46.533Q-5.026 46.263-5.134 46.202Q-5.241 46.140-5.552 46.140L-5.552 45.860L-4.524 45.785L-4.524 46.492Q-4.394 46.184-4.151 45.985Q-3.908 45.785-3.591 45.785Q-3.372 45.785-3.201 45.909Q-3.030 46.034-3.030 46.246Q-3.030 46.383-3.129 46.482Q-3.228 46.581-3.362 46.581Q-3.498 46.581-3.597 46.482Q-3.696 46.383-3.696 46.246Q-3.696 46.106-3.597 46.007Q-3.888 46.007-4.088 46.203Q-4.288 46.400-4.380 46.694Q-4.472 46.988-4.472 47.268L-4.472 48.382Q-4.472 48.591-3.816 48.591L-3.816 48.871M-2.487 47.336Q-2.487 47.015-2.362 46.726Q-2.237 46.437-2.011 46.214Q-1.786 45.990-1.490 45.870Q-1.195 45.750-0.877 45.750Q-0.549 45.750-0.287 45.850Q-0.026 45.949 0.150 46.131Q0.326 46.314 0.420 46.572Q0.514 46.830 0.514 47.162Q0.514 47.254 0.432 47.275L-1.823 47.275L-1.823 47.336Q-1.823 47.924-1.540 48.307Q-1.256 48.690-0.689 48.690Q-0.367 48.690-0.099 48.497Q0.169 48.304 0.258 47.989Q0.265 47.948 0.340 47.934L0.432 47.934Q0.514 47.958 0.514 48.030Q0.514 48.037 0.508 48.064Q0.395 48.461 0.024 48.700Q-0.347 48.939-0.771 48.939Q-1.208 48.939-1.608 48.731Q-2.008 48.522-2.247 48.155Q-2.487 47.788-2.487 47.336M-1.817 47.066L-0.002 47.066Q-0.002 46.789-0.099 46.537Q-0.196 46.284-0.395 46.128Q-0.593 45.973-0.877 45.973Q-1.154 45.973-1.367 46.131Q-1.581 46.290-1.699 46.545Q-1.817 46.800-1.817 47.066M1.061 47.336Q1.061 47.015 1.186 46.726Q1.311 46.437 1.536 46.214Q1.762 45.990 2.058 45.870Q2.353 45.750 2.671 45.750Q2.999 45.750 3.261 45.850Q3.522 45.949 3.698 46.131Q3.874 46.314 3.968 46.572Q4.062 46.830 4.062 47.162Q4.062 47.254 3.980 47.275L1.724 47.275L1.724 47.336Q1.724 47.924 2.008 48.307Q2.292 48.690 2.859 48.690Q3.180 48.690 3.449 48.497Q3.717 48.304 3.806 47.989Q3.813 47.948 3.888 47.934L3.980 47.934Q4.062 47.958 4.062 48.030Q4.062 48.037 4.055 48.064Q3.943 48.461 3.572 48.700Q3.201 48.939 2.777 48.939Q2.340 48.939 1.940 48.731Q1.540 48.522 1.301 48.155Q1.061 47.788 1.061 47.336M1.731 47.066L3.546 47.066Q3.546 46.789 3.449 46.537Q3.351 46.284 3.153 46.128Q2.955 45.973 2.671 45.973Q2.394 45.973 2.181 46.131Q1.967 46.290 1.849 46.545Q1.731 46.800 1.731 47.066M4.650 47.360Q4.650 47.022 4.790 46.731Q4.930 46.441 5.175 46.227Q5.419 46.014 5.723 45.899Q6.028 45.785 6.352 45.785Q6.622 45.785 6.886 45.884Q7.149 45.983 7.340 46.161L7.340 44.763Q7.340 44.493 7.232 44.431Q7.125 44.370 6.814 44.370L6.814 44.089L7.890 44.014L7.890 48.198Q7.890 48.386 7.945 48.469Q8 48.553 8.101 48.572Q8.201 48.591 8.417 48.591L8.417 48.871L7.309 48.939L7.309 48.522Q6.892 48.939 6.267 48.939Q5.836 48.939 5.464 48.727Q5.091 48.516 4.871 48.155Q4.650 47.794 4.650 47.360M6.325 48.717Q6.533 48.717 6.720 48.645Q6.906 48.574 7.060 48.437Q7.214 48.300 7.309 48.122L7.309 46.513Q7.224 46.366 7.079 46.246Q6.933 46.126 6.764 46.067Q6.595 46.007 6.414 46.007Q5.853 46.007 5.585 46.396Q5.317 46.786 5.317 47.367Q5.317 47.938 5.551 48.328Q5.785 48.717 6.325 48.717M9.401 50.006Q9.531 50.074 9.668 50.074Q9.839 50.074 9.989 49.985Q10.139 49.896 10.251 49.751Q10.362 49.606 10.440 49.438L10.703 48.871L9.534 46.345Q9.459 46.198 9.329 46.166Q9.200 46.133 8.967 46.133L8.967 45.853L10.488 45.853L10.488 46.133Q10.139 46.133 10.139 46.280Q10.143 46.301 10.145 46.318Q10.146 46.335 10.146 46.345L11.004 48.204L11.777 46.533Q11.811 46.465 11.811 46.386Q11.811 46.273 11.727 46.203Q11.643 46.133 11.531 46.133L11.531 45.853L12.727 45.853L12.727 46.133Q12.508 46.133 12.335 46.237Q12.163 46.342 12.071 46.533L10.734 49.438Q10.563 49.808 10.293 50.054Q10.023 50.300 9.668 50.300Q9.398 50.300 9.179 50.134Q8.960 49.968 8.960 49.705Q8.960 49.568 9.053 49.479Q9.145 49.391 9.285 49.391Q9.422 49.391 9.511 49.479Q9.599 49.568 9.599 49.705Q9.599 49.808 9.546 49.886Q9.493 49.965 9.401 50.006\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(167.655 -12.945)\">\u003Cpath d=\"M-40.795 48.871L-42.398 48.871L-42.398 48.591Q-42.172 48.591-42.023 48.557Q-41.875 48.522-41.875 48.382L-41.875 44.763Q-41.875 44.493-41.982 44.431Q-42.090 44.370-42.398 44.370L-42.398 44.089L-41.321 44.014L-41.321 48.382Q-41.321 48.519-41.171 48.555Q-41.020 48.591-40.795 48.591L-40.795 48.871M-40.142 48.143Q-40.142 47.811-39.918 47.584Q-39.694 47.357-39.350 47.229Q-39.007 47.100-38.634 47.048Q-38.262 46.995-37.958 46.995L-37.958 46.742Q-37.958 46.537-38.065 46.357Q-38.173 46.178-38.354 46.075Q-38.535 45.973-38.744 45.973Q-39.151 45.973-39.386 46.065Q-39.298 46.102-39.251 46.186Q-39.205 46.270-39.205 46.372Q-39.205 46.468-39.251 46.547Q-39.298 46.625-39.378 46.670Q-39.458 46.714-39.547 46.714Q-39.697 46.714-39.798 46.617Q-39.899 46.519-39.899 46.372Q-39.899 45.750-38.744 45.750Q-38.532 45.750-38.282 45.814Q-38.033 45.877-37.831 45.996Q-37.630 46.116-37.503 46.301Q-37.377 46.485-37.377 46.728L-37.377 48.304Q-37.377 48.420-37.315 48.516Q-37.254 48.611-37.141 48.611Q-37.031 48.611-36.966 48.517Q-36.902 48.423-36.902 48.304L-36.902 47.856L-36.635 47.856L-36.635 48.304Q-36.635 48.574-36.862 48.739Q-37.090 48.905-37.370 48.905Q-37.578 48.905-37.715 48.751Q-37.852 48.598-37.876 48.382Q-38.023 48.649-38.305 48.794Q-38.587 48.939-38.911 48.939Q-39.188 48.939-39.472 48.864Q-39.756 48.789-39.949 48.610Q-40.142 48.430-40.142 48.143M-39.527 48.143Q-39.527 48.317-39.426 48.447Q-39.325 48.577-39.169 48.647Q-39.014 48.717-38.850 48.717Q-38.631 48.717-38.423 48.620Q-38.214 48.522-38.086 48.341Q-37.958 48.160-37.958 47.934L-37.958 47.206Q-38.282 47.206-38.648 47.297Q-39.014 47.388-39.270 47.600Q-39.527 47.811-39.527 48.143M-34.536 48.871L-36.170 48.871L-36.170 48.591Q-35.941 48.591-35.792 48.557Q-35.644 48.522-35.644 48.382L-35.644 46.533Q-35.644 46.263-35.751 46.202Q-35.859 46.140-36.170 46.140L-36.170 45.860L-35.110 45.785L-35.110 46.434Q-34.940 46.126-34.635 45.955Q-34.331 45.785-33.986 45.785Q-33.586 45.785-33.309 45.925Q-33.032 46.065-32.947 46.413Q-32.779 46.120-32.480 45.952Q-32.181 45.785-31.836 45.785Q-31.330 45.785-31.047 46.008Q-30.763 46.232-30.763 46.728L-30.763 48.382Q-30.763 48.519-30.614 48.555Q-30.465 48.591-30.240 48.591L-30.240 48.871L-31.870 48.871L-31.870 48.591Q-31.645 48.591-31.494 48.555Q-31.344 48.519-31.344 48.382L-31.344 46.742Q-31.344 46.407-31.464 46.207Q-31.583 46.007-31.898 46.007Q-32.168 46.007-32.402 46.143Q-32.636 46.280-32.774 46.514Q-32.913 46.748-32.913 47.022L-32.913 48.382Q-32.913 48.519-32.764 48.555Q-32.615 48.591-32.390 48.591L-32.390 48.871L-34.020 48.871L-34.020 48.591Q-33.791 48.591-33.642 48.557Q-33.494 48.522-33.494 48.382L-33.494 46.742Q-33.494 46.407-33.613 46.207Q-33.733 46.007-34.048 46.007Q-34.318 46.007-34.552 46.143Q-34.786 46.280-34.924 46.514Q-35.063 46.748-35.063 47.022L-35.063 48.382Q-35.063 48.519-34.912 48.555Q-34.762 48.591-34.536 48.591\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(167.655 -12.945)\">\u003Cpath d=\"M-29.052 48.871L-29.319 48.871L-29.319 44.763Q-29.319 44.493-29.426 44.431Q-29.534 44.370-29.845 44.370L-29.845 44.089L-28.765 44.014L-28.765 46.184Q-28.556 45.993-28.271 45.889Q-27.985 45.785-27.688 45.785Q-27.370 45.785-27.073 45.906Q-26.776 46.027-26.553 46.243Q-26.331 46.458-26.205 46.743Q-26.078 47.029-26.078 47.360Q-26.078 47.805-26.318 48.169Q-26.557 48.533-26.950 48.736Q-27.343 48.939-27.787 48.939Q-27.982 48.939-28.172 48.883Q-28.361 48.827-28.522 48.722Q-28.683 48.618-28.823 48.457L-29.052 48.871M-28.737 46.526L-28.737 48.143Q-28.601 48.403-28.360 48.560Q-28.119 48.717-27.842 48.717Q-27.548 48.717-27.336 48.610Q-27.124 48.502-26.991 48.310Q-26.858 48.119-26.799 47.880Q-26.741 47.641-26.741 47.360Q-26.741 47.001-26.835 46.697Q-26.929 46.393-27.157 46.200Q-27.384 46.007-27.750 46.007Q-28.050 46.007-28.317 46.143Q-28.584 46.280-28.737 46.526\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(167.655 -12.945)\">\u003Cpath d=\"M-25.227 47.360Q-25.227 47.022-25.086 46.731Q-24.946 46.441-24.702 46.227Q-24.458 46.014-24.153 45.899Q-23.849 45.785-23.524 45.785Q-23.254 45.785-22.991 45.884Q-22.728 45.983-22.537 46.161L-22.537 44.763Q-22.537 44.493-22.644 44.431Q-22.752 44.370-23.063 44.370L-23.063 44.089L-21.986 44.014L-21.986 48.198Q-21.986 48.386-21.932 48.469Q-21.877 48.553-21.776 48.572Q-21.675 48.591-21.460 48.591L-21.460 48.871L-22.567 48.939L-22.567 48.522Q-22.984 48.939-23.610 48.939Q-24.041 48.939-24.413 48.727Q-24.786 48.516-25.006 48.155Q-25.227 47.794-25.227 47.360M-23.552 48.717Q-23.343 48.717-23.157 48.645Q-22.971 48.574-22.817 48.437Q-22.663 48.300-22.567 48.122L-22.567 46.513Q-22.653 46.366-22.798 46.246Q-22.943 46.126-23.113 46.067Q-23.282 46.007-23.463 46.007Q-24.023 46.007-24.292 46.396Q-24.560 46.786-24.560 47.367Q-24.560 47.938-24.326 48.328Q-24.092 48.717-23.552 48.717M-20.752 48.143Q-20.752 47.811-20.529 47.584Q-20.305 47.357-19.961 47.229Q-19.618 47.100-19.245 47.048Q-18.873 46.995-18.568 46.995L-18.568 46.742Q-18.568 46.537-18.676 46.357Q-18.784 46.178-18.965 46.075Q-19.146 45.973-19.354 45.973Q-19.761 45.973-19.997 46.065Q-19.908 46.102-19.862 46.186Q-19.816 46.270-19.816 46.372Q-19.816 46.468-19.862 46.547Q-19.908 46.625-19.989 46.670Q-20.069 46.714-20.158 46.714Q-20.308 46.714-20.409 46.617Q-20.510 46.519-20.510 46.372Q-20.510 45.750-19.354 45.750Q-19.143 45.750-18.893 45.814Q-18.644 45.877-18.442 45.996Q-18.240 46.116-18.114 46.301Q-17.987 46.485-17.987 46.728L-17.987 48.304Q-17.987 48.420-17.926 48.516Q-17.864 48.611-17.751 48.611Q-17.642 48.611-17.577 48.517Q-17.512 48.423-17.512 48.304L-17.512 47.856L-17.246 47.856L-17.246 48.304Q-17.246 48.574-17.473 48.739Q-17.700 48.905-17.980 48.905Q-18.189 48.905-18.326 48.751Q-18.462 48.598-18.486 48.382Q-18.633 48.649-18.915 48.794Q-19.197 48.939-19.522 48.939Q-19.799 48.939-20.083 48.864Q-20.366 48.789-20.559 48.610Q-20.752 48.430-20.752 48.143M-20.137 48.143Q-20.137 48.317-20.036 48.447Q-19.936 48.577-19.780 48.647Q-19.624 48.717-19.460 48.717Q-19.242 48.717-19.033 48.620Q-18.825 48.522-18.697 48.341Q-18.568 48.160-18.568 47.934L-18.568 47.206Q-18.893 47.206-19.259 47.297Q-19.624 47.388-19.881 47.600Q-20.137 47.811-20.137 48.143\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(167.655 -12.945)\">\u003Cpath d=\"M-13.603 48.030L-13.603 46.133L-14.242 46.133L-14.242 45.911Q-13.924 45.911-13.707 45.701Q-13.490 45.491-13.390 45.181Q-13.289 44.872-13.289 44.564L-13.022 44.564L-13.022 45.853L-11.945 45.853L-11.945 46.133L-13.022 46.133L-13.022 48.017Q-13.022 48.293-12.918 48.492Q-12.814 48.690-12.554 48.690Q-12.397 48.690-12.291 48.586Q-12.185 48.481-12.135 48.328Q-12.086 48.174-12.086 48.017L-12.086 47.603L-11.819 47.603L-11.819 48.030Q-11.819 48.256-11.918 48.466Q-12.017 48.676-12.202 48.808Q-12.386 48.939-12.615 48.939Q-13.053 48.939-13.328 48.702Q-13.603 48.464-13.603 48.030M-11.050 47.388Q-11.050 47.046-10.915 46.747Q-10.780 46.448-10.541 46.224Q-10.301 46-9.984 45.875Q-9.666 45.750-9.334 45.750Q-8.890 45.750-8.490 45.966Q-8.090 46.181-7.856 46.559Q-7.622 46.936-7.622 47.388Q-7.622 47.729-7.764 48.013Q-7.905 48.297-8.150 48.504Q-8.394 48.710-8.704 48.825Q-9.013 48.939-9.334 48.939Q-9.765 48.939-10.166 48.738Q-10.568 48.536-10.809 48.184Q-11.050 47.832-11.050 47.388M-9.334 48.690Q-8.733 48.690-8.509 48.312Q-8.285 47.934-8.285 47.302Q-8.285 46.690-8.519 46.331Q-8.753 45.973-9.334 45.973Q-10.387 45.973-10.387 47.302Q-10.387 47.934-10.161 48.312Q-9.936 48.690-9.334 48.690\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(167.655 -12.945)\">\u003Cpath d=\"M-3.797 48.030L-3.797 46.133L-4.436 46.133L-4.436 45.911Q-4.118 45.911-3.901 45.701Q-3.684 45.491-3.584 45.181Q-3.483 44.872-3.483 44.564L-3.216 44.564L-3.216 45.853L-2.139 45.853L-2.139 46.133L-3.216 46.133L-3.216 48.017Q-3.216 48.293-3.112 48.492Q-3.008 48.690-2.748 48.690Q-2.591 48.690-2.485 48.586Q-2.379 48.481-2.329 48.328Q-2.280 48.174-2.280 48.017L-2.280 47.603L-2.013 47.603L-2.013 48.030Q-2.013 48.256-2.112 48.466Q-2.211 48.676-2.396 48.808Q-2.580 48.939-2.809 48.939Q-3.247 48.939-3.522 48.702Q-3.797 48.464-3.797 48.030M0.479 48.871L-1.155 48.871L-1.155 48.591Q-0.926 48.591-0.777 48.557Q-0.629 48.522-0.629 48.382L-0.629 44.763Q-0.629 44.493-0.736 44.431Q-0.844 44.370-1.155 44.370L-1.155 44.089L-0.075 44.014L-0.075 46.400Q0.031 46.215 0.209 46.073Q0.386 45.932 0.595 45.858Q0.803 45.785 1.029 45.785Q1.535 45.785 1.819 46.008Q2.102 46.232 2.102 46.728L2.102 48.382Q2.102 48.519 2.251 48.555Q2.400 48.591 2.625 48.591L2.625 48.871L0.995 48.871L0.995 48.591Q1.224 48.591 1.372 48.557Q1.521 48.522 1.521 48.382L1.521 46.742Q1.521 46.407 1.402 46.207Q1.282 46.007 0.967 46.007Q0.697 46.007 0.463 46.143Q0.229 46.280 0.091 46.514Q-0.048 46.748-0.048 47.022L-0.048 48.382Q-0.048 48.519 0.103 48.555Q0.253 48.591 0.479 48.591L0.479 48.871M3.172 47.336Q3.172 47.015 3.297 46.726Q3.422 46.437 3.647 46.214Q3.873 45.990 4.168 45.870Q4.464 45.750 4.782 45.750Q5.110 45.750 5.372 45.850Q5.633 45.949 5.809 46.131Q5.985 46.314 6.079 46.572Q6.173 46.830 6.173 47.162Q6.173 47.254 6.091 47.275L3.835 47.275L3.835 47.336Q3.835 47.924 4.119 48.307Q4.403 48.690 4.970 48.690Q5.291 48.690 5.560 48.497Q5.828 48.304 5.917 47.989Q5.924 47.948 5.999 47.934L6.091 47.934Q6.173 47.958 6.173 48.030Q6.173 48.037 6.166 48.064Q6.053 48.461 5.683 48.700Q5.312 48.939 4.888 48.939Q4.450 48.939 4.050 48.731Q3.651 48.522 3.411 48.155Q3.172 47.788 3.172 47.336M3.842 47.066L5.657 47.066Q5.657 46.789 5.560 46.537Q5.462 46.284 5.264 46.128Q5.066 45.973 4.782 45.973Q4.505 45.973 4.291 46.131Q4.078 46.290 3.960 46.545Q3.842 46.800 3.842 47.066\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(167.655 -12.945)\">\u003Cpath d=\"M11.150 48.871L9.516 48.871L9.516 48.591Q9.745 48.591 9.894 48.557Q10.043 48.522 10.043 48.382L10.043 46.533Q10.043 46.263 9.935 46.202Q9.827 46.140 9.516 46.140L9.516 45.860L10.576 45.785L10.576 46.434Q10.747 46.126 11.051 45.955Q11.355 45.785 11.700 45.785Q12.206 45.785 12.490 46.008Q12.774 46.232 12.774 46.728L12.774 48.382Q12.774 48.519 12.922 48.555Q13.071 48.591 13.297 48.591L13.297 48.871L11.666 48.871L11.666 48.591Q11.895 48.591 12.044 48.557Q12.193 48.522 12.193 48.382L12.193 46.742Q12.193 46.407 12.073 46.207Q11.953 46.007 11.639 46.007Q11.369 46.007 11.135 46.143Q10.901 46.280 10.762 46.514Q10.624 46.748 10.624 47.022L10.624 48.382Q10.624 48.519 10.774 48.555Q10.925 48.591 11.150 48.591L11.150 48.871M15.768 47.617L13.710 47.617L13.710 47.114L15.768 47.114L15.768 47.617M19.582 48.871L17.053 48.871L17.053 48.591Q18.020 48.591 18.020 48.382L18.020 44.763Q17.627 44.951 17.005 44.951L17.005 44.670Q17.422 44.670 17.786 44.569Q18.150 44.469 18.406 44.223L18.533 44.223Q18.598 44.240 18.615 44.308L18.615 48.382Q18.615 48.591 19.582 48.591\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">Backup diagram for Watkins's Q(lambda). Component backups of increasing length are weighted (1-lambda), (1-lambda)lambda, (1-lambda)lambda^2, ..., as in any lambda-return; the series ends either at episode termination (weight lambda^{T-t-1}, the box) or at the first non-greedy action, whichever comes first. After that first non-greedy branch the trace is cut.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:318.188px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 238.641 133.974\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg stroke=\"none\" font-family=\"cmr9\" font-size=\"9\">\u003Cg transform=\"translate(58.415 -116.748)\">\u003Cpath d=\"M-41.053 54.561L-44.489 54.561L-44.489 54.245Q-43.628 54.245-43.413 54.192Q-43.325 54.179-43.259 54.108Q-43.193 54.038-43.193 53.950L-43.193 49.024Q-43.193 48.831-43.298 48.780Q-43.404 48.729-43.654 48.729L-44.059 48.729Q-44.415 48.729-44.674 48.793Q-44.933 48.857-45.131 49.055Q-45.298 49.217-45.362 49.527Q-45.425 49.837-45.491 50.474L-45.777 50.474L-45.597 48.413L-39.954 48.413L-39.774 50.474L-40.064 50.474Q-40.086 50.193-40.123 49.907Q-40.161 49.622-40.231 49.397Q-40.301 49.173-40.424 49.055Q-40.736 48.729-41.492 48.729L-41.888 48.729Q-42.138 48.729-42.244 48.778Q-42.349 48.826-42.349 49.024L-42.349 53.950Q-42.349 54.038-42.283 54.108Q-42.217 54.179-42.134 54.192Q-41.919 54.245-41.053 54.245\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(58.415 -116.748)\">\u003Cpath d=\"M-37.718 54.561L-39.951 54.561L-39.951 54.245Q-39.639 54.245-39.447 54.192Q-39.256 54.139-39.256 53.950L-39.256 51.502Q-39.256 51.261-39.326 51.153Q-39.397 51.045-39.531 51.021Q-39.665 50.997-39.951 50.997L-39.951 50.681L-38.637 50.584L-38.637 51.445Q-38.474 51.054-38.206 50.819Q-37.938 50.584-37.547 50.584Q-37.274 50.584-37.059 50.747Q-36.844 50.909-36.844 51.168Q-36.844 51.344-36.962 51.463Q-37.081 51.582-37.257 51.582Q-37.437 51.582-37.555 51.463Q-37.674 51.344-37.674 51.168Q-37.674 50.953-37.520 50.843L-37.538 50.843Q-37.916 50.843-38.149 51.105Q-38.382 51.366-38.481 51.753Q-38.579 52.140-38.579 52.500L-38.579 53.950Q-38.579 54.139-38.322 54.192Q-38.065 54.245-37.718 54.245L-37.718 54.561M-34.273 54.662Q-34.831 54.662-35.303 54.379Q-35.776 54.095-36.050 53.618Q-36.325 53.142-36.325 52.588Q-36.325 52.192-36.182 51.817Q-36.039 51.441-35.782 51.153Q-35.525 50.865-35.167 50.696Q-34.809 50.527-34.405 50.527Q-33.860 50.527-33.488 50.764Q-33.117 51.001-32.930 51.419Q-32.743 51.836-32.743 52.373Q-32.743 52.425-32.768 52.463Q-32.792 52.500-32.840 52.500L-35.512 52.500L-35.512 52.579Q-35.512 53.326-35.200 53.849Q-34.888 54.372-34.189 54.372Q-33.785 54.372-33.464 54.115Q-33.143 53.858-33.020 53.454Q-33.003 53.374-32.919 53.374L-32.840 53.374Q-32.801 53.374-32.772 53.405Q-32.743 53.436-32.743 53.480L-32.743 53.515Q-32.849 53.858-33.071 54.117Q-33.293 54.376-33.607 54.519Q-33.921 54.662-34.273 54.662M-35.503 52.249L-33.389 52.249Q-33.389 51.981-33.442 51.735Q-33.495 51.489-33.616 51.267Q-33.737 51.045-33.934 50.918Q-34.132 50.790-34.405 50.790Q-34.747 50.790-35 51.015Q-35.253 51.239-35.378 51.577Q-35.503 51.915-35.503 52.249M-30.159 54.662Q-30.718 54.662-31.190 54.379Q-31.662 54.095-31.937 53.618Q-32.212 53.142-32.212 52.588Q-32.212 52.192-32.069 51.817Q-31.926 51.441-31.669 51.153Q-31.412 50.865-31.054 50.696Q-30.696 50.527-30.291 50.527Q-29.746 50.527-29.375 50.764Q-29.004 51.001-28.817 51.419Q-28.630 51.836-28.630 52.373Q-28.630 52.425-28.654 52.463Q-28.679 52.500-28.727 52.500L-31.399 52.500L-31.399 52.579Q-31.399 53.326-31.087 53.849Q-30.775 54.372-30.076 54.372Q-29.672 54.372-29.351 54.115Q-29.030 53.858-28.907 53.454Q-28.889 53.374-28.806 53.374L-28.727 53.374Q-28.687 53.374-28.659 53.405Q-28.630 53.436-28.630 53.480L-28.630 53.515Q-28.736 53.858-28.958 54.117Q-29.180 54.376-29.494 54.519Q-29.808 54.662-30.159 54.662M-31.390 52.249L-29.276 52.249Q-29.276 51.981-29.329 51.735Q-29.382 51.489-29.503 51.267Q-29.623 51.045-29.821 50.918Q-30.019 50.790-30.291 50.790Q-30.634 50.790-30.887 51.015Q-31.139 51.239-31.265 51.577Q-31.390 51.915-31.390 52.249\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(58.415 -116.748)\">\u003Cpath d=\"M-21.293 54.561L-24.918 54.561L-24.918 54.245Q-24 54.245-24 53.950L-24 49.024Q-24 48.729-24.918 48.729L-24.918 48.413L-21.552 48.413Q-21.086 48.413-20.607 48.591Q-20.128 48.769-19.814 49.114Q-19.500 49.459-19.500 49.942Q-19.500 50.492-19.961 50.863Q-20.423 51.234-21.020 51.349Q-20.726 51.349-20.410 51.472Q-20.093 51.595-19.830 51.808Q-19.566 52.021-19.403 52.307Q-19.241 52.592-19.241 52.904Q-19.241 53.405-19.548 53.785Q-19.856 54.165-20.333 54.363Q-20.810 54.561-21.293 54.561M-23.209 51.494L-23.209 53.950Q-23.209 54.144-23.103 54.194Q-22.998 54.245-22.752 54.245L-21.552 54.245Q-21.192 54.245-20.869 54.064Q-20.546 53.884-20.357 53.579Q-20.168 53.273-20.168 52.904Q-20.168 52.561-20.326 52.232Q-20.484 51.902-20.777 51.698Q-21.069 51.494-21.429 51.494L-23.209 51.494M-23.209 49.024L-23.209 51.230L-21.807 51.230Q-21.258 51.230-20.829 50.859Q-20.401 50.487-20.401 49.942Q-20.401 49.617-20.552 49.342Q-20.704 49.068-20.968 48.899Q-21.231 48.729-21.552 48.729L-22.752 48.729Q-22.998 48.729-23.103 48.778Q-23.209 48.826-23.209 49.024M-18.357 53.651Q-18.357 53.111-17.925 52.777Q-17.492 52.443-16.885 52.304Q-16.279 52.166-15.747 52.166L-15.747 51.832Q-15.747 51.573-15.866 51.329Q-15.984 51.085-16.193 50.938Q-16.402 50.790-16.674 50.790Q-17.237 50.790-17.549 50.988Q-17.399 51.015-17.307 51.133Q-17.215 51.252-17.215 51.410Q-17.215 51.586-17.340 51.716Q-17.465 51.845-17.645 51.845Q-17.834 51.845-17.962 51.718Q-18.089 51.590-18.089 51.410Q-18.089 50.940-17.650 50.733Q-17.210 50.527-16.674 50.527Q-16.384 50.527-16.096 50.615Q-15.809 50.703-15.576 50.863Q-15.343 51.023-15.193 51.263Q-15.044 51.502-15.044 51.797L-15.044 53.832Q-15.044 53.985-14.969 54.124Q-14.895 54.262-14.749 54.262Q-14.596 54.262-14.523 54.126Q-14.451 53.990-14.451 53.832L-14.451 53.256L-14.165 53.256L-14.165 53.832Q-14.165 54.165-14.413 54.390Q-14.662 54.614-14.991 54.614Q-15.250 54.614-15.433 54.420Q-15.615 54.227-15.659 53.950Q-15.826 54.271-16.160 54.467Q-16.494 54.662-16.863 54.662Q-17.417 54.662-17.887 54.414Q-18.357 54.165-18.357 53.651M-17.602 53.651Q-17.602 53.963-17.360 54.181Q-17.118 54.398-16.802 54.398Q-16.367 54.398-16.057 54.089Q-15.747 53.779-15.747 53.348L-15.747 52.421Q-16.165 52.421-16.591 52.548Q-17.017 52.676-17.309 52.953Q-17.602 53.229-17.602 53.651M-11.796 54.662Q-12.346 54.662-12.805 54.383Q-13.264 54.104-13.532 53.634Q-13.800 53.164-13.800 52.619Q-13.800 52.206-13.651 51.828Q-13.501 51.450-13.227 51.155Q-12.952 50.861-12.583 50.694Q-12.214 50.527-11.796 50.527Q-11.462 50.527-11.142 50.593Q-10.821 50.659-10.592 50.845Q-10.364 51.032-10.364 51.357Q-10.364 51.533-10.491 51.661Q-10.619 51.788-10.794 51.788Q-10.979 51.788-11.109 51.663Q-11.238 51.538-11.238 51.357Q-11.238 51.221-11.164 51.109Q-11.089 50.997-10.957 50.944Q-11.265 50.817-11.796 50.817Q-12.231 50.817-12.499 51.094Q-12.768 51.371-12.880 51.786Q-12.992 52.201-12.992 52.619Q-12.992 53.049-12.853 53.451Q-12.715 53.853-12.420 54.113Q-12.126 54.372-11.687 54.372Q-11.265 54.372-10.961 54.122Q-10.658 53.871-10.553 53.462Q-10.544 53.432-10.520 53.407Q-10.496 53.383-10.465 53.383L-10.355 53.383Q-10.267 53.383-10.267 53.498Q-10.412 54.034-10.825 54.348Q-11.238 54.662-11.796 54.662\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(58.415 -116.748)\">\u003Cpath d=\"M-7.955 54.561L-9.990 54.561L-9.990 54.245Q-9.678 54.245-9.486 54.192Q-9.295 54.139-9.295 53.950L-9.295 49.235Q-9.295 48.993-9.365 48.885Q-9.436 48.778-9.570 48.754Q-9.704 48.729-9.990 48.729L-9.990 48.413L-8.618 48.316L-8.618 52.561L-7.414 51.546Q-7.208 51.375-7.208 51.195Q-7.208 51.098-7.276 51.048Q-7.344 50.997-7.441 50.997L-7.441 50.681L-5.731 50.681L-5.731 50.997Q-6.329 50.997-6.975 51.546L-7.621 52.096L-6.461 53.695Q-6.285 53.933-6.182 54.038Q-6.078 54.144-5.927 54.194Q-5.775 54.245-5.525 54.245L-5.525 54.561L-7.353 54.561L-7.353 54.245Q-7.054 54.245-7.054 54.056Q-7.054 53.937-7.225 53.695L-8.100 52.500L-8.649 52.970L-8.649 53.950Q-8.649 54.139-8.456 54.192Q-8.262 54.245-7.955 54.245L-7.955 54.561M-4.373 53.489L-4.373 51.502Q-4.373 51.261-4.444 51.153Q-4.514 51.045-4.648 51.021Q-4.782 50.997-5.063 50.997L-5.063 50.681L-3.670 50.584L-3.670 53.454Q-3.670 53.832-3.624 54.018Q-3.578 54.205-3.407 54.302Q-3.235 54.398-2.879 54.398Q-2.545 54.398-2.306 54.207Q-2.066 54.016-1.941 53.713Q-1.816 53.410-1.816 53.093L-1.816 51.502Q-1.816 51.261-1.886 51.153Q-1.956 51.045-2.090 51.021Q-2.224 50.997-2.510 50.997L-2.510 50.681L-1.113 50.584L-1.113 53.744Q-1.113 53.981-1.042 54.089Q-0.972 54.196-0.838 54.220Q-0.704 54.245-0.423 54.245L-0.423 54.561L-1.789 54.662L-1.789 53.941Q-1.956 54.267-2.262 54.464Q-2.567 54.662-2.923 54.662Q-3.582 54.662-3.978 54.392Q-4.373 54.122-4.373 53.489M2.113 56.306L0.026 56.306L0.026 55.994Q0.338 55.994 0.529 55.943Q0.720 55.893 0.720 55.695L0.720 51.357Q0.720 51.116 0.546 51.056Q0.373 50.997 0.026 50.997L0.026 50.681L1.397 50.584L1.397 51.124Q1.656 50.856 1.990 50.720Q2.324 50.584 2.693 50.584Q3.093 50.584 3.449 50.751Q3.805 50.918 4.060 51.199Q4.315 51.480 4.458 51.845Q4.600 52.210 4.600 52.619Q4.600 53.181 4.323 53.647Q4.047 54.113 3.572 54.387Q3.097 54.662 2.548 54.662Q2.236 54.662 1.946 54.528Q1.656 54.394 1.423 54.148L1.423 55.695Q1.423 55.893 1.614 55.943Q1.805 55.994 2.113 55.994L2.113 56.306M1.423 51.538L1.423 53.669Q1.581 53.994 1.865 54.196Q2.148 54.398 2.491 54.398Q2.904 54.398 3.198 54.122Q3.493 53.845 3.640 53.427Q3.787 53.010 3.787 52.619Q3.787 52.241 3.653 51.832Q3.519 51.423 3.245 51.146Q2.970 50.870 2.592 50.870Q2.350 50.870 2.135 50.946Q1.920 51.023 1.729 51.184Q1.537 51.344 1.423 51.538M7.764 56.802Q7.259 56.415 6.890 55.910Q6.521 55.405 6.277 54.805Q6.033 54.205 5.919 53.588Q5.804 52.970 5.804 52.311Q5.804 51.652 5.919 51.037Q6.033 50.421 6.272 49.828Q6.512 49.235 6.885 48.725Q7.259 48.215 7.764 47.829Q7.800 47.811 7.822 47.811L7.901 47.811Q7.989 47.811 7.989 47.912Q7.989 47.947 7.953 47.982Q7.391 48.505 7.046 49.211Q6.701 49.916 6.554 50.698Q6.406 51.480 6.406 52.311Q6.406 52.935 6.486 53.519Q6.565 54.104 6.743 54.671Q6.921 55.238 7.219 55.739Q7.518 56.240 7.953 56.648Q7.989 56.684 7.989 56.723Q7.989 56.820 7.901 56.820L7.822 56.820Q7.800 56.820 7.764 56.802M10.871 54.561L8.810 54.561L8.810 54.245Q9.118 54.245 9.309 54.192Q9.500 54.139 9.500 53.950L9.500 49.235Q9.500 48.993 9.430 48.885Q9.360 48.778 9.226 48.754Q9.092 48.729 8.810 48.729L8.810 48.413L10.177 48.316L10.177 53.950Q10.177 54.139 10.370 54.192Q10.564 54.245 10.871 54.245L10.871 54.561M11.438 53.651Q11.438 53.111 11.871 52.777Q12.304 52.443 12.910 52.304Q13.517 52.166 14.049 52.166L14.049 51.832Q14.049 51.573 13.930 51.329Q13.811 51.085 13.603 50.938Q13.394 50.790 13.121 50.790Q12.559 50.790 12.247 50.988Q12.396 51.015 12.489 51.133Q12.581 51.252 12.581 51.410Q12.581 51.586 12.456 51.716Q12.330 51.845 12.150 51.845Q11.961 51.845 11.834 51.718Q11.706 51.590 11.706 51.410Q11.706 50.940 12.146 50.733Q12.585 50.527 13.121 50.527Q13.411 50.527 13.699 50.615Q13.987 50.703 14.220 50.863Q14.453 51.023 14.602 51.263Q14.752 51.502 14.752 51.797L14.752 53.832Q14.752 53.985 14.826 54.124Q14.901 54.262 15.046 54.262Q15.200 54.262 15.272 54.126Q15.345 53.990 15.345 53.832L15.345 53.256L15.631 53.256L15.631 53.832Q15.631 54.165 15.382 54.390Q15.134 54.614 14.804 54.614Q14.545 54.614 14.363 54.420Q14.180 54.227 14.136 53.950Q13.969 54.271 13.635 54.467Q13.302 54.662 12.932 54.662Q12.379 54.662 11.908 54.414Q11.438 54.165 11.438 53.651M12.194 53.651Q12.194 53.963 12.436 54.181Q12.677 54.398 12.994 54.398Q13.429 54.398 13.739 54.089Q14.049 53.779 14.049 53.348L14.049 52.421Q13.631 52.421 13.205 52.548Q12.779 52.676 12.486 52.953Q12.194 53.229 12.194 53.651M18.074 54.561L15.987 54.561L15.987 54.245Q16.294 54.245 16.485 54.192Q16.677 54.139 16.677 53.950L16.677 51.502Q16.677 51.261 16.606 51.153Q16.536 51.045 16.402 51.021Q16.268 50.997 15.987 50.997L15.987 50.681L17.327 50.584L17.327 51.419Q17.525 51.041 17.885 50.812Q18.245 50.584 18.667 50.584Q19.713 50.584 19.898 51.393Q20.100 51.023 20.458 50.804Q20.816 50.584 21.234 50.584Q21.858 50.584 22.183 50.878Q22.508 51.173 22.508 51.797L22.508 53.950Q22.508 54.139 22.701 54.192Q22.895 54.245 23.202 54.245L23.202 54.561L21.115 54.561L21.115 54.245Q21.423 54.245 21.616 54.192Q21.809 54.139 21.809 53.950L21.809 51.832Q21.809 51.401 21.682 51.122Q21.554 50.843 21.168 50.843Q20.825 50.843 20.541 51.032Q20.258 51.221 20.102 51.533Q19.946 51.845 19.946 52.192L19.946 53.950Q19.946 54.139 20.137 54.192Q20.328 54.245 20.636 54.245L20.636 54.561L18.549 54.561L18.549 54.245Q18.861 54.245 19.052 54.192Q19.243 54.139 19.243 53.950L19.243 51.832Q19.243 51.573 19.199 51.351Q19.155 51.129 19.010 50.986Q18.865 50.843 18.606 50.843Q18.070 50.843 17.725 51.250Q17.380 51.656 17.380 52.192L17.380 53.950Q17.380 54.139 17.573 54.192Q17.766 54.245 18.074 54.245\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(58.415 -116.748)\">\u003Cpath d=\"M24.398 54.561L24.108 54.561L24.108 49.235Q24.108 48.993 24.038 48.885Q23.967 48.778 23.833 48.754Q23.699 48.729 23.413 48.729L23.413 48.413L24.785 48.316L24.785 51.116Q25.031 50.865 25.365 50.725Q25.699 50.584 26.050 50.584Q26.450 50.584 26.813 50.747Q27.175 50.909 27.432 51.188Q27.689 51.467 27.839 51.845Q27.988 52.223 27.988 52.619Q27.988 53.181 27.711 53.647Q27.434 54.113 26.960 54.387Q26.485 54.662 25.936 54.662Q25.571 54.662 25.250 54.493Q24.930 54.324 24.710 54.029L24.398 54.561M24.811 51.529L24.811 53.660Q24.969 53.994 25.253 54.196Q25.536 54.398 25.879 54.398Q26.569 54.398 26.872 53.878Q27.175 53.357 27.175 52.619Q27.175 51.894 26.909 51.368Q26.643 50.843 25.980 50.843Q25.620 50.843 25.305 51.028Q24.991 51.212 24.811 51.529\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(58.415 -116.748)\">\u003Cpath d=\"M30.808 54.662Q30.263 54.662 29.819 54.379Q29.375 54.095 29.120 53.623Q28.866 53.150 28.866 52.619Q28.866 52.065 29.142 51.599Q29.419 51.133 29.892 50.859Q30.364 50.584 30.909 50.584Q31.243 50.584 31.546 50.716Q31.850 50.848 32.069 51.085L32.069 49.235Q32.069 48.993 31.999 48.885Q31.929 48.778 31.795 48.754Q31.661 48.729 31.375 48.729L31.375 48.413L32.742 48.316L32.742 53.744Q32.742 53.981 32.812 54.089Q32.882 54.196 33.018 54.220Q33.155 54.245 33.436 54.245L33.436 54.561L32.043 54.662L32.043 54.113Q31.792 54.376 31.474 54.519Q31.155 54.662 30.808 54.662M30.870 54.398Q31.239 54.398 31.548 54.187Q31.858 53.977 32.043 53.634L32.043 51.502Q31.871 51.199 31.590 51.021Q31.309 50.843 30.971 50.843Q30.281 50.843 29.977 51.364Q29.674 51.885 29.674 52.627Q29.674 53.348 29.944 53.873Q30.215 54.398 30.870 54.398M34.064 53.651Q34.064 53.111 34.497 52.777Q34.930 52.443 35.537 52.304Q36.143 52.166 36.675 52.166L36.675 51.832Q36.675 51.573 36.556 51.329Q36.437 51.085 36.229 50.938Q36.020 50.790 35.747 50.790Q35.185 50.790 34.873 50.988Q35.022 51.015 35.115 51.133Q35.207 51.252 35.207 51.410Q35.207 51.586 35.082 51.716Q34.956 51.845 34.776 51.845Q34.587 51.845 34.460 51.718Q34.332 51.590 34.332 51.410Q34.332 50.940 34.772 50.733Q35.211 50.527 35.747 50.527Q36.037 50.527 36.325 50.615Q36.613 50.703 36.846 50.863Q37.079 51.023 37.228 51.263Q37.378 51.502 37.378 51.797L37.378 53.832Q37.378 53.985 37.453 54.124Q37.527 54.262 37.672 54.262Q37.826 54.262 37.899 54.126Q37.971 53.990 37.971 53.832L37.971 53.256L38.257 53.256L38.257 53.832Q38.257 54.165 38.008 54.390Q37.760 54.614 37.431 54.614Q37.171 54.614 36.989 54.420Q36.807 54.227 36.763 53.950Q36.596 54.271 36.262 54.467Q35.928 54.662 35.558 54.662Q35.005 54.662 34.535 54.414Q34.064 54.165 34.064 53.651M34.820 53.651Q34.820 53.963 35.062 54.181Q35.304 54.398 35.620 54.398Q36.055 54.398 36.365 54.089Q36.675 53.779 36.675 53.348L36.675 52.421Q36.257 52.421 35.831 52.548Q35.405 52.676 35.112 52.953Q34.820 53.229 34.820 53.651M38.999 56.820L38.916 56.820Q38.828 56.820 38.828 56.723Q38.828 56.684 38.863 56.648Q39.698 55.875 40.058 54.741Q40.419 53.607 40.419 52.311Q40.419 51.691 40.340 51.100Q40.261 50.509 40.080 49.951Q39.900 49.393 39.599 48.885Q39.298 48.378 38.863 47.982Q38.828 47.947 38.828 47.912Q38.828 47.811 38.916 47.811L38.999 47.811Q39.017 47.811 39.052 47.829Q39.558 48.211 39.931 48.725Q40.305 49.239 40.542 49.817Q40.779 50.395 40.896 51.023Q41.012 51.652 41.012 52.311Q41.012 52.970 40.896 53.601Q40.779 54.231 40.540 54.816Q40.300 55.400 39.929 55.910Q39.558 56.420 39.052 56.802Q39.017 56.820 38.999 56.820\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.270\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath d=\"M-43.834-42.178a2.276 2.276 0 1 0-4.553 0 2.276 2.276 0 0 0 4.553 0Zm-2.277 0\"\u002F>\u003Cpath fill=\"none\" d=\"M-41.558-25.106a4.552 4.552 0 1 0-9.105 0 4.552 4.552 0 0 0 9.105 0ZM-46.11-39.701v9.842\"\u002F>\u003Cpath d=\"M-54.362-12.303a1.707 1.707 0 1 0-3.414 0 1.707 1.707 0 0 0 3.414 0ZM-34.445-12.303a1.707 1.707 0 1 0-3.414 0 1.707 1.707 0 0 0 3.414 0Zm-1.707 0\"\u002F>\u003Cpath fill=\"none\" d=\"m-49.028-21.355-7.041 9.052M-43.193-21.355l7.041 9.052\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-16.16 -47.089)\">\u003Cpath d=\"M-42.785 54.561L-45.315 54.561L-45.315 54.281Q-44.347 54.281-44.347 54.072L-44.347 50.453Q-44.740 50.641-45.362 50.641L-45.362 50.360Q-44.945 50.360-44.581 50.259Q-44.217 50.159-43.961 49.913L-43.835 49.913Q-43.770 49.930-43.753 49.998L-43.753 54.072Q-43.753 54.281-42.785 54.281L-42.785 54.561M-39.931 53.307L-41.989 53.307L-41.989 52.804L-39.931 52.804L-39.931 53.307M-37.460 54.561L-39.063 54.561L-39.063 54.281Q-38.838 54.281-38.689 54.247Q-38.540 54.212-38.540 54.072L-38.540 50.453Q-38.540 50.183-38.648 50.121Q-38.756 50.060-39.063 50.060L-39.063 49.779L-37.986 49.704L-37.986 54.072Q-37.986 54.209-37.836 54.245Q-37.686 54.281-37.460 54.281L-37.460 54.561M-36.807 53.833Q-36.807 53.501-36.583 53.274Q-36.360 53.047-36.016 52.919Q-35.673 52.790-35.300 52.738Q-34.927 52.685-34.623 52.685L-34.623 52.432Q-34.623 52.227-34.731 52.047Q-34.839 51.868-35.020 51.765Q-35.201 51.663-35.409 51.663Q-35.816 51.663-36.052 51.755Q-35.963 51.792-35.917 51.876Q-35.871 51.960-35.871 52.062Q-35.871 52.158-35.917 52.237Q-35.963 52.315-36.043 52.360Q-36.124 52.404-36.213 52.404Q-36.363 52.404-36.464 52.307Q-36.565 52.209-36.565 52.062Q-36.565 51.440-35.409 51.440Q-35.197 51.440-34.948 51.504Q-34.698 51.567-34.497 51.686Q-34.295 51.806-34.169 51.991Q-34.042 52.175-34.042 52.418L-34.042 53.994Q-34.042 54.110-33.981 54.206Q-33.919 54.301-33.806 54.301Q-33.697 54.301-33.632 54.207Q-33.567 54.113-33.567 53.994L-33.567 53.546L-33.300 53.546L-33.300 53.994Q-33.300 54.264-33.528 54.429Q-33.755 54.595-34.035 54.595Q-34.244 54.595-34.381 54.441Q-34.517 54.288-34.541 54.072Q-34.688 54.339-34.970 54.484Q-35.252 54.629-35.577 54.629Q-35.854 54.629-36.137 54.554Q-36.421 54.479-36.614 54.300Q-36.807 54.120-36.807 53.833M-36.192 53.833Q-36.192 54.007-36.091 54.137Q-35.990 54.267-35.835 54.337Q-35.679 54.407-35.515 54.407Q-35.297 54.407-35.088 54.310Q-34.880 54.212-34.751 54.031Q-34.623 53.850-34.623 53.624L-34.623 52.896Q-34.948 52.896-35.314 52.987Q-35.679 53.078-35.936 53.290Q-36.192 53.501-36.192 53.833M-31.202 54.561L-32.836 54.561L-32.836 54.281Q-32.607 54.281-32.458 54.247Q-32.309 54.212-32.309 54.072L-32.309 52.223Q-32.309 51.953-32.417 51.892Q-32.525 51.830-32.836 51.830L-32.836 51.550L-31.776 51.475L-31.776 52.124Q-31.605 51.816-31.301 51.645Q-30.997 51.475-30.652 51.475Q-30.252 51.475-29.975 51.615Q-29.698 51.755-29.612 52.103Q-29.445 51.810-29.146 51.642Q-28.847 51.475-28.502 51.475Q-27.996 51.475-27.712 51.698Q-27.428 51.922-27.428 52.418L-27.428 54.072Q-27.428 54.209-27.280 54.245Q-27.131 54.281-26.905 54.281L-26.905 54.561L-28.536 54.561L-28.536 54.281Q-28.310 54.281-28.160 54.245Q-28.009 54.209-28.009 54.072L-28.009 52.432Q-28.009 52.097-28.129 51.897Q-28.249 51.697-28.563 51.697Q-28.833 51.697-29.067 51.833Q-29.301 51.970-29.440 52.204Q-29.578 52.438-29.578 52.712L-29.578 54.072Q-29.578 54.209-29.430 54.245Q-29.281 54.281-29.055 54.281L-29.055 54.561L-30.686 54.561L-30.686 54.281Q-30.457 54.281-30.308 54.247Q-30.159 54.212-30.159 54.072L-30.159 52.432Q-30.159 52.097-30.279 51.897Q-30.399 51.697-30.713 51.697Q-30.983 51.697-31.217 51.833Q-31.451 51.970-31.590 52.204Q-31.728 52.438-31.728 52.712L-31.728 54.072Q-31.728 54.209-31.578 54.245Q-31.427 54.281-31.202 54.281\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-16.16 -47.089)\">\u003Cpath d=\"M-25.705 54.561L-25.972 54.561L-25.972 50.453Q-25.972 50.183-26.079 50.121Q-26.187 50.060-26.498 50.060L-26.498 49.779L-25.418 49.704L-25.418 51.874Q-25.209 51.683-24.924 51.579Q-24.639 51.475-24.341 51.475Q-24.023 51.475-23.726 51.596Q-23.429 51.717-23.206 51.933Q-22.984 52.148-22.858 52.433Q-22.731 52.719-22.731 53.050Q-22.731 53.495-22.971 53.859Q-23.210 54.223-23.603 54.426Q-23.996 54.629-24.440 54.629Q-24.635 54.629-24.825 54.573Q-25.014 54.517-25.175 54.412Q-25.336 54.308-25.476 54.147L-25.705 54.561M-25.390 52.216L-25.390 53.833Q-25.254 54.093-25.013 54.250Q-24.772 54.407-24.495 54.407Q-24.201 54.407-23.989 54.300Q-23.777 54.192-23.644 54Q-23.511 53.809-23.452 53.570Q-23.394 53.331-23.394 53.050Q-23.394 52.691-23.488 52.387Q-23.582 52.083-23.810 51.890Q-24.037 51.697-24.403 51.697Q-24.703 51.697-24.970 51.833Q-25.237 51.970-25.390 52.216\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-16.16 -47.089)\">\u003Cpath d=\"M-21.880 53.050Q-21.880 52.712-21.739 52.421Q-21.599 52.131-21.355 51.917Q-21.111 51.704-20.806 51.589Q-20.502 51.475-20.177 51.475Q-19.907 51.475-19.644 51.574Q-19.381 51.673-19.190 51.851L-19.190 50.453Q-19.190 50.183-19.297 50.121Q-19.405 50.060-19.716 50.060L-19.716 49.779L-18.639 49.704L-18.639 53.888Q-18.639 54.076-18.585 54.159Q-18.530 54.243-18.429 54.262Q-18.328 54.281-18.113 54.281L-18.113 54.561L-19.220 54.629L-19.220 54.212Q-19.637 54.629-20.263 54.629Q-20.694 54.629-21.066 54.417Q-21.439 54.206-21.659 53.845Q-21.880 53.484-21.880 53.050M-20.205 54.407Q-19.996 54.407-19.810 54.335Q-19.624 54.264-19.470 54.127Q-19.316 53.990-19.220 53.812L-19.220 52.203Q-19.306 52.056-19.451 51.936Q-19.596 51.816-19.766 51.757Q-19.935 51.697-20.116 51.697Q-20.676 51.697-20.945 52.086Q-21.213 52.476-21.213 53.057Q-21.213 53.628-20.979 54.018Q-20.745 54.407-20.205 54.407M-17.405 53.833Q-17.405 53.501-17.182 53.274Q-16.958 53.047-16.614 52.919Q-16.271 52.790-15.898 52.738Q-15.526 52.685-15.221 52.685L-15.221 52.432Q-15.221 52.227-15.329 52.047Q-15.437 51.868-15.618 51.765Q-15.799 51.663-16.007 51.663Q-16.414 51.663-16.650 51.755Q-16.561 51.792-16.515 51.876Q-16.469 51.960-16.469 52.062Q-16.469 52.158-16.515 52.237Q-16.561 52.315-16.642 52.360Q-16.722 52.404-16.811 52.404Q-16.961 52.404-17.062 52.307Q-17.163 52.209-17.163 52.062Q-17.163 51.440-16.007 51.440Q-15.796 51.440-15.546 51.504Q-15.297 51.567-15.095 51.686Q-14.893 51.806-14.767 51.991Q-14.640 52.175-14.640 52.418L-14.640 53.994Q-14.640 54.110-14.579 54.206Q-14.517 54.301-14.404 54.301Q-14.295 54.301-14.230 54.207Q-14.165 54.113-14.165 53.994L-14.165 53.546L-13.899 53.546L-13.899 53.994Q-13.899 54.264-14.126 54.429Q-14.353 54.595-14.633 54.595Q-14.842 54.595-14.979 54.441Q-15.115 54.288-15.139 54.072Q-15.286 54.339-15.568 54.484Q-15.850 54.629-16.175 54.629Q-16.452 54.629-16.736 54.554Q-17.019 54.479-17.212 54.300Q-17.405 54.120-17.405 53.833M-16.790 53.833Q-16.790 54.007-16.689 54.137Q-16.589 54.267-16.433 54.337Q-16.277 54.407-16.113 54.407Q-15.895 54.407-15.686 54.310Q-15.478 54.212-15.350 54.031Q-15.221 53.850-15.221 53.624L-15.221 52.896Q-15.546 52.896-15.912 52.987Q-16.277 53.078-16.534 53.290Q-16.790 53.501-16.790 53.833\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath d=\"M-4-42.178a2.276 2.276 0 1 0-4.553 0 2.276 2.276 0 0 0 4.552 0Zm-2.277 0\"\u002F>\u003Cpath fill=\"none\" d=\"M-1.724-25.106a4.552 4.552 0 1 0-9.105 0 4.552 4.552 0 0 0 9.105 0Zm-4.553 0\"\u002F>\u003Cpath d=\"M-14.528-13.725a1.707 1.707 0 1 0-3.414 0 1.707 1.707 0 0 0 3.414 0ZM5.389-13.725a1.707 1.707 0 1 0-3.414 0 1.707 1.707 0 0 0 3.414 0Zm-1.707 0\"\u002F>\u003Cpath fill=\"none\" d=\"m-9.406-21.53-6.83 7.805M-3.148-21.53l6.83 7.805\"\u002F>\u003Cpath d=\"M-4-10.88a2.276 2.276 0 1 0-4.553 0 2.276 2.276 0 0 0 4.552 0Zm-2.277 0\"\u002F>\u003Cpath fill=\"none\" d=\"M-1.724 6.192a4.552 4.552 0 1 0-9.105 0 4.552 4.552 0 0 0 9.105 0Zm-4.553 0\"\u002F>\u003Cpath d=\"M-14.528 17.573a1.707 1.707 0 1 0-3.414 0 1.707 1.707 0 0 0 3.414 0ZM5.389 17.573a1.707 1.707 0 1 0-3.414 0 1.707 1.707 0 0 0 3.414 0Zm-1.707 0\"\u002F>\u003Cpath fill=\"none\" d=\"m-9.406 9.768-6.83 7.805M-3.148 9.768l6.83 7.805M-6.277-39.701v9.842m0 9.505v6.998m0 4.952V1.44\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(7.73 -18.247)\">\u003Cpath d=\"M-43.667 56.311Q-44.217 55.911-44.588 55.356Q-44.959 54.800-45.140 54.154Q-45.321 53.508-45.321 52.811Q-45.321 52.298-45.221 51.803Q-45.120 51.307-44.915 50.856Q-44.710 50.405-44.397 50.013Q-44.084 49.622-43.667 49.318Q-43.657 49.314-43.650 49.313Q-43.643 49.311-43.633 49.311L-43.565 49.311Q-43.530 49.311-43.508 49.335Q-43.486 49.359-43.486 49.396Q-43.486 49.441-43.513 49.458Q-43.862 49.759-44.115 50.143Q-44.368 50.528-44.520 50.969Q-44.672 51.410-44.744 51.866Q-44.816 52.322-44.816 52.811Q-44.816 53.812-44.506 54.699Q-44.197 55.586-43.513 56.171Q-43.486 56.188-43.486 56.232Q-43.486 56.270-43.508 56.294Q-43.530 56.318-43.565 56.318L-43.633 56.318Q-43.640 56.314-43.648 56.313Q-43.657 56.311-43.667 56.311M-39.665 54.561L-42.194 54.561L-42.194 54.281Q-41.227 54.281-41.227 54.072L-41.227 50.453Q-41.620 50.641-42.242 50.641L-42.242 50.360Q-41.825 50.360-41.461 50.259Q-41.097 50.159-40.840 49.913L-40.714 49.913Q-40.649 49.930-40.632 49.998L-40.632 54.072Q-40.632 54.281-39.665 54.281L-39.665 54.561M-36.811 53.307L-38.868 53.307L-38.868 52.804L-36.811 52.804L-36.811 53.307M-34.340 54.561L-35.943 54.561L-35.943 54.281Q-35.717 54.281-35.568 54.247Q-35.420 54.212-35.420 54.072L-35.420 50.453Q-35.420 50.183-35.527 50.121Q-35.635 50.060-35.943 50.060L-35.943 49.779L-34.866 49.704L-34.866 54.072Q-34.866 54.209-34.715 54.245Q-34.565 54.281-34.340 54.281L-34.340 54.561M-33.687 53.833Q-33.687 53.501-33.463 53.274Q-33.239 53.047-32.895 52.919Q-32.552 52.790-32.179 52.738Q-31.807 52.685-31.503 52.685L-31.503 52.432Q-31.503 52.227-31.610 52.047Q-31.718 51.868-31.899 51.765Q-32.080 51.663-32.289 51.663Q-32.695 51.663-32.931 51.755Q-32.842 51.792-32.796 51.876Q-32.750 51.960-32.750 52.062Q-32.750 52.158-32.796 52.237Q-32.842 52.315-32.923 52.360Q-33.003 52.404-33.092 52.404Q-33.242 52.404-33.343 52.307Q-33.444 52.209-33.444 52.062Q-33.444 51.440-32.289 51.440Q-32.077 51.440-31.827 51.504Q-31.578 51.567-31.376 51.686Q-31.174 51.806-31.048 51.991Q-30.922 52.175-30.922 52.418L-30.922 53.994Q-30.922 54.110-30.860 54.206Q-30.798 54.301-30.686 54.301Q-30.576 54.301-30.511 54.207Q-30.446 54.113-30.446 53.994L-30.446 53.546L-30.180 53.546L-30.180 53.994Q-30.180 54.264-30.407 54.429Q-30.634 54.595-30.915 54.595Q-31.123 54.595-31.260 54.441Q-31.397 54.288-31.421 54.072Q-31.568 54.339-31.850 54.484Q-32.132 54.629-32.456 54.629Q-32.733 54.629-33.017 54.554Q-33.300 54.479-33.494 54.300Q-33.687 54.120-33.687 53.833M-33.071 53.833Q-33.071 54.007-32.971 54.137Q-32.870 54.267-32.714 54.337Q-32.559 54.407-32.395 54.407Q-32.176 54.407-31.967 54.310Q-31.759 54.212-31.631 54.031Q-31.503 53.850-31.503 53.624L-31.503 52.896Q-31.827 52.896-32.193 52.987Q-32.559 53.078-32.815 53.290Q-33.071 53.501-33.071 53.833M-28.081 54.561L-29.715 54.561L-29.715 54.281Q-29.486 54.281-29.337 54.247Q-29.189 54.212-29.189 54.072L-29.189 52.223Q-29.189 51.953-29.296 51.892Q-29.404 51.830-29.715 51.830L-29.715 51.550L-28.655 51.475L-28.655 52.124Q-28.485 51.816-28.180 51.645Q-27.876 51.475-27.531 51.475Q-27.131 51.475-26.854 51.615Q-26.577 51.755-26.492 52.103Q-26.324 51.810-26.025 51.642Q-25.726 51.475-25.381 51.475Q-24.875 51.475-24.591 51.698Q-24.308 51.922-24.308 52.418L-24.308 54.072Q-24.308 54.209-24.159 54.245Q-24.010 54.281-23.785 54.281L-23.785 54.561L-25.415 54.561L-25.415 54.281Q-25.190 54.281-25.039 54.245Q-24.889 54.209-24.889 54.072L-24.889 52.432Q-24.889 52.097-25.008 51.897Q-25.128 51.697-25.443 51.697Q-25.713 51.697-25.947 51.833Q-26.181 51.970-26.319 52.204Q-26.458 52.438-26.458 52.712L-26.458 54.072Q-26.458 54.209-26.309 54.245Q-26.160 54.281-25.935 54.281L-25.935 54.561L-27.565 54.561L-27.565 54.281Q-27.336 54.281-27.187 54.247Q-27.039 54.212-27.039 54.072L-27.039 52.432Q-27.039 52.097-27.158 51.897Q-27.278 51.697-27.592 51.697Q-27.862 51.697-28.097 51.833Q-28.331 51.970-28.469 52.204Q-28.608 52.438-28.608 52.712L-28.608 54.072Q-28.608 54.209-28.457 54.245Q-28.307 54.281-28.081 54.281\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(7.73 -18.247)\">\u003Cpath d=\"M-22.580 54.561L-22.847 54.561L-22.847 50.453Q-22.847 50.183-22.954 50.121Q-23.062 50.060-23.373 50.060L-23.373 49.779L-22.293 49.704L-22.293 51.874Q-22.084 51.683-21.799 51.579Q-21.514 51.475-21.216 51.475Q-20.898 51.475-20.601 51.596Q-20.304 51.717-20.081 51.933Q-19.859 52.148-19.733 52.433Q-19.606 52.719-19.606 53.050Q-19.606 53.495-19.846 53.859Q-20.085 54.223-20.478 54.426Q-20.871 54.629-21.315 54.629Q-21.510 54.629-21.700 54.573Q-21.889 54.517-22.050 54.412Q-22.211 54.308-22.351 54.147L-22.580 54.561M-22.265 52.216L-22.265 53.833Q-22.129 54.093-21.888 54.250Q-21.647 54.407-21.370 54.407Q-21.076 54.407-20.864 54.300Q-20.652 54.192-20.519 54Q-20.386 53.809-20.327 53.570Q-20.269 53.331-20.269 53.050Q-20.269 52.691-20.363 52.387Q-20.457 52.083-20.685 51.890Q-20.912 51.697-21.278 51.697Q-21.578 51.697-21.845 51.833Q-22.112 51.970-22.265 52.216\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(7.73 -18.247)\">\u003Cpath d=\"M-18.755 53.050Q-18.755 52.712-18.614 52.421Q-18.474 52.131-18.230 51.917Q-17.986 51.704-17.681 51.589Q-17.377 51.475-17.052 51.475Q-16.782 51.475-16.519 51.574Q-16.256 51.673-16.065 51.851L-16.065 50.453Q-16.065 50.183-16.172 50.121Q-16.280 50.060-16.591 50.060L-16.591 49.779L-15.514 49.704L-15.514 53.888Q-15.514 54.076-15.460 54.159Q-15.405 54.243-15.304 54.262Q-15.203 54.281-14.988 54.281L-14.988 54.561L-16.095 54.629L-16.095 54.212Q-16.512 54.629-17.138 54.629Q-17.569 54.629-17.941 54.417Q-18.314 54.206-18.534 53.845Q-18.755 53.484-18.755 53.050M-17.080 54.407Q-16.871 54.407-16.685 54.335Q-16.499 54.264-16.345 54.127Q-16.191 53.990-16.095 53.812L-16.095 52.203Q-16.181 52.056-16.326 51.936Q-16.471 51.816-16.641 51.757Q-16.810 51.697-16.991 51.697Q-17.551 51.697-17.820 52.086Q-18.088 52.476-18.088 53.057Q-18.088 53.628-17.854 54.018Q-17.620 54.407-17.080 54.407M-14.280 53.833Q-14.280 53.501-14.057 53.274Q-13.833 53.047-13.489 52.919Q-13.146 52.790-12.773 52.738Q-12.401 52.685-12.096 52.685L-12.096 52.432Q-12.096 52.227-12.204 52.047Q-12.312 51.868-12.493 51.765Q-12.674 51.663-12.882 51.663Q-13.289 51.663-13.525 51.755Q-13.436 51.792-13.390 51.876Q-13.344 51.960-13.344 52.062Q-13.344 52.158-13.390 52.237Q-13.436 52.315-13.517 52.360Q-13.597 52.404-13.686 52.404Q-13.836 52.404-13.937 52.307Q-14.038 52.209-14.038 52.062Q-14.038 51.440-12.882 51.440Q-12.671 51.440-12.421 51.504Q-12.172 51.567-11.970 51.686Q-11.768 51.806-11.642 51.991Q-11.515 52.175-11.515 52.418L-11.515 53.994Q-11.515 54.110-11.454 54.206Q-11.392 54.301-11.279 54.301Q-11.170 54.301-11.105 54.207Q-11.040 54.113-11.040 53.994L-11.040 53.546L-10.774 53.546L-10.774 53.994Q-10.774 54.264-11.001 54.429Q-11.228 54.595-11.508 54.595Q-11.717 54.595-11.854 54.441Q-11.990 54.288-12.014 54.072Q-12.161 54.339-12.443 54.484Q-12.725 54.629-13.050 54.629Q-13.327 54.629-13.611 54.554Q-13.894 54.479-14.087 54.300Q-14.280 54.120-14.280 53.833M-13.665 53.833Q-13.665 54.007-13.564 54.137Q-13.464 54.267-13.308 54.337Q-13.152 54.407-12.988 54.407Q-12.770 54.407-12.561 54.310Q-12.353 54.212-12.225 54.031Q-12.096 53.850-12.096 53.624L-12.096 52.896Q-12.421 52.896-12.787 52.987Q-13.152 53.078-13.409 53.290Q-13.665 53.501-13.665 53.833M-10.035 56.318L-10.104 56.318Q-10.138 56.318-10.160 56.292Q-10.182 56.267-10.182 56.232Q-10.182 56.188-10.152 56.171Q-9.796 55.867-9.547 55.477Q-9.297 55.087-9.145 54.655Q-8.993 54.223-8.923 53.754Q-8.853 53.286-8.853 52.811Q-8.853 52.332-8.923 51.866Q-8.993 51.399-9.147 50.964Q-9.300 50.528-9.552 50.140Q-9.803 49.752-10.152 49.458Q-10.182 49.441-10.182 49.396Q-10.182 49.362-10.160 49.337Q-10.138 49.311-10.104 49.311L-10.035 49.311Q-10.025 49.311-10.017 49.313Q-10.008 49.314-9.998 49.318Q-9.454 49.718-9.082 50.271Q-8.709 50.825-8.528 51.471Q-8.347 52.117-8.347 52.811Q-8.347 53.512-8.528 54.159Q-8.709 54.807-9.083 55.361Q-9.458 55.915-9.998 56.311Q-10.008 56.311-10.017 56.313Q-10.025 56.314-10.035 56.318M-5.568 54.561L-7.171 54.561L-7.171 54.281Q-6.945 54.281-6.797 54.247Q-6.648 54.212-6.648 54.072L-6.648 50.453Q-6.648 50.183-6.756 50.121Q-6.863 50.060-7.171 50.060L-7.171 49.779L-6.094 49.704L-6.094 54.072Q-6.094 54.209-5.944 54.245Q-5.794 54.281-5.568 54.281L-5.568 54.561M-4.915 53.833Q-4.915 53.501-4.691 53.274Q-4.467 53.047-4.124 52.919Q-3.780 52.790-3.408 52.738Q-3.035 52.685-2.731 52.685L-2.731 52.432Q-2.731 52.227-2.839 52.047Q-2.946 51.868-3.128 51.765Q-3.309 51.663-3.517 51.663Q-3.924 51.663-4.160 51.755Q-4.071 51.792-4.025 51.876Q-3.979 51.960-3.979 52.062Q-3.979 52.158-4.025 52.237Q-4.071 52.315-4.151 52.360Q-4.232 52.404-4.320 52.404Q-4.471 52.404-4.572 52.307Q-4.673 52.209-4.673 52.062Q-4.673 51.440-3.517 51.440Q-3.305 51.440-3.056 51.504Q-2.806 51.567-2.605 51.686Q-2.403 51.806-2.277 51.991Q-2.150 52.175-2.150 52.418L-2.150 53.994Q-2.150 54.110-2.089 54.206Q-2.027 54.301-1.914 54.301Q-1.805 54.301-1.740 54.207Q-1.675 54.113-1.675 53.994L-1.675 53.546L-1.408 53.546L-1.408 53.994Q-1.408 54.264-1.636 54.429Q-1.863 54.595-2.143 54.595Q-2.352 54.595-2.488 54.441Q-2.625 54.288-2.649 54.072Q-2.796 54.339-3.078 54.484Q-3.360 54.629-3.685 54.629Q-3.962 54.629-4.245 54.554Q-4.529 54.479-4.722 54.300Q-4.915 54.120-4.915 53.833M-4.300 53.833Q-4.300 54.007-4.199 54.137Q-4.098 54.267-3.943 54.337Q-3.787 54.407-3.623 54.407Q-3.404 54.407-3.196 54.310Q-2.987 54.212-2.859 54.031Q-2.731 53.850-2.731 53.624L-2.731 52.896Q-3.056 52.896-3.422 52.987Q-3.787 53.078-4.044 53.290Q-4.300 53.501-4.300 53.833M0.690 54.561L-0.944 54.561L-0.944 54.281Q-0.715 54.281-0.566 54.247Q-0.417 54.212-0.417 54.072L-0.417 52.223Q-0.417 51.953-0.525 51.892Q-0.632 51.830-0.944 51.830L-0.944 51.550L0.116 51.475L0.116 52.124Q0.287 51.816 0.591 51.645Q0.895 51.475 1.241 51.475Q1.640 51.475 1.917 51.615Q2.194 51.755 2.280 52.103Q2.447 51.810 2.746 51.642Q3.045 51.475 3.390 51.475Q3.896 51.475 4.180 51.698Q4.464 51.922 4.464 52.418L4.464 54.072Q4.464 54.209 4.612 54.245Q4.761 54.281 4.987 54.281L4.987 54.561L3.356 54.561L3.356 54.281Q3.582 54.281 3.732 54.245Q3.883 54.209 3.883 54.072L3.883 52.432Q3.883 52.097 3.763 51.897Q3.643 51.697 3.329 51.697Q3.059 51.697 2.825 51.833Q2.591 51.970 2.452 52.204Q2.314 52.438 2.314 52.712L2.314 54.072Q2.314 54.209 2.462 54.245Q2.611 54.281 2.837 54.281L2.837 54.561L1.206 54.561L1.206 54.281Q1.435 54.281 1.584 54.247Q1.733 54.212 1.733 54.072L1.733 52.432Q1.733 52.097 1.613 51.897Q1.494 51.697 1.179 51.697Q0.909 51.697 0.675 51.833Q0.441 51.970 0.302 52.204Q0.164 52.438 0.164 52.712L0.164 54.072Q0.164 54.209 0.314 54.245Q0.465 54.281 0.690 54.281\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(7.73 -18.247)\">\u003Cpath d=\"M6.184 54.561L5.917 54.561L5.917 50.453Q5.917 50.183 5.810 50.121Q5.702 50.060 5.391 50.060L5.391 49.779L6.471 49.704L6.471 51.874Q6.680 51.683 6.965 51.579Q7.250 51.475 7.548 51.475Q7.866 51.475 8.163 51.596Q8.460 51.717 8.683 51.933Q8.905 52.148 9.031 52.433Q9.158 52.719 9.158 53.050Q9.158 53.495 8.918 53.859Q8.679 54.223 8.286 54.426Q7.893 54.629 7.449 54.629Q7.254 54.629 7.064 54.573Q6.875 54.517 6.714 54.412Q6.553 54.308 6.413 54.147L6.184 54.561M6.499 52.216L6.499 53.833Q6.635 54.093 6.876 54.250Q7.117 54.407 7.394 54.407Q7.688 54.407 7.900 54.300Q8.112 54.192 8.245 54Q8.378 53.809 8.437 53.570Q8.495 53.331 8.495 53.050Q8.495 52.691 8.401 52.387Q8.307 52.083 8.079 51.890Q7.852 51.697 7.486 51.697Q7.186 51.697 6.919 51.833Q6.652 51.970 6.499 52.216\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(7.73 -18.247)\">\u003Cpath d=\"M10.009 53.050Q10.009 52.712 10.150 52.421Q10.290 52.131 10.534 51.917Q10.778 51.704 11.083 51.589Q11.387 51.475 11.712 51.475Q11.982 51.475 12.245 51.574Q12.508 51.673 12.699 51.851L12.699 50.453Q12.699 50.183 12.592 50.121Q12.484 50.060 12.173 50.060L12.173 49.779L13.250 49.704L13.250 53.888Q13.250 54.076 13.304 54.159Q13.359 54.243 13.460 54.262Q13.561 54.281 13.776 54.281L13.776 54.561L12.669 54.629L12.669 54.212Q12.252 54.629 11.626 54.629Q11.195 54.629 10.823 54.417Q10.450 54.206 10.230 53.845Q10.009 53.484 10.009 53.050M11.684 54.407Q11.893 54.407 12.079 54.335Q12.265 54.264 12.419 54.127Q12.573 53.990 12.669 53.812L12.669 52.203Q12.583 52.056 12.438 51.936Q12.293 51.816 12.123 51.757Q11.954 51.697 11.773 51.697Q11.213 51.697 10.944 52.086Q10.676 52.476 10.676 53.057Q10.676 53.628 10.910 54.018Q11.144 54.407 11.684 54.407M14.484 53.833Q14.484 53.501 14.707 53.274Q14.931 53.047 15.275 52.919Q15.618 52.790 15.991 52.738Q16.363 52.685 16.668 52.685L16.668 52.432Q16.668 52.227 16.560 52.047Q16.452 51.868 16.271 51.765Q16.090 51.663 15.882 51.663Q15.475 51.663 15.239 51.755Q15.328 51.792 15.374 51.876Q15.420 51.960 15.420 52.062Q15.420 52.158 15.374 52.237Q15.328 52.315 15.247 52.360Q15.167 52.404 15.078 52.404Q14.928 52.404 14.827 52.307Q14.726 52.209 14.726 52.062Q14.726 51.440 15.882 51.440Q16.093 51.440 16.343 51.504Q16.592 51.567 16.794 51.686Q16.996 51.806 17.122 51.991Q17.249 52.175 17.249 52.418L17.249 53.994Q17.249 54.110 17.310 54.206Q17.372 54.301 17.485 54.301Q17.594 54.301 17.659 54.207Q17.724 54.113 17.724 53.994L17.724 53.546L17.990 53.546L17.990 53.994Q17.990 54.264 17.763 54.429Q17.536 54.595 17.256 54.595Q17.047 54.595 16.910 54.441Q16.774 54.288 16.750 54.072Q16.603 54.339 16.321 54.484Q16.039 54.629 15.714 54.629Q15.437 54.629 15.153 54.554Q14.870 54.479 14.677 54.300Q14.484 54.120 14.484 53.833M15.099 53.833Q15.099 54.007 15.200 54.137Q15.300 54.267 15.456 54.337Q15.612 54.407 15.776 54.407Q15.994 54.407 16.203 54.310Q16.411 54.212 16.539 54.031Q16.668 53.850 16.668 53.624L16.668 52.896Q16.343 52.896 15.977 52.987Q15.612 53.078 15.355 53.290Q15.099 53.501 15.099 53.833\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg transform=\"translate(84.808 -67.884)\">\u003Cpath d=\"M-45.397 54.141Q-45.397 53.973-45.274 53.850Q-45.151 53.727-44.976 53.727Q-44.809 53.727-44.686 53.850Q-44.563 53.973-44.563 54.141Q-44.563 54.315-44.686 54.438Q-44.809 54.561-44.976 54.561Q-45.151 54.561-45.274 54.438Q-45.397 54.315-45.397 54.141M-43.134 54.141Q-43.134 53.973-43.011 53.850Q-42.888 53.727-42.714 53.727Q-42.546 53.727-42.423 53.850Q-42.300 53.973-42.300 54.141Q-42.300 54.315-42.423 54.438Q-42.546 54.561-42.714 54.561Q-42.888 54.561-43.011 54.438Q-43.134 54.315-43.134 54.141M-40.871 54.141Q-40.871 53.973-40.748 53.850Q-40.625 53.727-40.451 53.727Q-40.283 53.727-40.160 53.850Q-40.037 53.973-40.037 54.141Q-40.037 54.315-40.160 54.438Q-40.283 54.561-40.451 54.561Q-40.625 54.561-40.748 54.438Q-40.871 54.315-40.871 54.141\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath d=\"M87.048-42.178a2.276 2.276 0 1 0-4.552 0 2.276 2.276 0 0 0 4.552 0Zm-2.276 0\"\u002F>\u003Cpath fill=\"none\" d=\"M89.325-25.106a4.552 4.552 0 1 0-9.105 0 4.552 4.552 0 0 0 9.105 0Zm-4.553 0\"\u002F>\u003Cpath d=\"M76.52-13.725a1.707 1.707 0 1 0-3.414 0 1.707 1.707 0 0 0 3.415 0ZM96.438-13.725a1.707 1.707 0 1 0-3.415 0 1.707 1.707 0 0 0 3.415 0Zm-1.708 0\"\u002F>\u003Cpath fill=\"none\" d=\"m81.643-21.53-6.83 7.805M87.901-21.53l6.83 7.805\"\u002F>\u003Cpath d=\"M87.048-10.88a2.276 2.276 0 1 0-4.552 0 2.276 2.276 0 0 0 4.552 0Zm-2.276 0\"\u002F>\u003Cpath fill=\"none\" d=\"M89.325 6.192a4.552 4.552 0 1 0-9.105 0 4.552 4.552 0 0 0 9.105 0Zm-4.553 0\"\u002F>\u003Cpath d=\"M76.52 16.15a1.707 1.707 0 1 0-3.414 0 1.707 1.707 0 0 0 3.415 0ZM96.438 16.15a1.707 1.707 0 1 0-3.415 0 1.707 1.707 0 0 0 3.415 0Zm-1.708 0\"\u002F>\u003Cpath fill=\"none\" d=\"m81.412 9.552-6.598 6.598M88.133 9.552l6.597 6.598\"\u002F>\u003Cg fill=\"var(--tk-soft-neutral)\">\u003Cpath d=\"M79.082 36.067h11.38V24.686h-11.38Z\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M84.772-39.701v9.842m0 9.505v6.998m0 4.952V1.44m0 9.505v13.542\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(95.813 -8.678)\">\u003Cpath d=\"M-44.129 54.561L-45.732 54.561L-45.732 54.281Q-45.506 54.281-45.357 54.247Q-45.209 54.212-45.209 54.072L-45.209 50.453Q-45.209 50.183-45.316 50.121Q-45.424 50.060-45.732 50.060L-45.732 49.779L-44.655 49.704L-44.655 54.072Q-44.655 54.209-44.505 54.245Q-44.354 54.281-44.129 54.281L-44.129 54.561M-43.476 53.833Q-43.476 53.501-43.252 53.274Q-43.028 53.047-42.684 52.919Q-42.341 52.790-41.968 52.738Q-41.596 52.685-41.292 52.685L-41.292 52.432Q-41.292 52.227-41.399 52.047Q-41.507 51.868-41.688 51.765Q-41.869 51.663-42.078 51.663Q-42.485 51.663-42.720 51.755Q-42.632 51.792-42.585 51.876Q-42.539 51.960-42.539 52.062Q-42.539 52.158-42.585 52.237Q-42.632 52.315-42.712 52.360Q-42.792 52.404-42.881 52.404Q-43.031 52.404-43.132 52.307Q-43.233 52.209-43.233 52.062Q-43.233 51.440-42.078 51.440Q-41.866 51.440-41.616 51.504Q-41.367 51.567-41.165 51.686Q-40.964 51.806-40.837 51.991Q-40.711 52.175-40.711 52.418L-40.711 53.994Q-40.711 54.110-40.649 54.206Q-40.588 54.301-40.475 54.301Q-40.365 54.301-40.300 54.207Q-40.236 54.113-40.236 53.994L-40.236 53.546L-39.969 53.546L-39.969 53.994Q-39.969 54.264-40.196 54.429Q-40.423 54.595-40.704 54.595Q-40.912 54.595-41.049 54.441Q-41.186 54.288-41.210 54.072Q-41.357 54.339-41.639 54.484Q-41.921 54.629-42.245 54.629Q-42.522 54.629-42.806 54.554Q-43.090 54.479-43.283 54.300Q-43.476 54.120-43.476 53.833M-42.861 53.833Q-42.861 54.007-42.760 54.137Q-42.659 54.267-42.503 54.337Q-42.348 54.407-42.184 54.407Q-41.965 54.407-41.757 54.310Q-41.548 54.212-41.420 54.031Q-41.292 53.850-41.292 53.624L-41.292 52.896Q-41.616 52.896-41.982 52.987Q-42.348 53.078-42.604 53.290Q-42.861 53.501-42.861 53.833M-37.870 54.561L-39.504 54.561L-39.504 54.281Q-39.275 54.281-39.126 54.247Q-38.978 54.212-38.978 54.072L-38.978 52.223Q-38.978 51.953-39.085 51.892Q-39.193 51.830-39.504 51.830L-39.504 51.550L-38.444 51.475L-38.444 52.124Q-38.274 51.816-37.969 51.645Q-37.665 51.475-37.320 51.475Q-36.920 51.475-36.643 51.615Q-36.366 51.755-36.281 52.103Q-36.113 51.810-35.814 51.642Q-35.515 51.475-35.170 51.475Q-34.664 51.475-34.381 51.698Q-34.097 51.922-34.097 52.418L-34.097 54.072Q-34.097 54.209-33.948 54.245Q-33.799 54.281-33.574 54.281L-33.574 54.561L-35.204 54.561L-35.204 54.281Q-34.979 54.281-34.828 54.245Q-34.678 54.209-34.678 54.072L-34.678 52.432Q-34.678 52.097-34.798 51.897Q-34.917 51.697-35.232 51.697Q-35.502 51.697-35.736 51.833Q-35.970 51.970-36.108 52.204Q-36.247 52.438-36.247 52.712L-36.247 54.072Q-36.247 54.209-36.098 54.245Q-35.949 54.281-35.724 54.281L-35.724 54.561L-37.354 54.561L-37.354 54.281Q-37.125 54.281-36.976 54.247Q-36.828 54.212-36.828 54.072L-36.828 52.432Q-36.828 52.097-36.947 51.897Q-37.067 51.697-37.382 51.697Q-37.652 51.697-37.886 51.833Q-38.120 51.970-38.258 52.204Q-38.397 52.438-38.397 52.712L-38.397 54.072Q-38.397 54.209-38.246 54.245Q-38.096 54.281-37.870 54.281\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(95.813 -8.678)\">\u003Cpath d=\"M-32.385 54.561L-32.652 54.561L-32.652 50.453Q-32.652 50.183-32.759 50.121Q-32.867 50.060-33.178 50.060L-33.178 49.779L-32.098 49.704L-32.098 51.874Q-31.889 51.683-31.604 51.579Q-31.319 51.475-31.021 51.475Q-30.703 51.475-30.406 51.596Q-30.109 51.717-29.886 51.933Q-29.664 52.148-29.538 52.433Q-29.411 52.719-29.411 53.050Q-29.411 53.495-29.651 53.859Q-29.890 54.223-30.283 54.426Q-30.676 54.629-31.120 54.629Q-31.315 54.629-31.505 54.573Q-31.694 54.517-31.855 54.412Q-32.016 54.308-32.156 54.147L-32.385 54.561M-32.070 52.216L-32.070 53.833Q-31.934 54.093-31.693 54.250Q-31.452 54.407-31.175 54.407Q-30.881 54.407-30.669 54.300Q-30.457 54.192-30.324 54Q-30.191 53.809-30.132 53.570Q-30.074 53.331-30.074 53.050Q-30.074 52.691-30.168 52.387Q-30.262 52.083-30.490 51.890Q-30.717 51.697-31.083 51.697Q-31.383 51.697-31.650 51.833Q-31.917 51.970-32.070 52.216\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(95.813 -8.678)\">\u003Cpath d=\"M-28.560 53.050Q-28.560 52.712-28.419 52.421Q-28.279 52.131-28.035 51.917Q-27.791 51.704-27.486 51.589Q-27.182 51.475-26.857 51.475Q-26.587 51.475-26.324 51.574Q-26.061 51.673-25.870 51.851L-25.870 50.453Q-25.870 50.183-25.977 50.121Q-26.085 50.060-26.396 50.060L-26.396 49.779L-25.319 49.704L-25.319 53.888Q-25.319 54.076-25.265 54.159Q-25.210 54.243-25.109 54.262Q-25.008 54.281-24.793 54.281L-24.793 54.561L-25.900 54.629L-25.900 54.212Q-26.317 54.629-26.943 54.629Q-27.374 54.629-27.746 54.417Q-28.119 54.206-28.339 53.845Q-28.560 53.484-28.560 53.050M-26.885 54.407Q-26.676 54.407-26.490 54.335Q-26.304 54.264-26.150 54.127Q-25.996 53.990-25.900 53.812L-25.900 52.203Q-25.986 52.056-26.131 51.936Q-26.276 51.816-26.446 51.757Q-26.615 51.697-26.796 51.697Q-27.356 51.697-27.625 52.086Q-27.893 52.476-27.893 53.057Q-27.893 53.628-27.659 54.018Q-27.425 54.407-26.885 54.407M-24.085 53.833Q-24.085 53.501-23.862 53.274Q-23.638 53.047-23.294 52.919Q-22.951 52.790-22.578 52.738Q-22.206 52.685-21.901 52.685L-21.901 52.432Q-21.901 52.227-22.009 52.047Q-22.117 51.868-22.298 51.765Q-22.479 51.663-22.687 51.663Q-23.094 51.663-23.330 51.755Q-23.241 51.792-23.195 51.876Q-23.149 51.960-23.149 52.062Q-23.149 52.158-23.195 52.237Q-23.241 52.315-23.322 52.360Q-23.402 52.404-23.491 52.404Q-23.641 52.404-23.742 52.307Q-23.843 52.209-23.843 52.062Q-23.843 51.440-22.687 51.440Q-22.476 51.440-22.226 51.504Q-21.977 51.567-21.775 51.686Q-21.573 51.806-21.447 51.991Q-21.320 52.175-21.320 52.418L-21.320 53.994Q-21.320 54.110-21.259 54.206Q-21.197 54.301-21.084 54.301Q-20.975 54.301-20.910 54.207Q-20.845 54.113-20.845 53.994L-20.845 53.546L-20.579 53.546L-20.579 53.994Q-20.579 54.264-20.806 54.429Q-21.033 54.595-21.313 54.595Q-21.522 54.595-21.659 54.441Q-21.795 54.288-21.819 54.072Q-21.966 54.339-22.248 54.484Q-22.530 54.629-22.855 54.629Q-23.132 54.629-23.416 54.554Q-23.699 54.479-23.892 54.300Q-24.085 54.120-24.085 53.833M-23.470 53.833Q-23.470 54.007-23.369 54.137Q-23.269 54.267-23.113 54.337Q-22.957 54.407-22.793 54.407Q-22.575 54.407-22.366 54.310Q-22.158 54.212-22.030 54.031Q-21.901 53.850-21.901 53.624L-21.901 52.896Q-22.226 52.896-22.592 52.987Q-22.957 53.078-23.214 53.290Q-23.470 53.501-23.470 53.833\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(95.813 -8.678)\">\u003Cpath d=\"M-16.936 53.720L-16.936 51.823L-17.575 51.823L-17.575 51.601Q-17.257 51.601-17.040 51.391Q-16.823 51.181-16.723 50.871Q-16.622 50.562-16.622 50.254L-16.355 50.254L-16.355 51.543L-15.278 51.543L-15.278 51.823L-16.355 51.823L-16.355 53.707Q-16.355 53.983-16.251 54.182Q-16.147 54.380-15.887 54.380Q-15.730 54.380-15.624 54.276Q-15.518 54.171-15.468 54.018Q-15.419 53.864-15.419 53.707L-15.419 53.293L-15.152 53.293L-15.152 53.720Q-15.152 53.946-15.251 54.156Q-15.350 54.366-15.535 54.498Q-15.719 54.629-15.948 54.629Q-16.386 54.629-16.661 54.392Q-16.936 54.154-16.936 53.720M-14.383 53.078Q-14.383 52.736-14.248 52.437Q-14.113 52.138-13.874 51.914Q-13.634 51.690-13.317 51.565Q-12.999 51.440-12.667 51.440Q-12.223 51.440-11.823 51.656Q-11.423 51.871-11.189 52.249Q-10.955 52.626-10.955 53.078Q-10.955 53.419-11.097 53.703Q-11.238 53.987-11.483 54.194Q-11.727 54.400-12.037 54.515Q-12.346 54.629-12.667 54.629Q-13.098 54.629-13.499 54.428Q-13.901 54.226-14.142 53.874Q-14.383 53.522-14.383 53.078M-12.667 54.380Q-12.066 54.380-11.842 54.002Q-11.618 53.624-11.618 52.992Q-11.618 52.380-11.852 52.021Q-12.086 51.663-12.667 51.663Q-13.720 51.663-13.720 52.992Q-13.720 53.624-13.494 54.002Q-13.269 54.380-12.667 54.380\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(95.813 -8.678)\">\u003Cpath d=\"M-7.131 53.720L-7.131 51.823L-7.770 51.823L-7.770 51.601Q-7.452 51.601-7.235 51.391Q-7.018 51.181-6.918 50.871Q-6.817 50.562-6.817 50.254L-6.550 50.254L-6.550 51.543L-5.473 51.543L-5.473 51.823L-6.550 51.823L-6.550 53.707Q-6.550 53.983-6.446 54.182Q-6.342 54.380-6.082 54.380Q-5.925 54.380-5.819 54.276Q-5.713 54.171-5.663 54.018Q-5.614 53.864-5.614 53.707L-5.614 53.293L-5.347 53.293L-5.347 53.720Q-5.347 53.946-5.446 54.156Q-5.545 54.366-5.730 54.498Q-5.914 54.629-6.143 54.629Q-6.581 54.629-6.856 54.392Q-7.131 54.154-7.131 53.720M-2.855 54.561L-4.489 54.561L-4.489 54.281Q-4.260 54.281-4.111 54.247Q-3.963 54.212-3.963 54.072L-3.963 50.453Q-3.963 50.183-4.070 50.121Q-4.178 50.060-4.489 50.060L-4.489 49.779L-3.409 49.704L-3.409 52.090Q-3.303 51.905-3.125 51.763Q-2.948 51.622-2.739 51.548Q-2.531 51.475-2.305 51.475Q-1.799 51.475-1.515 51.698Q-1.232 51.922-1.232 52.418L-1.232 54.072Q-1.232 54.209-1.083 54.245Q-0.934 54.281-0.709 54.281L-0.709 54.561L-2.339 54.561L-2.339 54.281Q-2.110 54.281-1.962 54.247Q-1.813 54.212-1.813 54.072L-1.813 52.432Q-1.813 52.097-1.932 51.897Q-2.052 51.697-2.367 51.697Q-2.637 51.697-2.871 51.833Q-3.105 51.970-3.243 52.204Q-3.382 52.438-3.382 52.712L-3.382 54.072Q-3.382 54.209-3.231 54.245Q-3.081 54.281-2.855 54.281L-2.855 54.561M-0.162 53.026Q-0.162 52.705-0.037 52.416Q0.088 52.127 0.313 51.904Q0.539 51.680 0.834 51.560Q1.130 51.440 1.448 51.440Q1.776 51.440 2.038 51.540Q2.299 51.639 2.475 51.821Q2.651 52.004 2.745 52.262Q2.839 52.520 2.839 52.852Q2.839 52.944 2.757 52.965L0.501 52.965L0.501 53.026Q0.501 53.614 0.785 53.997Q1.069 54.380 1.636 54.380Q1.957 54.380 2.226 54.187Q2.494 53.994 2.583 53.679Q2.590 53.638 2.665 53.624L2.757 53.624Q2.839 53.648 2.839 53.720Q2.839 53.727 2.832 53.754Q2.719 54.151 2.349 54.390Q1.978 54.629 1.554 54.629Q1.116 54.629 0.716 54.421Q0.317 54.212 0.077 53.845Q-0.162 53.478-0.162 53.026M0.508 52.756L2.323 52.756Q2.323 52.479 2.226 52.227Q2.128 51.974 1.930 51.818Q1.732 51.663 1.448 51.663Q1.171 51.663 0.957 51.821Q0.744 51.980 0.626 52.235Q0.508 52.490 0.508 52.756\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(95.813 -8.678)\">\u003Cpath d=\"M10.041 54.561L7.303 54.561L7.303 54.281Q7.652 54.281 7.989 54.245Q8.325 54.209 8.325 54.072L8.325 50.271Q8.325 50.128 8.237 50.094Q8.148 50.060 7.963 50.060L7.604 50.060Q7.303 50.060 7.088 50.107Q6.873 50.155 6.716 50.312Q6.579 50.446 6.519 50.724Q6.459 51.003 6.422 51.416L6.155 51.416L6.302 49.779L11.036 49.779L11.183 51.416L10.916 51.416Q10.879 51.003 10.822 50.726Q10.766 50.449 10.622 50.312Q10.462 50.152 10.250 50.106Q10.038 50.060 9.734 50.060L9.382 50.060Q9.197 50.060 9.108 50.094Q9.019 50.128 9.019 50.271L9.019 54.072Q9.019 54.209 9.356 54.245Q9.693 54.281 10.041 54.281L10.041 54.561M13.722 53.307L11.665 53.307L11.665 52.804L13.722 52.804L13.722 53.307M15.052 53.720L15.052 51.823L14.413 51.823L14.413 51.601Q14.731 51.601 14.948 51.391Q15.165 51.181 15.266 50.871Q15.366 50.562 15.366 50.254L15.633 50.254L15.633 51.543L16.710 51.543L16.710 51.823L15.633 51.823L15.633 53.707Q15.633 53.983 15.737 54.182Q15.841 54.380 16.101 54.380Q16.258 54.380 16.364 54.276Q16.470 54.171 16.520 54.018Q16.570 53.864 16.570 53.707L16.570 53.293L16.836 53.293L16.836 53.720Q16.836 53.946 16.737 54.156Q16.638 54.366 16.453 54.498Q16.269 54.629 16.040 54.629Q15.602 54.629 15.327 54.392Q15.052 54.154 15.052 53.720M19.529 53.307L17.472 53.307L17.472 52.804L19.529 52.804L19.529 53.307M23.344 54.561L20.815 54.561L20.815 54.281Q21.782 54.281 21.782 54.072L21.782 50.453Q21.389 50.641 20.767 50.641L20.767 50.360Q21.184 50.360 21.548 50.259Q21.912 50.159 22.168 49.913L22.295 49.913Q22.360 49.930 22.377 49.998L22.377 54.072Q22.377 54.281 23.344 54.281\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(157.178 -95.028)\">\u003Cpath d=\"M-45.622 54.623L-45.622 53.050Q-45.622 53.023-45.597 52.997Q-45.571 52.972-45.544 52.972L-45.431 52.972Q-45.403 52.972-45.380 52.999Q-45.356 53.026-45.356 53.050Q-45.356 53.395-45.224 53.659Q-45.092 53.922-44.863 54.091Q-44.634 54.260-44.332 54.341Q-44.029 54.421-43.688 54.421Q-43.421 54.421-43.185 54.293Q-42.949 54.165-42.804 53.942Q-42.659 53.720-42.659 53.454Q-42.659 53.231-42.765 53.035Q-42.871 52.838-43.052 52.703Q-43.233 52.568-43.459 52.517L-44.487 52.285Q-44.798 52.213-45.058 52.027Q-45.318 51.840-45.470 51.569Q-45.622 51.297-45.622 50.982Q-45.622 50.596-45.409 50.289Q-45.195 49.981-44.848 49.810Q-44.501 49.639-44.122 49.639Q-43.893 49.639-43.664 49.692Q-43.435 49.745-43.236 49.853Q-43.038 49.960-42.884 50.124L-42.590 49.684Q-42.567 49.639-42.526 49.639L-42.478 49.639Q-42.447 49.639-42.425 49.665Q-42.403 49.690-42.403 49.718L-42.403 51.293Q-42.403 51.314-42.426 51.341Q-42.450 51.369-42.478 51.369L-42.590 51.369Q-42.652 51.369-42.666 51.293Q-42.707 50.880-42.888 50.560Q-43.069 50.241-43.380 50.066Q-43.691 49.892-44.122 49.892Q-44.371 49.892-44.611 50.003Q-44.850 50.114-45 50.312Q-45.151 50.511-45.151 50.774Q-45.151 50.986-45.043 51.167Q-44.935 51.348-44.759 51.468Q-44.583 51.587-44.375 51.628L-43.346 51.857Q-43.028 51.929-42.761 52.134Q-42.495 52.339-42.343 52.633Q-42.191 52.927-42.191 53.259Q-42.191 53.652-42.396 53.987Q-42.601 54.322-42.946 54.511Q-43.291 54.701-43.688 54.701Q-44.108 54.701-44.487 54.588Q-44.867 54.476-45.137 54.226L-45.431 54.660Q-45.458 54.701-45.496 54.701L-45.544 54.701Q-45.571 54.701-45.597 54.676Q-45.622 54.650-45.622 54.623M-40.854 53.720L-40.854 51.823L-41.493 51.823L-41.493 51.601Q-41.175 51.601-40.958 51.391Q-40.741 51.181-40.641 50.871Q-40.540 50.562-40.540 50.254L-40.273 50.254L-40.273 51.543L-39.196 51.543L-39.196 51.823L-40.273 51.823L-40.273 53.707Q-40.273 53.983-40.169 54.182Q-40.065 54.380-39.805 54.380Q-39.648 54.380-39.542 54.276Q-39.436 54.171-39.386 54.018Q-39.337 53.864-39.337 53.707L-39.337 53.293L-39.070 53.293L-39.070 53.720Q-39.070 53.946-39.169 54.156Q-39.268 54.366-39.453 54.498Q-39.637 54.629-39.866 54.629Q-40.304 54.629-40.579 54.392Q-40.854 54.154-40.854 53.720M-37.761 55.791Q-37.761 55.757-37.734 55.730Q-37.464 55.501-37.315 55.178Q-37.166 54.855-37.166 54.499L-37.166 54.462Q-37.276 54.561-37.440 54.561Q-37.621 54.561-37.740 54.441Q-37.860 54.322-37.860 54.141Q-37.860 53.966-37.740 53.847Q-37.621 53.727-37.440 53.727Q-37.183 53.727-37.064 53.966Q-36.944 54.206-36.944 54.499Q-36.944 54.899-37.113 55.270Q-37.282 55.641-37.580 55.897Q-37.611 55.918-37.638 55.918Q-37.679 55.918-37.720 55.877Q-37.761 55.836-37.761 55.791\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(157.178 -95.028)\">\u003Cpath d=\"M-31.707 54.561L-33.297 54.561L-33.297 54.281Q-32.654 54.281-32.497 53.881L-30.853 49.666Q-30.819 49.571-30.706 49.571L-30.624 49.571Q-30.514 49.571-30.473 49.666L-28.754 54.072Q-28.686 54.212-28.496 54.247Q-28.306 54.281-28.033 54.281L-28.033 54.561L-30.032 54.561L-30.032 54.281Q-29.468 54.281-29.468 54.106Q-29.468 54.089-29.470 54.082Q-29.472 54.076-29.475 54.072L-29.896 53.006L-31.854 53.006L-32.196 53.881Q-32.210 53.881-32.210 53.959Q-32.210 54.120-32.047 54.200Q-31.885 54.281-31.707 54.281L-31.707 54.561M-30.873 50.487L-31.741 52.726L-29.998 52.726\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(157.178 -95.028)\">\u003Cpath d=\"M-27.082 53.720L-27.082 51.823L-27.721 51.823L-27.721 51.601Q-27.403 51.601-27.186 51.391Q-26.969 51.181-26.869 50.871Q-26.768 50.562-26.768 50.254L-26.501 50.254L-26.501 51.543L-25.424 51.543L-25.424 51.823L-26.501 51.823L-26.501 53.707Q-26.501 53.983-26.397 54.182Q-26.293 54.380-26.033 54.380Q-25.876 54.380-25.770 54.276Q-25.664 54.171-25.614 54.018Q-25.565 53.864-25.565 53.707L-25.565 53.293L-25.298 53.293L-25.298 53.720Q-25.298 53.946-25.397 54.156Q-25.496 54.366-25.681 54.498Q-25.865 54.629-26.094 54.629Q-26.532 54.629-26.807 54.392Q-27.082 54.154-27.082 53.720\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg transform=\"translate(157.178 -77.693)\">\u003Cpath d=\"M-45.622 54.623L-45.622 53.050Q-45.622 53.023-45.597 52.997Q-45.571 52.972-45.544 52.972L-45.431 52.972Q-45.403 52.972-45.380 52.999Q-45.356 53.026-45.356 53.050Q-45.356 53.395-45.224 53.659Q-45.092 53.922-44.863 54.091Q-44.634 54.260-44.332 54.341Q-44.029 54.421-43.688 54.421Q-43.421 54.421-43.185 54.293Q-42.949 54.165-42.804 53.942Q-42.659 53.720-42.659 53.454Q-42.659 53.231-42.765 53.035Q-42.871 52.838-43.052 52.703Q-43.233 52.568-43.459 52.517L-44.487 52.285Q-44.798 52.213-45.058 52.027Q-45.318 51.840-45.470 51.569Q-45.622 51.297-45.622 50.982Q-45.622 50.596-45.409 50.289Q-45.195 49.981-44.848 49.810Q-44.501 49.639-44.122 49.639Q-43.893 49.639-43.664 49.692Q-43.435 49.745-43.236 49.853Q-43.038 49.960-42.884 50.124L-42.590 49.684Q-42.567 49.639-42.526 49.639L-42.478 49.639Q-42.447 49.639-42.425 49.665Q-42.403 49.690-42.403 49.718L-42.403 51.293Q-42.403 51.314-42.426 51.341Q-42.450 51.369-42.478 51.369L-42.590 51.369Q-42.652 51.369-42.666 51.293Q-42.707 50.880-42.888 50.560Q-43.069 50.241-43.380 50.066Q-43.691 49.892-44.122 49.892Q-44.371 49.892-44.611 50.003Q-44.850 50.114-45 50.312Q-45.151 50.511-45.151 50.774Q-45.151 50.986-45.043 51.167Q-44.935 51.348-44.759 51.468Q-44.583 51.587-44.375 51.628L-43.346 51.857Q-43.028 51.929-42.761 52.134Q-42.495 52.339-42.343 52.633Q-42.191 52.927-42.191 53.259Q-42.191 53.652-42.396 53.987Q-42.601 54.322-42.946 54.511Q-43.291 54.701-43.688 54.701Q-44.108 54.701-44.487 54.588Q-44.867 54.476-45.137 54.226L-45.431 54.660Q-45.458 54.701-45.496 54.701L-45.544 54.701Q-45.571 54.701-45.597 54.676Q-45.622 54.650-45.622 54.623M-40.854 53.720L-40.854 51.823L-41.493 51.823L-41.493 51.601Q-41.175 51.601-40.958 51.391Q-40.741 51.181-40.641 50.871Q-40.540 50.562-40.540 50.254L-40.273 50.254L-40.273 51.543L-39.196 51.543L-39.196 51.823L-40.273 51.823L-40.273 53.707Q-40.273 53.983-40.169 54.182Q-40.065 54.380-39.805 54.380Q-39.648 54.380-39.542 54.276Q-39.436 54.171-39.386 54.018Q-39.337 53.864-39.337 53.707L-39.337 53.293L-39.070 53.293L-39.070 53.720Q-39.070 53.946-39.169 54.156Q-39.268 54.366-39.453 54.498Q-39.637 54.629-39.866 54.629Q-40.304 54.629-40.579 54.392Q-40.854 54.154-40.854 53.720M-35.676 55.241L-35.676 52.985L-37.925 52.985Q-37.993 52.975-38.039 52.929Q-38.086 52.883-38.086 52.811Q-38.086 52.667-37.925 52.644L-35.676 52.644L-35.676 50.388Q-35.666 50.319-35.620 50.273Q-35.573 50.227-35.502 50.227Q-35.358 50.227-35.334 50.388L-35.334 52.644L-33.092 52.644Q-32.931 52.667-32.931 52.811Q-32.931 52.883-32.977 52.929Q-33.024 52.975-33.092 52.985L-35.334 52.985L-35.334 55.241Q-35.358 55.402-35.502 55.402Q-35.573 55.402-35.620 55.356Q-35.666 55.310-35.676 55.241M-29.110 54.561L-31.639 54.561L-31.639 54.281Q-30.672 54.281-30.672 54.072L-30.672 50.453Q-31.065 50.641-31.687 50.641L-31.687 50.360Q-31.270 50.360-30.906 50.259Q-30.542 50.159-30.286 49.913L-30.159 49.913Q-30.094 49.930-30.077 49.998L-30.077 54.072Q-30.077 54.281-29.110 54.281\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(157.178 -46.395)\">\u003Cpath d=\"M-45.622 54.623L-45.622 53.050Q-45.622 53.023-45.597 52.997Q-45.571 52.972-45.544 52.972L-45.431 52.972Q-45.403 52.972-45.380 52.999Q-45.356 53.026-45.356 53.050Q-45.356 53.395-45.224 53.659Q-45.092 53.922-44.863 54.091Q-44.634 54.260-44.332 54.341Q-44.029 54.421-43.688 54.421Q-43.421 54.421-43.185 54.293Q-42.949 54.165-42.804 53.942Q-42.659 53.720-42.659 53.454Q-42.659 53.231-42.765 53.035Q-42.871 52.838-43.052 52.703Q-43.233 52.568-43.459 52.517L-44.487 52.285Q-44.798 52.213-45.058 52.027Q-45.318 51.840-45.470 51.569Q-45.622 51.297-45.622 50.982Q-45.622 50.596-45.409 50.289Q-45.195 49.981-44.848 49.810Q-44.501 49.639-44.122 49.639Q-43.893 49.639-43.664 49.692Q-43.435 49.745-43.236 49.853Q-43.038 49.960-42.884 50.124L-42.590 49.684Q-42.567 49.639-42.526 49.639L-42.478 49.639Q-42.447 49.639-42.425 49.665Q-42.403 49.690-42.403 49.718L-42.403 51.293Q-42.403 51.314-42.426 51.341Q-42.450 51.369-42.478 51.369L-42.590 51.369Q-42.652 51.369-42.666 51.293Q-42.707 50.880-42.888 50.560Q-43.069 50.241-43.380 50.066Q-43.691 49.892-44.122 49.892Q-44.371 49.892-44.611 50.003Q-44.850 50.114-45 50.312Q-45.151 50.511-45.151 50.774Q-45.151 50.986-45.043 51.167Q-44.935 51.348-44.759 51.468Q-44.583 51.587-44.375 51.628L-43.346 51.857Q-43.028 51.929-42.761 52.134Q-42.495 52.339-42.343 52.633Q-42.191 52.927-42.191 53.259Q-42.191 53.652-42.396 53.987Q-42.601 54.322-42.946 54.511Q-43.291 54.701-43.688 54.701Q-44.108 54.701-44.487 54.588Q-44.867 54.476-45.137 54.226L-45.431 54.660Q-45.458 54.701-45.496 54.701L-45.544 54.701Q-45.571 54.701-45.597 54.676Q-45.622 54.650-45.622 54.623M-40.854 53.720L-40.854 51.823L-41.493 51.823L-41.493 51.601Q-41.175 51.601-40.958 51.391Q-40.741 51.181-40.641 50.871Q-40.540 50.562-40.540 50.254L-40.273 50.254L-40.273 51.543L-39.196 51.543L-39.196 51.823L-40.273 51.823L-40.273 53.707Q-40.273 53.983-40.169 54.182Q-40.065 54.380-39.805 54.380Q-39.648 54.380-39.542 54.276Q-39.436 54.171-39.386 54.018Q-39.337 53.864-39.337 53.707L-39.337 53.293L-39.070 53.293L-39.070 53.720Q-39.070 53.946-39.169 54.156Q-39.268 54.366-39.453 54.498Q-39.637 54.629-39.866 54.629Q-40.304 54.629-40.579 54.392Q-40.854 54.154-40.854 53.720M-35.676 55.241L-35.676 52.985L-37.925 52.985Q-37.993 52.975-38.039 52.929Q-38.086 52.883-38.086 52.811Q-38.086 52.667-37.925 52.644L-35.676 52.644L-35.676 50.388Q-35.666 50.319-35.620 50.273Q-35.573 50.227-35.502 50.227Q-35.358 50.227-35.334 50.388L-35.334 52.644L-33.092 52.644Q-32.931 52.667-32.931 52.811Q-32.931 52.883-32.977 52.929Q-33.024 52.975-33.092 52.985L-35.334 52.985L-35.334 55.241Q-35.358 55.402-35.502 55.402Q-35.573 55.402-35.620 55.356Q-35.666 55.310-35.676 55.241M-29.110 54.561L-31.995 54.561L-31.995 54.359Q-31.995 54.329-31.967 54.301L-30.720 53.084Q-30.648 53.009-30.605 52.967Q-30.563 52.924-30.484 52.845Q-30.070 52.432-29.840 52.074Q-29.609 51.717-29.609 51.293Q-29.609 51.061-29.688 50.858Q-29.766 50.654-29.908 50.504Q-30.050 50.353-30.245 50.273Q-30.440 50.193-30.672 50.193Q-30.983 50.193-31.241 50.352Q-31.499 50.511-31.629 50.788L-31.609 50.788Q-31.441 50.788-31.333 50.899Q-31.226 51.010-31.226 51.174Q-31.226 51.331-31.335 51.444Q-31.444 51.557-31.609 51.557Q-31.769 51.557-31.882 51.444Q-31.995 51.331-31.995 51.174Q-31.995 50.798-31.786 50.511Q-31.578 50.224-31.243 50.068Q-30.908 49.913-30.552 49.913Q-30.129 49.913-29.749 50.071Q-29.370 50.230-29.136 50.547Q-28.902 50.863-28.902 51.293Q-28.902 51.604-29.042 51.873Q-29.182 52.141-29.387 52.346Q-29.592 52.551-29.954 52.833Q-30.317 53.115-30.426 53.211L-31.280 53.939L-30.638 53.939Q-30.375 53.939-30.086 53.937Q-29.797 53.936-29.578 53.927Q-29.360 53.918-29.342 53.901Q-29.281 53.836-29.243 53.669Q-29.206 53.501-29.168 53.259L-28.902 53.259\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(157.178 -26.061)\">\u003Cpath d=\"M-45.622 54.623L-45.622 53.050Q-45.622 53.023-45.597 52.997Q-45.571 52.972-45.544 52.972L-45.431 52.972Q-45.403 52.972-45.380 52.999Q-45.356 53.026-45.356 53.050Q-45.356 53.395-45.224 53.659Q-45.092 53.922-44.863 54.091Q-44.634 54.260-44.332 54.341Q-44.029 54.421-43.688 54.421Q-43.421 54.421-43.185 54.293Q-42.949 54.165-42.804 53.942Q-42.659 53.720-42.659 53.454Q-42.659 53.231-42.765 53.035Q-42.871 52.838-43.052 52.703Q-43.233 52.568-43.459 52.517L-44.487 52.285Q-44.798 52.213-45.058 52.027Q-45.318 51.840-45.470 51.569Q-45.622 51.297-45.622 50.982Q-45.622 50.596-45.409 50.289Q-45.195 49.981-44.848 49.810Q-44.501 49.639-44.122 49.639Q-43.893 49.639-43.664 49.692Q-43.435 49.745-43.236 49.853Q-43.038 49.960-42.884 50.124L-42.590 49.684Q-42.567 49.639-42.526 49.639L-42.478 49.639Q-42.447 49.639-42.425 49.665Q-42.403 49.690-42.403 49.718L-42.403 51.293Q-42.403 51.314-42.426 51.341Q-42.450 51.369-42.478 51.369L-42.590 51.369Q-42.652 51.369-42.666 51.293Q-42.707 50.880-42.888 50.560Q-43.069 50.241-43.380 50.066Q-43.691 49.892-44.122 49.892Q-44.371 49.892-44.611 50.003Q-44.850 50.114-45 50.312Q-45.151 50.511-45.151 50.774Q-45.151 50.986-45.043 51.167Q-44.935 51.348-44.759 51.468Q-44.583 51.587-44.375 51.628L-43.346 51.857Q-43.028 51.929-42.761 52.134Q-42.495 52.339-42.343 52.633Q-42.191 52.927-42.191 53.259Q-42.191 53.652-42.396 53.987Q-42.601 54.322-42.946 54.511Q-43.291 54.701-43.688 54.701Q-44.108 54.701-44.487 54.588Q-44.867 54.476-45.137 54.226L-45.431 54.660Q-45.458 54.701-45.496 54.701L-45.544 54.701Q-45.571 54.701-45.597 54.676Q-45.622 54.650-45.622 54.623M-37.474 54.561L-40.212 54.561L-40.212 54.281Q-39.863 54.281-39.526 54.245Q-39.190 54.209-39.190 54.072L-39.190 50.271Q-39.190 50.128-39.278 50.094Q-39.367 50.060-39.552 50.060L-39.911 50.060Q-40.212 50.060-40.427 50.107Q-40.642 50.155-40.799 50.312Q-40.936 50.446-40.996 50.724Q-41.056 51.003-41.093 51.416L-41.360 51.416L-41.213 49.779L-36.479 49.779L-36.332 51.416L-36.599 51.416Q-36.636 51.003-36.693 50.726Q-36.749 50.449-36.893 50.312Q-37.053 50.152-37.265 50.106Q-37.477 50.060-37.781 50.060L-38.133 50.060Q-38.318 50.060-38.407 50.094Q-38.496 50.128-38.496 50.271L-38.496 54.072Q-38.496 54.209-38.159 54.245Q-37.822 54.281-37.474 54.281\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath d=\"M-38.144 48.871a2.276 2.276 0 1 0-4.552 0 2.276 2.276 0 0 0 4.552 0Zm-2.276 0\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(13.492 -3.26)\">\u003Cpath d=\"M-45.797 54.554L-45.797 53.491Q-45.797 53.467-45.769 53.440Q-45.742 53.413-45.718 53.413L-45.609 53.413Q-45.544 53.413-45.530 53.471Q-45.434 53.905-45.188 54.156Q-44.942 54.407-44.528 54.407Q-44.187 54.407-43.934 54.274Q-43.681 54.141-43.681 53.833Q-43.681 53.676-43.775 53.561Q-43.869 53.447-44.007 53.378Q-44.146 53.310-44.313 53.272L-44.894 53.173Q-45.250 53.105-45.523 52.884Q-45.797 52.664-45.797 52.322Q-45.797 52.073-45.685 51.898Q-45.574 51.724-45.388 51.625Q-45.202 51.526-44.986 51.483Q-44.771 51.440-44.528 51.440Q-44.115 51.440-43.835 51.622L-43.619 51.447Q-43.609 51.444-43.602 51.442Q-43.595 51.440-43.585 51.440L-43.534 51.440Q-43.507 51.440-43.483 51.464Q-43.459 51.488-43.459 51.516L-43.459 52.363Q-43.459 52.384-43.483 52.411Q-43.507 52.438-43.534 52.438L-43.647 52.438Q-43.674 52.438-43.700 52.413Q-43.725 52.387-43.725 52.363Q-43.725 52.127-43.831 51.963Q-43.937 51.799-44.120 51.717Q-44.303 51.635-44.535 51.635Q-44.863 51.635-45.120 51.738Q-45.376 51.840-45.376 52.117Q-45.376 52.312-45.193 52.421Q-45.010 52.531-44.781 52.572L-44.207 52.678Q-43.961 52.726-43.747 52.854Q-43.534 52.982-43.397 53.185Q-43.260 53.389-43.260 53.638Q-43.260 54.151-43.626 54.390Q-43.992 54.629-44.528 54.629Q-45.024 54.629-45.356 54.335L-45.622 54.609Q-45.643 54.629-45.670 54.629L-45.718 54.629Q-45.742 54.629-45.769 54.602Q-45.797 54.575-45.797 54.554M-42.673 53.026Q-42.673 52.705-42.548 52.416Q-42.423 52.127-42.197 51.904Q-41.972 51.680-41.676 51.560Q-41.381 51.440-41.063 51.440Q-40.735 51.440-40.473 51.540Q-40.212 51.639-40.036 51.821Q-39.860 52.004-39.766 52.262Q-39.672 52.520-39.672 52.852Q-39.672 52.944-39.754 52.965L-42.009 52.965L-42.009 53.026Q-42.009 53.614-41.726 53.997Q-41.442 54.380-40.875 54.380Q-40.553 54.380-40.285 54.187Q-40.017 53.994-39.928 53.679Q-39.921 53.638-39.846 53.624L-39.754 53.624Q-39.672 53.648-39.672 53.720Q-39.672 53.727-39.678 53.754Q-39.791 54.151-40.162 54.390Q-40.533 54.629-40.957 54.629Q-41.394 54.629-41.794 54.421Q-42.194 54.212-42.433 53.845Q-42.673 53.478-42.673 53.026M-42.003 52.756L-40.188 52.756Q-40.188 52.479-40.285 52.227Q-40.382 51.974-40.581 51.818Q-40.779 51.663-41.063 51.663Q-41.340 51.663-41.553 51.821Q-41.767 51.980-41.885 52.235Q-42.003 52.490-42.003 52.756M-37.416 54.561L-39.019 54.561L-39.019 54.281Q-38.793 54.281-38.644 54.247Q-38.496 54.212-38.496 54.072L-38.496 50.453Q-38.496 50.183-38.603 50.121Q-38.711 50.060-39.019 50.060L-39.019 49.779L-37.942 49.704L-37.942 54.072Q-37.942 54.209-37.792 54.245Q-37.641 54.281-37.416 54.281L-37.416 54.561M-36.862 53.026Q-36.862 52.705-36.737 52.416Q-36.612 52.127-36.387 51.904Q-36.161 51.680-35.866 51.560Q-35.570 51.440-35.252 51.440Q-34.924 51.440-34.663 51.540Q-34.401 51.639-34.225 51.821Q-34.049 52.004-33.955 52.262Q-33.861 52.520-33.861 52.852Q-33.861 52.944-33.943 52.965L-36.199 52.965L-36.199 53.026Q-36.199 53.614-35.915 53.997Q-35.632 54.380-35.064 54.380Q-34.743 54.380-34.475 54.187Q-34.206 53.994-34.117 53.679Q-34.111 53.638-34.035 53.624L-33.943 53.624Q-33.861 53.648-33.861 53.720Q-33.861 53.727-33.868 53.754Q-33.981 54.151-34.351 54.390Q-34.722 54.629-35.146 54.629Q-35.584 54.629-35.984 54.421Q-36.383 54.212-36.623 53.845Q-36.862 53.478-36.862 53.026M-36.192 52.756L-34.377 52.756Q-34.377 52.479-34.475 52.227Q-34.572 51.974-34.770 51.818Q-34.968 51.663-35.252 51.663Q-35.529 51.663-35.743 51.821Q-35.956 51.980-36.074 52.235Q-36.192 52.490-36.192 52.756M-33.273 53.050Q-33.273 52.722-33.138 52.421Q-33.003 52.121-32.767 51.900Q-32.531 51.680-32.227 51.560Q-31.923 51.440-31.598 51.440Q-31.092 51.440-30.744 51.543Q-30.395 51.645-30.395 52.021Q-30.395 52.168-30.493 52.269Q-30.590 52.370-30.737 52.370Q-30.891 52.370-30.990 52.271Q-31.089 52.172-31.089 52.021Q-31.089 51.833-30.949 51.741Q-31.151 51.690-31.591 51.690Q-31.947 51.690-32.176 51.886Q-32.405 52.083-32.506 52.392Q-32.607 52.702-32.607 53.050Q-32.607 53.399-32.480 53.705Q-32.354 54.011-32.099 54.195Q-31.844 54.380-31.489 54.380Q-31.267 54.380-31.082 54.296Q-30.898 54.212-30.763 54.057Q-30.628 53.901-30.569 53.693Q-30.556 53.638-30.501 53.638L-30.388 53.638Q-30.358 53.638-30.335 53.662Q-30.313 53.686-30.313 53.720L-30.313 53.741Q-30.399 54.028-30.587 54.226Q-30.775 54.424-31.039 54.527Q-31.304 54.629-31.598 54.629Q-32.029 54.629-32.417 54.423Q-32.805 54.216-33.039 53.853Q-33.273 53.491-33.273 53.050M-29.199 53.720L-29.199 51.823L-29.838 51.823L-29.838 51.601Q-29.520 51.601-29.303 51.391Q-29.086 51.181-28.985 50.871Q-28.884 50.562-28.884 50.254L-28.618 50.254L-28.618 51.543L-27.541 51.543L-27.541 51.823L-28.618 51.823L-28.618 53.707Q-28.618 53.983-28.514 54.182Q-28.409 54.380-28.150 54.380Q-27.992 54.380-27.886 54.276Q-27.780 54.171-27.731 54.018Q-27.681 53.864-27.681 53.707L-27.681 53.293L-27.415 53.293L-27.415 53.720Q-27.415 53.946-27.514 54.156Q-27.613 54.366-27.798 54.498Q-27.982 54.629-28.211 54.629Q-28.649 54.629-28.924 54.392Q-29.199 54.154-29.199 53.720M-26.646 53.026Q-26.646 52.705-26.521 52.416Q-26.396 52.127-26.171 51.904Q-25.945 51.680-25.649 51.560Q-25.354 51.440-25.036 51.440Q-24.708 51.440-24.446 51.540Q-24.185 51.639-24.009 51.821Q-23.833 52.004-23.739 52.262Q-23.645 52.520-23.645 52.852Q-23.645 52.944-23.727 52.965L-25.983 52.965L-25.983 53.026Q-25.983 53.614-25.699 53.997Q-25.415 54.380-24.848 54.380Q-24.527 54.380-24.258 54.187Q-23.990 53.994-23.901 53.679Q-23.894 53.638-23.819 53.624L-23.727 53.624Q-23.645 53.648-23.645 53.720Q-23.645 53.727-23.652 53.754Q-23.764 54.151-24.135 54.390Q-24.506 54.629-24.930 54.629Q-25.367 54.629-25.767 54.421Q-26.167 54.212-26.406 53.845Q-26.646 53.478-26.646 53.026M-25.976 52.756L-24.161 52.756Q-24.161 52.479-24.258 52.227Q-24.356 51.974-24.554 51.818Q-24.752 51.663-25.036 51.663Q-25.313 51.663-25.526 51.821Q-25.740 51.980-25.858 52.235Q-25.976 52.490-25.976 52.756M-23.057 53.050Q-23.057 52.712-22.917 52.421Q-22.777 52.131-22.532 51.917Q-22.288 51.704-21.984 51.589Q-21.679 51.475-21.355 51.475Q-21.085 51.475-20.821 51.574Q-20.558 51.673-20.367 51.851L-20.367 50.453Q-20.367 50.183-20.475 50.121Q-20.582 50.060-20.893 50.060L-20.893 49.779L-19.817 49.704L-19.817 53.888Q-19.817 54.076-19.762 54.159Q-19.707 54.243-19.606 54.262Q-19.506 54.281-19.290 54.281L-19.290 54.561L-20.398 54.629L-20.398 54.212Q-20.815 54.629-21.440 54.629Q-21.871 54.629-22.243 54.417Q-22.616 54.206-22.836 53.845Q-23.057 53.484-23.057 53.050M-21.382 54.407Q-21.173 54.407-20.987 54.335Q-20.801 54.264-20.647 54.127Q-20.493 53.990-20.398 53.812L-20.398 52.203Q-20.483 52.056-20.628 51.936Q-20.774 51.816-20.943 51.757Q-21.112 51.697-21.293 51.697Q-21.854 51.697-22.122 52.086Q-22.390 52.476-22.390 53.057Q-22.390 53.628-22.156 54.018Q-21.922 54.407-21.382 54.407\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(13.492 -3.26)\">\u003Cpath d=\"M-15.847 53.833Q-15.847 53.501-15.624 53.274Q-15.400 53.047-15.056 52.919Q-14.713 52.790-14.340 52.738Q-13.968 52.685-13.663 52.685L-13.663 52.432Q-13.663 52.227-13.771 52.047Q-13.879 51.868-14.060 51.765Q-14.241 51.663-14.449 51.663Q-14.856 51.663-15.092 51.755Q-15.003 51.792-14.957 51.876Q-14.911 51.960-14.911 52.062Q-14.911 52.158-14.957 52.237Q-15.003 52.315-15.084 52.360Q-15.164 52.404-15.253 52.404Q-15.403 52.404-15.504 52.307Q-15.605 52.209-15.605 52.062Q-15.605 51.440-14.449 51.440Q-14.238 51.440-13.988 51.504Q-13.739 51.567-13.537 51.686Q-13.335 51.806-13.209 51.991Q-13.082 52.175-13.082 52.418L-13.082 53.994Q-13.082 54.110-13.021 54.206Q-12.959 54.301-12.846 54.301Q-12.737 54.301-12.672 54.207Q-12.607 54.113-12.607 53.994L-12.607 53.546L-12.341 53.546L-12.341 53.994Q-12.341 54.264-12.568 54.429Q-12.795 54.595-13.075 54.595Q-13.284 54.595-13.421 54.441Q-13.557 54.288-13.581 54.072Q-13.728 54.339-14.010 54.484Q-14.292 54.629-14.617 54.629Q-14.894 54.629-15.178 54.554Q-15.461 54.479-15.654 54.300Q-15.847 54.120-15.847 53.833M-15.232 53.833Q-15.232 54.007-15.131 54.137Q-15.031 54.267-14.875 54.337Q-14.720 54.407-14.555 54.407Q-14.337 54.407-14.128 54.310Q-13.920 54.212-13.792 54.031Q-13.663 53.850-13.663 53.624L-13.663 52.896Q-13.988 52.896-14.354 52.987Q-14.720 53.078-14.976 53.290Q-15.232 53.501-15.232 53.833M-11.924 53.050Q-11.924 52.722-11.789 52.421Q-11.654 52.121-11.418 51.900Q-11.182 51.680-10.878 51.560Q-10.574 51.440-10.249 51.440Q-9.743 51.440-9.394 51.543Q-9.046 51.645-9.046 52.021Q-9.046 52.168-9.143 52.269Q-9.241 52.370-9.387 52.370Q-9.541 52.370-9.640 52.271Q-9.740 52.172-9.740 52.021Q-9.740 51.833-9.599 51.741Q-9.801 51.690-10.242 51.690Q-10.597 51.690-10.826 51.886Q-11.055 52.083-11.156 52.392Q-11.257 52.702-11.257 53.050Q-11.257 53.399-11.131 53.705Q-11.004 54.011-10.750 54.195Q-10.495 54.380-10.139 54.380Q-9.917 54.380-9.733 54.296Q-9.548 54.212-9.413 54.057Q-9.278 53.901-9.220 53.693Q-9.206 53.638-9.152 53.638L-9.039 53.638Q-9.008 53.638-8.986 53.662Q-8.964 53.686-8.964 53.720L-8.964 53.741Q-9.049 54.028-9.237 54.226Q-9.425 54.424-9.690 54.527Q-9.955 54.629-10.249 54.629Q-10.679 54.629-11.067 54.423Q-11.455 54.216-11.689 53.853Q-11.924 53.491-11.924 53.050M-7.849 53.720L-7.849 51.823L-8.489 51.823L-8.489 51.601Q-8.171 51.601-7.954 51.391Q-7.737 51.181-7.636 50.871Q-7.535 50.562-7.535 50.254L-7.268 50.254L-7.268 51.543L-6.192 51.543L-6.192 51.823L-7.268 51.823L-7.268 53.707Q-7.268 53.983-7.164 54.182Q-7.060 54.380-6.800 54.380Q-6.643 54.380-6.537 54.276Q-6.431 54.171-6.381 54.018Q-6.332 53.864-6.332 53.707L-6.332 53.293L-6.065 53.293L-6.065 53.720Q-6.065 53.946-6.164 54.156Q-6.263 54.366-6.448 54.498Q-6.633 54.629-6.862 54.629Q-7.299 54.629-7.574 54.392Q-7.849 54.154-7.849 53.720M-3.638 54.561L-5.190 54.561L-5.190 54.281Q-4.965 54.281-4.816 54.247Q-4.667 54.212-4.667 54.072L-4.667 52.223Q-4.667 52.035-4.715 51.951Q-4.763 51.868-4.860 51.849Q-4.958 51.830-5.170 51.830L-5.170 51.550L-4.114 51.475L-4.114 54.072Q-4.114 54.212-3.982 54.247Q-3.850 54.281-3.638 54.281L-3.638 54.561M-4.910 50.254Q-4.910 50.083-4.787 49.964Q-4.664 49.844-4.493 49.844Q-4.325 49.844-4.202 49.964Q-4.079 50.083-4.079 50.254Q-4.079 50.429-4.202 50.552Q-4.325 50.675-4.493 50.675Q-4.664 50.675-4.787 50.552Q-4.910 50.429-4.910 50.254M-3.033 53.078Q-3.033 52.736-2.898 52.437Q-2.763 52.138-2.524 51.914Q-2.285 51.690-1.967 51.565Q-1.649 51.440-1.318 51.440Q-0.873 51.440-0.473 51.656Q-0.074 51.871 0.161 52.249Q0.395 52.626 0.395 53.078Q0.395 53.419 0.253 53.703Q0.111 53.987-0.133 54.194Q-0.378 54.400-0.687 54.515Q-0.996 54.629-1.318 54.629Q-1.748 54.629-2.150 54.428Q-2.552 54.226-2.793 53.874Q-3.033 53.522-3.033 53.078M-1.318 54.380Q-0.716 54.380-0.492 54.002Q-0.268 53.624-0.268 52.992Q-0.268 52.380-0.502 52.021Q-0.737 51.663-1.318 51.663Q-2.370 51.663-2.370 52.992Q-2.370 53.624-2.145 54.002Q-1.919 54.380-1.318 54.380M2.671 54.561L1.037 54.561L1.037 54.281Q1.266 54.281 1.415 54.247Q1.564 54.212 1.564 54.072L1.564 52.223Q1.564 51.953 1.456 51.892Q1.348 51.830 1.037 51.830L1.037 51.550L2.097 51.475L2.097 52.124Q2.268 51.816 2.572 51.645Q2.876 51.475 3.221 51.475Q3.727 51.475 4.011 51.698Q4.295 51.922 4.295 52.418L4.295 54.072Q4.295 54.209 4.443 54.245Q4.592 54.281 4.818 54.281L4.818 54.561L3.187 54.561L3.187 54.281Q3.416 54.281 3.565 54.247Q3.714 54.212 3.714 54.072L3.714 52.432Q3.714 52.097 3.594 51.897Q3.474 51.697 3.160 51.697Q2.890 51.697 2.656 51.833Q2.422 51.970 2.283 52.204Q2.145 52.438 2.145 52.712L2.145 54.072Q2.145 54.209 2.295 54.245Q2.446 54.281 2.671 54.281\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath d=\"M40.955 48.871a1.707 1.707 0 1 0-3.414 0 1.707 1.707 0 0 0 3.414 0Zm-1.707 0\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(93.159 -3.94)\">\u003Cpath d=\"M-45.222 53.727L-45.222 52.223Q-45.222 51.953-45.330 51.892Q-45.438 51.830-45.749 51.830L-45.749 51.550L-44.641 51.475L-44.641 53.707L-44.641 53.727Q-44.641 54.007-44.590 54.151Q-44.539 54.294-44.397 54.351Q-44.255 54.407-43.968 54.407Q-43.715 54.407-43.510 54.267Q-43.305 54.127-43.189 53.901Q-43.072 53.676-43.072 53.426L-43.072 52.223Q-43.072 51.953-43.180 51.892Q-43.288 51.830-43.599 51.830L-43.599 51.550L-42.491 51.475L-42.491 53.888Q-42.491 54.079-42.438 54.161Q-42.385 54.243-42.285 54.262Q-42.184 54.281-41.968 54.281L-41.968 54.561L-43.045 54.629L-43.045 54.065Q-43.154 54.247-43.300 54.370Q-43.445 54.493-43.631 54.561Q-43.818 54.629-44.019 54.629Q-45.222 54.629-45.222 53.727M-39.699 54.561L-41.333 54.561L-41.333 54.281Q-41.104 54.281-40.955 54.247Q-40.806 54.212-40.806 54.072L-40.806 52.223Q-40.806 51.953-40.914 51.892Q-41.022 51.830-41.333 51.830L-41.333 51.550L-40.273 51.475L-40.273 52.124Q-40.102 51.816-39.798 51.645Q-39.494 51.475-39.149 51.475Q-38.643 51.475-38.359 51.698Q-38.075 51.922-38.075 52.418L-38.075 54.072Q-38.075 54.209-37.927 54.245Q-37.778 54.281-37.552 54.281L-37.552 54.561L-39.183 54.561L-39.183 54.281Q-38.954 54.281-38.805 54.247Q-38.656 54.212-38.656 54.072L-38.656 52.432Q-38.656 52.097-38.776 51.897Q-38.896 51.697-39.210 51.697Q-39.480 51.697-39.714 51.833Q-39.948 51.970-40.087 52.204Q-40.225 52.438-40.225 52.712L-40.225 54.072Q-40.225 54.209-40.075 54.245Q-39.924 54.281-39.699 54.281L-39.699 54.561M-36.965 54.554L-36.965 53.491Q-36.965 53.467-36.937 53.440Q-36.910 53.413-36.886 53.413L-36.777 53.413Q-36.712 53.413-36.698 53.471Q-36.602 53.905-36.356 54.156Q-36.110 54.407-35.696 54.407Q-35.355 54.407-35.102 54.274Q-34.849 54.141-34.849 53.833Q-34.849 53.676-34.943 53.561Q-35.037 53.447-35.175 53.378Q-35.314 53.310-35.481 53.272L-36.062 53.173Q-36.418 53.105-36.691 52.884Q-36.965 52.664-36.965 52.322Q-36.965 52.073-36.853 51.898Q-36.742 51.724-36.556 51.625Q-36.370 51.526-36.154 51.483Q-35.939 51.440-35.696 51.440Q-35.283 51.440-35.003 51.622L-34.787 51.447Q-34.777 51.444-34.770 51.442Q-34.763 51.440-34.753 51.440L-34.702 51.440Q-34.674 51.440-34.651 51.464Q-34.627 51.488-34.627 51.516L-34.627 52.363Q-34.627 52.384-34.651 52.411Q-34.674 52.438-34.702 52.438L-34.815 52.438Q-34.842 52.438-34.868 52.413Q-34.893 52.387-34.893 52.363Q-34.893 52.127-34.999 51.963Q-35.105 51.799-35.288 51.717Q-35.471 51.635-35.703 51.635Q-36.031 51.635-36.288 51.738Q-36.544 51.840-36.544 52.117Q-36.544 52.312-36.361 52.421Q-36.178 52.531-35.949 52.572L-35.375 52.678Q-35.129 52.726-34.915 52.854Q-34.702 52.982-34.565 53.185Q-34.428 53.389-34.428 53.638Q-34.428 54.151-34.794 54.390Q-35.160 54.629-35.696 54.629Q-36.192 54.629-36.524 54.335L-36.790 54.609Q-36.811 54.629-36.838 54.629L-36.886 54.629Q-36.910 54.629-36.937 54.602Q-36.965 54.575-36.965 54.554M-33.840 53.026Q-33.840 52.705-33.716 52.416Q-33.591 52.127-33.365 51.904Q-33.140 51.680-32.844 51.560Q-32.548 51.440-32.231 51.440Q-31.903 51.440-31.641 51.540Q-31.380 51.639-31.204 51.821Q-31.028 52.004-30.934 52.262Q-30.840 52.520-30.840 52.852Q-30.840 52.944-30.922 52.965L-33.177 52.965L-33.177 53.026Q-33.177 53.614-32.894 53.997Q-32.610 54.380-32.043 54.380Q-31.721 54.380-31.453 54.187Q-31.185 53.994-31.096 53.679Q-31.089 53.638-31.014 53.624L-30.922 53.624Q-30.840 53.648-30.840 53.720Q-30.840 53.727-30.846 53.754Q-30.959 54.151-31.330 54.390Q-31.701 54.629-32.125 54.629Q-32.562 54.629-32.962 54.421Q-33.362 54.212-33.601 53.845Q-33.840 53.478-33.840 53.026M-33.171 52.756L-31.356 52.756Q-31.356 52.479-31.453 52.227Q-31.550 51.974-31.749 51.818Q-31.947 51.663-32.231 51.663Q-32.507 51.663-32.721 51.821Q-32.935 51.980-33.053 52.235Q-33.171 52.490-33.171 52.756M-28.584 54.561L-30.187 54.561L-30.187 54.281Q-29.961 54.281-29.812 54.247Q-29.664 54.212-29.664 54.072L-29.664 50.453Q-29.664 50.183-29.771 50.121Q-29.879 50.060-30.187 50.060L-30.187 49.779L-29.110 49.704L-29.110 54.072Q-29.110 54.209-28.960 54.245Q-28.809 54.281-28.584 54.281L-28.584 54.561M-28.030 53.026Q-28.030 52.705-27.905 52.416Q-27.780 52.127-27.555 51.904Q-27.329 51.680-27.034 51.560Q-26.738 51.440-26.420 51.440Q-26.092 51.440-25.830 51.540Q-25.569 51.639-25.393 51.821Q-25.217 52.004-25.123 52.262Q-25.029 52.520-25.029 52.852Q-25.029 52.944-25.111 52.965L-27.367 52.965L-27.367 53.026Q-27.367 53.614-27.083 53.997Q-26.799 54.380-26.232 54.380Q-25.911 54.380-25.642 54.187Q-25.374 53.994-25.285 53.679Q-25.278 53.638-25.203 53.624L-25.111 53.624Q-25.029 53.648-25.029 53.720Q-25.029 53.727-25.036 53.754Q-25.149 54.151-25.519 54.390Q-25.890 54.629-26.314 54.629Q-26.752 54.629-27.152 54.421Q-27.551 54.212-27.791 53.845Q-28.030 53.478-28.030 53.026M-27.360 52.756L-25.545 52.756Q-25.545 52.479-25.642 52.227Q-25.740 51.974-25.938 51.818Q-26.136 51.663-26.420 51.663Q-26.697 51.663-26.911 51.821Q-27.124 51.980-27.242 52.235Q-27.360 52.490-27.360 52.756M-24.441 53.050Q-24.441 52.722-24.306 52.421Q-24.171 52.121-23.935 51.900Q-23.699 51.680-23.395 51.560Q-23.091 51.440-22.766 51.440Q-22.260 51.440-21.912 51.543Q-21.563 51.645-21.563 52.021Q-21.563 52.168-21.661 52.269Q-21.758 52.370-21.905 52.370Q-22.059 52.370-22.158 52.271Q-22.257 52.172-22.257 52.021Q-22.257 51.833-22.117 51.741Q-22.319 51.690-22.759 51.690Q-23.115 51.690-23.344 51.886Q-23.573 52.083-23.674 52.392Q-23.775 52.702-23.775 53.050Q-23.775 53.399-23.648 53.705Q-23.522 54.011-23.267 54.195Q-23.012 54.380-22.657 54.380Q-22.435 54.380-22.250 54.296Q-22.066 54.212-21.931 54.057Q-21.796 53.901-21.737 53.693Q-21.724 53.638-21.669 53.638L-21.556 53.638Q-21.526 53.638-21.503 53.662Q-21.481 53.686-21.481 53.720L-21.481 53.741Q-21.567 54.028-21.755 54.226Q-21.943 54.424-22.207 54.527Q-22.472 54.629-22.766 54.629Q-23.197 54.629-23.585 54.423Q-23.973 54.216-24.207 53.853Q-24.441 53.491-24.441 53.050M-20.367 53.720L-20.367 51.823L-21.006 51.823L-21.006 51.601Q-20.688 51.601-20.471 51.391Q-20.254 51.181-20.153 50.871Q-20.052 50.562-20.052 50.254L-19.786 50.254L-19.786 51.543L-18.709 51.543L-18.709 51.823L-19.786 51.823L-19.786 53.707Q-19.786 53.983-19.682 54.182Q-19.577 54.380-19.318 54.380Q-19.160 54.380-19.054 54.276Q-18.948 54.171-18.899 54.018Q-18.849 53.864-18.849 53.707L-18.849 53.293L-18.583 53.293L-18.583 53.720Q-18.583 53.946-18.682 54.156Q-18.781 54.366-18.965 54.498Q-19.150 54.629-19.379 54.629Q-19.817 54.629-20.092 54.392Q-20.367 54.154-20.367 53.720M-17.814 53.026Q-17.814 52.705-17.689 52.416Q-17.564 52.127-17.339 51.904Q-17.113 51.680-16.817 51.560Q-16.522 51.440-16.204 51.440Q-15.876 51.440-15.614 51.540Q-15.353 51.639-15.177 51.821Q-15.001 52.004-14.907 52.262Q-14.813 52.520-14.813 52.852Q-14.813 52.944-14.895 52.965L-17.151 52.965L-17.151 53.026Q-17.151 53.614-16.867 53.997Q-16.583 54.380-16.016 54.380Q-15.694 54.380-15.426 54.187Q-15.158 53.994-15.069 53.679Q-15.062 53.638-14.987 53.624L-14.895 53.624Q-14.813 53.648-14.813 53.720Q-14.813 53.727-14.819 53.754Q-14.932 54.151-15.303 54.390Q-15.674 54.629-16.098 54.629Q-16.535 54.629-16.935 54.421Q-17.335 54.212-17.574 53.845Q-17.814 53.478-17.814 53.026M-17.144 52.756L-15.329 52.756Q-15.329 52.479-15.426 52.227Q-15.524 51.974-15.722 51.818Q-15.920 51.663-16.204 51.663Q-16.481 51.663-16.694 51.821Q-16.908 51.980-17.026 52.235Q-17.144 52.490-17.144 52.756M-14.225 53.050Q-14.225 52.712-14.085 52.421Q-13.944 52.131-13.700 51.917Q-13.456 51.704-13.152 51.589Q-12.847 51.475-12.523 51.475Q-12.253 51.475-11.989 51.574Q-11.726 51.673-11.535 51.851L-11.535 50.453Q-11.535 50.183-11.642 50.121Q-11.750 50.060-12.061 50.060L-12.061 49.779L-10.985 49.704L-10.985 53.888Q-10.985 54.076-10.930 54.159Q-10.875 54.243-10.774 54.262Q-10.673 54.281-10.458 54.281L-10.458 54.561L-11.566 54.629L-11.566 54.212Q-11.983 54.629-12.608 54.629Q-13.039 54.629-13.411 54.417Q-13.784 54.206-14.004 53.845Q-14.225 53.484-14.225 53.050M-12.550 54.407Q-12.341 54.407-12.155 54.335Q-11.969 54.264-11.815 54.127Q-11.661 53.990-11.566 53.812L-11.566 52.203Q-11.651 52.056-11.796 51.936Q-11.942 51.816-12.111 51.757Q-12.280 51.697-12.461 51.697Q-13.022 51.697-13.290 52.086Q-13.558 52.476-13.558 53.057Q-13.558 53.628-13.324 54.018Q-13.090 54.407-12.550 54.407M-9.310 55.791Q-9.310 55.757-9.282 55.730Q-9.012 55.501-8.864 55.178Q-8.715 54.855-8.715 54.499L-8.715 54.462Q-8.824 54.561-8.988 54.561Q-9.170 54.561-9.289 54.441Q-9.409 54.322-9.409 54.141Q-9.409 53.966-9.289 53.847Q-9.170 53.727-8.988 53.727Q-8.732 53.727-8.612 53.966Q-8.493 54.206-8.493 54.499Q-8.493 54.899-8.662 55.270Q-8.831 55.641-9.129 55.897Q-9.159 55.918-9.187 55.918Q-9.228 55.918-9.269 55.877Q-9.310 55.836-9.310 55.791\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(93.159 -3.94)\">\u003Cpath d=\"M-3.421 54.534L-4.402 52.035Q-4.463 51.892-4.581 51.857Q-4.699 51.823-4.915 51.823L-4.915 51.543L-3.435 51.543L-3.435 51.823Q-3.814 51.823-3.814 51.984Q-3.814 51.994-3.800 52.035L-3.086 53.867L-2.413 52.162Q-2.443 52.090-2.443 52.062Q-2.443 52.035-2.471 52.035Q-2.532 51.888-2.650 51.856Q-2.768 51.823-2.980 51.823L-2.980 51.543L-1.582 51.543L-1.582 51.823Q-1.958 51.823-1.958 51.984Q-1.958 52.015-1.951 52.035L-1.196 53.973L-0.509 52.223Q-0.488 52.172-0.488 52.117Q-0.488 51.977-0.601 51.900Q-0.714 51.823-0.854 51.823L-0.854 51.543L0.366 51.543L0.366 51.823Q0.161 51.823 0.006 51.929Q-0.150 52.035-0.222 52.223L-1.127 54.534Q-1.162 54.629-1.274 54.629L-1.343 54.629Q-1.452 54.629-1.490 54.534L-2.272 52.531L-3.059 54.534Q-3.093 54.629-3.206 54.629L-3.274 54.629Q-3.383 54.629-3.421 54.534\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(93.159 -3.94)\">\u003Cpath d=\"M0.643 53.026Q0.643 52.705 0.768 52.416Q0.893 52.127 1.119 51.904Q1.344 51.680 1.640 51.560Q1.935 51.440 2.253 51.440Q2.581 51.440 2.843 51.540Q3.104 51.639 3.280 51.821Q3.456 52.004 3.550 52.262Q3.644 52.520 3.644 52.852Q3.644 52.944 3.562 52.965L1.307 52.965L1.307 53.026Q1.307 53.614 1.590 53.997Q1.874 54.380 2.441 54.380Q2.763 54.380 3.031 54.187Q3.299 53.994 3.388 53.679Q3.395 53.638 3.470 53.624L3.562 53.624Q3.644 53.648 3.644 53.720Q3.644 53.727 3.638 53.754Q3.525 54.151 3.154 54.390Q2.783 54.629 2.359 54.629Q1.922 54.629 1.522 54.421Q1.122 54.212 0.883 53.845Q0.643 53.478 0.643 53.026M1.313 52.756L3.128 52.756Q3.128 52.479 3.031 52.227Q2.933 51.974 2.735 51.818Q2.537 51.663 2.253 51.663Q1.976 51.663 1.763 51.821Q1.549 51.980 1.431 52.235Q1.313 52.490 1.313 52.756M5.849 54.561L4.297 54.561L4.297 54.281Q4.523 54.281 4.672 54.247Q4.820 54.212 4.820 54.072L4.820 52.223Q4.820 52.035 4.772 51.951Q4.724 51.868 4.627 51.849Q4.530 51.830 4.318 51.830L4.318 51.550L5.374 51.475L5.374 54.072Q5.374 54.212 5.505 54.247Q5.637 54.281 5.849 54.281L5.849 54.561M4.578 50.254Q4.578 50.083 4.701 49.964Q4.824 49.844 4.995 49.844Q5.162 49.844 5.285 49.964Q5.408 50.083 5.408 50.254Q5.408 50.429 5.285 50.552Q5.162 50.675 4.995 50.675Q4.824 50.675 4.701 50.552Q4.578 50.429 4.578 50.254M6.454 55.094Q6.454 54.848 6.651 54.664Q6.847 54.479 7.103 54.400Q6.967 54.288 6.895 54.127Q6.823 53.966 6.823 53.785Q6.823 53.464 7.035 53.218Q6.700 52.920 6.700 52.510Q6.700 52.049 7.090 51.762Q7.479 51.475 7.958 51.475Q8.430 51.475 8.765 51.721Q8.939 51.567 9.149 51.485Q9.359 51.403 9.588 51.403Q9.752 51.403 9.874 51.510Q9.995 51.618 9.995 51.782Q9.995 51.878 9.923 51.950Q9.851 52.021 9.759 52.021Q9.660 52.021 9.590 51.948Q9.520 51.874 9.520 51.775Q9.520 51.721 9.534 51.690L9.540 51.676Q9.547 51.656 9.556 51.645Q9.564 51.635 9.568 51.628Q9.212 51.628 8.925 51.851Q9.212 52.144 9.212 52.510Q9.212 52.825 9.028 53.057Q8.843 53.290 8.554 53.418Q8.266 53.546 7.958 53.546Q7.756 53.546 7.565 53.496Q7.373 53.447 7.196 53.337Q7.103 53.464 7.103 53.607Q7.103 53.789 7.232 53.924Q7.360 54.059 7.544 54.059L8.177 54.059Q8.624 54.059 8.994 54.130Q9.363 54.202 9.622 54.431Q9.882 54.660 9.882 55.094Q9.882 55.415 9.587 55.617Q9.291 55.819 8.888 55.908Q8.484 55.997 8.170 55.997Q7.852 55.997 7.449 55.908Q7.045 55.819 6.750 55.617Q6.454 55.415 6.454 55.094M6.909 55.094Q6.909 55.323 7.127 55.472Q7.346 55.621 7.638 55.689Q7.931 55.757 8.170 55.757Q8.334 55.757 8.542 55.721Q8.751 55.686 8.958 55.605Q9.164 55.525 9.296 55.397Q9.428 55.269 9.428 55.094Q9.428 54.742 9.047 54.648Q8.665 54.554 8.163 54.554L7.544 54.554Q7.305 54.554 7.107 54.705Q6.909 54.855 6.909 55.094M7.958 53.307Q8.624 53.307 8.624 52.510Q8.624 51.710 7.958 51.710Q7.288 51.710 7.288 52.510Q7.288 53.307 7.958 53.307M12.159 54.561L10.525 54.561L10.525 54.281Q10.754 54.281 10.902 54.247Q11.051 54.212 11.051 54.072L11.051 50.453Q11.051 50.183 10.943 50.121Q10.836 50.060 10.525 50.060L10.525 49.779L11.605 49.704L11.605 52.090Q11.711 51.905 11.889 51.763Q12.066 51.622 12.275 51.548Q12.483 51.475 12.709 51.475Q13.215 51.475 13.498 51.698Q13.782 51.922 13.782 52.418L13.782 54.072Q13.782 54.209 13.931 54.245Q14.079 54.281 14.305 54.281L14.305 54.561L12.675 54.561L12.675 54.281Q12.904 54.281 13.052 54.247Q13.201 54.212 13.201 54.072L13.201 52.432Q13.201 52.097 13.081 51.897Q12.962 51.697 12.647 51.697Q12.377 51.697 12.143 51.833Q11.909 51.970 11.771 52.204Q11.632 52.438 11.632 52.712L11.632 54.072Q11.632 54.209 11.783 54.245Q11.933 54.281 12.159 54.281\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(93.159 -3.94)\">\u003Cpath d=\"M15.218 53.720L15.218 51.823L14.579 51.823L14.579 51.601Q14.897 51.601 15.114 51.391Q15.331 51.181 15.431 50.871Q15.532 50.562 15.532 50.254L15.799 50.254L15.799 51.543L16.876 51.543L16.876 51.823L15.799 51.823L15.799 53.707Q15.799 53.983 15.903 54.182Q16.007 54.380 16.267 54.380Q16.424 54.380 16.530 54.276Q16.636 54.171 16.686 54.018Q16.735 53.864 16.735 53.707L16.735 53.293L17.002 53.293L17.002 53.720Q17.002 53.946 16.903 54.156Q16.804 54.366 16.619 54.498Q16.435 54.629 16.206 54.629Q15.768 54.629 15.493 54.392Q15.218 54.154 15.218 53.720\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(93.159 -3.94)\">\u003Cpath d=\"M22.155 55.918L20.525 55.918L20.525 55.638Q20.754 55.638 20.903 55.603Q21.051 55.569 21.051 55.429L21.051 52.083Q21.051 51.912 20.915 51.871Q20.778 51.830 20.525 51.830L20.525 51.550L21.605 51.475L21.605 51.881Q21.827 51.680 22.114 51.577Q22.402 51.475 22.709 51.475Q23.136 51.475 23.500 51.688Q23.864 51.902 24.078 52.266Q24.292 52.630 24.292 53.050Q24.292 53.495 24.052 53.859Q23.813 54.223 23.420 54.426Q23.027 54.629 22.583 54.629Q22.316 54.629 22.068 54.529Q21.821 54.428 21.633 54.247L21.633 55.429Q21.633 55.566 21.781 55.602Q21.930 55.638 22.155 55.638L22.155 55.918M21.633 52.230L21.633 53.840Q21.766 54.093 22.009 54.250Q22.251 54.407 22.528 54.407Q22.856 54.407 23.109 54.206Q23.362 54.004 23.495 53.686Q23.629 53.368 23.629 53.050Q23.629 52.821 23.564 52.592Q23.499 52.363 23.371 52.165Q23.242 51.967 23.048 51.847Q22.853 51.728 22.620 51.728Q22.326 51.728 22.058 51.857Q21.790 51.987 21.633 52.230M26.544 54.561L24.992 54.561L24.992 54.281Q25.218 54.281 25.367 54.247Q25.515 54.212 25.515 54.072L25.515 52.223Q25.515 52.035 25.468 51.951Q25.420 51.868 25.322 51.849Q25.225 51.830 25.013 51.830L25.013 51.550L26.069 51.475L26.069 54.072Q26.069 54.212 26.201 54.247Q26.332 54.281 26.544 54.281L26.544 54.561M25.273 50.254Q25.273 50.083 25.396 49.964Q25.519 49.844 25.690 49.844Q25.857 49.844 25.980 49.964Q26.103 50.083 26.103 50.254Q26.103 50.429 25.980 50.552Q25.857 50.675 25.690 50.675Q25.519 50.675 25.396 50.552Q25.273 50.429 25.273 50.254M29.320 56.311Q28.769 55.911 28.398 55.356Q28.028 54.800 27.846 54.154Q27.665 53.508 27.665 52.811Q27.665 52.298 27.766 51.803Q27.867 51.307 28.072 50.856Q28.277 50.405 28.590 50.013Q28.903 49.622 29.320 49.318Q29.330 49.314 29.337 49.313Q29.343 49.311 29.354 49.311L29.422 49.311Q29.456 49.311 29.478 49.335Q29.501 49.359 29.501 49.396Q29.501 49.441 29.473 49.458Q29.125 49.759 28.872 50.143Q28.619 50.528 28.467 50.969Q28.315 51.410 28.243 51.866Q28.171 52.322 28.171 52.811Q28.171 53.812 28.480 54.699Q28.790 55.586 29.473 56.171Q29.501 56.188 29.501 56.232Q29.501 56.270 29.478 56.294Q29.456 56.318 29.422 56.318L29.354 56.318Q29.347 56.314 29.338 56.313Q29.330 56.311 29.320 56.311M30.369 53.833Q30.369 53.501 30.593 53.274Q30.817 53.047 31.160 52.919Q31.504 52.790 31.876 52.738Q32.249 52.685 32.553 52.685L32.553 52.432Q32.553 52.227 32.445 52.047Q32.338 51.868 32.156 51.765Q31.975 51.663 31.767 51.663Q31.360 51.663 31.124 51.755Q31.213 51.792 31.259 51.876Q31.305 51.960 31.305 52.062Q31.305 52.158 31.259 52.237Q31.213 52.315 31.133 52.360Q31.052 52.404 30.964 52.404Q30.813 52.404 30.712 52.307Q30.612 52.209 30.612 52.062Q30.612 51.440 31.767 51.440Q31.979 51.440 32.228 51.504Q32.478 51.567 32.679 51.686Q32.881 51.806 33.008 51.991Q33.134 52.175 33.134 52.418L33.134 53.994Q33.134 54.110 33.196 54.206Q33.257 54.301 33.370 54.301Q33.479 54.301 33.544 54.207Q33.609 54.113 33.609 53.994L33.609 53.546L33.876 53.546L33.876 53.994Q33.876 54.264 33.648 54.429Q33.421 54.595 33.141 54.595Q32.932 54.595 32.796 54.441Q32.659 54.288 32.635 54.072Q32.488 54.339 32.206 54.484Q31.924 54.629 31.599 54.629Q31.322 54.629 31.039 54.554Q30.755 54.479 30.562 54.300Q30.369 54.120 30.369 53.833M30.984 53.833Q30.984 54.007 31.085 54.137Q31.186 54.267 31.341 54.337Q31.497 54.407 31.661 54.407Q31.880 54.407 32.088 54.310Q32.297 54.212 32.425 54.031Q32.553 53.850 32.553 53.624L32.553 52.896Q32.228 52.896 31.863 52.987Q31.497 53.078 31.240 53.290Q30.984 53.501 30.984 53.833\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(93.159 -3.94)\">\u003Cpath d=\"M36.956 55.094Q36.956 54.848 37.153 54.664Q37.350 54.479 37.606 54.400Q37.469 54.288 37.397 54.127Q37.326 53.966 37.326 53.785Q37.326 53.464 37.537 53.218Q37.203 52.920 37.203 52.510Q37.203 52.049 37.592 51.762Q37.982 51.475 38.460 51.475Q38.932 51.475 39.267 51.721Q39.441 51.567 39.652 51.485Q39.862 51.403 40.091 51.403Q40.255 51.403 40.376 51.510Q40.497 51.618 40.497 51.782Q40.497 51.878 40.426 51.950Q40.354 52.021 40.262 52.021Q40.162 52.021 40.092 51.948Q40.022 51.874 40.022 51.775Q40.022 51.721 40.036 51.690L40.043 51.676Q40.050 51.656 40.058 51.645Q40.067 51.635 40.070 51.628Q39.715 51.628 39.428 51.851Q39.715 52.144 39.715 52.510Q39.715 52.825 39.530 53.057Q39.346 53.290 39.057 53.418Q38.768 53.546 38.460 53.546Q38.259 53.546 38.067 53.496Q37.876 53.447 37.698 53.337Q37.606 53.464 37.606 53.607Q37.606 53.789 37.734 53.924Q37.862 54.059 38.047 54.059L38.679 54.059Q39.127 54.059 39.496 54.130Q39.865 54.202 40.125 54.431Q40.385 54.660 40.385 55.094Q40.385 55.415 40.089 55.617Q39.793 55.819 39.390 55.908Q38.987 55.997 38.672 55.997Q38.354 55.997 37.951 55.908Q37.548 55.819 37.252 55.617Q36.956 55.415 36.956 55.094M37.411 55.094Q37.411 55.323 37.630 55.472Q37.849 55.621 38.141 55.689Q38.433 55.757 38.672 55.757Q38.836 55.757 39.045 55.721Q39.253 55.686 39.460 55.605Q39.667 55.525 39.798 55.397Q39.930 55.269 39.930 55.094Q39.930 54.742 39.549 54.648Q39.168 54.554 38.665 54.554L38.047 54.554Q37.808 54.554 37.609 54.705Q37.411 54.855 37.411 55.094M38.460 53.307Q39.127 53.307 39.127 52.510Q39.127 51.710 38.460 51.710Q37.790 51.710 37.790 52.510Q37.790 53.307 38.460 53.307M42.596 54.561L41.044 54.561L41.044 54.281Q41.270 54.281 41.419 54.247Q41.567 54.212 41.567 54.072L41.567 52.223Q41.567 52.035 41.519 51.951Q41.472 51.868 41.374 51.849Q41.277 51.830 41.065 51.830L41.065 51.550L42.121 51.475L42.121 54.072Q42.121 54.212 42.253 54.247Q42.384 54.281 42.596 54.281L42.596 54.561M41.325 50.254Q41.325 50.083 41.448 49.964Q41.571 49.844 41.742 49.844Q41.909 49.844 42.032 49.964Q42.155 50.083 42.155 50.254Q42.155 50.429 42.032 50.552Q41.909 50.675 41.742 50.675Q41.571 50.675 41.448 50.552Q41.325 50.429 41.325 50.254M44.831 54.534L43.704 52.035Q43.632 51.888 43.502 51.856Q43.372 51.823 43.143 51.823L43.143 51.543L44.657 51.543L44.657 51.823Q44.305 51.823 44.305 51.970Q44.305 52.015 44.315 52.035L45.180 53.953L45.959 52.223Q45.994 52.155 45.994 52.076Q45.994 51.963 45.910 51.893Q45.826 51.823 45.706 51.823L45.706 51.543L46.903 51.543L46.903 51.823Q46.684 51.823 46.513 51.926Q46.342 52.028 46.253 52.223L45.218 54.534Q45.170 54.629 45.064 54.629L44.985 54.629Q44.879 54.629 44.831 54.534\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(93.159 -3.94)\">\u003Cpath d=\"M47.192 53.026Q47.192 52.705 47.317 52.416Q47.442 52.127 47.668 51.904Q47.893 51.680 48.189 51.560Q48.484 51.440 48.802 51.440Q49.130 51.440 49.392 51.540Q49.653 51.639 49.829 51.821Q50.005 52.004 50.099 52.262Q50.193 52.520 50.193 52.852Q50.193 52.944 50.111 52.965L47.856 52.965L47.856 53.026Q47.856 53.614 48.139 53.997Q48.423 54.380 48.990 54.380Q49.312 54.380 49.580 54.187Q49.848 53.994 49.937 53.679Q49.944 53.638 50.019 53.624L50.111 53.624Q50.193 53.648 50.193 53.720Q50.193 53.727 50.187 53.754Q50.074 54.151 49.703 54.390Q49.332 54.629 48.908 54.629Q48.471 54.629 48.071 54.421Q47.671 54.212 47.432 53.845Q47.192 53.478 47.192 53.026M47.862 52.756L49.677 52.756Q49.677 52.479 49.580 52.227Q49.482 51.974 49.284 51.818Q49.086 51.663 48.802 51.663Q48.525 51.663 48.312 51.821Q48.098 51.980 47.980 52.235Q47.862 52.490 47.862 52.756M52.463 54.561L50.829 54.561L50.829 54.281Q51.058 54.281 51.207 54.247Q51.356 54.212 51.356 54.072L51.356 52.223Q51.356 51.953 51.248 51.892Q51.140 51.830 50.829 51.830L50.829 51.550L51.889 51.475L51.889 52.124Q52.060 51.816 52.364 51.645Q52.668 51.475 53.013 51.475Q53.519 51.475 53.803 51.698Q54.086 51.922 54.086 52.418L54.086 54.072Q54.086 54.209 54.235 54.245Q54.384 54.281 54.609 54.281L54.609 54.561L52.979 54.561L52.979 54.281Q53.208 54.281 53.357 54.247Q53.505 54.212 53.505 54.072L53.505 52.432Q53.505 52.097 53.386 51.897Q53.266 51.697 52.952 51.697Q52.682 51.697 52.448 51.833Q52.213 51.970 52.075 52.204Q51.937 52.438 51.937 52.712L51.937 54.072Q51.937 54.209 52.087 54.245Q52.237 54.281 52.463 54.281\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(93.159 -3.94)\">\u003Cpath d=\"M57.900 54.554L57.900 53.491Q57.900 53.467 57.928 53.440Q57.955 53.413 57.979 53.413L58.088 53.413Q58.153 53.413 58.167 53.471Q58.263 53.905 58.509 54.156Q58.755 54.407 59.169 54.407Q59.510 54.407 59.763 54.274Q60.016 54.141 60.016 53.833Q60.016 53.676 59.922 53.561Q59.828 53.447 59.690 53.378Q59.551 53.310 59.384 53.272L58.803 53.173Q58.447 53.105 58.174 52.884Q57.900 52.664 57.900 52.322Q57.900 52.073 58.012 51.898Q58.123 51.724 58.309 51.625Q58.495 51.526 58.711 51.483Q58.926 51.440 59.169 51.440Q59.582 51.440 59.862 51.622L60.078 51.447Q60.088 51.444 60.095 51.442Q60.102 51.440 60.112 51.440L60.163 51.440Q60.190 51.440 60.214 51.464Q60.238 51.488 60.238 51.516L60.238 52.363Q60.238 52.384 60.214 52.411Q60.190 52.438 60.163 52.438L60.050 52.438Q60.023 52.438 59.997 52.413Q59.972 52.387 59.972 52.363Q59.972 52.127 59.866 51.963Q59.760 51.799 59.577 51.717Q59.394 51.635 59.162 51.635Q58.834 51.635 58.577 51.738Q58.321 51.840 58.321 52.117Q58.321 52.312 58.504 52.421Q58.687 52.531 58.916 52.572L59.490 52.678Q59.736 52.726 59.950 52.854Q60.163 52.982 60.300 53.185Q60.437 53.389 60.437 53.638Q60.437 54.151 60.071 54.390Q59.705 54.629 59.169 54.629Q58.673 54.629 58.341 54.335L58.075 54.609Q58.054 54.629 58.027 54.629L57.979 54.629Q57.955 54.629 57.928 54.602Q57.900 54.575 57.900 54.554M61.387 56.318L61.318 56.318Q61.284 56.318 61.262 56.292Q61.240 56.267 61.240 56.232Q61.240 56.188 61.271 56.171Q61.626 55.867 61.876 55.477Q62.125 55.087 62.277 54.655Q62.429 54.223 62.499 53.754Q62.569 53.286 62.569 52.811Q62.569 52.332 62.499 51.866Q62.429 51.399 62.275 50.964Q62.122 50.528 61.870 50.140Q61.619 49.752 61.271 49.458Q61.240 49.441 61.240 49.396Q61.240 49.362 61.262 49.337Q61.284 49.311 61.318 49.311L61.387 49.311Q61.397 49.311 61.406 49.313Q61.414 49.314 61.424 49.318Q61.968 49.718 62.340 50.271Q62.713 50.825 62.894 51.471Q63.075 52.117 63.075 52.811Q63.075 53.512 62.894 54.159Q62.713 54.807 62.339 55.361Q61.964 55.915 61.424 56.311Q61.414 56.311 61.406 56.313Q61.397 56.314 61.387 56.318\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">Backup diagram for Tree-Backup(lambda). Every backup fans out to all actions at each level: the selected action (solid, continuing down) plus the unselected actions (leaf dots), each weighted by its target-policy probability pi(a|s). No importance sampling is needed — the target policy enters only through these branch weights. Component lengths are weighted 1-lambda, (1-lambda)lambda, ... exactly as before, terminating at the box.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:466.423px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 349.817 105.985\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cpath fill=\"none\" d=\"M-62.07-46.463h85.358V-72.07H-62.07Z\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-25.286 -55.155)\">\u003Cpath d=\"M-18.902-2.299L-18.902-3.872Q-18.902-3.899-18.877-3.925Q-18.851-3.950-18.824-3.950L-18.711-3.950Q-18.683-3.950-18.660-3.923Q-18.636-3.896-18.636-3.872Q-18.636-3.527-18.504-3.263Q-18.372-3-18.143-2.831Q-17.914-2.662-17.612-2.581Q-17.309-2.501-16.968-2.501Q-16.701-2.501-16.465-2.629Q-16.229-2.757-16.084-2.980Q-15.939-3.202-15.939-3.468Q-15.939-3.691-16.045-3.887Q-16.151-4.084-16.332-4.219Q-16.513-4.354-16.739-4.405L-17.767-4.637Q-18.078-4.709-18.338-4.895Q-18.598-5.082-18.750-5.353Q-18.902-5.625-18.902-5.940Q-18.902-6.326-18.689-6.633Q-18.475-6.941-18.128-7.112Q-17.781-7.283-17.402-7.283Q-17.173-7.283-16.944-7.230Q-16.715-7.177-16.516-7.069Q-16.318-6.962-16.164-6.798L-15.870-7.238Q-15.847-7.283-15.806-7.283L-15.758-7.283Q-15.727-7.283-15.705-7.257Q-15.683-7.232-15.683-7.204L-15.683-5.629Q-15.683-5.608-15.706-5.581Q-15.730-5.553-15.758-5.553L-15.870-5.553Q-15.932-5.553-15.946-5.629Q-15.987-6.042-16.168-6.362Q-16.349-6.681-16.660-6.856Q-16.971-7.030-17.402-7.030Q-17.651-7.030-17.891-6.919Q-18.130-6.808-18.280-6.610Q-18.431-6.411-18.431-6.148Q-18.431-5.936-18.323-5.755Q-18.215-5.574-18.039-5.454Q-17.863-5.335-17.655-5.294L-16.626-5.065Q-16.308-4.993-16.041-4.788Q-15.775-4.583-15.623-4.289Q-15.471-3.995-15.471-3.663Q-15.471-3.270-15.676-2.935Q-15.881-2.600-16.226-2.411Q-16.571-2.221-16.968-2.221Q-17.388-2.221-17.767-2.334Q-18.147-2.446-18.417-2.696L-18.711-2.262Q-18.738-2.221-18.776-2.221L-18.824-2.221Q-18.851-2.221-18.877-2.246Q-18.902-2.272-18.902-2.299M-14.602-3.089Q-14.602-3.421-14.379-3.648Q-14.155-3.875-13.811-4.003Q-13.468-4.132-13.095-4.184Q-12.723-4.237-12.418-4.237L-12.418-4.490Q-12.418-4.695-12.526-4.875Q-12.634-5.054-12.815-5.157Q-12.996-5.259-13.204-5.259Q-13.611-5.259-13.847-5.167Q-13.758-5.130-13.712-5.046Q-13.666-4.962-13.666-4.860Q-13.666-4.764-13.712-4.685Q-13.758-4.607-13.839-4.562Q-13.919-4.518-14.008-4.518Q-14.158-4.518-14.259-4.615Q-14.360-4.713-14.360-4.860Q-14.360-5.482-13.204-5.482Q-12.993-5.482-12.743-5.418Q-12.494-5.355-12.292-5.236Q-12.090-5.116-11.964-4.931Q-11.837-4.747-11.837-4.504L-11.837-2.928Q-11.837-2.812-11.776-2.716Q-11.714-2.621-11.601-2.621Q-11.492-2.621-11.427-2.715Q-11.362-2.809-11.362-2.928L-11.362-3.376L-11.096-3.376L-11.096-2.928Q-11.096-2.658-11.323-2.493Q-11.550-2.327-11.830-2.327Q-12.039-2.327-12.176-2.481Q-12.312-2.634-12.336-2.850Q-12.483-2.583-12.765-2.438Q-13.047-2.293-13.372-2.293Q-13.649-2.293-13.933-2.368Q-14.216-2.443-14.409-2.622Q-14.602-2.802-14.602-3.089M-13.987-3.089Q-13.987-2.915-13.886-2.785Q-13.786-2.655-13.630-2.585Q-13.474-2.515-13.310-2.515Q-13.092-2.515-12.883-2.612Q-12.675-2.710-12.547-2.891Q-12.418-3.072-12.418-3.298L-12.418-4.026Q-12.743-4.026-13.109-3.935Q-13.474-3.844-13.731-3.632Q-13.987-3.421-13.987-3.089M-8.929-2.361L-10.665-2.361L-10.665-2.641Q-10.436-2.641-10.287-2.675Q-10.139-2.710-10.139-2.850L-10.139-4.699Q-10.139-4.969-10.246-5.030Q-10.354-5.092-10.665-5.092L-10.665-5.372L-9.636-5.447L-9.636-4.740Q-9.506-5.048-9.264-5.247Q-9.021-5.447-8.703-5.447Q-8.484-5.447-8.313-5.323Q-8.142-5.198-8.142-4.986Q-8.142-4.849-8.242-4.750Q-8.341-4.651-8.474-4.651Q-8.611-4.651-8.710-4.750Q-8.809-4.849-8.809-4.986Q-8.809-5.126-8.710-5.225Q-9-5.225-9.200-5.029Q-9.400-4.832-9.493-4.538Q-9.585-4.244-9.585-3.964L-9.585-2.850Q-9.585-2.641-8.929-2.641L-8.929-2.361M-7.558-2.368L-7.558-3.431Q-7.558-3.455-7.531-3.482Q-7.503-3.509-7.479-3.509L-7.370-3.509Q-7.305-3.509-7.291-3.451Q-7.196-3.017-6.950-2.766Q-6.703-2.515-6.290-2.515Q-5.948-2.515-5.695-2.648Q-5.442-2.781-5.442-3.089Q-5.442-3.246-5.536-3.361Q-5.630-3.475-5.769-3.544Q-5.907-3.612-6.075-3.650L-6.656-3.749Q-7.011-3.817-7.285-4.038Q-7.558-4.258-7.558-4.600Q-7.558-4.849-7.447-5.024Q-7.336-5.198-7.150-5.297Q-6.963-5.396-6.748-5.439Q-6.533-5.482-6.290-5.482Q-5.876-5.482-5.596-5.300L-5.381-5.475Q-5.370-5.478-5.364-5.480Q-5.357-5.482-5.347-5.482L-5.295-5.482Q-5.268-5.482-5.244-5.458Q-5.220-5.434-5.220-5.406L-5.220-4.559Q-5.220-4.538-5.244-4.511Q-5.268-4.484-5.295-4.484L-5.408-4.484Q-5.435-4.484-5.461-4.509Q-5.487-4.535-5.487-4.559Q-5.487-4.795-5.593-4.959Q-5.699-5.123-5.881-5.205Q-6.064-5.287-6.297-5.287Q-6.625-5.287-6.881-5.184Q-7.138-5.082-7.138-4.805Q-7.138-4.610-6.955-4.501Q-6.772-4.391-6.543-4.350L-5.969-4.244Q-5.723-4.196-5.509-4.068Q-5.295-3.940-5.159-3.737Q-5.022-3.533-5.022-3.284Q-5.022-2.771-5.388-2.532Q-5.753-2.293-6.290-2.293Q-6.786-2.293-7.117-2.587L-7.384-2.313Q-7.404-2.293-7.432-2.293L-7.479-2.293Q-7.503-2.293-7.531-2.320Q-7.558-2.347-7.558-2.368M-4.335-3.089Q-4.335-3.421-4.111-3.648Q-3.887-3.875-3.544-4.003Q-3.200-4.132-2.828-4.184Q-2.455-4.237-2.151-4.237L-2.151-4.490Q-2.151-4.695-2.258-4.875Q-2.366-5.054-2.547-5.157Q-2.728-5.259-2.937-5.259Q-3.344-5.259-3.579-5.167Q-3.491-5.130-3.444-5.046Q-3.398-4.962-3.398-4.860Q-3.398-4.764-3.444-4.685Q-3.491-4.607-3.571-4.562Q-3.651-4.518-3.740-4.518Q-3.891-4.518-3.991-4.615Q-4.092-4.713-4.092-4.860Q-4.092-5.482-2.937-5.482Q-2.725-5.482-2.475-5.418Q-2.226-5.355-2.024-5.236Q-1.823-5.116-1.696-4.931Q-1.570-4.747-1.570-4.504L-1.570-2.928Q-1.570-2.812-1.508-2.716Q-1.447-2.621-1.334-2.621Q-1.224-2.621-1.160-2.715Q-1.095-2.809-1.095-2.928L-1.095-3.376L-0.828-3.376L-0.828-2.928Q-0.828-2.658-1.055-2.493Q-1.283-2.327-1.563-2.327Q-1.771-2.327-1.908-2.481Q-2.045-2.634-2.069-2.850Q-2.216-2.583-2.498-2.438Q-2.780-2.293-3.104-2.293Q-3.381-2.293-3.665-2.368Q-3.949-2.443-4.142-2.622Q-4.335-2.802-4.335-3.089M-3.720-3.089Q-3.720-2.915-3.619-2.785Q-3.518-2.655-3.362-2.585Q-3.207-2.515-3.043-2.515Q-2.824-2.515-2.616-2.612Q-2.407-2.710-2.279-2.891Q-2.151-3.072-2.151-3.298L-2.151-4.026Q-2.475-4.026-2.841-3.935Q-3.207-3.844-3.463-3.632Q-3.720-3.421-3.720-3.089M1.718-0.611Q1.168-1.011 0.797-1.566Q0.426-2.122 0.245-2.768Q0.064-3.414 0.064-4.111Q0.064-4.624 0.165-5.119Q0.266-5.615 0.471-6.066Q0.676-6.517 0.989-6.909Q1.301-7.300 1.718-7.604Q1.729-7.608 1.735-7.609Q1.742-7.611 1.753-7.611L1.821-7.611Q1.855-7.611 1.877-7.587Q1.900-7.563 1.900-7.526Q1.900-7.481 1.872-7.464Q1.524-7.163 1.271-6.779Q1.018-6.394 0.866-5.953Q0.713-5.512 0.642-5.056Q0.570-4.600 0.570-4.111Q0.570-3.110 0.879-2.223Q1.189-1.336 1.872-0.751Q1.900-0.734 1.900-0.690Q1.900-0.652 1.877-0.628Q1.855-0.604 1.821-0.604L1.753-0.604Q1.746-0.608 1.737-0.609Q1.729-0.611 1.718-0.611M4.378-2.361L2.775-2.361L2.775-2.641Q3-2.641 3.149-2.675Q3.297-2.710 3.297-2.850L3.297-6.469Q3.297-6.739 3.190-6.801Q3.082-6.862 2.775-6.862L2.775-7.143L3.851-7.218L3.851-2.850Q3.851-2.713 4.002-2.677Q4.152-2.641 4.378-2.641L4.378-2.361M5.030-3.089Q5.030-3.421 5.254-3.648Q5.478-3.875 5.822-4.003Q6.165-4.132 6.538-4.184Q6.910-4.237 7.214-4.237L7.214-4.490Q7.214-4.695 7.107-4.875Q6.999-5.054 6.818-5.157Q6.637-5.259 6.428-5.259Q6.022-5.259 5.786-5.167Q5.875-5.130 5.921-5.046Q5.967-4.962 5.967-4.860Q5.967-4.764 5.921-4.685Q5.875-4.607 5.794-4.562Q5.714-4.518 5.625-4.518Q5.475-4.518 5.374-4.615Q5.273-4.713 5.273-4.860Q5.273-5.482 6.428-5.482Q6.640-5.482 6.890-5.418Q7.139-5.355 7.341-5.236Q7.543-5.116 7.669-4.931Q7.796-4.747 7.796-4.504L7.796-2.928Q7.796-2.812 7.857-2.716Q7.919-2.621 8.031-2.621Q8.141-2.621 8.206-2.715Q8.271-2.809 8.271-2.928L8.271-3.376L8.537-3.376L8.537-2.928Q8.537-2.658 8.310-2.493Q8.083-2.327 7.802-2.327Q7.594-2.327 7.457-2.481Q7.320-2.634 7.297-2.850Q7.150-2.583 6.868-2.438Q6.586-2.293 6.261-2.293Q5.984-2.293 5.700-2.368Q5.417-2.443 5.224-2.622Q5.030-2.802 5.030-3.089M5.646-3.089Q5.646-2.915 5.746-2.785Q5.847-2.655 6.003-2.585Q6.158-2.515 6.322-2.515Q6.541-2.515 6.750-2.612Q6.958-2.710 7.086-2.891Q7.214-3.072 7.214-3.298L7.214-4.026Q6.890-4.026 6.524-3.935Q6.158-3.844 5.902-3.632Q5.646-3.421 5.646-3.089M10.636-2.361L9.002-2.361L9.002-2.641Q9.231-2.641 9.380-2.675Q9.528-2.710 9.528-2.850L9.528-4.699Q9.528-4.969 9.421-5.030Q9.313-5.092 9.002-5.092L9.002-5.372L10.062-5.447L10.062-4.798Q10.233-5.106 10.537-5.277Q10.841-5.447 11.186-5.447Q11.586-5.447 11.863-5.307Q12.140-5.167 12.225-4.819Q12.393-5.112 12.692-5.280Q12.991-5.447 13.336-5.447Q13.842-5.447 14.126-5.224Q14.409-5 14.409-4.504L14.409-2.850Q14.409-2.713 14.558-2.677Q14.707-2.641 14.932-2.641L14.932-2.361L13.302-2.361L13.302-2.641Q13.527-2.641 13.678-2.677Q13.828-2.713 13.828-2.850L13.828-4.490Q13.828-4.825 13.709-5.025Q13.589-5.225 13.275-5.225Q13.005-5.225 12.770-5.089Q12.536-4.952 12.398-4.718Q12.259-4.484 12.259-4.210L12.259-2.850Q12.259-2.713 12.408-2.677Q12.557-2.641 12.782-2.641L12.782-2.361L11.152-2.361L11.152-2.641Q11.381-2.641 11.530-2.675Q11.678-2.710 11.678-2.850L11.678-4.490Q11.678-4.825 11.559-5.025Q11.439-5.225 11.125-5.225Q10.855-5.225 10.620-5.089Q10.386-4.952 10.248-4.718Q10.109-4.484 10.109-4.210L10.109-2.850Q10.109-2.713 10.260-2.677Q10.410-2.641 10.636-2.641\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-25.286 -55.155)\">\u003Cpath d=\"M16.141-2.361L15.874-2.361L15.874-6.469Q15.874-6.739 15.767-6.801Q15.659-6.862 15.348-6.862L15.348-7.143L16.428-7.218L16.428-5.048Q16.637-5.239 16.922-5.343Q17.207-5.447 17.505-5.447Q17.823-5.447 18.120-5.326Q18.417-5.205 18.640-4.989Q18.862-4.774 18.988-4.489Q19.115-4.203 19.115-3.872Q19.115-3.427 18.875-3.063Q18.636-2.699 18.243-2.496Q17.850-2.293 17.406-2.293Q17.211-2.293 17.021-2.349Q16.832-2.405 16.671-2.510Q16.510-2.614 16.370-2.775L16.141-2.361M16.456-4.706L16.456-3.089Q16.592-2.829 16.833-2.672Q17.074-2.515 17.351-2.515Q17.645-2.515 17.857-2.622Q18.069-2.730 18.202-2.922Q18.335-3.113 18.394-3.352Q18.452-3.591 18.452-3.872Q18.452-4.231 18.358-4.535Q18.264-4.839 18.036-5.032Q17.809-5.225 17.443-5.225Q17.143-5.225 16.876-5.089Q16.609-4.952 16.456-4.706\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-25.286 -55.155)\">\u003Cpath d=\"M19.966-3.872Q19.966-4.210 20.107-4.501Q20.247-4.791 20.491-5.005Q20.735-5.218 21.040-5.333Q21.344-5.447 21.669-5.447Q21.939-5.447 22.202-5.348Q22.465-5.249 22.656-5.071L22.656-6.469Q22.656-6.739 22.549-6.801Q22.441-6.862 22.130-6.862L22.130-7.143L23.207-7.218L23.207-3.034Q23.207-2.846 23.261-2.763Q23.316-2.679 23.417-2.660Q23.518-2.641 23.733-2.641L23.733-2.361L22.626-2.293L22.626-2.710Q22.209-2.293 21.583-2.293Q21.152-2.293 20.780-2.505Q20.407-2.716 20.187-3.077Q19.966-3.438 19.966-3.872M21.641-2.515Q21.850-2.515 22.036-2.587Q22.222-2.658 22.376-2.795Q22.530-2.932 22.626-3.110L22.626-4.719Q22.540-4.866 22.395-4.986Q22.250-5.106 22.080-5.165Q21.911-5.225 21.730-5.225Q21.170-5.225 20.901-4.836Q20.633-4.446 20.633-3.865Q20.633-3.294 20.867-2.904Q21.101-2.515 21.641-2.515M24.441-3.089Q24.441-3.421 24.664-3.648Q24.888-3.875 25.232-4.003Q25.575-4.132 25.948-4.184Q26.320-4.237 26.625-4.237L26.625-4.490Q26.625-4.695 26.517-4.875Q26.409-5.054 26.228-5.157Q26.047-5.259 25.839-5.259Q25.432-5.259 25.196-5.167Q25.285-5.130 25.331-5.046Q25.377-4.962 25.377-4.860Q25.377-4.764 25.331-4.685Q25.285-4.607 25.204-4.562Q25.124-4.518 25.035-4.518Q24.885-4.518 24.784-4.615Q24.683-4.713 24.683-4.860Q24.683-5.482 25.839-5.482Q26.050-5.482 26.300-5.418Q26.549-5.355 26.751-5.236Q26.953-5.116 27.079-4.931Q27.206-4.747 27.206-4.504L27.206-2.928Q27.206-2.812 27.267-2.716Q27.329-2.621 27.442-2.621Q27.551-2.621 27.616-2.715Q27.681-2.809 27.681-2.928L27.681-3.376L27.947-3.376L27.947-2.928Q27.947-2.658 27.720-2.493Q27.493-2.327 27.213-2.327Q27.004-2.327 26.867-2.481Q26.731-2.634 26.707-2.850Q26.560-2.583 26.278-2.438Q25.996-2.293 25.671-2.293Q25.394-2.293 25.110-2.368Q24.827-2.443 24.634-2.622Q24.441-2.802 24.441-3.089M25.056-3.089Q25.056-2.915 25.157-2.785Q25.257-2.655 25.413-2.585Q25.569-2.515 25.733-2.515Q25.951-2.515 26.160-2.612Q26.368-2.710 26.496-2.891Q26.625-3.072 26.625-3.298L26.625-4.026Q26.300-4.026 25.934-3.935Q25.569-3.844 25.312-3.632Q25.056-3.421 25.056-3.089M28.686-0.604L28.617-0.604Q28.583-0.604 28.561-0.630Q28.539-0.655 28.539-0.690Q28.539-0.734 28.569-0.751Q28.925-1.055 29.174-1.445Q29.424-1.835 29.576-2.267Q29.728-2.699 29.798-3.168Q29.868-3.636 29.868-4.111Q29.868-4.590 29.798-5.056Q29.728-5.523 29.574-5.958Q29.421-6.394 29.169-6.782Q28.918-7.170 28.569-7.464Q28.539-7.481 28.539-7.526Q28.539-7.560 28.561-7.585Q28.583-7.611 28.617-7.611L28.686-7.611Q28.696-7.611 28.704-7.609Q28.713-7.608 28.723-7.604Q29.267-7.204 29.639-6.651Q30.012-6.097 30.193-5.451Q30.374-4.805 30.374-4.111Q30.374-3.410 30.193-2.763Q30.012-2.115 29.638-1.561Q29.263-1.007 28.723-0.611Q28.713-0.611 28.704-0.609Q28.696-0.608 28.686-0.604\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-62.07-9.474h85.358V-35.08H-62.07Z\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-21.55 -18.167)\">\u003Cpath d=\"M-16.845-2.361L-19.049-2.361L-19.049-2.641Q-18.290-2.641-18.290-2.850L-18.290-6.651Q-18.290-6.862-19.049-6.862L-19.049-7.143L-16.845-7.143L-16.845-6.862Q-17.600-6.862-17.600-6.651L-17.600-2.850Q-17.600-2.641-16.845-2.641L-16.845-2.361M-16.017-2.299L-16.017-3.872Q-16.017-3.899-15.992-3.925Q-15.966-3.950-15.939-3.950L-15.826-3.950Q-15.799-3.950-15.775-3.923Q-15.751-3.896-15.751-3.872Q-15.751-3.527-15.619-3.263Q-15.488-3-15.259-2.831Q-15.030-2.662-14.727-2.581Q-14.425-2.501-14.083-2.501Q-13.816-2.501-13.580-2.629Q-13.345-2.757-13.199-2.980Q-13.054-3.202-13.054-3.468Q-13.054-3.691-13.160-3.887Q-13.266-4.084-13.447-4.219Q-13.628-4.354-13.854-4.405L-14.883-4.637Q-15.194-4.709-15.453-4.895Q-15.713-5.082-15.865-5.353Q-16.017-5.625-16.017-5.940Q-16.017-6.326-15.804-6.633Q-15.590-6.941-15.243-7.112Q-14.896-7.283-14.517-7.283Q-14.288-7.283-14.059-7.230Q-13.830-7.177-13.632-7.069Q-13.433-6.962-13.280-6.798L-12.986-7.238Q-12.962-7.283-12.921-7.283L-12.873-7.283Q-12.842-7.283-12.820-7.257Q-12.798-7.232-12.798-7.204L-12.798-5.629Q-12.798-5.608-12.822-5.581Q-12.846-5.553-12.873-5.553L-12.986-5.553Q-13.047-5.553-13.061-5.629Q-13.102-6.042-13.283-6.362Q-13.464-6.681-13.775-6.856Q-14.086-7.030-14.517-7.030Q-14.766-7.030-15.006-6.919Q-15.245-6.808-15.395-6.610Q-15.546-6.411-15.546-6.148Q-15.546-5.936-15.438-5.755Q-15.330-5.574-15.154-5.454Q-14.978-5.335-14.770-5.294L-13.741-5.065Q-13.423-4.993-13.157-4.788Q-12.890-4.583-12.738-4.289Q-12.586-3.995-12.586-3.663Q-12.586-3.270-12.791-2.935Q-12.996-2.600-13.341-2.411Q-13.686-2.221-14.083-2.221Q-14.503-2.221-14.883-2.334Q-15.262-2.446-15.532-2.696L-15.826-2.262Q-15.853-2.221-15.891-2.221L-15.939-2.221Q-15.966-2.221-15.992-2.246Q-16.017-2.272-16.017-2.299\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-21.55 -18.167)\">\u003Cpath d=\"M-9.118-3.844Q-9.118-4.186-8.983-4.485Q-8.848-4.784-8.608-5.008Q-8.369-5.232-8.051-5.357Q-7.733-5.482-7.402-5.482Q-6.957-5.482-6.558-5.266Q-6.158-5.051-5.923-4.673Q-5.689-4.296-5.689-3.844Q-5.689-3.503-5.831-3.219Q-5.973-2.935-6.217-2.728Q-6.462-2.522-6.771-2.407Q-7.080-2.293-7.402-2.293Q-7.832-2.293-8.234-2.494Q-8.636-2.696-8.877-3.048Q-9.118-3.400-9.118-3.844M-7.402-2.542Q-6.800-2.542-6.576-2.920Q-6.352-3.298-6.352-3.930Q-6.352-4.542-6.587-4.901Q-6.821-5.259-7.402-5.259Q-8.454-5.259-8.454-3.930Q-8.454-3.298-8.229-2.920Q-8.003-2.542-7.402-2.542M-3.297-2.361L-5.030-2.361L-5.030-2.641Q-4.804-2.641-4.655-2.675Q-4.507-2.710-4.507-2.850L-4.507-5.099L-5.095-5.099L-5.095-5.379L-4.507-5.379L-4.507-6.196Q-4.507-6.514-4.329-6.762Q-4.151-7.009-3.861-7.150Q-3.570-7.290-3.259-7.290Q-3.003-7.290-2.799-7.148Q-2.596-7.006-2.596-6.763Q-2.596-6.627-2.695-6.528Q-2.794-6.428-2.931-6.428Q-3.068-6.428-3.167-6.528Q-3.266-6.627-3.266-6.763Q-3.266-6.944-3.126-7.037Q-3.204-7.064-3.304-7.064Q-3.512-7.064-3.666-6.931Q-3.820-6.798-3.900-6.594Q-3.980-6.391-3.980-6.182L-3.980-5.379L-3.092-5.379L-3.092-5.099L-3.953-5.099L-3.953-2.850Q-3.953-2.641-3.297-2.641\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-21.55 -18.167)\">\u003Cpath d=\"M-0.254-2.361L-1.987-2.361L-1.987-2.641Q-1.761-2.641-1.612-2.675Q-1.464-2.710-1.464-2.850L-1.464-5.099L-2.052-5.099L-2.052-5.379L-1.464-5.379L-1.464-6.196Q-1.464-6.514-1.286-6.762Q-1.108-7.009-0.818-7.150Q-0.527-7.290-0.216-7.290Q0.040-7.290 0.244-7.148Q0.447-7.006 0.447-6.763Q0.447-6.627 0.348-6.528Q0.249-6.428 0.112-6.428Q-0.025-6.428-0.124-6.528Q-0.223-6.627-0.223-6.763Q-0.223-6.944-0.083-7.037Q-0.161-7.064-0.261-7.064Q-0.469-7.064-0.623-6.931Q-0.777-6.798-0.857-6.594Q-0.937-6.391-0.937-6.182L-0.937-5.379L-0.049-5.379L-0.049-5.099L-0.910-5.099L-0.910-2.850Q-0.910-2.641-0.254-2.641L-0.254-2.361M2.310-3.615L0.252-3.615L0.252-4.118L2.310-4.118L2.310-3.615M4.757-1.004L3.127-1.004L3.127-1.284Q3.356-1.284 3.504-1.319Q3.653-1.353 3.653-1.493L3.653-4.839Q3.653-5.010 3.516-5.051Q3.380-5.092 3.127-5.092L3.127-5.372L4.207-5.447L4.207-5.041Q4.429-5.242 4.716-5.345Q5.003-5.447 5.311-5.447Q5.738-5.447 6.102-5.234Q6.466-5.020 6.680-4.656Q6.893-4.292 6.893-3.872Q6.893-3.427 6.654-3.063Q6.415-2.699 6.022-2.496Q5.629-2.293 5.184-2.293Q4.918-2.293 4.670-2.393Q4.422-2.494 4.234-2.675L4.234-1.493Q4.234-1.356 4.383-1.320Q4.531-1.284 4.757-1.284L4.757-1.004M4.234-4.692L4.234-3.082Q4.367-2.829 4.610-2.672Q4.853-2.515 5.130-2.515Q5.458-2.515 5.711-2.716Q5.964-2.918 6.097-3.236Q6.230-3.554 6.230-3.872Q6.230-4.101 6.165-4.330Q6.100-4.559 5.972-4.757Q5.844-4.955 5.649-5.075Q5.454-5.194 5.222-5.194Q4.928-5.194 4.660-5.065Q4.391-4.935 4.234-4.692\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-21.55 -18.167)\">\u003Cpath d=\"M7.712-3.844Q7.712-4.186 7.847-4.485Q7.982-4.784 8.222-5.008Q8.461-5.232 8.779-5.357Q9.097-5.482 9.428-5.482Q9.873-5.482 10.272-5.266Q10.672-5.051 10.907-4.673Q11.141-4.296 11.141-3.844Q11.141-3.503 10.999-3.219Q10.857-2.935 10.613-2.728Q10.368-2.522 10.059-2.407Q9.750-2.293 9.428-2.293Q8.998-2.293 8.596-2.494Q8.194-2.696 7.953-3.048Q7.712-3.400 7.712-3.844M9.428-2.542Q10.030-2.542 10.254-2.920Q10.478-3.298 10.478-3.930Q10.478-4.542 10.243-4.901Q10.009-5.259 9.428-5.259Q8.376-5.259 8.376-3.930Q8.376-3.298 8.601-2.920Q8.827-2.542 9.428-2.542M13.403-2.361L11.800-2.361L11.800-2.641Q12.026-2.641 12.175-2.675Q12.323-2.710 12.323-2.850L12.323-6.469Q12.323-6.739 12.216-6.801Q12.108-6.862 11.800-6.862L11.800-7.143L12.877-7.218L12.877-2.850Q12.877-2.713 13.027-2.677Q13.178-2.641 13.403-2.641L13.403-2.361M15.615-2.361L14.063-2.361L14.063-2.641Q14.289-2.641 14.437-2.675Q14.586-2.710 14.586-2.850L14.586-4.699Q14.586-4.887 14.538-4.971Q14.490-5.054 14.393-5.073Q14.295-5.092 14.084-5.092L14.084-5.372L15.140-5.447L15.140-2.850Q15.140-2.710 15.271-2.675Q15.403-2.641 15.615-2.641L15.615-2.361M14.343-6.668Q14.343-6.839 14.466-6.958Q14.589-7.078 14.760-7.078Q14.928-7.078 15.051-6.958Q15.174-6.839 15.174-6.668Q15.174-6.493 15.051-6.370Q14.928-6.247 14.760-6.247Q14.589-6.247 14.466-6.370Q14.343-6.493 14.343-6.668M16.261-3.872Q16.261-4.200 16.396-4.501Q16.531-4.801 16.767-5.022Q17.002-5.242 17.307-5.362Q17.611-5.482 17.936-5.482Q18.441-5.482 18.790-5.379Q19.139-5.277 19.139-4.901Q19.139-4.754 19.041-4.653Q18.944-4.552 18.797-4.552Q18.643-4.552 18.544-4.651Q18.445-4.750 18.445-4.901Q18.445-5.089 18.585-5.181Q18.383-5.232 17.942-5.232Q17.587-5.232 17.358-5.036Q17.129-4.839 17.028-4.530Q16.927-4.220 16.927-3.872Q16.927-3.523 17.054-3.217Q17.180-2.911 17.435-2.727Q17.689-2.542 18.045-2.542Q18.267-2.542 18.452-2.626Q18.636-2.710 18.771-2.865Q18.906-3.021 18.964-3.229Q18.978-3.284 19.033-3.284L19.146-3.284Q19.176-3.284 19.199-3.260Q19.221-3.236 19.221-3.202L19.221-3.181Q19.135-2.894 18.947-2.696Q18.759-2.498 18.494-2.395Q18.230-2.293 17.936-2.293Q17.505-2.293 17.117-2.499Q16.729-2.706 16.495-3.069Q16.261-3.431 16.261-3.872M20.144-1.226Q20.273-1.158 20.410-1.158Q20.581-1.158 20.731-1.247Q20.882-1.336 20.993-1.481Q21.104-1.626 21.183-1.794L21.446-2.361L20.277-4.887Q20.202-5.034 20.072-5.066Q19.942-5.099 19.710-5.099L19.710-5.379L21.231-5.379L21.231-5.099Q20.882-5.099 20.882-4.952Q20.885-4.931 20.887-4.914Q20.889-4.897 20.889-4.887L21.747-3.028L22.519-4.699Q22.553-4.767 22.553-4.846Q22.553-4.959 22.470-5.029Q22.386-5.099 22.273-5.099L22.273-5.379L23.469-5.379L23.469-5.099Q23.251-5.099 23.078-4.995Q22.905-4.890 22.813-4.699L21.477-1.794Q21.306-1.424 21.036-1.178Q20.766-0.932 20.410-0.932Q20.140-0.932 19.921-1.098Q19.703-1.264 19.703-1.527Q19.703-1.664 19.795-1.753Q19.887-1.841 20.027-1.841Q20.164-1.841 20.253-1.753Q20.342-1.664 20.342-1.527Q20.342-1.424 20.289-1.346Q20.236-1.267 20.144-1.226\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"var(--tk-accent)\">\u003Cpath fill=\"none\" d=\"M-62.07 27.515h85.358V1.907H-62.07Z\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-37.917 18.822)\">\u003Cpath d=\"M-15.170-2.361L-17.908-2.361L-17.908-2.641Q-17.559-2.641-17.222-2.677Q-16.886-2.713-16.886-2.850L-16.886-6.651Q-16.886-6.794-16.974-6.828Q-17.063-6.862-17.248-6.862L-17.607-6.862Q-17.908-6.862-18.123-6.815Q-18.338-6.767-18.495-6.610Q-18.632-6.476-18.692-6.198Q-18.752-5.919-18.789-5.506L-19.056-5.506L-18.909-7.143L-14.175-7.143L-14.028-5.506L-14.295-5.506Q-14.332-5.919-14.389-6.196Q-14.445-6.473-14.589-6.610Q-14.749-6.770-14.961-6.816Q-15.173-6.862-15.477-6.862L-15.829-6.862Q-16.014-6.862-16.103-6.828Q-16.192-6.794-16.192-6.651L-16.192-2.850Q-16.192-2.713-15.855-2.677Q-15.518-2.641-15.170-2.641\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-37.917 18.822)\">\u003Cpath d=\"M-12.264-2.361L-14-2.361L-14-2.641Q-13.771-2.641-13.622-2.675Q-13.474-2.710-13.474-2.850L-13.474-4.699Q-13.474-4.969-13.581-5.030Q-13.689-5.092-14-5.092L-14-5.372L-12.971-5.447L-12.971-4.740Q-12.841-5.048-12.599-5.247Q-12.356-5.447-12.038-5.447Q-11.819-5.447-11.648-5.323Q-11.477-5.198-11.477-4.986Q-11.477-4.849-11.577-4.750Q-11.676-4.651-11.809-4.651Q-11.946-4.651-12.045-4.750Q-12.144-4.849-12.144-4.986Q-12.144-5.126-12.045-5.225Q-12.335-5.225-12.535-5.029Q-12.735-4.832-12.828-4.538Q-12.920-4.244-12.920-3.964L-12.920-2.850Q-12.920-2.641-12.264-2.641L-12.264-2.361M-10.934-3.896Q-10.934-4.217-10.809-4.506Q-10.684-4.795-10.459-5.018Q-10.233-5.242-9.938-5.362Q-9.642-5.482-9.324-5.482Q-8.996-5.482-8.734-5.382Q-8.473-5.283-8.297-5.101Q-8.121-4.918-8.027-4.660Q-7.933-4.402-7.933-4.070Q-7.933-3.978-8.015-3.957L-10.271-3.957L-10.271-3.896Q-10.271-3.308-9.987-2.925Q-9.703-2.542-9.136-2.542Q-8.815-2.542-8.547-2.735Q-8.278-2.928-8.189-3.243Q-8.182-3.284-8.107-3.298L-8.015-3.298Q-7.933-3.274-7.933-3.202Q-7.933-3.195-7.940-3.168Q-8.053-2.771-8.423-2.532Q-8.794-2.293-9.218-2.293Q-9.656-2.293-10.056-2.501Q-10.455-2.710-10.695-3.077Q-10.934-3.444-10.934-3.896M-10.264-4.166L-8.449-4.166Q-8.449-4.443-8.547-4.695Q-8.644-4.948-8.842-5.104Q-9.040-5.259-9.324-5.259Q-9.601-5.259-9.815-5.101Q-10.028-4.942-10.146-4.687Q-10.264-4.432-10.264-4.166M-7.386-3.896Q-7.386-4.217-7.261-4.506Q-7.137-4.795-6.911-5.018Q-6.685-5.242-6.390-5.362Q-6.094-5.482-5.776-5.482Q-5.448-5.482-5.187-5.382Q-4.925-5.283-4.749-5.101Q-4.573-4.918-4.479-4.660Q-4.385-4.402-4.385-4.070Q-4.385-3.978-4.467-3.957L-6.723-3.957L-6.723-3.896Q-6.723-3.308-6.439-2.925Q-6.156-2.542-5.588-2.542Q-5.267-2.542-4.999-2.735Q-4.730-2.928-4.641-3.243Q-4.635-3.284-4.559-3.298L-4.467-3.298Q-4.385-3.274-4.385-3.202Q-4.385-3.195-4.392-3.168Q-4.505-2.771-4.876-2.532Q-5.246-2.293-5.670-2.293Q-6.108-2.293-6.508-2.501Q-6.908-2.710-7.147-3.077Q-7.386-3.444-7.386-3.896M-6.716-4.166L-4.901-4.166Q-4.901-4.443-4.999-4.695Q-5.096-4.948-5.294-5.104Q-5.493-5.259-5.776-5.259Q-6.053-5.259-6.267-5.101Q-6.480-4.942-6.598-4.687Q-6.716-4.432-6.716-4.166M-1.914-3.615L-3.972-3.615L-3.972-4.118L-1.914-4.118L-1.914-3.615M1.955-2.361L-1.018-2.361L-1.018-2.641Q-0.297-2.641-0.297-2.850L-0.297-6.651Q-0.297-6.862-1.018-6.862L-1.018-7.143L1.740-7.143Q2.006-7.143 2.312-7.068Q2.618-6.992 2.878-6.845Q3.138-6.698 3.300-6.469Q3.463-6.240 3.463-5.946Q3.463-5.516 3.076-5.234Q2.690-4.952 2.208-4.860Q2.516-4.860 2.864-4.695Q3.213-4.531 3.442-4.253Q3.671-3.974 3.671-3.656Q3.671-3.250 3.408-2.956Q3.145-2.662 2.746-2.511Q2.348-2.361 1.955-2.361M0.345-4.733L0.345-2.850Q0.345-2.710 0.434-2.675Q0.523-2.641 0.711-2.641L1.740-2.641Q2.027-2.641 2.304-2.768Q2.581-2.894 2.752-3.128Q2.922-3.362 2.922-3.656Q2.922-3.872 2.840-4.068Q2.758-4.265 2.608-4.415Q2.458-4.566 2.261-4.649Q2.065-4.733 1.849-4.733L0.345-4.733M0.345-6.651L0.345-4.959L1.528-4.959Q1.815-4.959 2.095-5.078Q2.376-5.198 2.555-5.425Q2.735-5.653 2.735-5.946Q2.735-6.196 2.596-6.410Q2.458-6.623 2.229-6.743Q2-6.862 1.740-6.862L0.711-6.862Q0.523-6.862 0.434-6.828Q0.345-6.794 0.345-6.651M4.532-3.089Q4.532-3.421 4.756-3.648Q4.980-3.875 5.324-4.003Q5.667-4.132 6.040-4.184Q6.412-4.237 6.716-4.237L6.716-4.490Q6.716-4.695 6.609-4.875Q6.501-5.054 6.320-5.157Q6.139-5.259 5.930-5.259Q5.524-5.259 5.288-5.167Q5.377-5.130 5.423-5.046Q5.469-4.962 5.469-4.860Q5.469-4.764 5.423-4.685Q5.377-4.607 5.296-4.562Q5.216-4.518 5.127-4.518Q4.977-4.518 4.876-4.615Q4.775-4.713 4.775-4.860Q4.775-5.482 5.930-5.482Q6.142-5.482 6.392-5.418Q6.641-5.355 6.843-5.236Q7.045-5.116 7.171-4.931Q7.297-4.747 7.297-4.504L7.297-2.928Q7.297-2.812 7.359-2.716Q7.421-2.621 7.533-2.621Q7.643-2.621 7.708-2.715Q7.773-2.809 7.773-2.928L7.773-3.376L8.039-3.376L8.039-2.928Q8.039-2.658 7.812-2.493Q7.585-2.327 7.304-2.327Q7.096-2.327 6.959-2.481Q6.822-2.634 6.798-2.850Q6.651-2.583 6.370-2.438Q6.088-2.293 5.763-2.293Q5.486-2.293 5.202-2.368Q4.919-2.443 4.725-2.622Q4.532-2.802 4.532-3.089M5.148-3.089Q5.148-2.915 5.248-2.785Q5.349-2.655 5.505-2.585Q5.660-2.515 5.824-2.515Q6.043-2.515 6.252-2.612Q6.460-2.710 6.588-2.891Q6.716-3.072 6.716-3.298L6.716-4.026Q6.392-4.026 6.026-3.935Q5.660-3.844 5.404-3.632Q5.148-3.421 5.148-3.089M8.456-3.872Q8.456-4.200 8.591-4.501Q8.726-4.801 8.962-5.022Q9.198-5.242 9.502-5.362Q9.806-5.482 10.131-5.482Q10.637-5.482 10.985-5.379Q11.334-5.277 11.334-4.901Q11.334-4.754 11.237-4.653Q11.139-4.552 10.992-4.552Q10.839-4.552 10.739-4.651Q10.640-4.750 10.640-4.901Q10.640-5.089 10.780-5.181Q10.579-5.232 10.138-5.232Q9.782-5.232 9.553-5.036Q9.324-4.839 9.224-4.530Q9.123-4.220 9.123-3.872Q9.123-3.523 9.249-3.217Q9.376-2.911 9.630-2.727Q9.885-2.542 10.240-2.542Q10.463-2.542 10.647-2.626Q10.832-2.710 10.967-2.865Q11.102-3.021 11.160-3.229Q11.173-3.284 11.228-3.284L11.341-3.284Q11.372-3.284 11.394-3.260Q11.416-3.236 11.416-3.202L11.416-3.181Q11.331-2.894 11.143-2.696Q10.955-2.498 10.690-2.395Q10.425-2.293 10.131-2.293Q9.700-2.293 9.312-2.499Q8.924-2.706 8.690-3.069Q8.456-3.431 8.456-3.872\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-37.917 18.822)\">\u003Cpath d=\"M13.430-2.361L11.847-2.361L11.847-2.641Q12.076-2.641 12.225-2.675Q12.373-2.710 12.373-2.850L12.373-6.469Q12.373-6.739 12.266-6.801Q12.158-6.862 11.847-6.862L11.847-7.143L12.927-7.218L12.927-3.930L13.912-4.699Q14.117-4.836 14.117-4.986Q14.117-5.030 14.076-5.065Q14.035-5.099 13.990-5.099L13.990-5.379L15.354-5.379L15.354-5.099Q14.865-5.099 14.346-4.699L13.789-4.265L14.766-3.041Q14.968-2.795 15.101-2.718Q15.234-2.641 15.521-2.641L15.521-2.361L14.089-2.361L14.089-2.641Q14.277-2.641 14.277-2.754Q14.277-2.850 14.123-3.041L13.389-3.950L12.907-3.571L12.907-2.850Q12.907-2.713 13.055-2.677Q13.204-2.641 13.430-2.641L13.430-2.361M16.608-3.195L16.608-4.699Q16.608-4.969 16.501-5.030Q16.393-5.092 16.082-5.092L16.082-5.372L17.189-5.447L17.189-3.215L17.189-3.195Q17.189-2.915 17.241-2.771Q17.292-2.628 17.434-2.571Q17.576-2.515 17.863-2.515Q18.116-2.515 18.321-2.655Q18.526-2.795 18.642-3.021Q18.758-3.246 18.758-3.496L18.758-4.699Q18.758-4.969 18.651-5.030Q18.543-5.092 18.232-5.092L18.232-5.372L19.339-5.447L19.339-3.034Q19.339-2.843 19.392-2.761Q19.445-2.679 19.546-2.660Q19.647-2.641 19.862-2.641L19.862-2.361L18.786-2.293L18.786-2.857Q18.676-2.675 18.531-2.552Q18.386-2.429 18.199-2.361Q18.013-2.293 17.811-2.293Q16.608-2.293 16.608-3.195M22.094-1.004L20.464-1.004L20.464-1.284Q20.693-1.284 20.842-1.319Q20.990-1.353 20.990-1.493L20.990-4.839Q20.990-5.010 20.853-5.051Q20.717-5.092 20.464-5.092L20.464-5.372L21.544-5.447L21.544-5.041Q21.766-5.242 22.053-5.345Q22.340-5.447 22.648-5.447Q23.075-5.447 23.439-5.234Q23.803-5.020 24.017-4.656Q24.230-4.292 24.230-3.872Q24.230-3.427 23.991-3.063Q23.752-2.699 23.359-2.496Q22.966-2.293 22.521-2.293Q22.255-2.293 22.007-2.393Q21.759-2.494 21.571-2.675L21.571-1.493Q21.571-1.356 21.720-1.320Q21.869-1.284 22.094-1.284L22.094-1.004M21.571-4.692L21.571-3.082Q21.705-2.829 21.947-2.672Q22.190-2.515 22.467-2.515Q22.795-2.515 23.048-2.716Q23.301-2.918 23.434-3.236Q23.567-3.554 23.567-3.872Q23.567-4.101 23.502-4.330Q23.437-4.559 23.309-4.757Q23.181-4.955 22.986-5.075Q22.791-5.194 22.559-5.194Q22.265-5.194 21.997-5.065Q21.728-4.935 21.571-4.692M26.996-0.611Q26.445-1.011 26.074-1.566Q25.704-2.122 25.522-2.768Q25.341-3.414 25.341-4.111Q25.341-4.624 25.442-5.119Q25.543-5.615 25.748-6.066Q25.953-6.517 26.266-6.909Q26.579-7.300 26.996-7.604Q27.006-7.608 27.013-7.609Q27.019-7.611 27.030-7.611L27.098-7.611Q27.132-7.611 27.154-7.587Q27.177-7.563 27.177-7.526Q27.177-7.481 27.149-7.464Q26.801-7.163 26.548-6.779Q26.295-6.394 26.143-5.953Q25.991-5.512 25.919-5.056Q25.847-4.600 25.847-4.111Q25.847-3.110 26.156-2.223Q26.466-1.336 27.149-0.751Q27.177-0.734 27.177-0.690Q27.177-0.652 27.154-0.628Q27.132-0.604 27.098-0.604L27.030-0.604Q27.023-0.608 27.014-0.609Q27.006-0.611 26.996-0.611M29.655-2.361L28.052-2.361L28.052-2.641Q28.277-2.641 28.426-2.675Q28.575-2.710 28.575-2.850L28.575-6.469Q28.575-6.739 28.467-6.801Q28.359-6.862 28.052-6.862L28.052-7.143L29.128-7.218L29.128-2.850Q29.128-2.713 29.279-2.677Q29.429-2.641 29.655-2.641L29.655-2.361M30.308-3.089Q30.308-3.421 30.531-3.648Q30.755-3.875 31.099-4.003Q31.442-4.132 31.815-4.184Q32.187-4.237 32.492-4.237L32.492-4.490Q32.492-4.695 32.384-4.875Q32.276-5.054 32.095-5.157Q31.914-5.259 31.706-5.259Q31.299-5.259 31.063-5.167Q31.152-5.130 31.198-5.046Q31.244-4.962 31.244-4.860Q31.244-4.764 31.198-4.685Q31.152-4.607 31.071-4.562Q30.991-4.518 30.902-4.518Q30.752-4.518 30.651-4.615Q30.550-4.713 30.550-4.860Q30.550-5.482 31.706-5.482Q31.917-5.482 32.167-5.418Q32.416-5.355 32.618-5.236Q32.820-5.116 32.946-4.931Q33.073-4.747 33.073-4.504L33.073-2.928Q33.073-2.812 33.134-2.716Q33.196-2.621 33.309-2.621Q33.418-2.621 33.483-2.715Q33.548-2.809 33.548-2.928L33.548-3.376L33.814-3.376L33.814-2.928Q33.814-2.658 33.587-2.493Q33.360-2.327 33.080-2.327Q32.871-2.327 32.734-2.481Q32.598-2.634 32.574-2.850Q32.427-2.583 32.145-2.438Q31.863-2.293 31.538-2.293Q31.261-2.293 30.977-2.368Q30.694-2.443 30.501-2.622Q30.308-2.802 30.308-3.089M30.923-3.089Q30.923-2.915 31.024-2.785Q31.124-2.655 31.280-2.585Q31.436-2.515 31.600-2.515Q31.818-2.515 32.027-2.612Q32.235-2.710 32.363-2.891Q32.492-3.072 32.492-3.298L32.492-4.026Q32.167-4.026 31.801-3.935Q31.436-3.844 31.179-3.632Q30.923-3.421 30.923-3.089M35.913-2.361L34.279-2.361L34.279-2.641Q34.508-2.641 34.657-2.675Q34.806-2.710 34.806-2.850L34.806-4.699Q34.806-4.969 34.698-5.030Q34.590-5.092 34.279-5.092L34.279-5.372L35.339-5.447L35.339-4.798Q35.510-5.106 35.814-5.277Q36.118-5.447 36.463-5.447Q36.863-5.447 37.140-5.307Q37.417-5.167 37.502-4.819Q37.670-5.112 37.969-5.280Q38.268-5.447 38.613-5.447Q39.119-5.447 39.403-5.224Q39.686-5 39.686-4.504L39.686-2.850Q39.686-2.713 39.835-2.677Q39.984-2.641 40.209-2.641L40.209-2.361L38.579-2.361L38.579-2.641Q38.805-2.641 38.955-2.677Q39.105-2.713 39.105-2.850L39.105-4.490Q39.105-4.825 38.986-5.025Q38.866-5.225 38.552-5.225Q38.282-5.225 38.048-5.089Q37.813-4.952 37.675-4.718Q37.537-4.484 37.537-4.210L37.537-2.850Q37.537-2.713 37.685-2.677Q37.834-2.641 38.060-2.641L38.060-2.361L36.429-2.361L36.429-2.641Q36.658-2.641 36.807-2.675Q36.956-2.710 36.956-2.850L36.956-4.490Q36.956-4.825 36.836-5.025Q36.716-5.225 36.402-5.225Q36.132-5.225 35.898-5.089Q35.664-4.952 35.525-4.718Q35.387-4.484 35.387-4.210L35.387-2.850Q35.387-2.713 35.537-2.677Q35.687-2.641 35.913-2.641\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-37.917 18.822)\">\u003Cpath d=\"M41.404-2.361L41.137-2.361L41.137-6.469Q41.137-6.739 41.030-6.801Q40.922-6.862 40.611-6.862L40.611-7.143L41.691-7.218L41.691-5.048Q41.900-5.239 42.185-5.343Q42.471-5.447 42.768-5.447Q43.086-5.447 43.383-5.326Q43.680-5.205 43.903-4.989Q44.125-4.774 44.251-4.489Q44.378-4.203 44.378-3.872Q44.378-3.427 44.138-3.063Q43.899-2.699 43.506-2.496Q43.113-2.293 42.669-2.293Q42.474-2.293 42.284-2.349Q42.095-2.405 41.934-2.510Q41.773-2.614 41.633-2.775L41.404-2.361M41.719-4.706L41.719-3.089Q41.855-2.829 42.096-2.672Q42.337-2.515 42.614-2.515Q42.908-2.515 43.120-2.622Q43.332-2.730 43.465-2.922Q43.598-3.113 43.657-3.352Q43.715-3.591 43.715-3.872Q43.715-4.231 43.621-4.535Q43.527-4.839 43.299-5.032Q43.072-5.225 42.706-5.225Q42.406-5.225 42.139-5.089Q41.872-4.952 41.719-4.706\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-37.917 18.822)\">\u003Cpath d=\"M45.229-3.872Q45.229-4.210 45.370-4.501Q45.510-4.791 45.754-5.005Q45.998-5.218 46.303-5.333Q46.607-5.447 46.932-5.447Q47.202-5.447 47.465-5.348Q47.728-5.249 47.919-5.071L47.919-6.469Q47.919-6.739 47.812-6.801Q47.704-6.862 47.393-6.862L47.393-7.143L48.470-7.218L48.470-3.034Q48.470-2.846 48.524-2.763Q48.579-2.679 48.680-2.660Q48.781-2.641 48.996-2.641L48.996-2.361L47.889-2.293L47.889-2.710Q47.472-2.293 46.846-2.293Q46.415-2.293 46.043-2.505Q45.670-2.716 45.450-3.077Q45.229-3.438 45.229-3.872M46.904-2.515Q47.113-2.515 47.299-2.587Q47.485-2.658 47.639-2.795Q47.793-2.932 47.889-3.110L47.889-4.719Q47.803-4.866 47.658-4.986Q47.513-5.106 47.343-5.165Q47.174-5.225 46.993-5.225Q46.433-5.225 46.164-4.836Q45.896-4.446 45.896-3.865Q45.896-3.294 46.130-2.904Q46.364-2.515 46.904-2.515M49.704-3.089Q49.704-3.421 49.927-3.648Q50.151-3.875 50.495-4.003Q50.838-4.132 51.211-4.184Q51.583-4.237 51.888-4.237L51.888-4.490Q51.888-4.695 51.780-4.875Q51.672-5.054 51.491-5.157Q51.310-5.259 51.102-5.259Q50.695-5.259 50.459-5.167Q50.548-5.130 50.594-5.046Q50.640-4.962 50.640-4.860Q50.640-4.764 50.594-4.685Q50.548-4.607 50.467-4.562Q50.387-4.518 50.298-4.518Q50.148-4.518 50.047-4.615Q49.946-4.713 49.946-4.860Q49.946-5.482 51.102-5.482Q51.313-5.482 51.563-5.418Q51.812-5.355 52.014-5.236Q52.216-5.116 52.342-4.931Q52.469-4.747 52.469-4.504L52.469-2.928Q52.469-2.812 52.530-2.716Q52.592-2.621 52.705-2.621Q52.814-2.621 52.879-2.715Q52.944-2.809 52.944-2.928L52.944-3.376L53.210-3.376L53.210-2.928Q53.210-2.658 52.983-2.493Q52.756-2.327 52.476-2.327Q52.267-2.327 52.130-2.481Q51.994-2.634 51.970-2.850Q51.823-2.583 51.541-2.438Q51.259-2.293 50.934-2.293Q50.657-2.293 50.373-2.368Q50.090-2.443 49.897-2.622Q49.704-2.802 49.704-3.089M50.319-3.089Q50.319-2.915 50.420-2.785Q50.520-2.655 50.676-2.585Q50.832-2.515 50.996-2.515Q51.214-2.515 51.423-2.612Q51.631-2.710 51.759-2.891Q51.888-3.072 51.888-3.298L51.888-4.026Q51.563-4.026 51.197-3.935Q50.832-3.844 50.575-3.632Q50.319-3.421 50.319-3.089M53.949-0.604L53.880-0.604Q53.846-0.604 53.824-0.630Q53.802-0.655 53.802-0.690Q53.802-0.734 53.832-0.751Q54.188-1.055 54.437-1.445Q54.687-1.835 54.839-2.267Q54.991-2.699 55.061-3.168Q55.131-3.636 55.131-4.111Q55.131-4.590 55.061-5.056Q54.991-5.523 54.837-5.958Q54.684-6.394 54.432-6.782Q54.181-7.170 53.832-7.464Q53.802-7.481 53.802-7.526Q53.802-7.560 53.824-7.585Q53.846-7.611 53.880-7.611L53.949-7.611Q53.959-7.611 53.967-7.609Q53.976-7.608 53.986-7.604Q54.530-7.204 54.902-6.651Q55.275-6.097 55.456-5.451Q55.637-4.805 55.637-4.111Q55.637-3.410 55.456-2.763Q55.275-2.115 54.901-1.561Q54.526-1.007 53.986-0.611Q53.976-0.611 53.967-0.609Q53.959-0.608 53.949-0.604\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(68.974 -55.155)\">\u003Cpath d=\"M-19.077-3.872Q-19.077-4.210-18.936-4.501Q-18.796-4.791-18.552-5.005Q-18.308-5.218-18.003-5.333Q-17.699-5.447-17.374-5.447Q-17.104-5.447-16.841-5.348Q-16.578-5.249-16.387-5.071L-16.387-6.469Q-16.387-6.739-16.494-6.801Q-16.602-6.862-16.913-6.862L-16.913-7.143L-15.836-7.218L-15.836-3.034Q-15.836-2.846-15.782-2.763Q-15.727-2.679-15.626-2.660Q-15.525-2.641-15.310-2.641L-15.310-2.361L-16.417-2.293L-16.417-2.710Q-16.834-2.293-17.460-2.293Q-17.891-2.293-18.263-2.505Q-18.636-2.716-18.856-3.077Q-19.077-3.438-19.077-3.872M-17.402-2.515Q-17.193-2.515-17.007-2.587Q-16.821-2.658-16.667-2.795Q-16.513-2.932-16.417-3.110L-16.417-4.719Q-16.503-4.866-16.648-4.986Q-16.793-5.106-16.963-5.165Q-17.132-5.225-17.313-5.225Q-17.873-5.225-18.142-4.836Q-18.410-4.446-18.410-3.865Q-18.410-3.294-18.176-2.904Q-17.942-2.515-17.402-2.515M-14.702-3.896Q-14.702-4.217-14.577-4.506Q-14.452-4.795-14.226-5.018Q-14.001-5.242-13.705-5.362Q-13.410-5.482-13.092-5.482Q-12.764-5.482-12.502-5.382Q-12.241-5.283-12.065-5.101Q-11.889-4.918-11.795-4.660Q-11.701-4.402-11.701-4.070Q-11.701-3.978-11.783-3.957L-14.038-3.957L-14.038-3.896Q-14.038-3.308-13.755-2.925Q-13.471-2.542-12.904-2.542Q-12.582-2.542-12.314-2.735Q-12.046-2.928-11.957-3.243Q-11.950-3.284-11.875-3.298L-11.783-3.298Q-11.701-3.274-11.701-3.202Q-11.701-3.195-11.707-3.168Q-11.820-2.771-12.191-2.532Q-12.562-2.293-12.986-2.293Q-13.423-2.293-13.823-2.501Q-14.223-2.710-14.462-3.077Q-14.702-3.444-14.702-3.896M-14.032-4.166L-12.217-4.166Q-12.217-4.443-12.314-4.695Q-12.412-4.948-12.610-5.104Q-12.808-5.259-13.092-5.259Q-13.369-5.259-13.582-5.101Q-13.796-4.942-13.914-4.687Q-14.032-4.432-14.032-4.166M-11.113-3.872Q-11.113-4.200-10.978-4.501Q-10.843-4.801-10.607-5.022Q-10.371-5.242-10.067-5.362Q-9.763-5.482-9.438-5.482Q-8.932-5.482-8.583-5.379Q-8.235-5.277-8.235-4.901Q-8.235-4.754-8.332-4.653Q-8.430-4.552-8.577-4.552Q-8.730-4.552-8.829-4.651Q-8.929-4.750-8.929-4.901Q-8.929-5.089-8.788-5.181Q-8.990-5.232-9.431-5.232Q-9.787-5.232-10.016-5.036Q-10.245-4.839-10.345-4.530Q-10.446-4.220-10.446-3.872Q-10.446-3.523-10.320-3.217Q-10.193-2.911-9.939-2.727Q-9.684-2.542-9.328-2.542Q-9.106-2.542-8.922-2.626Q-8.737-2.710-8.602-2.865Q-8.467-3.021-8.409-3.229Q-8.395-3.284-8.341-3.284L-8.228-3.284Q-8.197-3.284-8.175-3.260Q-8.153-3.236-8.153-3.202L-8.153-3.181Q-8.238-2.894-8.426-2.696Q-8.614-2.498-8.879-2.395Q-9.144-2.293-9.438-2.293Q-9.869-2.293-10.256-2.499Q-10.644-2.706-10.879-3.069Q-11.113-3.431-11.113-3.872M-7.507-3.089Q-7.507-3.421-7.283-3.648Q-7.059-3.875-6.715-4.003Q-6.372-4.132-5.999-4.184Q-5.627-4.237-5.323-4.237L-5.323-4.490Q-5.323-4.695-5.430-4.875Q-5.538-5.054-5.719-5.157Q-5.900-5.259-6.109-5.259Q-6.516-5.259-6.751-5.167Q-6.662-5.130-6.616-5.046Q-6.570-4.962-6.570-4.860Q-6.570-4.764-6.616-4.685Q-6.662-4.607-6.743-4.562Q-6.823-4.518-6.912-4.518Q-7.062-4.518-7.163-4.615Q-7.264-4.713-7.264-4.860Q-7.264-5.482-6.109-5.482Q-5.897-5.482-5.647-5.418Q-5.398-5.355-5.196-5.236Q-4.995-5.116-4.868-4.931Q-4.742-4.747-4.742-4.504L-4.742-2.928Q-4.742-2.812-4.680-2.716Q-4.619-2.621-4.506-2.621Q-4.396-2.621-4.331-2.715Q-4.266-2.809-4.266-2.928L-4.266-3.376L-4-3.376L-4-2.928Q-4-2.658-4.227-2.493Q-4.454-2.327-4.735-2.327Q-4.943-2.327-5.080-2.481Q-5.217-2.634-5.241-2.850Q-5.388-2.583-5.670-2.438Q-5.952-2.293-6.276-2.293Q-6.553-2.293-6.837-2.368Q-7.120-2.443-7.314-2.622Q-7.507-2.802-7.507-3.089M-6.891-3.089Q-6.891-2.915-6.791-2.785Q-6.690-2.655-6.534-2.585Q-6.379-2.515-6.215-2.515Q-5.996-2.515-5.787-2.612Q-5.579-2.710-5.451-2.891Q-5.323-3.072-5.323-3.298L-5.323-4.026Q-5.647-4.026-6.013-3.935Q-6.379-3.844-6.635-3.632Q-6.891-3.421-6.891-3.089\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 -55.155)\">\u003Cpath d=\"M-3.443-1.226Q-3.313-1.158-3.176-1.158Q-3.005-1.158-2.855-1.247Q-2.704-1.336-2.593-1.481Q-2.482-1.626-2.404-1.794L-2.140-2.361L-3.309-4.887Q-3.384-5.034-3.514-5.066Q-3.644-5.099-3.877-5.099L-3.877-5.379L-2.356-5.379L-2.356-5.099Q-2.704-5.099-2.704-4.952Q-2.701-4.931-2.699-4.914Q-2.697-4.897-2.697-4.887L-1.840-3.028L-1.067-4.699Q-1.033-4.767-1.033-4.846Q-1.033-4.959-1.117-5.029Q-1.200-5.099-1.313-5.099L-1.313-5.379L-0.117-5.379L-0.117-5.099Q-0.336-5.099-0.508-4.995Q-0.681-4.890-0.773-4.699L-2.110-1.794Q-2.280-1.424-2.550-1.178Q-2.821-0.932-3.176-0.932Q-3.446-0.932-3.665-1.098Q-3.884-1.264-3.884-1.527Q-3.884-1.664-3.791-1.753Q-3.699-1.841-3.559-1.841Q-3.422-1.841-3.333-1.753Q-3.244-1.664-3.244-1.527Q-3.244-1.424-3.297-1.346Q-3.350-1.267-3.443-1.226\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 -55.155)\">\u003Cpath d=\"M8.286-3.168L3.453-3.168Q3.385-3.178 3.339-3.224Q3.293-3.270 3.293-3.342Q3.293-3.407 3.339-3.453Q3.385-3.499 3.453-3.509L8.286-3.509Q8.355-3.499 8.401-3.453Q8.447-3.407 8.447-3.342Q8.447-3.270 8.401-3.224Q8.355-3.178 8.286-3.168M8.286-4.706L3.453-4.706Q3.385-4.716 3.339-4.762Q3.293-4.808 3.293-4.880Q3.293-5.024 3.453-5.048L8.286-5.048Q8.447-5.024 8.447-4.880Q8.447-4.808 8.401-4.762Q8.355-4.716 8.286-4.706\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 -55.155)\">\u003Cpath d=\"M11.910-1.828Q11.910-2.074 12.107-2.258Q12.304-2.443 12.560-2.522Q12.423-2.634 12.351-2.795Q12.280-2.956 12.280-3.137Q12.280-3.458 12.491-3.704Q12.157-4.002 12.157-4.412Q12.157-4.873 12.546-5.160Q12.936-5.447 13.414-5.447Q13.886-5.447 14.221-5.201Q14.395-5.355 14.606-5.437Q14.816-5.519 15.045-5.519Q15.209-5.519 15.330-5.412Q15.451-5.304 15.451-5.140Q15.451-5.044 15.380-4.972Q15.308-4.901 15.216-4.901Q15.116-4.901 15.046-4.974Q14.976-5.048 14.976-5.147Q14.976-5.201 14.990-5.232L14.997-5.246Q15.004-5.266 15.012-5.277Q15.021-5.287 15.024-5.294Q14.669-5.294 14.382-5.071Q14.669-4.778 14.669-4.412Q14.669-4.097 14.484-3.865Q14.300-3.632 14.011-3.504Q13.722-3.376 13.414-3.376Q13.213-3.376 13.021-3.426Q12.830-3.475 12.652-3.585Q12.560-3.458 12.560-3.315Q12.560-3.133 12.688-2.998Q12.816-2.863 13.001-2.863L13.633-2.863Q14.081-2.863 14.450-2.792Q14.819-2.720 15.079-2.491Q15.339-2.262 15.339-1.828Q15.339-1.507 15.043-1.305Q14.747-1.103 14.344-1.014Q13.941-0.925 13.626-0.925Q13.308-0.925 12.905-1.014Q12.502-1.103 12.206-1.305Q11.910-1.507 11.910-1.828M12.365-1.828Q12.365-1.599 12.584-1.450Q12.803-1.301 13.095-1.233Q13.387-1.165 13.626-1.165Q13.790-1.165 13.999-1.201Q14.207-1.236 14.414-1.317Q14.621-1.397 14.752-1.525Q14.884-1.653 14.884-1.828Q14.884-2.180 14.503-2.274Q14.122-2.368 13.619-2.368L13.001-2.368Q12.762-2.368 12.563-2.217Q12.365-2.067 12.365-1.828M13.414-3.615Q14.081-3.615 14.081-4.412Q14.081-5.212 13.414-5.212Q12.744-5.212 12.744-4.412Q12.744-3.615 13.414-3.615M15.991-3.089Q15.991-3.421 16.215-3.648Q16.439-3.875 16.783-4.003Q17.126-4.132 17.499-4.184Q17.871-4.237 18.176-4.237L18.176-4.490Q18.176-4.695 18.068-4.875Q17.960-5.054 17.779-5.157Q17.598-5.259 17.389-5.259Q16.983-5.259 16.747-5.167Q16.836-5.130 16.882-5.046Q16.928-4.962 16.928-4.860Q16.928-4.764 16.882-4.685Q16.836-4.607 16.755-4.562Q16.675-4.518 16.586-4.518Q16.436-4.518 16.335-4.615Q16.234-4.713 16.234-4.860Q16.234-5.482 17.389-5.482Q17.601-5.482 17.851-5.418Q18.100-5.355 18.302-5.236Q18.504-5.116 18.630-4.931Q18.757-4.747 18.757-4.504L18.757-2.928Q18.757-2.812 18.818-2.716Q18.880-2.621 18.992-2.621Q19.102-2.621 19.167-2.715Q19.232-2.809 19.232-2.928L19.232-3.376L19.498-3.376L19.498-2.928Q19.498-2.658 19.271-2.493Q19.044-2.327 18.763-2.327Q18.555-2.327 18.418-2.481Q18.282-2.634 18.258-2.850Q18.111-2.583 17.829-2.438Q17.547-2.293 17.222-2.293Q16.945-2.293 16.661-2.368Q16.378-2.443 16.185-2.622Q15.991-2.802 15.991-3.089M16.607-3.089Q16.607-2.915 16.708-2.785Q16.808-2.655 16.964-2.585Q17.119-2.515 17.283-2.515Q17.502-2.515 17.711-2.612Q17.919-2.710 18.047-2.891Q18.176-3.072 18.176-3.298L18.176-4.026Q17.851-4.026 17.485-3.935Q17.119-3.844 16.863-3.632Q16.607-3.421 16.607-3.089M21.597-2.361L19.963-2.361L19.963-2.641Q20.192-2.641 20.341-2.675Q20.490-2.710 20.490-2.850L20.490-4.699Q20.490-4.969 20.382-5.030Q20.274-5.092 19.963-5.092L19.963-5.372L21.023-5.447L21.023-4.798Q21.194-5.106 21.498-5.277Q21.802-5.447 22.147-5.447Q22.547-5.447 22.824-5.307Q23.101-5.167 23.186-4.819Q23.354-5.112 23.653-5.280Q23.952-5.447 24.297-5.447Q24.803-5.447 25.087-5.224Q25.370-5 25.370-4.504L25.370-2.850Q25.370-2.713 25.519-2.677Q25.668-2.641 25.893-2.641L25.893-2.361L24.263-2.361L24.263-2.641Q24.489-2.641 24.639-2.677Q24.789-2.713 24.789-2.850L24.789-4.490Q24.789-4.825 24.670-5.025Q24.550-5.225 24.236-5.225Q23.966-5.225 23.731-5.089Q23.497-4.952 23.359-4.718Q23.220-4.484 23.220-4.210L23.220-2.850Q23.220-2.713 23.369-2.677Q23.518-2.641 23.743-2.641L23.743-2.361L22.113-2.361L22.113-2.641Q22.342-2.641 22.491-2.675Q22.639-2.710 22.639-2.850L22.639-4.490Q22.639-4.825 22.520-5.025Q22.400-5.225 22.086-5.225Q21.816-5.225 21.582-5.089Q21.347-4.952 21.209-4.718Q21.071-4.484 21.071-4.210L21.071-2.850Q21.071-2.713 21.221-2.677Q21.371-2.641 21.597-2.641L21.597-2.361M28.163-2.361L26.529-2.361L26.529-2.641Q26.758-2.641 26.907-2.675Q27.055-2.710 27.055-2.850L27.055-4.699Q27.055-4.969 26.948-5.030Q26.840-5.092 26.529-5.092L26.529-5.372L27.589-5.447L27.589-4.798Q27.760-5.106 28.064-5.277Q28.368-5.447 28.713-5.447Q29.113-5.447 29.390-5.307Q29.667-5.167 29.752-4.819Q29.920-5.112 30.219-5.280Q30.518-5.447 30.863-5.447Q31.369-5.447 31.653-5.224Q31.936-5 31.936-4.504L31.936-2.850Q31.936-2.713 32.085-2.677Q32.234-2.641 32.459-2.641L32.459-2.361L30.829-2.361L30.829-2.641Q31.054-2.641 31.205-2.677Q31.355-2.713 31.355-2.850L31.355-4.490Q31.355-4.825 31.236-5.025Q31.116-5.225 30.802-5.225Q30.532-5.225 30.297-5.089Q30.063-4.952 29.925-4.718Q29.786-4.484 29.786-4.210L29.786-2.850Q29.786-2.713 29.935-2.677Q30.084-2.641 30.309-2.641L30.309-2.361L28.679-2.361L28.679-2.641Q28.908-2.641 29.057-2.675Q29.205-2.710 29.205-2.850L29.205-4.490Q29.205-4.825 29.086-5.025Q28.966-5.225 28.652-5.225Q28.382-5.225 28.147-5.089Q27.913-4.952 27.775-4.718Q27.637-4.484 27.637-4.210L27.637-2.850Q27.637-2.713 27.787-2.677Q27.937-2.641 28.163-2.641L28.163-2.361M33.105-3.089Q33.105-3.421 33.329-3.648Q33.553-3.875 33.897-4.003Q34.240-4.132 34.613-4.184Q34.985-4.237 35.289-4.237L35.289-4.490Q35.289-4.695 35.182-4.875Q35.074-5.054 34.893-5.157Q34.712-5.259 34.503-5.259Q34.096-5.259 33.861-5.167Q33.950-5.130 33.996-5.046Q34.042-4.962 34.042-4.860Q34.042-4.764 33.996-4.685Q33.950-4.607 33.869-4.562Q33.789-4.518 33.700-4.518Q33.550-4.518 33.449-4.615Q33.348-4.713 33.348-4.860Q33.348-5.482 34.503-5.482Q34.715-5.482 34.965-5.418Q35.214-5.355 35.416-5.236Q35.617-5.116 35.744-4.931Q35.870-4.747 35.870-4.504L35.870-2.928Q35.870-2.812 35.932-2.716Q35.993-2.621 36.106-2.621Q36.216-2.621 36.281-2.715Q36.345-2.809 36.345-2.928L36.345-3.376L36.612-3.376L36.612-2.928Q36.612-2.658 36.385-2.493Q36.158-2.327 35.877-2.327Q35.669-2.327 35.532-2.481Q35.395-2.634 35.371-2.850Q35.224-2.583 34.942-2.438Q34.660-2.293 34.336-2.293Q34.059-2.293 33.775-2.368Q33.491-2.443 33.298-2.622Q33.105-2.802 33.105-3.089M33.720-3.089Q33.720-2.915 33.821-2.785Q33.922-2.655 34.078-2.585Q34.233-2.515 34.397-2.515Q34.616-2.515 34.825-2.612Q35.033-2.710 35.161-2.891Q35.289-3.072 35.289-3.298L35.289-4.026Q34.965-4.026 34.599-3.935Q34.233-3.844 33.977-3.632Q33.720-3.421 33.720-3.089\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 -55.155)\">\u003Cpath d=\"M41.411-2.361L39.808-2.361L39.808-2.641Q40.034-2.641 40.183-2.675Q40.331-2.710 40.331-2.850L40.331-6.469Q40.331-6.739 40.224-6.801Q40.116-6.862 39.808-6.862L39.808-7.143L40.885-7.218L40.885-2.850Q40.885-2.713 41.035-2.677Q41.186-2.641 41.411-2.641L41.411-2.361M42.064-3.089Q42.064-3.421 42.288-3.648Q42.512-3.875 42.856-4.003Q43.199-4.132 43.572-4.184Q43.944-4.237 44.248-4.237L44.248-4.490Q44.248-4.695 44.141-4.875Q44.033-5.054 43.852-5.157Q43.671-5.259 43.462-5.259Q43.055-5.259 42.820-5.167Q42.908-5.130 42.955-5.046Q43.001-4.962 43.001-4.860Q43.001-4.764 42.955-4.685Q42.908-4.607 42.828-4.562Q42.748-4.518 42.659-4.518Q42.509-4.518 42.408-4.615Q42.307-4.713 42.307-4.860Q42.307-5.482 43.462-5.482Q43.674-5.482 43.924-5.418Q44.173-5.355 44.375-5.236Q44.576-5.116 44.703-4.931Q44.829-4.747 44.829-4.504L44.829-2.928Q44.829-2.812 44.891-2.716Q44.952-2.621 45.065-2.621Q45.175-2.621 45.240-2.715Q45.304-2.809 45.304-2.928L45.304-3.376L45.571-3.376L45.571-2.928Q45.571-2.658 45.344-2.493Q45.117-2.327 44.836-2.327Q44.628-2.327 44.491-2.481Q44.354-2.634 44.330-2.850Q44.183-2.583 43.901-2.438Q43.619-2.293 43.295-2.293Q43.018-2.293 42.734-2.368Q42.450-2.443 42.257-2.622Q42.064-2.802 42.064-3.089M42.679-3.089Q42.679-2.915 42.780-2.785Q42.881-2.655 43.037-2.585Q43.192-2.515 43.356-2.515Q43.575-2.515 43.783-2.612Q43.992-2.710 44.120-2.891Q44.248-3.072 44.248-3.298L44.248-4.026Q43.924-4.026 43.558-3.935Q43.192-3.844 42.936-3.632Q42.679-3.421 42.679-3.089M47.670-2.361L46.036-2.361L46.036-2.641Q46.265-2.641 46.414-2.675Q46.562-2.710 46.562-2.850L46.562-4.699Q46.562-4.969 46.455-5.030Q46.347-5.092 46.036-5.092L46.036-5.372L47.096-5.447L47.096-4.798Q47.266-5.106 47.571-5.277Q47.875-5.447 48.220-5.447Q48.620-5.447 48.897-5.307Q49.174-5.167 49.259-4.819Q49.427-5.112 49.726-5.280Q50.025-5.447 50.370-5.447Q50.876-5.447 51.159-5.224Q51.443-5 51.443-4.504L51.443-2.850Q51.443-2.713 51.592-2.677Q51.741-2.641 51.966-2.641L51.966-2.361L50.336-2.361L50.336-2.641Q50.561-2.641 50.712-2.677Q50.862-2.713 50.862-2.850L50.862-4.490Q50.862-4.825 50.742-5.025Q50.623-5.225 50.308-5.225Q50.038-5.225 49.804-5.089Q49.570-4.952 49.432-4.718Q49.293-4.484 49.293-4.210L49.293-2.850Q49.293-2.713 49.442-2.677Q49.591-2.641 49.816-2.641L49.816-2.361L48.186-2.361L48.186-2.641Q48.415-2.641 48.564-2.675Q48.712-2.710 48.712-2.850L48.712-4.490Q48.712-4.825 48.593-5.025Q48.473-5.225 48.158-5.225Q47.888-5.225 47.654-5.089Q47.420-4.952 47.282-4.718Q47.143-4.484 47.143-4.210L47.143-2.850Q47.143-2.713 47.294-2.677Q47.444-2.641 47.670-2.641\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 -55.155)\">\u003Cpath d=\"M53.154-2.361L52.887-2.361L52.887-6.469Q52.887-6.739 52.780-6.801Q52.672-6.862 52.361-6.862L52.361-7.143L53.441-7.218L53.441-5.048Q53.650-5.239 53.935-5.343Q54.221-5.447 54.518-5.447Q54.836-5.447 55.133-5.326Q55.430-5.205 55.653-4.989Q55.875-4.774 56.001-4.489Q56.128-4.203 56.128-3.872Q56.128-3.427 55.888-3.063Q55.649-2.699 55.256-2.496Q54.863-2.293 54.419-2.293Q54.224-2.293 54.034-2.349Q53.845-2.405 53.684-2.510Q53.523-2.614 53.383-2.775L53.154-2.361M53.469-4.706L53.469-3.089Q53.605-2.829 53.846-2.672Q54.087-2.515 54.364-2.515Q54.658-2.515 54.870-2.622Q55.082-2.730 55.215-2.922Q55.348-3.113 55.407-3.352Q55.465-3.591 55.465-3.872Q55.465-4.231 55.371-4.535Q55.277-4.839 55.049-5.032Q54.822-5.225 54.456-5.225Q54.156-5.225 53.889-5.089Q53.622-4.952 53.469-4.706\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 -55.155)\">\u003Cpath d=\"M56.979-3.872Q56.979-4.210 57.120-4.501Q57.260-4.791 57.504-5.005Q57.748-5.218 58.053-5.333Q58.357-5.447 58.682-5.447Q58.952-5.447 59.215-5.348Q59.478-5.249 59.669-5.071L59.669-6.469Q59.669-6.739 59.562-6.801Q59.454-6.862 59.143-6.862L59.143-7.143L60.220-7.218L60.220-3.034Q60.220-2.846 60.274-2.763Q60.329-2.679 60.430-2.660Q60.531-2.641 60.746-2.641L60.746-2.361L59.639-2.293L59.639-2.710Q59.222-2.293 58.596-2.293Q58.165-2.293 57.793-2.505Q57.420-2.716 57.200-3.077Q56.979-3.438 56.979-3.872M58.654-2.515Q58.863-2.515 59.049-2.587Q59.235-2.658 59.389-2.795Q59.543-2.932 59.639-3.110L59.639-4.719Q59.553-4.866 59.408-4.986Q59.263-5.106 59.093-5.165Q58.924-5.225 58.743-5.225Q58.183-5.225 57.914-4.836Q57.646-4.446 57.646-3.865Q57.646-3.294 57.880-2.904Q58.114-2.515 58.654-2.515M61.454-3.089Q61.454-3.421 61.677-3.648Q61.901-3.875 62.245-4.003Q62.588-4.132 62.961-4.184Q63.333-4.237 63.638-4.237L63.638-4.490Q63.638-4.695 63.530-4.875Q63.422-5.054 63.241-5.157Q63.060-5.259 62.852-5.259Q62.445-5.259 62.209-5.167Q62.298-5.130 62.344-5.046Q62.390-4.962 62.390-4.860Q62.390-4.764 62.344-4.685Q62.298-4.607 62.217-4.562Q62.137-4.518 62.048-4.518Q61.898-4.518 61.797-4.615Q61.696-4.713 61.696-4.860Q61.696-5.482 62.852-5.482Q63.063-5.482 63.313-5.418Q63.562-5.355 63.764-5.236Q63.966-5.116 64.092-4.931Q64.219-4.747 64.219-4.504L64.219-2.928Q64.219-2.812 64.280-2.716Q64.342-2.621 64.455-2.621Q64.564-2.621 64.629-2.715Q64.694-2.809 64.694-2.928L64.694-3.376L64.960-3.376L64.960-2.928Q64.960-2.658 64.733-2.493Q64.506-2.327 64.226-2.327Q64.017-2.327 63.880-2.481Q63.744-2.634 63.720-2.850Q63.573-2.583 63.291-2.438Q63.009-2.293 62.684-2.293Q62.407-2.293 62.123-2.368Q61.840-2.443 61.647-2.622Q61.454-2.802 61.454-3.089M62.069-3.089Q62.069-2.915 62.170-2.785Q62.270-2.655 62.426-2.585Q62.582-2.515 62.746-2.515Q62.964-2.515 63.173-2.612Q63.381-2.710 63.509-2.891Q63.638-3.072 63.638-3.298L63.638-4.026Q63.313-4.026 62.947-3.935Q62.582-3.844 62.325-3.632Q62.069-3.421 62.069-3.089\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(68.974 -18.167)\">\u003Cpath d=\"M-19.077-3.872Q-19.077-4.210-18.936-4.501Q-18.796-4.791-18.552-5.005Q-18.308-5.218-18.003-5.333Q-17.699-5.447-17.374-5.447Q-17.104-5.447-16.841-5.348Q-16.578-5.249-16.387-5.071L-16.387-6.469Q-16.387-6.739-16.494-6.801Q-16.602-6.862-16.913-6.862L-16.913-7.143L-15.836-7.218L-15.836-3.034Q-15.836-2.846-15.782-2.763Q-15.727-2.679-15.626-2.660Q-15.525-2.641-15.310-2.641L-15.310-2.361L-16.417-2.293L-16.417-2.710Q-16.834-2.293-17.460-2.293Q-17.891-2.293-18.263-2.505Q-18.636-2.716-18.856-3.077Q-19.077-3.438-19.077-3.872M-17.402-2.515Q-17.193-2.515-17.007-2.587Q-16.821-2.658-16.667-2.795Q-16.513-2.932-16.417-3.110L-16.417-4.719Q-16.503-4.866-16.648-4.986Q-16.793-5.106-16.963-5.165Q-17.132-5.225-17.313-5.225Q-17.873-5.225-18.142-4.836Q-18.410-4.446-18.410-3.865Q-18.410-3.294-18.176-2.904Q-17.942-2.515-17.402-2.515M-14.702-3.896Q-14.702-4.217-14.577-4.506Q-14.452-4.795-14.226-5.018Q-14.001-5.242-13.705-5.362Q-13.410-5.482-13.092-5.482Q-12.764-5.482-12.502-5.382Q-12.241-5.283-12.065-5.101Q-11.889-4.918-11.795-4.660Q-11.701-4.402-11.701-4.070Q-11.701-3.978-11.783-3.957L-14.038-3.957L-14.038-3.896Q-14.038-3.308-13.755-2.925Q-13.471-2.542-12.904-2.542Q-12.582-2.542-12.314-2.735Q-12.046-2.928-11.957-3.243Q-11.950-3.284-11.875-3.298L-11.783-3.298Q-11.701-3.274-11.701-3.202Q-11.701-3.195-11.707-3.168Q-11.820-2.771-12.191-2.532Q-12.562-2.293-12.986-2.293Q-13.423-2.293-13.823-2.501Q-14.223-2.710-14.462-3.077Q-14.702-3.444-14.702-3.896M-14.032-4.166L-12.217-4.166Q-12.217-4.443-12.314-4.695Q-12.412-4.948-12.610-5.104Q-12.808-5.259-13.092-5.259Q-13.369-5.259-13.582-5.101Q-13.796-4.942-13.914-4.687Q-14.032-4.432-14.032-4.166M-11.113-3.872Q-11.113-4.200-10.978-4.501Q-10.843-4.801-10.607-5.022Q-10.371-5.242-10.067-5.362Q-9.763-5.482-9.438-5.482Q-8.932-5.482-8.583-5.379Q-8.235-5.277-8.235-4.901Q-8.235-4.754-8.332-4.653Q-8.430-4.552-8.577-4.552Q-8.730-4.552-8.829-4.651Q-8.929-4.750-8.929-4.901Q-8.929-5.089-8.788-5.181Q-8.990-5.232-9.431-5.232Q-9.787-5.232-10.016-5.036Q-10.245-4.839-10.345-4.530Q-10.446-4.220-10.446-3.872Q-10.446-3.523-10.320-3.217Q-10.193-2.911-9.939-2.727Q-9.684-2.542-9.328-2.542Q-9.106-2.542-8.922-2.626Q-8.737-2.710-8.602-2.865Q-8.467-3.021-8.409-3.229Q-8.395-3.284-8.341-3.284L-8.228-3.284Q-8.197-3.284-8.175-3.260Q-8.153-3.236-8.153-3.202L-8.153-3.181Q-8.238-2.894-8.426-2.696Q-8.614-2.498-8.879-2.395Q-9.144-2.293-9.438-2.293Q-9.869-2.293-10.256-2.499Q-10.644-2.706-10.879-3.069Q-11.113-3.431-11.113-3.872M-7.507-3.089Q-7.507-3.421-7.283-3.648Q-7.059-3.875-6.715-4.003Q-6.372-4.132-5.999-4.184Q-5.627-4.237-5.323-4.237L-5.323-4.490Q-5.323-4.695-5.430-4.875Q-5.538-5.054-5.719-5.157Q-5.900-5.259-6.109-5.259Q-6.516-5.259-6.751-5.167Q-6.662-5.130-6.616-5.046Q-6.570-4.962-6.570-4.860Q-6.570-4.764-6.616-4.685Q-6.662-4.607-6.743-4.562Q-6.823-4.518-6.912-4.518Q-7.062-4.518-7.163-4.615Q-7.264-4.713-7.264-4.860Q-7.264-5.482-6.109-5.482Q-5.897-5.482-5.647-5.418Q-5.398-5.355-5.196-5.236Q-4.995-5.116-4.868-4.931Q-4.742-4.747-4.742-4.504L-4.742-2.928Q-4.742-2.812-4.680-2.716Q-4.619-2.621-4.506-2.621Q-4.396-2.621-4.331-2.715Q-4.266-2.809-4.266-2.928L-4.266-3.376L-4-3.376L-4-2.928Q-4-2.658-4.227-2.493Q-4.454-2.327-4.735-2.327Q-4.943-2.327-5.080-2.481Q-5.217-2.634-5.241-2.850Q-5.388-2.583-5.670-2.438Q-5.952-2.293-6.276-2.293Q-6.553-2.293-6.837-2.368Q-7.120-2.443-7.314-2.622Q-7.507-2.802-7.507-3.089M-6.891-3.089Q-6.891-2.915-6.791-2.785Q-6.690-2.655-6.534-2.585Q-6.379-2.515-6.215-2.515Q-5.996-2.515-5.787-2.612Q-5.579-2.710-5.451-2.891Q-5.323-3.072-5.323-3.298L-5.323-4.026Q-5.647-4.026-6.013-3.935Q-6.379-3.844-6.635-3.632Q-6.891-3.421-6.891-3.089\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 -18.167)\">\u003Cpath d=\"M-3.443-1.226Q-3.313-1.158-3.176-1.158Q-3.005-1.158-2.855-1.247Q-2.704-1.336-2.593-1.481Q-2.482-1.626-2.404-1.794L-2.140-2.361L-3.309-4.887Q-3.384-5.034-3.514-5.066Q-3.644-5.099-3.877-5.099L-3.877-5.379L-2.356-5.379L-2.356-5.099Q-2.704-5.099-2.704-4.952Q-2.701-4.931-2.699-4.914Q-2.697-4.897-2.697-4.887L-1.840-3.028L-1.067-4.699Q-1.033-4.767-1.033-4.846Q-1.033-4.959-1.117-5.029Q-1.200-5.099-1.313-5.099L-1.313-5.379L-0.117-5.379L-0.117-5.099Q-0.336-5.099-0.508-4.995Q-0.681-4.890-0.773-4.699L-2.110-1.794Q-2.280-1.424-2.550-1.178Q-2.821-0.932-3.176-0.932Q-3.446-0.932-3.665-1.098Q-3.884-1.264-3.884-1.527Q-3.884-1.664-3.791-1.753Q-3.699-1.841-3.559-1.841Q-3.422-1.841-3.333-1.753Q-3.244-1.664-3.244-1.527Q-3.244-1.424-3.297-1.346Q-3.350-1.267-3.443-1.226\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 -18.167)\">\u003Cpath d=\"M8.286-3.168L3.453-3.168Q3.385-3.178 3.339-3.224Q3.293-3.270 3.293-3.342Q3.293-3.407 3.339-3.453Q3.385-3.499 3.453-3.509L8.286-3.509Q8.355-3.499 8.401-3.453Q8.447-3.407 8.447-3.342Q8.447-3.270 8.401-3.224Q8.355-3.178 8.286-3.168M8.286-4.706L3.453-4.706Q3.385-4.716 3.339-4.762Q3.293-4.808 3.293-4.880Q3.293-5.024 3.453-5.048L8.286-5.048Q8.447-5.024 8.447-4.880Q8.447-4.808 8.401-4.762Q8.355-4.716 8.286-4.706\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 -18.167)\">\u003Cpath d=\"M11.910-1.828Q11.910-2.074 12.107-2.258Q12.304-2.443 12.560-2.522Q12.423-2.634 12.351-2.795Q12.280-2.956 12.280-3.137Q12.280-3.458 12.491-3.704Q12.157-4.002 12.157-4.412Q12.157-4.873 12.546-5.160Q12.936-5.447 13.414-5.447Q13.886-5.447 14.221-5.201Q14.395-5.355 14.606-5.437Q14.816-5.519 15.045-5.519Q15.209-5.519 15.330-5.412Q15.451-5.304 15.451-5.140Q15.451-5.044 15.380-4.972Q15.308-4.901 15.216-4.901Q15.116-4.901 15.046-4.974Q14.976-5.048 14.976-5.147Q14.976-5.201 14.990-5.232L14.997-5.246Q15.004-5.266 15.012-5.277Q15.021-5.287 15.024-5.294Q14.669-5.294 14.382-5.071Q14.669-4.778 14.669-4.412Q14.669-4.097 14.484-3.865Q14.300-3.632 14.011-3.504Q13.722-3.376 13.414-3.376Q13.213-3.376 13.021-3.426Q12.830-3.475 12.652-3.585Q12.560-3.458 12.560-3.315Q12.560-3.133 12.688-2.998Q12.816-2.863 13.001-2.863L13.633-2.863Q14.081-2.863 14.450-2.792Q14.819-2.720 15.079-2.491Q15.339-2.262 15.339-1.828Q15.339-1.507 15.043-1.305Q14.747-1.103 14.344-1.014Q13.941-0.925 13.626-0.925Q13.308-0.925 12.905-1.014Q12.502-1.103 12.206-1.305Q11.910-1.507 11.910-1.828M12.365-1.828Q12.365-1.599 12.584-1.450Q12.803-1.301 13.095-1.233Q13.387-1.165 13.626-1.165Q13.790-1.165 13.999-1.201Q14.207-1.236 14.414-1.317Q14.621-1.397 14.752-1.525Q14.884-1.653 14.884-1.828Q14.884-2.180 14.503-2.274Q14.122-2.368 13.619-2.368L13.001-2.368Q12.762-2.368 12.563-2.217Q12.365-2.067 12.365-1.828M13.414-3.615Q14.081-3.615 14.081-4.412Q14.081-5.212 13.414-5.212Q12.744-5.212 12.744-4.412Q12.744-3.615 13.414-3.615M15.991-3.089Q15.991-3.421 16.215-3.648Q16.439-3.875 16.783-4.003Q17.126-4.132 17.499-4.184Q17.871-4.237 18.176-4.237L18.176-4.490Q18.176-4.695 18.068-4.875Q17.960-5.054 17.779-5.157Q17.598-5.259 17.389-5.259Q16.983-5.259 16.747-5.167Q16.836-5.130 16.882-5.046Q16.928-4.962 16.928-4.860Q16.928-4.764 16.882-4.685Q16.836-4.607 16.755-4.562Q16.675-4.518 16.586-4.518Q16.436-4.518 16.335-4.615Q16.234-4.713 16.234-4.860Q16.234-5.482 17.389-5.482Q17.601-5.482 17.851-5.418Q18.100-5.355 18.302-5.236Q18.504-5.116 18.630-4.931Q18.757-4.747 18.757-4.504L18.757-2.928Q18.757-2.812 18.818-2.716Q18.880-2.621 18.992-2.621Q19.102-2.621 19.167-2.715Q19.232-2.809 19.232-2.928L19.232-3.376L19.498-3.376L19.498-2.928Q19.498-2.658 19.271-2.493Q19.044-2.327 18.763-2.327Q18.555-2.327 18.418-2.481Q18.282-2.634 18.258-2.850Q18.111-2.583 17.829-2.438Q17.547-2.293 17.222-2.293Q16.945-2.293 16.661-2.368Q16.378-2.443 16.185-2.622Q15.991-2.802 15.991-3.089M16.607-3.089Q16.607-2.915 16.708-2.785Q16.808-2.655 16.964-2.585Q17.119-2.515 17.283-2.515Q17.502-2.515 17.711-2.612Q17.919-2.710 18.047-2.891Q18.176-3.072 18.176-3.298L18.176-4.026Q17.851-4.026 17.485-3.935Q17.119-3.844 16.863-3.632Q16.607-3.421 16.607-3.089M21.597-2.361L19.963-2.361L19.963-2.641Q20.192-2.641 20.341-2.675Q20.490-2.710 20.490-2.850L20.490-4.699Q20.490-4.969 20.382-5.030Q20.274-5.092 19.963-5.092L19.963-5.372L21.023-5.447L21.023-4.798Q21.194-5.106 21.498-5.277Q21.802-5.447 22.147-5.447Q22.547-5.447 22.824-5.307Q23.101-5.167 23.186-4.819Q23.354-5.112 23.653-5.280Q23.952-5.447 24.297-5.447Q24.803-5.447 25.087-5.224Q25.370-5 25.370-4.504L25.370-2.850Q25.370-2.713 25.519-2.677Q25.668-2.641 25.893-2.641L25.893-2.361L24.263-2.361L24.263-2.641Q24.489-2.641 24.639-2.677Q24.789-2.713 24.789-2.850L24.789-4.490Q24.789-4.825 24.670-5.025Q24.550-5.225 24.236-5.225Q23.966-5.225 23.731-5.089Q23.497-4.952 23.359-4.718Q23.220-4.484 23.220-4.210L23.220-2.850Q23.220-2.713 23.369-2.677Q23.518-2.641 23.743-2.641L23.743-2.361L22.113-2.361L22.113-2.641Q22.342-2.641 22.491-2.675Q22.639-2.710 22.639-2.850L22.639-4.490Q22.639-4.825 22.520-5.025Q22.400-5.225 22.086-5.225Q21.816-5.225 21.582-5.089Q21.347-4.952 21.209-4.718Q21.071-4.484 21.071-4.210L21.071-2.850Q21.071-2.713 21.221-2.677Q21.371-2.641 21.597-2.641L21.597-2.361M28.163-2.361L26.529-2.361L26.529-2.641Q26.758-2.641 26.907-2.675Q27.055-2.710 27.055-2.850L27.055-4.699Q27.055-4.969 26.948-5.030Q26.840-5.092 26.529-5.092L26.529-5.372L27.589-5.447L27.589-4.798Q27.760-5.106 28.064-5.277Q28.368-5.447 28.713-5.447Q29.113-5.447 29.390-5.307Q29.667-5.167 29.752-4.819Q29.920-5.112 30.219-5.280Q30.518-5.447 30.863-5.447Q31.369-5.447 31.653-5.224Q31.936-5 31.936-4.504L31.936-2.850Q31.936-2.713 32.085-2.677Q32.234-2.641 32.459-2.641L32.459-2.361L30.829-2.361L30.829-2.641Q31.054-2.641 31.205-2.677Q31.355-2.713 31.355-2.850L31.355-4.490Q31.355-4.825 31.236-5.025Q31.116-5.225 30.802-5.225Q30.532-5.225 30.297-5.089Q30.063-4.952 29.925-4.718Q29.786-4.484 29.786-4.210L29.786-2.850Q29.786-2.713 29.935-2.677Q30.084-2.641 30.309-2.641L30.309-2.361L28.679-2.361L28.679-2.641Q28.908-2.641 29.057-2.675Q29.205-2.710 29.205-2.850L29.205-4.490Q29.205-4.825 29.086-5.025Q28.966-5.225 28.652-5.225Q28.382-5.225 28.147-5.089Q27.913-4.952 27.775-4.718Q27.637-4.484 27.637-4.210L27.637-2.850Q27.637-2.713 27.787-2.677Q27.937-2.641 28.163-2.641L28.163-2.361M33.105-3.089Q33.105-3.421 33.329-3.648Q33.553-3.875 33.897-4.003Q34.240-4.132 34.613-4.184Q34.985-4.237 35.289-4.237L35.289-4.490Q35.289-4.695 35.182-4.875Q35.074-5.054 34.893-5.157Q34.712-5.259 34.503-5.259Q34.096-5.259 33.861-5.167Q33.950-5.130 33.996-5.046Q34.042-4.962 34.042-4.860Q34.042-4.764 33.996-4.685Q33.950-4.607 33.869-4.562Q33.789-4.518 33.700-4.518Q33.550-4.518 33.449-4.615Q33.348-4.713 33.348-4.860Q33.348-5.482 34.503-5.482Q34.715-5.482 34.965-5.418Q35.214-5.355 35.416-5.236Q35.617-5.116 35.744-4.931Q35.870-4.747 35.870-4.504L35.870-2.928Q35.870-2.812 35.932-2.716Q35.993-2.621 36.106-2.621Q36.216-2.621 36.281-2.715Q36.345-2.809 36.345-2.928L36.345-3.376L36.612-3.376L36.612-2.928Q36.612-2.658 36.385-2.493Q36.158-2.327 35.877-2.327Q35.669-2.327 35.532-2.481Q35.395-2.634 35.371-2.850Q35.224-2.583 34.942-2.438Q34.660-2.293 34.336-2.293Q34.059-2.293 33.775-2.368Q33.491-2.443 33.298-2.622Q33.105-2.802 33.105-3.089M33.720-3.089Q33.720-2.915 33.821-2.785Q33.922-2.655 34.078-2.585Q34.233-2.515 34.397-2.515Q34.616-2.515 34.825-2.612Q35.033-2.710 35.161-2.891Q35.289-3.072 35.289-3.298L35.289-4.026Q34.965-4.026 34.599-3.935Q34.233-3.844 33.977-3.632Q33.720-3.421 33.720-3.089\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 -18.167)\">\u003Cpath d=\"M41.411-2.361L39.808-2.361L39.808-2.641Q40.034-2.641 40.183-2.675Q40.331-2.710 40.331-2.850L40.331-6.469Q40.331-6.739 40.224-6.801Q40.116-6.862 39.808-6.862L39.808-7.143L40.885-7.218L40.885-2.850Q40.885-2.713 41.035-2.677Q41.186-2.641 41.411-2.641L41.411-2.361M42.064-3.089Q42.064-3.421 42.288-3.648Q42.512-3.875 42.856-4.003Q43.199-4.132 43.572-4.184Q43.944-4.237 44.248-4.237L44.248-4.490Q44.248-4.695 44.141-4.875Q44.033-5.054 43.852-5.157Q43.671-5.259 43.462-5.259Q43.055-5.259 42.820-5.167Q42.908-5.130 42.955-5.046Q43.001-4.962 43.001-4.860Q43.001-4.764 42.955-4.685Q42.908-4.607 42.828-4.562Q42.748-4.518 42.659-4.518Q42.509-4.518 42.408-4.615Q42.307-4.713 42.307-4.860Q42.307-5.482 43.462-5.482Q43.674-5.482 43.924-5.418Q44.173-5.355 44.375-5.236Q44.576-5.116 44.703-4.931Q44.829-4.747 44.829-4.504L44.829-2.928Q44.829-2.812 44.891-2.716Q44.952-2.621 45.065-2.621Q45.175-2.621 45.240-2.715Q45.304-2.809 45.304-2.928L45.304-3.376L45.571-3.376L45.571-2.928Q45.571-2.658 45.344-2.493Q45.117-2.327 44.836-2.327Q44.628-2.327 44.491-2.481Q44.354-2.634 44.330-2.850Q44.183-2.583 43.901-2.438Q43.619-2.293 43.295-2.293Q43.018-2.293 42.734-2.368Q42.450-2.443 42.257-2.622Q42.064-2.802 42.064-3.089M42.679-3.089Q42.679-2.915 42.780-2.785Q42.881-2.655 43.037-2.585Q43.192-2.515 43.356-2.515Q43.575-2.515 43.783-2.612Q43.992-2.710 44.120-2.891Q44.248-3.072 44.248-3.298L44.248-4.026Q43.924-4.026 43.558-3.935Q43.192-3.844 42.936-3.632Q42.679-3.421 42.679-3.089M47.670-2.361L46.036-2.361L46.036-2.641Q46.265-2.641 46.414-2.675Q46.562-2.710 46.562-2.850L46.562-4.699Q46.562-4.969 46.455-5.030Q46.347-5.092 46.036-5.092L46.036-5.372L47.096-5.447L47.096-4.798Q47.266-5.106 47.571-5.277Q47.875-5.447 48.220-5.447Q48.620-5.447 48.897-5.307Q49.174-5.167 49.259-4.819Q49.427-5.112 49.726-5.280Q50.025-5.447 50.370-5.447Q50.876-5.447 51.159-5.224Q51.443-5 51.443-4.504L51.443-2.850Q51.443-2.713 51.592-2.677Q51.741-2.641 51.966-2.641L51.966-2.361L50.336-2.361L50.336-2.641Q50.561-2.641 50.712-2.677Q50.862-2.713 50.862-2.850L50.862-4.490Q50.862-4.825 50.742-5.025Q50.623-5.225 50.308-5.225Q50.038-5.225 49.804-5.089Q49.570-4.952 49.432-4.718Q49.293-4.484 49.293-4.210L49.293-2.850Q49.293-2.713 49.442-2.677Q49.591-2.641 49.816-2.641L49.816-2.361L48.186-2.361L48.186-2.641Q48.415-2.641 48.564-2.675Q48.712-2.710 48.712-2.850L48.712-4.490Q48.712-4.825 48.593-5.025Q48.473-5.225 48.158-5.225Q47.888-5.225 47.654-5.089Q47.420-4.952 47.282-4.718Q47.143-4.484 47.143-4.210L47.143-2.850Q47.143-2.713 47.294-2.677Q47.444-2.641 47.670-2.641\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 -18.167)\">\u003Cpath d=\"M53.154-2.361L52.887-2.361L52.887-6.469Q52.887-6.739 52.780-6.801Q52.672-6.862 52.361-6.862L52.361-7.143L53.441-7.218L53.441-5.048Q53.650-5.239 53.935-5.343Q54.221-5.447 54.518-5.447Q54.836-5.447 55.133-5.326Q55.430-5.205 55.653-4.989Q55.875-4.774 56.001-4.489Q56.128-4.203 56.128-3.872Q56.128-3.427 55.888-3.063Q55.649-2.699 55.256-2.496Q54.863-2.293 54.419-2.293Q54.224-2.293 54.034-2.349Q53.845-2.405 53.684-2.510Q53.523-2.614 53.383-2.775L53.154-2.361M53.469-4.706L53.469-3.089Q53.605-2.829 53.846-2.672Q54.087-2.515 54.364-2.515Q54.658-2.515 54.870-2.622Q55.082-2.730 55.215-2.922Q55.348-3.113 55.407-3.352Q55.465-3.591 55.465-3.872Q55.465-4.231 55.371-4.535Q55.277-4.839 55.049-5.032Q54.822-5.225 54.456-5.225Q54.156-5.225 53.889-5.089Q53.622-4.952 53.469-4.706\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 -18.167)\">\u003Cpath d=\"M56.979-3.872Q56.979-4.210 57.120-4.501Q57.260-4.791 57.504-5.005Q57.748-5.218 58.053-5.333Q58.357-5.447 58.682-5.447Q58.952-5.447 59.215-5.348Q59.478-5.249 59.669-5.071L59.669-6.469Q59.669-6.739 59.562-6.801Q59.454-6.862 59.143-6.862L59.143-7.143L60.220-7.218L60.220-3.034Q60.220-2.846 60.274-2.763Q60.329-2.679 60.430-2.660Q60.531-2.641 60.746-2.641L60.746-2.361L59.639-2.293L59.639-2.710Q59.222-2.293 58.596-2.293Q58.165-2.293 57.793-2.505Q57.420-2.716 57.200-3.077Q56.979-3.438 56.979-3.872M58.654-2.515Q58.863-2.515 59.049-2.587Q59.235-2.658 59.389-2.795Q59.543-2.932 59.639-3.110L59.639-4.719Q59.553-4.866 59.408-4.986Q59.263-5.106 59.093-5.165Q58.924-5.225 58.743-5.225Q58.183-5.225 57.914-4.836Q57.646-4.446 57.646-3.865Q57.646-3.294 57.880-2.904Q58.114-2.515 58.654-2.515M61.454-3.089Q61.454-3.421 61.677-3.648Q61.901-3.875 62.245-4.003Q62.588-4.132 62.961-4.184Q63.333-4.237 63.638-4.237L63.638-4.490Q63.638-4.695 63.530-4.875Q63.422-5.054 63.241-5.157Q63.060-5.259 62.852-5.259Q62.445-5.259 62.209-5.167Q62.298-5.130 62.344-5.046Q62.390-4.962 62.390-4.860Q62.390-4.764 62.344-4.685Q62.298-4.607 62.217-4.562Q62.137-4.518 62.048-4.518Q61.898-4.518 61.797-4.615Q61.696-4.713 61.696-4.860Q61.696-5.482 62.852-5.482Q63.063-5.482 63.313-5.418Q63.562-5.355 63.764-5.236Q63.966-5.116 64.092-4.931Q64.219-4.747 64.219-4.504L64.219-2.928Q64.219-2.812 64.280-2.716Q64.342-2.621 64.455-2.621Q64.564-2.621 64.629-2.715Q64.694-2.809 64.694-2.928L64.694-3.376L64.960-3.376L64.960-2.928Q64.960-2.658 64.733-2.493Q64.506-2.327 64.226-2.327Q64.017-2.327 63.880-2.481Q63.744-2.634 63.720-2.850Q63.573-2.583 63.291-2.438Q63.009-2.293 62.684-2.293Q62.407-2.293 62.123-2.368Q61.840-2.443 61.647-2.622Q61.454-2.802 61.454-3.089M62.069-3.089Q62.069-2.915 62.170-2.785Q62.270-2.655 62.426-2.585Q62.582-2.515 62.746-2.515Q62.964-2.515 63.173-2.612Q63.381-2.710 63.509-2.891Q63.638-3.072 63.638-3.298L63.638-4.026Q63.313-4.026 62.947-3.935Q62.582-3.844 62.325-3.632Q62.069-3.421 62.069-3.089\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 -18.167)\">\u003Cpath d=\"M69.826-2.361L68.090-2.361L68.090-2.641Q68.319-2.641 68.468-2.675Q68.616-2.710 68.616-2.850L68.616-4.699Q68.616-4.969 68.509-5.030Q68.401-5.092 68.090-5.092L68.090-5.372L69.119-5.447L69.119-4.740Q69.249-5.048 69.491-5.247Q69.734-5.447 70.052-5.447Q70.271-5.447 70.442-5.323Q70.613-5.198 70.613-4.986Q70.613-4.849 70.513-4.750Q70.414-4.651 70.281-4.651Q70.144-4.651 70.045-4.750Q69.946-4.849 69.946-4.986Q69.946-5.126 70.045-5.225Q69.755-5.225 69.555-5.029Q69.355-4.832 69.262-4.538Q69.170-4.244 69.170-3.964L69.170-2.850Q69.170-2.641 69.826-2.641L69.826-2.361M72.879-2.361L71.245-2.361L71.245-2.641Q71.474-2.641 71.623-2.675Q71.771-2.710 71.771-2.850L71.771-6.469Q71.771-6.739 71.664-6.801Q71.556-6.862 71.245-6.862L71.245-7.143L72.325-7.218L72.325-4.832Q72.431-5.017 72.609-5.159Q72.786-5.300 72.995-5.374Q73.203-5.447 73.429-5.447Q73.935-5.447 74.219-5.224Q74.502-5 74.502-4.504L74.502-2.850Q74.502-2.713 74.651-2.677Q74.800-2.641 75.025-2.641L75.025-2.361L73.395-2.361L73.395-2.641Q73.624-2.641 73.772-2.675Q73.921-2.710 73.921-2.850L73.921-4.490Q73.921-4.825 73.802-5.025Q73.682-5.225 73.367-5.225Q73.097-5.225 72.863-5.089Q72.629-4.952 72.491-4.718Q72.352-4.484 72.352-4.210L72.352-2.850Q72.352-2.713 72.503-2.677Q72.653-2.641 72.879-2.641L72.879-2.361M75.572-3.844Q75.572-4.186 75.707-4.485Q75.842-4.784 76.081-5.008Q76.321-5.232 76.638-5.357Q76.956-5.482 77.288-5.482Q77.732-5.482 78.132-5.266Q78.532-5.051 78.766-4.673Q79-4.296 79-3.844Q79-3.503 78.858-3.219Q78.717-2.935 78.472-2.728Q78.228-2.522 77.918-2.407Q77.609-2.293 77.288-2.293Q76.857-2.293 76.456-2.494Q76.054-2.696 75.813-3.048Q75.572-3.400 75.572-3.844M77.288-2.542Q77.889-2.542 78.113-2.920Q78.337-3.298 78.337-3.930Q78.337-4.542 78.103-4.901Q77.869-5.259 77.288-5.259Q76.235-5.259 76.235-3.930Q76.235-3.298 76.461-2.920Q76.686-2.542 77.288-2.542\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(68.974 18.822)\">\u003Cpath d=\"M-19.077-3.872Q-19.077-4.210-18.936-4.501Q-18.796-4.791-18.552-5.005Q-18.308-5.218-18.003-5.333Q-17.699-5.447-17.374-5.447Q-17.104-5.447-16.841-5.348Q-16.578-5.249-16.387-5.071L-16.387-6.469Q-16.387-6.739-16.494-6.801Q-16.602-6.862-16.913-6.862L-16.913-7.143L-15.836-7.218L-15.836-3.034Q-15.836-2.846-15.782-2.763Q-15.727-2.679-15.626-2.660Q-15.525-2.641-15.310-2.641L-15.310-2.361L-16.417-2.293L-16.417-2.710Q-16.834-2.293-17.460-2.293Q-17.891-2.293-18.263-2.505Q-18.636-2.716-18.856-3.077Q-19.077-3.438-19.077-3.872M-17.402-2.515Q-17.193-2.515-17.007-2.587Q-16.821-2.658-16.667-2.795Q-16.513-2.932-16.417-3.110L-16.417-4.719Q-16.503-4.866-16.648-4.986Q-16.793-5.106-16.963-5.165Q-17.132-5.225-17.313-5.225Q-17.873-5.225-18.142-4.836Q-18.410-4.446-18.410-3.865Q-18.410-3.294-18.176-2.904Q-17.942-2.515-17.402-2.515M-14.702-3.896Q-14.702-4.217-14.577-4.506Q-14.452-4.795-14.226-5.018Q-14.001-5.242-13.705-5.362Q-13.410-5.482-13.092-5.482Q-12.764-5.482-12.502-5.382Q-12.241-5.283-12.065-5.101Q-11.889-4.918-11.795-4.660Q-11.701-4.402-11.701-4.070Q-11.701-3.978-11.783-3.957L-14.038-3.957L-14.038-3.896Q-14.038-3.308-13.755-2.925Q-13.471-2.542-12.904-2.542Q-12.582-2.542-12.314-2.735Q-12.046-2.928-11.957-3.243Q-11.950-3.284-11.875-3.298L-11.783-3.298Q-11.701-3.274-11.701-3.202Q-11.701-3.195-11.707-3.168Q-11.820-2.771-12.191-2.532Q-12.562-2.293-12.986-2.293Q-13.423-2.293-13.823-2.501Q-14.223-2.710-14.462-3.077Q-14.702-3.444-14.702-3.896M-14.032-4.166L-12.217-4.166Q-12.217-4.443-12.314-4.695Q-12.412-4.948-12.610-5.104Q-12.808-5.259-13.092-5.259Q-13.369-5.259-13.582-5.101Q-13.796-4.942-13.914-4.687Q-14.032-4.432-14.032-4.166M-11.113-3.872Q-11.113-4.200-10.978-4.501Q-10.843-4.801-10.607-5.022Q-10.371-5.242-10.067-5.362Q-9.763-5.482-9.438-5.482Q-8.932-5.482-8.583-5.379Q-8.235-5.277-8.235-4.901Q-8.235-4.754-8.332-4.653Q-8.430-4.552-8.577-4.552Q-8.730-4.552-8.829-4.651Q-8.929-4.750-8.929-4.901Q-8.929-5.089-8.788-5.181Q-8.990-5.232-9.431-5.232Q-9.787-5.232-10.016-5.036Q-10.245-4.839-10.345-4.530Q-10.446-4.220-10.446-3.872Q-10.446-3.523-10.320-3.217Q-10.193-2.911-9.939-2.727Q-9.684-2.542-9.328-2.542Q-9.106-2.542-8.922-2.626Q-8.737-2.710-8.602-2.865Q-8.467-3.021-8.409-3.229Q-8.395-3.284-8.341-3.284L-8.228-3.284Q-8.197-3.284-8.175-3.260Q-8.153-3.236-8.153-3.202L-8.153-3.181Q-8.238-2.894-8.426-2.696Q-8.614-2.498-8.879-2.395Q-9.144-2.293-9.438-2.293Q-9.869-2.293-10.256-2.499Q-10.644-2.706-10.879-3.069Q-11.113-3.431-11.113-3.872M-7.507-3.089Q-7.507-3.421-7.283-3.648Q-7.059-3.875-6.715-4.003Q-6.372-4.132-5.999-4.184Q-5.627-4.237-5.323-4.237L-5.323-4.490Q-5.323-4.695-5.430-4.875Q-5.538-5.054-5.719-5.157Q-5.900-5.259-6.109-5.259Q-6.516-5.259-6.751-5.167Q-6.662-5.130-6.616-5.046Q-6.570-4.962-6.570-4.860Q-6.570-4.764-6.616-4.685Q-6.662-4.607-6.743-4.562Q-6.823-4.518-6.912-4.518Q-7.062-4.518-7.163-4.615Q-7.264-4.713-7.264-4.860Q-7.264-5.482-6.109-5.482Q-5.897-5.482-5.647-5.418Q-5.398-5.355-5.196-5.236Q-4.995-5.116-4.868-4.931Q-4.742-4.747-4.742-4.504L-4.742-2.928Q-4.742-2.812-4.680-2.716Q-4.619-2.621-4.506-2.621Q-4.396-2.621-4.331-2.715Q-4.266-2.809-4.266-2.928L-4.266-3.376L-4-3.376L-4-2.928Q-4-2.658-4.227-2.493Q-4.454-2.327-4.735-2.327Q-4.943-2.327-5.080-2.481Q-5.217-2.634-5.241-2.850Q-5.388-2.583-5.670-2.438Q-5.952-2.293-6.276-2.293Q-6.553-2.293-6.837-2.368Q-7.120-2.443-7.314-2.622Q-7.507-2.802-7.507-3.089M-6.891-3.089Q-6.891-2.915-6.791-2.785Q-6.690-2.655-6.534-2.585Q-6.379-2.515-6.215-2.515Q-5.996-2.515-5.787-2.612Q-5.579-2.710-5.451-2.891Q-5.323-3.072-5.323-3.298L-5.323-4.026Q-5.647-4.026-6.013-3.935Q-6.379-3.844-6.635-3.632Q-6.891-3.421-6.891-3.089\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 18.822)\">\u003Cpath d=\"M-3.443-1.226Q-3.313-1.158-3.176-1.158Q-3.005-1.158-2.855-1.247Q-2.704-1.336-2.593-1.481Q-2.482-1.626-2.404-1.794L-2.140-2.361L-3.309-4.887Q-3.384-5.034-3.514-5.066Q-3.644-5.099-3.877-5.099L-3.877-5.379L-2.356-5.379L-2.356-5.099Q-2.704-5.099-2.704-4.952Q-2.701-4.931-2.699-4.914Q-2.697-4.897-2.697-4.887L-1.840-3.028L-1.067-4.699Q-1.033-4.767-1.033-4.846Q-1.033-4.959-1.117-5.029Q-1.200-5.099-1.313-5.099L-1.313-5.379L-0.117-5.379L-0.117-5.099Q-0.336-5.099-0.508-4.995Q-0.681-4.890-0.773-4.699L-2.110-1.794Q-2.280-1.424-2.550-1.178Q-2.821-0.932-3.176-0.932Q-3.446-0.932-3.665-1.098Q-3.884-1.264-3.884-1.527Q-3.884-1.664-3.791-1.753Q-3.699-1.841-3.559-1.841Q-3.422-1.841-3.333-1.753Q-3.244-1.664-3.244-1.527Q-3.244-1.424-3.297-1.346Q-3.350-1.267-3.443-1.226\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 18.822)\">\u003Cpath d=\"M8.286-3.168L3.453-3.168Q3.385-3.178 3.339-3.224Q3.293-3.270 3.293-3.342Q3.293-3.407 3.339-3.453Q3.385-3.499 3.453-3.509L8.286-3.509Q8.355-3.499 8.401-3.453Q8.447-3.407 8.447-3.342Q8.447-3.270 8.401-3.224Q8.355-3.178 8.286-3.168M8.286-4.706L3.453-4.706Q3.385-4.716 3.339-4.762Q3.293-4.808 3.293-4.880Q3.293-5.024 3.453-5.048L8.286-5.048Q8.447-5.024 8.447-4.880Q8.447-4.808 8.401-4.762Q8.355-4.716 8.286-4.706\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 18.822)\">\u003Cpath d=\"M11.910-1.828Q11.910-2.074 12.107-2.258Q12.304-2.443 12.560-2.522Q12.423-2.634 12.351-2.795Q12.280-2.956 12.280-3.137Q12.280-3.458 12.491-3.704Q12.157-4.002 12.157-4.412Q12.157-4.873 12.546-5.160Q12.936-5.447 13.414-5.447Q13.886-5.447 14.221-5.201Q14.395-5.355 14.606-5.437Q14.816-5.519 15.045-5.519Q15.209-5.519 15.330-5.412Q15.451-5.304 15.451-5.140Q15.451-5.044 15.380-4.972Q15.308-4.901 15.216-4.901Q15.116-4.901 15.046-4.974Q14.976-5.048 14.976-5.147Q14.976-5.201 14.990-5.232L14.997-5.246Q15.004-5.266 15.012-5.277Q15.021-5.287 15.024-5.294Q14.669-5.294 14.382-5.071Q14.669-4.778 14.669-4.412Q14.669-4.097 14.484-3.865Q14.300-3.632 14.011-3.504Q13.722-3.376 13.414-3.376Q13.213-3.376 13.021-3.426Q12.830-3.475 12.652-3.585Q12.560-3.458 12.560-3.315Q12.560-3.133 12.688-2.998Q12.816-2.863 13.001-2.863L13.633-2.863Q14.081-2.863 14.450-2.792Q14.819-2.720 15.079-2.491Q15.339-2.262 15.339-1.828Q15.339-1.507 15.043-1.305Q14.747-1.103 14.344-1.014Q13.941-0.925 13.626-0.925Q13.308-0.925 12.905-1.014Q12.502-1.103 12.206-1.305Q11.910-1.507 11.910-1.828M12.365-1.828Q12.365-1.599 12.584-1.450Q12.803-1.301 13.095-1.233Q13.387-1.165 13.626-1.165Q13.790-1.165 13.999-1.201Q14.207-1.236 14.414-1.317Q14.621-1.397 14.752-1.525Q14.884-1.653 14.884-1.828Q14.884-2.180 14.503-2.274Q14.122-2.368 13.619-2.368L13.001-2.368Q12.762-2.368 12.563-2.217Q12.365-2.067 12.365-1.828M13.414-3.615Q14.081-3.615 14.081-4.412Q14.081-5.212 13.414-5.212Q12.744-5.212 12.744-4.412Q12.744-3.615 13.414-3.615M15.991-3.089Q15.991-3.421 16.215-3.648Q16.439-3.875 16.783-4.003Q17.126-4.132 17.499-4.184Q17.871-4.237 18.176-4.237L18.176-4.490Q18.176-4.695 18.068-4.875Q17.960-5.054 17.779-5.157Q17.598-5.259 17.389-5.259Q16.983-5.259 16.747-5.167Q16.836-5.130 16.882-5.046Q16.928-4.962 16.928-4.860Q16.928-4.764 16.882-4.685Q16.836-4.607 16.755-4.562Q16.675-4.518 16.586-4.518Q16.436-4.518 16.335-4.615Q16.234-4.713 16.234-4.860Q16.234-5.482 17.389-5.482Q17.601-5.482 17.851-5.418Q18.100-5.355 18.302-5.236Q18.504-5.116 18.630-4.931Q18.757-4.747 18.757-4.504L18.757-2.928Q18.757-2.812 18.818-2.716Q18.880-2.621 18.992-2.621Q19.102-2.621 19.167-2.715Q19.232-2.809 19.232-2.928L19.232-3.376L19.498-3.376L19.498-2.928Q19.498-2.658 19.271-2.493Q19.044-2.327 18.763-2.327Q18.555-2.327 18.418-2.481Q18.282-2.634 18.258-2.850Q18.111-2.583 17.829-2.438Q17.547-2.293 17.222-2.293Q16.945-2.293 16.661-2.368Q16.378-2.443 16.185-2.622Q15.991-2.802 15.991-3.089M16.607-3.089Q16.607-2.915 16.708-2.785Q16.808-2.655 16.964-2.585Q17.119-2.515 17.283-2.515Q17.502-2.515 17.711-2.612Q17.919-2.710 18.047-2.891Q18.176-3.072 18.176-3.298L18.176-4.026Q17.851-4.026 17.485-3.935Q17.119-3.844 16.863-3.632Q16.607-3.421 16.607-3.089M21.597-2.361L19.963-2.361L19.963-2.641Q20.192-2.641 20.341-2.675Q20.490-2.710 20.490-2.850L20.490-4.699Q20.490-4.969 20.382-5.030Q20.274-5.092 19.963-5.092L19.963-5.372L21.023-5.447L21.023-4.798Q21.194-5.106 21.498-5.277Q21.802-5.447 22.147-5.447Q22.547-5.447 22.824-5.307Q23.101-5.167 23.186-4.819Q23.354-5.112 23.653-5.280Q23.952-5.447 24.297-5.447Q24.803-5.447 25.087-5.224Q25.370-5 25.370-4.504L25.370-2.850Q25.370-2.713 25.519-2.677Q25.668-2.641 25.893-2.641L25.893-2.361L24.263-2.361L24.263-2.641Q24.489-2.641 24.639-2.677Q24.789-2.713 24.789-2.850L24.789-4.490Q24.789-4.825 24.670-5.025Q24.550-5.225 24.236-5.225Q23.966-5.225 23.731-5.089Q23.497-4.952 23.359-4.718Q23.220-4.484 23.220-4.210L23.220-2.850Q23.220-2.713 23.369-2.677Q23.518-2.641 23.743-2.641L23.743-2.361L22.113-2.361L22.113-2.641Q22.342-2.641 22.491-2.675Q22.639-2.710 22.639-2.850L22.639-4.490Q22.639-4.825 22.520-5.025Q22.400-5.225 22.086-5.225Q21.816-5.225 21.582-5.089Q21.347-4.952 21.209-4.718Q21.071-4.484 21.071-4.210L21.071-2.850Q21.071-2.713 21.221-2.677Q21.371-2.641 21.597-2.641L21.597-2.361M28.163-2.361L26.529-2.361L26.529-2.641Q26.758-2.641 26.907-2.675Q27.055-2.710 27.055-2.850L27.055-4.699Q27.055-4.969 26.948-5.030Q26.840-5.092 26.529-5.092L26.529-5.372L27.589-5.447L27.589-4.798Q27.760-5.106 28.064-5.277Q28.368-5.447 28.713-5.447Q29.113-5.447 29.390-5.307Q29.667-5.167 29.752-4.819Q29.920-5.112 30.219-5.280Q30.518-5.447 30.863-5.447Q31.369-5.447 31.653-5.224Q31.936-5 31.936-4.504L31.936-2.850Q31.936-2.713 32.085-2.677Q32.234-2.641 32.459-2.641L32.459-2.361L30.829-2.361L30.829-2.641Q31.054-2.641 31.205-2.677Q31.355-2.713 31.355-2.850L31.355-4.490Q31.355-4.825 31.236-5.025Q31.116-5.225 30.802-5.225Q30.532-5.225 30.297-5.089Q30.063-4.952 29.925-4.718Q29.786-4.484 29.786-4.210L29.786-2.850Q29.786-2.713 29.935-2.677Q30.084-2.641 30.309-2.641L30.309-2.361L28.679-2.361L28.679-2.641Q28.908-2.641 29.057-2.675Q29.205-2.710 29.205-2.850L29.205-4.490Q29.205-4.825 29.086-5.025Q28.966-5.225 28.652-5.225Q28.382-5.225 28.147-5.089Q27.913-4.952 27.775-4.718Q27.637-4.484 27.637-4.210L27.637-2.850Q27.637-2.713 27.787-2.677Q27.937-2.641 28.163-2.641L28.163-2.361M33.105-3.089Q33.105-3.421 33.329-3.648Q33.553-3.875 33.897-4.003Q34.240-4.132 34.613-4.184Q34.985-4.237 35.289-4.237L35.289-4.490Q35.289-4.695 35.182-4.875Q35.074-5.054 34.893-5.157Q34.712-5.259 34.503-5.259Q34.096-5.259 33.861-5.167Q33.950-5.130 33.996-5.046Q34.042-4.962 34.042-4.860Q34.042-4.764 33.996-4.685Q33.950-4.607 33.869-4.562Q33.789-4.518 33.700-4.518Q33.550-4.518 33.449-4.615Q33.348-4.713 33.348-4.860Q33.348-5.482 34.503-5.482Q34.715-5.482 34.965-5.418Q35.214-5.355 35.416-5.236Q35.617-5.116 35.744-4.931Q35.870-4.747 35.870-4.504L35.870-2.928Q35.870-2.812 35.932-2.716Q35.993-2.621 36.106-2.621Q36.216-2.621 36.281-2.715Q36.345-2.809 36.345-2.928L36.345-3.376L36.612-3.376L36.612-2.928Q36.612-2.658 36.385-2.493Q36.158-2.327 35.877-2.327Q35.669-2.327 35.532-2.481Q35.395-2.634 35.371-2.850Q35.224-2.583 34.942-2.438Q34.660-2.293 34.336-2.293Q34.059-2.293 33.775-2.368Q33.491-2.443 33.298-2.622Q33.105-2.802 33.105-3.089M33.720-3.089Q33.720-2.915 33.821-2.785Q33.922-2.655 34.078-2.585Q34.233-2.515 34.397-2.515Q34.616-2.515 34.825-2.612Q35.033-2.710 35.161-2.891Q35.289-3.072 35.289-3.298L35.289-4.026Q34.965-4.026 34.599-3.935Q34.233-3.844 33.977-3.632Q33.720-3.421 33.720-3.089\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 18.822)\">\u003Cpath d=\"M41.411-2.361L39.808-2.361L39.808-2.641Q40.034-2.641 40.183-2.675Q40.331-2.710 40.331-2.850L40.331-6.469Q40.331-6.739 40.224-6.801Q40.116-6.862 39.808-6.862L39.808-7.143L40.885-7.218L40.885-2.850Q40.885-2.713 41.035-2.677Q41.186-2.641 41.411-2.641L41.411-2.361M42.064-3.089Q42.064-3.421 42.288-3.648Q42.512-3.875 42.856-4.003Q43.199-4.132 43.572-4.184Q43.944-4.237 44.248-4.237L44.248-4.490Q44.248-4.695 44.141-4.875Q44.033-5.054 43.852-5.157Q43.671-5.259 43.462-5.259Q43.055-5.259 42.820-5.167Q42.908-5.130 42.955-5.046Q43.001-4.962 43.001-4.860Q43.001-4.764 42.955-4.685Q42.908-4.607 42.828-4.562Q42.748-4.518 42.659-4.518Q42.509-4.518 42.408-4.615Q42.307-4.713 42.307-4.860Q42.307-5.482 43.462-5.482Q43.674-5.482 43.924-5.418Q44.173-5.355 44.375-5.236Q44.576-5.116 44.703-4.931Q44.829-4.747 44.829-4.504L44.829-2.928Q44.829-2.812 44.891-2.716Q44.952-2.621 45.065-2.621Q45.175-2.621 45.240-2.715Q45.304-2.809 45.304-2.928L45.304-3.376L45.571-3.376L45.571-2.928Q45.571-2.658 45.344-2.493Q45.117-2.327 44.836-2.327Q44.628-2.327 44.491-2.481Q44.354-2.634 44.330-2.850Q44.183-2.583 43.901-2.438Q43.619-2.293 43.295-2.293Q43.018-2.293 42.734-2.368Q42.450-2.443 42.257-2.622Q42.064-2.802 42.064-3.089M42.679-3.089Q42.679-2.915 42.780-2.785Q42.881-2.655 43.037-2.585Q43.192-2.515 43.356-2.515Q43.575-2.515 43.783-2.612Q43.992-2.710 44.120-2.891Q44.248-3.072 44.248-3.298L44.248-4.026Q43.924-4.026 43.558-3.935Q43.192-3.844 42.936-3.632Q42.679-3.421 42.679-3.089M47.670-2.361L46.036-2.361L46.036-2.641Q46.265-2.641 46.414-2.675Q46.562-2.710 46.562-2.850L46.562-4.699Q46.562-4.969 46.455-5.030Q46.347-5.092 46.036-5.092L46.036-5.372L47.096-5.447L47.096-4.798Q47.266-5.106 47.571-5.277Q47.875-5.447 48.220-5.447Q48.620-5.447 48.897-5.307Q49.174-5.167 49.259-4.819Q49.427-5.112 49.726-5.280Q50.025-5.447 50.370-5.447Q50.876-5.447 51.159-5.224Q51.443-5 51.443-4.504L51.443-2.850Q51.443-2.713 51.592-2.677Q51.741-2.641 51.966-2.641L51.966-2.361L50.336-2.361L50.336-2.641Q50.561-2.641 50.712-2.677Q50.862-2.713 50.862-2.850L50.862-4.490Q50.862-4.825 50.742-5.025Q50.623-5.225 50.308-5.225Q50.038-5.225 49.804-5.089Q49.570-4.952 49.432-4.718Q49.293-4.484 49.293-4.210L49.293-2.850Q49.293-2.713 49.442-2.677Q49.591-2.641 49.816-2.641L49.816-2.361L48.186-2.361L48.186-2.641Q48.415-2.641 48.564-2.675Q48.712-2.710 48.712-2.850L48.712-4.490Q48.712-4.825 48.593-5.025Q48.473-5.225 48.158-5.225Q47.888-5.225 47.654-5.089Q47.420-4.952 47.282-4.718Q47.143-4.484 47.143-4.210L47.143-2.850Q47.143-2.713 47.294-2.677Q47.444-2.641 47.670-2.641\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 18.822)\">\u003Cpath d=\"M53.154-2.361L52.887-2.361L52.887-6.469Q52.887-6.739 52.780-6.801Q52.672-6.862 52.361-6.862L52.361-7.143L53.441-7.218L53.441-5.048Q53.650-5.239 53.935-5.343Q54.221-5.447 54.518-5.447Q54.836-5.447 55.133-5.326Q55.430-5.205 55.653-4.989Q55.875-4.774 56.001-4.489Q56.128-4.203 56.128-3.872Q56.128-3.427 55.888-3.063Q55.649-2.699 55.256-2.496Q54.863-2.293 54.419-2.293Q54.224-2.293 54.034-2.349Q53.845-2.405 53.684-2.510Q53.523-2.614 53.383-2.775L53.154-2.361M53.469-4.706L53.469-3.089Q53.605-2.829 53.846-2.672Q54.087-2.515 54.364-2.515Q54.658-2.515 54.870-2.622Q55.082-2.730 55.215-2.922Q55.348-3.113 55.407-3.352Q55.465-3.591 55.465-3.872Q55.465-4.231 55.371-4.535Q55.277-4.839 55.049-5.032Q54.822-5.225 54.456-5.225Q54.156-5.225 53.889-5.089Q53.622-4.952 53.469-4.706\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 18.822)\">\u003Cpath d=\"M56.979-3.872Q56.979-4.210 57.120-4.501Q57.260-4.791 57.504-5.005Q57.748-5.218 58.053-5.333Q58.357-5.447 58.682-5.447Q58.952-5.447 59.215-5.348Q59.478-5.249 59.669-5.071L59.669-6.469Q59.669-6.739 59.562-6.801Q59.454-6.862 59.143-6.862L59.143-7.143L60.220-7.218L60.220-3.034Q60.220-2.846 60.274-2.763Q60.329-2.679 60.430-2.660Q60.531-2.641 60.746-2.641L60.746-2.361L59.639-2.293L59.639-2.710Q59.222-2.293 58.596-2.293Q58.165-2.293 57.793-2.505Q57.420-2.716 57.200-3.077Q56.979-3.438 56.979-3.872M58.654-2.515Q58.863-2.515 59.049-2.587Q59.235-2.658 59.389-2.795Q59.543-2.932 59.639-3.110L59.639-4.719Q59.553-4.866 59.408-4.986Q59.263-5.106 59.093-5.165Q58.924-5.225 58.743-5.225Q58.183-5.225 57.914-4.836Q57.646-4.446 57.646-3.865Q57.646-3.294 57.880-2.904Q58.114-2.515 58.654-2.515M61.454-3.089Q61.454-3.421 61.677-3.648Q61.901-3.875 62.245-4.003Q62.588-4.132 62.961-4.184Q63.333-4.237 63.638-4.237L63.638-4.490Q63.638-4.695 63.530-4.875Q63.422-5.054 63.241-5.157Q63.060-5.259 62.852-5.259Q62.445-5.259 62.209-5.167Q62.298-5.130 62.344-5.046Q62.390-4.962 62.390-4.860Q62.390-4.764 62.344-4.685Q62.298-4.607 62.217-4.562Q62.137-4.518 62.048-4.518Q61.898-4.518 61.797-4.615Q61.696-4.713 61.696-4.860Q61.696-5.482 62.852-5.482Q63.063-5.482 63.313-5.418Q63.562-5.355 63.764-5.236Q63.966-5.116 64.092-4.931Q64.219-4.747 64.219-4.504L64.219-2.928Q64.219-2.812 64.280-2.716Q64.342-2.621 64.455-2.621Q64.564-2.621 64.629-2.715Q64.694-2.809 64.694-2.928L64.694-3.376L64.960-3.376L64.960-2.928Q64.960-2.658 64.733-2.493Q64.506-2.327 64.226-2.327Q64.017-2.327 63.880-2.481Q63.744-2.634 63.720-2.850Q63.573-2.583 63.291-2.438Q63.009-2.293 62.684-2.293Q62.407-2.293 62.123-2.368Q61.840-2.443 61.647-2.622Q61.454-2.802 61.454-3.089M62.069-3.089Q62.069-2.915 62.170-2.785Q62.270-2.655 62.426-2.585Q62.582-2.515 62.746-2.515Q62.964-2.515 63.173-2.612Q63.381-2.710 63.509-2.891Q63.638-3.072 63.638-3.298L63.638-4.026Q63.313-4.026 62.947-3.935Q62.582-3.844 62.325-3.632Q62.069-3.421 62.069-3.089\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(68.974 18.822)\">\u003Cpath d=\"M69.720-1.004L68.090-1.004L68.090-1.284Q68.319-1.284 68.468-1.319Q68.616-1.353 68.616-1.493L68.616-4.839Q68.616-5.010 68.480-5.051Q68.343-5.092 68.090-5.092L68.090-5.372L69.170-5.447L69.170-5.041Q69.392-5.242 69.679-5.345Q69.967-5.447 70.274-5.447Q70.701-5.447 71.065-5.234Q71.429-5.020 71.643-4.656Q71.857-4.292 71.857-3.872Q71.857-3.427 71.617-3.063Q71.378-2.699 70.985-2.496Q70.592-2.293 70.148-2.293Q69.881-2.293 69.633-2.393Q69.386-2.494 69.198-2.675L69.198-1.493Q69.198-1.356 69.346-1.320Q69.495-1.284 69.720-1.284L69.720-1.004M69.198-4.692L69.198-3.082Q69.331-2.829 69.574-2.672Q69.816-2.515 70.093-2.515Q70.421-2.515 70.674-2.716Q70.927-2.918 71.060-3.236Q71.194-3.554 71.194-3.872Q71.194-4.101 71.129-4.330Q71.064-4.559 70.936-4.757Q70.807-4.955 70.613-5.075Q70.418-5.194 70.185-5.194Q69.891-5.194 69.623-5.065Q69.355-4.935 69.198-4.692M74.109-2.361L72.557-2.361L72.557-2.641Q72.783-2.641 72.932-2.675Q73.080-2.710 73.080-2.850L73.080-4.699Q73.080-4.887 73.033-4.971Q72.985-5.054 72.887-5.073Q72.790-5.092 72.578-5.092L72.578-5.372L73.634-5.447L73.634-2.850Q73.634-2.710 73.766-2.675Q73.897-2.641 74.109-2.641L74.109-2.361M72.838-6.668Q72.838-6.839 72.961-6.958Q73.084-7.078 73.255-7.078Q73.422-7.078 73.545-6.958Q73.668-6.839 73.668-6.668Q73.668-6.493 73.545-6.370Q73.422-6.247 73.255-6.247Q73.084-6.247 72.961-6.370Q72.838-6.493 72.838-6.668M76.885-0.611Q76.334-1.011 75.963-1.566Q75.593-2.122 75.411-2.768Q75.230-3.414 75.230-4.111Q75.230-4.624 75.331-5.119Q75.432-5.615 75.637-6.066Q75.842-6.517 76.155-6.909Q76.468-7.300 76.885-7.604Q76.895-7.608 76.902-7.609Q76.908-7.611 76.919-7.611L76.987-7.611Q77.021-7.611 77.043-7.587Q77.066-7.563 77.066-7.526Q77.066-7.481 77.038-7.464Q76.690-7.163 76.437-6.779Q76.184-6.394 76.032-5.953Q75.880-5.512 75.808-5.056Q75.736-4.600 75.736-4.111Q75.736-3.110 76.045-2.223Q76.355-1.336 77.038-0.751Q77.066-0.734 77.066-0.690Q77.066-0.652 77.043-0.628Q77.021-0.604 76.987-0.604L76.919-0.604Q76.912-0.608 76.903-0.609Q76.895-0.611 76.885-0.611M79.465-2.361L77.876-2.361L77.876-2.641Q78.518-2.641 78.676-3.041L80.320-7.256Q80.354-7.351 80.467-7.351L80.549-7.351Q80.658-7.351 80.699-7.256L82.418-2.850Q82.487-2.710 82.676-2.675Q82.866-2.641 83.139-2.641L83.139-2.361L81.140-2.361L81.140-2.641Q81.704-2.641 81.704-2.816Q81.704-2.833 81.702-2.840Q81.700-2.846 81.697-2.850L81.277-3.916L79.318-3.916L78.976-3.041Q78.963-3.041 78.963-2.963Q78.963-2.802 79.125-2.722Q79.287-2.641 79.465-2.641L79.465-2.361M80.299-6.435L79.431-4.196L81.174-4.196L80.299-6.435M84.175-2.781Q84.175-2.949 84.298-3.072Q84.421-3.195 84.595-3.195Q84.763-3.195 84.886-3.072Q85.009-2.949 85.009-2.781Q85.009-2.607 84.886-2.484Q84.763-2.361 84.595-2.361Q84.421-2.361 84.298-2.484Q84.175-2.607 84.175-2.781M84.175-4.965Q84.175-5.133 84.298-5.256Q84.421-5.379 84.595-5.379Q84.763-5.379 84.886-5.256Q85.009-5.133 85.009-4.965Q85.009-4.791 84.886-4.668Q84.763-4.545 84.595-4.545Q84.421-4.545 84.298-4.668Q84.175-4.791 84.175-4.965M86.212-2.299L86.212-3.872Q86.212-3.899 86.238-3.925Q86.263-3.950 86.291-3.950L86.404-3.950Q86.431-3.950 86.455-3.923Q86.479-3.896 86.479-3.872Q86.479-3.527 86.610-3.263Q86.742-3 86.971-2.831Q87.200-2.662 87.502-2.581Q87.805-2.501 88.147-2.501Q88.413-2.501 88.649-2.629Q88.885-2.757 89.030-2.980Q89.176-3.202 89.176-3.468Q89.176-3.691 89.070-3.887Q88.964-4.084 88.783-4.219Q88.601-4.354 88.376-4.405L87.347-4.637Q87.036-4.709 86.776-4.895Q86.516-5.082 86.364-5.353Q86.212-5.625 86.212-5.940Q86.212-6.326 86.426-6.633Q86.639-6.941 86.986-7.112Q87.333-7.283 87.713-7.283Q87.942-7.283 88.171-7.230Q88.400-7.177 88.598-7.069Q88.796-6.962 88.950-6.798L89.244-7.238Q89.268-7.283 89.309-7.283L89.357-7.283Q89.387-7.283 89.410-7.257Q89.432-7.232 89.432-7.204L89.432-5.629Q89.432-5.608 89.408-5.581Q89.384-5.553 89.357-5.553L89.244-5.553Q89.182-5.553 89.169-5.629Q89.128-6.042 88.947-6.362Q88.765-6.681 88.454-6.856Q88.143-7.030 87.713-7.030Q87.463-7.030 87.224-6.919Q86.985-6.808 86.834-6.610Q86.684-6.411 86.684-6.148Q86.684-5.936 86.792-5.755Q86.899-5.574 87.075-5.454Q87.251-5.335 87.460-5.294L88.489-5.065Q88.806-4.993 89.073-4.788Q89.340-4.583 89.492-4.289Q89.644-3.995 89.644-3.663Q89.644-3.270 89.439-2.935Q89.234-2.600 88.888-2.411Q88.543-2.221 88.147-2.221Q87.726-2.221 87.347-2.334Q86.968-2.446 86.698-2.696L86.404-2.262Q86.376-2.221 86.339-2.221L86.291-2.221Q86.263-2.221 86.238-2.246Q86.212-2.272 86.212-2.299M90.775-0.604L90.707-0.604Q90.673-0.604 90.650-0.630Q90.628-0.655 90.628-0.690Q90.628-0.734 90.659-0.751Q91.014-1.055 91.264-1.445Q91.513-1.835 91.666-2.267Q91.818-2.699 91.888-3.168Q91.958-3.636 91.958-4.111Q91.958-4.590 91.888-5.056Q91.818-5.523 91.664-5.958Q91.510-6.394 91.259-6.782Q91.008-7.170 90.659-7.464Q90.628-7.481 90.628-7.526Q90.628-7.560 90.650-7.585Q90.673-7.611 90.707-7.611L90.775-7.611Q90.785-7.611 90.794-7.609Q90.803-7.608 90.813-7.604Q91.356-7.204 91.729-6.651Q92.101-6.097 92.283-5.451Q92.464-4.805 92.464-4.111Q92.464-3.410 92.283-2.763Q92.101-2.115 91.727-1.561Q91.353-1.007 90.813-0.611Q90.803-0.611 90.794-0.609Q90.785-0.608 90.775-0.604\" 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-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(202.702 -18.167)\">\u003Cpath d=\"M-17.327-2.361L-19.063-2.361L-19.063-2.641Q-18.834-2.641-18.685-2.675Q-18.537-2.710-18.537-2.850L-18.537-4.699Q-18.537-4.969-18.644-5.030Q-18.752-5.092-19.063-5.092L-19.063-5.372L-18.034-5.447L-18.034-4.740Q-17.904-5.048-17.662-5.247Q-17.419-5.447-17.101-5.447Q-16.882-5.447-16.711-5.323Q-16.540-5.198-16.540-4.986Q-16.540-4.849-16.640-4.750Q-16.739-4.651-16.872-4.651Q-17.009-4.651-17.108-4.750Q-17.207-4.849-17.207-4.986Q-17.207-5.126-17.108-5.225Q-17.398-5.225-17.598-5.029Q-17.798-4.832-17.891-4.538Q-17.983-4.244-17.983-3.964L-17.983-2.850Q-17.983-2.641-17.327-2.641L-17.327-2.361M-14.274-2.361L-15.908-2.361L-15.908-2.641Q-15.679-2.641-15.530-2.675Q-15.382-2.710-15.382-2.850L-15.382-6.469Q-15.382-6.739-15.489-6.801Q-15.597-6.862-15.908-6.862L-15.908-7.143L-14.828-7.218L-14.828-4.832Q-14.722-5.017-14.544-5.159Q-14.367-5.300-14.158-5.374Q-13.950-5.447-13.724-5.447Q-13.218-5.447-12.934-5.224Q-12.651-5-12.651-4.504L-12.651-2.850Q-12.651-2.713-12.502-2.677Q-12.353-2.641-12.128-2.641L-12.128-2.361L-13.758-2.361L-13.758-2.641Q-13.529-2.641-13.381-2.675Q-13.232-2.710-13.232-2.850L-13.232-4.490Q-13.232-4.825-13.351-5.025Q-13.471-5.225-13.786-5.225Q-14.056-5.225-14.290-5.089Q-14.524-4.952-14.662-4.718Q-14.801-4.484-14.801-4.210L-14.801-2.850Q-14.801-2.713-14.650-2.677Q-14.500-2.641-14.274-2.641L-14.274-2.361M-11.581-3.844Q-11.581-4.186-11.446-4.485Q-11.311-4.784-11.072-5.008Q-10.832-5.232-10.515-5.357Q-10.197-5.482-9.865-5.482Q-9.421-5.482-9.021-5.266Q-8.621-5.051-8.387-4.673Q-8.153-4.296-8.153-3.844Q-8.153-3.503-8.295-3.219Q-8.436-2.935-8.681-2.728Q-8.925-2.522-9.235-2.407Q-9.544-2.293-9.865-2.293Q-10.296-2.293-10.697-2.494Q-11.099-2.696-11.340-3.048Q-11.581-3.400-11.581-3.844M-9.865-2.542Q-9.264-2.542-9.040-2.920Q-8.816-3.298-8.816-3.930Q-8.816-4.542-9.050-4.901Q-9.284-5.259-9.865-5.259Q-10.918-5.259-10.918-3.930Q-10.918-3.298-10.692-2.920Q-10.467-2.542-9.865-2.542\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(202.702 -18.167)\">\u003Cpath d=\"M-4.855-3.872Q-4.855-4.200-4.720-4.501Q-4.585-4.801-4.349-5.022Q-4.113-5.242-3.809-5.362Q-3.504-5.482-3.180-5.482Q-2.674-5.482-2.325-5.379Q-1.977-5.277-1.977-4.901Q-1.977-4.754-2.074-4.653Q-2.171-4.552-2.318-4.552Q-2.472-4.552-2.571-4.651Q-2.670-4.750-2.670-4.901Q-2.670-5.089-2.530-5.181Q-2.732-5.232-3.173-5.232Q-3.528-5.232-3.757-5.036Q-3.986-4.839-4.087-4.530Q-4.188-4.220-4.188-3.872Q-4.188-3.523-4.062-3.217Q-3.935-2.911-3.680-2.727Q-3.426-2.542-3.070-2.542Q-2.848-2.542-2.664-2.626Q-2.479-2.710-2.344-2.865Q-2.209-3.021-2.151-3.229Q-2.137-3.284-2.083-3.284L-1.970-3.284Q-1.939-3.284-1.917-3.260Q-1.895-3.236-1.895-3.202L-1.895-3.181Q-1.980-2.894-2.168-2.696Q-2.356-2.498-2.621-2.395Q-2.886-2.293-3.180-2.293Q-3.610-2.293-3.998-2.499Q-4.386-2.706-4.620-3.069Q-4.855-3.431-4.855-3.872M-1.249-3.089Q-1.249-3.421-1.025-3.648Q-0.801-3.875-0.457-4.003Q-0.114-4.132 0.259-4.184Q0.631-4.237 0.935-4.237L0.935-4.490Q0.935-4.695 0.828-4.875Q0.720-5.054 0.539-5.157Q0.358-5.259 0.149-5.259Q-0.257-5.259-0.493-5.167Q-0.404-5.130-0.358-5.046Q-0.312-4.962-0.312-4.860Q-0.312-4.764-0.358-4.685Q-0.404-4.607-0.485-4.562Q-0.565-4.518-0.654-4.518Q-0.804-4.518-0.905-4.615Q-1.006-4.713-1.006-4.860Q-1.006-5.482 0.149-5.482Q0.361-5.482 0.611-5.418Q0.860-5.355 1.062-5.236Q1.264-5.116 1.390-4.931Q1.517-4.747 1.517-4.504L1.517-2.928Q1.517-2.812 1.578-2.716Q1.640-2.621 1.752-2.621Q1.862-2.621 1.927-2.715Q1.992-2.809 1.992-2.928L1.992-3.376L2.258-3.376L2.258-2.928Q2.258-2.658 2.031-2.493Q1.804-2.327 1.523-2.327Q1.315-2.327 1.178-2.481Q1.041-2.634 1.018-2.850Q0.871-2.583 0.589-2.438Q0.307-2.293-0.018-2.293Q-0.295-2.293-0.579-2.368Q-0.862-2.443-1.055-2.622Q-1.249-2.802-1.249-3.089M-0.633-3.089Q-0.633-2.915-0.533-2.785Q-0.432-2.655-0.276-2.585Q-0.121-2.515 0.043-2.515Q0.262-2.515 0.471-2.612Q0.679-2.710 0.807-2.891Q0.935-3.072 0.935-3.298L0.935-4.026Q0.611-4.026 0.245-3.935Q-0.121-3.844-0.377-3.632Q-0.633-3.421-0.633-3.089M4.357-2.361L2.723-2.361L2.723-2.641Q2.952-2.641 3.101-2.675Q3.249-2.710 3.249-2.850L3.249-4.699Q3.249-4.969 3.142-5.030Q3.034-5.092 2.723-5.092L2.723-5.372L3.783-5.447L3.783-4.798Q3.954-5.106 4.258-5.277Q4.562-5.447 4.907-5.447Q5.413-5.447 5.697-5.224Q5.980-5 5.980-4.504L5.980-2.850Q5.980-2.713 6.129-2.677Q6.278-2.641 6.503-2.641L6.503-2.361L4.873-2.361L4.873-2.641Q5.102-2.641 5.251-2.675Q5.399-2.710 5.399-2.850L5.399-4.490Q5.399-4.825 5.280-5.025Q5.160-5.225 4.846-5.225Q4.576-5.225 4.341-5.089Q4.107-4.952 3.969-4.718Q3.831-4.484 3.831-4.210L3.831-2.850Q3.831-2.713 3.981-2.677Q4.131-2.641 4.357-2.641\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(202.702 -18.167)\">\u003Cpath d=\"M10.605-2.361L10.338-2.361L10.338-6.469Q10.338-6.739 10.231-6.801Q10.123-6.862 9.812-6.862L9.812-7.143L10.892-7.218L10.892-5.048Q11.101-5.239 11.386-5.343Q11.672-5.447 11.969-5.447Q12.287-5.447 12.584-5.326Q12.881-5.205 13.104-4.989Q13.326-4.774 13.452-4.489Q13.579-4.203 13.579-3.872Q13.579-3.427 13.339-3.063Q13.100-2.699 12.707-2.496Q12.314-2.293 11.870-2.293Q11.675-2.293 11.485-2.349Q11.296-2.405 11.135-2.510Q10.974-2.614 10.834-2.775L10.605-2.361M10.920-4.706L10.920-3.089Q11.056-2.829 11.297-2.672Q11.538-2.515 11.815-2.515Q12.109-2.515 12.321-2.622Q12.533-2.730 12.666-2.922Q12.799-3.113 12.858-3.352Q12.916-3.591 12.916-3.872Q12.916-4.231 12.822-4.535Q12.728-4.839 12.500-5.032Q12.273-5.225 11.907-5.225Q11.607-5.225 11.340-5.089Q11.073-4.952 10.920-4.706\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(202.702 -18.167)\">\u003Cpath d=\"M14.389-3.896Q14.389-4.217 14.514-4.506Q14.639-4.795 14.865-5.018Q15.090-5.242 15.386-5.362Q15.681-5.482 15.999-5.482Q16.327-5.482 16.589-5.382Q16.850-5.283 17.026-5.101Q17.202-4.918 17.296-4.660Q17.390-4.402 17.390-4.070Q17.390-3.978 17.308-3.957L15.053-3.957L15.053-3.896Q15.053-3.308 15.336-2.925Q15.620-2.542 16.187-2.542Q16.509-2.542 16.777-2.735Q17.045-2.928 17.134-3.243Q17.141-3.284 17.216-3.298L17.308-3.298Q17.390-3.274 17.390-3.202Q17.390-3.195 17.384-3.168Q17.271-2.771 16.900-2.532Q16.529-2.293 16.105-2.293Q15.668-2.293 15.268-2.501Q14.868-2.710 14.629-3.077Q14.389-3.444 14.389-3.896M15.059-4.166L16.874-4.166Q16.874-4.443 16.777-4.695Q16.679-4.948 16.481-5.104Q16.283-5.259 15.999-5.259Q15.722-5.259 15.509-5.101Q15.295-4.942 15.177-4.687Q15.059-4.432 15.059-4.166\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(202.702 -18.167)\">\u003Cpath d=\"M21.487-2.361L21.220-2.361L21.220-6.469Q21.220-6.739 21.113-6.801Q21.005-6.862 20.694-6.862L20.694-7.143L21.774-7.218L21.774-5.048Q21.983-5.239 22.268-5.343Q22.553-5.447 22.851-5.447Q23.169-5.447 23.466-5.326Q23.763-5.205 23.986-4.989Q24.208-4.774 24.334-4.489Q24.461-4.203 24.461-3.872Q24.461-3.427 24.221-3.063Q23.982-2.699 23.589-2.496Q23.196-2.293 22.752-2.293Q22.557-2.293 22.367-2.349Q22.178-2.405 22.017-2.510Q21.856-2.614 21.716-2.775L21.487-2.361M21.802-4.706L21.802-3.089Q21.938-2.829 22.179-2.672Q22.420-2.515 22.697-2.515Q22.991-2.515 23.203-2.622Q23.415-2.730 23.548-2.922Q23.681-3.113 23.740-3.352Q23.798-3.591 23.798-3.872Q23.798-4.231 23.704-4.535Q23.610-4.839 23.382-5.032Q23.155-5.225 22.789-5.225Q22.489-5.225 22.222-5.089Q21.955-4.952 21.802-4.706M26.713-2.361L25.161-2.361L25.161-2.641Q25.387-2.641 25.536-2.675Q25.684-2.710 25.684-2.850L25.684-4.699Q25.684-4.887 25.637-4.971Q25.589-5.054 25.491-5.073Q25.394-5.092 25.182-5.092L25.182-5.372L26.238-5.447L26.238-2.850Q26.238-2.710 26.370-2.675Q26.501-2.641 26.713-2.641L26.713-2.361M25.442-6.668Q25.442-6.839 25.565-6.958Q25.688-7.078 25.859-7.078Q26.026-7.078 26.149-6.958Q26.272-6.839 26.272-6.668Q26.272-6.493 26.149-6.370Q26.026-6.247 25.859-6.247Q25.688-6.247 25.565-6.370Q25.442-6.493 25.442-6.668M27.318-1.828Q27.318-2.074 27.515-2.258Q27.711-2.443 27.968-2.522Q27.831-2.634 27.759-2.795Q27.687-2.956 27.687-3.137Q27.687-3.458 27.899-3.704Q27.564-4.002 27.564-4.412Q27.564-4.873 27.954-5.160Q28.344-5.447 28.822-5.447Q29.294-5.447 29.629-5.201Q29.803-5.355 30.013-5.437Q30.223-5.519 30.452-5.519Q30.616-5.519 30.738-5.412Q30.859-5.304 30.859-5.140Q30.859-5.044 30.787-4.972Q30.716-4.901 30.623-4.901Q30.524-4.901 30.454-4.974Q30.384-5.048 30.384-5.147Q30.384-5.201 30.398-5.232L30.405-5.246Q30.411-5.266 30.420-5.277Q30.428-5.287 30.432-5.294Q30.076-5.294 29.789-5.071Q30.076-4.778 30.076-4.412Q30.076-4.097 29.892-3.865Q29.707-3.632 29.418-3.504Q29.130-3.376 28.822-3.376Q28.620-3.376 28.429-3.426Q28.238-3.475 28.060-3.585Q27.968-3.458 27.968-3.315Q27.968-3.133 28.096-2.998Q28.224-2.863 28.408-2.863L29.041-2.863Q29.489-2.863 29.858-2.792Q30.227-2.720 30.487-2.491Q30.746-2.262 30.746-1.828Q30.746-1.507 30.451-1.305Q30.155-1.103 29.752-1.014Q29.348-0.925 29.034-0.925Q28.716-0.925 28.313-1.014Q27.909-1.103 27.614-1.305Q27.318-1.507 27.318-1.828M27.773-1.828Q27.773-1.599 27.991-1.450Q28.210-1.301 28.502-1.233Q28.795-1.165 29.034-1.165Q29.198-1.165 29.407-1.201Q29.615-1.236 29.822-1.317Q30.029-1.397 30.160-1.525Q30.292-1.653 30.292-1.828Q30.292-2.180 29.911-2.274Q29.530-2.368 29.027-2.368L28.408-2.368Q28.169-2.368 27.971-2.217Q27.773-2.067 27.773-1.828M28.822-3.615Q29.489-3.615 29.489-4.412Q29.489-5.212 28.822-5.212Q28.152-5.212 28.152-4.412Q28.152-3.615 28.822-3.615M31.741-2.781Q31.741-2.949 31.864-3.072Q31.987-3.195 32.161-3.195Q32.329-3.195 32.452-3.072Q32.575-2.949 32.575-2.781Q32.575-2.607 32.452-2.484Q32.329-2.361 32.161-2.361Q31.987-2.361 31.864-2.484Q31.741-2.607 31.741-2.781M31.741-4.965Q31.741-5.133 31.864-5.256Q31.987-5.379 32.161-5.379Q32.329-5.379 32.452-5.256Q32.575-5.133 32.575-4.965Q32.575-4.791 32.452-4.668Q32.329-4.545 32.161-4.545Q31.987-4.545 31.864-4.668Q31.741-4.791 31.741-4.965\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(202.702 -18.167)\">\u003Cpath d=\"M38.756-2.388L37.628-4.887Q37.556-5.034 37.426-5.066Q37.296-5.099 37.067-5.099L37.067-5.379L38.581-5.379L38.581-5.099Q38.229-5.099 38.229-4.952Q38.229-4.907 38.240-4.887L39.104-2.969L39.884-4.699Q39.918-4.767 39.918-4.846Q39.918-4.959 39.834-5.029Q39.750-5.099 39.631-5.099L39.631-5.379L40.827-5.379L40.827-5.099Q40.608-5.099 40.437-4.996Q40.267-4.894 40.178-4.699L39.142-2.388Q39.094-2.293 38.988-2.293L38.910-2.293Q38.804-2.293 38.756-2.388\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(202.702 -18.167)\">\u003Cpath d=\"M40.996-3.089Q40.996-3.421 41.219-3.648Q41.443-3.875 41.787-4.003Q42.130-4.132 42.503-4.184Q42.875-4.237 43.180-4.237L43.180-4.490Q43.180-4.695 43.072-4.875Q42.964-5.054 42.783-5.157Q42.602-5.259 42.394-5.259Q41.987-5.259 41.751-5.167Q41.840-5.130 41.886-5.046Q41.932-4.962 41.932-4.860Q41.932-4.764 41.886-4.685Q41.840-4.607 41.759-4.562Q41.679-4.518 41.590-4.518Q41.440-4.518 41.339-4.615Q41.238-4.713 41.238-4.860Q41.238-5.482 42.394-5.482Q42.605-5.482 42.855-5.418Q43.104-5.355 43.306-5.236Q43.508-5.116 43.634-4.931Q43.761-4.747 43.761-4.504L43.761-2.928Q43.761-2.812 43.822-2.716Q43.884-2.621 43.997-2.621Q44.106-2.621 44.171-2.715Q44.236-2.809 44.236-2.928L44.236-3.376L44.502-3.376L44.502-2.928Q44.502-2.658 44.275-2.493Q44.048-2.327 43.768-2.327Q43.559-2.327 43.422-2.481Q43.286-2.634 43.262-2.850Q43.115-2.583 42.833-2.438Q42.551-2.293 42.226-2.293Q41.949-2.293 41.665-2.368Q41.382-2.443 41.189-2.622Q40.996-2.802 40.996-3.089M41.611-3.089Q41.611-2.915 41.712-2.785Q41.812-2.655 41.968-2.585Q42.123-2.515 42.288-2.515Q42.506-2.515 42.715-2.612Q42.923-2.710 43.051-2.891Q43.180-3.072 43.180-3.298L43.180-4.026Q42.855-4.026 42.489-3.935Q42.123-3.844 41.867-3.632Q41.611-3.421 41.611-3.089M46.669-2.361L44.933-2.361L44.933-2.641Q45.162-2.641 45.311-2.675Q45.459-2.710 45.459-2.850L45.459-4.699Q45.459-4.969 45.352-5.030Q45.244-5.092 44.933-5.092L44.933-5.372L45.962-5.447L45.962-4.740Q46.092-5.048 46.334-5.247Q46.577-5.447 46.895-5.447Q47.114-5.447 47.285-5.323Q47.456-5.198 47.456-4.986Q47.456-4.849 47.356-4.750Q47.257-4.651 47.124-4.651Q46.987-4.651 46.888-4.750Q46.789-4.849 46.789-4.986Q46.789-5.126 46.888-5.225Q46.598-5.225 46.398-5.029Q46.198-4.832 46.105-4.538Q46.013-4.244 46.013-3.964L46.013-2.850Q46.013-2.641 46.669-2.641L46.669-2.361M49.657-2.361L48.105-2.361L48.105-2.641Q48.331-2.641 48.479-2.675Q48.628-2.710 48.628-2.850L48.628-4.699Q48.628-4.887 48.580-4.971Q48.532-5.054 48.435-5.073Q48.337-5.092 48.125-5.092L48.125-5.372L49.182-5.447L49.182-2.850Q49.182-2.710 49.313-2.675Q49.445-2.641 49.657-2.641L49.657-2.361M48.385-6.668Q48.385-6.839 48.508-6.958Q48.631-7.078 48.802-7.078Q48.970-7.078 49.093-6.958Q49.216-6.839 49.216-6.668Q49.216-6.493 49.093-6.370Q48.970-6.247 48.802-6.247Q48.631-6.247 48.508-6.370Q48.385-6.493 48.385-6.668M50.361-3.089Q50.361-3.421 50.585-3.648Q50.809-3.875 51.152-4.003Q51.496-4.132 51.868-4.184Q52.241-4.237 52.545-4.237L52.545-4.490Q52.545-4.695 52.437-4.875Q52.330-5.054 52.148-5.157Q51.967-5.259 51.759-5.259Q51.352-5.259 51.116-5.167Q51.205-5.130 51.251-5.046Q51.297-4.962 51.297-4.860Q51.297-4.764 51.251-4.685Q51.205-4.607 51.125-4.562Q51.044-4.518 50.956-4.518Q50.805-4.518 50.704-4.615Q50.603-4.713 50.603-4.860Q50.603-5.482 51.759-5.482Q51.971-5.482 52.220-5.418Q52.470-5.355 52.671-5.236Q52.873-5.116 52.999-4.931Q53.126-4.747 53.126-4.504L53.126-2.928Q53.126-2.812 53.187-2.716Q53.249-2.621 53.362-2.621Q53.471-2.621 53.536-2.715Q53.601-2.809 53.601-2.928L53.601-3.376L53.868-3.376L53.868-2.928Q53.868-2.658 53.640-2.493Q53.413-2.327 53.133-2.327Q52.924-2.327 52.788-2.481Q52.651-2.634 52.627-2.850Q52.480-2.583 52.198-2.438Q51.916-2.293 51.591-2.293Q51.314-2.293 51.031-2.368Q50.747-2.443 50.554-2.622Q50.361-2.802 50.361-3.089M50.976-3.089Q50.976-2.915 51.077-2.785Q51.178-2.655 51.333-2.585Q51.489-2.515 51.653-2.515Q51.872-2.515 52.080-2.612Q52.289-2.710 52.417-2.891Q52.545-3.072 52.545-3.298L52.545-4.026Q52.220-4.026 51.854-3.935Q51.489-3.844 51.232-3.632Q50.976-3.421 50.976-3.089M55.966-2.361L54.332-2.361L54.332-2.641Q54.561-2.641 54.710-2.675Q54.859-2.710 54.859-2.850L54.859-4.699Q54.859-4.969 54.751-5.030Q54.644-5.092 54.332-5.092L54.332-5.372L55.392-5.447L55.392-4.798Q55.563-5.106 55.867-5.277Q56.171-5.447 56.517-5.447Q57.022-5.447 57.306-5.224Q57.590-5 57.590-4.504L57.590-2.850Q57.590-2.713 57.738-2.677Q57.887-2.641 58.113-2.641L58.113-2.361L56.482-2.361L56.482-2.641Q56.711-2.641 56.860-2.675Q57.009-2.710 57.009-2.850L57.009-4.490Q57.009-4.825 56.889-5.025Q56.769-5.225 56.455-5.225Q56.185-5.225 55.951-5.089Q55.717-4.952 55.578-4.718Q55.440-4.484 55.440-4.210L55.440-2.850Q55.440-2.713 55.590-2.677Q55.741-2.641 55.966-2.641L55.966-2.361M58.701-3.872Q58.701-4.200 58.836-4.501Q58.971-4.801 59.206-5.022Q59.442-5.242 59.747-5.362Q60.051-5.482 60.375-5.482Q60.881-5.482 61.230-5.379Q61.579-5.277 61.579-4.901Q61.579-4.754 61.481-4.653Q61.384-4.552 61.237-4.552Q61.083-4.552 60.984-4.651Q60.885-4.750 60.885-4.901Q60.885-5.089 61.025-5.181Q60.823-5.232 60.382-5.232Q60.027-5.232 59.798-5.036Q59.569-4.839 59.468-4.530Q59.367-4.220 59.367-3.872Q59.367-3.523 59.494-3.217Q59.620-2.911 59.875-2.727Q60.129-2.542 60.485-2.542Q60.707-2.542 60.892-2.626Q61.076-2.710 61.211-2.865Q61.346-3.021 61.404-3.229Q61.418-3.284 61.473-3.284L61.585-3.284Q61.616-3.284 61.638-3.260Q61.661-3.236 61.661-3.202L61.661-3.181Q61.575-2.894 61.387-2.696Q61.199-2.498 60.934-2.395Q60.669-2.293 60.375-2.293Q59.945-2.293 59.557-2.499Q59.169-2.706 58.935-3.069Q58.701-3.431 58.701-3.872M62.207-3.896Q62.207-4.217 62.332-4.506Q62.457-4.795 62.683-5.018Q62.908-5.242 63.204-5.362Q63.499-5.482 63.817-5.482Q64.145-5.482 64.407-5.382Q64.668-5.283 64.844-5.101Q65.020-4.918 65.114-4.660Q65.208-4.402 65.208-4.070Q65.208-3.978 65.126-3.957L62.871-3.957L62.871-3.896Q62.871-3.308 63.154-2.925Q63.438-2.542 64.005-2.542Q64.327-2.542 64.595-2.735Q64.863-2.928 64.952-3.243Q64.959-3.284 65.034-3.298L65.126-3.298Q65.208-3.274 65.208-3.202Q65.208-3.195 65.202-3.168Q65.089-2.771 64.718-2.532Q64.347-2.293 63.923-2.293Q63.486-2.293 63.086-2.501Q62.686-2.710 62.447-3.077Q62.207-3.444 62.207-3.896M62.877-4.166L64.692-4.166Q64.692-4.443 64.595-4.695Q64.498-4.948 64.299-5.104Q64.101-5.259 63.817-5.259Q63.540-5.259 63.327-5.101Q63.113-4.942 62.995-4.687Q62.877-4.432 62.877-4.166\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\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(202.702 18.822)\">\u003Cpath d=\"M-17.433-1.004L-19.063-1.004L-19.063-1.284Q-18.834-1.284-18.685-1.319Q-18.537-1.353-18.537-1.493L-18.537-4.839Q-18.537-5.010-18.673-5.051Q-18.810-5.092-19.063-5.092L-19.063-5.372L-17.983-5.447L-17.983-5.041Q-17.761-5.242-17.474-5.345Q-17.186-5.447-16.879-5.447Q-16.452-5.447-16.088-5.234Q-15.724-5.020-15.510-4.656Q-15.296-4.292-15.296-3.872Q-15.296-3.427-15.536-3.063Q-15.775-2.699-16.168-2.496Q-16.561-2.293-17.005-2.293Q-17.272-2.293-17.520-2.393Q-17.767-2.494-17.955-2.675L-17.955-1.493Q-17.955-1.356-17.807-1.320Q-17.658-1.284-17.433-1.284L-17.433-1.004M-17.955-4.692L-17.955-3.082Q-17.822-2.829-17.579-2.672Q-17.337-2.515-17.060-2.515Q-16.732-2.515-16.479-2.716Q-16.226-2.918-16.093-3.236Q-15.959-3.554-15.959-3.872Q-15.959-4.101-16.024-4.330Q-16.089-4.559-16.217-4.757Q-16.346-4.955-16.540-5.075Q-16.735-5.194-16.968-5.194Q-17.262-5.194-17.530-5.065Q-17.798-4.935-17.955-4.692M-13.044-2.361L-14.596-2.361L-14.596-2.641Q-14.370-2.641-14.221-2.675Q-14.073-2.710-14.073-2.850L-14.073-4.699Q-14.073-4.887-14.120-4.971Q-14.168-5.054-14.266-5.073Q-14.363-5.092-14.575-5.092L-14.575-5.372L-13.519-5.447L-13.519-2.850Q-13.519-2.710-13.387-2.675Q-13.256-2.641-13.044-2.641L-13.044-2.361M-14.315-6.668Q-14.315-6.839-14.192-6.958Q-14.069-7.078-13.898-7.078Q-13.731-7.078-13.608-6.958Q-13.485-6.839-13.485-6.668Q-13.485-6.493-13.608-6.370Q-13.731-6.247-13.898-6.247Q-14.069-6.247-14.192-6.370Q-14.315-6.493-14.315-6.668\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(202.702 18.822)\">\u003Cpath d=\"M-8.085-2.361L-9.637-2.361L-9.637-2.641Q-9.411-2.641-9.262-2.675Q-9.114-2.710-9.114-2.850L-9.114-4.699Q-9.114-4.887-9.162-4.971Q-9.209-5.054-9.307-5.073Q-9.404-5.092-9.616-5.092L-9.616-5.372L-8.560-5.447L-8.560-2.850Q-8.560-2.710-8.428-2.675Q-8.297-2.641-8.085-2.641L-8.085-2.361M-9.356-6.668Q-9.356-6.839-9.233-6.958Q-9.110-7.078-8.939-7.078Q-8.772-7.078-8.649-6.958Q-8.526-6.839-8.526-6.668Q-8.526-6.493-8.649-6.370Q-8.772-6.247-8.939-6.247Q-9.110-6.247-9.233-6.370Q-9.356-6.493-9.356-6.668M-5.757-2.361L-7.391-2.361L-7.391-2.641Q-7.162-2.641-7.013-2.675Q-6.865-2.710-6.865-2.850L-6.865-4.699Q-6.865-4.969-6.972-5.030Q-7.080-5.092-7.391-5.092L-7.391-5.372L-6.331-5.447L-6.331-4.798Q-6.161-5.106-5.856-5.277Q-5.552-5.447-5.207-5.447Q-4.701-5.447-4.417-5.224Q-4.134-5-4.134-4.504L-4.134-2.850Q-4.134-2.713-3.985-2.677Q-3.836-2.641-3.611-2.641L-3.611-2.361L-5.241-2.361L-5.241-2.641Q-5.012-2.641-4.863-2.675Q-4.715-2.710-4.715-2.850L-4.715-4.490Q-4.715-4.825-4.834-5.025Q-4.954-5.225-5.268-5.225Q-5.538-5.225-5.773-5.089Q-6.007-4.952-6.145-4.718Q-6.284-4.484-6.284-4.210L-6.284-2.850Q-6.284-2.713-6.133-2.677Q-5.983-2.641-5.757-2.641\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(202.702 18.822)\">\u003Cpath d=\"M1.348-2.221Q0.713-2.221 0.349-2.566Q-0.016-2.911-0.151-3.436Q-0.286-3.961-0.286-4.586Q-0.286-5.611 0.070-6.310Q0.425-7.009 1.348-7.009Q2.275-7.009 2.627-6.310Q2.979-5.611 2.979-4.586Q2.979-3.961 2.844-3.436Q2.709-2.911 2.346-2.566Q1.984-2.221 1.348-2.221M1.348-2.446Q1.786-2.446 1.999-2.821Q2.213-3.195 2.263-3.662Q2.312-4.128 2.312-4.706Q2.312-5.259 2.263-5.687Q2.213-6.114 2.001-6.449Q1.789-6.784 1.348-6.784Q1.006-6.784 0.803-6.577Q0.600-6.370 0.513-6.058Q0.425-5.745 0.403-5.429Q0.381-5.112 0.381-4.706Q0.381-4.289 0.403-3.947Q0.425-3.605 0.514-3.257Q0.603-2.908 0.808-2.677Q1.013-2.446 1.348-2.446M4.055-2.781Q4.055-2.949 4.178-3.072Q4.301-3.195 4.476-3.195Q4.643-3.195 4.766-3.072Q4.889-2.949 4.889-2.781Q4.889-2.607 4.766-2.484Q4.643-2.361 4.476-2.361Q4.301-2.361 4.178-2.484Q4.055-2.607 4.055-2.781M6.318-2.781Q6.318-2.949 6.441-3.072Q6.564-3.195 6.738-3.195Q6.906-3.195 7.029-3.072Q7.152-2.949 7.152-2.781Q7.152-2.607 7.029-2.484Q6.906-2.361 6.738-2.361Q6.564-2.361 6.441-2.484Q6.318-2.607 6.318-2.781M11.192-2.361L8.663-2.361L8.663-2.641Q9.630-2.641 9.630-2.850L9.630-6.469Q9.237-6.281 8.615-6.281L8.615-6.562Q9.032-6.562 9.396-6.663Q9.760-6.763 10.016-7.009L10.143-7.009Q10.208-6.992 10.225-6.924L10.225-2.850Q10.225-2.641 11.192-2.641L11.192-2.361M12.563-2.781Q12.563-2.949 12.686-3.072Q12.809-3.195 12.983-3.195Q13.151-3.195 13.274-3.072Q13.397-2.949 13.397-2.781Q13.397-2.607 13.274-2.484Q13.151-2.361 12.983-2.361Q12.809-2.361 12.686-2.484Q12.563-2.607 12.563-2.781M12.563-4.965Q12.563-5.133 12.686-5.256Q12.809-5.379 12.983-5.379Q13.151-5.379 13.274-5.256Q13.397-5.133 13.397-4.965Q13.397-4.791 13.274-4.668Q13.151-4.545 12.983-4.545Q12.809-4.545 12.686-4.668Q12.563-4.791 12.563-4.965\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(202.702 18.822)\">\u003Cpath d=\"M17.952-3.844Q17.952-4.186 18.087-4.485Q18.222-4.784 18.462-5.008Q18.701-5.232 19.019-5.357Q19.337-5.482 19.668-5.482Q20.113-5.482 20.512-5.266Q20.912-5.051 21.147-4.673Q21.381-4.296 21.381-3.844Q21.381-3.503 21.239-3.219Q21.097-2.935 20.853-2.728Q20.608-2.522 20.299-2.407Q19.990-2.293 19.668-2.293Q19.238-2.293 18.836-2.494Q18.434-2.696 18.193-3.048Q17.952-3.400 17.952-3.844M19.668-2.542Q20.270-2.542 20.494-2.920Q20.718-3.298 20.718-3.930Q20.718-4.542 20.483-4.901Q20.249-5.259 19.668-5.259Q18.616-5.259 18.616-3.930Q18.616-3.298 18.841-2.920Q19.067-2.542 19.668-2.542M23.657-2.361L22.023-2.361L22.023-2.641Q22.252-2.641 22.401-2.675Q22.550-2.710 22.550-2.850L22.550-4.699Q22.550-4.969 22.442-5.030Q22.334-5.092 22.023-5.092L22.023-5.372L23.083-5.447L23.083-4.798Q23.254-5.106 23.558-5.277Q23.862-5.447 24.207-5.447Q24.713-5.447 24.997-5.224Q25.281-5 25.281-4.504L25.281-2.850Q25.281-2.713 25.429-2.677Q25.578-2.641 25.804-2.641L25.804-2.361L24.173-2.361L24.173-2.641Q24.402-2.641 24.551-2.675Q24.700-2.710 24.700-2.850L24.700-4.490Q24.700-4.825 24.580-5.025Q24.460-5.225 24.146-5.225Q23.876-5.225 23.642-5.089Q23.408-4.952 23.269-4.718Q23.131-4.484 23.131-4.210L23.131-2.850Q23.131-2.713 23.281-2.677Q23.431-2.641 23.657-2.641L23.657-2.361M28.059-2.361L26.456-2.361L26.456-2.641Q26.682-2.641 26.831-2.675Q26.979-2.710 26.979-2.850L26.979-6.469Q26.979-6.739 26.872-6.801Q26.764-6.862 26.456-6.862L26.456-7.143L27.533-7.218L27.533-2.850Q27.533-2.713 27.683-2.677Q27.834-2.641 28.059-2.641L28.059-2.361M28.989-1.226Q29.119-1.158 29.256-1.158Q29.427-1.158 29.577-1.247Q29.727-1.336 29.838-1.481Q29.950-1.626 30.028-1.794L30.291-2.361L29.122-4.887Q29.047-5.034 28.917-5.066Q28.787-5.099 28.555-5.099L28.555-5.379L30.076-5.379L30.076-5.099Q29.727-5.099 29.727-4.952Q29.731-4.931 29.732-4.914Q29.734-4.897 29.734-4.887L30.592-3.028L31.365-4.699Q31.399-4.767 31.399-4.846Q31.399-4.959 31.315-5.029Q31.231-5.099 31.118-5.099L31.118-5.379L32.315-5.379L32.315-5.099Q32.096-5.099 31.923-4.995Q31.751-4.890 31.658-4.699L30.322-1.794Q30.151-1.424 29.881-1.178Q29.611-0.932 29.256-0.932Q28.986-0.932 28.767-1.098Q28.548-1.264 28.548-1.527Q28.548-1.664 28.640-1.753Q28.733-1.841 28.873-1.841Q29.010-1.841 29.098-1.753Q29.187-1.664 29.187-1.527Q29.187-1.424 29.134-1.346Q29.081-1.267 28.989-1.226\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(202.702 18.822)\">\u003Cpath d=\"M35.555-3.872Q35.555-4.210 35.696-4.501Q35.836-4.791 36.080-5.005Q36.324-5.218 36.629-5.333Q36.933-5.447 37.258-5.447Q37.528-5.447 37.791-5.348Q38.054-5.249 38.245-5.071L38.245-6.469Q38.245-6.739 38.138-6.801Q38.030-6.862 37.719-6.862L37.719-7.143L38.796-7.218L38.796-3.034Q38.796-2.846 38.850-2.763Q38.905-2.679 39.006-2.660Q39.107-2.641 39.322-2.641L39.322-2.361L38.215-2.293L38.215-2.710Q37.798-2.293 37.172-2.293Q36.741-2.293 36.369-2.505Q35.996-2.716 35.776-3.077Q35.555-3.438 35.555-3.872M37.230-2.515Q37.439-2.515 37.625-2.587Q37.811-2.658 37.965-2.795Q38.119-2.932 38.215-3.110L38.215-4.719Q38.129-4.866 37.984-4.986Q37.839-5.106 37.669-5.165Q37.500-5.225 37.319-5.225Q36.759-5.225 36.490-4.836Q36.222-4.446 36.222-3.865Q36.222-3.294 36.456-2.904Q36.690-2.515 37.230-2.515M40.030-3.089Q40.030-3.421 40.253-3.648Q40.477-3.875 40.821-4.003Q41.164-4.132 41.537-4.184Q41.909-4.237 42.214-4.237L42.214-4.490Q42.214-4.695 42.106-4.875Q41.998-5.054 41.817-5.157Q41.636-5.259 41.428-5.259Q41.021-5.259 40.785-5.167Q40.874-5.130 40.920-5.046Q40.966-4.962 40.966-4.860Q40.966-4.764 40.920-4.685Q40.874-4.607 40.793-4.562Q40.713-4.518 40.624-4.518Q40.474-4.518 40.373-4.615Q40.272-4.713 40.272-4.860Q40.272-5.482 41.428-5.482Q41.639-5.482 41.889-5.418Q42.138-5.355 42.340-5.236Q42.542-5.116 42.668-4.931Q42.795-4.747 42.795-4.504L42.795-2.928Q42.795-2.812 42.856-2.716Q42.918-2.621 43.031-2.621Q43.140-2.621 43.205-2.715Q43.270-2.809 43.270-2.928L43.270-3.376L43.536-3.376L43.536-2.928Q43.536-2.658 43.309-2.493Q43.082-2.327 42.802-2.327Q42.593-2.327 42.456-2.481Q42.320-2.634 42.296-2.850Q42.149-2.583 41.867-2.438Q41.585-2.293 41.260-2.293Q40.983-2.293 40.699-2.368Q40.416-2.443 40.223-2.622Q40.030-2.802 40.030-3.089M40.645-3.089Q40.645-2.915 40.746-2.785Q40.846-2.655 41.002-2.585Q41.158-2.515 41.322-2.515Q41.540-2.515 41.749-2.612Q41.957-2.710 42.085-2.891Q42.214-3.072 42.214-3.298L42.214-4.026Q41.889-4.026 41.523-3.935Q41.158-3.844 40.901-3.632Q40.645-3.421 40.645-3.089M45.635-2.361L44.001-2.361L44.001-2.641Q44.230-2.641 44.379-2.675Q44.528-2.710 44.528-2.850L44.528-4.699Q44.528-4.969 44.420-5.030Q44.312-5.092 44.001-5.092L44.001-5.372L45.061-5.447L45.061-4.798Q45.232-5.106 45.536-5.277Q45.840-5.447 46.185-5.447Q46.585-5.447 46.862-5.307Q47.139-5.167 47.224-4.819Q47.392-5.112 47.691-5.280Q47.990-5.447 48.335-5.447Q48.841-5.447 49.125-5.224Q49.408-5 49.408-4.504L49.408-2.850Q49.408-2.713 49.557-2.677Q49.706-2.641 49.931-2.641L49.931-2.361L48.301-2.361L48.301-2.641Q48.527-2.641 48.677-2.677Q48.827-2.713 48.827-2.850L48.827-4.490Q48.827-4.825 48.708-5.025Q48.588-5.225 48.274-5.225Q48.004-5.225 47.770-5.089Q47.535-4.952 47.397-4.718Q47.259-4.484 47.259-4.210L47.259-2.850Q47.259-2.713 47.407-2.677Q47.556-2.641 47.782-2.641L47.782-2.361L46.151-2.361L46.151-2.641Q46.380-2.641 46.529-2.675Q46.678-2.710 46.678-2.850L46.678-4.490Q46.678-4.825 46.558-5.025Q46.438-5.225 46.124-5.225Q45.854-5.225 45.620-5.089Q45.386-4.952 45.247-4.718Q45.109-4.484 45.109-4.210L45.109-2.850Q45.109-2.713 45.259-2.677Q45.409-2.641 45.635-2.641L45.635-2.361M52.163-1.004L50.533-1.004L50.533-1.284Q50.762-1.284 50.911-1.319Q51.059-1.353 51.059-1.493L51.059-4.839Q51.059-5.010 50.923-5.051Q50.786-5.092 50.533-5.092L50.533-5.372L51.613-5.447L51.613-5.041Q51.835-5.242 52.122-5.345Q52.409-5.447 52.717-5.447Q53.144-5.447 53.508-5.234Q53.872-5.020 54.086-4.656Q54.300-4.292 54.300-3.872Q54.300-3.427 54.060-3.063Q53.821-2.699 53.428-2.496Q53.035-2.293 52.591-2.293Q52.324-2.293 52.076-2.393Q51.828-2.494 51.640-2.675L51.640-1.493Q51.640-1.356 51.789-1.320Q51.938-1.284 52.163-1.284L52.163-1.004M51.640-4.692L51.640-3.082Q51.774-2.829 52.016-2.672Q52.259-2.515 52.536-2.515Q52.864-2.515 53.117-2.716Q53.370-2.918 53.503-3.236Q53.637-3.554 53.637-3.872Q53.637-4.101 53.572-4.330Q53.507-4.559 53.378-4.757Q53.250-4.955 53.055-5.075Q52.861-5.194 52.628-5.194Q52.334-5.194 52.066-5.065Q51.798-4.935 51.640-4.692M54.935-2.368L54.935-3.431Q54.935-3.455 54.963-3.482Q54.990-3.509 55.014-3.509L55.123-3.509Q55.188-3.509 55.202-3.451Q55.298-3.017 55.544-2.766Q55.790-2.515 56.203-2.515Q56.545-2.515 56.798-2.648Q57.051-2.781 57.051-3.089Q57.051-3.246 56.957-3.361Q56.863-3.475 56.725-3.544Q56.586-3.612 56.419-3.650L55.838-3.749Q55.482-3.817 55.209-4.038Q54.935-4.258 54.935-4.600Q54.935-4.849 55.046-5.024Q55.158-5.198 55.344-5.297Q55.530-5.396 55.745-5.439Q55.961-5.482 56.203-5.482Q56.617-5.482 56.897-5.300L57.113-5.475Q57.123-5.478 57.130-5.480Q57.137-5.482 57.147-5.482L57.198-5.482Q57.225-5.482 57.249-5.458Q57.273-5.434 57.273-5.406L57.273-4.559Q57.273-4.538 57.249-4.511Q57.225-4.484 57.198-4.484L57.085-4.484Q57.058-4.484 57.032-4.509Q57.007-4.535 57.007-4.559Q57.007-4.795 56.901-4.959Q56.795-5.123 56.612-5.205Q56.429-5.287 56.197-5.287Q55.868-5.287 55.612-5.184Q55.356-5.082 55.356-4.805Q55.356-4.610 55.539-4.501Q55.721-4.391 55.950-4.350L56.525-4.244Q56.771-4.196 56.984-4.068Q57.198-3.940 57.335-3.737Q57.471-3.533 57.471-3.284Q57.471-2.771 57.106-2.532Q56.740-2.293 56.203-2.293Q55.708-2.293 55.376-2.587L55.110-2.313Q55.089-2.293 55.062-2.293L55.014-2.293Q54.990-2.293 54.963-2.320Q54.935-2.347 54.935-2.368\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">The three action-value traces share the base decay \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\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0556em;\">γ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0556em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">λ\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> and differ only in the off-policy factor. Sarsa(\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\">λ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>) uses none; the importance-sampling trace multiplies by \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\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:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>, which can spike far above 1; TB(\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\">λ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>) multiplies by \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">A\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">S\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.2806em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">t\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∈\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">[\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003Cspan class=\"mclose\">]\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>, which only ever damps the trace. Watkins's Q(\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\">λ\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>) is TB with a hard 0\u002F1 gate.\u003C\u002Ffigcaption>",1785117792444]