[{"data":1,"prerenderedAt":20545},["ShallowReactive",2],{"lesson:\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback":3,"course-wordcounts":14771,"ref-card-index":15682,"nav:deep-learning":20364,"tikz:81660259609149aa0eb6da07ed33fce61439a06ae8f03b6fbf5ba7aaaccd50df":20537,"tikz:fdd3ef414f543c3f9ae396a091c4a71f3e18ad24bf5371b1974b618ebe7b9da4":20538,"tikz:c1c1d901226794f0f116f03fee2974bedc74da17e08a2c646774a62ddf2b58cb":20539,"tikz:36109c1744978e93f156602bcbf6bcc273e8c8afdb382fccd08080837fdda192":20540,"tikz:75d710db88248a95f01f3f7605ef3066857a45ad2c834993514a3d49401fb16d":20541,"tikz:1efe3ce7d256e884f632abec1c60542ed0ece7a8516f411b5b7d5443d75b7f80":20542,"tikz:a128bede09311dd5157c1d4a5afbea05279bb41aec257e075ad47cd896341110":20543,"tikz:666c0d5d224dfe32ce350a59876110d22652adb4980ddd4e4d3e2dbb7a135367":20544},{"id":4,"title":5,"blurb":6,"body":7,"brief":14744,"category":14745,"description":14746,"draft":14747,"extension":14748,"meta":14749,"module":14751,"navigation":14752,"path":14753,"practice":14754,"rawbody":14755,"readingTime":14756,"seo":14761,"sources":14762,"status":14766,"stem":14767,"summary":14768,"topics":14769,"__hash__":14770},"course\u002F06.deep-learning\u002F11.reinforcement-learning\u002F05.rl-from-human-feedback.md","Reinforcement Learning from Human Feedback","",{"type":8,"value":9,"toc":14727},"minimark",[10,30,35,43,777,820,890,894,1144,1148,1151,1544,1548,1777,2556,3097,3100,3105,3641,3933,3936,4545,4822,5656,5781,6021,6177,6181,6294,6952,7083,7086,7090,7164,7996,8018,8021,8025,8259,8347,8351,8358,8646,8665,8668,8695,8699,8709,9586,12868,12871,12874,13227,13265,13268,13271,13514,13518,13521,13612,13630,13640,13650,13761,13765,13826,13836,13889,13895,13898,13902,14618,14723],[11,12,13,14,19,20,24,25,29],"p",{},"The ",[15,16,18],"a",{"href":17},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models","large language model","\nlesson sketched RLHF in a paragraph: fit a reward model, optimize with PPO, regularize\nwith a KL term. This lesson is the derivation. We build the reward model from the\nBradley-Terry likelihood, write the KL-regularized objective the policy actually\nmaximizes, solve it in closed form, and then show that the same closed form, inverted,\nturns the entire reinforcement-learning loop into a single supervised loss. The\n",[15,21,23],{"href":22},"\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic","policy gradients and PPO","\nlesson supplies the optimizer; the ",[15,26,28],{"href":27},"\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning","foundations of RL","\nlesson supplies the language of policies, rewards, and returns we reuse throughout.",[31,32,34],"h2",{"id":33},"the-specification-problem","The specification problem",[11,36,37,38,42],{},"The objectives we want from an assistant, that it be helpful, harmless, and honest,\nresist direct specification. There is no loss function for ",[39,40,41],"q",{},"helpful."," But a human\nshown two responses to the same prompt can reliably say which is better, even when\nthey could not have written down a rule that produces the better one. RLHF is built on\nexactly this asymmetry.",[44,45,47],"callout",{"type":46},"definition",[11,48,49,53,54,79,80,203,204,321,322,429,430,445,446,462,463,776],{},[50,51,52],"strong",{},"Definition (Preference comparison)."," For a prompt ",[55,56,59],"span",{"className":57},[58],"katex",[55,60,64],{"className":61,"ariaHidden":63},[62],"katex-html","true",[55,65,68,73],{"className":66},[67],"base",[55,69],{"className":70,"style":72},[71],"strut","height:0.4306em;",[55,74,78],{"className":75},[76,77],"mord","mathnormal","x"," and two candidate\nresponses ",[55,81,83],{"className":82},[58],[55,84,86],{"className":85,"ariaHidden":63},[62],[55,87,89,93,152,157,162],{"className":88},[67],[55,90],{"className":91,"style":92},[71],"height:0.625em;vertical-align:-0.1944em;",[55,94,96,101],{"className":95},[76],[55,97,100],{"className":98,"style":99},[76,77],"margin-right:0.0359em;","y",[55,102,105],{"className":103},[104],"msupsub",[55,106,110,143],{"className":107},[108,109],"vlist-t","vlist-t2",[55,111,114,138],{"className":112},[113],"vlist-r",[55,115,119],{"className":116,"style":118},[117],"vlist","height:0.3011em;",[55,120,122,127],{"style":121},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[55,123],{"className":124,"style":126},[125],"pstrut","height:2.7em;",[55,128,134],{"className":129},[130,131,132,133],"sizing","reset-size6","size3","mtight",[55,135,137],{"className":136},[76,133],"1",[55,139,142],{"className":140},[141],"vlist-s","​",[55,144,146],{"className":145},[113],[55,147,150],{"className":148,"style":149},[117],"height:0.15em;",[55,151],{},[55,153,156],{"className":154},[155],"mpunct",",",[55,158],{"className":159,"style":161},[160],"mspace","margin-right:0.1667em;",[55,163,165,168],{"className":164},[76],[55,166,100],{"className":167,"style":99},[76,77],[55,169,171],{"className":170},[104],[55,172,174,195],{"className":173},[108,109],[55,175,177,192],{"className":176},[113],[55,178,180],{"className":179,"style":118},[117],[55,181,182,185],{"style":121},[55,183],{"className":184,"style":126},[125],[55,186,188],{"className":187},[130,131,132,133],[55,189,191],{"className":190},[76,133],"2",[55,193,142],{"className":194},[141],[55,196,198],{"className":197},[113],[55,199,201],{"className":200,"style":149},[117],[55,202],{},", a human annotator returns a binary preference ",[55,205,207],{"className":206},[58],[55,208,210,272],{"className":209,"ariaHidden":63},[62],[55,211,213,217,260,264,269],{"className":212},[67],[55,214],{"className":215,"style":216},[71],"height:0.7335em;vertical-align:-0.1944em;",[55,218,220,223],{"className":219},[76],[55,221,100],{"className":222,"style":99},[76,77],[55,224,226],{"className":225},[104],[55,227,229,252],{"className":228},[108,109],[55,230,232,249],{"className":231},[113],[55,233,236],{"className":234,"style":235},[117],"height:0.1514em;",[55,237,238,241],{"style":121},[55,239],{"className":240,"style":126},[125],[55,242,244],{"className":243},[130,131,132,133],[55,245,248],{"className":246,"style":247},[76,77,133],"margin-right:0.0269em;","w",[55,250,142],{"className":251},[141],[55,253,255],{"className":254},[113],[55,256,258],{"className":257,"style":149},[117],[55,259],{},[55,261],{"className":262,"style":263},[160],"margin-right:0.2778em;",[55,265,268],{"className":266},[267],"mrel","≻",[55,270],{"className":271,"style":263},[160],[55,273,275,278],{"className":274},[67],[55,276],{"className":277,"style":92},[71],[55,279,281,284],{"className":280},[76],[55,282,100],{"className":283,"style":99},[76,77],[55,285,287],{"className":286},[104],[55,288,290,313],{"className":289},[108,109],[55,291,293,310],{"className":292},[113],[55,294,297],{"className":295,"style":296},[117],"height:0.3361em;",[55,298,299,302],{"style":121},[55,300],{"className":301,"style":126},[125],[55,303,305],{"className":304},[130,131,132,133],[55,306,309],{"className":307,"style":308},[76,77,133],"margin-right:0.0197em;","l",[55,311,142],{"className":312},[141],[55,314,316],{"className":315},[113],[55,317,319],{"className":318,"style":149},[117],[55,320],{},"\n(",[39,323,324,376,377],{},[55,325,327],{"className":326},[58],[55,328,330],{"className":329,"ariaHidden":63},[62],[55,331,333,336],{"className":332},[67],[55,334],{"className":335,"style":92},[71],[55,337,339,342],{"className":338},[76],[55,340,100],{"className":341,"style":99},[76,77],[55,343,345],{"className":344},[104],[55,346,348,368],{"className":347},[108,109],[55,349,351,365],{"className":350},[113],[55,352,354],{"className":353,"style":235},[117],[55,355,356,359],{"style":121},[55,357],{"className":358,"style":126},[125],[55,360,362],{"className":361},[130,131,132,133],[55,363,248],{"className":364,"style":247},[76,77,133],[55,366,142],{"className":367},[141],[55,369,371],{"className":370},[113],[55,372,374],{"className":373,"style":149},[117],[55,375],{}," is preferred to ",[55,378,380],{"className":379},[58],[55,381,383],{"className":382,"ariaHidden":63},[62],[55,384,386,389],{"className":385},[67],[55,387],{"className":388,"style":92},[71],[55,390,392,395],{"className":391},[76],[55,393,100],{"className":394,"style":99},[76,77],[55,396,398],{"className":397},[104],[55,399,401,421],{"className":400},[108,109],[55,402,404,418],{"className":403},[113],[55,405,407],{"className":406,"style":296},[117],[55,408,409,412],{"style":121},[55,410],{"className":411,"style":126},[125],[55,413,415],{"className":414},[130,131,132,133],[55,416,309],{"className":417,"style":308},[76,77,133],[55,419,142],{"className":420},[141],[55,422,424],{"className":423},[113],[55,425,427],{"className":426,"style":149},[117],[55,428],{},"), where ",[55,431,433],{"className":432},[58],[55,434,436],{"className":435,"ariaHidden":63},[62],[55,437,439,442],{"className":438},[67],[55,440],{"className":441,"style":72},[71],[55,443,248],{"className":444,"style":247},[76,77]," marks the winner and ",[55,447,449],{"className":448},[58],[55,450,452],{"className":451,"ariaHidden":63},[62],[55,453,455,459],{"className":454},[67],[55,456],{"className":457,"style":458},[71],"height:0.6944em;",[55,460,309],{"className":461,"style":308},[76,77]," the loser. A\npreference dataset is a collection ",[55,464,466],{"className":465},[58],[55,467,469,492],{"className":468,"ariaHidden":63},[62],[55,470,472,476,482,485,489],{"className":471},[67],[55,473],{"className":474,"style":475},[71],"height:0.6833em;",[55,477,481],{"className":478,"style":480},[76,479],"mathcal","margin-right:0.0278em;","D",[55,483],{"className":484,"style":263},[160],[55,486,488],{"className":487},[267],"=",[55,490],{"className":491,"style":263},[160],[55,493,495,499],{"className":494},[67],[55,496],{"className":497,"style":498},[71],"height:2.081em;vertical-align:-0.6997em;",[55,500,503,713],{"className":501},[502],"minner",[55,504,506,518,522,565,568,571,635,638,641,703,706],{"className":505},[502],[55,507,512],{"className":508,"style":511},[509,510],"mopen","delimcenter","top:0em;",[55,513,517],{"className":514},[515,516],"delimsizing","size2","{",[55,519,521],{"className":520},[509],"(",[55,523,525,528],{"className":524},[76],[55,526,78],{"className":527},[76,77],[55,529,531],{"className":530},[104],[55,532,534],{"className":533},[108],[55,535,537],{"className":536},[113],[55,538,541],{"className":539,"style":540},[117],"height:0.888em;",[55,542,544,547],{"style":543},"top:-3.063em;margin-right:0.05em;",[55,545],{"className":546,"style":126},[125],[55,548,550],{"className":549},[130,131,132,133],[55,551,553,556,560],{"className":552},[76,133],[55,554,521],{"className":555},[509,133],[55,557,559],{"className":558},[76,77,133],"i",[55,561,564],{"className":562},[563,133],"mclose",")",[55,566,156],{"className":567},[155],[55,569],{"className":570,"style":161},[160],[55,572,574,577],{"className":573},[76],[55,575,100],{"className":576,"style":99},[76,77],[55,578,580],{"className":579},[104],[55,581,583,626],{"className":582},[108,109],[55,584,586,623],{"className":585},[113],[55,587,590,602],{"className":588,"style":589},[117],"height:1.0448em;",[55,591,593,596],{"style":592},"top:-2.5834em;margin-left:-0.0359em;margin-right:0.05em;",[55,594],{"className":595,"style":126},[125],[55,597,599],{"className":598},[130,131,132,133],[55,600,248],{"className":601,"style":247},[76,77,133],[55,603,605,608],{"style":604},"top:-3.2198em;margin-right:0.05em;",[55,606],{"className":607,"style":126},[125],[55,609,611],{"className":610},[130,131,132,133],[55,612,614,617,620],{"className":613},[76,133],[55,615,521],{"className":616},[509,133],[55,618,559],{"className":619},[76,77,133],[55,621,564],{"className":622},[563,133],[55,624,142],{"className":625},[141],[55,627,629],{"className":628},[113],[55,630,633],{"className":631,"style":632},[117],"height:0.1166em;",[55,634],{},[55,636,156],{"className":637},[155],[55,639],{"className":640,"style":161},[160],[55,642,644,647],{"className":643},[76],[55,645,100],{"className":646,"style":99},[76,77],[55,648,650],{"className":649},[104],[55,651,653,694],{"className":652},[108,109],[55,654,656,691],{"className":655},[113],[55,657,659,671],{"className":658,"style":589},[117],[55,660,662,665],{"style":661},"top:-2.3987em;margin-left:-0.0359em;margin-right:0.05em;",[55,663],{"className":664,"style":126},[125],[55,666,668],{"className":667},[130,131,132,133],[55,669,309],{"className":670,"style":308},[76,77,133],[55,672,673,676],{"style":604},[55,674],{"className":675,"style":126},[125],[55,677,679],{"className":678},[130,131,132,133],[55,680,682,685,688],{"className":681},[76,133],[55,683,521],{"className":684},[509,133],[55,686,559],{"className":687},[76,77,133],[55,689,564],{"className":690},[563,133],[55,692,142],{"className":693},[141],[55,695,697],{"className":696},[113],[55,698,701],{"className":699,"style":700},[117],"height:0.3013em;",[55,702],{},[55,704,564],{"className":705},[563],[55,707,709],{"className":708,"style":511},[563,510],[55,710,712],{"className":711},[515,516],"}",[55,714,716],{"className":715},[104],[55,717,719,767],{"className":718},[108,109],[55,720,722,764],{"className":721},[113],[55,723,726,747],{"className":724,"style":725},[117],"height:1.3812em;",[55,727,729,732],{"style":728},"top:-2.0003em;margin-right:0.05em;",[55,730],{"className":731,"style":126},[125],[55,733,735],{"className":734},[130,131,132,133],[55,736,738,741,744],{"className":737},[76,133],[55,739,559],{"className":740},[76,77,133],[55,742,488],{"className":743},[267,133],[55,745,137],{"className":746},[76,133],[55,748,750,753],{"style":749},"top:-3.6029em;margin-right:0.05em;",[55,751],{"className":752,"style":126},[125],[55,754,756],{"className":755},[130,131,132,133],[55,757,759],{"className":758},[76,133],[55,760,763],{"className":761,"style":762},[76,77,133],"margin-right:0.109em;","N",[55,765,142],{"className":766},[141],[55,768,770],{"className":769},[113],[55,771,774],{"className":772,"style":773},[117],"height:0.6997em;",[55,775],{},"\nof such judgments.",[11,778,779,780,783,784,819],{},"The plan is to learn a scalar ",[50,781,782],{},"reward"," ",[55,785,787],{"className":786},[58],[55,788,790],{"className":789,"ariaHidden":63},[62],[55,791,793,797,801,804,807,810,813,816],{"className":792},[67],[55,794],{"className":795,"style":796},[71],"height:1em;vertical-align:-0.25em;",[55,798,800],{"className":799,"style":480},[76,77],"r",[55,802,521],{"className":803},[509],[55,805,78],{"className":806},[76,77],[55,808,156],{"className":809},[155],[55,811],{"className":812,"style":161},[160],[55,814,100],{"className":815,"style":99},[76,77],[55,817,564],{"className":818},[563]," that ranks responses the way humans\ndo, then push a language model toward high-reward responses. The reward converts a\nset of discrete comparisons into a differentiable signal a policy can ascend.",[821,822,823,839],"table",{},[824,825,826],"thead",{},[827,828,829,833,836],"tr",{},[830,831,832],"th",{},"Quantity",[830,834,835],{},"Hard to specify",[830,837,838],{},"Easy to elicit",[840,841,842,854,864,874],"tbody",{},[827,843,844,848,851],{},[845,846,847],"td",{},"Absolute quality of one response",[845,849,850],{},"yes (no scalar in the annotator's head)",[845,852,853],{},"no",[827,855,856,859,861],{},[845,857,858],{},"Which of two responses is better",[845,860,853],{},[845,862,863],{},"yes (a single click)",[827,865,866,869,872],{},[845,867,868],{},"A rule that generates good responses",[845,870,871],{},"yes",[845,873,853],{},[827,875,876,884,887],{},[845,877,878,879,883],{},"A reward that ",[880,881,882],"em",{},"ranks"," responses",[845,885,886],{},"learned from comparisons",[845,888,889],{},"--",[31,891,893],{"id":892},"the-three-stage-pipeline","The three-stage pipeline",[11,895,896,897,900,901,904,905,967,968,783,971,1027,1028,1031,1032,1084,1085,1143],{},"RLHF as deployed in modern instruction-tuned systems is three stages run in sequence. First,\n",[50,898,899],{},"supervised fine-tuning"," (SFT) on curated demonstrations teaches the base model the\n",[880,902,903],{},"format"," of an answer and gives a reference policy ",[55,906,908],{"className":907},[58],[55,909,911],{"className":910,"ariaHidden":63},[62],[55,912,914,918],{"className":913},[67],[55,915],{"className":916,"style":917},[71],"height:0.5806em;vertical-align:-0.15em;",[55,919,921,925],{"className":920},[76],[55,922,924],{"className":923,"style":99},[76,77],"π",[55,926,928],{"className":927},[104],[55,929,931,959],{"className":930},[108,109],[55,932,934,956],{"className":933},[113],[55,935,937],{"className":936,"style":296},[117],[55,938,939,942],{"style":121},[55,940],{"className":941,"style":126},[125],[55,943,945],{"className":944},[130,131,132,133],[55,946,948],{"className":947},[76,133],[55,949,952],{"className":950},[76,951,133],"text",[55,953,955],{"className":954},[76,133],"ref",[55,957,142],{"className":958},[141],[55,960,962],{"className":961},[113],[55,963,965],{"className":964,"style":149},[117],[55,966],{},". Second, a\n",[50,969,970],{},"reward model",[55,972,974],{"className":973},[58],[55,975,977],{"className":976,"ariaHidden":63},[62],[55,978,980,984],{"className":979},[67],[55,981],{"className":982,"style":983},[71],"height:0.7167em;vertical-align:-0.2861em;",[55,985,987,990],{"className":986},[76],[55,988,800],{"className":989,"style":480},[76,77],[55,991,993],{"className":992},[104],[55,994,996,1018],{"className":995},[108,109],[55,997,999,1015],{"className":998},[113],[55,1000,1002],{"className":1001,"style":296},[117],[55,1003,1005,1008],{"style":1004},"top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;",[55,1006],{"className":1007,"style":126},[125],[55,1009,1011],{"className":1010},[130,131,132,133],[55,1012,1014],{"className":1013},[76,77,133],"ϕ",[55,1016,142],{"className":1017},[141],[55,1019,1021],{"className":1020},[113],[55,1022,1025],{"className":1023,"style":1024},[117],"height:0.2861em;",[55,1026],{}," is fit to human comparisons. Third, the policy is\n",[50,1029,1030],{},"RL-fine-tuned"," to maximize ",[55,1033,1035],{"className":1034},[58],[55,1036,1038],{"className":1037,"ariaHidden":63},[62],[55,1039,1041,1044],{"className":1040},[67],[55,1042],{"className":1043,"style":983},[71],[55,1045,1047,1050],{"className":1046},[76],[55,1048,800],{"className":1049,"style":480},[76,77],[55,1051,1053],{"className":1052},[104],[55,1054,1056,1076],{"className":1055},[108,109],[55,1057,1059,1073],{"className":1058},[113],[55,1060,1062],{"className":1061,"style":296},[117],[55,1063,1064,1067],{"style":1004},[55,1065],{"className":1066,"style":126},[125],[55,1068,1070],{"className":1069},[130,131,132,133],[55,1071,1014],{"className":1072},[76,77,133],[55,1074,142],{"className":1075},[141],[55,1077,1079],{"className":1078},[113],[55,1080,1082],{"className":1081,"style":1024},[117],[55,1083],{}," while a KL penalty holds it near\n",[55,1086,1088],{"className":1087},[58],[55,1089,1091],{"className":1090,"ariaHidden":63},[62],[55,1092,1094,1097],{"className":1093},[67],[55,1095],{"className":1096,"style":917},[71],[55,1098,1100,1103],{"className":1099},[76],[55,1101,924],{"className":1102,"style":99},[76,77],[55,1104,1106],{"className":1105},[104],[55,1107,1109,1135],{"className":1108},[108,109],[55,1110,1112,1132],{"className":1111},[113],[55,1113,1115],{"className":1114,"style":296},[117],[55,1116,1117,1120],{"style":121},[55,1118],{"className":1119,"style":126},[125],[55,1121,1123],{"className":1122},[130,131,132,133],[55,1124,1126],{"className":1125},[76,133],[55,1127,1129],{"className":1128},[76,951,133],[55,1130,955],{"className":1131},[76,133],[55,1133,142],{"className":1134},[141],[55,1136,1138],{"className":1137},[113],[55,1139,1141],{"className":1140,"style":149},[117],[55,1142],{},".",[1145,1146],"tikz-figure",{"hash":1147},"81660259609149aa0eb6da07ed33fce61439a06ae8f03b6fbf5ba7aaaccd50df",[11,1149,1150],{},"The three stages consume three different kinds of data, and that is the point: each\nstage extracts the signal that is cheapest to collect for what it must learn.",[821,1152,1153,1172],{},[824,1154,1155],{},[827,1156,1157,1160,1163,1166,1169],{},[830,1158,1159],{},"Stage",[830,1161,1162],{},"Data",[830,1164,1165],{},"Trains",[830,1167,1168],{},"Loss type",[830,1170,1171],{},"Output",[840,1173,1174,1278,1459],{},[827,1175,1176,1179,1212,1215,1218],{},[845,1177,1178],{},"1. SFT",[845,1180,1181,1211],{},[55,1182,1184],{"className":1183},[58],[55,1185,1187],{"className":1186,"ariaHidden":63},[62],[55,1188,1190,1193,1196,1199,1202,1205,1208],{"className":1189},[67],[55,1191],{"className":1192,"style":796},[71],[55,1194,521],{"className":1195},[509],[55,1197,78],{"className":1198},[76,77],[55,1200,156],{"className":1201},[155],[55,1203],{"className":1204,"style":161},[160],[55,1206,100],{"className":1207,"style":99},[76,77],[55,1209,564],{"className":1210},[563]," demonstrations",[845,1213,1214],{},"base LM",[845,1216,1217],{},"supervised cross-entropy",[845,1219,1220],{},[55,1221,1223],{"className":1222},[58],[55,1224,1226],{"className":1225,"ariaHidden":63},[62],[55,1227,1229,1232],{"className":1228},[67],[55,1230],{"className":1231,"style":917},[71],[55,1233,1235,1238],{"className":1234},[76],[55,1236,924],{"className":1237,"style":99},[76,77],[55,1239,1241],{"className":1240},[104],[55,1242,1244,1270],{"className":1243},[108,109],[55,1245,1247,1267],{"className":1246},[113],[55,1248,1250],{"className":1249,"style":296},[117],[55,1251,1252,1255],{"style":121},[55,1253],{"className":1254,"style":126},[125],[55,1256,1258],{"className":1257},[130,131,132,133],[55,1259,1261],{"className":1260},[76,133],[55,1262,1264],{"className":1263},[76,951,133],[55,1265,955],{"className":1266},[76,133],[55,1268,142],{"className":1269},[141],[55,1271,1273],{"className":1272},[113],[55,1274,1276],{"className":1275,"style":149},[117],[55,1277],{},[827,1279,1280,1283,1399,1402,1405],{},[845,1281,1282],{},"2. Reward model",[845,1284,1285,1398],{},[55,1286,1288],{"className":1287},[58],[55,1289,1291],{"className":1290,"ariaHidden":63},[62],[55,1292,1294,1297,1300,1303,1306,1309,1349,1352,1355,1395],{"className":1293},[67],[55,1295],{"className":1296,"style":796},[71],[55,1298,521],{"className":1299},[509],[55,1301,78],{"className":1302},[76,77],[55,1304,156],{"className":1305},[155],[55,1307],{"className":1308,"style":161},[160],[55,1310,1312,1315],{"className":1311},[76],[55,1313,100],{"className":1314,"style":99},[76,77],[55,1316,1318],{"className":1317},[104],[55,1319,1321,1341],{"className":1320},[108,109],[55,1322,1324,1338],{"className":1323},[113],[55,1325,1327],{"className":1326,"style":235},[117],[55,1328,1329,1332],{"style":121},[55,1330],{"className":1331,"style":126},[125],[55,1333,1335],{"className":1334},[130,131,132,133],[55,1336,248],{"className":1337,"style":247},[76,77,133],[55,1339,142],{"className":1340},[141],[55,1342,1344],{"className":1343},[113],[55,1345,1347],{"className":1346,"style":149},[117],[55,1348],{},[55,1350,156],{"className":1351},[155],[55,1353],{"className":1354,"style":161},[160],[55,1356,1358,1361],{"className":1357},[76],[55,1359,100],{"className":1360,"style":99},[76,77],[55,1362,1364],{"className":1363},[104],[55,1365,1367,1387],{"className":1366},[108,109],[55,1368,1370,1384],{"className":1369},[113],[55,1371,1373],{"className":1372,"style":296},[117],[55,1374,1375,1378],{"style":121},[55,1376],{"className":1377,"style":126},[125],[55,1379,1381],{"className":1380},[130,131,132,133],[55,1382,309],{"className":1383,"style":308},[76,77,133],[55,1385,142],{"className":1386},[141],[55,1388,1390],{"className":1389},[113],[55,1391,1393],{"className":1392,"style":149},[117],[55,1394],{},[55,1396,564],{"className":1397},[563]," comparisons",[845,1400,1401],{},"a scalar head",[845,1403,1404],{},"Bradley-Terry log-loss",[845,1406,1407],{},[55,1408,1410],{"className":1409},[58],[55,1411,1413],{"className":1412,"ariaHidden":63},[62],[55,1414,1416,1419],{"className":1415},[67],[55,1417],{"className":1418,"style":983},[71],[55,1420,1422,1425],{"className":1421},[76],[55,1423,800],{"className":1424,"style":480},[76,77],[55,1426,1428],{"className":1427},[104],[55,1429,1431,1451],{"className":1430},[108,109],[55,1432,1434,1448],{"className":1433},[113],[55,1435,1437],{"className":1436,"style":296},[117],[55,1438,1439,1442],{"style":1004},[55,1440],{"className":1441,"style":126},[125],[55,1443,1445],{"className":1444},[130,131,132,133],[55,1446,1014],{"className":1447},[76,77,133],[55,1449,142],{"className":1450},[141],[55,1452,1454],{"className":1453},[113],[55,1455,1457],{"className":1456,"style":1024},[117],[55,1458],{},[827,1460,1461,1464,1483,1486,1489],{},[845,1462,1463],{},"3. RL fine-tune",[845,1465,1466,1467,1482],{},"prompts ",[55,1468,1470],{"className":1469},[58],[55,1471,1473],{"className":1472,"ariaHidden":63},[62],[55,1474,1476,1479],{"className":1475},[67],[55,1477],{"className":1478,"style":72},[71],[55,1480,78],{"className":1481},[76,77]," only",[845,1484,1485],{},"the policy",[845,1487,1488],{},"KL-regularized reward",[845,1490,1491],{},[55,1492,1494],{"className":1493},[58],[55,1495,1497],{"className":1496,"ariaHidden":63},[62],[55,1498,1500,1503],{"className":1499},[67],[55,1501],{"className":1502,"style":917},[71],[55,1504,1506,1509],{"className":1505},[76],[55,1507,924],{"className":1508,"style":99},[76,77],[55,1510,1512],{"className":1511},[104],[55,1513,1515,1536],{"className":1514},[108,109],[55,1516,1518,1533],{"className":1517},[113],[55,1519,1521],{"className":1520,"style":296},[117],[55,1522,1523,1526],{"style":121},[55,1524],{"className":1525,"style":126},[125],[55,1527,1529],{"className":1528},[130,131,132,133],[55,1530,1532],{"className":1531,"style":480},[76,77,133],"θ",[55,1534,142],{"className":1535},[141],[55,1537,1539],{"className":1538},[113],[55,1540,1542],{"className":1541,"style":149},[117],[55,1543],{},[31,1545,1547],{"id":1546},"reward-modeling-from-pairwise-preferences","Reward modeling from pairwise preferences",[11,1549,1550,1551,1664,1665,1772,1773,1776],{},"We need a model that, given a comparison ",[55,1552,1554],{"className":1553},[58],[55,1555,1557],{"className":1556,"ariaHidden":63},[62],[55,1558,1560,1563,1566,1569,1572,1575,1615,1618,1621,1661],{"className":1559},[67],[55,1561],{"className":1562,"style":796},[71],[55,1564,521],{"className":1565},[509],[55,1567,78],{"className":1568},[76,77],[55,1570,156],{"className":1571},[155],[55,1573],{"className":1574,"style":161},[160],[55,1576,1578,1581],{"className":1577},[76],[55,1579,100],{"className":1580,"style":99},[76,77],[55,1582,1584],{"className":1583},[104],[55,1585,1587,1607],{"className":1586},[108,109],[55,1588,1590,1604],{"className":1589},[113],[55,1591,1593],{"className":1592,"style":235},[117],[55,1594,1595,1598],{"style":121},[55,1596],{"className":1597,"style":126},[125],[55,1599,1601],{"className":1600},[130,131,132,133],[55,1602,248],{"className":1603,"style":247},[76,77,133],[55,1605,142],{"className":1606},[141],[55,1608,1610],{"className":1609},[113],[55,1611,1613],{"className":1612,"style":149},[117],[55,1614],{},[55,1616,156],{"className":1617},[155],[55,1619],{"className":1620,"style":161},[160],[55,1622,1624,1627],{"className":1623},[76],[55,1625,100],{"className":1626,"style":99},[76,77],[55,1628,1630],{"className":1629},[104],[55,1631,1633,1653],{"className":1632},[108,109],[55,1634,1636,1650],{"className":1635},[113],[55,1637,1639],{"className":1638,"style":296},[117],[55,1640,1641,1644],{"style":121},[55,1642],{"className":1643,"style":126},[125],[55,1645,1647],{"className":1646},[130,131,132,133],[55,1648,309],{"className":1649,"style":308},[76,77,133],[55,1651,142],{"className":1652},[141],[55,1654,1656],{"className":1655},[113],[55,1657,1659],{"className":1658,"style":149},[117],[55,1660],{},[55,1662,564],{"className":1663},[563],", assigns a high probability to\nthe event ",[39,1666,1667,1719,1720,1143],{},[55,1668,1670],{"className":1669},[58],[55,1671,1673],{"className":1672,"ariaHidden":63},[62],[55,1674,1676,1679],{"className":1675},[67],[55,1677],{"className":1678,"style":92},[71],[55,1680,1682,1685],{"className":1681},[76],[55,1683,100],{"className":1684,"style":99},[76,77],[55,1686,1688],{"className":1687},[104],[55,1689,1691,1711],{"className":1690},[108,109],[55,1692,1694,1708],{"className":1693},[113],[55,1695,1697],{"className":1696,"style":235},[117],[55,1698,1699,1702],{"style":121},[55,1700],{"className":1701,"style":126},[125],[55,1703,1705],{"className":1704},[130,131,132,133],[55,1706,248],{"className":1707,"style":247},[76,77,133],[55,1709,142],{"className":1710},[141],[55,1712,1714],{"className":1713},[113],[55,1715,1717],{"className":1716,"style":149},[117],[55,1718],{}," beats ",[55,1721,1723],{"className":1722},[58],[55,1724,1726],{"className":1725,"ariaHidden":63},[62],[55,1727,1729,1732],{"className":1728},[67],[55,1730],{"className":1731,"style":92},[71],[55,1733,1735,1738],{"className":1734},[76],[55,1736,100],{"className":1737,"style":99},[76,77],[55,1739,1741],{"className":1740},[104],[55,1742,1744,1764],{"className":1743},[108,109],[55,1745,1747,1761],{"className":1746},[113],[55,1748,1750],{"className":1749,"style":296},[117],[55,1751,1752,1755],{"style":121},[55,1753],{"className":1754,"style":126},[125],[55,1756,1758],{"className":1757},[130,131,132,133],[55,1759,309],{"className":1760,"style":308},[76,77,133],[55,1762,142],{"className":1763},[141],[55,1765,1767],{"className":1766},[113],[55,1768,1770],{"className":1769,"style":149},[117],[55,1771],{}," The ",[50,1774,1775],{},"Bradley-Terry"," model is the canonical choice: it\nposits a latent scalar score per item and makes the probability of one beating another\na logistic function of their score difference.",[44,1778,1779,1946,2523],{"type":46},[11,1780,1781,1784,1785,1840,1841,376,1893,1945],{},[50,1782,1783],{},"Definition (Bradley-Terry preference model)."," Each response carries a latent\nreward ",[55,1786,1788],{"className":1787},[58],[55,1789,1791,1828],{"className":1790,"ariaHidden":63},[62],[55,1792,1794,1797,1800,1803,1806,1809,1812,1815,1818,1821,1825],{"className":1793},[67],[55,1795],{"className":1796,"style":796},[71],[55,1798,800],{"className":1799,"style":480},[76,77],[55,1801,521],{"className":1802},[509],[55,1804,78],{"className":1805},[76,77],[55,1807,156],{"className":1808},[155],[55,1810],{"className":1811,"style":161},[160],[55,1813,100],{"className":1814,"style":99},[76,77],[55,1816,564],{"className":1817},[563],[55,1819],{"className":1820,"style":263},[160],[55,1822,1824],{"className":1823},[267],"∈",[55,1826],{"className":1827,"style":263},[160],[55,1829,1831,1835],{"className":1830},[67],[55,1832],{"className":1833,"style":1834},[71],"height:0.6889em;",[55,1836,1839],{"className":1837},[76,1838],"mathbb","R",", and the probability that ",[55,1842,1844],{"className":1843},[58],[55,1845,1847],{"className":1846,"ariaHidden":63},[62],[55,1848,1850,1853],{"className":1849},[67],[55,1851],{"className":1852,"style":92},[71],[55,1854,1856,1859],{"className":1855},[76],[55,1857,100],{"className":1858,"style":99},[76,77],[55,1860,1862],{"className":1861},[104],[55,1863,1865,1885],{"className":1864},[108,109],[55,1866,1868,1882],{"className":1867},[113],[55,1869,1871],{"className":1870,"style":235},[117],[55,1872,1873,1876],{"style":121},[55,1874],{"className":1875,"style":126},[125],[55,1877,1879],{"className":1878},[130,131,132,133],[55,1880,248],{"className":1881,"style":247},[76,77,133],[55,1883,142],{"className":1884},[141],[55,1886,1888],{"className":1887},[113],[55,1889,1891],{"className":1890,"style":149},[117],[55,1892],{},[55,1894,1896],{"className":1895},[58],[55,1897,1899],{"className":1898,"ariaHidden":63},[62],[55,1900,1902,1905],{"className":1901},[67],[55,1903],{"className":1904,"style":92},[71],[55,1906,1908,1911],{"className":1907},[76],[55,1909,100],{"className":1910,"style":99},[76,77],[55,1912,1914],{"className":1913},[104],[55,1915,1917,1937],{"className":1916},[108,109],[55,1918,1920,1934],{"className":1919},[113],[55,1921,1923],{"className":1922,"style":296},[117],[55,1924,1925,1928],{"style":121},[55,1926],{"className":1927,"style":126},[125],[55,1929,1931],{"className":1930},[130,131,132,133],[55,1932,309],{"className":1933,"style":308},[76,77,133],[55,1935,142],{"className":1936},[141],[55,1938,1940],{"className":1939},[113],[55,1941,1943],{"className":1942,"style":149},[117],[55,1944],{},"\nis the logistic of the reward gap,",[55,1947,1950],{"className":1948},[1949],"katex-display",[55,1951,1953],{"className":1952},[58],[55,1954,1956,2019,2075,2102,2271],{"className":1955,"ariaHidden":63},[62],[55,1957,1959,1962,1967,1970,2010,2013,2016],{"className":1958},[67],[55,1960],{"className":1961,"style":796},[71],[55,1963,1966],{"className":1964,"style":1965},[76,77],"margin-right:0.1389em;","P",[55,1968,521],{"className":1969},[509],[55,1971,1973,1976],{"className":1972},[76],[55,1974,100],{"className":1975,"style":99},[76,77],[55,1977,1979],{"className":1978},[104],[55,1980,1982,2002],{"className":1981},[108,109],[55,1983,1985,1999],{"className":1984},[113],[55,1986,1988],{"className":1987,"style":235},[117],[55,1989,1990,1993],{"style":121},[55,1991],{"className":1992,"style":126},[125],[55,1994,1996],{"className":1995},[130,131,132,133],[55,1997,248],{"className":1998,"style":247},[76,77,133],[55,2000,142],{"className":2001},[141],[55,2003,2005],{"className":2004},[113],[55,2006,2008],{"className":2007,"style":149},[117],[55,2009],{},[55,2011],{"className":2012,"style":263},[160],[55,2014,268],{"className":2015},[267],[55,2017],{"className":2018,"style":263},[160],[55,2020,2022,2025,2065,2068,2072],{"className":2021},[67],[55,2023],{"className":2024,"style":796},[71],[55,2026,2028,2031],{"className":2027},[76],[55,2029,100],{"className":2030,"style":99},[76,77],[55,2032,2034],{"className":2033},[104],[55,2035,2037,2057],{"className":2036},[108,109],[55,2038,2040,2054],{"className":2039},[113],[55,2041,2043],{"className":2042,"style":296},[117],[55,2044,2045,2048],{"style":121},[55,2046],{"className":2047,"style":126},[125],[55,2049,2051],{"className":2050},[130,131,132,133],[55,2052,309],{"className":2053,"style":308},[76,77,133],[55,2055,142],{"className":2056},[141],[55,2058,2060],{"className":2059},[113],[55,2061,2063],{"className":2062,"style":149},[117],[55,2064],{},[55,2066],{"className":2067,"style":263},[160],[55,2069,2071],{"className":2070},[267],"∣",[55,2073],{"className":2074,"style":263},[160],[55,2076,2078,2081,2084,2087,2090,2093,2096,2099],{"className":2077},[67],[55,2079],{"className":2080,"style":796},[71],[55,2082,78],{"className":2083},[76,77],[55,2085,564],{"className":2086},[563],[55,2088],{"className":2089,"style":263},[160],[55,2091],{"className":2092,"style":263},[160],[55,2094,488],{"className":2095},[267],[55,2097],{"className":2098,"style":263},[160],[55,2100],{"className":2101,"style":263},[160],[55,2103,2105,2108,2112,2116,2119,2256,2259,2262,2265,2268],{"className":2104},[67],[55,2106],{"className":2107,"style":796},[71],[55,2109,2111],{"className":2110,"style":99},[76,77],"σ",[55,2113],{"className":2114,"style":2115},[160],"margin-right:-0.1667em;",[55,2117],{"className":2118,"style":161},[160],[55,2120,2122,2125,2128,2131,2134,2137,2140,2180,2183,2187,2192,2195,2198,2201,2204,2207,2210,2250,2253],{"className":2121},[502],[55,2123,521],{"className":2124,"style":511},[509,510],[55,2126,800],{"className":2127,"style":480},[76,77],[55,2129,521],{"className":2130},[509],[55,2132,78],{"className":2133},[76,77],[55,2135,156],{"className":2136},[155],[55,2138],{"className":2139,"style":161},[160],[55,2141,2143,2146],{"className":2142},[76],[55,2144,100],{"className":2145,"style":99},[76,77],[55,2147,2149],{"className":2148},[104],[55,2150,2152,2172],{"className":2151},[108,109],[55,2153,2155,2169],{"className":2154},[113],[55,2156,2158],{"className":2157,"style":235},[117],[55,2159,2160,2163],{"style":121},[55,2161],{"className":2162,"style":126},[125],[55,2164,2166],{"className":2165},[130,131,132,133],[55,2167,248],{"className":2168,"style":247},[76,77,133],[55,2170,142],{"className":2171},[141],[55,2173,2175],{"className":2174},[113],[55,2176,2178],{"className":2177,"style":149},[117],[55,2179],{},[55,2181,564],{"className":2182},[563],[55,2184],{"className":2185,"style":2186},[160],"margin-right:0.2222em;",[55,2188,2191],{"className":2189},[2190],"mbin","−",[55,2193],{"className":2194,"style":2186},[160],[55,2196,800],{"className":2197,"style":480},[76,77],[55,2199,521],{"className":2200},[509],[55,2202,78],{"className":2203},[76,77],[55,2205,156],{"className":2206},[155],[55,2208],{"className":2209,"style":161},[160],[55,2211,2213,2216],{"className":2212},[76],[55,2214,100],{"className":2215,"style":99},[76,77],[55,2217,2219],{"className":2218},[104],[55,2220,2222,2242],{"className":2221},[108,109],[55,2223,2225,2239],{"className":2224},[113],[55,2226,2228],{"className":2227,"style":296},[117],[55,2229,2230,2233],{"style":121},[55,2231],{"className":2232,"style":126},[125],[55,2234,2236],{"className":2235},[130,131,132,133],[55,2237,309],{"className":2238,"style":308},[76,77,133],[55,2240,142],{"className":2241},[141],[55,2243,2245],{"className":2244},[113],[55,2246,2248],{"className":2247,"style":149},[117],[55,2249],{},[55,2251,564],{"className":2252},[563],[55,2254,564],{"className":2255,"style":511},[563,510],[55,2257],{"className":2258,"style":263},[160],[55,2260],{"className":2261,"style":263},[160],[55,2263,488],{"className":2264},[267],[55,2266],{"className":2267,"style":263},[160],[55,2269],{"className":2270,"style":263},[160],[55,2272,2274,2278,2520],{"className":2273},[67],[55,2275],{"className":2276,"style":2277},[71],"height:2.2574em;vertical-align:-0.936em;",[55,2279,2281,2285,2517],{"className":2280},[76],[55,2282],{"className":2283},[509,2284],"nulldelimiter",[55,2286,2289],{"className":2287},[2288],"mfrac",[55,2290,2292,2508],{"className":2291},[108,109],[55,2293,2295,2505],{"className":2294},[113],[55,2296,2299,2482,2493],{"className":2297,"style":2298},[117],"height:1.3214em;",[55,2300,2302,2306],{"style":2301},"top:-2.314em;",[55,2303],{"className":2304,"style":2305},[125],"height:3em;",[55,2307,2309,2312,2315,2319,2322,2327,2330,2333],{"className":2308},[76],[55,2310,137],{"className":2311},[76],[55,2313],{"className":2314,"style":2186},[160],[55,2316,2318],{"className":2317},[2190],"+",[55,2320],{"className":2321,"style":2186},[160],[55,2323,2326],{"className":2324},[2325],"mop","exp",[55,2328],{"className":2329,"style":2115},[160],[55,2331],{"className":2332,"style":161},[160],[55,2334,2336,2339,2342,2345,2479],{"className":2335},[502],[55,2337,521],{"className":2338,"style":511},[509,510],[55,2340,2191],{"className":2341},[76],[55,2343],{"className":2344,"style":161},[160],[55,2346,2348,2351,2354,2357,2360,2363,2366,2406,2409,2412,2415,2418,2421,2424,2427,2430,2433,2473,2476],{"className":2347},[502],[55,2349,521],{"className":2350,"style":511},[509,510],[55,2352,800],{"className":2353,"style":480},[76,77],[55,2355,521],{"className":2356},[509],[55,2358,78],{"className":2359},[76,77],[55,2361,156],{"className":2362},[155],[55,2364],{"className":2365,"style":161},[160],[55,2367,2369,2372],{"className":2368},[76],[55,2370,100],{"className":2371,"style":99},[76,77],[55,2373,2375],{"className":2374},[104],[55,2376,2378,2398],{"className":2377},[108,109],[55,2379,2381,2395],{"className":2380},[113],[55,2382,2384],{"className":2383,"style":235},[117],[55,2385,2386,2389],{"style":121},[55,2387],{"className":2388,"style":126},[125],[55,2390,2392],{"className":2391},[130,131,132,133],[55,2393,248],{"className":2394,"style":247},[76,77,133],[55,2396,142],{"className":2397},[141],[55,2399,2401],{"className":2400},[113],[55,2402,2404],{"className":2403,"style":149},[117],[55,2405],{},[55,2407,564],{"className":2408},[563],[55,2410],{"className":2411,"style":2186},[160],[55,2413,2191],{"className":2414},[2190],[55,2416],{"className":2417,"style":2186},[160],[55,2419,800],{"className":2420,"style":480},[76,77],[55,2422,521],{"className":2423},[509],[55,2425,78],{"className":2426},[76,77],[55,2428,156],{"className":2429},[155],[55,2431],{"className":2432,"style":161},[160],[55,2434,2436,2439],{"className":2435},[76],[55,2437,100],{"className":2438,"style":99},[76,77],[55,2440,2442],{"className":2441},[104],[55,2443,2445,2465],{"className":2444},[108,109],[55,2446,2448,2462],{"className":2447},[113],[55,2449,2451],{"className":2450,"style":296},[117],[55,2452,2453,2456],{"style":121},[55,2454],{"className":2455,"style":126},[125],[55,2457,2459],{"className":2458},[130,131,132,133],[55,2460,309],{"className":2461,"style":308},[76,77,133],[55,2463,142],{"className":2464},[141],[55,2466,2468],{"className":2467},[113],[55,2469,2471],{"className":2470,"style":149},[117],[55,2472],{},[55,2474,564],{"className":2475},[563],[55,2477,564],{"className":2478,"style":511},[563,510],[55,2480,564],{"className":2481,"style":511},[563,510],[55,2483,2485,2488],{"style":2484},"top:-3.23em;",[55,2486],{"className":2487,"style":2305},[125],[55,2489],{"className":2490,"style":2492},[2491],"frac-line","border-bottom-width:0.04em;",[55,2494,2496,2499],{"style":2495},"top:-3.677em;",[55,2497],{"className":2498,"style":2305},[125],[55,2500,2502],{"className":2501},[76],[55,2503,137],{"className":2504},[76],[55,2506,142],{"className":2507},[141],[55,2509,2511],{"className":2510},[113],[55,2512,2515],{"className":2513,"style":2514},[117],"height:0.936em;",[55,2516],{},[55,2518],{"className":2519},[563,2284],[55,2521,1143],{"className":2522},[76],[11,2524,2525,2526,2529,2530,2555],{},"Only the ",[880,2527,2528],{},"difference"," of rewards is identified: adding any function ",[55,2531,2533],{"className":2532},[58],[55,2534,2536],{"className":2535,"ariaHidden":63},[62],[55,2537,2539,2542,2546,2549,2552],{"className":2538},[67],[55,2540],{"className":2541,"style":796},[71],[55,2543,2545],{"className":2544},[76,77],"c",[55,2547,521],{"className":2548},[509],[55,2550,78],{"className":2551},[76,77],[55,2553,564],{"className":2554},[563]," to every\nreward leaves all preference probabilities unchanged.",[11,2557,2558,2559,2703,2704,2739,2740,2843,2844,1143],{},"The model has the right limits. When ",[55,2560,2562],{"className":2561},[58],[55,2563,2565,2639],{"className":2564,"ariaHidden":63},[62],[55,2566,2568,2571,2574,2577,2580,2583,2586,2626,2629,2632,2636],{"className":2567},[67],[55,2569],{"className":2570,"style":796},[71],[55,2572,800],{"className":2573,"style":480},[76,77],[55,2575,521],{"className":2576},[509],[55,2578,78],{"className":2579},[76,77],[55,2581,156],{"className":2582},[155],[55,2584],{"className":2585,"style":161},[160],[55,2587,2589,2592],{"className":2588},[76],[55,2590,100],{"className":2591,"style":99},[76,77],[55,2593,2595],{"className":2594},[104],[55,2596,2598,2618],{"className":2597},[108,109],[55,2599,2601,2615],{"className":2600},[113],[55,2602,2604],{"className":2603,"style":235},[117],[55,2605,2606,2609],{"style":121},[55,2607],{"className":2608,"style":126},[125],[55,2610,2612],{"className":2611},[130,131,132,133],[55,2613,248],{"className":2614,"style":247},[76,77,133],[55,2616,142],{"className":2617},[141],[55,2619,2621],{"className":2620},[113],[55,2622,2624],{"className":2623,"style":149},[117],[55,2625],{},[55,2627,564],{"className":2628},[563],[55,2630],{"className":2631,"style":263},[160],[55,2633,2635],{"className":2634},[267],"≫",[55,2637],{"className":2638,"style":263},[160],[55,2640,2642,2645,2648,2651,2654,2657,2660,2700],{"className":2641},[67],[55,2643],{"className":2644,"style":796},[71],[55,2646,800],{"className":2647,"style":480},[76,77],[55,2649,521],{"className":2650},[509],[55,2652,78],{"className":2653},[76,77],[55,2655,156],{"className":2656},[155],[55,2658],{"className":2659,"style":161},[160],[55,2661,2663,2666],{"className":2662},[76],[55,2664,100],{"className":2665,"style":99},[76,77],[55,2667,2669],{"className":2668},[104],[55,2670,2672,2692],{"className":2671},[108,109],[55,2673,2675,2689],{"className":2674},[113],[55,2676,2678],{"className":2677,"style":296},[117],[55,2679,2680,2683],{"style":121},[55,2681],{"className":2682,"style":126},[125],[55,2684,2686],{"className":2685},[130,131,132,133],[55,2687,309],{"className":2688,"style":308},[76,77,133],[55,2690,142],{"className":2691},[141],[55,2693,2695],{"className":2694},[113],[55,2696,2698],{"className":2697,"style":149},[117],[55,2699],{},[55,2701,564],{"className":2702},[563]," the gap is large and\npositive, so ",[55,2705,2707],{"className":2706},[58],[55,2708,2710,2729],{"className":2709,"ariaHidden":63},[62],[55,2711,2713,2716,2719,2722,2726],{"className":2712},[67],[55,2714],{"className":2715,"style":475},[71],[55,2717,1966],{"className":2718,"style":1965},[76,77],[55,2720],{"className":2721,"style":263},[160],[55,2723,2725],{"className":2724},[267],"→",[55,2727],{"className":2728,"style":263},[160],[55,2730,2732,2736],{"className":2731},[67],[55,2733],{"className":2734,"style":2735},[71],"height:0.6444em;",[55,2737,137],{"className":2738},[76],"; when the two rewards are equal the gap is zero and ",[55,2741,2743],{"className":2742},[58],[55,2744,2746,2764],{"className":2745,"ariaHidden":63},[62],[55,2747,2749,2752,2755,2758,2761],{"className":2748},[67],[55,2750],{"className":2751,"style":475},[71],[55,2753,1966],{"className":2754,"style":1965},[76,77],[55,2756],{"className":2757,"style":263},[160],[55,2759,488],{"className":2760},[267],[55,2762],{"className":2763,"style":263},[160],[55,2765,2767,2771],{"className":2766},[67],[55,2768],{"className":2769,"style":2770},[71],"height:1.1901em;vertical-align:-0.345em;",[55,2772,2774,2777,2840],{"className":2773},[76],[55,2775],{"className":2776},[509,2284],[55,2778,2780],{"className":2779},[2288],[55,2781,2783,2831],{"className":2782},[108,109],[55,2784,2786,2828],{"className":2785},[113],[55,2787,2790,2805,2813],{"className":2788,"style":2789},[117],"height:0.8451em;",[55,2791,2793,2796],{"style":2792},"top:-2.655em;",[55,2794],{"className":2795,"style":2305},[125],[55,2797,2799],{"className":2798},[130,131,132,133],[55,2800,2802],{"className":2801},[76,133],[55,2803,191],{"className":2804},[76,133],[55,2806,2807,2810],{"style":2484},[55,2808],{"className":2809,"style":2305},[125],[55,2811],{"className":2812,"style":2492},[2491],[55,2814,2816,2819],{"style":2815},"top:-3.394em;",[55,2817],{"className":2818,"style":2305},[125],[55,2820,2822],{"className":2821},[130,131,132,133],[55,2823,2825],{"className":2824},[76,133],[55,2826,137],{"className":2827},[76,133],[55,2829,142],{"className":2830},[141],[55,2832,2834],{"className":2833},[113],[55,2835,2838],{"className":2836,"style":2837},[117],"height:0.345em;",[55,2839],{},[55,2841],{"className":2842},[563,2284],",\na coin flip; and the curve is symmetric, so ",[55,2845,2847],{"className":2846},[58],[55,2848,2850,2911,2969,3030,3088],{"className":2849,"ariaHidden":63},[62],[55,2851,2853,2856,2859,2862,2902,2905,2908],{"className":2852},[67],[55,2854],{"className":2855,"style":796},[71],[55,2857,1966],{"className":2858,"style":1965},[76,77],[55,2860,521],{"className":2861},[509],[55,2863,2865,2868],{"className":2864},[76],[55,2866,100],{"className":2867,"style":99},[76,77],[55,2869,2871],{"className":2870},[104],[55,2872,2874,2894],{"className":2873},[108,109],[55,2875,2877,2891],{"className":2876},[113],[55,2878,2880],{"className":2879,"style":235},[117],[55,2881,2882,2885],{"style":121},[55,2883],{"className":2884,"style":126},[125],[55,2886,2888],{"className":2887},[130,131,132,133],[55,2889,248],{"className":2890,"style":247},[76,77,133],[55,2892,142],{"className":2893},[141],[55,2895,2897],{"className":2896},[113],[55,2898,2900],{"className":2899,"style":149},[117],[55,2901],{},[55,2903],{"className":2904,"style":263},[160],[55,2906,268],{"className":2907},[267],[55,2909],{"className":2910,"style":263},[160],[55,2912,2914,2917,2957,2960,2963,2966],{"className":2913},[67],[55,2915],{"className":2916,"style":796},[71],[55,2918,2920,2923],{"className":2919},[76],[55,2921,100],{"className":2922,"style":99},[76,77],[55,2924,2926],{"className":2925},[104],[55,2927,2929,2949],{"className":2928},[108,109],[55,2930,2932,2946],{"className":2931},[113],[55,2933,2935],{"className":2934,"style":296},[117],[55,2936,2937,2940],{"style":121},[55,2938],{"className":2939,"style":126},[125],[55,2941,2943],{"className":2942},[130,131,132,133],[55,2944,309],{"className":2945,"style":308},[76,77,133],[55,2947,142],{"className":2948},[141],[55,2950,2952],{"className":2951},[113],[55,2953,2955],{"className":2954,"style":149},[117],[55,2956],{},[55,2958,564],{"className":2959},[563],[55,2961],{"className":2962,"style":2186},[160],[55,2964,2318],{"className":2965},[2190],[55,2967],{"className":2968,"style":2186},[160],[55,2970,2972,2975,2978,2981,3021,3024,3027],{"className":2971},[67],[55,2973],{"className":2974,"style":796},[71],[55,2976,1966],{"className":2977,"style":1965},[76,77],[55,2979,521],{"className":2980},[509],[55,2982,2984,2987],{"className":2983},[76],[55,2985,100],{"className":2986,"style":99},[76,77],[55,2988,2990],{"className":2989},[104],[55,2991,2993,3013],{"className":2992},[108,109],[55,2994,2996,3010],{"className":2995},[113],[55,2997,2999],{"className":2998,"style":296},[117],[55,3000,3001,3004],{"style":121},[55,3002],{"className":3003,"style":126},[125],[55,3005,3007],{"className":3006},[130,131,132,133],[55,3008,309],{"className":3009,"style":308},[76,77,133],[55,3011,142],{"className":3012},[141],[55,3014,3016],{"className":3015},[113],[55,3017,3019],{"className":3018,"style":149},[117],[55,3020],{},[55,3022],{"className":3023,"style":263},[160],[55,3025,268],{"className":3026},[267],[55,3028],{"className":3029,"style":263},[160],[55,3031,3033,3036,3076,3079,3082,3085],{"className":3032},[67],[55,3034],{"className":3035,"style":796},[71],[55,3037,3039,3042],{"className":3038},[76],[55,3040,100],{"className":3041,"style":99},[76,77],[55,3043,3045],{"className":3044},[104],[55,3046,3048,3068],{"className":3047},[108,109],[55,3049,3051,3065],{"className":3050},[113],[55,3052,3054],{"className":3053,"style":235},[117],[55,3055,3056,3059],{"style":121},[55,3057],{"className":3058,"style":126},[125],[55,3060,3062],{"className":3061},[130,131,132,133],[55,3063,248],{"className":3064,"style":247},[76,77,133],[55,3066,142],{"className":3067},[141],[55,3069,3071],{"className":3070},[113],[55,3072,3074],{"className":3073,"style":149},[117],[55,3075],{},[55,3077,564],{"className":3078},[563],[55,3080],{"className":3081,"style":263},[160],[55,3083,488],{"className":3084},[267],[55,3086],{"className":3087,"style":263},[160],[55,3089,3091,3094],{"className":3090},[67],[55,3092],{"className":3093,"style":2735},[71],[55,3095,137],{"className":3096},[76],[1145,3098],{"hash":3099},"fdd3ef414f543c3f9ae396a091c4a71f3e18ad24bf5371b1974b618ebe7b9da4",[3101,3102,3104],"h3",{"id":3103},"deriving-the-reward-loss","Deriving the reward loss",[11,3106,3107,3108,3160,3161,3164,3165,3366,3367,3640],{},"We fit ",[55,3109,3111],{"className":3110},[58],[55,3112,3114],{"className":3113,"ariaHidden":63},[62],[55,3115,3117,3120],{"className":3116},[67],[55,3118],{"className":3119,"style":983},[71],[55,3121,3123,3126],{"className":3122},[76],[55,3124,800],{"className":3125,"style":480},[76,77],[55,3127,3129],{"className":3128},[104],[55,3130,3132,3152],{"className":3131},[108,109],[55,3133,3135,3149],{"className":3134},[113],[55,3136,3138],{"className":3137,"style":296},[117],[55,3139,3140,3143],{"style":1004},[55,3141],{"className":3142,"style":126},[125],[55,3144,3146],{"className":3145},[130,131,132,133],[55,3147,1014],{"className":3148},[76,77,133],[55,3150,142],{"className":3151},[141],[55,3153,3155],{"className":3154},[113],[55,3156,3158],{"className":3157,"style":1024},[117],[55,3159],{}," by maximum likelihood on the comparison dataset. Treat each comparison\nas one Bernoulli observation whose outcome is ",[39,3162,3163],{},"the labeled winner won."," The likelihood\nof a single comparison is ",[55,3166,3168],{"className":3167},[58],[55,3169,3171,3232,3287,3308],{"className":3170,"ariaHidden":63},[62],[55,3172,3174,3177,3180,3183,3223,3226,3229],{"className":3173},[67],[55,3175],{"className":3176,"style":796},[71],[55,3178,1966],{"className":3179,"style":1965},[76,77],[55,3181,521],{"className":3182},[509],[55,3184,3186,3189],{"className":3185},[76],[55,3187,100],{"className":3188,"style":99},[76,77],[55,3190,3192],{"className":3191},[104],[55,3193,3195,3215],{"className":3194},[108,109],[55,3196,3198,3212],{"className":3197},[113],[55,3199,3201],{"className":3200,"style":235},[117],[55,3202,3203,3206],{"style":121},[55,3204],{"className":3205,"style":126},[125],[55,3207,3209],{"className":3208},[130,131,132,133],[55,3210,248],{"className":3211,"style":247},[76,77,133],[55,3213,142],{"className":3214},[141],[55,3216,3218],{"className":3217},[113],[55,3219,3221],{"className":3220,"style":149},[117],[55,3222],{},[55,3224],{"className":3225,"style":263},[160],[55,3227,268],{"className":3228},[267],[55,3230],{"className":3231,"style":263},[160],[55,3233,3235,3238,3278,3281,3284],{"className":3234},[67],[55,3236],{"className":3237,"style":796},[71],[55,3239,3241,3244],{"className":3240},[76],[55,3242,100],{"className":3243,"style":99},[76,77],[55,3245,3247],{"className":3246},[104],[55,3248,3250,3270],{"className":3249},[108,109],[55,3251,3253,3267],{"className":3252},[113],[55,3254,3256],{"className":3255,"style":296},[117],[55,3257,3258,3261],{"style":121},[55,3259],{"className":3260,"style":126},[125],[55,3262,3264],{"className":3263},[130,131,132,133],[55,3265,309],{"className":3266,"style":308},[76,77,133],[55,3268,142],{"className":3269},[141],[55,3271,3273],{"className":3272},[113],[55,3274,3276],{"className":3275,"style":149},[117],[55,3277],{},[55,3279],{"className":3280,"style":263},[160],[55,3282,2071],{"className":3283},[267],[55,3285],{"className":3286,"style":263},[160],[55,3288,3290,3293,3296,3299,3302,3305],{"className":3289},[67],[55,3291],{"className":3292,"style":796},[71],[55,3294,78],{"className":3295},[76,77],[55,3297,564],{"className":3298},[563],[55,3300],{"className":3301,"style":263},[160],[55,3303,488],{"className":3304},[267],[55,3306],{"className":3307,"style":263},[160],[55,3309,3311,3315,3318,3321,3363],{"className":3310},[67],[55,3312],{"className":3313,"style":3314},[71],"height:1.0361em;vertical-align:-0.2861em;",[55,3316,2111],{"className":3317,"style":99},[76,77],[55,3319,521],{"className":3320},[509],[55,3322,3324,3328],{"className":3323},[76],[55,3325,3327],{"className":3326},[76],"Δ",[55,3329,3331],{"className":3330},[104],[55,3332,3334,3355],{"className":3333},[108,109],[55,3335,3337,3352],{"className":3336},[113],[55,3338,3340],{"className":3339,"style":296},[117],[55,3341,3343,3346],{"style":3342},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[55,3344],{"className":3345,"style":126},[125],[55,3347,3349],{"className":3348},[130,131,132,133],[55,3350,1014],{"className":3351},[76,77,133],[55,3353,142],{"className":3354},[141],[55,3356,3358],{"className":3357},[113],[55,3359,3361],{"className":3360,"style":1024},[117],[55,3362],{},[55,3364,564],{"className":3365},[563]," with the\nshorthand ",[55,3368,3370],{"className":3369},[58],[55,3371,3373,3429,3539],{"className":3372,"ariaHidden":63},[62],[55,3374,3376,3380,3420,3423,3426],{"className":3375},[67],[55,3377],{"className":3378,"style":3379},[71],"height:0.9694em;vertical-align:-0.2861em;",[55,3381,3383,3386],{"className":3382},[76],[55,3384,3327],{"className":3385},[76],[55,3387,3389],{"className":3388},[104],[55,3390,3392,3412],{"className":3391},[108,109],[55,3393,3395,3409],{"className":3394},[113],[55,3396,3398],{"className":3397,"style":296},[117],[55,3399,3400,3403],{"style":3342},[55,3401],{"className":3402,"style":126},[125],[55,3404,3406],{"className":3405},[130,131,132,133],[55,3407,1014],{"className":3408},[76,77,133],[55,3410,142],{"className":3411},[141],[55,3413,3415],{"className":3414},[113],[55,3416,3418],{"className":3417,"style":1024},[117],[55,3419],{},[55,3421],{"className":3422,"style":263},[160],[55,3424,488],{"className":3425},[267],[55,3427],{"className":3428,"style":263},[160],[55,3430,3432,3435,3475,3478,3481,3484,3487,3527,3530,3533,3536],{"className":3431},[67],[55,3433],{"className":3434,"style":3314},[71],[55,3436,3438,3441],{"className":3437},[76],[55,3439,800],{"className":3440,"style":480},[76,77],[55,3442,3444],{"className":3443},[104],[55,3445,3447,3467],{"className":3446},[108,109],[55,3448,3450,3464],{"className":3449},[113],[55,3451,3453],{"className":3452,"style":296},[117],[55,3454,3455,3458],{"style":1004},[55,3456],{"className":3457,"style":126},[125],[55,3459,3461],{"className":3460},[130,131,132,133],[55,3462,1014],{"className":3463},[76,77,133],[55,3465,142],{"className":3466},[141],[55,3468,3470],{"className":3469},[113],[55,3471,3473],{"className":3472,"style":1024},[117],[55,3474],{},[55,3476,521],{"className":3477},[509],[55,3479,78],{"className":3480},[76,77],[55,3482,156],{"className":3483},[155],[55,3485],{"className":3486,"style":161},[160],[55,3488,3490,3493],{"className":3489},[76],[55,3491,100],{"className":3492,"style":99},[76,77],[55,3494,3496],{"className":3495},[104],[55,3497,3499,3519],{"className":3498},[108,109],[55,3500,3502,3516],{"className":3501},[113],[55,3503,3505],{"className":3504,"style":235},[117],[55,3506,3507,3510],{"style":121},[55,3508],{"className":3509,"style":126},[125],[55,3511,3513],{"className":3512},[130,131,132,133],[55,3514,248],{"className":3515,"style":247},[76,77,133],[55,3517,142],{"className":3518},[141],[55,3520,3522],{"className":3521},[113],[55,3523,3525],{"className":3524,"style":149},[117],[55,3526],{},[55,3528,564],{"className":3529},[563],[55,3531],{"className":3532,"style":2186},[160],[55,3534,2191],{"className":3535},[2190],[55,3537],{"className":3538,"style":2186},[160],[55,3540,3542,3545,3585,3588,3591,3594,3597,3637],{"className":3541},[67],[55,3543],{"className":3544,"style":3314},[71],[55,3546,3548,3551],{"className":3547},[76],[55,3549,800],{"className":3550,"style":480},[76,77],[55,3552,3554],{"className":3553},[104],[55,3555,3557,3577],{"className":3556},[108,109],[55,3558,3560,3574],{"className":3559},[113],[55,3561,3563],{"className":3562,"style":296},[117],[55,3564,3565,3568],{"style":1004},[55,3566],{"className":3567,"style":126},[125],[55,3569,3571],{"className":3570},[130,131,132,133],[55,3572,1014],{"className":3573},[76,77,133],[55,3575,142],{"className":3576},[141],[55,3578,3580],{"className":3579},[113],[55,3581,3583],{"className":3582,"style":1024},[117],[55,3584],{},[55,3586,521],{"className":3587},[509],[55,3589,78],{"className":3590},[76,77],[55,3592,156],{"className":3593},[155],[55,3595],{"className":3596,"style":161},[160],[55,3598,3600,3603],{"className":3599},[76],[55,3601,100],{"className":3602,"style":99},[76,77],[55,3604,3606],{"className":3605},[104],[55,3607,3609,3629],{"className":3608},[108,109],[55,3610,3612,3626],{"className":3611},[113],[55,3613,3615],{"className":3614,"style":296},[117],[55,3616,3617,3620],{"style":121},[55,3618],{"className":3619,"style":126},[125],[55,3621,3623],{"className":3622},[130,131,132,133],[55,3624,309],{"className":3625,"style":308},[76,77,133],[55,3627,142],{"className":3628},[141],[55,3630,3632],{"className":3631},[113],[55,3633,3635],{"className":3634,"style":149},[117],[55,3636],{},[55,3638,564],{"className":3639},[563],". The dataset log-likelihood is\nthe sum over independent comparisons,",[55,3642,3644],{"className":3643},[1949],[55,3645,3647],{"className":3646},[58],[55,3648,3650,3684],{"className":3649,"ariaHidden":63},[62],[55,3651,3653,3656,3660,3663,3666,3669,3672,3675,3678,3681],{"className":3652},[67],[55,3654],{"className":3655,"style":796},[71],[55,3657,3659],{"className":3658},[76],"ℓ",[55,3661,521],{"className":3662},[509],[55,3664,1014],{"className":3665},[76,77],[55,3667,564],{"className":3668},[563],[55,3670],{"className":3671,"style":263},[160],[55,3673],{"className":3674,"style":263},[160],[55,3676,488],{"className":3677},[267],[55,3679],{"className":3680,"style":263},[160],[55,3682],{"className":3683,"style":263},[160],[55,3685,3687,3691,3854,3857,3866,3869,3872,3875,3878,3927,3930],{"className":3686},[67],[55,3688],{"className":3689,"style":3690},[71],"height:2.566em;vertical-align:-1.516em;",[55,3692,3695],{"className":3693},[2325,3694],"op-limits",[55,3696,3698,3845],{"className":3697},[108,109],[55,3699,3701,3842],{"className":3700},[113],[55,3702,3705,3828],{"className":3703,"style":3704},[117],"height:1.05em;",[55,3706,3708,3712],{"style":3707},"top:-1.809em;margin-left:0em;",[55,3709],{"className":3710,"style":3711},[125],"height:3.05em;",[55,3713,3715],{"className":3714},[130,131,132,133],[55,3716,3718,3721,3724,3727,3773,3776,3819,3822,3825],{"className":3717},[76,133],[55,3719,521],{"className":3720},[509,133],[55,3722,78],{"className":3723},[76,77,133],[55,3725,156],{"className":3726},[155,133],[55,3728,3730,3733],{"className":3729},[76,133],[55,3731,100],{"className":3732,"style":99},[76,77,133],[55,3734,3736],{"className":3735},[104],[55,3737,3739,3764],{"className":3738},[108,109],[55,3740,3742,3761],{"className":3741},[113],[55,3743,3746],{"className":3744,"style":3745},[117],"height:0.1645em;",[55,3747,3749,3753],{"style":3748},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[55,3750],{"className":3751,"style":3752},[125],"height:2.5em;",[55,3754,3758],{"className":3755},[130,3756,3757,133],"reset-size3","size1",[55,3759,248],{"className":3760,"style":247},[76,77,133],[55,3762,142],{"className":3763},[141],[55,3765,3767],{"className":3766},[113],[55,3768,3771],{"className":3769,"style":3770},[117],"height:0.143em;",[55,3772],{},[55,3774,156],{"className":3775},[155,133],[55,3777,3779,3782],{"className":3778},[76,133],[55,3780,100],{"className":3781,"style":99},[76,77,133],[55,3783,3785],{"className":3784},[104],[55,3786,3788,3810],{"className":3787},[108,109],[55,3789,3791,3807],{"className":3790},[113],[55,3792,3795],{"className":3793,"style":3794},[117],"height:0.3448em;",[55,3796,3798,3801],{"style":3797},"top:-2.3488em;margin-left:-0.0359em;margin-right:0.0714em;",[55,3799],{"className":3800,"style":3752},[125],[55,3802,3804],{"className":3803},[130,3756,3757,133],[55,3805,309],{"className":3806,"style":308},[76,77,133],[55,3808,142],{"className":3809},[141],[55,3811,3813],{"className":3812},[113],[55,3814,3817],{"className":3815,"style":3816},[117],"height:0.1512em;",[55,3818],{},[55,3820,564],{"className":3821},[563,133],[55,3823,1824],{"className":3824},[267,133],[55,3826,481],{"className":3827,"style":480},[76,479,133],[55,3829,3831,3834],{"style":3830},"top:-3.05em;",[55,3832],{"className":3833,"style":3711},[125],[55,3835,3836],{},[55,3837,3841],{"className":3838},[2325,3839,3840],"op-symbol","large-op","∑",[55,3843,142],{"className":3844},[141],[55,3846,3848],{"className":3847},[113],[55,3849,3852],{"className":3850,"style":3851},[117],"height:1.516em;",[55,3853],{},[55,3855],{"className":3856,"style":161},[160],[55,3858,3860],{"className":3859},[2325],[55,3861,3865],{"className":3862,"style":3864},[76,3863],"mathrm","margin-right:0.0139em;","log",[55,3867],{"className":3868,"style":161},[160],[55,3870,2111],{"className":3871,"style":99},[76,77],[55,3873],{"className":3874,"style":2115},[160],[55,3876],{"className":3877,"style":161},[160],[55,3879,3881,3884,3924],{"className":3880},[502],[55,3882,521],{"className":3883,"style":511},[509,510],[55,3885,3887,3890],{"className":3886},[76],[55,3888,3327],{"className":3889},[76],[55,3891,3893],{"className":3892},[104],[55,3894,3896,3916],{"className":3895},[108,109],[55,3897,3899,3913],{"className":3898},[113],[55,3900,3902],{"className":3901,"style":296},[117],[55,3903,3904,3907],{"style":3342},[55,3905],{"className":3906,"style":126},[125],[55,3908,3910],{"className":3909},[130,131,132,133],[55,3911,1014],{"className":3912},[76,77,133],[55,3914,142],{"className":3915},[141],[55,3917,3919],{"className":3918},[113],[55,3920,3922],{"className":3921,"style":1024},[117],[55,3923],{},[55,3925,564],{"className":3926,"style":511},[563,510],[55,3928],{"className":3929,"style":161},[160],[55,3931,1143],{"className":3932},[76],[11,3934,3935],{},"Maximizing the log-likelihood is minimizing its negative, normalized to an expectation\nover the dataset. This is the reward-model objective.",[44,3937,3938,3997,4489],{"type":46},[11,3939,3940,3943,3944,3996],{},[50,3941,3942],{},"Definition (Reward-model loss)."," The reward model ",[55,3945,3947],{"className":3946},[58],[55,3948,3950],{"className":3949,"ariaHidden":63},[62],[55,3951,3953,3956],{"className":3952},[67],[55,3954],{"className":3955,"style":983},[71],[55,3957,3959,3962],{"className":3958},[76],[55,3960,800],{"className":3961,"style":480},[76,77],[55,3963,3965],{"className":3964},[104],[55,3966,3968,3988],{"className":3967},[108,109],[55,3969,3971,3985],{"className":3970},[113],[55,3972,3974],{"className":3973,"style":296},[117],[55,3975,3976,3979],{"style":1004},[55,3977],{"className":3978,"style":126},[125],[55,3980,3982],{"className":3981},[130,131,132,133],[55,3983,1014],{"className":3984},[76,77,133],[55,3986,142],{"className":3987},[141],[55,3989,3991],{"className":3990},[113],[55,3992,3994],{"className":3993,"style":1024},[117],[55,3995],{}," minimizes the negative\nlog-likelihood of the observed preferences under Bradley-Terry,",[55,3998,4000],{"className":3999},[1949],[55,4001,4003],{"className":4002},[58],[55,4004,4006,4085],{"className":4005,"ariaHidden":63},[62],[55,4007,4009,4012,4061,4064,4067,4070,4073,4076,4079,4082],{"className":4008},[67],[55,4010],{"className":4011,"style":796},[71],[55,4013,4015,4019],{"className":4014},[76],[55,4016,4018],{"className":4017},[76,479],"L",[55,4020,4022],{"className":4021},[104],[55,4023,4025,4053],{"className":4024},[108,109],[55,4026,4028,4050],{"className":4027},[113],[55,4029,4032],{"className":4030,"style":4031},[117],"height:0.3283em;",[55,4033,4034,4037],{"style":3342},[55,4035],{"className":4036,"style":126},[125],[55,4038,4040],{"className":4039},[130,131,132,133],[55,4041,4043],{"className":4042},[76,133],[55,4044,4046],{"className":4045},[76,951,133],[55,4047,4049],{"className":4048},[76,133],"RM",[55,4051,142],{"className":4052},[141],[55,4054,4056],{"className":4055},[113],[55,4057,4059],{"className":4058,"style":149},[117],[55,4060],{},[55,4062,521],{"className":4063},[509],[55,4065,1014],{"className":4066},[76,77],[55,4068,564],{"className":4069},[563],[55,4071],{"className":4072,"style":263},[160],[55,4074],{"className":4075,"style":263},[160],[55,4077,488],{"className":4078},[267],[55,4080],{"className":4081,"style":263},[160],[55,4083],{"className":4084,"style":263},[160],[55,4086,4088,4092,4095,4098,4243,4246,4483,4486],{"className":4087},[67],[55,4089],{"className":4090,"style":4091},[71],"height:1.1052em;vertical-align:-0.3552em;",[55,4093,2191],{"className":4094},[76],[55,4096],{"className":4097,"style":161},[160],[55,4099,4101,4105],{"className":4100},[76],[55,4102,4104],{"className":4103},[76,1838],"E",[55,4106,4108],{"className":4107},[104],[55,4109,4111,4234],{"className":4110},[108,109],[55,4112,4114,4231],{"className":4113},[113],[55,4115,4117],{"className":4116,"style":3794},[117],[55,4118,4120,4123],{"style":4119},"top:-2.5198em;margin-left:0em;margin-right:0.05em;",[55,4121],{"className":4122,"style":126},[125],[55,4124,4126],{"className":4125},[130,131,132,133],[55,4127,4129,4132,4135,4138,4178,4181,4221,4224,4228],{"className":4128},[76,133],[55,4130,521],{"className":4131},[509,133],[55,4133,78],{"className":4134},[76,77,133],[55,4136,156],{"className":4137},[155,133],[55,4139,4141,4144],{"className":4140},[76,133],[55,4142,100],{"className":4143,"style":99},[76,77,133],[55,4145,4147],{"className":4146},[104],[55,4148,4150,4170],{"className":4149},[108,109],[55,4151,4153,4167],{"className":4152},[113],[55,4154,4156],{"className":4155,"style":3745},[117],[55,4157,4158,4161],{"style":3748},[55,4159],{"className":4160,"style":3752},[125],[55,4162,4164],{"className":4163},[130,3756,3757,133],[55,4165,248],{"className":4166,"style":247},[76,77,133],[55,4168,142],{"className":4169},[141],[55,4171,4173],{"className":4172},[113],[55,4174,4176],{"className":4175,"style":3770},[117],[55,4177],{},[55,4179,156],{"className":4180},[155,133],[55,4182,4184,4187],{"className":4183},[76,133],[55,4185,100],{"className":4186,"style":99},[76,77,133],[55,4188,4190],{"className":4189},[104],[55,4191,4193,4213],{"className":4192},[108,109],[55,4194,4196,4210],{"className":4195},[113],[55,4197,4199],{"className":4198,"style":3794},[117],[55,4200,4201,4204],{"style":3797},[55,4202],{"className":4203,"style":3752},[125],[55,4205,4207],{"className":4206},[130,3756,3757,133],[55,4208,309],{"className":4209,"style":308},[76,77,133],[55,4211,142],{"className":4212},[141],[55,4214,4216],{"className":4215},[113],[55,4217,4219],{"className":4218,"style":3816},[117],[55,4220],{},[55,4222,564],{"className":4223},[563,133],[55,4225,4227],{"className":4226},[267,133],"∼",[55,4229,481],{"className":4230,"style":480},[76,479,133],[55,4232,142],{"className":4233},[141],[55,4235,4237],{"className":4236},[113],[55,4238,4241],{"className":4239,"style":4240},[117],"height:0.3552em;",[55,4242],{},[55,4244],{"className":4245,"style":161},[160],[55,4247,4249,4253,4259,4262,4265,4268,4271,4479],{"className":4248},[502],[55,4250,4252],{"className":4251,"style":511},[509,510],"[",[55,4254,4256],{"className":4255},[2325],[55,4257,3865],{"className":4258,"style":3864},[76,3863],[55,4260],{"className":4261,"style":161},[160],[55,4263,2111],{"className":4264,"style":99},[76,77],[55,4266],{"className":4267,"style":2115},[160],[55,4269],{"className":4270,"style":161},[160],[55,4272,4274,4277,4317,4320,4323,4326,4329,4369,4372,4375,4378,4381,4421,4424,4427,4430,4433,4473,4476],{"className":4273},[502],[55,4275,521],{"className":4276,"style":511},[509,510],[55,4278,4280,4283],{"className":4279},[76],[55,4281,800],{"className":4282,"style":480},[76,77],[55,4284,4286],{"className":4285},[104],[55,4287,4289,4309],{"className":4288},[108,109],[55,4290,4292,4306],{"className":4291},[113],[55,4293,4295],{"className":4294,"style":296},[117],[55,4296,4297,4300],{"style":1004},[55,4298],{"className":4299,"style":126},[125],[55,4301,4303],{"className":4302},[130,131,132,133],[55,4304,1014],{"className":4305},[76,77,133],[55,4307,142],{"className":4308},[141],[55,4310,4312],{"className":4311},[113],[55,4313,4315],{"className":4314,"style":1024},[117],[55,4316],{},[55,4318,521],{"className":4319},[509],[55,4321,78],{"className":4322},[76,77],[55,4324,156],{"className":4325},[155],[55,4327],{"className":4328,"style":161},[160],[55,4330,4332,4335],{"className":4331},[76],[55,4333,100],{"className":4334,"style":99},[76,77],[55,4336,4338],{"className":4337},[104],[55,4339,4341,4361],{"className":4340},[108,109],[55,4342,4344,4358],{"className":4343},[113],[55,4345,4347],{"className":4346,"style":235},[117],[55,4348,4349,4352],{"style":121},[55,4350],{"className":4351,"style":126},[125],[55,4353,4355],{"className":4354},[130,131,132,133],[55,4356,248],{"className":4357,"style":247},[76,77,133],[55,4359,142],{"className":4360},[141],[55,4362,4364],{"className":4363},[113],[55,4365,4367],{"className":4366,"style":149},[117],[55,4368],{},[55,4370,564],{"className":4371},[563],[55,4373],{"className":4374,"style":2186},[160],[55,4376,2191],{"className":4377},[2190],[55,4379],{"className":4380,"style":2186},[160],[55,4382,4384,4387],{"className":4383},[76],[55,4385,800],{"className":4386,"style":480},[76,77],[55,4388,4390],{"className":4389},[104],[55,4391,4393,4413],{"className":4392},[108,109],[55,4394,4396,4410],{"className":4395},[113],[55,4397,4399],{"className":4398,"style":296},[117],[55,4400,4401,4404],{"style":1004},[55,4402],{"className":4403,"style":126},[125],[55,4405,4407],{"className":4406},[130,131,132,133],[55,4408,1014],{"className":4409},[76,77,133],[55,4411,142],{"className":4412},[141],[55,4414,4416],{"className":4415},[113],[55,4417,4419],{"className":4418,"style":1024},[117],[55,4420],{},[55,4422,521],{"className":4423},[509],[55,4425,78],{"className":4426},[76,77],[55,4428,156],{"className":4429},[155],[55,4431],{"className":4432,"style":161},[160],[55,4434,4436,4439],{"className":4435},[76],[55,4437,100],{"className":4438,"style":99},[76,77],[55,4440,4442],{"className":4441},[104],[55,4443,4445,4465],{"className":4444},[108,109],[55,4446,4448,4462],{"className":4447},[113],[55,4449,4451],{"className":4450,"style":296},[117],[55,4452,4453,4456],{"style":121},[55,4454],{"className":4455,"style":126},[125],[55,4457,4459],{"className":4458},[130,131,132,133],[55,4460,309],{"className":4461,"style":308},[76,77,133],[55,4463,142],{"className":4464},[141],[55,4466,4468],{"className":4467},[113],[55,4469,4471],{"className":4470,"style":149},[117],[55,4472],{},[55,4474,564],{"className":4475},[563],[55,4477,564],{"className":4478,"style":511},[563,510],[55,4480,4482],{"className":4481,"style":511},[563,510],"]",[55,4484],{"className":4485,"style":161},[160],[55,4487,1143],{"className":4488},[76],[11,4490,4491,4492,4544],{},"Architecturally, ",[55,4493,4495],{"className":4494},[58],[55,4496,4498],{"className":4497,"ariaHidden":63},[62],[55,4499,4501,4504],{"className":4500},[67],[55,4502],{"className":4503,"style":983},[71],[55,4505,4507,4510],{"className":4506},[76],[55,4508,800],{"className":4509,"style":480},[76,77],[55,4511,4513],{"className":4512},[104],[55,4514,4516,4536],{"className":4515},[108,109],[55,4517,4519,4533],{"className":4518},[113],[55,4520,4522],{"className":4521,"style":296},[117],[55,4523,4524,4527],{"style":1004},[55,4525],{"className":4526,"style":126},[125],[55,4528,4530],{"className":4529},[130,131,132,133],[55,4531,1014],{"className":4532},[76,77,133],[55,4534,142],{"className":4535},[141],[55,4537,4539],{"className":4538},[113],[55,4540,4542],{"className":4541,"style":1024},[117],[55,4543],{}," is the SFT model with the next-token head replaced by a\nscalar head reading the final hidden state of the last token.",[11,4546,4547,4548,4669,4670,4821],{},"The gradient explains what the loss does mechanically. Using ",[55,4549,4551],{"className":4550},[58],[55,4552,4554,4615],{"className":4553,"ariaHidden":63},[62],[55,4555,4557,4561,4595,4598,4603,4606,4609,4612],{"className":4556},[67],[55,4558],{"className":4559,"style":4560},[71],"height:1.0019em;vertical-align:-0.25em;",[55,4562,4564,4567],{"className":4563},[76],[55,4565,2111],{"className":4566,"style":99},[76,77],[55,4568,4570],{"className":4569},[104],[55,4571,4573],{"className":4572},[108],[55,4574,4576],{"className":4575},[113],[55,4577,4580],{"className":4578,"style":4579},[117],"height:0.7519em;",[55,4581,4582,4585],{"style":543},[55,4583],{"className":4584,"style":126},[125],[55,4586,4588],{"className":4587},[130,131,132,133],[55,4589,4591],{"className":4590},[76,133],[55,4592,4594],{"className":4593},[76,133],"′",[55,4596,521],{"className":4597},[509],[55,4599,4602],{"className":4600,"style":4601},[76,77],"margin-right:0.044em;","z",[55,4604,564],{"className":4605},[563],[55,4607],{"className":4608,"style":263},[160],[55,4610,488],{"className":4611},[267],[55,4613],{"className":4614,"style":263},[160],[55,4616,4618,4621,4624,4627,4630,4633,4636],{"className":4617},[67],[55,4619],{"className":4620,"style":796},[71],[55,4622,2111],{"className":4623,"style":99},[76,77],[55,4625,521],{"className":4626},[509],[55,4628,4602],{"className":4629,"style":4601},[76,77],[55,4631,564],{"className":4632},[563],[55,4634],{"className":4635,"style":161},[160],[55,4637,4639,4642,4645,4648,4651,4654,4657,4660,4663,4666],{"className":4638},[502],[55,4640,521],{"className":4641,"style":511},[509,510],[55,4643,137],{"className":4644},[76],[55,4646],{"className":4647,"style":2186},[160],[55,4649,2191],{"className":4650},[2190],[55,4652],{"className":4653,"style":2186},[160],[55,4655,2111],{"className":4656,"style":99},[76,77],[55,4658,521],{"className":4659},[509],[55,4661,4602],{"className":4662,"style":4601},[76,77],[55,4664,564],{"className":4665},[563],[55,4667,564],{"className":4668,"style":511},[563,510],"\nand ",[55,4671,4673],{"className":4672},[58],[55,4674,4676,4784,4803],{"className":4675,"ariaHidden":63},[62],[55,4677,4679,4683,4751,4754,4760,4763,4766,4769,4772,4775,4778,4781],{"className":4678},[67],[55,4680],{"className":4681,"style":4682},[71],"height:2.0574em;vertical-align:-0.686em;",[55,4684,4686,4689,4748],{"className":4685},[76],[55,4687],{"className":4688},[509,2284],[55,4690,4692],{"className":4691},[2288],[55,4693,4695,4739],{"className":4694},[108,109],[55,4696,4698,4736],{"className":4697},[113],[55,4699,4702,4717,4725],{"className":4700,"style":4701},[117],"height:1.3714em;",[55,4703,4704,4707],{"style":2301},[55,4705],{"className":4706,"style":2305},[125],[55,4708,4710,4714],{"className":4709},[76],[55,4711,4713],{"className":4712},[76,77],"d",[55,4715,4602],{"className":4716,"style":4601},[76,77],[55,4718,4719,4722],{"style":2484},[55,4720],{"className":4721,"style":2305},[125],[55,4723],{"className":4724,"style":2492},[2491],[55,4726,4727,4730],{"style":2495},[55,4728],{"className":4729,"style":2305},[125],[55,4731,4733],{"className":4732},[76],[55,4734,4713],{"className":4735},[76,77],[55,4737,142],{"className":4738},[141],[55,4740,4742],{"className":4741},[113],[55,4743,4746],{"className":4744,"style":4745},[117],"height:0.686em;",[55,4747],{},[55,4749],{"className":4750},[563,2284],[55,4752],{"className":4753,"style":161},[160],[55,4755,4757],{"className":4756},[2325],[55,4758,3865],{"className":4759,"style":3864},[76,3863],[55,4761],{"className":4762,"style":161},[160],[55,4764,2111],{"className":4765,"style":99},[76,77],[55,4767,521],{"className":4768},[509],[55,4770,4602],{"className":4771,"style":4601},[76,77],[55,4773,564],{"className":4774},[563],[55,4776],{"className":4777,"style":263},[160],[55,4779,488],{"className":4780},[267],[55,4782],{"className":4783,"style":263},[160],[55,4785,4787,4791,4794,4797,4800],{"className":4786},[67],[55,4788],{"className":4789,"style":4790},[71],"height:0.7278em;vertical-align:-0.0833em;",[55,4792,137],{"className":4793},[76],[55,4795],{"className":4796,"style":2186},[160],[55,4798,2191],{"className":4799},[2190],[55,4801],{"className":4802,"style":2186},[160],[55,4804,4806,4809,4812,4815,4818],{"className":4805},[67],[55,4807],{"className":4808,"style":796},[71],[55,4810,2111],{"className":4811,"style":99},[76,77],[55,4813,521],{"className":4814},[509],[55,4816,4602],{"className":4817,"style":4601},[76,77],[55,4819,564],{"className":4820},[563],", the per-comparison gradient is",[55,4823,4825],{"className":4824},[1949],[55,4826,4828],{"className":4827},[58],[55,4829,4831,4972,5155],{"className":4830,"ariaHidden":63},[62],[55,4832,4834,4837,4878,4881,4884,4957,4960,4963,4966,4969],{"className":4833},[67],[55,4835],{"className":4836,"style":3314},[71],[55,4838,4840,4844],{"className":4839},[76],[55,4841,4843],{"className":4842},[76],"∇",[55,4845,4847],{"className":4846},[104],[55,4848,4850,4870],{"className":4849},[108,109],[55,4851,4853,4867],{"className":4852},[113],[55,4854,4856],{"className":4855,"style":296},[117],[55,4857,4858,4861],{"style":3342},[55,4859],{"className":4860,"style":126},[125],[55,4862,4864],{"className":4863},[130,131,132,133],[55,4865,1014],{"className":4866},[76,77,133],[55,4868,142],{"className":4869},[141],[55,4871,4873],{"className":4872},[113],[55,4874,4876],{"className":4875,"style":1024},[117],[55,4877],{},[55,4879],{"className":4880,"style":2115},[160],[55,4882],{"className":4883,"style":161},[160],[55,4885,4887,4890,4893,4896,4902,4905,4908,4911,4951,4954],{"className":4886},[502],[55,4888,4252],{"className":4889,"style":511},[509,510],[55,4891,2191],{"className":4892},[76],[55,4894],{"className":4895,"style":161},[160],[55,4897,4899],{"className":4898},[2325],[55,4900,3865],{"className":4901,"style":3864},[76,3863],[55,4903],{"className":4904,"style":161},[160],[55,4906,2111],{"className":4907,"style":99},[76,77],[55,4909,521],{"className":4910},[509],[55,4912,4914,4917],{"className":4913},[76],[55,4915,3327],{"className":4916},[76],[55,4918,4920],{"className":4919},[104],[55,4921,4923,4943],{"className":4922},[108,109],[55,4924,4926,4940],{"className":4925},[113],[55,4927,4929],{"className":4928,"style":296},[117],[55,4930,4931,4934],{"style":3342},[55,4932],{"className":4933,"style":126},[125],[55,4935,4937],{"className":4936},[130,131,132,133],[55,4938,1014],{"className":4939},[76,77,133],[55,4941,142],{"className":4942},[141],[55,4944,4946],{"className":4945},[113],[55,4947,4949],{"className":4948,"style":1024},[117],[55,4950],{},[55,4952,564],{"className":4953},[563],[55,4955,4482],{"className":4956,"style":511},[563,510],[55,4958],{"className":4959,"style":263},[160],[55,4961],{"className":4962,"style":263},[160],[55,4964,488],{"className":4965},[267],[55,4967],{"className":4968,"style":263},[160],[55,4970],{"className":4971,"style":263},[160],[55,4973,4975,4978,4981,4984,5054,5057,5060,5100,5140,5143,5146,5149,5152],{"className":4974},[67],[55,4976],{"className":4977,"style":3314},[71],[55,4979,2191],{"className":4980},[76],[55,4982],{"className":4983,"style":161},[160],[55,4985,4987,4990,4993,4996,4999,5002,5005,5008,5048,5051],{"className":4986},[502],[55,4988,521],{"className":4989,"style":511},[509,510],[55,4991,137],{"className":4992},[76],[55,4994],{"className":4995,"style":2186},[160],[55,4997,2191],{"className":4998},[2190],[55,5000],{"className":5001,"style":2186},[160],[55,5003,2111],{"className":5004,"style":99},[76,77],[55,5006,521],{"className":5007},[509],[55,5009,5011,5014],{"className":5010},[76],[55,5012,3327],{"className":5013},[76],[55,5015,5017],{"className":5016},[104],[55,5018,5020,5040],{"className":5019},[108,109],[55,5021,5023,5037],{"className":5022},[113],[55,5024,5026],{"className":5025,"style":296},[117],[55,5027,5028,5031],{"style":3342},[55,5029],{"className":5030,"style":126},[125],[55,5032,5034],{"className":5033},[130,131,132,133],[55,5035,1014],{"className":5036},[76,77,133],[55,5038,142],{"className":5039},[141],[55,5041,5043],{"className":5042},[113],[55,5044,5046],{"className":5045,"style":1024},[117],[55,5047],{},[55,5049,564],{"className":5050},[563],[55,5052,564],{"className":5053,"style":511},[563,510],[55,5055],{"className":5056,"style":161},[160],[55,5058],{"className":5059,"style":161},[160],[55,5061,5063,5066],{"className":5062},[76],[55,5064,4843],{"className":5065},[76],[55,5067,5069],{"className":5068},[104],[55,5070,5072,5092],{"className":5071},[108,109],[55,5073,5075,5089],{"className":5074},[113],[55,5076,5078],{"className":5077,"style":296},[117],[55,5079,5080,5083],{"style":3342},[55,5081],{"className":5082,"style":126},[125],[55,5084,5086],{"className":5085},[130,131,132,133],[55,5087,1014],{"className":5088},[76,77,133],[55,5090,142],{"className":5091},[141],[55,5093,5095],{"className":5094},[113],[55,5096,5098],{"className":5097,"style":1024},[117],[55,5099],{},[55,5101,5103,5106],{"className":5102},[76],[55,5104,3327],{"className":5105},[76],[55,5107,5109],{"className":5108},[104],[55,5110,5112,5132],{"className":5111},[108,109],[55,5113,5115,5129],{"className":5114},[113],[55,5116,5118],{"className":5117,"style":296},[117],[55,5119,5120,5123],{"style":3342},[55,5121],{"className":5122,"style":126},[125],[55,5124,5126],{"className":5125},[130,131,132,133],[55,5127,1014],{"className":5128},[76,77,133],[55,5130,142],{"className":5131},[141],[55,5133,5135],{"className":5134},[113],[55,5136,5138],{"className":5137,"style":1024},[117],[55,5139],{},[55,5141],{"className":5142,"style":263},[160],[55,5144],{"className":5145,"style":263},[160],[55,5147,488],{"className":5148},[267],[55,5150],{"className":5151,"style":263},[160],[55,5153],{"className":5154,"style":263},[160],[55,5156,5158,5162,5165,5168,5171,5356,5359,5362,5650,5653],{"className":5157},[67],[55,5159],{"className":5160,"style":5161},[71],"height:2.5063em;vertical-align:-1.7563em;",[55,5163,2191],{"className":5164},[76],[55,5166],{"className":5167,"style":161},[160],[55,5169],{"className":5170,"style":161},[160],[55,5172,5175],{"className":5173},[502,5174],"munder",[55,5176,5178,5347],{"className":5177},[108,109],[55,5179,5181,5344],{"className":5180},[113],[55,5182,5185,5204],{"className":5183,"style":5184},[117],"height:0.75em;",[55,5186,5188,5191],{"style":5187},"top:-1.3798em;",[55,5189],{"className":5190,"style":2305},[125],[55,5192,5194],{"className":5193},[130,131,132,133],[55,5195,5197],{"className":5196},[76,133],[55,5198,5200],{"className":5199},[76,951,133],[55,5201,5203],{"className":5202},[76,133],"prob. of being wrong",[55,5205,5207,5210],{"style":5206},"top:-3em;",[55,5208],{"className":5209,"style":2305},[125],[55,5211,5213],{"className":5212},[502,5174],[55,5214,5216,5335],{"className":5215},[108,109],[55,5217,5219,5332],{"className":5218},[113],[55,5220,5222,5272],{"className":5221,"style":5184},[117],[55,5223,5227,5230],{"className":5224,"style":5226},[5225],"svg-align","top:-2.0659em;",[55,5228],{"className":5229,"style":2305},[125],[55,5231,5235,5252,5262],{"className":5232,"style":5234},[5233],"stretchy","height:0.548em;min-width:1.6em;",[55,5236,5240],{"className":5237,"style":5239},[5238],"brace-left","height:0.548em;",[5241,5242,5248],"svg",{"xmlns":5243,"width":5244,"height":5245,"viewBox":5246,"preserveAspectRatio":5247},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","0.548em","0 0 400000 548","xMinYMin slice",[5249,5250],"path",{"d":5251},"M0 6l6-6h17c12.688 0 19.313.3 20 1 4 4 7.313 8.3 10 13\n 35.313 51.3 80.813 93.8 136.5 127.5 55.688 33.7 117.188 55.8 184.5 66.5.688\n 0 2 .3 4 1 18.688 2.7 76 4.3 172 5h399450v120H429l-6-1c-124.688-8-235-61.7\n-331-161C60.687 138.7 32.312 99.3 7 54L0 41V6z",[55,5253,5256],{"className":5254,"style":5239},[5255],"brace-center",[5241,5257,5259],{"xmlns":5243,"width":5244,"height":5245,"viewBox":5246,"preserveAspectRatio":5258},"xMidYMin slice",[5249,5260],{"d":5261},"M199572 214\nc100.7 8.3 195.3 44 280 108 55.3 42 101.7 93 139 153l9 14c2.7-4 5.7-8.7 9-14\n 53.3-86.7 123.7-153 211-199 66.7-36 137.3-56.3 212-62h199568v120H200432c-178.3\n 11.7-311.7 78.3-403 201-6 8-9.7 12-11 12-.7.7-6.7 1-18 1s-17.3-.3-18-1c-1.3 0\n-5-4-11-12-44.7-59.3-101.3-106.3-170-141s-145.3-54.3-229-60H0V214z",[55,5263,5266],{"className":5264,"style":5239},[5265],"brace-right",[5241,5267,5269],{"xmlns":5243,"width":5244,"height":5245,"viewBox":5246,"preserveAspectRatio":5268},"xMaxYMin slice",[5249,5270],{"d":5271},"M399994 0l6 6v35l-6 11c-56 104-135.3 181.3-238 232-57.3\n 28.7-117 45-179 50H-300V214h399897c43.3-7 81-15 113-26 100.7-33 179.7-91 237\n-174 2.7-5 6-9 10-13 .7-1 7.3-1 20-1h17z",[55,5273,5274,5277],{"style":5206},[55,5275],{"className":5276,"style":2305},[125],[55,5278,5280,5283,5286,5289,5329],{"className":5279},[76],[55,5281,2111],{"className":5282,"style":99},[76,77],[55,5284,521],{"className":5285},[509],[55,5287,2191],{"className":5288},[76],[55,5290,5292,5295],{"className":5291},[76],[55,5293,3327],{"className":5294},[76],[55,5296,5298],{"className":5297},[104],[55,5299,5301,5321],{"className":5300},[108,109],[55,5302,5304,5318],{"className":5303},[113],[55,5305,5307],{"className":5306,"style":296},[117],[55,5308,5309,5312],{"style":3342},[55,5310],{"className":5311,"style":126},[125],[55,5313,5315],{"className":5314},[130,131,132,133],[55,5316,1014],{"className":5317},[76,77,133],[55,5319,142],{"className":5320},[141],[55,5322,5324],{"className":5323},[113],[55,5325,5327],{"className":5326,"style":1024},[117],[55,5328],{},[55,5330,564],{"className":5331},[563],[55,5333,142],{"className":5334},[141],[55,5336,5338],{"className":5337},[113],[55,5339,5342],{"className":5340,"style":5341},[117],"height:0.9341em;",[55,5343],{},[55,5345,142],{"className":5346},[141],[55,5348,5350],{"className":5349},[113],[55,5351,5354],{"className":5352,"style":5353},[117],"height:1.7563em;",[55,5355],{},[55,5357],{"className":5358,"style":161},[160],[55,5360],{"className":5361,"style":161},[160],[55,5363,5365,5368,5408,5448,5451,5454,5457,5460,5500,5503,5506,5509,5512,5552,5592,5595,5598,5601,5604,5644,5647],{"className":5364},[502],[55,5366,521],{"className":5367,"style":511},[509,510],[55,5369,5371,5374],{"className":5370},[76],[55,5372,4843],{"className":5373},[76],[55,5375,5377],{"className":5376},[104],[55,5378,5380,5400],{"className":5379},[108,109],[55,5381,5383,5397],{"className":5382},[113],[55,5384,5386],{"className":5385,"style":296},[117],[55,5387,5388,5391],{"style":3342},[55,5389],{"className":5390,"style":126},[125],[55,5392,5394],{"className":5393},[130,131,132,133],[55,5395,1014],{"className":5396},[76,77,133],[55,5398,142],{"className":5399},[141],[55,5401,5403],{"className":5402},[113],[55,5404,5406],{"className":5405,"style":1024},[117],[55,5407],{},[55,5409,5411,5414],{"className":5410},[76],[55,5412,800],{"className":5413,"style":480},[76,77],[55,5415,5417],{"className":5416},[104],[55,5418,5420,5440],{"className":5419},[108,109],[55,5421,5423,5437],{"className":5422},[113],[55,5424,5426],{"className":5425,"style":296},[117],[55,5427,5428,5431],{"style":1004},[55,5429],{"className":5430,"style":126},[125],[55,5432,5434],{"className":5433},[130,131,132,133],[55,5435,1014],{"className":5436},[76,77,133],[55,5438,142],{"className":5439},[141],[55,5441,5443],{"className":5442},[113],[55,5444,5446],{"className":5445,"style":1024},[117],[55,5447],{},[55,5449,521],{"className":5450},[509],[55,5452,78],{"className":5453},[76,77],[55,5455,156],{"className":5456},[155],[55,5458],{"className":5459,"style":161},[160],[55,5461,5463,5466],{"className":5462},[76],[55,5464,100],{"className":5465,"style":99},[76,77],[55,5467,5469],{"className":5468},[104],[55,5470,5472,5492],{"className":5471},[108,109],[55,5473,5475,5489],{"className":5474},[113],[55,5476,5478],{"className":5477,"style":235},[117],[55,5479,5480,5483],{"style":121},[55,5481],{"className":5482,"style":126},[125],[55,5484,5486],{"className":5485},[130,131,132,133],[55,5487,248],{"className":5488,"style":247},[76,77,133],[55,5490,142],{"className":5491},[141],[55,5493,5495],{"className":5494},[113],[55,5496,5498],{"className":5497,"style":149},[117],[55,5499],{},[55,5501,564],{"className":5502},[563],[55,5504],{"className":5505,"style":2186},[160],[55,5507,2191],{"className":5508},[2190],[55,5510],{"className":5511,"style":2186},[160],[55,5513,5515,5518],{"className":5514},[76],[55,5516,4843],{"className":5517},[76],[55,5519,5521],{"className":5520},[104],[55,5522,5524,5544],{"className":5523},[108,109],[55,5525,5527,5541],{"className":5526},[113],[55,5528,5530],{"className":5529,"style":296},[117],[55,5531,5532,5535],{"style":3342},[55,5533],{"className":5534,"style":126},[125],[55,5536,5538],{"className":5537},[130,131,132,133],[55,5539,1014],{"className":5540},[76,77,133],[55,5542,142],{"className":5543},[141],[55,5545,5547],{"className":5546},[113],[55,5548,5550],{"className":5549,"style":1024},[117],[55,5551],{},[55,5553,5555,5558],{"className":5554},[76],[55,5556,800],{"className":5557,"style":480},[76,77],[55,5559,5561],{"className":5560},[104],[55,5562,5564,5584],{"className":5563},[108,109],[55,5565,5567,5581],{"className":5566},[113],[55,5568,5570],{"className":5569,"style":296},[117],[55,5571,5572,5575],{"style":1004},[55,5573],{"className":5574,"style":126},[125],[55,5576,5578],{"className":5577},[130,131,132,133],[55,5579,1014],{"className":5580},[76,77,133],[55,5582,142],{"className":5583},[141],[55,5585,5587],{"className":5586},[113],[55,5588,5590],{"className":5589,"style":1024},[117],[55,5591],{},[55,5593,521],{"className":5594},[509],[55,5596,78],{"className":5597},[76,77],[55,5599,156],{"className":5600},[155],[55,5602],{"className":5603,"style":161},[160],[55,5605,5607,5610],{"className":5606},[76],[55,5608,100],{"className":5609,"style":99},[76,77],[55,5611,5613],{"className":5612},[104],[55,5614,5616,5636],{"className":5615},[108,109],[55,5617,5619,5633],{"className":5618},[113],[55,5620,5622],{"className":5621,"style":296},[117],[55,5623,5624,5627],{"style":121},[55,5625],{"className":5626,"style":126},[125],[55,5628,5630],{"className":5629},[130,131,132,133],[55,5631,309],{"className":5632,"style":308},[76,77,133],[55,5634,142],{"className":5635},[141],[55,5637,5639],{"className":5638},[113],[55,5640,5642],{"className":5641,"style":149},[117],[55,5643],{},[55,5645,564],{"className":5646},[563],[55,5648,564],{"className":5649,"style":511},[563,510],[55,5651],{"className":5652,"style":161},[160],[55,5654,1143],{"className":5655},[76],[11,5657,5658,5659,5723,5724,5727,5728,5780],{},"The update raises the winner's reward and lowers the loser's, scaled by ",[55,5660,5662],{"className":5661},[58],[55,5663,5665],{"className":5664,"ariaHidden":63},[62],[55,5666,5668,5671,5674,5677,5680,5720],{"className":5667},[67],[55,5669],{"className":5670,"style":3314},[71],[55,5672,2111],{"className":5673,"style":99},[76,77],[55,5675,521],{"className":5676},[509],[55,5678,2191],{"className":5679},[76],[55,5681,5683,5686],{"className":5682},[76],[55,5684,3327],{"className":5685},[76],[55,5687,5689],{"className":5688},[104],[55,5690,5692,5712],{"className":5691},[108,109],[55,5693,5695,5709],{"className":5694},[113],[55,5696,5698],{"className":5697,"style":296},[117],[55,5699,5700,5703],{"style":3342},[55,5701],{"className":5702,"style":126},[125],[55,5704,5706],{"className":5705},[130,131,132,133],[55,5707,1014],{"className":5708},[76,77,133],[55,5710,142],{"className":5711},[141],[55,5713,5715],{"className":5714},[113],[55,5716,5718],{"className":5717,"style":1024},[117],[55,5719],{},[55,5721,564],{"className":5722},[563],",\nthe model's current probability of ",[880,5725,5726],{},"mis","-ranking the pair. Already-correct comparisons\n(large ",[55,5729,5731],{"className":5730},[58],[55,5732,5734],{"className":5733,"ariaHidden":63},[62],[55,5735,5737,5740],{"className":5736},[67],[55,5738],{"className":5739,"style":3379},[71],[55,5741,5743,5746],{"className":5742},[76],[55,5744,3327],{"className":5745},[76],[55,5747,5749],{"className":5748},[104],[55,5750,5752,5772],{"className":5751},[108,109],[55,5753,5755,5769],{"className":5754},[113],[55,5756,5758],{"className":5757,"style":296},[117],[55,5759,5760,5763],{"style":3342},[55,5761],{"className":5762,"style":126},[125],[55,5764,5766],{"className":5765},[130,131,132,133],[55,5767,1014],{"className":5768},[76,77,133],[55,5770,142],{"className":5771},[141],[55,5773,5775],{"className":5774},[113],[55,5776,5778],{"className":5777,"style":1024},[117],[55,5779],{},") get a vanishing gradient; only the contested pairs move the\nweights. This is the same self-weighting that makes logistic regression focus on the\nmargin.",[11,5782,5783,5784,5855,5856,5903,5904,5855,5977,6020],{},"For example, suppose the model already ranks a pair\ncorrectly with ",[55,5785,5787],{"className":5786},[58],[55,5788,5790,5845],{"className":5789,"ariaHidden":63},[62],[55,5791,5793,5796,5836,5839,5842],{"className":5792},[67],[55,5794],{"className":5795,"style":3379},[71],[55,5797,5799,5802],{"className":5798},[76],[55,5800,3327],{"className":5801},[76],[55,5803,5805],{"className":5804},[104],[55,5806,5808,5828],{"className":5807},[108,109],[55,5809,5811,5825],{"className":5810},[113],[55,5812,5814],{"className":5813,"style":296},[117],[55,5815,5816,5819],{"style":3342},[55,5817],{"className":5818,"style":126},[125],[55,5820,5822],{"className":5821},[130,131,132,133],[55,5823,1014],{"className":5824},[76,77,133],[55,5826,142],{"className":5827},[141],[55,5829,5831],{"className":5830},[113],[55,5832,5834],{"className":5833,"style":1024},[117],[55,5835],{},[55,5837],{"className":5838,"style":263},[160],[55,5840,488],{"className":5841},[267],[55,5843],{"className":5844,"style":263},[160],[55,5846,5848,5851],{"className":5847},[67],[55,5849],{"className":5850,"style":2735},[71],[55,5852,5854],{"className":5853},[76],"4",": then ",[55,5857,5859],{"className":5858},[58],[55,5860,5862,5893],{"className":5861,"ariaHidden":63},[62],[55,5863,5865,5868,5871,5874,5877,5880,5883,5886,5890],{"className":5864},[67],[55,5866],{"className":5867,"style":796},[71],[55,5869,2111],{"className":5870,"style":99},[76,77],[55,5872,521],{"className":5873},[509],[55,5875,2191],{"className":5876},[76],[55,5878,5854],{"className":5879},[76],[55,5881,564],{"className":5882},[563],[55,5884],{"className":5885,"style":263},[160],[55,5887,5889],{"className":5888},[267],"≈",[55,5891],{"className":5892,"style":263},[160],[55,5894,5896,5899],{"className":5895},[67],[55,5897],{"className":5898,"style":2735},[71],[55,5900,5902],{"className":5901},[76],"0.018",", so the pair's gradient\nis scaled by under two percent and it barely nudges the weights. Suppose instead the\nmodel ranks a pair backwards with ",[55,5905,5907],{"className":5906},[58],[55,5908,5910,5965],{"className":5909,"ariaHidden":63},[62],[55,5911,5913,5916,5956,5959,5962],{"className":5912},[67],[55,5914],{"className":5915,"style":3379},[71],[55,5917,5919,5922],{"className":5918},[76],[55,5920,3327],{"className":5921},[76],[55,5923,5925],{"className":5924},[104],[55,5926,5928,5948],{"className":5927},[108,109],[55,5929,5931,5945],{"className":5930},[113],[55,5932,5934],{"className":5933,"style":296},[117],[55,5935,5936,5939],{"style":3342},[55,5937],{"className":5938,"style":126},[125],[55,5940,5942],{"className":5941},[130,131,132,133],[55,5943,1014],{"className":5944},[76,77,133],[55,5946,142],{"className":5947},[141],[55,5949,5951],{"className":5950},[113],[55,5952,5954],{"className":5953,"style":1024},[117],[55,5955],{},[55,5957],{"className":5958,"style":263},[160],[55,5960,488],{"className":5961},[267],[55,5963],{"className":5964,"style":263},[160],[55,5966,5968,5971,5974],{"className":5967},[67],[55,5969],{"className":5970,"style":4790},[71],[55,5972,2191],{"className":5973},[76],[55,5975,191],{"className":5976},[76],[55,5978,5980],{"className":5979},[58],[55,5981,5983,6010],{"className":5982,"ariaHidden":63},[62],[55,5984,5986,5989,5992,5995,5998,6001,6004,6007],{"className":5985},[67],[55,5987],{"className":5988,"style":796},[71],[55,5990,2111],{"className":5991,"style":99},[76,77],[55,5993,521],{"className":5994},[509],[55,5996,191],{"className":5997},[76],[55,5999,564],{"className":6000},[563],[55,6002],{"className":6003,"style":263},[160],[55,6005,5889],{"className":6006},[267],[55,6008],{"className":6009,"style":263},[160],[55,6011,6013,6016],{"className":6012},[67],[55,6014],{"className":6015,"style":2735},[71],[55,6017,6019],{"className":6018},[76],"0.88",", an\neighty-eight percent weight, and the update pushes hard to reverse the ordering. The loss\nconcentrates its gradient on currently-misranked pairs and contributes almost nothing\nfor pairs already ranked confidently, which is the correct behavior for a ranking\nobjective.",[11,6022,6023,6024,6027,6028,6080,6081,6105,6106,6176],{},"The identifiability caveat carries a practical consequence. Because only reward\n",[880,6025,6026],{},"differences"," enter the loss, ",[55,6029,6031],{"className":6030},[58],[55,6032,6034],{"className":6033,"ariaHidden":63},[62],[55,6035,6037,6040],{"className":6036},[67],[55,6038],{"className":6039,"style":983},[71],[55,6041,6043,6046],{"className":6042},[76],[55,6044,800],{"className":6045,"style":480},[76,77],[55,6047,6049],{"className":6048},[104],[55,6050,6052,6072],{"className":6051},[108,109],[55,6053,6055,6069],{"className":6054},[113],[55,6056,6058],{"className":6057,"style":296},[117],[55,6059,6060,6063],{"style":1004},[55,6061],{"className":6062,"style":126},[125],[55,6064,6066],{"className":6065},[130,131,132,133],[55,6067,1014],{"className":6068},[76,77,133],[55,6070,142],{"className":6071},[141],[55,6073,6075],{"className":6074},[113],[55,6076,6078],{"className":6077,"style":1024},[117],[55,6079],{}," is pinned only up to a per-prompt additive constant\n",[55,6082,6084],{"className":6083},[58],[55,6085,6087],{"className":6086,"ariaHidden":63},[62],[55,6088,6090,6093,6096,6099,6102],{"className":6089},[67],[55,6091],{"className":6092,"style":796},[71],[55,6094,2545],{"className":6095},[76,77],[55,6097,521],{"className":6098},[509],[55,6100,78],{"className":6101},[76,77],[55,6103,564],{"className":6104},[563],": the scale of the scores is fixed by the logistic, but their offset is free. When\nthe reward feeds the RL step this is harmless, since the KL term and the advantage\nbaseline both subtract off any prompt-level constant. It does mean reward magnitudes are\nnot comparable across prompts, so a raw ",[55,6107,6109],{"className":6108},[58],[55,6110,6112],{"className":6111,"ariaHidden":63},[62],[55,6113,6115,6118,6158,6161,6164,6167,6170,6173],{"className":6114},[67],[55,6116],{"className":6117,"style":3314},[71],[55,6119,6121,6124],{"className":6120},[76],[55,6122,800],{"className":6123,"style":480},[76,77],[55,6125,6127],{"className":6126},[104],[55,6128,6130,6150],{"className":6129},[108,109],[55,6131,6133,6147],{"className":6132},[113],[55,6134,6136],{"className":6135,"style":296},[117],[55,6137,6138,6141],{"style":1004},[55,6139],{"className":6140,"style":126},[125],[55,6142,6144],{"className":6143},[130,131,132,133],[55,6145,1014],{"className":6146},[76,77,133],[55,6148,142],{"className":6149},[141],[55,6151,6153],{"className":6152},[113],[55,6154,6156],{"className":6155,"style":1024},[117],[55,6157],{},[55,6159,521],{"className":6160},[509],[55,6162,78],{"className":6163},[76,77],[55,6165,156],{"className":6166},[155],[55,6168],{"className":6169,"style":161},[160],[55,6171,100],{"className":6172,"style":99},[76,77],[55,6174,564],{"className":6175},[563]," is a within-prompt ranking signal,\nnot an absolute quality score.",[31,6178,6180],{"id":6179},"the-rl-fine-tuning-step","The RL fine-tuning step",[11,6182,6183,6184,6236,6237,6289,6290,6293],{},"With ",[55,6185,6187],{"className":6186},[58],[55,6188,6190],{"className":6189,"ariaHidden":63},[62],[55,6191,6193,6196],{"className":6192},[67],[55,6194],{"className":6195,"style":983},[71],[55,6197,6199,6202],{"className":6198},[76],[55,6200,800],{"className":6201,"style":480},[76,77],[55,6203,6205],{"className":6204},[104],[55,6206,6208,6228],{"className":6207},[108,109],[55,6209,6211,6225],{"className":6210},[113],[55,6212,6214],{"className":6213,"style":296},[117],[55,6215,6216,6219],{"style":1004},[55,6217],{"className":6218,"style":126},[125],[55,6220,6222],{"className":6221},[130,131,132,133],[55,6223,1014],{"className":6224},[76,77,133],[55,6226,142],{"className":6227},[141],[55,6229,6231],{"className":6230},[113],[55,6232,6234],{"className":6233,"style":1024},[117],[55,6235],{}," fixed, stage three optimizes the policy. Naively maximizing\nexpected reward fails: the reward model is an imperfect proxy fit on finite data,\nand an unconstrained optimizer will find responses that score highly under ",[55,6238,6240],{"className":6239},[58],[55,6241,6243],{"className":6242,"ariaHidden":63},[62],[55,6244,6246,6249],{"className":6245},[67],[55,6247],{"className":6248,"style":983},[71],[55,6250,6252,6255],{"className":6251},[76],[55,6253,800],{"className":6254,"style":480},[76,77],[55,6256,6258],{"className":6257},[104],[55,6259,6261,6281],{"className":6260},[108,109],[55,6262,6264,6278],{"className":6263},[113],[55,6265,6267],{"className":6266,"style":296},[117],[55,6268,6269,6272],{"style":1004},[55,6270],{"className":6271,"style":126},[125],[55,6273,6275],{"className":6274},[130,131,132,133],[55,6276,1014],{"className":6277},[76,77,133],[55,6279,142],{"className":6280},[141],[55,6282,6284],{"className":6283},[113],[55,6285,6287],{"className":6286,"style":1024},[117],[55,6288],{}," yet\nare degenerate to a human. To address this, ",[880,6291,6292],{},"anchor"," the policy to the SFT reference with a\nKL penalty.",[44,6295,6296,6377,6785],{"type":46},[11,6297,6298,6301,6302,6317,6318,6376],{},[50,6299,6300],{},"Definition (KL-regularized RL objective)."," Given a reward ",[55,6303,6305],{"className":6304},[58],[55,6306,6308],{"className":6307,"ariaHidden":63},[62],[55,6309,6311,6314],{"className":6310},[67],[55,6312],{"className":6313,"style":72},[71],[55,6315,800],{"className":6316,"style":480},[76,77]," and reference policy\n",[55,6319,6321],{"className":6320},[58],[55,6322,6324],{"className":6323,"ariaHidden":63},[62],[55,6325,6327,6330],{"className":6326},[67],[55,6328],{"className":6329,"style":917},[71],[55,6331,6333,6336],{"className":6332},[76],[55,6334,924],{"className":6335,"style":99},[76,77],[55,6337,6339],{"className":6338},[104],[55,6340,6342,6368],{"className":6341},[108,109],[55,6343,6345,6365],{"className":6344},[113],[55,6346,6348],{"className":6347,"style":296},[117],[55,6349,6350,6353],{"style":121},[55,6351],{"className":6352,"style":126},[125],[55,6354,6356],{"className":6355},[130,131,132,133],[55,6357,6359],{"className":6358},[76,133],[55,6360,6362],{"className":6361},[76,951,133],[55,6363,955],{"className":6364},[76,133],[55,6366,142],{"className":6367},[141],[55,6369,6371],{"className":6370},[113],[55,6372,6374],{"className":6373,"style":149},[117],[55,6375],{},", the RLHF policy maximizes expected reward minus a KL penalty,",[55,6378,6380],{"className":6379},[1949],[55,6381,6383],{"className":6382},[58],[55,6384,6386,6581],{"className":6385,"ariaHidden":63},[62],[55,6387,6389,6393,6446,6449,6452,6533,6536,6566,6569,6572,6575,6578],{"className":6388},[67],[55,6390],{"className":6391,"style":6392},[71],"height:1.45em;vertical-align:-0.7em;",[55,6394,6396],{"className":6395},[2325,3694],[55,6397,6399,6437],{"className":6398},[108,109],[55,6400,6402,6434],{"className":6401},[113],[55,6403,6405,6420],{"className":6404,"style":72},[117],[55,6406,6408,6411],{"style":6407},"top:-2.4em;margin-left:0em;",[55,6409],{"className":6410,"style":2305},[125],[55,6412,6414],{"className":6413},[130,131,132,133],[55,6415,6417],{"className":6416},[76,133],[55,6418,924],{"className":6419,"style":99},[76,77,133],[55,6421,6422,6425],{"style":5206},[55,6423],{"className":6424,"style":2305},[125],[55,6426,6427],{},[55,6428,6430],{"className":6429},[2325],[55,6431,6433],{"className":6432},[76,3863],"max",[55,6435,142],{"className":6436},[141],[55,6438,6440],{"className":6439},[113],[55,6441,6444],{"className":6442,"style":6443},[117],"height:0.7em;",[55,6445],{},[55,6447],{"className":6448,"style":263},[160],[55,6450],{"className":6451,"style":161},[160],[55,6453,6455,6458],{"className":6454},[76],[55,6456,4104],{"className":6457},[76,1838],[55,6459,6461],{"className":6460},[104],[55,6462,6464,6525],{"className":6463},[108,109],[55,6465,6467,6522],{"className":6466},[113],[55,6468,6470],{"className":6469,"style":3794},[117],[55,6471,6472,6475],{"style":4119},[55,6473],{"className":6474,"style":126},[125],[55,6476,6478],{"className":6477},[130,131,132,133],[55,6479,6481,6484,6487,6490,6493,6497,6500,6503,6506,6509,6513,6516,6519],{"className":6480},[76,133],[55,6482,78],{"className":6483},[76,77,133],[55,6485,4227],{"className":6486},[267,133],[55,6488,481],{"className":6489,"style":480},[76,479,133],[55,6491,156],{"className":6492},[155,133],[55,6494],{"className":6495,"style":6496},[160,133],"margin-right:0.1952em;",[55,6498,100],{"className":6499,"style":99},[76,77,133],[55,6501,4227],{"className":6502},[267,133],[55,6504,924],{"className":6505,"style":99},[76,77,133],[55,6507,521],{"className":6508},[509,133],[55,6510,6512],{"className":6511},[76,133],"⋅",[55,6514,2071],{"className":6515},[267,133],[55,6517,78],{"className":6518},[76,77,133],[55,6520,564],{"className":6521},[563,133],[55,6523,142],{"className":6524},[141],[55,6526,6528],{"className":6527},[113],[55,6529,6531],{"className":6530,"style":4240},[117],[55,6532],{},[55,6534],{"className":6535,"style":161},[160],[55,6537,6539,6542,6545,6548,6551,6554,6557,6560,6563],{"className":6538},[502],[55,6540,4252],{"className":6541,"style":511},[509,510],[55,6543,800],{"className":6544,"style":480},[76,77],[55,6546,521],{"className":6547},[509],[55,6549,78],{"className":6550},[76,77],[55,6552,156],{"className":6553},[155],[55,6555],{"className":6556,"style":161},[160],[55,6558,100],{"className":6559,"style":99},[76,77],[55,6561,564],{"className":6562},[563],[55,6564,4482],{"className":6565,"style":511},[563,510],[55,6567],{"className":6568,"style":263},[160],[55,6570],{"className":6571,"style":2186},[160],[55,6573,2191],{"className":6574},[2190],[55,6576],{"className":6577,"style":263},[160],[55,6579],{"className":6580,"style":2186},[160],[55,6582,6584,6587,6592,6595,6644,6647,6779,6782],{"className":6583},[67],[55,6585],{"className":6586,"style":796},[71],[55,6588,6591],{"className":6589,"style":6590},[76,77],"margin-right:0.0528em;","β",[55,6593],{"className":6594,"style":161},[160],[55,6596,6598,6601],{"className":6597},[76],[55,6599,4104],{"className":6600},[76,1838],[55,6602,6604],{"className":6603},[104],[55,6605,6607,6636],{"className":6606},[108,109],[55,6608,6610,6633],{"className":6609},[113],[55,6611,6613],{"className":6612,"style":4031},[117],[55,6614,6615,6618],{"style":3342},[55,6616],{"className":6617,"style":126},[125],[55,6619,6621],{"className":6620},[130,131,132,133],[55,6622,6624,6627,6630],{"className":6623},[76,133],[55,6625,78],{"className":6626},[76,77,133],[55,6628,4227],{"className":6629},[267,133],[55,6631,481],{"className":6632,"style":480},[76,479,133],[55,6634,142],{"className":6635},[141],[55,6637,6639],{"className":6638},[113],[55,6640,6642],{"className":6641,"style":149},[117],[55,6643],{},[55,6645],{"className":6646,"style":161},[160],[55,6648,6650,6653,6660,6663,6666,6776],{"className":6649},[502],[55,6651,4252],{"className":6652,"style":511},[509,510],[55,6654,6656],{"className":6655},[2325],[55,6657,6659],{"className":6658},[76,3863],"KL",[55,6661],{"className":6662,"style":2115},[160],[55,6664],{"className":6665,"style":161},[160],[55,6667,6669,6672,6675,6678,6681,6684,6687,6690,6693,6696,6699,6703,6706,6752,6755,6758,6761,6764,6767,6770,6773],{"className":6668},[502],[55,6670,521],{"className":6671,"style":511},[509,510],[55,6673,924],{"className":6674,"style":99},[76,77],[55,6676,521],{"className":6677},[509],[55,6679,6512],{"className":6680},[76],[55,6682],{"className":6683,"style":263},[160],[55,6685,2071],{"className":6686},[267],[55,6688],{"className":6689,"style":263},[160],[55,6691,78],{"className":6692},[76,77],[55,6694,564],{"className":6695},[563],[55,6697],{"className":6698,"style":161},[160],[55,6700,6702],{"className":6701},[76],"∥",[55,6704],{"className":6705,"style":161},[160],[55,6707,6709,6712],{"className":6708},[76],[55,6710,924],{"className":6711,"style":99},[76,77],[55,6713,6715],{"className":6714},[104],[55,6716,6718,6744],{"className":6717},[108,109],[55,6719,6721,6741],{"className":6720},[113],[55,6722,6724],{"className":6723,"style":296},[117],[55,6725,6726,6729],{"style":121},[55,6727],{"className":6728,"style":126},[125],[55,6730,6732],{"className":6731},[130,131,132,133],[55,6733,6735],{"className":6734},[76,133],[55,6736,6738],{"className":6737},[76,951,133],[55,6739,955],{"className":6740},[76,133],[55,6742,142],{"className":6743},[141],[55,6745,6747],{"className":6746},[113],[55,6748,6750],{"className":6749,"style":149},[117],[55,6751],{},[55,6753,521],{"className":6754},[509],[55,6756,6512],{"className":6757},[76],[55,6759],{"className":6760,"style":263},[160],[55,6762,2071],{"className":6763},[267],[55,6765],{"className":6766,"style":263},[160],[55,6768,78],{"className":6769},[76,77],[55,6771,564],{"className":6772},[563],[55,6774,564],{"className":6775,"style":511},[563,510],[55,6777,4482],{"className":6778,"style":511},[563,510],[55,6780],{"className":6781,"style":161},[160],[55,6783,156],{"className":6784},[155],[11,6786,6787,6788,6824,6825,6858,6859,6893,6894,1143],{},"where ",[55,6789,6791],{"className":6790},[58],[55,6792,6794,6814],{"className":6793,"ariaHidden":63},[62],[55,6795,6797,6801,6804,6807,6811],{"className":6796},[67],[55,6798],{"className":6799,"style":6800},[71],"height:0.8889em;vertical-align:-0.1944em;",[55,6802,6591],{"className":6803,"style":6590},[76,77],[55,6805],{"className":6806,"style":263},[160],[55,6808,6810],{"className":6809},[267],">",[55,6812],{"className":6813,"style":263},[160],[55,6815,6817,6820],{"className":6816},[67],[55,6818],{"className":6819,"style":2735},[71],[55,6821,6823],{"className":6822},[76],"0"," sets the strength of the leash. As ",[55,6826,6828],{"className":6827},[58],[55,6829,6831,6849],{"className":6830,"ariaHidden":63},[62],[55,6832,6834,6837,6840,6843,6846],{"className":6833},[67],[55,6835],{"className":6836,"style":6800},[71],[55,6838,6591],{"className":6839,"style":6590},[76,77],[55,6841],{"className":6842,"style":263},[160],[55,6844,2725],{"className":6845},[267],[55,6847],{"className":6848,"style":263},[160],[55,6850,6852,6855],{"className":6851},[67],[55,6853],{"className":6854,"style":2735},[71],[55,6856,6823],{"className":6857},[76]," the policy maximizes\nreward unconstrained; as ",[55,6860,6862],{"className":6861},[58],[55,6863,6865,6883],{"className":6864,"ariaHidden":63},[62],[55,6866,6868,6871,6874,6877,6880],{"className":6867},[67],[55,6869],{"className":6870,"style":6800},[71],[55,6872,6591],{"className":6873,"style":6590},[76,77],[55,6875],{"className":6876,"style":263},[160],[55,6878,2725],{"className":6879},[267],[55,6881],{"className":6882,"style":263},[160],[55,6884,6886,6889],{"className":6885},[67],[55,6887],{"className":6888,"style":72},[71],[55,6890,6892],{"className":6891},[76],"∞"," it is pinned to ",[55,6895,6897],{"className":6896},[58],[55,6898,6900],{"className":6899,"ariaHidden":63},[62],[55,6901,6903,6906],{"className":6902},[67],[55,6904],{"className":6905,"style":917},[71],[55,6907,6909,6912],{"className":6908},[76],[55,6910,924],{"className":6911,"style":99},[76,77],[55,6913,6915],{"className":6914},[104],[55,6916,6918,6944],{"className":6917},[108,109],[55,6919,6921,6941],{"className":6920},[113],[55,6922,6924],{"className":6923,"style":296},[117],[55,6925,6926,6929],{"style":121},[55,6927],{"className":6928,"style":126},[125],[55,6930,6932],{"className":6931},[130,131,132,133],[55,6933,6935],{"className":6934},[76,133],[55,6936,6938],{"className":6937},[76,951,133],[55,6939,955],{"className":6940},[76,133],[55,6942,142],{"className":6943},[141],[55,6945,6947],{"className":6946},[113],[55,6948,6950],{"className":6949,"style":149},[117],[55,6951],{},[11,6953,6954,6955,7013,7014,7066,7067,7082],{},"The KL term serves two purposes. It keeps the policy in the region where the reward model\nwas trained and is therefore trustworthy, and it preserves the fluency and breadth the\nbase model already has, which the sparse reward would otherwise erode. Without the anchor\nthree failures follow directly. The optimizer exploits the reward model's blind spots and\nfinds high-scoring gibberish (reward hacking); it collapses onto the single\nhighest-reward response and loses diversity (mode collapse); and it drifts so far from the\ndata ",[55,6956,6958],{"className":6957},[58],[55,6959,6961],{"className":6960,"ariaHidden":63},[62],[55,6962,6964,6967],{"className":6963},[67],[55,6965],{"className":6966,"style":917},[71],[55,6968,6970,6973],{"className":6969},[76],[55,6971,924],{"className":6972,"style":99},[76,77],[55,6974,6976],{"className":6975},[104],[55,6977,6979,7005],{"className":6978},[108,109],[55,6980,6982,7002],{"className":6981},[113],[55,6983,6985],{"className":6984,"style":296},[117],[55,6986,6987,6990],{"style":121},[55,6988],{"className":6989,"style":126},[125],[55,6991,6993],{"className":6992},[130,131,132,133],[55,6994,6996],{"className":6995},[76,133],[55,6997,6999],{"className":6998},[76,951,133],[55,7000,955],{"className":7001},[76,133],[55,7003,142],{"className":7004},[141],[55,7006,7008],{"className":7007},[113],[55,7009,7011],{"className":7010,"style":149},[117],[55,7012],{}," was fit on that ",[55,7015,7017],{"className":7016},[58],[55,7018,7020],{"className":7019,"ariaHidden":63},[62],[55,7021,7023,7026],{"className":7022},[67],[55,7024],{"className":7025,"style":983},[71],[55,7027,7029,7032],{"className":7028},[76],[55,7030,800],{"className":7031,"style":480},[76,77],[55,7033,7035],{"className":7034},[104],[55,7036,7038,7058],{"className":7037},[108,109],[55,7039,7041,7055],{"className":7040},[113],[55,7042,7044],{"className":7043,"style":296},[117],[55,7045,7046,7049],{"style":1004},[55,7047],{"className":7048,"style":126},[125],[55,7050,7052],{"className":7051},[130,131,132,133],[55,7053,1014],{"className":7054},[76,77,133],[55,7056,142],{"className":7057},[141],[55,7059,7061],{"className":7060},[113],[55,7062,7064],{"className":7063,"style":1024},[117],[55,7065],{}," is extrapolating rather than judging\n(distribution drift). The two terms pull in opposite directions: reward pulls the policy\naway from the reference toward high-scoring responses, and the KL pulls it back toward the\nreference, with ",[55,7068,7070],{"className":7069},[58],[55,7071,7073],{"className":7072,"ariaHidden":63},[62],[55,7074,7076,7079],{"className":7075},[67],[55,7077],{"className":7078,"style":6800},[71],[55,7080,6591],{"className":7081,"style":6590},[76,77]," setting the balance point.",[1145,7084],{"hash":7085},"c1c1d901226794f0f116f03fee2974bedc74da17e08a2c646774a62ddf2b58cb",[3101,7087,7089],{"id":7088},"per-token-reward-shaping","Per-token reward shaping",[11,7091,7092,7093,7163],{},"Generation is a token-level Markov decision process: the state is the prompt plus the\ntokens emitted so far, the action is the next token, and the trajectory reward\n",[55,7094,7096],{"className":7095},[58],[55,7097,7099],{"className":7098,"ariaHidden":63},[62],[55,7100,7102,7105,7145,7148,7151,7154,7157,7160],{"className":7101},[67],[55,7103],{"className":7104,"style":3314},[71],[55,7106,7108,7111],{"className":7107},[76],[55,7109,800],{"className":7110,"style":480},[76,77],[55,7112,7114],{"className":7113},[104],[55,7115,7117,7137],{"className":7116},[108,109],[55,7118,7120,7134],{"className":7119},[113],[55,7121,7123],{"className":7122,"style":296},[117],[55,7124,7125,7128],{"style":1004},[55,7126],{"className":7127,"style":126},[125],[55,7129,7131],{"className":7130},[130,131,132,133],[55,7132,1014],{"className":7133},[76,77,133],[55,7135,142],{"className":7136},[141],[55,7138,7140],{"className":7139},[113],[55,7141,7143],{"className":7142,"style":1024},[117],[55,7144],{},[55,7146,521],{"className":7147},[509],[55,7149,78],{"className":7150},[76,77],[55,7152,156],{"className":7153},[155],[55,7155],{"className":7156,"style":161},[160],[55,7158,100],{"className":7159,"style":99},[76,77],[55,7161,564],{"className":7162},[563]," arrives only at the terminal token. To turn the sequence-level objective\ninto a dense per-step signal PPO can consume, the KL penalty is distributed across\ntokens, giving a per-token reward",[55,7165,7167],{"className":7166},[1949],[55,7168,7170],{"className":7169},[58],[55,7171,7173,7271,7487],{"className":7172,"ariaHidden":63},[62],[55,7174,7176,7180,7256,7259,7262,7265,7268],{"className":7175},[67],[55,7177],{"className":7178,"style":7179},[71],"height:0.8179em;vertical-align:-0.15em;",[55,7181,7183,7220],{"className":7182},[76],[55,7184,7187],{"className":7185},[76,7186],"accent",[55,7188,7190],{"className":7189},[108],[55,7191,7193],{"className":7192},[113],[55,7194,7197,7205],{"className":7195,"style":7196},[117],"height:0.6679em;",[55,7198,7199,7202],{"style":5206},[55,7200],{"className":7201,"style":2305},[125],[55,7203,800],{"className":7204,"style":480},[76,77],[55,7206,7208,7211],{"style":7207},"top:-3.35em;",[55,7209],{"className":7210,"style":2305},[125],[55,7212,7216],{"className":7213,"style":7215},[7214],"accent-body","left:-0.1944em;",[55,7217,7219],{"className":7218},[76],"~",[55,7221,7223],{"className":7222},[104],[55,7224,7226,7248],{"className":7225},[108,109],[55,7227,7229,7245],{"className":7228},[113],[55,7230,7233],{"className":7231,"style":7232},[117],"height:0.2806em;",[55,7234,7235,7238],{"style":1004},[55,7236],{"className":7237,"style":126},[125],[55,7239,7241],{"className":7240},[130,131,132,133],[55,7242,7244],{"className":7243},[76,77,133],"t",[55,7246,142],{"className":7247},[141],[55,7249,7251],{"className":7250},[113],[55,7252,7254],{"className":7253,"style":149},[117],[55,7255],{},[55,7257],{"className":7258,"style":263},[160],[55,7260],{"className":7261,"style":263},[160],[55,7263,488],{"className":7264},[267],[55,7266],{"className":7267,"style":263},[160],[55,7269],{"className":7270,"style":263},[160],[55,7272,7274,7278,7472,7475,7478,7481,7484],{"className":7273},[67],[55,7275],{"className":7276,"style":7277},[71],"height:2.3702em;vertical-align:-1.6202em;",[55,7279,7281],{"className":7280},[502,5174],[55,7282,7284,7463],{"className":7283},[108,109],[55,7285,7287,7460],{"className":7286},[113],[55,7288,7290,7308],{"className":7289,"style":5184},[117],[55,7291,7292,7295],{"style":5187},[55,7293],{"className":7294,"style":2305},[125],[55,7296,7298],{"className":7297},[130,131,132,133],[55,7299,7301],{"className":7300},[76,133],[55,7302,7304],{"className":7303},[76,951,133],[55,7305,7307],{"className":7306},[76,133],"terminal reward",[55,7309,7310,7313],{"style":5206},[55,7311],{"className":7312,"style":2305},[125],[55,7314,7316],{"className":7315},[502,5174],[55,7317,7319,7452],{"className":7318},[108,109],[55,7320,7322,7449],{"className":7321},[113],[55,7323,7325,7355],{"className":7324,"style":5184},[117],[55,7326,7328,7331],{"className":7327,"style":5226},[5225],[55,7329],{"className":7330,"style":2305},[125],[55,7332,7334,7341,7348],{"className":7333,"style":5234},[5233],[55,7335,7337],{"className":7336,"style":5239},[5238],[5241,7338,7339],{"xmlns":5243,"width":5244,"height":5245,"viewBox":5246,"preserveAspectRatio":5247},[5249,7340],{"d":5251},[55,7342,7344],{"className":7343,"style":5239},[5255],[5241,7345,7346],{"xmlns":5243,"width":5244,"height":5245,"viewBox":5246,"preserveAspectRatio":5258},[5249,7347],{"d":5261},[55,7349,7351],{"className":7350,"style":5239},[5265],[5241,7352,7353],{"xmlns":5243,"width":5244,"height":5245,"viewBox":5246,"preserveAspectRatio":5268},[5249,7354],{"d":5271},[55,7356,7357,7360],{"style":5206},[55,7358],{"className":7359,"style":2305},[125],[55,7361,7363,7403,7406,7409,7412,7415,7418,7421,7424,7427,7430,7433,7436,7439,7442,7446],{"className":7362},[76],[55,7364,7366,7369],{"className":7365},[76],[55,7367,800],{"className":7368,"style":480},[76,77],[55,7370,7372],{"className":7371},[104],[55,7373,7375,7395],{"className":7374},[108,109],[55,7376,7378,7392],{"className":7377},[113],[55,7379,7381],{"className":7380,"style":296},[117],[55,7382,7383,7386],{"style":1004},[55,7384],{"className":7385,"style":126},[125],[55,7387,7389],{"className":7388},[130,131,132,133],[55,7390,1014],{"className":7391},[76,77,133],[55,7393,142],{"className":7394},[141],[55,7396,7398],{"className":7397},[113],[55,7399,7401],{"className":7400,"style":1024},[117],[55,7402],{},[55,7404,521],{"className":7405},[509],[55,7407,78],{"className":7408},[76,77],[55,7410,156],{"className":7411},[155],[55,7413],{"className":7414,"style":161},[160],[55,7416,100],{"className":7417,"style":99},[76,77],[55,7419,564],{"className":7420},[563],[55,7422],{"className":7423,"style":161},[160],[55,7425,137],{"className":7426},[76],[55,7428,4252],{"className":7429},[509],[55,7431,7244],{"className":7432},[76,77],[55,7434],{"className":7435,"style":263},[160],[55,7437,488],{"className":7438},[267],[55,7440],{"className":7441,"style":263},[160],[55,7443,7445],{"className":7444,"style":1965},[76,77],"T",[55,7447,4482],{"className":7448},[563],[55,7450,142],{"className":7451},[141],[55,7453,7455],{"className":7454},[113],[55,7456,7458],{"className":7457,"style":5341},[117],[55,7459],{},[55,7461,142],{"className":7462},[141],[55,7464,7466],{"className":7465},[113],[55,7467,7470],{"className":7468,"style":7469},[117],"height:1.6202em;",[55,7471],{},[55,7473],{"className":7474,"style":263},[160],[55,7476],{"className":7477,"style":2186},[160],[55,7479,2191],{"className":7480},[2190],[55,7482],{"className":7483,"style":263},[160],[55,7485],{"className":7486,"style":2186},[160],[55,7488,7490,7494,7497,7500,7503,7990,7993],{"className":7489},[67],[55,7491],{"className":7492,"style":7493},[71],"height:3.8332em;vertical-align:-2.4062em;",[55,7495,6591],{"className":7496,"style":6590},[76,77],[55,7498],{"className":7499,"style":161},[160],[55,7501],{"className":7502,"style":161},[160],[55,7504,7506],{"className":7505},[502,5174],[55,7507,7509,7981],{"className":7508},[108,109],[55,7510,7512,7978],{"className":7511},[113],[55,7513,7516,7536],{"className":7514,"style":7515},[117],"height:1.427em;",[55,7517,7519,7523],{"style":7518},"top:-1.1569em;",[55,7520],{"className":7521,"style":7522},[125],"height:3.427em;",[55,7524,7526],{"className":7525},[130,131,132,133],[55,7527,7529],{"className":7528},[76,133],[55,7530,7532],{"className":7531},[76,951,133],[55,7533,7535],{"className":7534},[76,133],"per-token KL",[55,7537,7539,7542],{"style":7538},"top:-3.427em;",[55,7540],{"className":7541,"style":7522},[125],[55,7543,7545],{"className":7544},[502,5174],[55,7546,7548,7969],{"className":7547},[108,109],[55,7549,7551,7966],{"className":7550},[113],[55,7552,7554,7585],{"className":7553,"style":7515},[117],[55,7555,7558,7561],{"className":7556,"style":7557},[5225],"top:-1.843em;",[55,7559],{"className":7560,"style":7522},[125],[55,7562,7564,7571,7578],{"className":7563,"style":5234},[5233],[55,7565,7567],{"className":7566,"style":5239},[5238],[5241,7568,7569],{"xmlns":5243,"width":5244,"height":5245,"viewBox":5246,"preserveAspectRatio":5247},[5249,7570],{"d":5251},[55,7572,7574],{"className":7573,"style":5239},[5255],[5241,7575,7576],{"xmlns":5243,"width":5244,"height":5245,"viewBox":5246,"preserveAspectRatio":5258},[5249,7577],{"d":5261},[55,7579,7581],{"className":7580,"style":5239},[5265],[5241,7582,7583],{"xmlns":5243,"width":5244,"height":5245,"viewBox":5246,"preserveAspectRatio":5268},[5249,7584],{"d":5271},[55,7586,7587,7590],{"style":7538},[55,7588],{"className":7589,"style":7522},[125],[55,7591,7593,7599,7602],{"className":7592},[76],[55,7594,7596],{"className":7595},[2325],[55,7597,3865],{"className":7598,"style":3864},[76,3863],[55,7600],{"className":7601,"style":161},[160],[55,7603,7605,7608,7963],{"className":7604},[76],[55,7606],{"className":7607},[509,2284],[55,7609,7611],{"className":7610},[2288],[55,7612,7614,7955],{"className":7613},[108,109],[55,7615,7617,7952],{"className":7616},[113],[55,7618,7620,7786,7794],{"className":7619,"style":7515},[117],[55,7621,7622,7625],{"style":2301},[55,7623],{"className":7624,"style":2305},[125],[55,7626,7628,7674,7677,7717,7720,7723,7726,7729,7732,7735,7783],{"className":7627},[76],[55,7629,7631,7634],{"className":7630},[76],[55,7632,924],{"className":7633,"style":99},[76,77],[55,7635,7637],{"className":7636},[104],[55,7638,7640,7666],{"className":7639},[108,109],[55,7641,7643,7663],{"className":7642},[113],[55,7644,7646],{"className":7645,"style":296},[117],[55,7647,7648,7651],{"style":121},[55,7649],{"className":7650,"style":126},[125],[55,7652,7654],{"className":7653},[130,131,132,133],[55,7655,7657],{"className":7656},[76,133],[55,7658,7660],{"className":7659},[76,951,133],[55,7661,955],{"className":7662},[76,133],[55,7664,142],{"className":7665},[141],[55,7667,7669],{"className":7668},[113],[55,7670,7672],{"className":7671,"style":149},[117],[55,7673],{},[55,7675,521],{"className":7676},[509],[55,7678,7680,7683],{"className":7679},[76],[55,7681,100],{"className":7682,"style":99},[76,77],[55,7684,7686],{"className":7685},[104],[55,7687,7689,7709],{"className":7688},[108,109],[55,7690,7692,7706],{"className":7691},[113],[55,7693,7695],{"className":7694,"style":7232},[117],[55,7696,7697,7700],{"style":121},[55,7698],{"className":7699,"style":126},[125],[55,7701,7703],{"className":7702},[130,131,132,133],[55,7704,7244],{"className":7705},[76,77,133],[55,7707,142],{"className":7708},[141],[55,7710,7712],{"className":7711},[113],[55,7713,7715],{"className":7714,"style":149},[117],[55,7716],{},[55,7718],{"className":7719,"style":263},[160],[55,7721,2071],{"className":7722},[267],[55,7724],{"className":7725,"style":263},[160],[55,7727,78],{"className":7728},[76,77],[55,7730,156],{"className":7731},[155],[55,7733],{"className":7734,"style":161},[160],[55,7736,7738,7741],{"className":7737},[76],[55,7739,100],{"className":7740,"style":99},[76,77],[55,7742,7744],{"className":7743},[104],[55,7745,7747,7774],{"className":7746},[108,109],[55,7748,7750,7771],{"className":7749},[113],[55,7751,7753],{"className":7752,"style":7232},[117],[55,7754,7755,7758],{"style":121},[55,7756],{"className":7757,"style":126},[125],[55,7759,7761],{"className":7760},[130,131,132,133],[55,7762,7764,7768],{"className":7763},[76,133],[55,7765,7767],{"className":7766},[267,133],"\u003C",[55,7769,7244],{"className":7770},[76,77,133],[55,7772,142],{"className":7773},[141],[55,7775,7777],{"className":7776},[113],[55,7778,7781],{"className":7779,"style":7780},[117],"height:0.1774em;",[55,7782],{},[55,7784,564],{"className":7785},[563],[55,7787,7788,7791],{"style":2484},[55,7789],{"className":7790,"style":2305},[125],[55,7792],{"className":7793,"style":2492},[2491],[55,7795,7796,7799],{"style":2495},[55,7797],{"className":7798,"style":2305},[125],[55,7800,7802,7842,7845,7885,7888,7891,7894,7897,7900,7903,7949],{"className":7801},[76],[55,7803,7805,7808],{"className":7804},[76],[55,7806,924],{"className":7807,"style":99},[76,77],[55,7809,7811],{"className":7810},[104],[55,7812,7814,7834],{"className":7813},[108,109],[55,7815,7817,7831],{"className":7816},[113],[55,7818,7820],{"className":7819,"style":296},[117],[55,7821,7822,7825],{"style":121},[55,7823],{"className":7824,"style":126},[125],[55,7826,7828],{"className":7827},[130,131,132,133],[55,7829,1532],{"className":7830,"style":480},[76,77,133],[55,7832,142],{"className":7833},[141],[55,7835,7837],{"className":7836},[113],[55,7838,7840],{"className":7839,"style":149},[117],[55,7841],{},[55,7843,521],{"className":7844},[509],[55,7846,7848,7851],{"className":7847},[76],[55,7849,100],{"className":7850,"style":99},[76,77],[55,7852,7854],{"className":7853},[104],[55,7855,7857,7877],{"className":7856},[108,109],[55,7858,7860,7874],{"className":7859},[113],[55,7861,7863],{"className":7862,"style":7232},[117],[55,7864,7865,7868],{"style":121},[55,7866],{"className":7867,"style":126},[125],[55,7869,7871],{"className":7870},[130,131,132,133],[55,7872,7244],{"className":7873},[76,77,133],[55,7875,142],{"className":7876},[141],[55,7878,7880],{"className":7879},[113],[55,7881,7883],{"className":7882,"style":149},[117],[55,7884],{},[55,7886],{"className":7887,"style":263},[160],[55,7889,2071],{"className":7890},[267],[55,7892],{"className":7893,"style":263},[160],[55,7895,78],{"className":7896},[76,77],[55,7898,156],{"className":7899},[155],[55,7901],{"className":7902,"style":161},[160],[55,7904,7906,7909],{"className":7905},[76],[55,7907,100],{"className":7908,"style":99},[76,77],[55,7910,7912],{"className":7911},[104],[55,7913,7915,7941],{"className":7914},[108,109],[55,7916,7918,7938],{"className":7917},[113],[55,7919,7921],{"className":7920,"style":7232},[117],[55,7922,7923,7926],{"style":121},[55,7924],{"className":7925,"style":126},[125],[55,7927,7929],{"className":7928},[130,131,132,133],[55,7930,7932,7935],{"className":7931},[76,133],[55,7933,7767],{"className":7934},[267,133],[55,7936,7244],{"className":7937},[76,77,133],[55,7939,142],{"className":7940},[141],[55,7942,7944],{"className":7943},[113],[55,7945,7947],{"className":7946,"style":7780},[117],[55,7948],{},[55,7950,564],{"className":7951},[563],[55,7953,142],{"className":7954},[141],[55,7956,7958],{"className":7957},[113],[55,7959,7961],{"className":7960,"style":2514},[117],[55,7962],{},[55,7964],{"className":7965},[563,2284],[55,7967,142],{"className":7968},[141],[55,7970,7972],{"className":7971},[113],[55,7973,7976],{"className":7974,"style":7975},[117],"height:1.584em;",[55,7977],{},[55,7979,142],{"className":7980},[141],[55,7982,7984],{"className":7983},[113],[55,7985,7988],{"className":7986,"style":7987},[117],"height:2.4062em;",[55,7989],{},[55,7991],{"className":7992,"style":161},[160],[55,7994,1143],{"className":7995},[76],[11,7997,7998,7999,8017],{},"Summed over the trajectory, the second term telescopes into the sequence-level\n",[55,8000,8002],{"className":8001},[58],[55,8003,8005],{"className":8004,"ariaHidden":63},[62],[55,8006,8008,8011],{"className":8007},[67],[55,8009],{"className":8010,"style":475},[71],[55,8012,8014],{"className":8013},[2325],[55,8015,6659],{"className":8016},[76,3863],", so this shaping restates the objective above one step at a time.\nEvery token pays a small price for drifting from the reference, and the whole response\ncollects the terminal reward.",[1145,8019],{"hash":8020},"36109c1744978e93f156602bcbf6bcc273e8c8afdb382fccd08080837fdda192",[3101,8022,8024],{"id":8023},"ppo-as-the-optimizer","PPO as the optimizer",[11,8026,8027,8028,783,8031,8086,8087,8173,8174,8254,8255,8258],{},"The shaped reward feeds a standard actor-critic loop. A ",[50,8029,8030],{},"value head",[55,8032,8034],{"className":8033},[58],[55,8035,8037],{"className":8036,"ariaHidden":63},[62],[55,8038,8040,8043],{"className":8039},[67],[55,8041],{"className":8042,"style":3379},[71],[55,8044,8046,8050],{"className":8045},[76],[55,8047,8049],{"className":8048,"style":2186},[76,77],"V",[55,8051,8053],{"className":8052},[104],[55,8054,8056,8078],{"className":8055},[108,109],[55,8057,8059,8075],{"className":8058},[113],[55,8060,8062],{"className":8061,"style":296},[117],[55,8063,8065,8068],{"style":8064},"top:-2.55em;margin-left:-0.2222em;margin-right:0.05em;",[55,8066],{"className":8067,"style":126},[125],[55,8069,8071],{"className":8070},[130,131,132,133],[55,8072,8074],{"className":8073,"style":99},[76,77,133],"ψ",[55,8076,142],{"className":8077},[141],[55,8079,8081],{"className":8080},[113],[55,8082,8084],{"className":8083,"style":1024},[117],[55,8085],{}," is\nattached to the policy to estimate the expected return from each token, and the\nadvantage ",[55,8088,8090],{"className":8089},[58],[55,8091,8093],{"className":8092,"ariaHidden":63},[62],[55,8094,8096,8100],{"className":8095},[67],[55,8097],{"className":8098,"style":8099},[71],"height:1.0968em;vertical-align:-0.15em;",[55,8101,8103,8139],{"className":8102},[76],[55,8104,8106],{"className":8105},[76,7186],[55,8107,8109],{"className":8108},[108],[55,8110,8112],{"className":8111},[113],[55,8113,8116,8125],{"className":8114,"style":8115},[117],"height:0.9468em;",[55,8117,8118,8121],{"style":5206},[55,8119],{"className":8120,"style":2305},[125],[55,8122,8124],{"className":8123},[76,77],"A",[55,8126,8128,8131],{"style":8127},"top:-3.2523em;",[55,8129],{"className":8130,"style":2305},[125],[55,8132,8135],{"className":8133,"style":8134},[7214],"left:-0.1111em;",[55,8136,8138],{"className":8137},[76],"^",[55,8140,8142],{"className":8141},[104],[55,8143,8145,8165],{"className":8144},[108,109],[55,8146,8148,8162],{"className":8147},[113],[55,8149,8151],{"className":8150,"style":7232},[117],[55,8152,8153,8156],{"style":3342},[55,8154],{"className":8155,"style":126},[125],[55,8157,8159],{"className":8158},[130,131,132,133],[55,8160,7244],{"className":8161},[76,77,133],[55,8163,142],{"className":8164},[141],[55,8166,8168],{"className":8167},[113],[55,8169,8171],{"className":8170,"style":149},[117],[55,8172],{}," (typically by generalized advantage estimation) measures how much\nbetter the chosen token did than the value baseline. PPO then takes a clipped policy-gradient\nstep on ",[55,8175,8177],{"className":8176},[58],[55,8178,8180],{"className":8179,"ariaHidden":63},[62],[55,8181,8183,8186],{"className":8182},[67],[55,8184],{"className":8185,"style":8099},[71],[55,8187,8189,8220],{"className":8188},[76],[55,8190,8192],{"className":8191},[76,7186],[55,8193,8195],{"className":8194},[108],[55,8196,8198],{"className":8197},[113],[55,8199,8201,8209],{"className":8200,"style":8115},[117],[55,8202,8203,8206],{"style":5206},[55,8204],{"className":8205,"style":2305},[125],[55,8207,8124],{"className":8208},[76,77],[55,8210,8211,8214],{"style":8127},[55,8212],{"className":8213,"style":2305},[125],[55,8215,8217],{"className":8216,"style":8134},[7214],[55,8218,8138],{"className":8219},[76],[55,8221,8223],{"className":8222},[104],[55,8224,8226,8246],{"className":8225},[108,109],[55,8227,8229,8243],{"className":8228},[113],[55,8230,8232],{"className":8231,"style":7232},[117],[55,8233,8234,8237],{"style":3342},[55,8235],{"className":8236,"style":126},[125],[55,8238,8240],{"className":8239},[130,131,132,133],[55,8241,7244],{"className":8242},[76,77,133],[55,8244,142],{"className":8245},[141],[55,8247,8249],{"className":8248},[113],[55,8250,8252],{"className":8251,"style":149},[117],[55,8253],{},", which the\n",[15,8256,8257],{"href":22},"policy gradients lesson","\ndevelops in full. The clip is what keeps each update small enough that the\non-policy reward estimates stay valid.",[8260,8261,8265],"pre",{"className":8262,"code":8263,"language":8264,"meta":6,"style":6},"language-algorithm shiki shiki-themes Vesper Light - Orange Boost (Quick Open Adjusted) vesper","caption: $\\textsc{RLHF}(\\pi_{\\text{ref}}, \\mathcal{D}_{\\text{cmp}}, \\mathcal{D}_{\\text{prompt}}, \\beta)$ — align a policy from preferences\ntrain reward model $r_\\phi$ by minimizing $\\mathcal{L}_{\\text{RM}}$ on $\\mathcal{D}_{\\text{cmp}}$\n$\\pi_\\theta \\gets \\pi_{\\text{ref}}$; initialize value head $V_\\psi$\nrepeat\n  sample prompts $x \\sim \\mathcal{D}_{\\text{prompt}}$ and rollouts $y \\sim \\pi_\\theta(\\cdot \\mid x)$\n  score each rollout: $r \\gets r_\\phi(x, y)$ \u002F\u002F terminal reward\n  for each token $t$ do\n    $\\tilde r_t \\gets r\\cdot\\mathbb{1}[t{=}T] - \\beta\\log\\dfrac{\\pi_\\theta(y_t \\mid x, y_{\u003Ct})}{\\pi_{\\text{ref}}(y_t \\mid x, y_{\u003Ct})}$ \u002F\u002F KL-shaped reward\n  estimate advantages $\\hat A_t$ from $\\tilde r_t$ and $V_\\psi$ \u002F\u002F GAE\n  update $\\theta$ by the clipped PPO objective on $\\hat A_t$ \u002F\u002F policy step\n  update $\\psi$ by regressing $V_\\psi$ on the returns \u002F\u002F value step\nuntil reward plateaus or KL budget is spent\nreturn $\\pi_\\theta$\n","algorithm",[8266,8267,8268,8275,8281,8287,8293,8299,8305,8311,8317,8323,8329,8335,8341],"code",{"__ignoreMap":6},[55,8269,8272],{"class":8270,"line":8271},"line",1,[55,8273,8274],{},"caption: $\\textsc{RLHF}(\\pi_{\\text{ref}}, \\mathcal{D}_{\\text{cmp}}, \\mathcal{D}_{\\text{prompt}}, \\beta)$ — align a policy from preferences\n",[55,8276,8278],{"class":8270,"line":8277},2,[55,8279,8280],{},"train reward model $r_\\phi$ by minimizing $\\mathcal{L}_{\\text{RM}}$ on $\\mathcal{D}_{\\text{cmp}}$\n",[55,8282,8284],{"class":8270,"line":8283},3,[55,8285,8286],{},"$\\pi_\\theta \\gets \\pi_{\\text{ref}}$; initialize value head $V_\\psi$\n",[55,8288,8290],{"class":8270,"line":8289},4,[55,8291,8292],{},"repeat\n",[55,8294,8296],{"class":8270,"line":8295},5,[55,8297,8298],{},"  sample prompts $x \\sim \\mathcal{D}_{\\text{prompt}}$ and rollouts $y \\sim \\pi_\\theta(\\cdot \\mid x)$\n",[55,8300,8302],{"class":8270,"line":8301},6,[55,8303,8304],{},"  score each rollout: $r \\gets r_\\phi(x, y)$ \u002F\u002F terminal reward\n",[55,8306,8308],{"class":8270,"line":8307},7,[55,8309,8310],{},"  for each token $t$ do\n",[55,8312,8314],{"class":8270,"line":8313},8,[55,8315,8316],{},"    $\\tilde r_t \\gets r\\cdot\\mathbb{1}[t{=}T] - \\beta\\log\\dfrac{\\pi_\\theta(y_t \\mid x, y_{\u003Ct})}{\\pi_{\\text{ref}}(y_t \\mid x, y_{\u003Ct})}$ \u002F\u002F KL-shaped reward\n",[55,8318,8320],{"class":8270,"line":8319},9,[55,8321,8322],{},"  estimate advantages $\\hat A_t$ from $\\tilde r_t$ and $V_\\psi$ \u002F\u002F GAE\n",[55,8324,8326],{"class":8270,"line":8325},10,[55,8327,8328],{},"  update $\\theta$ by the clipped PPO objective on $\\hat A_t$ \u002F\u002F policy step\n",[55,8330,8332],{"class":8270,"line":8331},11,[55,8333,8334],{},"  update $\\psi$ by regressing $V_\\psi$ on the returns \u002F\u002F value step\n",[55,8336,8338],{"class":8270,"line":8337},12,[55,8339,8340],{},"until reward plateaus or KL budget is spent\n",[55,8342,8344],{"class":8270,"line":8343},13,[55,8345,8346],{},"return $\\pi_\\theta$\n",[31,8348,8350],{"id":8349},"failure-modes-reward-hacking-and-the-kl-leash","Failure modes: reward hacking and the KL leash",[11,8352,8353,8354,8357],{},"The reward model is a ",[880,8355,8356],{},"proxy",", not the true objective, and optimizing a proxy hard\nenough breaks it. This is Goodhart's law: a measure that is optimized ceases to measure\nwhat it once did.",[44,8359,8360],{"type":46},[11,8361,8362,8365,8366,8418,8419,8421,8422,8474,8475,8645],{},[50,8363,8364],{},"Definition (Reward over-optimization)."," As the policy is pushed to higher proxy\nreward ",[55,8367,8369],{"className":8368},[58],[55,8370,8372],{"className":8371,"ariaHidden":63},[62],[55,8373,8375,8378],{"className":8374},[67],[55,8376],{"className":8377,"style":983},[71],[55,8379,8381,8384],{"className":8380},[76],[55,8382,800],{"className":8383,"style":480},[76,77],[55,8385,8387],{"className":8386},[104],[55,8388,8390,8410],{"className":8389},[108,109],[55,8391,8393,8407],{"className":8392},[113],[55,8394,8396],{"className":8395,"style":296},[117],[55,8397,8398,8401],{"style":1004},[55,8399],{"className":8400,"style":126},[125],[55,8402,8404],{"className":8403},[130,131,132,133],[55,8405,1014],{"className":8406},[76,77,133],[55,8408,142],{"className":8409},[141],[55,8411,8413],{"className":8412},[113],[55,8414,8416],{"className":8415,"style":1024},[117],[55,8417],{},", the ",[880,8420,63],{}," human-judged quality first rises with it, then peaks and\nfalls, while ",[55,8423,8425],{"className":8424},[58],[55,8426,8428],{"className":8427,"ariaHidden":63},[62],[55,8429,8431,8434],{"className":8430},[67],[55,8432],{"className":8433,"style":983},[71],[55,8435,8437,8440],{"className":8436},[76],[55,8438,800],{"className":8439,"style":480},[76,77],[55,8441,8443],{"className":8442},[104],[55,8444,8446,8466],{"className":8445},[108,109],[55,8447,8449,8463],{"className":8448},[113],[55,8450,8452],{"className":8451,"style":296},[117],[55,8453,8454,8457],{"style":1004},[55,8455],{"className":8456,"style":126},[125],[55,8458,8460],{"className":8459},[130,131,132,133],[55,8461,1014],{"className":8462},[76,77,133],[55,8464,142],{"className":8465},[141],[55,8467,8469],{"className":8468},[113],[55,8470,8472],{"className":8471,"style":1024},[117],[55,8473],{}," keeps climbing. The policy has learned to exploit errors in the\nreward model rather than to improve. The gap typically widens with the KL distance\n",[55,8476,8478],{"className":8477},[58],[55,8479,8481],{"className":8480,"ariaHidden":63},[62],[55,8482,8484,8488],{"className":8483},[67],[55,8485],{"className":8486,"style":8487},[71],"height:1.24em;vertical-align:-0.305em;",[55,8489,8492],{"className":8490},[76,8491],"sqrt",[55,8493,8495,8636],{"className":8494},[108,109],[55,8496,8498,8633],{"className":8497},[113],[55,8499,8502,8615],{"className":8500,"style":8501},[117],"height:0.935em;",[55,8503,8506,8510],{"className":8504,"style":8505},[5225],"top:-3.2em;",[55,8507],{"className":8508,"style":8509},[125],"height:3.2em;",[55,8511,8514,8520,8523,8563,8566,8612],{"className":8512,"style":8513},[76],"padding-left:1em;",[55,8515,8517],{"className":8516},[2325],[55,8518,6659],{"className":8519},[76,3863],[55,8521,521],{"className":8522},[509],[55,8524,8526,8529],{"className":8525},[76],[55,8527,924],{"className":8528,"style":99},[76,77],[55,8530,8532],{"className":8531},[104],[55,8533,8535,8555],{"className":8534},[108,109],[55,8536,8538,8552],{"className":8537},[113],[55,8539,8541],{"className":8540,"style":296},[117],[55,8542,8543,8546],{"style":121},[55,8544],{"className":8545,"style":126},[125],[55,8547,8549],{"className":8548},[130,131,132,133],[55,8550,1532],{"className":8551,"style":480},[76,77,133],[55,8553,142],{"className":8554},[141],[55,8556,8558],{"className":8557},[113],[55,8559,8561],{"className":8560,"style":149},[117],[55,8562],{},[55,8564,6702],{"className":8565},[76],[55,8567,8569,8572],{"className":8568},[76],[55,8570,924],{"className":8571,"style":99},[76,77],[55,8573,8575],{"className":8574},[104],[55,8576,8578,8604],{"className":8577},[108,109],[55,8579,8581,8601],{"className":8580},[113],[55,8582,8584],{"className":8583,"style":296},[117],[55,8585,8586,8589],{"style":121},[55,8587],{"className":8588,"style":126},[125],[55,8590,8592],{"className":8591},[130,131,132,133],[55,8593,8595],{"className":8594},[76,133],[55,8596,8598],{"className":8597},[76,951,133],[55,8599,955],{"className":8600},[76,133],[55,8602,142],{"className":8603},[141],[55,8605,8607],{"className":8606},[113],[55,8608,8610],{"className":8609,"style":149},[117],[55,8611],{},[55,8613,564],{"className":8614},[563],[55,8616,8618,8621],{"style":8617},"top:-2.895em;",[55,8619],{"className":8620,"style":8509},[125],[55,8622,8626],{"className":8623,"style":8625},[8624],"hide-tail","min-width:1.02em;height:1.28em;",[5241,8627,8630],{"xmlns":5243,"width":5244,"height":8628,"viewBox":8629,"preserveAspectRatio":5247},"1.28em","0 0 400000 1296",[5249,8631],{"d":8632},"M263,681c0.7,0,18,39.7,52,119\nc34,79.3,68.167,158.7,102.5,238c34.3,79.3,51.8,119.3,52.5,120\nc340,-704.7,510.7,-1060.3,512,-1067\nl0 -0\nc4.7,-7.3,11,-11,19,-11\nH40000v40H1012.3\ns-271.3,567,-271.3,567c-38.7,80.7,-84,175,-136,283c-52,108,-89.167,185.3,-111.5,232\nc-22.3,46.7,-33.8,70.3,-34.5,71c-4.7,4.7,-12.3,7,-23,7s-12,-1,-12,-1\ns-109,-253,-109,-253c-72.7,-168,-109.3,-252,-110,-252c-10.7,8,-22,16.7,-34,26\nc-22,17.3,-33.3,26,-34,26s-26,-26,-26,-26s76,-59,76,-59s76,-60,76,-60z\nM1001 80h400000v40h-400000z",[55,8634,142],{"className":8635},[141],[55,8637,8639],{"className":8638},[113],[55,8640,8643],{"className":8641,"style":8642},[117],"height:0.305em;",[55,8644],{}," traveled from the reference.",[11,8647,8648,8649,8664],{},"The KL coefficient ",[55,8650,8652],{"className":8651},[58],[55,8653,8655],{"className":8654,"ariaHidden":63},[62],[55,8656,8658,8661],{"className":8657},[67],[55,8659],{"className":8660,"style":6800},[71],[55,8662,6591],{"className":8663,"style":6590},[76,77]," trades these two failure directions against\neach other. Too large, and the policy never moves; too small, and it over-optimizes the\nproxy into degeneracy.",[1145,8666],{"hash":8667},"75d710db88248a95f01f3f7605ef3066857a45ad2c834993514a3d49401fb16d",[11,8669,8670,8671,8673,8674,8677,8678,8694],{},"Two further levers control the same trade-off. The ",[50,8672,8030],{}," must be well-fit or\nthe advantage estimates are biased, which destabilizes the policy step; in practice the\nvalue and policy share the backbone but use separate heads. And ",[50,8675,8676],{},"advantage estimation","\nby GAE introduces a bias-variance parameter (",[55,8679,8681],{"className":8680},[58],[55,8682,8684],{"className":8683,"ariaHidden":63},[62],[55,8685,8687,8690],{"className":8686},[67],[55,8688],{"className":8689,"style":458},[71],[55,8691,8693],{"className":8692},[76,77],"λ",") that, set well, smooths the sparse\nterminal reward into a usable per-token learning signal without injecting too much\nvariance from long rollouts.",[31,8696,8698],{"id":8697},"direct-preference-optimization","Direct preference optimization",[11,8700,8701,8702,8704,8705,8708],{},"PPO-based RLHF is operationally heavy: a separate reward model, a value head, on-policy\nsampling, and a famously finicky RL loop. ",[50,8703,8698],{}," (DPO)\nremoves all of it by exploiting a fact about the KL-regularized objective: its optimum\nhas a closed form, and that closed form can be inverted to express the reward in terms\nof the policy. Substituting the inverted reward into the Bradley-Terry loss\nyields a ",[880,8706,8707],{},"supervised"," objective on the policy directly, with no reward model and no\nsampling.",[44,8710,8712,8734,9583],{"type":8711},"theorem",[11,8713,8714,8717,8718,8733],{},[50,8715,8716],{},"Theorem (DPO objective)."," The policy that maximizes the KL-regularized RL objective\nfor a reward ",[55,8719,8721],{"className":8720},[58],[55,8722,8724],{"className":8723,"ariaHidden":63},[62],[55,8725,8727,8730],{"className":8726},[67],[55,8728],{"className":8729,"style":72},[71],[55,8731,800],{"className":8732,"style":480},[76,77]," is the same policy that minimizes the supervised loss",[55,8735,8737],{"className":8736},[1949],[55,8738,8740],{"className":8739},[58],[55,8741,8743,8814],{"className":8742,"ariaHidden":63},[62],[55,8744,8746,8749,8796,8799,8802,8805,8808,8811],{"className":8745},[67],[55,8747],{"className":8748,"style":796},[71],[55,8750,8752,8755],{"className":8751},[76],[55,8753,4018],{"className":8754},[76,479],[55,8756,8758],{"className":8757},[104],[55,8759,8761,8788],{"className":8760},[108,109],[55,8762,8764,8785],{"className":8763},[113],[55,8765,8767],{"className":8766,"style":4031},[117],[55,8768,8769,8772],{"style":3342},[55,8770],{"className":8771,"style":126},[125],[55,8773,8775],{"className":8774},[130,131,132,133],[55,8776,8778],{"className":8777},[76,133],[55,8779,8781],{"className":8780},[76,951,133],[55,8782,8784],{"className":8783},[76,133],"DPO",[55,8786,142],{"className":8787},[141],[55,8789,8791],{"className":8790},[113],[55,8792,8794],{"className":8793,"style":149},[117],[55,8795],{},[55,8797,521],{"className":8798},[509],[55,8800,1532],{"className":8801,"style":480},[76,77],[55,8803,564],{"className":8804},[563],[55,8806],{"className":8807,"style":263},[160],[55,8809,488],{"className":8810},[267],[55,8812],{"className":8813,"style":263},[160],[55,8815,8817,8821,8824,8827,8968,8971,8974,9577,9580],{"className":8816},[67],[55,8818],{"className":8819,"style":8820},[71],"height:2.4em;vertical-align:-0.95em;",[55,8822,2191],{"className":8823},[76],[55,8825],{"className":8826,"style":161},[160],[55,8828,8830,8833],{"className":8829},[76],[55,8831,4104],{"className":8832},[76,1838],[55,8834,8836],{"className":8835},[104],[55,8837,8839,8960],{"className":8838},[108,109],[55,8840,8842,8957],{"className":8841},[113],[55,8843,8845],{"className":8844,"style":3794},[117],[55,8846,8847,8850],{"style":4119},[55,8848],{"className":8849,"style":126},[125],[55,8851,8853],{"className":8852},[130,131,132,133],[55,8854,8856,8859,8862,8865,8905,8908,8948,8951,8954],{"className":8855},[76,133],[55,8857,521],{"className":8858},[509,133],[55,8860,78],{"className":8861},[76,77,133],[55,8863,156],{"className":8864},[155,133],[55,8866,8868,8871],{"className":8867},[76,133],[55,8869,100],{"className":8870,"style":99},[76,77,133],[55,8872,8874],{"className":8873},[104],[55,8875,8877,8897],{"className":8876},[108,109],[55,8878,8880,8894],{"className":8879},[113],[55,8881,8883],{"className":8882,"style":3745},[117],[55,8884,8885,8888],{"style":3748},[55,8886],{"className":8887,"style":3752},[125],[55,8889,8891],{"className":8890},[130,3756,3757,133],[55,8892,248],{"className":8893,"style":247},[76,77,133],[55,8895,142],{"className":8896},[141],[55,8898,8900],{"className":8899},[113],[55,8901,8903],{"className":8902,"style":3770},[117],[55,8904],{},[55,8906,156],{"className":8907},[155,133],[55,8909,8911,8914],{"className":8910},[76,133],[55,8912,100],{"className":8913,"style":99},[76,77,133],[55,8915,8917],{"className":8916},[104],[55,8918,8920,8940],{"className":8919},[108,109],[55,8921,8923,8937],{"className":8922},[113],[55,8924,8926],{"className":8925,"style":3794},[117],[55,8927,8928,8931],{"style":3797},[55,8929],{"className":8930,"style":3752},[125],[55,8932,8934],{"className":8933},[130,3756,3757,133],[55,8935,309],{"className":8936,"style":308},[76,77,133],[55,8938,142],{"className":8939},[141],[55,8941,8943],{"className":8942},[113],[55,8944,8946],{"className":8945,"style":3816},[117],[55,8947],{},[55,8949,564],{"className":8950},[563,133],[55,8952,4227],{"className":8953},[267,133],[55,8955,481],{"className":8956,"style":480},[76,479,133],[55,8958,142],{"className":8959},[141],[55,8961,8963],{"className":8962},[113],[55,8964,8966],{"className":8965,"style":4240},[117],[55,8967],{},[55,8969],{"className":8970,"style":2115},[160],[55,8972],{"className":8973,"style":161},[160],[55,8975,8977,8983,8989,8992,8995,8998,9001,9571],{"className":8976},[502],[55,8978,8980],{"className":8979,"style":511},[509,510],[55,8981,4252],{"className":8982},[515,132],[55,8984,8986],{"className":8985},[2325],[55,8987,3865],{"className":8988,"style":3864},[76,3863],[55,8990],{"className":8991,"style":161},[160],[55,8993,2111],{"className":8994,"style":99},[76,77],[55,8996],{"className":8997,"style":2115},[160],[55,8999],{"className":9000,"style":161},[160],[55,9002,9004,9010,9013,9016,9022,9025,9283,9286,9289,9292,9295,9298,9304,9307,9565],{"className":9003},[502],[55,9005,9007],{"className":9006,"style":511},[509,510],[55,9008,521],{"className":9009},[515,132],[55,9011,6591],{"className":9012,"style":6590},[76,77],[55,9014],{"className":9015,"style":161},[160],[55,9017,9019],{"className":9018},[2325],[55,9020,3865],{"className":9021,"style":3864},[76,3863],[55,9023],{"className":9024,"style":161},[160],[55,9026,9028,9031,9280],{"className":9027},[76],[55,9029],{"className":9030},[509,2284],[55,9032,9034],{"className":9033},[2288],[55,9035,9037,9272],{"className":9036},[108,109],[55,9038,9040,9269],{"className":9039},[113],[55,9041,9043,9155,9163],{"className":9042,"style":7515},[117],[55,9044,9045,9048],{"style":2301},[55,9046],{"className":9047,"style":2305},[125],[55,9049,9051,9097,9100,9140,9143,9146,9149,9152],{"className":9050},[76],[55,9052,9054,9057],{"className":9053},[76],[55,9055,924],{"className":9056,"style":99},[76,77],[55,9058,9060],{"className":9059},[104],[55,9061,9063,9089],{"className":9062},[108,109],[55,9064,9066,9086],{"className":9065},[113],[55,9067,9069],{"className":9068,"style":296},[117],[55,9070,9071,9074],{"style":121},[55,9072],{"className":9073,"style":126},[125],[55,9075,9077],{"className":9076},[130,131,132,133],[55,9078,9080],{"className":9079},[76,133],[55,9081,9083],{"className":9082},[76,951,133],[55,9084,955],{"className":9085},[76,133],[55,9087,142],{"className":9088},[141],[55,9090,9092],{"className":9091},[113],[55,9093,9095],{"className":9094,"style":149},[117],[55,9096],{},[55,9098,521],{"className":9099},[509],[55,9101,9103,9106],{"className":9102},[76],[55,9104,100],{"className":9105,"style":99},[76,77],[55,9107,9109],{"className":9108},[104],[55,9110,9112,9132],{"className":9111},[108,109],[55,9113,9115,9129],{"className":9114},[113],[55,9116,9118],{"className":9117,"style":235},[117],[55,9119,9120,9123],{"style":121},[55,9121],{"className":9122,"style":126},[125],[55,9124,9126],{"className":9125},[130,131,132,133],[55,9127,248],{"className":9128,"style":247},[76,77,133],[55,9130,142],{"className":9131},[141],[55,9133,9135],{"className":9134},[113],[55,9136,9138],{"className":9137,"style":149},[117],[55,9139],{},[55,9141],{"className":9142,"style":263},[160],[55,9144,2071],{"className":9145},[267],[55,9147],{"className":9148,"style":263},[160],[55,9150,78],{"className":9151},[76,77],[55,9153,564],{"className":9154},[563],[55,9156,9157,9160],{"style":2484},[55,9158],{"className":9159,"style":2305},[125],[55,9161],{"className":9162,"style":2492},[2491],[55,9164,9165,9168],{"style":2495},[55,9166],{"className":9167,"style":2305},[125],[55,9169,9171,9211,9214,9254,9257,9260,9263,9266],{"className":9170},[76],[55,9172,9174,9177],{"className":9173},[76],[55,9175,924],{"className":9176,"style":99},[76,77],[55,9178,9180],{"className":9179},[104],[55,9181,9183,9203],{"className":9182},[108,109],[55,9184,9186,9200],{"className":9185},[113],[55,9187,9189],{"className":9188,"style":296},[117],[55,9190,9191,9194],{"style":121},[55,9192],{"className":9193,"style":126},[125],[55,9195,9197],{"className":9196},[130,131,132,133],[55,9198,1532],{"className":9199,"style":480},[76,77,133],[55,9201,142],{"className":9202},[141],[55,9204,9206],{"className":9205},[113],[55,9207,9209],{"className":9208,"style":149},[117],[55,9210],{},[55,9212,521],{"className":9213},[509],[55,9215,9217,9220],{"className":9216},[76],[55,9218,100],{"className":9219,"style":99},[76,77],[55,9221,9223],{"className":9222},[104],[55,9224,9226,9246],{"className":9225},[108,109],[55,9227,9229,9243],{"className":9228},[113],[55,9230,9232],{"className":9231,"style":235},[117],[55,9233,9234,9237],{"style":121},[55,9235],{"className":9236,"style":126},[125],[55,9238,9240],{"className":9239},[130,131,132,133],[55,9241,248],{"className":9242,"style":247},[76,77,133],[55,9244,142],{"className":9245},[141],[55,9247,9249],{"className":9248},[113],[55,9250,9252],{"className":9251,"style":149},[117],[55,9253],{},[55,9255],{"className":9256,"style":263},[160],[55,9258,2071],{"className":9259},[267],[55,9261],{"className":9262,"style":263},[160],[55,9264,78],{"className":9265},[76,77],[55,9267,564],{"className":9268},[563],[55,9270,142],{"className":9271},[141],[55,9273,9275],{"className":9274},[113],[55,9276,9278],{"className":9277,"style":2514},[117],[55,9279],{},[55,9281],{"className":9282},[563,2284],[55,9284],{"className":9285,"style":2186},[160],[55,9287,2191],{"className":9288},[2190],[55,9290],{"className":9291,"style":2186},[160],[55,9293,6591],{"className":9294,"style":6590},[76,77],[55,9296],{"className":9297,"style":161},[160],[55,9299,9301],{"className":9300},[2325],[55,9302,3865],{"className":9303,"style":3864},[76,3863],[55,9305],{"className":9306,"style":161},[160],[55,9308,9310,9313,9562],{"className":9309},[76],[55,9311],{"className":9312},[509,2284],[55,9314,9316],{"className":9315},[2288],[55,9317,9319,9554],{"className":9318},[108,109],[55,9320,9322,9551],{"className":9321},[113],[55,9323,9325,9437,9445],{"className":9324,"style":7515},[117],[55,9326,9327,9330],{"style":2301},[55,9328],{"className":9329,"style":2305},[125],[55,9331,9333,9379,9382,9422,9425,9428,9431,9434],{"className":9332},[76],[55,9334,9336,9339],{"className":9335},[76],[55,9337,924],{"className":9338,"style":99},[76,77],[55,9340,9342],{"className":9341},[104],[55,9343,9345,9371],{"className":9344},[108,109],[55,9346,9348,9368],{"className":9347},[113],[55,9349,9351],{"className":9350,"style":296},[117],[55,9352,9353,9356],{"style":121},[55,9354],{"className":9355,"style":126},[125],[55,9357,9359],{"className":9358},[130,131,132,133],[55,9360,9362],{"className":9361},[76,133],[55,9363,9365],{"className":9364},[76,951,133],[55,9366,955],{"className":9367},[76,133],[55,9369,142],{"className":9370},[141],[55,9372,9374],{"className":9373},[113],[55,9375,9377],{"className":9376,"style":149},[117],[55,9378],{},[55,9380,521],{"className":9381},[509],[55,9383,9385,9388],{"className":9384},[76],[55,9386,100],{"className":9387,"style":99},[76,77],[55,9389,9391],{"className":9390},[104],[55,9392,9394,9414],{"className":9393},[108,109],[55,9395,9397,9411],{"className":9396},[113],[55,9398,9400],{"className":9399,"style":296},[117],[55,9401,9402,9405],{"style":121},[55,9403],{"className":9404,"style":126},[125],[55,9406,9408],{"className":9407},[130,131,132,133],[55,9409,309],{"className":9410,"style":308},[76,77,133],[55,9412,142],{"className":9413},[141],[55,9415,9417],{"className":9416},[113],[55,9418,9420],{"className":9419,"style":149},[117],[55,9421],{},[55,9423],{"className":9424,"style":263},[160],[55,9426,2071],{"className":9427},[267],[55,9429],{"className":9430,"style":263},[160],[55,9432,78],{"className":9433},[76,77],[55,9435,564],{"className":9436},[563],[55,9438,9439,9442],{"style":2484},[55,9440],{"className":9441,"style":2305},[125],[55,9443],{"className":9444,"style":2492},[2491],[55,9446,9447,9450],{"style":2495},[55,9448],{"className":9449,"style":2305},[125],[55,9451,9453,9493,9496,9536,9539,9542,9545,9548],{"className":9452},[76],[55,9454,9456,9459],{"className":9455},[76],[55,9457,924],{"className":9458,"style":99},[76,77],[55,9460,9462],{"className":9461},[104],[55,9463,9465,9485],{"className":9464},[108,109],[55,9466,9468,9482],{"className":9467},[113],[55,9469,9471],{"className":9470,"style":296},[117],[55,9472,9473,9476],{"style":121},[55,9474],{"className":9475,"style":126},[125],[55,9477,9479],{"className":9478},[130,131,132,133],[55,9480,1532],{"className":9481,"style":480},[76,77,133],[55,9483,142],{"className":9484},[141],[55,9486,9488],{"className":9487},[113],[55,9489,9491],{"className":9490,"style":149},[117],[55,9492],{},[55,9494,521],{"className":9495},[509],[55,9497,9499,9502],{"className":9498},[76],[55,9500,100],{"className":9501,"style":99},[76,77],[55,9503,9505],{"className":9504},[104],[55,9506,9508,9528],{"className":9507},[108,109],[55,9509,9511,9525],{"className":9510},[113],[55,9512,9514],{"className":9513,"style":296},[117],[55,9515,9516,9519],{"style":121},[55,9517],{"className":9518,"style":126},[125],[55,9520,9522],{"className":9521},[130,131,132,133],[55,9523,309],{"className":9524,"style":308},[76,77,133],[55,9526,142],{"className":9527},[141],[55,9529,9531],{"className":9530},[113],[55,9532,9534],{"className":9533,"style":149},[117],[55,9535],{},[55,9537],{"className":9538,"style":263},[160],[55,9540,2071],{"className":9541},[267],[55,9543],{"className":9544,"style":263},[160],[55,9546,78],{"className":9547},[76,77],[55,9549,564],{"className":9550},[563],[55,9552,142],{"className":9553},[141],[55,9555,9557],{"className":9556},[113],[55,9558,9560],{"className":9559,"style":2514},[117],[55,9561],{},[55,9563],{"className":9564},[563,2284],[55,9566,9568],{"className":9567,"style":511},[563,510],[55,9569,564],{"className":9570},[515,132],[55,9572,9574],{"className":9573,"style":511},[563,510],[55,9575,4482],{"className":9576},[515,132],[55,9578],{"className":9579,"style":161},[160],[55,9581,156],{"className":9582},[155],[11,9584,9585],{},"a binary-classification loss over preference pairs that needs no explicit reward\nmodel, no value head, and no on-policy rollouts.",[44,9587,9589,9865,10184,10950,11324,11331,11630,11967,12676],{"type":9588},"proof",[11,9590,9591,9594,9595,9610,9611,9794,9795,9837,9838,9864],{},[50,9592,9593],{},"Proof."," Fix a prompt ",[55,9596,9598],{"className":9597},[58],[55,9599,9601],{"className":9600,"ariaHidden":63},[62],[55,9602,9604,9607],{"className":9603},[67],[55,9605],{"className":9606,"style":72},[71],[55,9608,78],{"className":9609},[76,77]," and maximize the per-prompt objective\n",[55,9612,9614],{"className":9613},[58],[55,9615,9617,9706],{"className":9616,"ariaHidden":63},[62],[55,9618,9620,9623,9672,9675,9678,9681,9684,9687,9690,9693,9697,9700,9703],{"className":9619},[67],[55,9621],{"className":9622,"style":3314},[71],[55,9624,9626,9629],{"className":9625},[76],[55,9627,4104],{"className":9628},[76,1838],[55,9630,9632],{"className":9631},[104],[55,9633,9635,9664],{"className":9634},[108,109],[55,9636,9638,9661],{"className":9637},[113],[55,9639,9641],{"className":9640,"style":235},[117],[55,9642,9643,9646],{"style":3342},[55,9644],{"className":9645,"style":126},[125],[55,9647,9649],{"className":9648},[130,131,132,133],[55,9650,9652,9655,9658],{"className":9651},[76,133],[55,9653,100],{"className":9654,"style":99},[76,77,133],[55,9656,4227],{"className":9657},[267,133],[55,9659,924],{"className":9660,"style":99},[76,77,133],[55,9662,142],{"className":9663},[141],[55,9665,9667],{"className":9666},[113],[55,9668,9670],{"className":9669,"style":1024},[117],[55,9671],{},[55,9673,4252],{"className":9674},[509],[55,9676,800],{"className":9677,"style":480},[76,77],[55,9679,521],{"className":9680},[509],[55,9682,78],{"className":9683},[76,77],[55,9685,156],{"className":9686},[155],[55,9688],{"className":9689,"style":161},[160],[55,9691,100],{"className":9692,"style":99},[76,77],[55,9694,9696],{"className":9695},[563],")]",[55,9698],{"className":9699,"style":2186},[160],[55,9701,2191],{"className":9702},[2190],[55,9704],{"className":9705,"style":2186},[160],[55,9707,9709,9712,9715,9718,9724,9727],{"className":9708},[67],[55,9710],{"className":9711,"style":796},[71],[55,9713,6591],{"className":9714,"style":6590},[76,77],[55,9716],{"className":9717,"style":161},[160],[55,9719,9721],{"className":9720},[2325],[55,9722,6659],{"className":9723},[76,3863],[55,9725],{"className":9726,"style":161},[160],[55,9728,9730,9733,9736,9739,9742,9745,9791],{"className":9729},[502],[55,9731,521],{"className":9732,"style":511},[509,510],[55,9734,924],{"className":9735,"style":99},[76,77],[55,9737],{"className":9738,"style":161},[160],[55,9740,6702],{"className":9741},[76],[55,9743],{"className":9744,"style":161},[160],[55,9746,9748,9751],{"className":9747},[76],[55,9749,924],{"className":9750,"style":99},[76,77],[55,9752,9754],{"className":9753},[104],[55,9755,9757,9783],{"className":9756},[108,109],[55,9758,9760,9780],{"className":9759},[113],[55,9761,9763],{"className":9762,"style":296},[117],[55,9764,9765,9768],{"style":121},[55,9766],{"className":9767,"style":126},[125],[55,9769,9771],{"className":9770},[130,131,132,133],[55,9772,9774],{"className":9773},[76,133],[55,9775,9777],{"className":9776},[76,951,133],[55,9778,955],{"className":9779},[76,133],[55,9781,142],{"className":9782},[141],[55,9784,9786],{"className":9785},[113],[55,9787,9789],{"className":9788,"style":149},[117],[55,9790],{},[55,9792,564],{"className":9793,"style":511},[563,510],"\nover distributions ",[55,9796,9798],{"className":9797},[58],[55,9799,9801,9825],{"className":9800,"ariaHidden":63},[62],[55,9802,9804,9807,9810,9813,9816,9819,9822],{"className":9803},[67],[55,9805],{"className":9806,"style":796},[71],[55,9808,924],{"className":9809,"style":99},[76,77],[55,9811,521],{"className":9812},[509],[55,9814,6512],{"className":9815},[76],[55,9817],{"className":9818,"style":263},[160],[55,9820,2071],{"className":9821},[267],[55,9823],{"className":9824,"style":263},[160],[55,9826,9828,9831,9834],{"className":9827},[67],[55,9829],{"className":9830,"style":796},[71],[55,9832,78],{"className":9833},[76,77],[55,9835,564],{"className":9836},[563],". Expanding the KL and dropping the constant\n",[55,9839,9841],{"className":9840},[58],[55,9842,9844],{"className":9843,"ariaHidden":63},[62],[55,9845,9847,9850,9856,9859],{"className":9846},[67],[55,9848],{"className":9849,"style":6800},[71],[55,9851,9853],{"className":9852},[2325],[55,9854,3865],{"className":9855,"style":3864},[76,3863],[55,9857],{"className":9858,"style":161},[160],[55,9860,9863],{"className":9861,"style":9862},[76,77],"margin-right:0.0715em;","Z"," that we introduce next, the objective equals",[55,9866,9868],{"className":9867},[1949],[55,9869,9871],{"className":9870},[58],[55,9872,9874,9957],{"className":9873,"ariaHidden":63},[62],[55,9875,9877,9881,9884,9887,9890,9936,9939,9942,9945,9948,9951,9954],{"className":9876},[67],[55,9878],{"className":9879,"style":9880},[71],"height:2.4361em;vertical-align:-1.3861em;",[55,9882,2191],{"className":9883},[76],[55,9885,6591],{"className":9886,"style":6590},[76,77],[55,9888],{"className":9889,"style":161},[160],[55,9891,9893],{"className":9892},[2325,3694],[55,9894,9896,9927],{"className":9895},[108,109],[55,9897,9899,9924],{"className":9898},[113],[55,9900,9902,9914],{"className":9901,"style":3704},[117],[55,9903,9905,9908],{"style":9904},"top:-1.9em;margin-left:0em;",[55,9906],{"className":9907,"style":3711},[125],[55,9909,9911],{"className":9910},[130,131,132,133],[55,9912,100],{"className":9913,"style":99},[76,77,133],[55,9915,9916,9919],{"style":3830},[55,9917],{"className":9918,"style":3711},[125],[55,9920,9921],{},[55,9922,3841],{"className":9923},[2325,3839,3840],[55,9925,142],{"className":9926},[141],[55,9928,9930],{"className":9929},[113],[55,9931,9934],{"className":9932,"style":9933},[117],"height:1.3861em;",[55,9935],{},[55,9937],{"className":9938,"style":161},[160],[55,9940,924],{"className":9941,"style":99},[76,77],[55,9943,521],{"className":9944},[509],[55,9946,100],{"className":9947,"style":99},[76,77],[55,9949],{"className":9950,"style":263},[160],[55,9952,2071],{"className":9953},[267],[55,9955],{"className":9956,"style":263},[160],[55,9958,9960,9964,9967,9970,9973,9976,9982,9985,10181],{"className":9959},[67],[55,9961],{"className":9962,"style":9963},[71],"height:2.363em;vertical-align:-0.936em;",[55,9965,78],{"className":9966},[76,77],[55,9968,564],{"className":9969},[563],[55,9971],{"className":9972,"style":161},[160],[55,9974],{"className":9975,"style":161},[160],[55,9977,9979],{"className":9978},[2325],[55,9980,3865],{"className":9981,"style":3864},[76,3863],[55,9983],{"className":9984,"style":161},[160],[55,9986,9988,9991,10178],{"className":9987},[76],[55,9989],{"className":9990},[509,2284],[55,9992,9994],{"className":9993},[2288],[55,9995,9997,10170],{"className":9996},[108,109],[55,9998,10000,10167],{"className":9999},[113],[55,10001,10003,10127,10135],{"className":10002,"style":7515},[117],[55,10004,10005,10008],{"style":2301},[55,10006],{"className":10007,"style":2305},[125],[55,10009,10011,10057,10060,10063,10066,10069,10072,10075,10078,10081,10084,10087,10090],{"className":10010},[76],[55,10012,10014,10017],{"className":10013},[76],[55,10015,924],{"className":10016,"style":99},[76,77],[55,10018,10020],{"className":10019},[104],[55,10021,10023,10049],{"className":10022},[108,109],[55,10024,10026,10046],{"className":10025},[113],[55,10027,10029],{"className":10028,"style":296},[117],[55,10030,10031,10034],{"style":121},[55,10032],{"className":10033,"style":126},[125],[55,10035,10037],{"className":10036},[130,131,132,133],[55,10038,10040],{"className":10039},[76,133],[55,10041,10043],{"className":10042},[76,951,133],[55,10044,955],{"className":10045},[76,133],[55,10047,142],{"className":10048},[141],[55,10050,10052],{"className":10051},[113],[55,10053,10055],{"className":10054,"style":149},[117],[55,10056],{},[55,10058,521],{"className":10059},[509],[55,10061,100],{"className":10062,"style":99},[76,77],[55,10064],{"className":10065,"style":263},[160],[55,10067,2071],{"className":10068},[267],[55,10070],{"className":10071,"style":263},[160],[55,10073,78],{"className":10074},[76,77],[55,10076,564],{"className":10077},[563],[55,10079],{"className":10080,"style":161},[160],[55,10082,2326],{"className":10083},[2325],[55,10085],{"className":10086,"style":2115},[160],[55,10088],{"className":10089,"style":161},[160],[55,10091,10093,10096,10099,10102,10105,10108,10111,10114,10117,10121,10124],{"className":10092},[502],[55,10094,521],{"className":10095,"style":511},[509,510],[55,10097,800],{"className":10098,"style":480},[76,77],[55,10100,521],{"className":10101},[509],[55,10103,78],{"className":10104},[76,77],[55,10106,156],{"className":10107},[155],[55,10109],{"className":10110,"style":161},[160],[55,10112,100],{"className":10113,"style":99},[76,77],[55,10115,564],{"className":10116},[563],[55,10118,10120],{"className":10119},[76],"\u002F",[55,10122,6591],{"className":10123,"style":6590},[76,77],[55,10125,564],{"className":10126,"style":511},[563,510],[55,10128,10129,10132],{"style":2484},[55,10130],{"className":10131,"style":2305},[125],[55,10133],{"className":10134,"style":2492},[2491],[55,10136,10137,10140],{"style":2495},[55,10138],{"className":10139,"style":2305},[125],[55,10141,10143,10146,10149,10152,10155,10158,10161,10164],{"className":10142},[76],[55,10144,924],{"className":10145,"style":99},[76,77],[55,10147,521],{"className":10148},[509],[55,10150,100],{"className":10151,"style":99},[76,77],[55,10153],{"className":10154,"style":263},[160],[55,10156,2071],{"className":10157},[267],[55,10159],{"className":10160,"style":263},[160],[55,10162,78],{"className":10163},[76,77],[55,10165,564],{"className":10166},[563],[55,10168,142],{"className":10169},[141],[55,10171,10173],{"className":10172},[113],[55,10174,10176],{"className":10175,"style":2514},[117],[55,10177],{},[55,10179],{"className":10180},[563,2284],[55,10182,1143],{"className":10183},[76],[11,10185,10186,10187,10396,10397,10685,10686,10710,10711,10836,10837,10870,10871,10886,10887,10949],{},"Define ",[55,10188,10190],{"className":10189},[58],[55,10191,10193,10220,10336],{"className":10192,"ariaHidden":63},[62],[55,10194,10196,10199,10202,10205,10208,10211,10214,10217],{"className":10195},[67],[55,10197],{"className":10198,"style":796},[71],[55,10200,9863],{"className":10201,"style":9862},[76,77],[55,10203,521],{"className":10204},[509],[55,10206,78],{"className":10207},[76,77],[55,10209,564],{"className":10210},[563],[55,10212],{"className":10213,"style":263},[160],[55,10215,488],{"className":10216},[267],[55,10218],{"className":10219,"style":263},[160],[55,10221,10223,10227,10272,10275,10321,10324,10327,10330,10333],{"className":10222},[67],[55,10224],{"className":10225,"style":10226},[71],"height:1.1858em;vertical-align:-0.4358em;",[55,10228,10230,10235],{"className":10229},[2325],[55,10231,3841],{"className":10232,"style":10234},[2325,3839,10233],"small-op","position:relative;top:0em;",[55,10236,10238],{"className":10237},[104],[55,10239,10241,10263],{"className":10240},[108,109],[55,10242,10244,10260],{"className":10243},[113],[55,10245,10248],{"className":10246,"style":10247},[117],"height:0.0017em;",[55,10249,10251,10254],{"style":10250},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[55,10252],{"className":10253,"style":126},[125],[55,10255,10257],{"className":10256},[130,131,132,133],[55,10258,100],{"className":10259,"style":99},[76,77,133],[55,10261,142],{"className":10262},[141],[55,10264,10266],{"className":10265},[113],[55,10267,10270],{"className":10268,"style":10269},[117],"height:0.4358em;",[55,10271],{},[55,10273],{"className":10274,"style":161},[160],[55,10276,10278,10281],{"className":10277},[76],[55,10279,924],{"className":10280,"style":99},[76,77],[55,10282,10284],{"className":10283},[104],[55,10285,10287,10313],{"className":10286},[108,109],[55,10288,10290,10310],{"className":10289},[113],[55,10291,10293],{"className":10292,"style":296},[117],[55,10294,10295,10298],{"style":121},[55,10296],{"className":10297,"style":126},[125],[55,10299,10301],{"className":10300},[130,131,132,133],[55,10302,10304],{"className":10303},[76,133],[55,10305,10307],{"className":10306},[76,951,133],[55,10308,955],{"className":10309},[76,133],[55,10311,142],{"className":10312},[141],[55,10314,10316],{"className":10315},[113],[55,10317,10319],{"className":10318,"style":149},[117],[55,10320],{},[55,10322,521],{"className":10323},[509],[55,10325,100],{"className":10326,"style":99},[76,77],[55,10328],{"className":10329,"style":263},[160],[55,10331,2071],{"className":10332},[267],[55,10334],{"className":10335,"style":263},[160],[55,10337,10339,10342,10345,10348,10351,10354,10357,10360],{"className":10338},[67],[55,10340],{"className":10341,"style":796},[71],[55,10343,78],{"className":10344},[76,77],[55,10346,564],{"className":10347},[563],[55,10349],{"className":10350,"style":161},[160],[55,10352,2326],{"className":10353},[2325],[55,10355],{"className":10356,"style":2115},[160],[55,10358],{"className":10359,"style":161},[160],[55,10361,10363,10366,10369,10372,10375,10378,10381,10384,10387,10390,10393],{"className":10362},[502],[55,10364,521],{"className":10365,"style":511},[509,510],[55,10367,800],{"className":10368,"style":480},[76,77],[55,10370,521],{"className":10371},[509],[55,10373,78],{"className":10374},[76,77],[55,10376,156],{"className":10377},[155],[55,10379],{"className":10380,"style":161},[160],[55,10382,100],{"className":10383,"style":99},[76,77],[55,10385,564],{"className":10386},[563],[55,10388,10120],{"className":10389},[76],[55,10391,6591],{"className":10392,"style":6590},[76,77],[55,10394,564],{"className":10395,"style":511},[563,510]," and the\nnormalized distribution\n",[55,10398,10400],{"className":10399},[58],[55,10401,10403,10458,10479,10625],{"className":10402,"ariaHidden":63},[62],[55,10404,10406,10409,10443,10446,10449,10452,10455],{"className":10405},[67],[55,10407],{"className":10408,"style":796},[71],[55,10410,10412,10415],{"className":10411},[76],[55,10413,924],{"className":10414,"style":99},[76,77],[55,10416,10418],{"className":10417},[104],[55,10419,10421],{"className":10420},[108],[55,10422,10424],{"className":10423},[113],[55,10425,10428],{"className":10426,"style":10427},[117],"height:0.6887em;",[55,10429,10430,10433],{"style":543},[55,10431],{"className":10432,"style":126},[125],[55,10434,10436],{"className":10435},[130,131,132,133],[55,10437,10439],{"className":10438},[76,133],[55,10440,10442],{"className":10441},[76,133],"∗",[55,10444,521],{"className":10445},[509],[55,10447,100],{"className":10448,"style":99},[76,77],[55,10450],{"className":10451,"style":263},[160],[55,10453,2071],{"className":10454},[267],[55,10456],{"className":10457,"style":263},[160],[55,10459,10461,10464,10467,10470,10473,10476],{"className":10460},[67],[55,10462],{"className":10463,"style":796},[71],[55,10465,78],{"className":10466},[76,77],[55,10468,564],{"className":10469},[563],[55,10471],{"className":10472,"style":263},[160],[55,10474,488],{"className":10475},[267],[55,10477],{"className":10478,"style":263},[160],[55,10480,10482,10486,10564,10610,10613,10616,10619,10622],{"className":10481},[67],[55,10483],{"className":10484,"style":10485},[71],"height:1.3651em;vertical-align:-0.52em;",[55,10487,10489,10492,10561],{"className":10488},[76],[55,10490],{"className":10491},[509,2284],[55,10493,10495],{"className":10494},[2288],[55,10496,10498,10552],{"className":10497},[108,109],[55,10499,10501,10549],{"className":10500},[113],[55,10502,10504,10527,10535],{"className":10503,"style":2789},[117],[55,10505,10506,10509],{"style":2792},[55,10507],{"className":10508,"style":2305},[125],[55,10510,10512],{"className":10511},[130,131,132,133],[55,10513,10515,10518,10521,10524],{"className":10514},[76,133],[55,10516,9863],{"className":10517,"style":9862},[76,77,133],[55,10519,521],{"className":10520},[509,133],[55,10522,78],{"className":10523},[76,77,133],[55,10525,564],{"className":10526},[563,133],[55,10528,10529,10532],{"style":2484},[55,10530],{"className":10531,"style":2305},[125],[55,10533],{"className":10534,"style":2492},[2491],[55,10536,10537,10540],{"style":2815},[55,10538],{"className":10539,"style":2305},[125],[55,10541,10543],{"className":10542},[130,131,132,133],[55,10544,10546],{"className":10545},[76,133],[55,10547,137],{"className":10548},[76,133],[55,10550,142],{"className":10551},[141],[55,10553,10555],{"className":10554},[113],[55,10556,10559],{"className":10557,"style":10558},[117],"height:0.52em;",[55,10560],{},[55,10562],{"className":10563},[563,2284],[55,10565,10567,10570],{"className":10566},[76],[55,10568,924],{"className":10569,"style":99},[76,77],[55,10571,10573],{"className":10572},[104],[55,10574,10576,10602],{"className":10575},[108,109],[55,10577,10579,10599],{"className":10578},[113],[55,10580,10582],{"className":10581,"style":296},[117],[55,10583,10584,10587],{"style":121},[55,10585],{"className":10586,"style":126},[125],[55,10588,10590],{"className":10589},[130,131,132,133],[55,10591,10593],{"className":10592},[76,133],[55,10594,10596],{"className":10595},[76,951,133],[55,10597,955],{"className":10598},[76,133],[55,10600,142],{"className":10601},[141],[55,10603,10605],{"className":10604},[113],[55,10606,10608],{"className":10607,"style":149},[117],[55,10609],{},[55,10611,521],{"className":10612},[509],[55,10614,100],{"className":10615,"style":99},[76,77],[55,10617],{"className":10618,"style":263},[160],[55,10620,2071],{"className":10621},[267],[55,10623],{"className":10624,"style":263},[160],[55,10626,10628,10631,10634,10637,10640,10643,10646,10649],{"className":10627},[67],[55,10629],{"className":10630,"style":796},[71],[55,10632,78],{"className":10633},[76,77],[55,10635,564],{"className":10636},[563],[55,10638],{"className":10639,"style":161},[160],[55,10641,2326],{"className":10642},[2325],[55,10644],{"className":10645,"style":2115},[160],[55,10647],{"className":10648,"style":161},[160],[55,10650,10652,10655,10658,10661,10664,10667,10670,10673,10676,10679,10682],{"className":10651},[502],[55,10653,521],{"className":10654,"style":511},[509,510],[55,10656,800],{"className":10657,"style":480},[76,77],[55,10659,521],{"className":10660},[509],[55,10662,78],{"className":10663},[76,77],[55,10665,156],{"className":10666},[155],[55,10668],{"className":10669,"style":161},[160],[55,10671,100],{"className":10672,"style":99},[76,77],[55,10674,564],{"className":10675},[563],[55,10677,10120],{"className":10678},[76],[55,10680,6591],{"className":10681,"style":6590},[76,77],[55,10683,564],{"className":10684,"style":511},[563,510],".\nMultiplying and dividing the argument of the log by ",[55,10687,10689],{"className":10688},[58],[55,10690,10692],{"className":10691,"ariaHidden":63},[62],[55,10693,10695,10698,10701,10704,10707],{"className":10694},[67],[55,10696],{"className":10697,"style":796},[71],[55,10699,9863],{"className":10700,"style":9862},[76,77],[55,10702,521],{"className":10703},[509],[55,10705,78],{"className":10706},[76,77],[55,10708,564],{"className":10709},[563]," rewrites the objective as\n",[55,10712,10714],{"className":10713},[58],[55,10715,10717,10803],{"className":10716,"ariaHidden":63},[62],[55,10718,10720,10723,10726,10729,10732,10738,10741,10794,10797,10800],{"className":10719},[67],[55,10721],{"className":10722,"style":796},[71],[55,10724,2191],{"className":10725},[76],[55,10727,6591],{"className":10728,"style":6590},[76,77],[55,10730],{"className":10731,"style":161},[160],[55,10733,10735],{"className":10734},[2325],[55,10736,6659],{"className":10737},[76,3863],[55,10739],{"className":10740,"style":161},[160],[55,10742,10744,10747,10750,10753,10756,10759,10791],{"className":10743},[502],[55,10745,521],{"className":10746,"style":511},[509,510],[55,10748,924],{"className":10749,"style":99},[76,77],[55,10751],{"className":10752,"style":161},[160],[55,10754,6702],{"className":10755},[76],[55,10757],{"className":10758,"style":161},[160],[55,10760,10762,10765],{"className":10761},[76],[55,10763,924],{"className":10764,"style":99},[76,77],[55,10766,10768],{"className":10767},[104],[55,10769,10771],{"className":10770},[108],[55,10772,10774],{"className":10773},[113],[55,10775,10777],{"className":10776,"style":10427},[117],[55,10778,10779,10782],{"style":543},[55,10780],{"className":10781,"style":126},[125],[55,10783,10785],{"className":10784},[130,131,132,133],[55,10786,10788],{"className":10787},[76,133],[55,10789,10442],{"className":10790},[76,133],[55,10792,564],{"className":10793,"style":511},[563,510],[55,10795],{"className":10796,"style":2186},[160],[55,10798,2318],{"className":10799},[2190],[55,10801],{"className":10802,"style":2186},[160],[55,10804,10806,10809,10812,10815,10821,10824,10827,10830,10833],{"className":10805},[67],[55,10807],{"className":10808,"style":796},[71],[55,10810,6591],{"className":10811,"style":6590},[76,77],[55,10813],{"className":10814,"style":161},[160],[55,10816,10818],{"className":10817},[2325],[55,10819,3865],{"className":10820,"style":3864},[76,3863],[55,10822],{"className":10823,"style":161},[160],[55,10825,9863],{"className":10826,"style":9862},[76,77],[55,10828,521],{"className":10829},[509],[55,10831,78],{"className":10832},[76,77],[55,10834,564],{"className":10835},[563],", and since\n",[55,10838,10840],{"className":10839},[58],[55,10841,10843],{"className":10842,"ariaHidden":63},[62],[55,10844,10846,10849,10855,10858,10861,10864,10867],{"className":10845},[67],[55,10847],{"className":10848,"style":796},[71],[55,10850,10852],{"className":10851},[2325],[55,10853,3865],{"className":10854,"style":3864},[76,3863],[55,10856],{"className":10857,"style":161},[160],[55,10859,9863],{"className":10860,"style":9862},[76,77],[55,10862,521],{"className":10863},[509],[55,10865,78],{"className":10866},[76,77],[55,10868,564],{"className":10869},[563]," is independent of ",[55,10872,10874],{"className":10873},[58],[55,10875,10877],{"className":10876,"ariaHidden":63},[62],[55,10878,10880,10883],{"className":10879},[67],[55,10881],{"className":10882,"style":72},[71],[55,10884,924],{"className":10885,"style":99},[76,77],", the objective is maximized exactly when the KL is\nzero, i.e. ",[55,10888,10890],{"className":10889},[58],[55,10891,10893,10911],{"className":10892,"ariaHidden":63},[62],[55,10894,10896,10899,10902,10905,10908],{"className":10895},[67],[55,10897],{"className":10898,"style":72},[71],[55,10900,924],{"className":10901,"style":99},[76,77],[55,10903],{"className":10904,"style":263},[160],[55,10906,488],{"className":10907},[267],[55,10909],{"className":10910,"style":263},[160],[55,10912,10914,10917],{"className":10913},[67],[55,10915],{"className":10916,"style":10427},[71],[55,10918,10920,10923],{"className":10919},[76],[55,10921,924],{"className":10922,"style":99},[76,77],[55,10924,10926],{"className":10925},[104],[55,10927,10929],{"className":10928},[108],[55,10930,10932],{"className":10931},[113],[55,10933,10935],{"className":10934,"style":10427},[117],[55,10936,10937,10940],{"style":543},[55,10938],{"className":10939,"style":126},[125],[55,10941,10943],{"className":10942},[130,131,132,133],[55,10944,10946],{"className":10945},[76,133],[55,10947,10442],{"className":10948},[76,133],":",[55,10951,10953],{"className":10952},[1949],[55,10954,10956],{"className":10955},[58],[55,10957,10959,11014,11041,11182],{"className":10958,"ariaHidden":63},[62],[55,10960,10962,10965,10999,11002,11005,11008,11011],{"className":10961},[67],[55,10963],{"className":10964,"style":796},[71],[55,10966,10968,10971],{"className":10967},[76],[55,10969,924],{"className":10970,"style":99},[76,77],[55,10972,10974],{"className":10973},[104],[55,10975,10977],{"className":10976},[108],[55,10978,10980],{"className":10979},[113],[55,10981,10984],{"className":10982,"style":10983},[117],"height:0.7387em;",[55,10985,10987,10990],{"style":10986},"top:-3.113em;margin-right:0.05em;",[55,10988],{"className":10989,"style":126},[125],[55,10991,10993],{"className":10992},[130,131,132,133],[55,10994,10996],{"className":10995},[76,133],[55,10997,10442],{"className":10998},[76,133],[55,11000,521],{"className":11001},[509],[55,11003,100],{"className":11004,"style":99},[76,77],[55,11006],{"className":11007,"style":263},[160],[55,11009,2071],{"className":11010},[267],[55,11012],{"className":11013,"style":263},[160],[55,11015,11017,11020,11023,11026,11029,11032,11035,11038],{"className":11016},[67],[55,11018],{"className":11019,"style":796},[71],[55,11021,78],{"className":11022},[76,77],[55,11024,564],{"className":11025},[563],[55,11027],{"className":11028,"style":263},[160],[55,11030],{"className":11031,"style":263},[160],[55,11033,488],{"className":11034},[267],[55,11036],{"className":11037,"style":263},[160],[55,11039],{"className":11040,"style":263},[160],[55,11042,11044,11047,11118,11121,11167,11170,11173,11176,11179],{"className":11043},[67],[55,11045],{"className":11046,"style":2277},[71],[55,11048,11050,11053,11115],{"className":11049},[76],[55,11051],{"className":11052},[509,2284],[55,11054,11056],{"className":11055},[2288],[55,11057,11059,11107],{"className":11058},[108,109],[55,11060,11062,11104],{"className":11061},[113],[55,11063,11065,11085,11093],{"className":11064,"style":2298},[117],[55,11066,11067,11070],{"style":2301},[55,11068],{"className":11069,"style":2305},[125],[55,11071,11073,11076,11079,11082],{"className":11072},[76],[55,11074,9863],{"className":11075,"style":9862},[76,77],[55,11077,521],{"className":11078},[509],[55,11080,78],{"className":11081},[76,77],[55,11083,564],{"className":11084},[563],[55,11086,11087,11090],{"style":2484},[55,11088],{"className":11089,"style":2305},[125],[55,11091],{"className":11092,"style":2492},[2491],[55,11094,11095,11098],{"style":2495},[55,11096],{"className":11097,"style":2305},[125],[55,11099,11101],{"className":11100},[76],[55,11102,137],{"className":11103},[76],[55,11105,142],{"className":11106},[141],[55,11108,11110],{"className":11109},[113],[55,11111,11113],{"className":11112,"style":2514},[117],[55,11114],{},[55,11116],{"className":11117},[563,2284],[55,11119],{"className":11120,"style":161},[160],[55,11122,11124,11127],{"className":11123},[76],[55,11125,924],{"className":11126,"style":99},[76,77],[55,11128,11130],{"className":11129},[104],[55,11131,11133,11159],{"className":11132},[108,109],[55,11134,11136,11156],{"className":11135},[113],[55,11137,11139],{"className":11138,"style":296},[117],[55,11140,11141,11144],{"style":121},[55,11142],{"className":11143,"style":126},[125],[55,11145,11147],{"className":11146},[130,131,132,133],[55,11148,11150],{"className":11149},[76,133],[55,11151,11153],{"className":11152},[76,951,133],[55,11154,955],{"className":11155},[76,133],[55,11157,142],{"className":11158},[141],[55,11160,11162],{"className":11161},[113],[55,11163,11165],{"className":11164,"style":149},[117],[55,11166],{},[55,11168,521],{"className":11169},[509],[55,11171,100],{"className":11172,"style":99},[76,77],[55,11174],{"className":11175,"style":263},[160],[55,11177,2071],{"className":11178},[267],[55,11180],{"className":11181,"style":263},[160],[55,11183,11185,11189,11192,11195,11198,11201,11204,11207,11210,11318,11321],{"className":11184},[67],[55,11186],{"className":11187,"style":11188},[71],"height:1.8em;vertical-align:-0.65em;",[55,11190,78],{"className":11191},[76,77],[55,11193,564],{"className":11194},[563],[55,11196],{"className":11197,"style":161},[160],[55,11199],{"className":11200,"style":161},[160],[55,11202,2326],{"className":11203},[2325],[55,11205],{"className":11206,"style":2115},[160],[55,11208],{"className":11209,"style":161},[160],[55,11211,11213,11219,11288,11291,11294,11297,11300,11303,11306,11309,11312],{"className":11212},[502],[55,11214,11216],{"className":11215,"style":511},[509,510],[55,11217,521],{"className":11218},[515,516],[55,11220,11222,11225,11285],{"className":11221},[76],[55,11223],{"className":11224},[509,2284],[55,11226,11228],{"className":11227},[2288],[55,11229,11231,11276],{"className":11230},[108,109],[55,11232,11234,11273],{"className":11233},[113],[55,11235,11237,11251,11259],{"className":11236,"style":2789},[117],[55,11238,11239,11242],{"style":2792},[55,11240],{"className":11241,"style":2305},[125],[55,11243,11245],{"className":11244},[130,131,132,133],[55,11246,11248],{"className":11247},[76,133],[55,11249,6591],{"className":11250,"style":6590},[76,77,133],[55,11252,11253,11256],{"style":2484},[55,11254],{"className":11255,"style":2305},[125],[55,11257],{"className":11258,"style":2492},[2491],[55,11260,11261,11264],{"style":2815},[55,11262],{"className":11263,"style":2305},[125],[55,11265,11267],{"className":11266},[130,131,132,133],[55,11268,11270],{"className":11269},[76,133],[55,11271,137],{"className":11272},[76,133],[55,11274,142],{"className":11275},[141],[55,11277,11279],{"className":11278},[113],[55,11280,11283],{"className":11281,"style":11282},[117],"height:0.4811em;",[55,11284],{},[55,11286],{"className":11287},[563,2284],[55,11289],{"className":11290,"style":161},[160],[55,11292,800],{"className":11293,"style":480},[76,77],[55,11295,521],{"className":11296},[509],[55,11298,78],{"className":11299},[76,77],[55,11301,156],{"className":11302},[155],[55,11304],{"className":11305,"style":161},[160],[55,11307,100],{"className":11308,"style":99},[76,77],[55,11310,564],{"className":11311},[563],[55,11313,11315],{"className":11314,"style":511},[563,510],[55,11316,564],{"className":11317},[515,516],[55,11319],{"className":11320,"style":161},[160],[55,11322,1143],{"className":11323},[76],[11,11325,11326,11327,11330],{},"This is the closed-form optimum. Now ",[880,11328,11329],{},"invert"," it: solve for the reward by taking the\nlog of both sides,",[55,11332,11334],{"className":11333},[1949],[55,11335,11337],{"className":11336},[58],[55,11338,11340,11382,11594],{"className":11339,"ariaHidden":63},[62],[55,11341,11343,11346,11349,11352,11355,11358,11361,11364,11367,11370,11373,11376,11379],{"className":11342},[67],[55,11344],{"className":11345,"style":796},[71],[55,11347,800],{"className":11348,"style":480},[76,77],[55,11350,521],{"className":11351},[509],[55,11353,78],{"className":11354},[76,77],[55,11356,156],{"className":11357},[155],[55,11359],{"className":11360,"style":161},[160],[55,11362,100],{"className":11363,"style":99},[76,77],[55,11365,564],{"className":11366},[563],[55,11368],{"className":11369,"style":263},[160],[55,11371],{"className":11372,"style":263},[160],[55,11374,488],{"className":11375},[267],[55,11377],{"className":11378,"style":263},[160],[55,11380],{"className":11381,"style":263},[160],[55,11383,11385,11388,11391,11394,11400,11403,11579,11582,11585,11588,11591],{"className":11384},[67],[55,11386],{"className":11387,"style":9963},[71],[55,11389,6591],{"className":11390,"style":6590},[76,77],[55,11392],{"className":11393,"style":161},[160],[55,11395,11397],{"className":11396},[2325],[55,11398,3865],{"className":11399,"style":3864},[76,3863],[55,11401],{"className":11402,"style":161},[160],[55,11404,11406,11409,11576],{"className":11405},[76],[55,11407],{"className":11408},[509,2284],[55,11410,11412],{"className":11411},[2288],[55,11413,11415,11568],{"className":11414},[108,109],[55,11416,11418,11565],{"className":11417},[113],[55,11419,11421,11496,11504],{"className":11420,"style":7515},[117],[55,11422,11423,11426],{"style":2301},[55,11424],{"className":11425,"style":2305},[125],[55,11427,11429,11475,11478,11481,11484,11487,11490,11493],{"className":11428},[76],[55,11430,11432,11435],{"className":11431},[76],[55,11433,924],{"className":11434,"style":99},[76,77],[55,11436,11438],{"className":11437},[104],[55,11439,11441,11467],{"className":11440},[108,109],[55,11442,11444,11464],{"className":11443},[113],[55,11445,11447],{"className":11446,"style":296},[117],[55,11448,11449,11452],{"style":121},[55,11450],{"className":11451,"style":126},[125],[55,11453,11455],{"className":11454},[130,131,132,133],[55,11456,11458],{"className":11457},[76,133],[55,11459,11461],{"className":11460},[76,951,133],[55,11462,955],{"className":11463},[76,133],[55,11465,142],{"className":11466},[141],[55,11468,11470],{"className":11469},[113],[55,11471,11473],{"className":11472,"style":149},[117],[55,11474],{},[55,11476,521],{"className":11477},[509],[55,11479,100],{"className":11480,"style":99},[76,77],[55,11482],{"className":11483,"style":263},[160],[55,11485,2071],{"className":11486},[267],[55,11488],{"className":11489,"style":263},[160],[55,11491,78],{"className":11492},[76,77],[55,11494,564],{"className":11495},[563],[55,11497,11498,11501],{"style":2484},[55,11499],{"className":11500,"style":2305},[125],[55,11502],{"className":11503,"style":2492},[2491],[55,11505,11506,11509],{"style":2495},[55,11507],{"className":11508,"style":2305},[125],[55,11510,11512,11544,11547,11550,11553,11556,11559,11562],{"className":11511},[76],[55,11513,11515,11518],{"className":11514},[76],[55,11516,924],{"className":11517,"style":99},[76,77],[55,11519,11521],{"className":11520},[104],[55,11522,11524],{"className":11523},[108],[55,11525,11527],{"className":11526},[113],[55,11528,11530],{"className":11529,"style":10427},[117],[55,11531,11532,11535],{"style":543},[55,11533],{"className":11534,"style":126},[125],[55,11536,11538],{"className":11537},[130,131,132,133],[55,11539,11541],{"className":11540},[76,133],[55,11542,10442],{"className":11543},[76,133],[55,11545,521],{"className":11546},[509],[55,11548,100],{"className":11549,"style":99},[76,77],[55,11551],{"className":11552,"style":263},[160],[55,11554,2071],{"className":11555},[267],[55,11557],{"className":11558,"style":263},[160],[55,11560,78],{"className":11561},[76,77],[55,11563,564],{"className":11564},[563],[55,11566,142],{"className":11567},[141],[55,11569,11571],{"className":11570},[113],[55,11572,11574],{"className":11573,"style":2514},[117],[55,11575],{},[55,11577],{"className":11578},[563,2284],[55,11580],{"className":11581,"style":263},[160],[55,11583],{"className":11584,"style":2186},[160],[55,11586,2318],{"className":11587},[2190],[55,11589],{"className":11590,"style":263},[160],[55,11592],{"className":11593,"style":2186},[160],[55,11595,11597,11600,11603,11606,11612,11615,11618,11621,11624,11627],{"className":11596},[67],[55,11598],{"className":11599,"style":796},[71],[55,11601,6591],{"className":11602,"style":6590},[76,77],[55,11604],{"className":11605,"style":161},[160],[55,11607,11609],{"className":11608},[2325],[55,11610,3865],{"className":11611,"style":3864},[76,3863],[55,11613],{"className":11614,"style":161},[160],[55,11616,9863],{"className":11617,"style":9862},[76,77],[55,11619,521],{"className":11620},[509],[55,11622,78],{"className":11623},[76,77],[55,11625,564],{"className":11626},[563],[55,11628,1143],{"className":11629},[76],[11,11631,11632,11633,11922,11923,11962,11963,11966],{},"Substitute this reward into the Bradley-Terry probability\n",[55,11634,11636],{"className":11635},[58],[55,11637,11639,11700,11755,11776],{"className":11638,"ariaHidden":63},[62],[55,11640,11642,11645,11648,11651,11691,11694,11697],{"className":11641},[67],[55,11643],{"className":11644,"style":796},[71],[55,11646,1966],{"className":11647,"style":1965},[76,77],[55,11649,521],{"className":11650},[509],[55,11652,11654,11657],{"className":11653},[76],[55,11655,100],{"className":11656,"style":99},[76,77],[55,11658,11660],{"className":11659},[104],[55,11661,11663,11683],{"className":11662},[108,109],[55,11664,11666,11680],{"className":11665},[113],[55,11667,11669],{"className":11668,"style":235},[117],[55,11670,11671,11674],{"style":121},[55,11672],{"className":11673,"style":126},[125],[55,11675,11677],{"className":11676},[130,131,132,133],[55,11678,248],{"className":11679,"style":247},[76,77,133],[55,11681,142],{"className":11682},[141],[55,11684,11686],{"className":11685},[113],[55,11687,11689],{"className":11688,"style":149},[117],[55,11690],{},[55,11692],{"className":11693,"style":263},[160],[55,11695,268],{"className":11696},[267],[55,11698],{"className":11699,"style":263},[160],[55,11701,11703,11706,11746,11749,11752],{"className":11702},[67],[55,11704],{"className":11705,"style":796},[71],[55,11707,11709,11712],{"className":11708},[76],[55,11710,100],{"className":11711,"style":99},[76,77],[55,11713,11715],{"className":11714},[104],[55,11716,11718,11738],{"className":11717},[108,109],[55,11719,11721,11735],{"className":11720},[113],[55,11722,11724],{"className":11723,"style":296},[117],[55,11725,11726,11729],{"style":121},[55,11727],{"className":11728,"style":126},[125],[55,11730,11732],{"className":11731},[130,131,132,133],[55,11733,309],{"className":11734,"style":308},[76,77,133],[55,11736,142],{"className":11737},[141],[55,11739,11741],{"className":11740},[113],[55,11742,11744],{"className":11743,"style":149},[117],[55,11745],{},[55,11747],{"className":11748,"style":263},[160],[55,11750,2071],{"className":11751},[267],[55,11753],{"className":11754,"style":263},[160],[55,11756,11758,11761,11764,11767,11770,11773],{"className":11757},[67],[55,11759],{"className":11760,"style":796},[71],[55,11762,78],{"className":11763},[76,77],[55,11765,564],{"className":11766},[563],[55,11768],{"className":11769,"style":263},[160],[55,11771,488],{"className":11772},[267],[55,11774],{"className":11775,"style":263},[160],[55,11777,11779,11782,11785,11788],{"className":11778},[67],[55,11780],{"className":11781,"style":796},[71],[55,11783,2111],{"className":11784,"style":99},[76,77],[55,11786],{"className":11787,"style":161},[160],[55,11789,11791,11794,11797,11800,11803,11806,11809,11849,11852,11855,11858,11861,11864,11867,11870,11873,11876,11916,11919],{"className":11790},[502],[55,11792,521],{"className":11793,"style":511},[509,510],[55,11795,800],{"className":11796,"style":480},[76,77],[55,11798,521],{"className":11799},[509],[55,11801,78],{"className":11802},[76,77],[55,11804,156],{"className":11805},[155],[55,11807],{"className":11808,"style":161},[160],[55,11810,11812,11815],{"className":11811},[76],[55,11813,100],{"className":11814,"style":99},[76,77],[55,11816,11818],{"className":11817},[104],[55,11819,11821,11841],{"className":11820},[108,109],[55,11822,11824,11838],{"className":11823},[113],[55,11825,11827],{"className":11826,"style":235},[117],[55,11828,11829,11832],{"style":121},[55,11830],{"className":11831,"style":126},[125],[55,11833,11835],{"className":11834},[130,131,132,133],[55,11836,248],{"className":11837,"style":247},[76,77,133],[55,11839,142],{"className":11840},[141],[55,11842,11844],{"className":11843},[113],[55,11845,11847],{"className":11846,"style":149},[117],[55,11848],{},[55,11850,564],{"className":11851},[563],[55,11853],{"className":11854,"style":2186},[160],[55,11856,2191],{"className":11857},[2190],[55,11859],{"className":11860,"style":2186},[160],[55,11862,800],{"className":11863,"style":480},[76,77],[55,11865,521],{"className":11866},[509],[55,11868,78],{"className":11869},[76,77],[55,11871,156],{"className":11872},[155],[55,11874],{"className":11875,"style":161},[160],[55,11877,11879,11882],{"className":11878},[76],[55,11880,100],{"className":11881,"style":99},[76,77],[55,11883,11885],{"className":11884},[104],[55,11886,11888,11908],{"className":11887},[108,109],[55,11889,11891,11905],{"className":11890},[113],[55,11892,11894],{"className":11893,"style":296},[117],[55,11895,11896,11899],{"style":121},[55,11897],{"className":11898,"style":126},[125],[55,11900,11902],{"className":11901},[130,131,132,133],[55,11903,309],{"className":11904,"style":308},[76,77,133],[55,11906,142],{"className":11907},[141],[55,11909,11911],{"className":11910},[113],[55,11912,11914],{"className":11913,"style":149},[117],[55,11915],{},[55,11917,564],{"className":11918},[563],[55,11920,564],{"className":11921,"style":511},[563,510],". The intractable\npartition term ",[55,11924,11926],{"className":11925},[58],[55,11927,11929],{"className":11928,"ariaHidden":63},[62],[55,11930,11932,11935,11938,11941,11947,11950,11953,11956,11959],{"className":11931},[67],[55,11933],{"className":11934,"style":796},[71],[55,11936,6591],{"className":11937,"style":6590},[76,77],[55,11939],{"className":11940,"style":161},[160],[55,11942,11944],{"className":11943},[2325],[55,11945,3865],{"className":11946,"style":3864},[76,3863],[55,11948],{"className":11949,"style":161},[160],[55,11951,9863],{"className":11952,"style":9862},[76,77],[55,11954,521],{"className":11955},[509],[55,11957,78],{"className":11958},[76,77],[55,11960,564],{"className":11961},[563]," appears in ",[880,11964,11965],{},"both"," rewards and cancels in the\ndifference, leaving",[55,11968,11970],{"className":11969},[1949],[55,11971,11973],{"className":11972},[58],[55,11974,11976,12049,12122,12402],{"className":11975,"ariaHidden":63},[62],[55,11977,11979,11982,11985,11988,11991,11994,11997,12037,12040,12043,12046],{"className":11978},[67],[55,11980],{"className":11981,"style":796},[71],[55,11983,800],{"className":11984,"style":480},[76,77],[55,11986,521],{"className":11987},[509],[55,11989,78],{"className":11990},[76,77],[55,11992,156],{"className":11993},[155],[55,11995],{"className":11996,"style":161},[160],[55,11998,12000,12003],{"className":11999},[76],[55,12001,100],{"className":12002,"style":99},[76,77],[55,12004,12006],{"className":12005},[104],[55,12007,12009,12029],{"className":12008},[108,109],[55,12010,12012,12026],{"className":12011},[113],[55,12013,12015],{"className":12014,"style":235},[117],[55,12016,12017,12020],{"style":121},[55,12018],{"className":12019,"style":126},[125],[55,12021,12023],{"className":12022},[130,131,132,133],[55,12024,248],{"className":12025,"style":247},[76,77,133],[55,12027,142],{"className":12028},[141],[55,12030,12032],{"className":12031},[113],[55,12033,12035],{"className":12034,"style":149},[117],[55,12036],{},[55,12038,564],{"className":12039},[563],[55,12041],{"className":12042,"style":2186},[160],[55,12044,2191],{"className":12045},[2190],[55,12047],{"className":12048,"style":2186},[160],[55,12050,12052,12055,12058,12061,12064,12067,12070,12110,12113,12116,12119],{"className":12051},[67],[55,12053],{"className":12054,"style":796},[71],[55,12056,800],{"className":12057,"style":480},[76,77],[55,12059,521],{"className":12060},[509],[55,12062,78],{"className":12063},[76,77],[55,12065,156],{"className":12066},[155],[55,12068],{"className":12069,"style":161},[160],[55,12071,12073,12076],{"className":12072},[76],[55,12074,100],{"className":12075,"style":99},[76,77],[55,12077,12079],{"className":12078},[104],[55,12080,12082,12102],{"className":12081},[108,109],[55,12083,12085,12099],{"className":12084},[113],[55,12086,12088],{"className":12087,"style":296},[117],[55,12089,12090,12093],{"style":121},[55,12091],{"className":12092,"style":126},[125],[55,12094,12096],{"className":12095},[130,131,132,133],[55,12097,309],{"className":12098,"style":308},[76,77,133],[55,12100,142],{"className":12101},[141],[55,12103,12105],{"className":12104},[113],[55,12106,12108],{"className":12107,"style":149},[117],[55,12109],{},[55,12111,564],{"className":12112},[563],[55,12114],{"className":12115,"style":263},[160],[55,12117,488],{"className":12118},[267],[55,12120],{"className":12121,"style":263},[160],[55,12123,12125,12128,12131,12134,12140,12143,12393,12396,12399],{"className":12124},[67],[55,12126],{"className":12127,"style":9963},[71],[55,12129,6591],{"className":12130,"style":6590},[76,77],[55,12132],{"className":12133,"style":161},[160],[55,12135,12137],{"className":12136},[2325],[55,12138,3865],{"className":12139,"style":3864},[76,3863],[55,12141],{"className":12142,"style":161},[160],[55,12144,12146,12149,12390],{"className":12145},[76],[55,12147],{"className":12148},[509,2284],[55,12150,12152],{"className":12151},[2288],[55,12153,12155,12382],{"className":12154},[108,109],[55,12156,12158,12379],{"className":12157},[113],[55,12159,12161,12273,12281],{"className":12160,"style":7515},[117],[55,12162,12163,12166],{"style":2301},[55,12164],{"className":12165,"style":2305},[125],[55,12167,12169,12215,12218,12258,12261,12264,12267,12270],{"className":12168},[76],[55,12170,12172,12175],{"className":12171},[76],[55,12173,924],{"className":12174,"style":99},[76,77],[55,12176,12178],{"className":12177},[104],[55,12179,12181,12207],{"className":12180},[108,109],[55,12182,12184,12204],{"className":12183},[113],[55,12185,12187],{"className":12186,"style":296},[117],[55,12188,12189,12192],{"style":121},[55,12190],{"className":12191,"style":126},[125],[55,12193,12195],{"className":12194},[130,131,132,133],[55,12196,12198],{"className":12197},[76,133],[55,12199,12201],{"className":12200},[76,951,133],[55,12202,955],{"className":12203},[76,133],[55,12205,142],{"className":12206},[141],[55,12208,12210],{"className":12209},[113],[55,12211,12213],{"className":12212,"style":149},[117],[55,12214],{},[55,12216,521],{"className":12217},[509],[55,12219,12221,12224],{"className":12220},[76],[55,12222,100],{"className":12223,"style":99},[76,77],[55,12225,12227],{"className":12226},[104],[55,12228,12230,12250],{"className":12229},[108,109],[55,12231,12233,12247],{"className":12232},[113],[55,12234,12236],{"className":12235,"style":235},[117],[55,12237,12238,12241],{"style":121},[55,12239],{"className":12240,"style":126},[125],[55,12242,12244],{"className":12243},[130,131,132,133],[55,12245,248],{"className":12246,"style":247},[76,77,133],[55,12248,142],{"className":12249},[141],[55,12251,12253],{"className":12252},[113],[55,12254,12256],{"className":12255,"style":149},[117],[55,12257],{},[55,12259],{"className":12260,"style":263},[160],[55,12262,2071],{"className":12263},[267],[55,12265],{"className":12266,"style":263},[160],[55,12268,78],{"className":12269},[76,77],[55,12271,564],{"className":12272},[563],[55,12274,12275,12278],{"style":2484},[55,12276],{"className":12277,"style":2305},[125],[55,12279],{"className":12280,"style":2492},[2491],[55,12282,12283,12286],{"style":2495},[55,12284],{"className":12285,"style":2305},[125],[55,12287,12289,12321,12324,12364,12367,12370,12373,12376],{"className":12288},[76],[55,12290,12292,12295],{"className":12291},[76],[55,12293,924],{"className":12294,"style":99},[76,77],[55,12296,12298],{"className":12297},[104],[55,12299,12301],{"className":12300},[108],[55,12302,12304],{"className":12303},[113],[55,12305,12307],{"className":12306,"style":10427},[117],[55,12308,12309,12312],{"style":543},[55,12310],{"className":12311,"style":126},[125],[55,12313,12315],{"className":12314},[130,131,132,133],[55,12316,12318],{"className":12317},[76,133],[55,12319,10442],{"className":12320},[76,133],[55,12322,521],{"className":12323},[509],[55,12325,12327,12330],{"className":12326},[76],[55,12328,100],{"className":12329,"style":99},[76,77],[55,12331,12333],{"className":12332},[104],[55,12334,12336,12356],{"className":12335},[108,109],[55,12337,12339,12353],{"className":12338},[113],[55,12340,12342],{"className":12341,"style":235},[117],[55,12343,12344,12347],{"style":121},[55,12345],{"className":12346,"style":126},[125],[55,12348,12350],{"className":12349},[130,131,132,133],[55,12351,248],{"className":12352,"style":247},[76,77,133],[55,12354,142],{"className":12355},[141],[55,12357,12359],{"className":12358},[113],[55,12360,12362],{"className":12361,"style":149},[117],[55,12363],{},[55,12365],{"className":12366,"style":263},[160],[55,12368,2071],{"className":12369},[267],[55,12371],{"className":12372,"style":263},[160],[55,12374,78],{"className":12375},[76,77],[55,12377,564],{"className":12378},[563],[55,12380,142],{"className":12381},[141],[55,12383,12385],{"className":12384},[113],[55,12386,12388],{"className":12387,"style":2514},[117],[55,12389],{},[55,12391],{"className":12392},[563,2284],[55,12394],{"className":12395,"style":2186},[160],[55,12397,2191],{"className":12398},[2190],[55,12400],{"className":12401,"style":2186},[160],[55,12403,12405,12408,12411,12414,12420,12423,12673],{"className":12404},[67],[55,12406],{"className":12407,"style":9963},[71],[55,12409,6591],{"className":12410,"style":6590},[76,77],[55,12412],{"className":12413,"style":161},[160],[55,12415,12417],{"className":12416},[2325],[55,12418,3865],{"className":12419,"style":3864},[76,3863],[55,12421],{"className":12422,"style":161},[160],[55,12424,12426,12429,12670],{"className":12425},[76],[55,12427],{"className":12428},[509,2284],[55,12430,12432],{"className":12431},[2288],[55,12433,12435,12662],{"className":12434},[108,109],[55,12436,12438,12659],{"className":12437},[113],[55,12439,12441,12553,12561],{"className":12440,"style":7515},[117],[55,12442,12443,12446],{"style":2301},[55,12444],{"className":12445,"style":2305},[125],[55,12447,12449,12495,12498,12538,12541,12544,12547,12550],{"className":12448},[76],[55,12450,12452,12455],{"className":12451},[76],[55,12453,924],{"className":12454,"style":99},[76,77],[55,12456,12458],{"className":12457},[104],[55,12459,12461,12487],{"className":12460},[108,109],[55,12462,12464,12484],{"className":12463},[113],[55,12465,12467],{"className":12466,"style":296},[117],[55,12468,12469,12472],{"style":121},[55,12470],{"className":12471,"style":126},[125],[55,12473,12475],{"className":12474},[130,131,132,133],[55,12476,12478],{"className":12477},[76,133],[55,12479,12481],{"className":12480},[76,951,133],[55,12482,955],{"className":12483},[76,133],[55,12485,142],{"className":12486},[141],[55,12488,12490],{"className":12489},[113],[55,12491,12493],{"className":12492,"style":149},[117],[55,12494],{},[55,12496,521],{"className":12497},[509],[55,12499,12501,12504],{"className":12500},[76],[55,12502,100],{"className":12503,"style":99},[76,77],[55,12505,12507],{"className":12506},[104],[55,12508,12510,12530],{"className":12509},[108,109],[55,12511,12513,12527],{"className":12512},[113],[55,12514,12516],{"className":12515,"style":296},[117],[55,12517,12518,12521],{"style":121},[55,12519],{"className":12520,"style":126},[125],[55,12522,12524],{"className":12523},[130,131,132,133],[55,12525,309],{"className":12526,"style":308},[76,77,133],[55,12528,142],{"className":12529},[141],[55,12531,12533],{"className":12532},[113],[55,12534,12536],{"className":12535,"style":149},[117],[55,12537],{},[55,12539],{"className":12540,"style":263},[160],[55,12542,2071],{"className":12543},[267],[55,12545],{"className":12546,"style":263},[160],[55,12548,78],{"className":12549},[76,77],[55,12551,564],{"className":12552},[563],[55,12554,12555,12558],{"style":2484},[55,12556],{"className":12557,"style":2305},[125],[55,12559],{"className":12560,"style":2492},[2491],[55,12562,12563,12566],{"style":2495},[55,12564],{"className":12565,"style":2305},[125],[55,12567,12569,12601,12604,12644,12647,12650,12653,12656],{"className":12568},[76],[55,12570,12572,12575],{"className":12571},[76],[55,12573,924],{"className":12574,"style":99},[76,77],[55,12576,12578],{"className":12577},[104],[55,12579,12581],{"className":12580},[108],[55,12582,12584],{"className":12583},[113],[55,12585,12587],{"className":12586,"style":10427},[117],[55,12588,12589,12592],{"style":543},[55,12590],{"className":12591,"style":126},[125],[55,12593,12595],{"className":12594},[130,131,132,133],[55,12596,12598],{"className":12597},[76,133],[55,12599,10442],{"className":12600},[76,133],[55,12602,521],{"className":12603},[509],[55,12605,12607,12610],{"className":12606},[76],[55,12608,100],{"className":12609,"style":99},[76,77],[55,12611,12613],{"className":12612},[104],[55,12614,12616,12636],{"className":12615},[108,109],[55,12617,12619,12633],{"className":12618},[113],[55,12620,12622],{"className":12621,"style":296},[117],[55,12623,12624,12627],{"style":121},[55,12625],{"className":12626,"style":126},[125],[55,12628,12630],{"className":12629},[130,131,132,133],[55,12631,309],{"className":12632,"style":308},[76,77,133],[55,12634,142],{"className":12635},[141],[55,12637,12639],{"className":12638},[113],[55,12640,12642],{"className":12641,"style":149},[117],[55,12643],{},[55,12645],{"className":12646,"style":263},[160],[55,12648,2071],{"className":12649},[267],[55,12651],{"className":12652,"style":263},[160],[55,12654,78],{"className":12655},[76,77],[55,12657,564],{"className":12658},[563],[55,12660,142],{"className":12661},[141],[55,12663,12665],{"className":12664},[113],[55,12666,12668],{"className":12667,"style":2514},[117],[55,12669],{},[55,12671],{"className":12672},[563,2284],[55,12674,1143],{"className":12675},[76],[11,12677,12678,12679,12723,12724,12776,12777,12844,12845],{},"Maximizing the Bradley-Terry likelihood of the observed preferences in ",[55,12680,12682],{"className":12681},[58],[55,12683,12685],{"className":12684,"ariaHidden":63},[62],[55,12686,12688,12691],{"className":12687},[67],[55,12689],{"className":12690,"style":10427},[71],[55,12692,12694,12697],{"className":12693},[76],[55,12695,924],{"className":12696,"style":99},[76,77],[55,12698,12700],{"className":12699},[104],[55,12701,12703],{"className":12702},[108],[55,12704,12706],{"className":12705},[113],[55,12707,12709],{"className":12708,"style":10427},[117],[55,12710,12711,12714],{"style":543},[55,12712],{"className":12713,"style":126},[125],[55,12715,12717],{"className":12716},[130,131,132,133],[55,12718,12720],{"className":12719},[76,133],[55,12721,10442],{"className":12722},[76,133],",\nparametrized as ",[55,12725,12727],{"className":12726},[58],[55,12728,12730],{"className":12729,"ariaHidden":63},[62],[55,12731,12733,12736],{"className":12732},[67],[55,12734],{"className":12735,"style":917},[71],[55,12737,12739,12742],{"className":12738},[76],[55,12740,924],{"className":12741,"style":99},[76,77],[55,12743,12745],{"className":12744},[104],[55,12746,12748,12768],{"className":12747},[108,109],[55,12749,12751,12765],{"className":12750},[113],[55,12752,12754],{"className":12753,"style":296},[117],[55,12755,12756,12759],{"style":121},[55,12757],{"className":12758,"style":126},[125],[55,12760,12762],{"className":12761},[130,131,132,133],[55,12763,1532],{"className":12764,"style":480},[76,77,133],[55,12766,142],{"className":12767},[141],[55,12769,12771],{"className":12770},[113],[55,12772,12774],{"className":12773,"style":149},[117],[55,12775],{},", is the same as minimizing the negative log-sigmoid of\nthis difference, which is ",[55,12778,12780],{"className":12779},[58],[55,12781,12783],{"className":12782,"ariaHidden":63},[62],[55,12784,12786,12789,12835,12838,12841],{"className":12785},[67],[55,12787],{"className":12788,"style":796},[71],[55,12790,12792,12795],{"className":12791},[76],[55,12793,4018],{"className":12794},[76,479],[55,12796,12798],{"className":12797},[104],[55,12799,12801,12827],{"className":12800},[108,109],[55,12802,12804,12824],{"className":12803},[113],[55,12805,12807],{"className":12806,"style":4031},[117],[55,12808,12809,12812],{"style":3342},[55,12810],{"className":12811,"style":126},[125],[55,12813,12815],{"className":12814},[130,131,132,133],[55,12816,12818],{"className":12817},[76,133],[55,12819,12821],{"className":12820},[76,951,133],[55,12822,8784],{"className":12823},[76,133],[55,12825,142],{"className":12826},[141],[55,12828,12830],{"className":12829},[113],[55,12831,12833],{"className":12832,"style":149},[117],[55,12834],{},[55,12836,521],{"className":12837},[509],[55,12839,1532],{"className":12840,"style":480},[76,77],[55,12842,564],{"className":12843},[563],". ",[55,12846,12848],{"className":12847},[58],[55,12849,12851],{"className":12850,"ariaHidden":63},[62],[55,12852,12854,12858],{"className":12853},[67],[55,12855],{"className":12856,"style":12857},[71],"height:0.675em;",[55,12859,12863],{"className":12860},[12861,12862],"enclosing","qed",[55,12864,12867],{"className":12865},[76,12866],"amsrm","□",[11,12869,12870],{},"The derivation is a change of variable: the reward and the optimal policy are two\nencodings of the same object, and DPO works in policy space so it never has to fit the\nreward at all. The implicit reward DPO assigns to a response is the quantity it pushes\nup or down.",[1145,12872],{"hash":12873},"1efe3ce7d256e884f632abec1c60542ed0ece7a8516f411b5b7d5443d75b7f80",[44,12875,12876],{"type":46},[11,12877,12878,12881,12882,12934,12935,13210,13211,13226],{},[50,12879,12880],{},"Definition (Implicit reward)."," DPO never instantiates ",[55,12883,12885],{"className":12884},[58],[55,12886,12888],{"className":12887,"ariaHidden":63},[62],[55,12889,12891,12894],{"className":12890},[67],[55,12892],{"className":12893,"style":983},[71],[55,12895,12897,12900],{"className":12896},[76],[55,12898,800],{"className":12899,"style":480},[76,77],[55,12901,12903],{"className":12902},[104],[55,12904,12906,12926],{"className":12905},[108,109],[55,12907,12909,12923],{"className":12908},[113],[55,12910,12912],{"className":12911,"style":296},[117],[55,12913,12914,12917],{"style":1004},[55,12915],{"className":12916,"style":126},[125],[55,12918,12920],{"className":12919},[130,131,132,133],[55,12921,1014],{"className":12922},[76,77,133],[55,12924,142],{"className":12925},[141],[55,12927,12929],{"className":12928},[113],[55,12930,12932],{"className":12931,"style":1024},[117],[55,12933],{},", but the optimum it\ntargets corresponds to the implicit reward\n",[55,12936,12938],{"className":12937},[58],[55,12939,12941,13005],{"className":12940,"ariaHidden":63},[62],[55,12942,12944,12947,12978,12981,12984,12987,12990,12993,12996,12999,13002],{"className":12943},[67],[55,12945],{"className":12946,"style":796},[71],[55,12948,12950],{"className":12949},[76,7186],[55,12951,12953],{"className":12952},[108],[55,12954,12956],{"className":12955},[113],[55,12957,12959,12967],{"className":12958,"style":458},[117],[55,12960,12961,12964],{"style":5206},[55,12962],{"className":12963,"style":2305},[125],[55,12965,800],{"className":12966,"style":480},[76,77],[55,12968,12969,12972],{"style":5206},[55,12970],{"className":12971,"style":2305},[125],[55,12973,12975],{"className":12974,"style":7215},[7214],[55,12976,8138],{"className":12977},[76],[55,12979,521],{"className":12980},[509],[55,12982,78],{"className":12983},[76,77],[55,12985,156],{"className":12986},[155],[55,12988],{"className":12989,"style":161},[160],[55,12991,100],{"className":12992,"style":99},[76,77],[55,12994,564],{"className":12995},[563],[55,12997],{"className":12998,"style":263},[160],[55,13000,488],{"className":13001},[267],[55,13003],{"className":13004,"style":263},[160],[55,13006,13008,13011,13014,13017,13023,13026],{"className":13007},[67],[55,13009],{"className":13010,"style":9963},[71],[55,13012,6591],{"className":13013,"style":6590},[76,77],[55,13015],{"className":13016,"style":161},[160],[55,13018,13020],{"className":13019},[2325],[55,13021,3865],{"className":13022,"style":3864},[76,3863],[55,13024],{"className":13025,"style":161},[160],[55,13027,13029,13032,13207],{"className":13028},[76],[55,13030],{"className":13031},[509,2284],[55,13033,13035],{"className":13034},[2288],[55,13036,13038,13199],{"className":13037},[108,109],[55,13039,13041,13196],{"className":13040},[113],[55,13042,13044,13119,13127],{"className":13043,"style":7515},[117],[55,13045,13046,13049],{"style":2301},[55,13047],{"className":13048,"style":2305},[125],[55,13050,13052,13098,13101,13104,13107,13110,13113,13116],{"className":13051},[76],[55,13053,13055,13058],{"className":13054},[76],[55,13056,924],{"className":13057,"style":99},[76,77],[55,13059,13061],{"className":13060},[104],[55,13062,13064,13090],{"className":13063},[108,109],[55,13065,13067,13087],{"className":13066},[113],[55,13068,13070],{"className":13069,"style":296},[117],[55,13071,13072,13075],{"style":121},[55,13073],{"className":13074,"style":126},[125],[55,13076,13078],{"className":13077},[130,131,132,133],[55,13079,13081],{"className":13080},[76,133],[55,13082,13084],{"className":13083},[76,951,133],[55,13085,955],{"className":13086},[76,133],[55,13088,142],{"className":13089},[141],[55,13091,13093],{"className":13092},[113],[55,13094,13096],{"className":13095,"style":149},[117],[55,13097],{},[55,13099,521],{"className":13100},[509],[55,13102,100],{"className":13103,"style":99},[76,77],[55,13105],{"className":13106,"style":263},[160],[55,13108,2071],{"className":13109},[267],[55,13111],{"className":13112,"style":263},[160],[55,13114,78],{"className":13115},[76,77],[55,13117,564],{"className":13118},[563],[55,13120,13121,13124],{"style":2484},[55,13122],{"className":13123,"style":2305},[125],[55,13125],{"className":13126,"style":2492},[2491],[55,13128,13129,13132],{"style":2495},[55,13130],{"className":13131,"style":2305},[125],[55,13133,13135,13175,13178,13181,13184,13187,13190,13193],{"className":13134},[76],[55,13136,13138,13141],{"className":13137},[76],[55,13139,924],{"className":13140,"style":99},[76,77],[55,13142,13144],{"className":13143},[104],[55,13145,13147,13167],{"className":13146},[108,109],[55,13148,13150,13164],{"className":13149},[113],[55,13151,13153],{"className":13152,"style":296},[117],[55,13154,13155,13158],{"style":121},[55,13156],{"className":13157,"style":126},[125],[55,13159,13161],{"className":13160},[130,131,132,133],[55,13162,1532],{"className":13163,"style":480},[76,77,133],[55,13165,142],{"className":13166},[141],[55,13168,13170],{"className":13169},[113],[55,13171,13173],{"className":13172,"style":149},[117],[55,13174],{},[55,13176,521],{"className":13177},[509],[55,13179,100],{"className":13180,"style":99},[76,77],[55,13182],{"className":13183,"style":263},[160],[55,13185,2071],{"className":13186},[267],[55,13188],{"className":13189,"style":263},[160],[55,13191,78],{"className":13192},[76,77],[55,13194,564],{"className":13195},[563],[55,13197,142],{"className":13198},[141],[55,13200,13202],{"className":13201},[113],[55,13203,13205],{"className":13204,"style":2514},[117],[55,13206],{},[55,13208],{"className":13209},[563,2284],",\nthe ",[55,13212,13214],{"className":13213},[58],[55,13215,13217],{"className":13216,"ariaHidden":63},[62],[55,13218,13220,13223],{"className":13219},[67],[55,13221],{"className":13222,"style":6800},[71],[55,13224,6591],{"className":13225,"style":6590},[76,77],"-scaled log-likelihood ratio of the trained policy to the reference. The\nDPO loss is precisely the Bradley-Terry loss on this implicit reward.",[11,13228,13229,13230,13233,13234,13264],{},"In words: the optimal policy is the reference\ndistribution ",[880,13231,13232],{},"reweighted"," by ",[55,13235,13237],{"className":13236},[58],[55,13238,13240],{"className":13239,"ariaHidden":63},[62],[55,13241,13243,13246,13249,13252,13255,13258,13261],{"className":13242},[67],[55,13244],{"className":13245,"style":796},[71],[55,13247,2326],{"className":13248},[2325],[55,13250,521],{"className":13251},[509],[55,13253,800],{"className":13254,"style":480},[76,77],[55,13256,10120],{"className":13257},[76],[55,13259,6591],{"className":13260,"style":6590},[76,77],[55,13262,564],{"className":13263},[563],", which tilts probability mass toward\nhigh-reward responses while the KL leash keeps the shape close to the reference.",[1145,13266],{"hash":13267},"a128bede09311dd5157c1d4a5afbea05279bb41aec257e075ad47cd896341110",[1145,13269],{"hash":13270},"666c0d5d224dfe32ce350a59876110d22652adb4980ddd4e4d3e2dbb7a135367",[821,13272,13273,13285],{},[824,13274,13275],{},[827,13276,13277,13280,13283],{},[830,13278,13279],{},"Property",[830,13281,13282],{},"RLHF (PPO)",[830,13284,8784],{},[840,13286,13287,13470,13481,13492,13503],{},[827,13288,13289,13292,13348],{},[845,13290,13291],{},"Reward model",[845,13293,13294,13295,13347],{},"explicit ",[55,13296,13298],{"className":13297},[58],[55,13299,13301],{"className":13300,"ariaHidden":63},[62],[55,13302,13304,13307],{"className":13303},[67],[55,13305],{"className":13306,"style":983},[71],[55,13308,13310,13313],{"className":13309},[76],[55,13311,800],{"className":13312,"style":480},[76,77],[55,13314,13316],{"className":13315},[104],[55,13317,13319,13339],{"className":13318},[108,109],[55,13320,13322,13336],{"className":13321},[113],[55,13323,13325],{"className":13324,"style":296},[117],[55,13326,13327,13330],{"style":1004},[55,13328],{"className":13329,"style":126},[125],[55,13331,13333],{"className":13332},[130,131,132,133],[55,13334,1014],{"className":13335},[76,77,133],[55,13337,142],{"className":13338},[141],[55,13340,13342],{"className":13341},[113],[55,13343,13345],{"className":13344,"style":1024},[117],[55,13346],{},", trained separately",[845,13349,13350,13351],{},"implicit, ",[55,13352,13354],{"className":13353},[58],[55,13355,13357],{"className":13356,"ariaHidden":63},[62],[55,13358,13360,13363,13366,13369,13375,13378,13418,13421,13467],{"className":13359},[67],[55,13361],{"className":13362,"style":796},[71],[55,13364,6591],{"className":13365,"style":6590},[76,77],[55,13367],{"className":13368,"style":161},[160],[55,13370,13372],{"className":13371},[2325],[55,13373,3865],{"className":13374,"style":3864},[76,3863],[55,13376,521],{"className":13377},[509],[55,13379,13381,13384],{"className":13380},[76],[55,13382,924],{"className":13383,"style":99},[76,77],[55,13385,13387],{"className":13386},[104],[55,13388,13390,13410],{"className":13389},[108,109],[55,13391,13393,13407],{"className":13392},[113],[55,13394,13396],{"className":13395,"style":296},[117],[55,13397,13398,13401],{"style":121},[55,13399],{"className":13400,"style":126},[125],[55,13402,13404],{"className":13403},[130,131,132,133],[55,13405,1532],{"className":13406,"style":480},[76,77,133],[55,13408,142],{"className":13409},[141],[55,13411,13413],{"className":13412},[113],[55,13414,13416],{"className":13415,"style":149},[117],[55,13417],{},[55,13419,10120],{"className":13420},[76],[55,13422,13424,13427],{"className":13423},[76],[55,13425,924],{"className":13426,"style":99},[76,77],[55,13428,13430],{"className":13429},[104],[55,13431,13433,13459],{"className":13432},[108,109],[55,13434,13436,13456],{"className":13435},[113],[55,13437,13439],{"className":13438,"style":296},[117],[55,13440,13441,13444],{"style":121},[55,13442],{"className":13443,"style":126},[125],[55,13445,13447],{"className":13446},[130,131,132,133],[55,13448,13450],{"className":13449},[76,133],[55,13451,13453],{"className":13452},[76,951,133],[55,13454,955],{"className":13455},[76,133],[55,13457,142],{"className":13458},[141],[55,13460,13462],{"className":13461},[113],[55,13463,13465],{"className":13464,"style":149},[117],[55,13466],{},[55,13468,564],{"className":13469},[563],[827,13471,13472,13475,13478],{},[845,13473,13474],{},"Sampling",[845,13476,13477],{},"on-policy rollouts each step",[845,13479,13480],{},"none; offline on fixed pairs",[827,13482,13483,13486,13489],{},[845,13484,13485],{},"Extra networks",[845,13487,13488],{},"reward model + value head",[845,13490,13491],{},"none beyond reference copy",[827,13493,13494,13497,13500],{},[845,13495,13496],{},"Loss",[845,13498,13499],{},"clipped policy gradient",[845,13501,13502],{},"supervised log-sigmoid",[827,13504,13505,13508,13511],{},[845,13506,13507],{},"Stability",[845,13509,13510],{},"sensitive to many knobs",[845,13512,13513],{},"stable, supervised-style",[31,13515,13517],{"id":13516},"variants","Variants",[11,13519,13520],{},"DPO opened a family of preference-optimization losses that change the link function, the\ndata format, or the optimizer while keeping the offline, RL-free spirit.",[11,13522,13523,13526,13527,13611],{},[50,13524,13525],{},"IPO"," (identity preference optimization) observes that DPO's log-sigmoid can drive the\nimplicit reward gap to infinity and overfit when a pair is always labeled the same way;\nit replaces the Bradley-Terry log-sigmoid with a squared loss that regresses the\nimplicit reward gap toward a finite target ",[55,13528,13530],{"className":13529},[58],[55,13531,13533],{"className":13532,"ariaHidden":63},[62],[55,13534,13536,13540],{"className":13535},[67],[55,13537],{"className":13538,"style":13539},[71],"height:1.3262em;vertical-align:-0.4811em;",[55,13541,13543,13546,13608],{"className":13542},[76],[55,13544],{"className":13545},[509,2284],[55,13547,13549],{"className":13548},[2288],[55,13550,13552,13600],{"className":13551},[108,109],[55,13553,13555,13597],{"className":13554},[113],[55,13556,13558,13575,13583],{"className":13557,"style":2789},[117],[55,13559,13560,13563],{"style":2792},[55,13561],{"className":13562,"style":2305},[125],[55,13564,13566],{"className":13565},[130,131,132,133],[55,13567,13569,13572],{"className":13568},[76,133],[55,13570,191],{"className":13571},[76,133],[55,13573,6591],{"className":13574,"style":6590},[76,77,133],[55,13576,13577,13580],{"style":2484},[55,13578],{"className":13579,"style":2305},[125],[55,13581],{"className":13582,"style":2492},[2491],[55,13584,13585,13588],{"style":2815},[55,13586],{"className":13587,"style":2305},[125],[55,13589,13591],{"className":13590},[130,131,132,133],[55,13592,13594],{"className":13593},[76,133],[55,13595,137],{"className":13596},[76,133],[55,13598,142],{"className":13599},[141],[55,13601,13603],{"className":13602},[113],[55,13604,13606],{"className":13605,"style":11282},[117],[55,13607],{},[55,13609],{"className":13610},[563,2284],", which bounds the\nsolution and resists over-fitting deterministic preferences.",[11,13613,13614,13617,13618,13621,13622,13625,13626,13629],{},[50,13615,13616],{},"KTO"," (Kahneman-Tversky optimization) drops the ",[880,13619,13620],{},"paired"," requirement entirely. Drawing\non prospect theory, it treats each response as an independent ",[39,13623,13624],{},"good"," or ",[39,13627,13628],{},"bad"," example\nand maximizes a human-utility function with reference-dependent gains and losses, so it\ntrains on unpaired thumbs-up \u002F thumbs-down data that is far cheaper to collect than\nmatched comparisons.",[11,13631,13632,13635,13636,13639],{},[50,13633,13634],{},"RLAIF and Constitutional AI"," replace the human labeler with a model. An AI critic,\nprompted with a written ",[39,13637,13638],{},"constitution"," of principles, generates the preference labels\n(and self-revisions), so the comparison dataset is produced without human annotation of\neach pair; humans write the principles once instead of judging thousands of\npairs. The reward-modeling and RL machinery is otherwise unchanged.",[11,13641,13642,13645,13646,13649],{},[50,13643,13644],{},"GRPO"," (group-relative policy optimization) keeps the RL loop but discards the value\nhead: for each prompt it samples a ",[880,13647,13648],{},"group"," of responses, scores them, and uses the\ngroup's mean reward as the baseline, so the advantage of a response is its reward minus\nthe group average. This removes the separate critic network while keeping on-policy\noptimization, trading the learned baseline for a Monte-Carlo one.",[821,13651,13652,13672],{},[824,13653,13654],{},[827,13655,13656,13659,13661,13664,13667,13669],{},[830,13657,13658],{},"Method",[830,13660,1162],{},[830,13662,13663],{},"Optimizer",[830,13665,13666],{},"Extra nets",[830,13668,13507],{},[830,13670,13671],{},"Compute",[840,13673,13674,13694,13711,13727,13742],{},[827,13675,13676,13679,13682,13685,13688,13691],{},[845,13677,13678],{},"RLHF-PPO",[845,13680,13681],{},"paired comparisons",[845,13683,13684],{},"RL (clipped PG)",[845,13686,13687],{},"reward + value",[845,13689,13690],{},"sensitive",[845,13692,13693],{},"high (sampling)",[827,13695,13696,13698,13700,13702,13705,13708],{},[845,13697,8784],{},[845,13699,13681],{},[845,13701,8707],{},[845,13703,13704],{},"reference only",[845,13706,13707],{},"stable",[845,13709,13710],{},"low (offline)",[827,13712,13713,13715,13717,13720,13722,13725],{},[845,13714,13525],{},[845,13716,13681],{},[845,13718,13719],{},"supervised (squared)",[845,13721,13704],{},[845,13723,13724],{},"stable, anti-overfit",[845,13726,13710],{},[827,13728,13729,13731,13734,13736,13738,13740],{},[845,13730,13616],{},[845,13732,13733],{},"unpaired labels",[845,13735,8707],{},[845,13737,13704],{},[845,13739,13707],{},[845,13741,13710],{},[827,13743,13744,13746,13749,13752,13755,13758],{},[845,13745,13644],{},[845,13747,13748],{},"sampled groups",[845,13750,13751],{},"RL (group baseline)",[845,13753,13754],{},"reward only",[845,13756,13757],{},"moderate",[845,13759,13760],{},"medium",[31,13762,13764],{"id":13763},"the-alignment-frontier","The alignment frontier",[11,13766,13767,13768,13777,13778,13785,13786,13794,13795,13802,13803,13811,13812,13820,13821,13825],{},"Chollet's chapter predates the alignment stack, so the citations are the primary papers: the RLHF\u002FInstructGPT pipeline,",[13769,13770,13771],"sup",{},[15,13772,137],{"href":13773,"ariaDescribedBy":13774,"dataFootnoteRef":6,"id":13776},"#user-content-fn-ouyang-rlhf",[13775],"footnote-label","user-content-fnref-ouyang-rlhf"," the Bradley-Terry model of paired comparisons,",[13769,13779,13780],{},[15,13781,191],{"href":13782,"ariaDescribedBy":13783,"dataFootnoteRef":6,"id":13784},"#user-content-fn-bradley-terry",[13775],"user-content-fnref-bradley-terry"," reward over-optimization,",[13769,13787,13788],{},[15,13789,13793],{"href":13790,"ariaDescribedBy":13791,"dataFootnoteRef":6,"id":13792},"#user-content-fn-gao-overopt",[13775],"user-content-fnref-gao-overopt","3"," DPO,",[13769,13796,13797],{},[15,13798,5854],{"href":13799,"ariaDescribedBy":13800,"dataFootnoteRef":6,"id":13801},"#user-content-fn-rafailov-dpo",[13775],"user-content-fnref-rafailov-dpo"," Constitutional AI \u002F RLAIF,",[13769,13804,13805],{},[15,13806,13810],{"href":13807,"ariaDescribedBy":13808,"dataFootnoteRef":6,"id":13809},"#user-content-fn-bai-cai",[13775],"user-content-fnref-bai-cai","5"," and GRPO.",[13769,13813,13814],{},[15,13815,13819],{"href":13816,"ariaDescribedBy":13817,"dataFootnoteRef":6,"id":13818},"#user-content-fn-shao-grpo",[13775],"user-content-fnref-shao-grpo","6"," Three threads connect this lesson to current practice and to the ",[15,13822,13824],{"href":13823},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment","large-language-model alignment"," lesson.",[11,13827,13828,13831,13832,13835],{},[50,13829,13830],{},"The reward model is the binding constraint."," Every failure in this lesson traces to the reward model being a finite-data proxy, and the biggest practical gains have come from making it better: larger preference datasets, ensembles that flag their own disagreement (high variance is a signal to stop optimizing), and process rewards that score ",[880,13833,13834],{},"each reasoning step"," rather than only the final answer. A better reward model pushes the peak of the over-optimization hump further out, which is worth more than any refinement of the RL loop.",[11,13837,13838,13841,13842,13845,13846,13864,13865,13880,13881,13885,13886,13888],{},[50,13839,13840],{},"Verifiable rewards sidestep the proxy entirely."," When the task has a checkable answer — a math problem, a passing test suite — the reward need not be ",[880,13843,13844],{},"learned"," at all: it is ",[55,13847,13849],{"className":13848},[58],[55,13850,13852],{"className":13851,"ariaHidden":63},[62],[55,13853,13855,13858,13861],{"className":13854},[67],[55,13856],{"className":13857,"style":4790},[71],[55,13859,2318],{"className":13860},[76],[55,13862,137],{"className":13863},[76]," if the answer is correct and ",[55,13866,13868],{"className":13867},[58],[55,13869,13871],{"className":13870,"ariaHidden":63},[62],[55,13872,13874,13877],{"className":13873},[67],[55,13875],{"className":13876,"style":2735},[71],[55,13878,6823],{"className":13879},[76]," otherwise. GRPO with such a verifiable reward is the training method behind the reasoning models of the ",[15,13882,13884],{"href":13883},"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents","AI-agents lesson",", and it removes the reward-hacking failure mode by construction, because a correct answer cannot be gamed. The Goodhart problem is a property of ",[880,13887,8356],{}," rewards; a verifiable reward is not a proxy.",[11,13890,13891,13894],{},[50,13892,13893],{},"DPO and PPO are converging in practice."," DPO's simplicity made it the default for offline preference data, but PPO-style online methods (and GRPO) remain stronger when fresh rollouts and a verifiable or strong reward model are available, because they optimize against the current policy's own mistakes rather than a fixed offline set. The field now mixes them: a DPO or SFT warm start, then online RL with a reward model or verifier for the final polish.",[11,13896,13897],{},"The reading is that RLHF turned an unspecifiable objective into a learnable one through the preference asymmetry, DPO showed the RL loop was optional for offline data, and the frontier has moved to making the reward trustworthy — better models, ensembles, process rewards, and, where possible, verifiable rewards that are not proxies at all.",[31,13899,13901],{"id":13900},"takeaways","Takeaways",[13903,13904,13905,13913,13919,14195,14327,14350,14603,14608],"ul",{},[13906,13907,13908,13909,13912],"li",{},"RLHF exploits an ",[50,13910,13911],{},"asymmetry",": helpfulness is hard to specify but easy to judge by\ncomparison, so we learn a reward from pairwise preferences instead of writing a loss.",[13906,13914,13,13915,13918],{},[50,13916,13917],{},"pipeline"," is three stages: SFT gives a reference policy, a reward model fits\nhuman comparisons, and RL fine-tuning maximizes that reward under a KL leash.",[13906,13920,13921,13922,13924,13925,14194],{},"Reward modeling uses ",[50,13923,1775],{},", ",[55,13926,13928],{"className":13927},[58],[55,13929,13931,13992,14050,14129],{"className":13930,"ariaHidden":63},[62],[55,13932,13934,13937,13940,13943,13983,13986,13989],{"className":13933},[67],[55,13935],{"className":13936,"style":796},[71],[55,13938,1966],{"className":13939,"style":1965},[76,77],[55,13941,521],{"className":13942},[509],[55,13944,13946,13949],{"className":13945},[76],[55,13947,100],{"className":13948,"style":99},[76,77],[55,13950,13952],{"className":13951},[104],[55,13953,13955,13975],{"className":13954},[108,109],[55,13956,13958,13972],{"className":13957},[113],[55,13959,13961],{"className":13960,"style":235},[117],[55,13962,13963,13966],{"style":121},[55,13964],{"className":13965,"style":126},[125],[55,13967,13969],{"className":13968},[130,131,132,133],[55,13970,248],{"className":13971,"style":247},[76,77,133],[55,13973,142],{"className":13974},[141],[55,13976,13978],{"className":13977},[113],[55,13979,13981],{"className":13980,"style":149},[117],[55,13982],{},[55,13984],{"className":13985,"style":263},[160],[55,13987,268],{"className":13988},[267],[55,13990],{"className":13991,"style":263},[160],[55,13993,13995,13998,14038,14041,14044,14047],{"className":13994},[67],[55,13996],{"className":13997,"style":796},[71],[55,13999,14001,14004],{"className":14000},[76],[55,14002,100],{"className":14003,"style":99},[76,77],[55,14005,14007],{"className":14006},[104],[55,14008,14010,14030],{"className":14009},[108,109],[55,14011,14013,14027],{"className":14012},[113],[55,14014,14016],{"className":14015,"style":296},[117],[55,14017,14018,14021],{"style":121},[55,14019],{"className":14020,"style":126},[125],[55,14022,14024],{"className":14023},[130,131,132,133],[55,14025,309],{"className":14026,"style":308},[76,77,133],[55,14028,142],{"className":14029},[141],[55,14031,14033],{"className":14032},[113],[55,14034,14036],{"className":14035,"style":149},[117],[55,14037],{},[55,14039,564],{"className":14040},[563],[55,14042],{"className":14043,"style":263},[160],[55,14045,488],{"className":14046},[267],[55,14048],{"className":14049,"style":263},[160],[55,14051,14053,14056,14059,14062,14065,14068,14071,14074,14077,14117,14120,14123,14126],{"className":14052},[67],[55,14054],{"className":14055,"style":796},[71],[55,14057,2111],{"className":14058,"style":99},[76,77],[55,14060,521],{"className":14061},[509],[55,14063,800],{"className":14064,"style":480},[76,77],[55,14066,521],{"className":14067},[509],[55,14069,78],{"className":14070},[76,77],[55,14072,156],{"className":14073},[155],[55,14075],{"className":14076,"style":161},[160],[55,14078,14080,14083],{"className":14079},[76],[55,14081,100],{"className":14082,"style":99},[76,77],[55,14084,14086],{"className":14085},[104],[55,14087,14089,14109],{"className":14088},[108,109],[55,14090,14092,14106],{"className":14091},[113],[55,14093,14095],{"className":14094,"style":235},[117],[55,14096,14097,14100],{"style":121},[55,14098],{"className":14099,"style":126},[125],[55,14101,14103],{"className":14102},[130,131,132,133],[55,14104,248],{"className":14105,"style":247},[76,77,133],[55,14107,142],{"className":14108},[141],[55,14110,14112],{"className":14111},[113],[55,14113,14115],{"className":14114,"style":149},[117],[55,14116],{},[55,14118,564],{"className":14119},[563],[55,14121],{"className":14122,"style":2186},[160],[55,14124,2191],{"className":14125},[2190],[55,14127],{"className":14128,"style":2186},[160],[55,14130,14132,14135,14138,14141,14144,14147,14150,14190],{"className":14131},[67],[55,14133],{"className":14134,"style":796},[71],[55,14136,800],{"className":14137,"style":480},[76,77],[55,14139,521],{"className":14140},[509],[55,14142,78],{"className":14143},[76,77],[55,14145,156],{"className":14146},[155],[55,14148],{"className":14149,"style":161},[160],[55,14151,14153,14156],{"className":14152},[76],[55,14154,100],{"className":14155,"style":99},[76,77],[55,14157,14159],{"className":14158},[104],[55,14160,14162,14182],{"className":14161},[108,109],[55,14163,14165,14179],{"className":14164},[113],[55,14166,14168],{"className":14167,"style":296},[117],[55,14169,14170,14173],{"style":121},[55,14171],{"className":14172,"style":126},[125],[55,14174,14176],{"className":14175},[130,131,132,133],[55,14177,309],{"className":14178,"style":308},[76,77,133],[55,14180,142],{"className":14181},[141],[55,14183,14185],{"className":14184},[113],[55,14186,14188],{"className":14187,"style":149},[117],[55,14189],{},[55,14191,14193],{"className":14192},[563],"))",",\nand the reward loss is the negative log-sigmoid of the reward gap; its gradient\nself-weights toward currently-misranked pairs.",[13906,14196,14197,14198,14322,14323,14326],{},"The RL step maximizes\n",[55,14199,14201],{"className":14200},[58],[55,14202,14204,14246],{"className":14203,"ariaHidden":63},[62],[55,14205,14207,14210,14213,14216,14219,14222,14225,14228,14231,14234,14237,14240,14243],{"className":14206},[67],[55,14208],{"className":14209,"style":796},[71],[55,14211,4104],{"className":14212},[76,1838],[55,14214,4252],{"className":14215},[509],[55,14217,800],{"className":14218,"style":480},[76,77],[55,14220,521],{"className":14221},[509],[55,14223,78],{"className":14224},[76,77],[55,14226,156],{"className":14227},[155],[55,14229],{"className":14230,"style":161},[160],[55,14232,100],{"className":14233,"style":99},[76,77],[55,14235,9696],{"className":14236},[563],[55,14238],{"className":14239,"style":2186},[160],[55,14241,2191],{"className":14242},[2190],[55,14244],{"className":14245,"style":2186},[160],[55,14247,14249,14252,14255,14258,14264,14267,14270,14273,14319],{"className":14248},[67],[55,14250],{"className":14251,"style":796},[71],[55,14253,6591],{"className":14254,"style":6590},[76,77],[55,14256],{"className":14257,"style":161},[160],[55,14259,14261],{"className":14260},[2325],[55,14262,6659],{"className":14263},[76,3863],[55,14265,521],{"className":14266},[509],[55,14268,924],{"className":14269,"style":99},[76,77],[55,14271,6702],{"className":14272},[76],[55,14274,14276,14279],{"className":14275},[76],[55,14277,924],{"className":14278,"style":99},[76,77],[55,14280,14282],{"className":14281},[104],[55,14283,14285,14311],{"className":14284},[108,109],[55,14286,14288,14308],{"className":14287},[113],[55,14289,14291],{"className":14290,"style":296},[117],[55,14292,14293,14296],{"style":121},[55,14294],{"className":14295,"style":126},[125],[55,14297,14299],{"className":14298},[130,131,132,133],[55,14300,14302],{"className":14301},[76,133],[55,14303,14305],{"className":14304},[76,951,133],[55,14306,955],{"className":14307},[76,133],[55,14309,142],{"className":14310},[141],[55,14312,14314],{"className":14313},[113],[55,14315,14317],{"className":14316,"style":149},[117],[55,14318],{},[55,14320,564],{"className":14321},[563],"; the KL penalty\nis distributed per token and the policy is optimized by ",[50,14324,14325],{},"PPO"," with a value head and\nadvantage estimation.",[13906,14328,14329,14330,14333,14334,14349],{},"The KL leash guards against ",[50,14331,14332],{},"reward over-optimization"," (Goodhart): proxy reward\nrises monotonically while true quality peaks and falls, and ",[55,14335,14337],{"className":14336},[58],[55,14338,14340],{"className":14339,"ariaHidden":63},[62],[55,14341,14343,14346],{"className":14342},[67],[55,14344],{"className":14345,"style":6800},[71],[55,14347,6591],{"className":14348,"style":6590},[76,77]," tunes the trade-off.",[13906,14351,14352,14354,14355,14482,14483,14602],{},[50,14353,8784],{}," solves the KL-regularized objective in closed form,\n",[55,14356,14358],{"className":14357},[58],[55,14359,14361,14409],{"className":14360,"ariaHidden":63},[62],[55,14362,14364,14367,14399,14402,14406],{"className":14363},[67],[55,14365],{"className":14366,"style":10427},[71],[55,14368,14370,14373],{"className":14369},[76],[55,14371,924],{"className":14372,"style":99},[76,77],[55,14374,14376],{"className":14375},[104],[55,14377,14379],{"className":14378},[108],[55,14380,14382],{"className":14381},[113],[55,14383,14385],{"className":14384,"style":10427},[117],[55,14386,14387,14390],{"style":543},[55,14388],{"className":14389,"style":126},[125],[55,14391,14393],{"className":14392},[130,131,132,133],[55,14394,14396],{"className":14395},[76,133],[55,14397,10442],{"className":14398},[76,133],[55,14400],{"className":14401,"style":263},[160],[55,14403,14405],{"className":14404},[267],"∝",[55,14407],{"className":14408,"style":263},[160],[55,14410,14412,14415,14461,14464,14467,14470,14473,14476,14479],{"className":14411},[67],[55,14413],{"className":14414,"style":796},[71],[55,14416,14418,14421],{"className":14417},[76],[55,14419,924],{"className":14420,"style":99},[76,77],[55,14422,14424],{"className":14423},[104],[55,14425,14427,14453],{"className":14426},[108,109],[55,14428,14430,14450],{"className":14429},[113],[55,14431,14433],{"className":14432,"style":296},[117],[55,14434,14435,14438],{"style":121},[55,14436],{"className":14437,"style":126},[125],[55,14439,14441],{"className":14440},[130,131,132,133],[55,14442,14444],{"className":14443},[76,133],[55,14445,14447],{"className":14446},[76,951,133],[55,14448,955],{"className":14449},[76,133],[55,14451,142],{"className":14452},[141],[55,14454,14456],{"className":14455},[113],[55,14457,14459],{"className":14458,"style":149},[117],[55,14460],{},[55,14462],{"className":14463,"style":161},[160],[55,14465,2326],{"className":14466},[2325],[55,14468,521],{"className":14469},[509],[55,14471,800],{"className":14472,"style":480},[76,77],[55,14474,10120],{"className":14475},[76],[55,14477,6591],{"className":14478,"style":6590},[76,77],[55,14480,564],{"className":14481},[563],", inverts it to an implicit reward\n",[55,14484,14486],{"className":14485},[58],[55,14487,14489],{"className":14488,"ariaHidden":63},[62],[55,14490,14492,14495,14498,14501,14507,14510,14550,14553,14599],{"className":14491},[67],[55,14493],{"className":14494,"style":796},[71],[55,14496,6591],{"className":14497,"style":6590},[76,77],[55,14499],{"className":14500,"style":161},[160],[55,14502,14504],{"className":14503},[2325],[55,14505,3865],{"className":14506,"style":3864},[76,3863],[55,14508,521],{"className":14509},[509],[55,14511,14513,14516],{"className":14512},[76],[55,14514,924],{"className":14515,"style":99},[76,77],[55,14517,14519],{"className":14518},[104],[55,14520,14522,14542],{"className":14521},[108,109],[55,14523,14525,14539],{"className":14524},[113],[55,14526,14528],{"className":14527,"style":296},[117],[55,14529,14530,14533],{"style":121},[55,14531],{"className":14532,"style":126},[125],[55,14534,14536],{"className":14535},[130,131,132,133],[55,14537,1532],{"className":14538,"style":480},[76,77,133],[55,14540,142],{"className":14541},[141],[55,14543,14545],{"className":14544},[113],[55,14546,14548],{"className":14547,"style":149},[117],[55,14549],{},[55,14551,10120],{"className":14552},[76],[55,14554,14556,14559],{"className":14555},[76],[55,14557,924],{"className":14558,"style":99},[76,77],[55,14560,14562],{"className":14561},[104],[55,14563,14565,14591],{"className":14564},[108,109],[55,14566,14568,14588],{"className":14567},[113],[55,14569,14571],{"className":14570,"style":296},[117],[55,14572,14573,14576],{"style":121},[55,14574],{"className":14575,"style":126},[125],[55,14577,14579],{"className":14578},[130,131,132,133],[55,14580,14582],{"className":14581},[76,133],[55,14583,14585],{"className":14584},[76,951,133],[55,14586,955],{"className":14587},[76,133],[55,14589,142],{"className":14590},[141],[55,14592,14594],{"className":14593},[113],[55,14595,14597],{"className":14596,"style":149},[117],[55,14598],{},[55,14600,564],{"className":14601},[563],", and substitutes into Bradley-Terry, collapsing\nthe pipeline into one supervised log-sigmoid loss.",[13906,14604,14605,14607],{},[50,14606,13517],{}," trade off data and machinery: IPO bounds the gap with a squared loss, KTO\nuses unpaired prospect-theory utilities, RLAIF \u002F Constitutional AI replace the human\nlabeler with a principled model critic, and GRPO drops the value head for a\ngroup-relative baseline.",[13906,14609,14610,14613,14614,14617],{},[50,14611,14612],{},"The frontier:"," the reward model is the binding constraint, so ensembles and\nprocess rewards matter more than the RL loop; ",[50,14615,14616],{},"verifiable rewards"," (correct-answer\nchecks) sidestep reward hacking entirely and drive GRPO-trained reasoning models.",[14619,14620,14623,14628],"section",{"className":14621,"dataFootnotes":6},[14622],"footnotes",[31,14624,14627],{"className":14625,"id":13775},[14626],"sr-only","Footnotes",[14629,14630,14631,14648,14663,14678,14693,14708],"ol",{},[13906,14632,14634,13924,14637,14640,14641],{"id":14633},"user-content-fn-ouyang-rlhf",[50,14635,14636],{},"Ouyang et al.",[880,14638,14639],{},"Training Language Models to Follow Instructions with Human Feedback"," (InstructGPT), NeurIPS 2022 — the three-stage RLHF pipeline: SFT, a Bradley-Terry reward model on comparisons, and PPO with a KL penalty. ",[15,14642,14647],{"href":14643,"ariaLabel":14644,"className":14645,"dataFootnoteBackref":6},"#user-content-fnref-ouyang-rlhf","Back to reference 1",[14646],"data-footnote-backref","↩",[13906,14649,14651,13924,14654,14657,14658],{"id":14650},"user-content-fn-bradley-terry",[50,14652,14653],{},"Bradley & Terry",[880,14655,14656],{},"Rank Analysis of Incomplete Block Designs",", Biometrika 1952 — the paired-comparison model in which the probability one item beats another is a logistic of their latent-score difference. ",[15,14659,14647],{"href":14660,"ariaLabel":14661,"className":14662,"dataFootnoteBackref":6},"#user-content-fnref-bradley-terry","Back to reference 2",[14646],[13906,14664,14666,13924,14669,14672,14673],{"id":14665},"user-content-fn-gao-overopt",[50,14667,14668],{},"Gao, Schulman & Hilton",[880,14670,14671],{},"Scaling Laws for Reward Model Overoptimization",", ICML 2023 — shows true quality peaks then falls as proxy reward is optimized, with the gap growing with KL distance from the reference. ",[15,14674,14647],{"href":14675,"ariaLabel":14676,"className":14677,"dataFootnoteBackref":6},"#user-content-fnref-gao-overopt","Back to reference 3",[14646],[13906,14679,14681,13924,14684,14687,14688],{"id":14680},"user-content-fn-rafailov-dpo",[50,14682,14683],{},"Rafailov et al.",[880,14685,14686],{},"Direct Preference Optimization: Your Language Model is Secretly a Reward Model",", NeurIPS 2023 — derives the closed-form KL-regularized optimum and inverts it into a supervised log-sigmoid loss on preference pairs. ",[15,14689,14647],{"href":14690,"ariaLabel":14691,"className":14692,"dataFootnoteBackref":6},"#user-content-fnref-rafailov-dpo","Back to reference 4",[14646],[13906,14694,14696,13924,14699,14702,14703],{"id":14695},"user-content-fn-bai-cai",[50,14697,14698],{},"Bai et al.",[880,14700,14701],{},"Constitutional AI: Harmlessness from AI Feedback",", 2022 — replaces human preference labels with an AI critic guided by a written set of principles (RLAIF), reducing human annotation to writing the constitution. ",[15,14704,14647],{"href":14705,"ariaLabel":14706,"className":14707,"dataFootnoteBackref":6},"#user-content-fnref-bai-cai","Back to reference 5",[14646],[13906,14709,14711,13924,14714,14717,14718],{"id":14710},"user-content-fn-shao-grpo",[50,14712,14713],{},"Shao et al.",[880,14715,14716],{},"DeepSeekMath: Pushing the Limits of Mathematical Reasoning"," (GRPO), 2024 — group-relative policy optimization uses a group of sampled responses' mean reward as the baseline, dropping the value head from the RL loop. ",[15,14719,14647],{"href":14720,"ariaLabel":14721,"className":14722,"dataFootnoteBackref":6},"#user-content-fnref-shao-grpo","Back to reference 6",[14646],[14724,14725,14726],"style",{},"html .default .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .shiki span {color: var(--shiki-default);background: var(--shiki-default-bg);font-style: var(--shiki-default-font-style);font-weight: var(--shiki-default-font-weight);text-decoration: var(--shiki-default-text-decoration);}html .dark-mode .shiki span {color: var(--shiki-dark-mode);background: var(--shiki-dark-mode-bg);font-style: var(--shiki-dark-mode-font-style);font-weight: var(--shiki-dark-mode-font-weight);text-decoration: var(--shiki-dark-mode-text-decoration);}html.dark-mode .shiki span {color: var(--shiki-dark-mode);background: var(--shiki-dark-mode-bg);font-style: var(--shiki-dark-mode-font-style);font-weight: var(--shiki-dark-mode-font-weight);text-decoration: var(--shiki-dark-mode-text-decoration);}",{"title":6,"searchDepth":8277,"depth":8277,"links":14728},[14729,14730,14731,14734,14738,14739,14740,14741,14742,14743],{"id":33,"depth":8277,"text":34},{"id":892,"depth":8277,"text":893},{"id":1546,"depth":8277,"text":1547,"children":14732},[14733],{"id":3103,"depth":8283,"text":3104},{"id":6179,"depth":8277,"text":6180,"children":14735},[14736,14737],{"id":7088,"depth":8283,"text":7089},{"id":8023,"depth":8283,"text":8024},{"id":8349,"depth":8277,"text":8350},{"id":8697,"depth":8277,"text":8698},{"id":13516,"depth":8277,"text":13517},{"id":13763,"depth":8277,"text":13764},{"id":13900,"depth":8277,"text":13901},{"id":13775,"depth":8277,"text":14627},[],"computer-science","The large language model\nlesson sketched RLHF in a paragraph: fit a reward model, optimize with PPO, regularize\nwith a KL term. This lesson is the derivation. We build the reward model from the\nBradley-Terry likelihood, write the KL-regularized objective the policy actually\nmaximizes, solve it in closed form, and then show that the same closed form, inverted,\nturns the entire reinforcement-learning loop into a single supervised loss. The\npolicy gradients and PPO\nlesson supplies the optimizer; the foundations of RL\nlesson supplies the language of policies, rewards, and returns we reuse throughout.",false,"md",{"moduleNumber":8331,"lessonNumber":8295,"order":14750},1105,"Reinforcement Learning",true,"\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback",[],"---\ntitle: Reinforcement Learning from Human Feedback\nmodule: Reinforcement Learning\nmoduleNumber: 11\nlessonNumber: 5\norder: 1105\nsummary: >\n  Many objectives we want from a model, that it be helpful and harmless, are\n  hard to write down but easy to judge by comparison. RLHF turns that asymmetry\n  into a training signal: fit a reward model to pairwise human preferences under\n  the Bradley-Terry likelihood, then fine-tune the policy to maximize that reward\n  under a KL penalty toward a reference. We derive the reward loss, the KL-regularized\n  RL objective and its closed-form optimum, then show how DPO inverts that optimum\n  to collapse the whole pipeline into one supervised log-sigmoid loss, and survey\n  IPO, KTO, RLAIF, and GRPO.\ntopics: [Reinforcement Learning]\nsources:\n  - book: Chollet\n    ref: \"Ch. 11 — aligning generative language models\"\n---\n\nThe [large language model](\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models)\nlesson sketched RLHF in a paragraph: fit a reward model, optimize with PPO, regularize\nwith a KL term. This lesson is the derivation. We build the reward model from the\nBradley-Terry likelihood, write the KL-regularized objective the policy actually\nmaximizes, solve it in closed form, and then show that the same closed form, inverted,\nturns the entire reinforcement-learning loop into a single supervised loss. The\n[policy gradients and PPO](\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic)\nlesson supplies the optimizer; the [foundations of RL](\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning)\nlesson supplies the language of policies, rewards, and returns we reuse throughout.\n\n## The specification problem\n\nThe objectives we want from an assistant, that it be helpful, harmless, and honest,\nresist direct specification. There is no loss function for \"helpful.\" But a human\nshown two responses to the same prompt can reliably say which is better, even when\nthey could not have written down a rule that produces the better one. RLHF is built on\nexactly this asymmetry.\n\n> **Definition (Preference comparison).** For a prompt $x$ and two candidate\n> responses $y_1, y_2$, a human annotator returns a binary preference $y_w \\succ y_l$\n> (\"$y_w$ is preferred to $y_l$\"), where $w$ marks the winner and $l$ the loser. A\n> preference dataset is a collection $\\mathcal{D} = \\set{(x^{(i)}, y_w^{(i)}, y_l^{(i)})}_{i=1}^{N}$\n> of such judgments.\n\nThe plan is to learn a scalar **reward** $r(x, y)$ that ranks responses the way humans\ndo, then push a language model toward high-reward responses. The reward converts a\nset of discrete comparisons into a differentiable signal a policy can ascend.\n\n| Quantity | Hard to specify | Easy to elicit |\n| --- | --- | --- |\n| Absolute quality of one response | yes (no scalar in the annotator's head) | no |\n| Which of two responses is better | no | yes (a single click) |\n| A rule that generates good responses | yes | no |\n| A reward that _ranks_ responses | learned from comparisons | -- |\n\n## The three-stage pipeline\n\nRLHF as deployed in modern instruction-tuned systems is three stages run in sequence. First,\n**supervised fine-tuning** (SFT) on curated demonstrations teaches the base model the\n_format_ of an answer and gives a reference policy $\\pi_{\\text{ref}}$. Second, a\n**reward model** $r_\\phi$ is fit to human comparisons. Third, the policy is\n**RL-fine-tuned** to maximize $r_\\phi$ while a KL penalty holds it near\n$\\pi_{\\text{ref}}$.\n\n$$\n% caption: The three-stage RLHF pipeline. SFT yields the reference policy; the reward\n% model learns from human comparisons; RL fine-tunes the policy under a KL leash.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize,\n  stage\u002F.style={draw=acc, text=acc, thick, fill=acc!12, minimum width=27mm, minimum height=13mm, align=center},\n  data\u002F.style={draw=black, thick, fill=black!8, minimum width=24mm, minimum height=11mm, align=center, font=\\scriptsize}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % data sources (top row)\n  \\node[data] (demo) at (0,2.2) {demonstrations};\n  \\node[data] (comp) at (4.6,2.2) {human compari-\\\\sons};\n  \\node[data] (prompts) at (9.2,2.2) {prompts only};\n  % stages (bottom row)\n  \\node[stage] (sft) at (0,0) {1. SFT};\n  \\node[stage] (rm) at (4.6,0) {2. reward model};\n  \\node[stage] (rl) at (9.2,0) {3. RL f\\\u002Fine-tune};\n  % vertical data -> stage\n  \\draw[->, black, thick] (demo) -- (sft);\n  \\draw[->, black, thick] (comp) -- (rm);\n  \\draw[->, black, thick] (prompts) -- (rl);\n  % horizontal flow\n  \\draw[->, acc, thick] (sft) -- (rm) node[midway, below, font=\\scriptsize, text=acc] {ref};\n  \\draw[->, acc, thick] (rm) -- (rl) node[midway, below, font=\\scriptsize, text=acc] {reward};\n  % KL leash back from rl to sft\n  \\draw[->, black, thick, dashed] (rl.east) .. controls (12.6,0.6) and (12.6,4.9) .. (4.6,4.9)\n    node[above, font=\\scriptsize, text=black] {KL leash to reference}\n    .. controls (-3.4,4.9) and (-3.4,0.6) .. (sft.west);\n\\end{tikzpicture}\n$$\n\nThe three stages consume three different kinds of data, and that is the point: each\nstage extracts the signal that is cheapest to collect for what it must learn.\n\n| Stage | Data | Trains | Loss type | Output |\n| --- | --- | --- | --- | --- |\n| 1. SFT | $(x, y)$ demonstrations | base LM | supervised cross-entropy | $\\pi_{\\text{ref}}$ |\n| 2. Reward model | $(x, y_w, y_l)$ comparisons | a scalar head | Bradley-Terry log-loss | $r_\\phi$ |\n| 3. RL fine-tune | prompts $x$ only | the policy | KL-regularized reward | $\\pi_\\theta$ |\n\n## Reward modeling from pairwise preferences\n\nWe need a model that, given a comparison $(x, y_w, y_l)$, assigns a high probability to\nthe event \"$y_w$ beats $y_l$.\" The **Bradley-Terry** model is the canonical choice: it\nposits a latent scalar score per item and makes the probability of one beating another\na logistic function of their score difference.\n\n> **Definition (Bradley-Terry preference model).** Each response carries a latent\n> reward $r(x, y) \\in \\mathbb{R}$, and the probability that $y_w$ is preferred to $y_l$\n> is the logistic of the reward gap,\n> $$\n> P(y_w \\succ y_l \\mid x) \\;=\\; \\sigma\\!\\parens{r(x, y_w) - r(x, y_l)}\n> \\;=\\; \\frac{1}{1 + \\exp\\!\\parens{-\\parens{r(x, y_w) - r(x, y_l)}}}.\n> $$\n> Only the _difference_ of rewards is identified: adding any function $c(x)$ to every\n> reward leaves all preference probabilities unchanged.\n\nThe model has the right limits. When $r(x, y_w) \\gg r(x, y_l)$ the gap is large and\npositive, so $P \\to 1$; when the two rewards are equal the gap is zero and $P = \\tfrac12$,\na coin flip; and the curve is symmetric, so $P(y_w \\succ y_l) + P(y_l \\succ y_w) = 1$.\n\n$$\n% caption: Bradley-Terry: preference probability is the logistic of the reward gap\n% $\\Delta = r(x,y_w) - r(x,y_l)$, passing through $\\tfrac12$ at $\\Delta = 0$.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize, x=1.0cm, y=1.0cm]\n  \\definecolor{acc}{HTML}{2348F2}\n  % axes\n  \\draw[->, thick] (-3.2,0) -- (3.4,0) node[right, font=\\scriptsize] {reward gap};\n  \\draw[->, thick] (0,-0.2) -- (0,2.6) node[above, font=\\scriptsize] {P(win)};\n  % half line (start after the label gap so it never runs over the tick text)\n  \\draw[black, dashed] (-2.8,1.1) -- (3.2,1.1);\n  \\node[black, anchor=east, font=\\scriptsize] at (-3.0,1.1) {1\u002F2};\n  % one line\n  \\draw[black, dashed] (-2.8,2.2) -- (3.2,2.2);\n  \\node[black, anchor=east, font=\\scriptsize] at (-3.0,2.2) {1};\n  % logistic curve, scaled: y = 2.2 * sigmoid(1.6 x)\n  \\draw[acc, very thick] plot[domain=-3.2:3.2, samples=80]\n    (\\x, {2.2\u002F(1+exp(-1.6*\\x))});\n  % marker at gap 0\n  \\fill[acc] (0,1.1) circle (2pt);\n  \\node[acc, anchor=north west, font=\\scriptsize] at (1.55,0.95) {logistic};\n\\end{tikzpicture}\n$$\n\n### Deriving the reward loss\n\nWe fit $r_\\phi$ by maximum likelihood on the comparison dataset. Treat each comparison\nas one Bernoulli observation whose outcome is \"the labeled winner won.\" The likelihood\nof a single comparison is $P(y_w \\succ y_l \\mid x) = \\sigma(\\Delta_\\phi)$ with the\nshorthand $\\Delta_\\phi = r_\\phi(x, y_w) - r_\\phi(x, y_l)$. The dataset log-likelihood is\nthe sum over independent comparisons,\n\n$$\n\\ell(\\phi) \\;=\\; \\sum_{(x, y_w, y_l) \\in \\mathcal{D}} \\log \\sigma\\!\\parens{\\Delta_\\phi}.\n$$\n\nMaximizing the log-likelihood is minimizing its negative, normalized to an expectation\nover the dataset. This is the reward-model objective.\n\n> **Definition (Reward-model loss).** The reward model $r_\\phi$ minimizes the negative\n> log-likelihood of the observed preferences under Bradley-Terry,\n> $$\n> \\mathcal{L}_{\\text{RM}}(\\phi)\n> \\;=\\; -\\,\\mathbb{E}_{(x, y_w, y_l) \\sim \\mathcal{D}}\n> \\brackets{\\log \\sigma\\!\\parens{r_\\phi(x, y_w) - r_\\phi(x, y_l)}}.\n> $$\n> Architecturally, $r_\\phi$ is the SFT model with the next-token head replaced by a\n> scalar head reading the final hidden state of the last token.\n\nThe gradient explains what the loss does mechanically. Using $\\sigma'(z) = \\sigma(z)\\parens{1 - \\sigma(z)}$\nand $\\dfrac{d}{dz}\\log\\sigma(z) = 1 - \\sigma(z)$, the per-comparison gradient is\n\n$$\n\\nabla_\\phi\\!\\brackets{-\\log\\sigma(\\Delta_\\phi)}\n\\;=\\; -\\parens{1 - \\sigma(\\Delta_\\phi)}\\,\\nabla_\\phi \\Delta_\\phi\n\\;=\\; -\\,\\underbrace{\\sigma(-\\Delta_\\phi)}_{\\text{prob. of being wrong}}\\,\n\\parens{\\nabla_\\phi r_\\phi(x, y_w) - \\nabla_\\phi r_\\phi(x, y_l)}.\n$$\n\nThe update raises the winner's reward and lowers the loser's, scaled by $\\sigma(-\\Delta_\\phi)$,\nthe model's current probability of _mis_-ranking the pair. Already-correct comparisons\n(large $\\Delta_\\phi$) get a vanishing gradient; only the contested pairs move the\nweights. This is the same self-weighting that makes logistic regression focus on the\nmargin.\n\nFor example, suppose the model already ranks a pair\ncorrectly with $\\Delta_\\phi = 4$: then $\\sigma(-4) \\approx 0.018$, so the pair's gradient\nis scaled by under two percent and it barely nudges the weights. Suppose instead the\nmodel ranks a pair backwards with $\\Delta_\\phi = -2$: then $\\sigma(2) \\approx 0.88$, an\neighty-eight percent weight, and the update pushes hard to reverse the ordering. The loss\nconcentrates its gradient on currently-misranked pairs and contributes almost nothing\nfor pairs already ranked confidently, which is the correct behavior for a ranking\nobjective.\n\nThe identifiability caveat carries a practical consequence. Because only reward\n_differences_ enter the loss, $r_\\phi$ is pinned only up to a per-prompt additive constant\n$c(x)$: the scale of the scores is fixed by the logistic, but their offset is free. When\nthe reward feeds the RL step this is harmless, since the KL term and the advantage\nbaseline both subtract off any prompt-level constant. It does mean reward magnitudes are\nnot comparable across prompts, so a raw $r_\\phi(x, y)$ is a within-prompt ranking signal,\nnot an absolute quality score.\n\n## The RL fine-tuning step\n\nWith $r_\\phi$ fixed, stage three optimizes the policy. Naively maximizing\nexpected reward fails: the reward model is an imperfect proxy fit on finite data,\nand an unconstrained optimizer will find responses that score highly under $r_\\phi$ yet\nare degenerate to a human. To address this, _anchor_ the policy to the SFT reference with a\nKL penalty.\n\n> **Definition (KL-regularized RL objective).** Given a reward $r$ and reference policy\n> $\\pi_{\\text{ref}}$, the RLHF policy maximizes expected reward minus a KL penalty,\n> $$\n> \\max_{\\pi}\\;\n> \\mathbb{E}_{x \\sim \\mathcal{D},\\, y \\sim \\pi(\\cdot \\mid x)}\n> \\brackets{ r(x, y) }\n> \\;-\\; \\beta\\, \\mathbb{E}_{x \\sim \\mathcal{D}}\n> \\brackets{ \\KL\\!\\parens{\\pi(\\cdot \\mid x) \\,\\|\\, \\pi_{\\text{ref}}(\\cdot \\mid x)} },\n> $$\n> where $\\beta > 0$ sets the strength of the leash. As $\\beta \\to 0$ the policy maximizes\n> reward unconstrained; as $\\beta \\to \\infty$ it is pinned to $\\pi_{\\text{ref}}$.\n\nThe KL term serves two purposes. It keeps the policy in the region where the reward model\nwas trained and is therefore trustworthy, and it preserves the fluency and breadth the\nbase model already has, which the sparse reward would otherwise erode. Without the anchor\nthree failures follow directly. The optimizer exploits the reward model's blind spots and\nfinds high-scoring gibberish (reward hacking); it collapses onto the single\nhighest-reward response and loses diversity (mode collapse); and it drifts so far from the\ndata $\\pi_{\\text{ref}}$ was fit on that $r_\\phi$ is extrapolating rather than judging\n(distribution drift). The two terms pull in opposite directions: reward pulls the policy\naway from the reference toward high-scoring responses, and the KL pulls it back toward the\nreference, with $\\beta$ setting the balance point.\n\n$$\n% caption: The KL-anchored objective as a tug-of-war. Reward pulls the policy away from\n% the reference toward high-scoring responses; the KL penalty pulls it back. The\n% coefficient $\\beta$ sets the balance point.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize,\n  anchor2\u002F.style={draw=black, thick, fill=black!8, minimum width=22mm, minimum height=12mm, align=center, font=\\scriptsize},\n  pol\u002F.style={draw=acc, text=acc, thick, fill=acc!12, minimum width=22mm, minimum height=12mm, align=center, font=\\scriptsize},\n  hi\u002F.style={draw=red, text=red, thick, fill=red!8, minimum width=22mm, minimum height=12mm, align=center, font=\\scriptsize}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  \\node[anchor2] (ref) at (0,0) {reference (SFT)};\n  \\node[pol] (pi) at (5.0,0) {policy};\n  \\node[hi] (hi) at (10.0,0) {high reward};\n  % KL pulls policy back to reference\n  \\draw[->, black, very thick] (pi.west) -- (ref.east)\n    node[midway, above, font=\\scriptsize, text=black] {KL pulls back};\n  % reward pulls policy toward high reward\n  \\draw[->, red, very thick] (pi.east) -- (hi.west)\n    node[midway, above, font=\\scriptsize, text=red] {reward pulls away};\n  % balance annotation\n  \\node[acc, anchor=north, font=\\scriptsize] at (5.0,-0.95) {balance set by beta};\n\\end{tikzpicture}\n$$\n\n### Per-token reward shaping\n\nGeneration is a token-level Markov decision process: the state is the prompt plus the\ntokens emitted so far, the action is the next token, and the trajectory reward\n$r_\\phi(x, y)$ arrives only at the terminal token. To turn the sequence-level objective\ninto a dense per-step signal PPO can consume, the KL penalty is distributed across\ntokens, giving a per-token reward\n\n$$\n\\tilde r_t\n\\;=\\;\n\\underbrace{r_\\phi(x, y)\\,\\mathbb{1}[t = T]}_{\\text{terminal reward}}\n\\;-\\;\n\\beta\\,\\underbrace{\\log \\frac{\\pi_\\theta(y_t \\mid x, y_{\u003Ct})}{\\pi_{\\text{ref}}(y_t \\mid x, y_{\u003Ct})}}_{\\text{per-token KL}}.\n$$\n\nSummed over the trajectory, the second term telescopes into the sequence-level\n$\\KL$, so this shaping restates the objective above one step at a time.\nEvery token pays a small price for drifting from the reference, and the whole response\ncollects the terminal reward.\n\n$$\n% caption: Per-token reward shaping. Each generated token pays a small KL penalty\n% (black) for drifting from the reference; the terminal token also collects the\n% sequence reward (blue).\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize,\n  tok\u002F.style={draw=black, thick, fill=black!8, minimum width=10mm, minimum height=8mm, inner sep=1pt, align=center},\n  endtok\u002F.style={draw=acc, text=acc, thick, fill=acc!12, minimum width=10mm, minimum height=8mm, inner sep=1pt, align=center}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % token chain\n  \\node[tok] (t1) at (0,0) {$y_1$};\n  \\node[tok] (t2) at (1.5,0) {$y_2$};\n  \\node[tok] (t3) at (3.0,0) {$y_3$};\n  \\node[font=\\footnotesize, black] (dots) at (4.3,0) {. . .};\n  \\node[endtok] (tT) at (5.6,0) {$y_T$};\n  \\draw[->, black, thick] (t1) -- (t2);\n  \\draw[->, black, thick] (t2) -- (t3);\n  \\draw[->, black, thick] (t3) -- (dots);\n  \\draw[->, black, thick] (dots) -- (tT);\n  % per-token KL penalty: each arrow points down AT a label that sits clear below it\n  \\foreach \\n in {t1,t2,t3,tT}\n    \\draw[->, black, thick] (\\n.south) -- ++(0,-0.5);\n  \\node[black, anchor=north, font=\\scriptsize] at (0,-1.02) {KL};\n  \\node[black, anchor=north, font=\\scriptsize] at (1.5,-1.02) {KL};\n  \\node[black, anchor=north, font=\\scriptsize] at (3.0,-1.02) {KL};\n  \\node[black, anchor=north, font=\\scriptsize] at (5.6,-1.02) {KL};\n  % terminal reward: arrow points up AT the label above it\n  \\draw[->, acc, thick] (tT.north) -- ++(0,0.55);\n  \\node[acc, anchor=south, font=\\scriptsize] at (5.6,1.12) {+ reward};\n\\end{tikzpicture}\n$$\n\n### PPO as the optimizer\n\nThe shaped reward feeds a standard actor-critic loop. A **value head** $V_\\psi$ is\nattached to the policy to estimate the expected return from each token, and the\nadvantage $\\hat A_t$ (typically by generalized advantage estimation) measures how much\nbetter the chosen token did than the value baseline. PPO then takes a clipped policy-gradient\nstep on $\\hat A_t$, which the\n[policy gradients lesson](\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic)\ndevelops in full. The clip is what keeps each update small enough that the\non-policy reward estimates stay valid.\n\n```algorithm\ncaption: $\\textsc{RLHF}(\\pi_{\\text{ref}}, \\mathcal{D}_{\\text{cmp}}, \\mathcal{D}_{\\text{prompt}}, \\beta)$ — align a policy from preferences\ntrain reward model $r_\\phi$ by minimizing $\\mathcal{L}_{\\text{RM}}$ on $\\mathcal{D}_{\\text{cmp}}$\n$\\pi_\\theta \\gets \\pi_{\\text{ref}}$; initialize value head $V_\\psi$\nrepeat\n  sample prompts $x \\sim \\mathcal{D}_{\\text{prompt}}$ and rollouts $y \\sim \\pi_\\theta(\\cdot \\mid x)$\n  score each rollout: $r \\gets r_\\phi(x, y)$ \u002F\u002F terminal reward\n  for each token $t$ do\n    $\\tilde r_t \\gets r\\cdot\\mathbb{1}[t{=}T] - \\beta\\log\\dfrac{\\pi_\\theta(y_t \\mid x, y_{\u003Ct})}{\\pi_{\\text{ref}}(y_t \\mid x, y_{\u003Ct})}$ \u002F\u002F KL-shaped reward\n  estimate advantages $\\hat A_t$ from $\\tilde r_t$ and $V_\\psi$ \u002F\u002F GAE\n  update $\\theta$ by the clipped PPO objective on $\\hat A_t$ \u002F\u002F policy step\n  update $\\psi$ by regressing $V_\\psi$ on the returns \u002F\u002F value step\nuntil reward plateaus or KL budget is spent\nreturn $\\pi_\\theta$\n```\n\n## Failure modes: reward hacking and the KL leash\n\nThe reward model is a _proxy_, not the true objective, and optimizing a proxy hard\nenough breaks it. This is Goodhart's law: a measure that is optimized ceases to measure\nwhat it once did.\n\n> **Definition (Reward over-optimization).** As the policy is pushed to higher proxy\n> reward $r_\\phi$, the _true_ human-judged quality first rises with it, then peaks and\n> falls, while $r_\\phi$ keeps climbing. The policy has learned to exploit errors in the\n> reward model rather than to improve. The gap typically widens with the KL distance\n> $\\sqrt{\\KL(\\pi_\\theta \\| \\pi_{\\text{ref}})}$ traveled from the reference.\n\nThe KL coefficient $\\beta$ trades these two failure directions against\neach other. Too large, and the policy never moves; too small, and it over-optimizes the\nproxy into degeneracy.\n\n$$\n% caption: Reward over-optimization. True quality (blue) rises then falls as the policy\n% travels in KL from the reference, while the proxy reward (red) keeps climbing.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize, x=1.0cm, y=1.0cm]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{red}{HTML}{C0392B}\n  \\draw[->, thick] (0,0) -- (6.6,0) node[right, font=\\scriptsize] {KL from reference};\n  \\draw[->, thick] (0,0) -- (0,3.4) node[above, font=\\scriptsize] {quality};\n  % proxy reward: monotone rising, concave\n  \\draw[red, very thick] plot[domain=0.1:6.2, samples=80]\n    (\\x, {3.0*(1 - exp(-0.55*\\x))});\n  \\node[red, anchor=south, font=\\scriptsize] at (4.9,1.45) {proxy reward};\n  % true quality: rise then fall (a hump)\n  \\draw[acc, very thick] plot[domain=0.1:6.2, samples=80]\n    (\\x, {2.7*exp(-0.5*(\\x-2.3)*(\\x-2.3)\u002F1.9)});\n  \\node[acc, anchor=south, font=\\scriptsize] at (2.3,2.85) {true quality};\n  % peak marker\n  \\draw[black, dashed] (2.3,0) -- (2.3,2.7);\n  \\node[black, anchor=north, font=\\scriptsize] at (2.3,-0.1) {best KL};\n\\end{tikzpicture}\n$$\n\nTwo further levers control the same trade-off. The **value head** must be well-fit or\nthe advantage estimates are biased, which destabilizes the policy step; in practice the\nvalue and policy share the backbone but use separate heads. And **advantage estimation**\nby GAE introduces a bias-variance parameter ($\\lambda$) that, set well, smooths the sparse\nterminal reward into a usable per-token learning signal without injecting too much\nvariance from long rollouts.\n\n## Direct preference optimization\n\nPPO-based RLHF is operationally heavy: a separate reward model, a value head, on-policy\nsampling, and a famously finicky RL loop. **Direct preference optimization** (DPO)\nremoves all of it by exploiting a fact about the KL-regularized objective: its optimum\nhas a closed form, and that closed form can be inverted to express the reward in terms\nof the policy. Substituting the inverted reward into the Bradley-Terry loss\nyields a _supervised_ objective on the policy directly, with no reward model and no\nsampling.\n\n> **Theorem (DPO objective).** The policy that maximizes the KL-regularized RL objective\n> for a reward $r$ is the same policy that minimizes the supervised loss\n> $$\n> \\mathcal{L}_{\\text{DPO}}(\\theta)\n> = -\\,\\mathbb{E}_{(x, y_w, y_l) \\sim \\mathcal{D}}\\!\n> \\brackets{\n> \\log \\sigma\\!\\parens{\n> \\beta \\log \\frac{\\pi_\\theta(y_w \\mid x)}{\\pi_{\\text{ref}}(y_w \\mid x)}\n> - \\beta \\log \\frac{\\pi_\\theta(y_l \\mid x)}{\\pi_{\\text{ref}}(y_l \\mid x)}\n> }\n> },\n> $$\n> a binary-classification loss over preference pairs that needs no explicit reward\n> model, no value head, and no on-policy rollouts.\n\n> **Proof.** Fix a prompt $x$ and maximize the per-prompt objective\n> $\\mathbb{E}_{y \\sim \\pi}[r(x,y)] - \\beta\\KL\\parens{\\pi \\,\\|\\, \\pi_{\\text{ref}}}$\n> over distributions $\\pi(\\cdot \\mid x)$. Expanding the KL and dropping the constant\n> $\\log Z$ that we introduce next, the objective equals\n> $$\n> -\\beta \\sum_y \\pi(y \\mid x)\\,\n> \\log \\frac{\\pi(y \\mid x)}{\\pi_{\\text{ref}}(y \\mid x)\\exp\\!\\parens{r(x,y)\u002F\\beta}}.\n> $$\n> Define $Z(x) = \\sum_y \\pi_{\\text{ref}}(y \\mid x)\\exp\\!\\parens{r(x,y)\u002F\\beta}$ and the\n> normalized distribution\n> $\\pi^{\\ast}(y \\mid x) = \\tfrac{1}{Z(x)}\\pi_{\\text{ref}}(y \\mid x)\\exp\\!\\parens{r(x,y)\u002F\\beta}$.\n> Multiplying and dividing the argument of the log by $Z(x)$ rewrites the objective as\n> $-\\beta\\KL\\parens{\\pi \\,\\|\\, \\pi^{\\ast}} + \\beta\\log Z(x)$, and since\n> $\\log Z(x)$ is independent of $\\pi$, the objective is maximized exactly when the KL is\n> zero, i.e. $\\pi = \\pi^{\\ast}$:\n> $$\n> \\pi^{\\ast}(y \\mid x)\n> \\;=\\; \\frac{1}{Z(x)}\\,\\pi_{\\text{ref}}(y \\mid x)\\,\n> \\exp\\!\\parens{\\tfrac{1}{\\beta}\\,r(x, y)}.\n> $$\n> This is the closed-form optimum. Now _invert_ it: solve for the reward by taking the\n> log of both sides,\n> $$\n> r(x, y)\n> \\;=\\; \\beta \\log \\frac{\\pi^{\\ast}(y \\mid x)}{\\pi_{\\text{ref}}(y \\mid x)} \\;+\\; \\beta \\log Z(x).\n> $$\n> Substitute this reward into the Bradley-Terry probability\n> $P(y_w \\succ y_l \\mid x) = \\sigma\\parens{r(x, y_w) - r(x, y_l)}$. The intractable\n> partition term $\\beta\\log Z(x)$ appears in _both_ rewards and cancels in the\n> difference, leaving\n> $$\n> r(x, y_w) - r(x, y_l)\n> = \\beta \\log \\frac{\\pi^{\\ast}(y_w \\mid x)}{\\pi_{\\text{ref}}(y_w \\mid x)}\n> - \\beta \\log \\frac{\\pi^{\\ast}(y_l \\mid x)}{\\pi_{\\text{ref}}(y_l \\mid x)}.\n> $$\n> Maximizing the Bradley-Terry likelihood of the observed preferences in $\\pi^{\\ast}$,\n> parametrized as $\\pi_\\theta$, is the same as minimizing the negative log-sigmoid of\n> this difference, which is $\\mathcal{L}_{\\text{DPO}}(\\theta)$. $\\qed$\n\nThe derivation is a change of variable: the reward and the optimal policy are two\nencodings of the same object, and DPO works in policy space so it never has to fit the\nreward at all. The implicit reward DPO assigns to a response is the quantity it pushes\nup or down.\n\n$$\n% caption: The DPO change of variable. The closed-form optimum maps a reward to a policy\n% ($\\pi^{*}\\propto\\pi_{\\text{ref}}\\exp(r\u002F\\beta)$); taking logs inverts it, mapping the\n% policy back to a reward ($r=\\beta\\log(\\pi\u002F\\pi_{\\text{ref}})+\\beta\\log Z$). DPO fits the\n% policy side, so the reward is never instantiated.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize,\n  spc\u002F.style={draw=black, thick, fill=black!8, minimum width=32mm, minimum height=15mm, align=center, font=\\scriptsize},\n  psp\u002F.style={draw=acc, text=acc, thick, fill=acc!12, minimum width=32mm, minimum height=15mm, align=center, font=\\scriptsize}]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\node[spc] (rew) at (0,0) {reward space\\\\reward model};\n  \\node[psp] (pol) at (7.0,0) {policy space\\\\policy vs reference};\n  % forward: reward -> policy (closed-form optimum)\n  \\draw[->, black, thick] (rew.north east) .. controls (3.5,1.5) .. (pol.north west)\n    node[midway, above, font=\\scriptsize, text=black] {optimum: reweight by exp(reward \u002F beta)};\n  % inverse: policy -> reward (take logs)\n  \\draw[->, acc, thick] (pol.south west) .. controls (3.5,-1.5) .. (rew.south east)\n    node[midway, below, font=\\scriptsize, text=acc] {invert: reward = beta log(policy \u002F ref)};\n  % DPO annotation on the policy side\n  \\node[acc, anchor=north, font=\\scriptsize, align=center] at (7.0,-1.15) {DPO trains here};\n\\end{tikzpicture}\n$$\n\n> **Definition (Implicit reward).** DPO never instantiates $r_\\phi$, but the optimum it\n> targets corresponds to the implicit reward\n> $\\hat r(x, y) = \\beta \\log \\dfrac{\\pi_\\theta(y \\mid x)}{\\pi_{\\text{ref}}(y \\mid x)}$,\n> the $\\beta$-scaled log-likelihood ratio of the trained policy to the reference. The\n> DPO loss is precisely the Bradley-Terry loss on this implicit reward.\n\nIn words: the optimal policy is the reference\ndistribution _reweighted_ by $\\exp(r\u002F\\beta)$, which tilts probability mass toward\nhigh-reward responses while the KL leash keeps the shape close to the reference.\n\n$$\n% caption: The closed-form optimum reweights the reference by $\\exp(r\u002F\\beta)$, tilting\n% mass from low-reward (left) toward high-reward (right) responses; smaller $\\beta$ tilts harder.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize, x=1.0cm, y=1.0cm]\n  \\definecolor{acc}{HTML}{2348F2}\n  \\definecolor{green}{HTML}{1F9D4D}\n  % axes\n  \\draw[->, thick] (-0.2,0) -- (6.6,0) node[right, font=\\scriptsize] {responses (low to high reward)};\n  \\draw[->, thick] (0,-0.2) -- (0,2.9) node[above, font=\\scriptsize] {prob};\n  % reference: symmetric bump centered at 3.0 (black)\n  \\draw[black, very thick] plot[domain=0.2:6.2, samples=80]\n    (\\x, {2.0*exp(-0.5*(\\x-3.0)*(\\x-3.0)\u002F1.1)});\n  \\node[black, anchor=south east, font=\\scriptsize] at (2.6,1.95) {reference};\n  % optimal: same bump shifted right + sharpened (green) = reweighted by exp(r\u002Fbeta)\n  \\draw[green, very thick] plot[domain=0.2:6.2, samples=80]\n    (\\x, {2.45*exp(-0.5*(\\x-4.0)*(\\x-4.0)\u002F0.7)});\n  \\node[green, anchor=south west, font=\\scriptsize] at (4.75,2.05) {optimal policy};\n  % tilt arrow\n  \\draw[->, acc, thick] (3.0,2.45) .. controls (3.5,2.7) .. (4.0,2.55);\n  \\node[acc, anchor=south, font=\\scriptsize] at (3.5,2.7) {tilt by exp(reward \u002F beta)};\n\\end{tikzpicture}\n$$\n\n$$\n% caption: RLHF versus DPO. RLHF fits a reward then optimizes through RL; DPO inverts\n% the same optimum to train the policy directly on preferences in one supervised step.\n\\begin{tikzpicture}[>=stealth, font=\\footnotesize,\n  box\u002F.style={draw=black, thick, fill=black!8, minimum width=24mm, minimum height=11mm, align=center, font=\\scriptsize},\n  acb\u002F.style={draw=acc, text=acc, thick, fill=acc!12, minimum width=24mm, minimum height=11mm, align=center, font=\\scriptsize}]\n  \\definecolor{acc}{HTML}{2348F2}\n  % shared start\n  \\node[box] (pref) at (0,0) {preferences\\\\$(x, y_w, y_l)$};\n  % ---- RLHF path (top) ----\n  \\node[box] (rm) at (4.2,1.6) {reward model};\n  \\node[box] (rl) at (8.4,1.6) {RL (PPO)\\\\+ KL leash};\n  \\node[acb] (pi1) at (12.4,1.6) {policy};\n  \\draw[->, black, thick] (pref.north) .. controls (1.8,1.6) .. (rm);\n  \\draw[->, black, thick] (rm) -- (rl);\n  \\draw[->, black, thick] (rl) -- (pi1);\n  \\node[black, font=\\scriptsize] at (6.3,2.6) {RLHF: two stages, sampling};\n  % ---- DPO path (bottom) ----\n  \\node[acb] (pi2) at (12.4,-1.6) {policy};\n  \\draw[->, acc, thick] (pref.south) .. controls (1.8,-1.6) and (8.4,-1.6) .. (pi2)\n    node[midway, below, font=\\scriptsize, text=acc] {DPO: one supervised log-sigmoid loss};\n  % link: implicit reward (Latin words only; no Greek in nodes)\n  \\node[black, font=\\scriptsize, align=center] at (8.4,-0.45)\n    {implicit reward = beta log(policy \u002F ref)};\n\\end{tikzpicture}\n$$\n\n| Property | RLHF (PPO) | DPO |\n| --- | --- | --- |\n| Reward model | explicit $r_\\phi$, trained separately | implicit, $\\beta\\log(\\pi_\\theta\u002F\\pi_{\\text{ref}})$ |\n| Sampling | on-policy rollouts each step | none; offline on fixed pairs |\n| Extra networks | reward model + value head | none beyond reference copy |\n| Loss | clipped policy gradient | supervised log-sigmoid |\n| Stability | sensitive to many knobs | stable, supervised-style |\n\n## Variants\n\nDPO opened a family of preference-optimization losses that change the link function, the\ndata format, or the optimizer while keeping the offline, RL-free spirit.\n\n**IPO** (identity preference optimization) observes that DPO's log-sigmoid can drive the\nimplicit reward gap to infinity and overfit when a pair is always labeled the same way;\nit replaces the Bradley-Terry log-sigmoid with a squared loss that regresses the\nimplicit reward gap toward a finite target $\\tfrac{1}{2\\beta}$, which bounds the\nsolution and resists over-fitting deterministic preferences.\n\n**KTO** (Kahneman-Tversky optimization) drops the _paired_ requirement entirely. Drawing\non prospect theory, it treats each response as an independent \"good\" or \"bad\" example\nand maximizes a human-utility function with reference-dependent gains and losses, so it\ntrains on unpaired thumbs-up \u002F thumbs-down data that is far cheaper to collect than\nmatched comparisons.\n\n**RLAIF and Constitutional AI** replace the human labeler with a model. An AI critic,\nprompted with a written \"constitution\" of principles, generates the preference labels\n(and self-revisions), so the comparison dataset is produced without human annotation of\neach pair; humans write the principles once instead of judging thousands of\npairs. The reward-modeling and RL machinery is otherwise unchanged.\n\n**GRPO** (group-relative policy optimization) keeps the RL loop but discards the value\nhead: for each prompt it samples a _group_ of responses, scores them, and uses the\ngroup's mean reward as the baseline, so the advantage of a response is its reward minus\nthe group average. This removes the separate critic network while keeping on-policy\noptimization, trading the learned baseline for a Monte-Carlo one.\n\n| Method | Data | Optimizer | Extra nets | Stability | Compute |\n| --- | --- | --- | --- | --- | --- |\n| RLHF-PPO | paired comparisons | RL (clipped PG) | reward + value | sensitive | high (sampling) |\n| DPO | paired comparisons | supervised | reference only | stable | low (offline) |\n| IPO | paired comparisons | supervised (squared) | reference only | stable, anti-overfit | low (offline) |\n| KTO | unpaired labels | supervised | reference only | stable | low (offline) |\n| GRPO | sampled groups | RL (group baseline) | reward only | moderate | medium |\n\n## The alignment frontier\n\nChollet's chapter predates the alignment stack, so the citations are the primary papers: the RLHF\u002FInstructGPT pipeline,[^ouyang-rlhf] the Bradley-Terry model of paired comparisons,[^bradley-terry] reward over-optimization,[^gao-overopt] DPO,[^rafailov-dpo] Constitutional AI \u002F RLAIF,[^bai-cai] and GRPO.[^shao-grpo] Three threads connect this lesson to current practice and to the [large-language-model alignment](\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment) lesson.\n\n**The reward model is the binding constraint.** Every failure in this lesson traces to the reward model being a finite-data proxy, and the biggest practical gains have come from making it better: larger preference datasets, ensembles that flag their own disagreement (high variance is a signal to stop optimizing), and process rewards that score _each reasoning step_ rather than only the final answer. A better reward model pushes the peak of the over-optimization hump further out, which is worth more than any refinement of the RL loop.\n\n**Verifiable rewards sidestep the proxy entirely.** When the task has a checkable answer — a math problem, a passing test suite — the reward need not be _learned_ at all: it is $+1$ if the answer is correct and $0$ otherwise. GRPO with such a verifiable reward is the training method behind the reasoning models of the [AI-agents lesson](\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents), and it removes the reward-hacking failure mode by construction, because a correct answer cannot be gamed. The Goodhart problem is a property of _proxy_ rewards; a verifiable reward is not a proxy.\n\n**DPO and PPO are converging in practice.** DPO's simplicity made it the default for offline preference data, but PPO-style online methods (and GRPO) remain stronger when fresh rollouts and a verifiable or strong reward model are available, because they optimize against the current policy's own mistakes rather than a fixed offline set. The field now mixes them: a DPO or SFT warm start, then online RL with a reward model or verifier for the final polish.\n\nThe reading is that RLHF turned an unspecifiable objective into a learnable one through the preference asymmetry, DPO showed the RL loop was optional for offline data, and the frontier has moved to making the reward trustworthy — better models, ensembles, process rewards, and, where possible, verifiable rewards that are not proxies at all.\n\n## Takeaways\n\n- RLHF exploits an **asymmetry**: helpfulness is hard to specify but easy to judge by\n  comparison, so we learn a reward from pairwise preferences instead of writing a loss.\n- The **pipeline** is three stages: SFT gives a reference policy, a reward model fits\n  human comparisons, and RL fine-tuning maximizes that reward under a KL leash.\n- Reward modeling uses **Bradley-Terry**, $P(y_w \\succ y_l) = \\sigma(r(x,y_w) - r(x,y_l))$,\n  and the reward loss is the negative log-sigmoid of the reward gap; its gradient\n  self-weights toward currently-misranked pairs.\n- The RL step maximizes\n  $\\mathbb{E}[r(x,y)] - \\beta\\KL(\\pi \\| \\pi_{\\text{ref}})$; the KL penalty\n  is distributed per token and the policy is optimized by **PPO** with a value head and\n  advantage estimation.\n- The KL leash guards against **reward over-optimization** (Goodhart): proxy reward\n  rises monotonically while true quality peaks and falls, and $\\beta$ tunes the trade-off.\n- **DPO** solves the KL-regularized objective in closed form,\n  $\\pi^{\\ast} \\propto \\pi_{\\text{ref}}\\exp(r\u002F\\beta)$, inverts it to an implicit reward\n  $\\beta\\log(\\pi_\\theta\u002F\\pi_{\\text{ref}})$, and substitutes into Bradley-Terry, collapsing\n  the pipeline into one supervised log-sigmoid loss.\n- **Variants** trade off data and machinery: IPO bounds the gap with a squared loss, KTO\n  uses unpaired prospect-theory utilities, RLAIF \u002F Constitutional AI replace the human\n  labeler with a principled model critic, and GRPO drops the value head for a\n  group-relative baseline.\n- **The frontier:** the reward model is the binding constraint, so ensembles and\n  process rewards matter more than the RL loop; **verifiable rewards** (correct-answer\n  checks) sidestep reward hacking entirely and drive GRPO-trained reasoning models.\n\n[^ouyang-rlhf]: **Ouyang et al.**, _Training Language Models to Follow Instructions with Human Feedback_ (InstructGPT), NeurIPS 2022 — the three-stage RLHF pipeline: SFT, a Bradley-Terry reward model on comparisons, and PPO with a KL penalty.\n[^bradley-terry]: **Bradley & Terry**, _Rank Analysis of Incomplete Block Designs_, Biometrika 1952 — the paired-comparison model in which the probability one item beats another is a logistic of their latent-score difference.\n[^gao-overopt]: **Gao, Schulman & Hilton**, _Scaling Laws for Reward Model Overoptimization_, ICML 2023 — shows true quality peaks then falls as proxy reward is optimized, with the gap growing with KL distance from the reference.\n[^rafailov-dpo]: **Rafailov et al.**, _Direct Preference Optimization: Your Language Model is Secretly a Reward Model_, NeurIPS 2023 — derives the closed-form KL-regularized optimum and inverts it into a supervised log-sigmoid loss on preference pairs.\n[^bai-cai]: **Bai et al.**, _Constitutional AI: Harmlessness from AI Feedback_, 2022 — replaces human preference labels with an AI critic guided by a written set of principles (RLAIF), reducing human annotation to writing the constitution.\n[^shao-grpo]: **Shao et al.**, _DeepSeekMath: Pushing the Limits of Mathematical Reasoning_ (GRPO), 2024 — group-relative policy optimization uses a group of sampled responses' mean reward as the baseline, dropping the value head from the RL loop.\n",{"text":14757,"minutes":14758,"time":14759,"words":14760},"15 min read",14.77,886200,2954,{"title":5,"description":14746},[14763],{"book":14764,"ref":14765},"Chollet","Ch. 11 — aligning generative language models","available","06.deep-learning\u002F11.reinforcement-learning\u002F05.rl-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",[14751],"Go65z6KX9fk6Y_bxXd9qgCubKt-rCSaWwcUXGg6ap3A",{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":14772,"\u002Falgorithms\u002Ffoundations\u002Fproof-techniques":14773,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":14774,"\u002Falgorithms\u002Ffoundations\u002Fgrowth-rates-and-loop-analysis":14775,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":14776,"\u002Falgorithms\u002Ffoundations\u002Famortized-analysis":14777,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":14778,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":14779,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":14780,"\u002Falgorithms\u002Fdivide-and-conquer\u002Ffast-multiplication":14781,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":14782,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":14783,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":14784,"\u002Falgorithms\u002Fsorting\u002Fexternal-sorting":14785,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":14786,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":14787,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":14788,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":14789,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":14790,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":14791,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":14792,"\u002Falgorithms\u002Fdata-structures\u002Fspatial-data-structures":14793,"\u002Falgorithms\u002Fdata-structures\u002Fskip-lists-and-probabilistic-structures":14794,"\u002Falgorithms\u002Fdata-structures\u002Fb-trees":14795,"\u002Falgorithms\u002Fdata-structures\u002Fdata-stream-algorithms":14796,"\u002Falgorithms\u002Fdata-structures\u002Fstreaming-sketches":14797,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":14798,"\u002Falgorithms\u002Fsequences\u002Fprefix-sums":14799,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":14800,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":14801,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":14802,"\u002Falgorithms\u002Fsequences\u002Fkmp-and-z-function":14803,"\u002Falgorithms\u002Fsequences\u002Ftries":14804,"\u002Falgorithms\u002Fsequences\u002Fsuffix-arrays-and-aho-corasick":14805,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":14806,"\u002Falgorithms\u002Fgraphs\u002Fdepth-first-search":14807,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":14808,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":14809,"\u002Falgorithms\u002Fgraphs\u002Fkruskal-and-prim":14810,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":14811,"\u002Falgorithms\u002Fgraphs\u002Fall-pairs-and-negative-weights":14812,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":14813,"\u002Falgorithms\u002Fgraphs\u002Fmax-flow-min-cut":14814,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":14815,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":14816,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":14817,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":14818,"\u002Falgorithms\u002Fgraphs\u002Fbipartite-matching":14819,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":14820,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":14821,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":14822,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":14823,"\u002Falgorithms\u002Fgreedy\u002Fstable-matching":14824,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":14825,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":14826,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":14827,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":14828,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":14829,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":14830,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":14831,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":14832,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":14833,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":14834,"\u002Falgorithms\u002Fdynamic-programming\u002Fdigit-and-probability-dp":14835,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":14836,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":14837,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":14838,"\u002Falgorithms\u002Fbacktracking\u002Fgraph-backtracking":14839,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":14840,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":14841,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":14842,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":14843,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmatrix-exponentiation":14844,"\u002Falgorithms\u002Fmathematical-algorithms\u002Ffast-fourier-transform":14845,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fgradient-descent":14846,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":14847,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":14848,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":14849,"\u002Falgorithms\u002Fcomputational-geometry\u002Fpolygons-and-proximity":14850,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":14851,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":14852,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":14853,"\u002Falgorithms\u002Fintractability\u002Fapproximation-algorithms":14854,"\u002Falgorithms":14855,"\u002Fcalculus\u002Flimits-and-continuity\u002Ffunctions-and-models":14856,"\u002Fcalculus\u002Flimits-and-continuity\u002Fthe-limit-of-a-function":14857,"\u002Fcalculus\u002Flimits-and-continuity\u002Flimit-laws-and-the-precise-definition":14858,"\u002Fcalculus\u002Flimits-and-continuity\u002Fcontinuity":14859,"\u002Fcalculus\u002Fderivatives\u002Fthe-derivative-and-rates-of-change":14860,"\u002Fcalculus\u002Fderivatives\u002Fdifferentiation-rules-and-the-chain-rule":14861,"\u002Fcalculus\u002Fderivatives\u002Fimplicit-differentiation-and-related-rates":14862,"\u002Fcalculus\u002Fderivatives\u002Flinear-approximations-and-differentials":14863,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fextrema-and-the-mean-value-theorem":14864,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fhow-derivatives-shape-a-graph":14865,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fcurve-sketching-and-optimization":14866,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fnewtons-method-and-antiderivatives":14867,"\u002Fcalculus\u002Fintegrals\u002Farea-and-the-definite-integral":14868,"\u002Fcalculus\u002Fintegrals\u002Fthe-fundamental-theorem-of-calculus":14869,"\u002Fcalculus\u002Fintegrals\u002Fthe-substitution-rule":14870,"\u002Fcalculus\u002Fapplications-of-integration\u002Fareas-and-volumes":14871,"\u002Fcalculus\u002Fapplications-of-integration\u002Fwork-average-value-and-arc-length":14872,"\u002Fcalculus\u002Fapplications-of-integration\u002Fphysics-economics-and-probability":14873,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Finverse-functions-logarithms-and-exponentials":14874,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Fgrowth-decay-inverse-trig-and-hyperbolic-functions":14875,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Flhospitals-rule":14876,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fintegration-by-parts":14877,"\u002Fcalculus\u002Ftechniques-of-integration\u002Ftrigonometric-integrals-and-substitution":14878,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fpartial-fractions-and-integration-strategy":14879,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fapproximate-and-improper-integrals":14880,"\u002Fcalculus\u002Fparametric-and-polar\u002Fparametric-curves-and-their-calculus":14881,"\u002Fcalculus\u002Fparametric-and-polar\u002Fpolar-coordinates":14882,"\u002Fcalculus\u002Fparametric-and-polar\u002Fconic-sections":14883,"\u002Fcalculus\u002Fsequences-and-series\u002Fsequences":14884,"\u002Fcalculus\u002Fsequences-and-series\u002Fseries-and-the-integral-test":14885,"\u002Fcalculus\u002Fsequences-and-series\u002Fthe-convergence-tests":14886,"\u002Fcalculus\u002Fsequences-and-series\u002Fpower-series":14887,"\u002Fcalculus\u002Fsequences-and-series\u002Ftaylor-and-maclaurin-series":14888,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvectors-and-the-dot-product":14889,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fthe-cross-product-lines-and-planes":14870,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fcylinders-and-quadric-surfaces":14890,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvector-functions-and-space-curves":14891,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Farc-length-curvature-and-motion":14892,"\u002Fcalculus\u002Fpartial-derivatives\u002Ffunctions-of-several-variables":14860,"\u002Fcalculus\u002Fpartial-derivatives\u002Fpartial-derivatives":14893,"\u002Fcalculus\u002Fpartial-derivatives\u002Ftangent-planes-and-the-chain-rule":14894,"\u002Fcalculus\u002Fpartial-derivatives\u002Fdirectional-derivatives-and-the-gradient":14895,"\u002Fcalculus\u002Fpartial-derivatives\u002Foptimization-and-lagrange-multipliers":14896,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fdouble-integrals":14897,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Ftriple-integrals-and-coordinate-systems":14898,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fvector-fields-and-line-integrals":14899,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fgreens-theorem-curl-and-divergence":14900,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fsurface-integrals":14901,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fstokes-and-the-divergence-theorem":14902,"\u002Fcalculus":14903,"\u002Fmechanics\u002Ffoundations\u002Fmeasurement-and-dimensions":14904,"\u002Fmechanics\u002Ffoundations\u002Fvector-algebra":14905,"\u002Fmechanics\u002Fkinematics\u002Fone-dimensional-motion":14906,"\u002Fmechanics\u002Fkinematics\u002Fmotion-graphs":14907,"\u002Fmechanics\u002Fkinematics\u002Fprojectile-motion":14908,"\u002Fmechanics\u002Fkinematics\u002Frelative-motion":14909,"\u002Fmechanics\u002Fkinematics\u002Fcircular-motion":14910,"\u002Fmechanics\u002Fdynamics\u002Fnewtons-laws":14911,"\u002Fmechanics\u002Fdynamics\u002Ffree-body-diagrams":14912,"\u002Fmechanics\u002Fdynamics\u002Ffriction-and-curved-motion":14913,"\u002Fmechanics\u002Fdynamics\u002Fnumerical-dynamics":14914,"\u002Fmechanics\u002Fdynamics\u002Fcenter-of-mass-systems":14915,"\u002Fmechanics\u002Fenergy\u002Fwork-and-kinetic-energy":14916,"\u002Fmechanics\u002Fenergy\u002Fpotential-energy":14917,"\u002Fmechanics\u002Fenergy\u002Fmultiparticle-work":14918,"\u002Fmechanics\u002Fenergy\u002Fmass-energy-and-binding":14919,"\u002Fmechanics\u002Fenergy\u002Fphotons-and-quantization":14920,"\u002Fmechanics\u002Fmomentum\u002Fmomentum-and-collisions":14921,"\u002Fmechanics\u002Fmomentum\u002Fcenter-of-mass-collisions":14922,"\u002Fmechanics\u002Fmomentum\u002Frocket-propulsion":14923,"\u002Fmechanics\u002Frotation\u002Frotational-inertia":14924,"\u002Fmechanics\u002Frotation\u002Frotational-dynamics":14925,"\u002Fmechanics\u002Frotation\u002Frolling-motion":14926,"\u002Fmechanics\u002Frotation\u002Fangular-momentum":14927,"\u002Fmechanics\u002Frotation\u002Frolling-resistance":14928,"\u002Fmechanics\u002Frotation\u002Fgyroscopic-precession":14929,"\u002Fmechanics\u002Fgravity-and-matter\u002Fkeplerian-orbits":14930,"\u002Fmechanics\u002Fgravity-and-matter\u002Fgravitational-fields":14931,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstatic-equilibrium":14932,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-statics":14933,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-flow":14934,"\u002Fmechanics\u002Fgravity-and-matter\u002Forbital-motion":14935,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstress-and-elasticity":14936,"\u002Fmechanics\u002Foscillations-waves\u002Fdamped-oscillators":14937,"\u002Fmechanics\u002Foscillations-waves\u002Ftravelling-waves":14938,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-superposition":14939,"\u002Fmechanics\u002Foscillations-waves\u002Fstanding-waves":14940,"\u002Fmechanics\u002Foscillations-waves\u002Fsound-waves":14941,"\u002Fmechanics\u002Foscillations-waves\u002Fdoppler-effect":14942,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-packets":14943,"\u002Fmechanics\u002Foscillations-waves\u002Fbeats-and-coupling":14944,"\u002Fmechanics\u002Foscillations-waves\u002Fsimple-harmonic-motion":14945,"\u002Fmechanics\u002Foscillations-waves\u002Fpendulum-motion":14946,"\u002Fmechanics\u002Foscillations-waves\u002Fdriven-oscillators":14947,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-boundaries":14948,"\u002Fmechanics\u002Fthermodynamics\u002Fkinetic-theory-of-ideal-gases":14949,"\u002Fmechanics\u002Fthermodynamics\u002Ffirst-law-of-thermodynamics":14950,"\u002Fmechanics\u002Fthermodynamics\u002Fentropy-and-the-second-law":14951,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-processes":14952,"\u002Fmechanics\u002Fthermodynamics\u002Fphase-changes":14953,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-machines":14954,"\u002Fmechanics":14955,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcharge-and-conductors":14956,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcoulombs-law":14957,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-and-force":14958,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-maps":14959,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-dipoles":14960,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fcontinuous-charge-fields":14961,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fgauss-law-and-conductors":14962,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpoint-charge-potential":14963,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpotential-gradients-and-equipotentials":14964,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Felectrostatic-energy-and-pressure":14965,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Flaplace-boundary-problems":14966,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fcontinuous-charge-potentials":14967,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitance-fundamentals":14944,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-networks":14968,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-energy-and-force":14969,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fdielectric-polarization-and-breakdown":14970,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fcurrent-and-resistance":14940,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fkirchhoff-network-analysis":14805,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Frc-transients":14971,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-trajectories":14931,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fhall-effect":14972,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-force-on-conductors":14973,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-dipoles":14974,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmass-spectrometry":14975,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmoving-charge-fields":14976,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fbiot-savart-law":14977,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fcircular-current-loops":14978,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Famperes-law":14979,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fgauss-law-for-magnetism":14980,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmagnetic-materials":14905,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-flux":14981,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Ffaradays-law":14982,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Flenzs-law":14983,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmotional-emf":14984,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Feddy-currents":14985,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fself-inductance":14986,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-energy":14987,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Frl-circuits":14988,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-fundamentals":14923,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Freactance":14922,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Frlc-resonance":14989,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-power":14990,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Ftransformers":14991,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdisplacement-current":14992,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-waves":14993,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-momentum":14994,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdipole-radiation":14995,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fpolarization":14996,"\u002Felectricity-and-magnetism\u002Foptics\u002Freflection-and-refraction":14997,"\u002Felectricity-and-magnetism\u002Foptics\u002Fthin-lenses":14949,"\u002Felectricity-and-magnetism\u002Foptics\u002Fspherical-mirrors":14947,"\u002Felectricity-and-magnetism":14998,"\u002Flinear-algebra\u002Flinear-systems\u002Fsystems-and-echelon-forms":14999,"\u002Flinear-algebra\u002Flinear-systems\u002Fvector-and-matrix-equations":15000,"\u002Flinear-algebra\u002Flinear-systems\u002Fsolution-sets-and-applications":15001,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-independence":15002,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-transformations":15003,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-operations":15004,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-inverse-and-invertibility":15005,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fpartitioned-matrices-and-lu":15006,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fsubspaces-dimension-rank":15007,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fapplications-leontief-and-graphics":14857,"\u002Flinear-algebra\u002Fdeterminants\u002Fdeterminants-and-cofactors":15008,"\u002Flinear-algebra\u002Fdeterminants\u002Fproperties-of-determinants":15009,"\u002Flinear-algebra\u002Fdeterminants\u002Fcramer-volume-and-area":14861,"\u002Flinear-algebra\u002Fvector-spaces\u002Fvector-spaces-and-subspaces":15010,"\u002Flinear-algebra\u002Fvector-spaces\u002Fnull-and-column-spaces":15011,"\u002Flinear-algebra\u002Fvector-spaces\u002Fbases-and-independent-sets":15012,"\u002Flinear-algebra\u002Fvector-spaces\u002Fcoordinate-systems":15013,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdimension-and-rank":15014,"\u002Flinear-algebra\u002Fvector-spaces\u002Fchange-of-basis":15015,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdifference-equations-and-markov":15016,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-eigenvalues":15017,"\u002Flinear-algebra\u002Feigenvalues\u002Fthe-characteristic-equation":15018,"\u002Flinear-algebra\u002Feigenvalues\u002Fdiagonalization":15019,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-linear-transformations":15020,"\u002Flinear-algebra\u002Feigenvalues\u002Fcomplex-eigenvalues":15021,"\u002Flinear-algebra\u002Feigenvalues\u002Fdynamical-systems":15022,"\u002Flinear-algebra\u002Feigenvalues\u002Fpower-method":15023,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-length-orthogonality":15024,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Forthogonal-sets-and-projections":15025,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fgram-schmidt-and-qr":15026,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-problems":15027,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-applications":15028,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-spaces":15029,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fdiagonalizing-symmetric-matrices":14896,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fquadratic-forms":15030,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fconstrained-optimization":15031,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsingular-value-decomposition":15032,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsvd-applications-pca-imaging":15033,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-thinking-and-matrix-computation":15034,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Flu-and-cholesky":15035,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fconditioning-and-floating-point":15036,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fstability-and-error-analysis":15037,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fqr-and-numerical-least-squares":15038,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-eigenvalues-and-svd":15039,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-combinations":15040,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-independence-and-barycentric-coordinates":15041,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fconvex-combinations-and-convex-sets":15042,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fhyperplanes-and-polytopes":15043,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fcurves-and-surfaces":15044,"\u002Flinear-algebra":15045,"\u002Ftheory-of-computation":15046,"\u002Fcomputer-architecture\u002Ffoundations\u002Fbits-bytes-and-words":15047,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-representation":15048,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-arithmetic":15049,"\u002Fcomputer-architecture\u002Ffoundations\u002Ffloating-point":15050,"\u002Fcomputer-architecture\u002Ffoundations\u002Fboolean-algebra-and-bit-manipulation":15051,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fthe-machines-view":15052,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fdata-movement":15053,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farithmetic-and-logic":15054,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fcontrol-flow":15055,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fprocedures":15056,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farrays-structs-and-alignment":15057,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fmemory-layout-and-buffer-overflows":15058,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fwhat-an-isa-is":15059,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Finstruction-formats-and-operands":15060,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Faddressing-modes":15061,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fthe-y86-64-instruction-set":15062,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fy86-64-programming":15063,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Ftransistors-gates-and-boolean-functions":15064,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fcombinational-logic-and-hcl":15065,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmultiplexers-decoders-and-the-alu":15066,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmemory-elements-latches-flip-flops-and-clocking":15067,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fregister-files-and-random-access-memory":15068,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-fetch-decode-execute-cycle":15069,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-seq-stages":15070,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fcontrol-logic-and-sequencing":15071,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fassembling-seq":15072,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Ftracing-a-program":15073,"\u002Fcomputer-architecture\u002Fpipelining\u002Fpipelining-principles":15074,"\u002Fcomputer-architecture\u002Fpipelining\u002Ffrom-seq-to-pipe":15075,"\u002Fcomputer-architecture\u002Fpipelining\u002Fdata-hazards-stalling-and-forwarding":15076,"\u002Fcomputer-architecture\u002Fpipelining\u002Fcontrol-hazards-and-branch-prediction":15077,"\u002Fcomputer-architecture\u002Fpipelining\u002Fthe-complete-pipe-processor":15078,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fstorage-technologies-and-the-latency-gap":15079,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Flocality":15080,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-memories-direct-mapped":15081,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fset-associative-and-write-policies":15082,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-performance-and-cache-friendly-code":15083,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Faddress-spaces-and-translation":15084,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fpage-tables-and-page-faults":15085,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fthe-tlb-and-multi-level-page-tables":15086,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Fexceptional-control-flow":15087,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Finterrupts-and-the-kernel":15088,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fprocesses-threads-and-parallelism":15089,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fhardware-multithreading":15090,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fcache-coherence":15091,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmemory-consistency-and-synchronization":15092,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmulticore-organization":15093,"\u002Fcomputer-architecture\u002Fcapstone\u002Fthe-whole-machine":15094,"\u002Fcomputer-architecture\u002Fcapstone\u002Fassembling-a-complete-cpu":15095,"\u002Fcomputer-architecture":15046,"\u002Fdifferential-equations\u002Ffoundations\u002Fmodels-and-direction-fields":15096,"\u002Fdifferential-equations\u002Ffoundations\u002Fclassification-and-terminology":15097,"\u002Fdifferential-equations\u002Ffirst-order\u002Flinear-first-order-integrating-factors":15098,"\u002Fdifferential-equations\u002Ffirst-order\u002Fseparable-and-exact":14861,"\u002Fdifferential-equations\u002Ffirst-order\u002Fmodeling-first-order":15099,"\u002Fdifferential-equations\u002Ffirst-order\u002Fautonomous-and-population-dynamics":14860,"\u002Fdifferential-equations\u002Ffirst-order\u002Fexistence-uniqueness-euler":14867,"\u002Fdifferential-equations\u002Ffirst-order\u002Ffirst-order-difference-equations":15100,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhomogeneous-constant-coefficients":15101,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fcomplex-and-repeated-roots":14901,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fnonhomogeneous-undetermined-coefficients":15102,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fvariation-of-parameters":15103,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fmechanical-electrical-vibrations":15104,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhigher-order-linear":15105,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fpower-series-ordinary-points":15106,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fregular-singular-frobenius":15107,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fbessel-and-special-functions":15108,"\u002Fdifferential-equations\u002Flaplace\u002Flaplace-definition-ivps":15109,"\u002Fdifferential-equations\u002Flaplace\u002Fstep-impulse-convolution":15110,"\u002Fdifferential-equations\u002Fsystems\u002Fmatrices-eigenvalues-review":15111,"\u002Fdifferential-equations\u002Fsystems\u002Fconstant-coefficient-systems-phase-portraits":15112,"\u002Fdifferential-equations\u002Fsystems\u002Frepeated-eigenvalues-fundamental-matrices":15113,"\u002Fdifferential-equations\u002Fnumerical\u002Feuler-and-runge-kutta":15108,"\u002Fdifferential-equations\u002Fnumerical\u002Fmultistep-systems-stability":15114,"\u002Fdifferential-equations\u002Fnonlinear\u002Fphase-plane-autonomous-stability":15115,"\u002Fdifferential-equations\u002Fnonlinear\u002Flocally-linear-and-liapunov":15116,"\u002Fdifferential-equations\u002Fnonlinear\u002Fcompeting-species-predator-prey-limit-cycles":15117,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Ffourier-series":15118,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fheat-wave-laplace-equations":15119,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fsturm-liouville":15120,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fcalculus-of-variations":15121,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fhistorical-notes":15122,"\u002Fdifferential-equations":15123,"\u002Frelativity\u002Ffoundations\u002Fspecial-relativity-postulates":15124,"\u002Frelativity\u002Ffoundations\u002Florentz-transformation-spacetime":15125,"\u002Frelativity\u002Ffoundations\u002Ftime-dilation-length-contraction":15126,"\u002Frelativity\u002Ffoundations\u002Frelativistic-momentum-energy":15127,"\u002Frelativity\u002Ffoundations\u002Fgeneral-relativity":15006,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fminkowski-spacetime-and-the-interval":15128,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Ffour-vectors-and-index-notation":15129,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fthe-lorentz-group-and-rapidity":15130,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fdoppler-aberration-and-appearance":15131,"\u002Frelativity\u002Frelativistic-dynamics\u002Ffour-momentum-force-and-accelerated-motion":15132,"\u002Frelativity\u002Frelativistic-dynamics\u002Fparticle-decays-and-two-body-kinematics":15133,"\u002Frelativity\u002Frelativistic-dynamics\u002Fcollisions-thresholds-and-the-cm-frame":14876,"\u002Frelativity\u002Frelativistic-dynamics\u002Fmandelstam-variables-and-invariants":15134,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ffour-current-and-the-four-potential":15135,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fthe-electromagnetic-field-tensor":15136,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ftransformation-of-electric-and-magnetic-fields":15137,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fcovariant-maxwell-and-the-stress-energy-tensor":15138,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-equivalence-principle-formalized":15139,"\u002Frelativity\u002Fcurved-spacetime\u002Fmanifolds-vectors-and-the-metric":15140,"\u002Frelativity\u002Fcurved-spacetime\u002Fcovariant-derivative-and-christoffel-symbols":15141,"\u002Frelativity\u002Fcurved-spacetime\u002Fgeodesics-and-the-geodesic-equation":15142,"\u002Frelativity\u002Fcurved-spacetime\u002Fcurvature-riemann-and-geodesic-deviation":15143,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-einstein-field-equations":15098,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fthe-schwarzschild-metric":15144,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fgeodesics-and-orbits-in-schwarzschild":15145,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Flight-bending-and-null-geodesics":15146,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fperihelion-precession-of-mercury":15147,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fdeflection-of-light-and-gravitational-lensing":15148,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fgravitational-redshift-and-shapiro-delay":15027,"\u002Frelativity\u002Ftests-of-general-relativity\u002Frelativity-in-technology-gps":15149,"\u002Frelativity\u002Fblack-holes\u002Fhorizons-and-coordinate-singularities":15150,"\u002Frelativity\u002Fblack-holes\u002Frotating-and-charged-black-holes":15037,"\u002Frelativity\u002Fblack-holes\u002Fblack-hole-thermodynamics":15151,"\u002Frelativity\u002Fgravitational-waves\u002Flinearized-gravity-and-wave-solutions":15152,"\u002Frelativity\u002Fgravitational-waves\u002Fgeneration-and-the-quadrupole-formula":15153,"\u002Frelativity\u002Fgravitational-waves\u002Fdetection-ligo-and-the-first-events":15154,"\u002Frelativity\u002Fcosmological-bridge\u002Fthe-cosmological-principle-and-flrw-metric":15155,"\u002Frelativity\u002Fcosmological-bridge\u002Ffriedmann-equations-and-cosmic-dynamics":15156,"\u002Frelativity":15157,"\u002Fphysical-computing":15046,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fblackbody-radiation-and-the-planck-quantum":15158,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-photoelectric-effect-and-the-photon":15137,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fx-rays-and-the-compton-effect":15159,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-old-quantum-theory-bohr-and-sommerfeld":15160,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fde-broglie-waves-and-electron-diffraction":15161,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fwave-packets-and-the-probability-interpretation":15162,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fthe-uncertainty-principle":15163,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-schrodinger-equation-in-one-dimension":15164,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-free-particle-and-wave-packet-dynamics":15165,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fparticle-in-infinite-and-finite-square-wells":15113,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Foperators-expectation-values-and-the-harmonic-oscillator":15037,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-dirac-delta-potential":15166,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fbarrier-penetration-and-quantum-tunneling":15167,"\u002Fquantum-mechanics\u002Fformalism\u002Fhilbert-space-and-dirac-notation":15168,"\u002Fquantum-mechanics\u002Fformalism\u002Fobservables-hermitian-operators-and-eigenvalues":15169,"\u002Fquantum-mechanics\u002Fformalism\u002Fthe-postulates-and-quantum-measurement":15165,"\u002Fquantum-mechanics\u002Fformalism\u002Fposition-momentum-and-continuous-spectra":15170,"\u002Fquantum-mechanics\u002Fformalism\u002Fcommutators-and-the-generalized-uncertainty-principle":15145,"\u002Fquantum-mechanics\u002Fformalism\u002Ftime-evolution-schrodinger-and-heisenberg-pictures":14898,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fladder-operators-and-the-number-states":15171,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fcoherent-and-squeezed-states":15172,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fsymmetries-generators-and-conservation-laws":14878,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fparity-time-reversal-and-discrete-symmetries":15173,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Forbital-angular-momentum-and-spherical-harmonics":15174,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Fthe-angular-momentum-algebra":15175,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Faddition-of-angular-momenta-and-clebsch-gordan":15176,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-schrodinger-equation-in-three-dimensions":15177,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-hydrogen-atom":15178,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-isotropic-oscillator-and-hidden-symmetry":15179,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-half-pauli-matrices-and-stern-gerlach":15180,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-in-a-magnetic-field-precession-and-resonance":15181,"\u002Fquantum-mechanics\u002Fspin\u002Ftwo-level-systems-and-the-bloch-sphere":15137,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fidentical-particles-and-exchange-symmetry":15182,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fthe-pauli-principle-atoms-and-the-periodic-table":15183,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ftime-independent-perturbation-theory":15184,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ffine-structure-and-the-real-hydrogen-atom":15171,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-zeeman-and-stark-effects":14859,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-variational-method":15185,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-wkb-approximation":15186,"\u002Fquantum-mechanics":15187,"\u002Freal-analysis\u002Ffoundations\u002Fsets-logic-functions":15121,"\u002Freal-analysis\u002Ffoundations\u002Fordered-fields-completeness":15188,"\u002Freal-analysis\u002Ffoundations\u002Fabsolute-value-bounds":15189,"\u002Freal-analysis\u002Ffoundations\u002Fintervals-uncountability":15010,"\u002Freal-analysis\u002Fsequences-series\u002Fsequences-limits":15190,"\u002Freal-analysis\u002Fsequences-series\u002Flimit-laws-monotone":14896,"\u002Freal-analysis\u002Fsequences-series\u002Flimsup-bolzano-weierstrass":15191,"\u002Freal-analysis\u002Fsequences-series\u002Fcauchy-completeness":15192,"\u002Freal-analysis\u002Fsequences-series\u002Fseries-convergence":15033,"\u002Freal-analysis\u002Fsequences-series\u002Fabsolute-conditional-rearrangement":15143,"\u002Freal-analysis\u002Fmetric-spaces\u002Fmetric-spaces-norms":15193,"\u002Freal-analysis\u002Fmetric-spaces\u002Fopen-closed-sets":15194,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconvergence-completeness":15195,"\u002Freal-analysis\u002Fmetric-spaces\u002Fcompactness":15196,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconnectedness":15197,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-of-functions":15170,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuous-functions":15198,"\u002Freal-analysis\u002Fcontinuity\u002Fevt-ivt":15003,"\u002Freal-analysis\u002Fcontinuity\u002Funiform-continuity":15199,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuity-metric-spaces":15200,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-infinity-monotone":14857,"\u002Freal-analysis\u002Fdifferentiation\u002Fthe-derivative":15201,"\u002Freal-analysis\u002Fdifferentiation\u002Fmean-value-theorem":15202,"\u002Freal-analysis\u002Fdifferentiation\u002Ftaylors-theorem":15159,"\u002Freal-analysis\u002Fdifferentiation\u002Finverse-function-1d":14886,"\u002Freal-analysis\u002Friemann-integration\u002Fdarboux-integral":15037,"\u002Freal-analysis\u002Friemann-integration\u002Fintegrability-classes":15203,"\u002Freal-analysis\u002Friemann-integration\u002Fproperties-of-the-integral":15204,"\u002Freal-analysis\u002Friemann-integration\u002Ffundamental-theorem":15022,"\u002Freal-analysis\u002Friemann-integration\u002Flog-exp-improper":15147,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpointwise-uniform-convergence":15205,"\u002Freal-analysis\u002Ffunction-sequences\u002Finterchange-of-limits":15206,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpower-series-weierstrass":15207,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpicard-ode":15042,"\u002Freal-analysis\u002Fseveral-variables\u002Fdifferentiability-rn":15208,"\u002Freal-analysis\u002Fseveral-variables\u002Fgradient-chain-rule":15209,"\u002Freal-analysis\u002Fseveral-variables\u002Fhigher-derivatives-taylor-extrema":15210,"\u002Freal-analysis\u002Fseveral-variables\u002Finverse-implicit-theorems":15210,"\u002Freal-analysis\u002Fseveral-variables\u002Fmultiple-integrals":15211,"\u002Freal-analysis":15212,"\u002Fabstract-algebra\u002Ffoundations\u002Fsets-functions-relations":15213,"\u002Fabstract-algebra\u002Ffoundations\u002Fintegers-and-modular-arithmetic":15214,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fgroup-axioms-and-first-examples":15215,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fdihedral-and-symmetric-groups":15216,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fmatrix-and-quaternion-groups":15217,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fhomomorphisms-and-group-actions":15218,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fsubgroups-and-substructures":15219,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcyclic-groups":15220,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fgeneration-and-subgroup-lattices":15221,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcosets-lagrange-and-normal-subgroups":15222,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fisomorphism-theorems":15186,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcomposition-series-and-the-alternating-group":15223,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Factions-and-cayleys-theorem":15215,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fconjugation-and-the-class-equation":15132,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fsylow-theorems":15224,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fautomorphisms-and-simple-groups":15225,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fdirect-products-and-finite-abelian-groups":15226,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fsemidirect-products":15227,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fnilpotent-and-solvable-groups":15228,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fclassifying-small-groups":15229,"\u002Fabstract-algebra\u002Fring-theory\u002Frings-definitions-and-examples":15230,"\u002Fabstract-algebra\u002Fring-theory\u002Fideals-quotients-and-homomorphisms":15231,"\u002Fabstract-algebra\u002Fring-theory\u002Ffractions-and-the-chinese-remainder-theorem":15225,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Feuclidean-domains-pids-ufds":15232,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fpolynomial-rings-over-fields":15201,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fgauss-lemma-and-unique-factorization":15233,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Firreducibility-criteria-and-groebner":15234,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fintroduction-to-modules":15235,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ffree-modules-and-direct-sums":15236,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ftensor-products-and-exact-sequences":15237,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fvector-spaces-and-linear-maps":15238,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fstructure-theorem-over-pids":15239,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Frational-canonical-form":15240,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fjordan-canonical-form":15241,"\u002Fabstract-algebra\u002Ffield-theory\u002Ffield-extensions-and-algebraic-elements":15242,"\u002Fabstract-algebra\u002Ffield-theory\u002Fstraightedge-and-compass-constructions":14875,"\u002Fabstract-algebra\u002Ffield-theory\u002Fsplitting-fields-and-algebraic-closure":15243,"\u002Fabstract-algebra\u002Ffield-theory\u002Fseparable-and-cyclotomic-extensions":15244,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fthe-galois-correspondence":15117,"\u002Fabstract-algebra\u002Fgalois-theory\u002Ffinite-fields":15245,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fcyclotomic-and-abelian-extensions":15246,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fgalois-groups-of-polynomials":15174,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fsolvability-by-radicals-and-the-quintic":15246,"\u002Fabstract-algebra\u002Fcapstone\u002Fcommutative-algebra-and-algebraic-geometry":15247,"\u002Fabstract-algebra\u002Fcapstone\u002Frepresentation-and-character-theory":15248,"\u002Fabstract-algebra":15249,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fatomic-spectra-rutherford":15250,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-model-hydrogen":15251,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fx-ray-spectra-franck-hertz":15252,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-sommerfeld-old-quantum-theory":15253,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fold-quantum-theory-limits-wkb":15254,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fschrodinger-3d-hydrogen":15166,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fhydrogen-wave-functions":15255,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fradial-equation-in-full":15256,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fsymmetry-degeneracy-runge-lenz":15257,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fexpectation-values-virial":15258,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fquantum-defects-alkali-spectra":15259,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Frydberg-atoms":15260,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Frelativistic-kinetic-correction":15261,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fspin-orbit-thomas-precession":15000,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdarwin-term-fine-structure-formula":15129,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdirac-equation-hydrogen":14875,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Flamb-shift-qed":15262,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fhyperfine-structure-21cm":14859,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fnuclear-effects-isotope-shift":15263,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fperiodic-table-atomic-spectra":15264,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fcentral-field-self-consistent":14897,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fidentical-particles-hartree-fock":15191,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhelium-two-electron-atom":15265,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fls-jj-coupling-term-symbols":15266,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhund-rules-ground-terms":15267,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fzeeman-effect":15268,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fpaschen-back-intermediate":15269,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fstark-effect-polarizability":14750,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Ftime-dependent-perturbation-golden-rule":15270,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fdipole-approximation-einstein-coefficients":15271,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fselection-rules-forbidden-transitions":15272,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Flifetimes-and-line-shapes":15273,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Flaser-principles":15274,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fspectroscopy-techniques":15275,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fline-catalog-nist-asd":15276,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Flaser-cooling-doppler":15277,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fsub-doppler-trapping":15222,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fbose-einstein-condensation":15278,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Foptical-clocks-precision":15279,"\u002Fatomic-physics":15280,"\u002Fdatabases":15046,"\u002Fcategory-theory\u002Ffoundations\u002Fwhat-is-a-category":15281,"\u002Fcategory-theory\u002Ffoundations\u002Fexamples-of-categories":15282,"\u002Fcategory-theory\u002Ffoundations\u002Fspecial-morphisms":15283,"\u002Fcategory-theory\u002Ffoundations\u002Ffunctors":15213,"\u002Fcategory-theory\u002Ffoundations\u002Fnatural-transformations":15284,"\u002Fcategory-theory\u002Ffoundations\u002Fsize-and-set-theory":15285,"\u002Fcategory-theory\u002Funiversal-properties\u002Funiversal-properties":15286,"\u002Fcategory-theory\u002Funiversal-properties\u002Fproducts-and-coproducts":15287,"\u002Fcategory-theory\u002Funiversal-properties\u002Fconstructions-on-categories":15288,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Frepresentable-functors":15289,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-lemma":15290,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-consequences":15291,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits":15292,"\u002Fcategory-theory\u002Flimits-colimits\u002Fproducts-equalizers-pullbacks":15293,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcolimits":15294,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcomputing-limits":15295,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits-and-functors":15296,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions":15297,"\u002Fcategory-theory\u002Fadjunctions\u002Funits-and-counits":15298,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions-via-universal-arrows":15299,"\u002Fcategory-theory\u002Fadjunctions\u002Ffree-forgetful-adjunctions":15300,"\u002Fcategory-theory\u002Fadjoints-limits\u002Flimits-via-adjoints":15301,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fpresheaf-limits-colimits":15302,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoints-preserve-limits":15303,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoint-functor-theorem":15294,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fmonads":15304,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-eilenberg-moore":15305,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fkleisli-and-programming":15306,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-for-endofunctors":15307,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Fcartesian-closed-categories":15308,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Flambda-calculus-correspondence":15254,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Ffixed-points-and-recursion":15309,"\u002Fcategory-theory":15310,"\u002Fdeep-learning\u002Fmathematical-background\u002Flinear-algebra-for-deep-learning":15311,"\u002Fdeep-learning\u002Fmathematical-background\u002Fprobability-and-information-theory":15312,"\u002Fdeep-learning\u002Fmathematical-background\u002Fnumerical-computation":15313,"\u002Fdeep-learning\u002Fmathematical-background\u002Fcalculus":15314,"\u002Fdeep-learning\u002Ffoundations\u002Fwhat-is-deep-learning":15315,"\u002Fdeep-learning\u002Ffoundations\u002Fmachine-learning-refresher":15316,"\u002Fdeep-learning\u002Ffoundations\u002Flinear-models-and-the-perceptron":15275,"\u002Fdeep-learning\u002Fneural-networks\u002Fthe-multilayer-perceptron":15317,"\u002Fdeep-learning\u002Fneural-networks\u002Factivation-functions":15318,"\u002Fdeep-learning\u002Fneural-networks\u002Funiversal-approximation":15319,"\u002Fdeep-learning\u002Fneural-networks\u002Fbackpropagation":15320,"\u002Fdeep-learning\u002Fneural-networks\u002Floss-functions-and-output-units":15321,"\u002Fdeep-learning\u002Foptimization\u002Fgradient-descent-and-sgd":15322,"\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods":15323,"\u002Fdeep-learning\u002Foptimization\u002Finitialization":15324,"\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape":15325,"\u002Fdeep-learning\u002Foptimization\u002Fsecond-order-and-approximate-methods":15326,"\u002Fdeep-learning\u002Fregularization\u002Fregularization-overview":15327,"\u002Fdeep-learning\u002Fregularization\u002Fdropout-and-data-augmentation":15328,"\u002Fdeep-learning\u002Fregularization\u002Fearly-stopping-and-parameter-sharing":15329,"\u002Fdeep-learning\u002Fregularization\u002Fnormalization":15330,"\u002Fdeep-learning\u002Farchitectures\u002Fconvolutional-networks":15331,"\u002Fdeep-learning\u002Farchitectures\u002Fcnn-architectures":15332,"\u002Fdeep-learning\u002Farchitectures\u002Frecurrent-networks":15333,"\u002Fdeep-learning\u002Farchitectures\u002Flstm-and-gru":15334,"\u002Fdeep-learning\u002Farchitectures\u002Fattention-and-transformers":15335,"\u002Fdeep-learning\u002Farchitectures\u002Fthe-transformer-architecture":15336,"\u002Fdeep-learning\u002Farchitectures\u002Ftransformers-in-practice":15337,"\u002Fdeep-learning\u002Farchitectures\u002Fgraph-neural-networks":15338,"\u002Fdeep-learning\u002Farchitectures\u002Fstate-space-models":15339,"\u002Fdeep-learning\u002Ftheory\u002Fgeneralization-theory":15340,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-robustness":15341,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-defenses":15342,"\u002Fdeep-learning\u002Ftheory\u002Fbayesian-and-ensemble-methods":15343,"\u002Fdeep-learning\u002Ftheory\u002Fdeep-equilibrium-models":15287,"\u002Fdeep-learning\u002Fgenerative-models\u002Flinear-factor-models":15344,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoencoders":15345,"\u002Fdeep-learning\u002Fgenerative-models\u002Fvariational-autoencoders":15346,"\u002Fdeep-learning\u002Fgenerative-models\u002Fgenerative-adversarial-networks":15347,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoregressive-and-normalizing-flows":15348,"\u002Fdeep-learning\u002Fgenerative-models\u002Fenergy-based-and-boltzmann-machines":15349,"\u002Fdeep-learning\u002Fgenerative-models\u002Fdiffusion-and-score-based-models":15350,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fstructured-probabilistic-models":14840,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fmonte-carlo-and-mcmc":15351,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fapproximate-inference":15352,"\u002Fdeep-learning\u002Fpractical\u002Fpractical-methodology":15076,"\u002Fdeep-learning\u002Fpractical\u002Fhyperparameters-and-debugging":15353,"\u002Fdeep-learning\u002Fpractical\u002Frepresentation-learning":15354,"\u002Fdeep-learning\u002Fpractical\u002Ftransfer-learning":15355,"\u002Fdeep-learning\u002Fpractical\u002Fapplications":15356,"\u002Fdeep-learning\u002Fpractical\u002Fmodel-compression-and-distillation":15357,"\u002Fdeep-learning\u002Fpractical\u002Fmeta-learning-and-few-shot":15358,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models":15359,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment":15360,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fseq2seq-pretraining-and-bart":15361,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ftext-to-text-transfer-and-conditional-generation":15362,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fspeech-and-audio-models":15363,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fself-supervised-speech-and-synthesis":15364,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents":15049,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fagent-memory-retrieval-and-orchestration":15365,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmixture-of-experts":15366,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmultimodal-models":15367,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ffusion-and-vision-language-models":15368,"\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning":15086,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fmodel-free-prediction-and-control":15369,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fdeep-q-networks":15370,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic":15371,"\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback":14760,"\u002Fdeep-learning":15046,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fequilibrium-state-variables-zeroth-law":15372,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Ffirst-law-heat-and-work":15171,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fsecond-law-entropy-and-the-carnot-bound":15373,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fthermodynamic-potentials-and-maxwell-relations":15374,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fstability-response-functions-and-the-third-law":15375,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fclassical-statistics-and-equipartition":15376,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fphase-space-and-liouvilles-theorem":15377,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fensembles-and-the-equal-probability-postulate":15378,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fstatistical-entropy-boltzmann-and-gibbs":15379,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fmicrocanonical-ensemble-and-entropy":15380,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fequilibrium-conditions-temperature-pressure-chemical-potential":15218,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fideal-gas-phase-space-and-the-sackur-tetrode-entropy":15381,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Ftwo-state-systems-paramagnets-and-negative-temperature":15382,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fcanonical-ensemble-and-the-boltzmann-distribution":15383,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fpartition-function-and-the-helmholtz-free-energy":15105,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fenergy-fluctuations-and-ensemble-equivalence":14898,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fthe-einstein-solid-and-harmonic-systems":15384,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fparamagnetism-and-the-schottky-anomaly":15385,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fideal-gas-partition-function-and-the-gibbs-paradox":15386,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fequipartition-and-the-virial-theorem":15032,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fmolecular-gases-rotation-and-vibration":15387,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fgrand-canonical-ensemble-and-the-grand-partition-function":15388,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fchemical-potential-fugacity-and-number-fluctuations":15110,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fensemble-summary-and-the-thermodynamic-web":15037,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fquantum-statistics-bose-einstein-and-fermi-dirac":15389,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fderiving-the-quantum-distributions":15006,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fthe-classical-limit-and-quantum-concentration":14876,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fideal-quantum-gases-general-framework":14892,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-and-the-fermion-gas":15390,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthe-photon-gas-and-plancks-radiation-law":15391,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fblackbody-thermodynamics-and-radiation-pressure":15392,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fphonons-and-the-debye-model":15024,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-derived":15393,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthermodynamics-of-the-bose-gas-and-superfluidity":14886,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fthe-ideal-fermi-gas-at-zero-temperature":15394,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fsommerfeld-expansion-and-electrons-in-metals":15395,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":15396,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fneutron-stars-and-nuclear-matter":15397,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-cluster-expansion-and-virial-coefficients":15220,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-van-der-waals-gas-and-liquid-gas-coexistence":15398,"\u002Fstatistical-mechanics\u002Finteractions\u002Fquantum-gases-with-interactions-and-exchange":15399,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fphases-coexistence-and-classification":15400,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-ising-model-and-exact-solutions":15401,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fmean-field-theory-and-the-weiss-model":15166,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fcritical-exponents-and-landau-theory":15402,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-renormalization-group-idea":15144,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fthermodynamic-fluctuations-and-response":15403,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fbrownian-motion-and-the-langevin-equation":15005,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Flinear-response-and-the-fluctuation-dissipation-theorem":15404,"\u002Fstatistical-mechanics":15405,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fbonding-mechanisms":15406,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fmolecular-orbitals-and-h2-plus":14871,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fhydrogen-molecule-and-exchange":15131,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fvan-der-waals-forces":15407,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Frotational-vibrational-spectra":15408,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Fanharmonicity-and-rovibrational-structure":15409,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Framan-and-electronic-bands":15410,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Flasers-and-masers":15411,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fstructure-of-solids":15412,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fbravais-lattices-and-crystal-systems":15004,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Freciprocal-lattice-and-brillouin-zones":15413,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fdiffraction-and-structure-factors":15414,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonon-dispersion":15415,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonons-quantization-and-dos":15416,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fdebye-einstein-heat-capacity":15147,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fanharmonicity-and-thermal-transport":15417,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ffree-electron-gas-and-conduction":15418,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fsommerfeld-model-and-heat-capacity":15419,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ftransport-and-the-hall-effect":15420,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fscreening-and-plasmons":15421,"\u002Fcondensed-matter\u002Fband-theory\u002Fblochs-theorem-and-energy-bands":15202,"\u002Fcondensed-matter\u002Fband-theory\u002Fnearly-free-electron-model":15146,"\u002Fcondensed-matter\u002Fband-theory\u002Ftight-binding-method":15422,"\u002Fcondensed-matter\u002Fband-theory\u002Ffermi-surfaces-and-semiclassical-dynamics":15423,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fsemiconductor-bands-and-junctions":15424,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fintrinsic-and-extrinsic-semiconductors":15425,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fcarrier-transport-and-recombination":15137,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fthe-pn-junction":15426,"\u002Fcondensed-matter\u002Fsemiconductors\u002Ftransistors-and-optoelectronics":15427,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fdielectrics-and-polarization":15372,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fferroelectrics-and-piezoelectrics":15273,"\u002Fcondensed-matter\u002Fmagnetism\u002Fdiamagnetism-and-paramagnetism":15001,"\u002Fcondensed-matter\u002Fmagnetism\u002Fexchange-and-ferromagnetism":15428,"\u002Fcondensed-matter\u002Fmagnetism\u002Fantiferromagnetism-and-domains":14867,"\u002Fcondensed-matter\u002Fmagnetism\u002Fspin-waves-and-magnons":15429,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fsuperconductivity-phenomenology":15430,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Flondon-theory-and-the-meissner-effect":15012,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fginzburg-landau-theory":15431,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fbcs-theory":15267,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fjosephson-and-high-tc":15432,"\u002Fcondensed-matter\u002Fnanostructures\u002Fquantum-wells-wires-and-dots":14859,"\u002Fcondensed-matter\u002Fnanostructures\u002Finteger-quantum-hall-effect":15433,"\u002Fcondensed-matter\u002Fnanostructures\u002Ffractional-quantum-hall-and-topology":14870,"\u002Fcondensed-matter\u002Fnanostructures\u002Fgraphene-and-dirac-materials":15434,"\u002Fcondensed-matter":15187,"\u002Flogic\u002Ffoundations\u002Flogic-as-a-mathematical-model":15435,"\u002Flogic\u002Fsentential-logic\u002Fformal-languages-and-well-formed-formulas":15436,"\u002Flogic\u002Fsentential-logic\u002Ftruth-assignments-and-tautologies":15437,"\u002Flogic\u002Fsentential-logic\u002Funique-readability-and-parsing":15438,"\u002Flogic\u002Fsentential-logic\u002Finduction-and-recursion":14884,"\u002Flogic\u002Fsentential-logic\u002Fexpressive-completeness-and-normal-forms":15439,"\u002Flogic\u002Fsentential-logic\u002Fboolean-circuits":15440,"\u002Flogic\u002Fsentential-logic\u002Fcompactness-and-effectiveness":14884,"\u002Flogic\u002Ffirst-order-languages\u002Ffirst-order-languages":15441,"\u002Flogic\u002Ffirst-order-languages\u002Fstructures-truth-and-satisfaction":15296,"\u002Flogic\u002Ffirst-order-languages\u002Fdefinability-and-elementary-equivalence":15442,"\u002Flogic\u002Ffirst-order-languages\u002Fterms-substitution-and-parsing":15443,"\u002Flogic\u002Fdeductive-calculus\u002Fa-deductive-calculus":15444,"\u002Flogic\u002Fdeductive-calculus\u002Fdeduction-theorem-and-derived-rules":15442,"\u002Flogic\u002Fdeductive-calculus\u002Fsoundness":15445,"\u002Flogic\u002Fdeductive-calculus\u002Fcompleteness-and-consistency":15446,"\u002Flogic\u002Fmodels-and-theories\u002Fcompactness-and-lowenheim-skolem":15447,"\u002Flogic\u002Fmodels-and-theories\u002Ftheories-elementary-classes-and-categoricity":15448,"\u002Flogic\u002Fmodels-and-theories\u002Finterpretations-between-theories":15449,"\u002Flogic\u002Fmodels-and-theories\u002Fnonstandard-analysis":15450,"\u002Flogic\u002Farithmetic-and-definability\u002Fdefinability-in-arithmetic":15451,"\u002Flogic\u002Farithmetic-and-definability\u002Fnatural-numbers-with-successor":15452,"\u002Flogic\u002Farithmetic-and-definability\u002Fpresburger-and-reducts":15373,"\u002Flogic\u002Farithmetic-and-definability\u002Fa-subtheory-and-representability":15453,"\u002Flogic\u002Fincompleteness\u002Farithmetization-of-syntax":15446,"\u002Flogic\u002Fincompleteness\u002Fincompleteness-and-undecidability":15454,"\u002Flogic\u002Fincompleteness\u002Fsecond-incompleteness-theorem":15455,"\u002Flogic\u002Fcomputability-and-representability\u002Frecursive-functions":15090,"\u002Flogic\u002Fcomputability-and-representability\u002Frepresenting-exponentiation":15456,"\u002Flogic\u002Fsecond-order-logic\u002Fsecond-order-languages":15274,"\u002Flogic\u002Fsecond-order-logic\u002Fskolem-functions-and-many-sorted-logic":15457,"\u002Flogic\u002Fsecond-order-logic\u002Fgeneral-structures":15458,"\u002Flogic":15459,"\u002Freinforcement-learning\u002Ffoundations\u002Fwhat-is-reinforcement-learning":15460,"\u002Freinforcement-learning\u002Ffoundations\u002Fa-brief-history-of-rl":15461,"\u002Freinforcement-learning\u002Ffoundations\u002Fmulti-armed-bandits":15076,"\u002Freinforcement-learning\u002Ffoundations\u002Fbandit-exploration-algorithms":15462,"\u002Freinforcement-learning\u002Ffoundations\u002Fmarkov-decision-processes":15463,"\u002Freinforcement-learning\u002Ffoundations\u002Fvalue-functions-and-optimality":15464,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdynamic-programming":15465,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdp-async-and-gpi":15455,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods":15466,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-off-policy":15467,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftemporal-difference-learning":15468,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftd-control-sarsa-and-q-learning":15367,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping":15469,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-off-policy-methods":15470,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-and-learning":15471,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-focusing-and-decision-time":15472,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdecision-time-planning":15473,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-tree-search":15093,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-prediction":15474,"\u002Freinforcement-learning\u002Fapproximation\u002Ffeature-construction-and-nonlinear":15475,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-control":15476,"\u002Freinforcement-learning\u002Fapproximation\u002Faverage-reward-control":15477,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad":15283,"\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td":15478,"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces":15454,"\u002Freinforcement-learning\u002Fapproximation\u002Ftrue-online-and-sarsa-lambda":15479,"\u002Freinforcement-learning\u002Fapproximation\u002Fpolicy-gradient-methods":15480,"\u002Freinforcement-learning\u002Fapproximation\u002Factor-critic-and-continuous-actions":15481,"\u002Freinforcement-learning\u002Fapproximation\u002Fleast-squares-and-memory-based-methods":15104,"\u002Freinforcement-learning\u002Fapproximation\u002Fmemory-and-kernel-methods":15482,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces":15234,"\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces":15483,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks":15484,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdqn-improvements":15066,"\u002Freinforcement-learning\u002Fdeep-rl\u002Factor-critic-and-ppo":15485,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fppo-and-continuous-control":15486,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fcase-studies":15487,"\u002Freinforcement-learning\u002Fdeep-rl\u002Frl-beyond-games":15488,"\u002Freinforcement-learning\u002Fdeep-rl\u002Ffrontiers":15489,"\u002Freinforcement-learning\u002Fdeep-rl\u002Freward-design-and-open-problems":15343,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow":15490,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow-part-2":15491,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control":15492,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control-part-2":15377,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl":15493,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl-part-2":15494,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration":15495,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration-part-2":15093,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl":15160,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2":15496,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl":15497,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl-part-2":15498,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl":15499,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl-part-2":15500,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl":15501,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl-part-2":15502,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Frlhf-and-language-models":15503,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps":15504,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps-part-2":15505,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl":15506,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl-part-2":15507,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmeta-rl-and-generalization":15508,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fpsychology-of-reinforcement":15509,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Finstrumental-conditioning-and-control":15510,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-and-td-error":15511,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-in-the-brain":15512,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fanimal-learning-and-cognition":15513,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fcognitive-maps-and-planning":15514,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fneuroscience-of-reinforcement":15515,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fseveral-learning-systems":15516,"\u002Freinforcement-learning":15046,"\u002Fartificial-intelligence\u002Ffoundations\u002Fwhat-is-ai":15517,"\u002Fartificial-intelligence\u002Ffoundations\u002Ffoundations-of-ai":15518,"\u002Fartificial-intelligence\u002Ffoundations\u002Fintelligent-agents":15519,"\u002Fartificial-intelligence\u002Ffoundations\u002Fagent-architectures":15520,"\u002Fartificial-intelligence\u002Fsearch\u002Funinformed-search":15521,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-strategies-compared":15522,"\u002Fartificial-intelligence\u002Fsearch\u002Finformed-search":15523,"\u002Fartificial-intelligence\u002Fsearch\u002Fheuristic-functions":15524,"\u002Fartificial-intelligence\u002Fsearch\u002Flocal-search":15525,"\u002Fartificial-intelligence\u002Fsearch\u002Fpopulation-and-continuous-search":15526,"\u002Fartificial-intelligence\u002Fsearch\u002Fadversarial-search":15527,"\u002Fartificial-intelligence\u002Fsearch\u002Fgames-of-chance-and-imperfect-information":15528,"\u002Fartificial-intelligence\u002Fsearch\u002Fconstraint-satisfaction":15529,"\u002Fartificial-intelligence\u002Fsearch\u002Fcsp-search-and-structure":14760,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-under-uncertainty":15228,"\u002Fartificial-intelligence\u002Fsearch\u002Fbelief-state-and-online-search":15530,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-logic":15531,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-inference":15532,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic":15533,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic-in-use":15534,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Finference-and-resolution":15535,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-resolution":15354,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fclassical-planning":15536,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-graphs-and-graphplan":15537,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-in-the-real-world":15538,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-under-uncertainty":15539,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fknowledge-representation":15540,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Freasoning-systems-and-defaults":15541,"\u002Fartificial-intelligence\u002Funcertainty\u002Fprobability-and-bayes":15542,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayes-rule-and-naive-bayes":15543,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayesian-networks":15544,"\u002Fartificial-intelligence\u002Funcertainty\u002Finference-in-bayesian-networks":15545,"\u002Fartificial-intelligence\u002Funcertainty\u002Freasoning-over-time":15546,"\u002Fartificial-intelligence\u002Funcertainty\u002Ftracking-and-data-association":15547,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmaking-decisions":15286,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmarkov-decision-processes":15537,"\u002Fartificial-intelligence\u002Funcertainty\u002Fdecision-networks-and-game-theory":15548,"\u002Fartificial-intelligence\u002Funcertainty\u002Fgame-theory-and-mechanism-design":14827,"\u002Fartificial-intelligence\u002Flearning\u002Flearning-from-examples":15549,"\u002Fartificial-intelligence\u002Flearning\u002Ftheory-and-model-families":15550,"\u002Fartificial-intelligence\u002Flearning\u002Fprobabilistic-learning":15551,"\u002Fartificial-intelligence\u002Flearning\u002Fexpectation-maximization":15552,"\u002Fartificial-intelligence\u002Flearning\u002Freinforcement-learning":15553,"\u002Fartificial-intelligence\u002Flearning\u002Fgeneralization-and-policy-search":15335,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-in-learning":15554,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-based-learning-methods":15555,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fvision-and-perception":15556,"\u002Fartificial-intelligence\u002Ffrontiers\u002Freconstructing-the-3d-world":15557,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobotics":15558,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobot-planning-and-control":15559,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnatural-language-in-ai":15560,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnlp-grammar-translation-and-speech":15561,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fphilosophy-and-future":15562,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fai-ethics-and-future":15563,"\u002Fartificial-intelligence":15046,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-constituents-nuclide-chart":15302,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-size-charge-distributions":15564,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-masses-binding-energy":14863,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fsemi-empirical-mass-formula":14861,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-moments-multipoles":15395,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnuclear-force-shell-overview":15565,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fthe-deuteron":14889,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnucleon-nucleon-scattering":15199,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fmeson-theory-isospin":15566,"\u002Fnuclear-physics\u002Fnuclear-models\u002Ffermi-gas-model":15567,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fliquid-drop-collective-coordinates":15568,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fshell-model-single-particle":15259,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fcollective-model-rotations-vibrations":15569,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-law-modes":15570,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-kinetics-equilibrium":15571,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-decay-gamow-theory":15456,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-fine-structure-hindrance":15572,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fbeta-decay-energetics-neutrino":15573,"\u002Fnuclear-physics\u002Fbeta-decay\u002Ffermi-theory-beta-decay":14873,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fweak-interaction-parity-violation":15102,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fdouble-beta-decay-neutrino-mass":15574,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fgamma-multipole-radiation":15203,"\u002Fnuclear-physics\u002Fgamma-decay\u002Finternal-conversion-isomers":15575,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fangular-correlations-mossbauer":15576,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Freaction-kinematics-cross-sections":15098,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fcompound-nucleus-resonances":15186,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fdirect-reactions-optical-model":15577,"\u002Fnuclear-physics\u002Ffission\u002Ffission-barrier-dynamics":15578,"\u002Fnuclear-physics\u002Ffission\u002Fchain-reactions-reactor-physics":15579,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Ffusion-reactions-confinement":14868,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fstellar-nucleosynthesis":15243,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fbig-bang-nucleosynthesis":15105,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fcharged-particle-stopping-power":15580,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fphoton-neutron-interactions":15153,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fradiation-detectors":15217,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fdosimetry-radiation-biology":15581,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fnuclear-applications-dating-medicine":15582,"\u002Fnuclear-physics":15583,"\u002Fnatural-language-processing\u002Ffoundations\u002Fwhat-is-nlp":15584,"\u002Fnatural-language-processing\u002Ffoundations\u002Fregex-and-text-normalization":15585,"\u002Fnatural-language-processing\u002Ffoundations\u002Fminimum-edit-distance":15224,"\u002Fnatural-language-processing\u002Ffoundations\u002Fn-gram-language-models":15586,"\u002Fnatural-language-processing\u002Ffoundations\u002Fsmoothing-and-backoff":15587,"\u002Fnatural-language-processing\u002Fclassification\u002Fnaive-bayes-and-sentiment":15588,"\u002Fnatural-language-processing\u002Fclassification\u002Fevaluating-classifiers":15072,"\u002Fnatural-language-processing\u002Fclassification\u002Flogistic-regression":15589,"\u002Fnatural-language-processing\u002Fclassification\u002Fsentiment-and-affect-lexicons":15590,"\u002Fnatural-language-processing\u002Fsemantics\u002Fvector-semantics-and-embeddings":15368,"\u002Fnatural-language-processing\u002Fsemantics\u002Fstatic-word-embeddings":15591,"\u002Fnatural-language-processing\u002Fsemantics\u002Fneural-language-models":15536,"\u002Fnatural-language-processing\u002Fsequences\u002Fsequence-labeling":15592,"\u002Fnatural-language-processing\u002Fsequences\u002Fcrfs-and-neural-taggers":15593,"\u002Fnatural-language-processing\u002Fsequences\u002Frnns-and-lstms":15594,"\u002Fnatural-language-processing\u002Ftransformers\u002Ftransformers-and-attention":15595,"\u002Fnatural-language-processing\u002Ftransformers\u002Fthe-transformer-architecture":15596,"\u002Fnatural-language-processing\u002Ftransformers\u002Flarge-language-models":15597,"\u002Fnatural-language-processing\u002Ftransformers\u002Fllm-pretraining-and-scaling":15598,"\u002Fnatural-language-processing\u002Ftransformers\u002Ffine-tuning-and-prompting":15050,"\u002Fnatural-language-processing\u002Ftransformers\u002Fprompting-and-alignment":15599,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-parsing":15600,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcky-scoring-and-evaluation":15542,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdependency-parsing":15601,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fgraph-based-and-neural-dependency-parsing":15602,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fword-senses-and-wsd":15603,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fwsd-in-practice-and-induction":15604,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-roles-and-information-extraction":15605,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Frelations-events-and-templates":15606,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoreference-and-discourse":15607,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoherence-and-discourse-structure":15608,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Flogical-semantics":15466,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcompositional-semantics-and-description-logics":15609,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-parsing":15610,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fneural-semantic-parsing":15611,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Finformation-extraction":15612,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftimes-events-and-templates":15613,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdiscourse-coherence":15614,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fentity-based-and-global-coherence":15615,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-grammars":15616,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftreebanks-and-lexicalized-grammars":15617,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation":15618,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation-decoding-and-evaluation":15619,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering":15620,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering-knowledge-and-llms":15321,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-and-chatbots":15487,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-systems-and-assistants":15078,"\u002Fnatural-language-processing\u002Fapplications\u002Ftext-summarization":15621,"\u002Fnatural-language-processing\u002Fapplications\u002Fabstractive-summarization-and-evaluation":15622,"\u002Fnatural-language-processing\u002Fspeech\u002Fphonetics":15623,"\u002Fnatural-language-processing\u002Fspeech\u002Facoustic-phonetics":15624,"\u002Fnatural-language-processing\u002Fspeech\u002Fautomatic-speech-recognition":15335,"\u002Fnatural-language-processing\u002Fspeech\u002Fasr-evaluation-and-applications":15625,"\u002Fnatural-language-processing":15046,"\u002Fparticle-physics\u002Ffoundations\u002Fhistorical-overview-particle-zoo":15626,"\u002Fparticle-physics\u002Ffoundations\u002Fparticle-physics-basic-concepts":15244,"\u002Fparticle-physics\u002Ffoundations\u002Ffundamental-interactions-force-carriers":15627,"\u002Fparticle-physics\u002Funits-kinematics\u002Fnatural-units-and-scales":15387,"\u002Fparticle-physics\u002Funits-kinematics\u002Ffour-vectors-invariant-mass":15628,"\u002Fparticle-physics\u002Funits-kinematics\u002Fdecay-scattering-kinematics-mandelstam":15629,"\u002Fparticle-physics\u002Funits-kinematics\u002Fcross-sections-golden-rule":15577,"\u002Fparticle-physics\u002Fsymmetries\u002Fconservation-laws-symmetries":15630,"\u002Fparticle-physics\u002Fsymmetries\u002Fdiscrete-symmetries-cpt":15631,"\u002Fparticle-physics\u002Fsymmetries\u002Fparity-violation-weak":15194,"\u002Fparticle-physics\u002Fsymmetries\u002Fsu2-su3-flavor-symmetry":14896,"\u002Fparticle-physics\u002Fquark-model\u002Feightfold-way-su3":15632,"\u002Fparticle-physics\u002Fquark-model\u002Fmeson-spectroscopy":15148,"\u002Fparticle-physics\u002Fquark-model\u002Fbaryon-spectroscopy":15633,"\u002Fparticle-physics\u002Fquark-model\u002Fcolor-confinement-exotics":15181,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fklein-gordon-equation":15634,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fdirac-equation-spinors":15433,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fantiparticles-hole-theory":15635,"\u002Fparticle-physics\u002Fqed\u002Ffeynman-rules-qed":15636,"\u002Fparticle-physics\u002Fqed\u002Fqed-tree-processes":15402,"\u002Fparticle-physics\u002Fqed\u002Frenormalization-running-coupling":15637,"\u002Fparticle-physics\u002Fqed\u002Felectron-g-2":14867,"\u002Fparticle-physics\u002Fweak-interaction\u002Fva-structure-weak":15638,"\u002Fparticle-physics\u002Fweak-interaction\u002Fw-z-bosons-decays":15639,"\u002Fparticle-physics\u002Fweak-interaction\u002Fckm-matrix":15640,"\u002Fparticle-physics\u002Fweak-interaction\u002Fcp-violation-kaons-b-mesons":15185,"\u002Fparticle-physics\u002Fqcd\u002Fcolor-su3-gluons":15399,"\u002Fparticle-physics\u002Fqcd\u002Fasymptotic-freedom-confinement":15641,"\u002Fparticle-physics\u002Fqcd\u002Fdeep-inelastic-scattering-partons":15642,"\u002Fparticle-physics\u002Fqcd\u002Fjets-hadronization":15260,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Felectroweak-su2-u1":15643,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fspontaneous-symmetry-breaking":15304,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-mechanism":15644,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-boson-discovery":15645,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fstandard-model":15194,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-oscillations":15227,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-mass-pmns":15646,"\u002Fparticle-physics\u002Fneutrinos\u002Fdirac-majorana-experiments":15647,"\u002Fparticle-physics\u002Fexperiment\u002Faccelerators-luminosity":15648,"\u002Fparticle-physics\u002Fexperiment\u002Fdetectors-subsystems":15282,"\u002Fparticle-physics\u002Fexperiment\u002Fhow-discoveries-are-made":15649,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fbeyond-standard-model":15024,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fgrand-unified-theories":15650,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fsupersymmetry":15127,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fhierarchy-problem-naturalness":15651,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fdark-matter-candidates":15652,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fmatter-antimatter-open-questions":15449,"\u002Fparticle-physics":15653,"\u002Fastrophysics-cosmology\u002Forientation\u002Fthe-sun-and-stars":15245,"\u002Fastrophysics-cosmology\u002Forientation\u002Fstellar-death-final-states":15397,"\u002Fastrophysics-cosmology\u002Forientation\u002Fgalaxies-and-cosmology":15654,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fmagnitudes-fluxes-and-the-distance-modulus":15184,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fstellar-spectra-and-spectral-classification":15655,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Ftelescopes-and-detectors-across-the-spectrum":15244,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fthe-cosmic-distance-ladder":15156,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fblackbody-radiation-and-specific-intensity":15656,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fradiative-transfer-and-the-transfer-equation":15657,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fspectral-line-formation-and-broadening":15658,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fopacity-and-the-rosseland-mean":15659,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fhydrostatic-equilibrium-and-the-virial-theorem":15660,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equations-of-stellar-structure":15448,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equation-of-state-and-polytropes":15379,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-standard-solar-model":15661,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fthermonuclear-reaction-rates-and-the-gamow-peak":15662,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhydrogen-burning-pp-chains-and-cno":15663,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhelium-burning-and-the-triple-alpha-process":15664,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fadvanced-burning-and-neutron-capture-nucleosynthesis":15576,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fphases-of-the-interstellar-medium":15665,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fmolecular-clouds-and-gravitational-collapse":15140,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fprotostars-and-the-pre-main-sequence":15203,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-main-sequence-and-its-structure":15185,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fpost-main-sequence-low-mass-evolution":15666,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-evolution-of-massive-stars":15269,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fstellar-pulsation-and-the-instability-strip":14885,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":15667,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fcore-collapse-supernovae":15668,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fthermonuclear-supernovae-type-ia":15398,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fneutron-stars-and-pulsars":15669,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fblack-holes-schwarzschild-and-kerr":15670,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fbinary-systems-and-mass-transfer":15671,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Faccreting-compact-objects":15672,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fgravitational-waves-from-inspiraling-binaries":14866,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fmultimessenger-astronomy-and-gamma-ray-bursts":15673,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fthe-milky-way":15674,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-morphology-and-classification":15127,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-rotation-curves-and-dark-matter":15675,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Factive-galactic-nuclei-and-supermassive-black-holes":15676,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-clusters-and-large-scale-structure":15391,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-expanding-universe-and-hubbles-law":15048,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-frw-metric-and-cosmological-redshift":15677,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-friedmann-equations-and-cosmic-dynamics":15678,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fcosmological-models-and-distances":15304,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fdark-energy-and-the-accelerating-universe":15290,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fthe-thermal-history-of-the-universe":15302,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fbig-bang-nucleosynthesis":15398,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Frecombination-and-the-cosmic-microwave-background":15679,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcmb-anisotropies-and-cosmological-parameters":14865,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcosmic-inflation":15057,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fstructure-formation-and-the-growth-of-perturbations":15680,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fdark-matter-dark-energy-and-open-questions":15144,"\u002Fastrophysics-cosmology":15249,"\u002Fcolophon":15681,"\u002F":15046},4250,4808,3626,2682,4109,4786,3878,3875,3751,3415,4067,3153,3000,4042,5461,5808,3961,3749,4327,5067,4246,4655,4154,5436,2640,4003,3601,2158,4331,4189,2273,3252,4633,4964,4172,3131,5524,3160,4031,2309,4207,3226,2648,4842,5340,3307,5701,4977,4039,2615,3472,4460,3848,4075,4400,3382,3010,3602,3737,3740,3707,3922,5191,4043,3804,4542,4214,5062,2850,4361,3443,3627,4044,3766,4140,3860,4006,5199,4334,5234,3651,5509,5680,153,1375,1073,1093,1125,1146,1014,1132,876,1541,1189,1173,984,1402,1301,950,1268,1063,1107,1408,1161,925,1012,866,964,1090,1142,1085,1020,1207,973,980,728,764,1225,1329,796,929,801,878,774,1044,1488,1175,1130,890,814,870,154,4073,5140,4961,5127,4870,5382,5195,4955,5369,4501,5576,3824,4132,4289,4307,4570,3403,5084,5105,5201,5116,5341,5175,5368,5188,5211,5499,5155,4981,5125,5415,5255,5304,5130,5167,5552,5164,5094,5239,5036,5190,5004,5099,5035,5159,5088,5026,4937,5023,5264,5244,133,5114,5078,5043,5312,5170,5342,5139,5151,5049,5212,5013,5068,5079,5102,5121,5081,5029,5379,5854,5110,2139,3798,5055,5364,4984,4935,4895,4972,5289,5112,5156,4987,5031,5025,5149,5302,5042,5002,4979,4922,4960,5279,126,1877,1180,1129,907,958,1112,1300,1053,1250,1181,1241,1234,966,1050,734,1190,484,1082,926,733,761,571,607,798,804,952,977,731,784,645,771,1017,742,1004,1000,1562,1254,1288,1101,1011,1486,1061,856,992,1169,988,137,0,2037,1782,2384,2254,2123,2332,1643,1714,2089,1751,1367,1660,2511,1998,1892,1854,1791,2438,2487,1917,2375,2525,2266,1845,2275,1810,1631,2310,2166,2233,2113,2505,2347,2672,2112,2473,2592,2380,3013,2513,3256,3218,2194,2173,2205,2326,2081,3342,3152,1799,1670,1027,960,1095,1291,986,897,1209,1055,1817,1801,1593,1465,1196,1464,1201,1230,1435,1684,1461,1926,1500,1409,1284,1774,1869,162,1487,1122,1188,1001,1351,982,1005,979,1325,1046,943,1279,824,1008,989,1798,1277,1025,987,1043,1211,1074,981,939,1002,739,1139,1108,1013,1070,978,1458,1317,157,1357,1077,2355,1116,1037,1178,1637,1314,1109,1056,1702,1474,1071,1158,832,993,1404,1024,1068,1339,1106,1264,1248,913,1848,1328,1633,1224,1143,135,1378,959,1028,998,911,1527,1203,1266,1483,1165,990,938,965,1257,1418,1099,942,1352,956,1035,1398,1003,1094,1292,138,1721,1827,1449,1354,1148,1184,1285,1281,1213,1290,1271,1252,1274,1778,1591,1503,1437,1571,1584,1957,1117,1781,1648,1342,1667,1510,1965,1607,1365,1849,1259,1303,1356,1238,2208,1564,173,1671,1286,1227,1638,1529,668,1078,918,709,865,880,940,1534,1015,874,922,841,794,1194,822,1658,1359,1296,1438,1921,1844,1570,1429,1324,1400,140,1787,1558,1654,1492,1747,2224,2002,2009,1323,1349,1785,1573,1722,1829,1353,1548,1552,1583,1624,1585,1245,1364,1514,1343,1397,1355,2211,1481,1770,160,2388,2293,2256,2552,2569,2478,2039,2496,2578,2814,2519,2461,2587,2492,2714,3278,2654,3050,2447,2849,2238,2369,2061,2214,2602,2563,2186,2985,2749,3364,2038,2282,2409,2126,2573,2206,2176,2268,2182,2402,2705,2633,2414,2213,2801,3313,3410,3195,1952,2017,1509,2537,2645,2027,2415,2838,2356,1906,3184,2950,2807,1683,1316,1034,1138,1763,1822,1705,1246,1701,1097,1104,1187,1032,1083,1228,916,1489,1033,1652,997,692,837,1023,888,864,1089,1231,1214,1675,1156,1075,1520,1309,139,1205,1051,735,1123,1072,915,567,768,825,1253,983,1007,762,1058,861,862,971,1208,1149,1145,1029,1084,927,810,838,857,807,936,949,2321,1622,1069,1113,1057,854,1958,1528,1618,2049,1432,1679,1796,1685,1346,1275,1476,1505,1610,2018,1599,1215,1838,1909,132,3902,2215,2240,3266,3208,3073,2454,2969,2451,1875,2728,1884,2371,2516,2842,1690,1904,2346,3146,1386,2607,1966,2668,1665,2885,1606,2577,3074,2869,2403,2433,2082,1939,1587,2460,2747,2032,2642,1619,3123,1993,2090,2339,3829,1737,2622,2340,2322,3828,4409,2305,3411,2510,4527,3030,3569,3043,2457,1946,2277,2044,2909,1693,1945,2093,2399,2115,2898,2742,2242,3895,3378,3376,2769,2223,3062,3262,2651,2949,2768,3128,2423,1977,2087,2866,3388,2830,2210,2489,2884,3945,2099,2713,3402,1692,2931,4195,3989,3206,4391,3004,3704,3494,2902,999,881,901,919,748,869,1018,1045,1049,1333,954,1092,1019,976,1771,1480,1396,953,1026,161,3533,2495,1818,3007,2595,3427,3537,2216,1895,2304,3396,1739,2073,1962,2203,1767,2666,2264,2276,2852,1807,3735,1560,4144,1669,1676,1972,2418,3291,1525,2040,2766,2337,2220,2800,3001,2078,1759,2836,1896,2026,1758,1543,1047,896,946,1060,1384,1482,815,1414,1322,1440,1240,1468,1098,1133,847,1009,1381,1052,1191,1258,1370,1712,1441,1199,957,1079,150,1262,1417,1368,1219,1136,1064,1463,1636,1059,931,1115,1736,1174,1376,1363,1411,1247,1746,1313,1299,1617,1102,1076,1495,1265,1193,1263,80,{"\u002Falgorithms\u002Ffoundations\u002Fwhat-is-an-algorithm":15683,"\u002Falgorithms\u002Ffoundations\u002Fproof-techniques":15688,"\u002Falgorithms\u002Ffoundations\u002Fasymptotic-analysis":15692,"\u002Falgorithms\u002Ffoundations\u002Fgrowth-rates-and-loop-analysis":15696,"\u002Falgorithms\u002Ffoundations\u002Frecurrences":15700,"\u002Falgorithms\u002Ffoundations\u002Famortized-analysis":15704,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fmergesort":15708,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fquicksort":15713,"\u002Falgorithms\u002Fdivide-and-conquer\u002Fselection":15717,"\u002Falgorithms\u002Fdivide-and-conquer\u002Ffast-multiplication":15721,"\u002Falgorithms\u002Fsorting\u002Fheaps-and-heapsort":15725,"\u002Falgorithms\u002Fsorting\u002Fsorting-lower-bounds":15730,"\u002Falgorithms\u002Fsorting\u002Flinear-time-sorting":15734,"\u002Falgorithms\u002Fsorting\u002Fexternal-sorting":15738,"\u002Falgorithms\u002Fdata-structures\u002Felementary-structures":15742,"\u002Falgorithms\u002Fdata-structures\u002Fhash-tables":15747,"\u002Falgorithms\u002Fdata-structures\u002Fbinary-search-trees":15751,"\u002Falgorithms\u002Fdata-structures\u002Favl-trees":15755,"\u002Falgorithms\u002Fdata-structures\u002Fbalanced-trees":15759,"\u002Falgorithms\u002Fdata-structures\u002Funion-find":15763,"\u002Falgorithms\u002Fdata-structures\u002Ffenwick-and-segment-trees":15767,"\u002Falgorithms\u002Fdata-structures\u002Fspatial-data-structures":15771,"\u002Falgorithms\u002Fdata-structures\u002Fskip-lists-and-probabilistic-structures":15775,"\u002Falgorithms\u002Fdata-structures\u002Fb-trees":15779,"\u002Falgorithms\u002Fdata-structures\u002Fdata-stream-algorithms":15783,"\u002Falgorithms\u002Fdata-structures\u002Fstreaming-sketches":15787,"\u002Falgorithms\u002Fsequences\u002Ftwo-pointers-and-windows":15791,"\u002Falgorithms\u002Fsequences\u002Fprefix-sums":15796,"\u002Falgorithms\u002Fsequences\u002Fmonotonic-stacks":15800,"\u002Falgorithms\u002Fsequences\u002Fbinary-search-on-the-answer":15804,"\u002Falgorithms\u002Fsequences\u002Fstring-matching":15808,"\u002Falgorithms\u002Fsequences\u002Fkmp-and-z-function":15812,"\u002Falgorithms\u002Fsequences\u002Ftries":15816,"\u002Falgorithms\u002Fsequences\u002Fsuffix-arrays-and-aho-corasick":15820,"\u002Falgorithms\u002Fgraphs\u002Frepresentations-and-traversal":15824,"\u002Falgorithms\u002Fgraphs\u002Fdepth-first-search":15829,"\u002Falgorithms\u002Fgraphs\u002Ftopological-sort-and-scc":15833,"\u002Falgorithms\u002Fgraphs\u002Fminimum-spanning-trees":15837,"\u002Falgorithms\u002Fgraphs\u002Fkruskal-and-prim":15841,"\u002Falgorithms\u002Fgraphs\u002Fshortest-paths":15845,"\u002Falgorithms\u002Fgraphs\u002Fall-pairs-and-negative-weights":15849,"\u002Falgorithms\u002Fgraphs\u002Fnetwork-flow":15853,"\u002Falgorithms\u002Fgraphs\u002Fmax-flow-min-cut":15857,"\u002Falgorithms\u002Fgraphs\u002Fbridges-and-articulation-points":15861,"\u002Falgorithms\u002Fgraphs\u002Flowest-common-ancestor":15865,"\u002Falgorithms\u002Fgraphs\u002Ftwo-sat":15869,"\u002Falgorithms\u002Fgraphs\u002Feulerian-tours":15873,"\u002Falgorithms\u002Fgraphs\u002Fbipartite-matching":15877,"\u002Falgorithms\u002Fgreedy\u002Fthe-greedy-method":15881,"\u002Falgorithms\u002Fgreedy\u002Fscheduling-and-intervals":15886,"\u002Falgorithms\u002Fgreedy\u002Fhuffman-codes":15890,"\u002Falgorithms\u002Fgreedy\u002Fmatroids":15894,"\u002Falgorithms\u002Fgreedy\u002Fstable-matching":15898,"\u002Falgorithms\u002Fdynamic-programming\u002Fprinciples":15902,"\u002Falgorithms\u002Fdynamic-programming\u002Fsequence-dp":15907,"\u002Falgorithms\u002Fdynamic-programming\u002Flongest-increasing-subsequence":15911,"\u002Falgorithms\u002Fdynamic-programming\u002Fknapsack":15915,"\u002Falgorithms\u002Fdynamic-programming\u002Fcoin-change-and-unbounded":15919,"\u002Falgorithms\u002Fdynamic-programming\u002Finterval-dp":15923,"\u002Falgorithms\u002Fdynamic-programming\u002Ftree-dp":15927,"\u002Falgorithms\u002Fdynamic-programming\u002Fbitmask-dp":15931,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-optimizations":15935,"\u002Falgorithms\u002Fdynamic-programming\u002Fdp-on-graphs":15939,"\u002Falgorithms\u002Fdynamic-programming\u002Fdigit-and-probability-dp":15943,"\u002Falgorithms\u002Fbacktracking\u002Fbacktracking-fundamentals":15947,"\u002Falgorithms\u002Fbacktracking\u002Fconstraint-search":15952,"\u002Falgorithms\u002Fbacktracking\u002Fbranch-and-bound":15956,"\u002Falgorithms\u002Fbacktracking\u002Fgraph-backtracking":15960,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fnumber-theory-basics":15964,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmodular-exponentiation-and-primality":15969,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fsieve-and-factorization":15973,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fcombinatorics":15977,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fmatrix-exponentiation":15981,"\u002Falgorithms\u002Fmathematical-algorithms\u002Ffast-fourier-transform":15985,"\u002Falgorithms\u002Fmathematical-algorithms\u002Fgradient-descent":15989,"\u002Falgorithms\u002Fcomputational-geometry\u002Fgeometric-primitives":15993,"\u002Falgorithms\u002Fcomputational-geometry\u002Fconvex-hull":15998,"\u002Falgorithms\u002Fcomputational-geometry\u002Fsweep-line":16002,"\u002Falgorithms\u002Fcomputational-geometry\u002Fpolygons-and-proximity":16006,"\u002Falgorithms\u002Fintractability\u002Fp-np-reductions":16010,"\u002Falgorithms\u002Fintractability\u002Fnp-completeness":16015,"\u002Falgorithms\u002Fintractability\u002Fcoping-with-hardness":16019,"\u002Falgorithms\u002Fintractability\u002Fapproximation-algorithms":16023,"\u002Falgorithms":16027,"\u002Fcalculus\u002Flimits-and-continuity\u002Ffunctions-and-models":16030,"\u002Fcalculus\u002Flimits-and-continuity\u002Fthe-limit-of-a-function":16035,"\u002Fcalculus\u002Flimits-and-continuity\u002Flimit-laws-and-the-precise-definition":16039,"\u002Fcalculus\u002Flimits-and-continuity\u002Fcontinuity":16043,"\u002Fcalculus\u002Fderivatives\u002Fthe-derivative-and-rates-of-change":16047,"\u002Fcalculus\u002Fderivatives\u002Fdifferentiation-rules-and-the-chain-rule":16052,"\u002Fcalculus\u002Fderivatives\u002Fimplicit-differentiation-and-related-rates":16056,"\u002Fcalculus\u002Fderivatives\u002Flinear-approximations-and-differentials":16060,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fextrema-and-the-mean-value-theorem":16064,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fhow-derivatives-shape-a-graph":16069,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fcurve-sketching-and-optimization":16073,"\u002Fcalculus\u002Fapplications-of-derivatives\u002Fnewtons-method-and-antiderivatives":16077,"\u002Fcalculus\u002Fintegrals\u002Farea-and-the-definite-integral":16081,"\u002Fcalculus\u002Fintegrals\u002Fthe-fundamental-theorem-of-calculus":16086,"\u002Fcalculus\u002Fintegrals\u002Fthe-substitution-rule":16090,"\u002Fcalculus\u002Fapplications-of-integration\u002Fareas-and-volumes":16094,"\u002Fcalculus\u002Fapplications-of-integration\u002Fwork-average-value-and-arc-length":16099,"\u002Fcalculus\u002Fapplications-of-integration\u002Fphysics-economics-and-probability":16103,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Finverse-functions-logarithms-and-exponentials":16107,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Fgrowth-decay-inverse-trig-and-hyperbolic-functions":16112,"\u002Fcalculus\u002Fexponential-logarithmic-and-inverse-functions\u002Flhospitals-rule":16116,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fintegration-by-parts":16120,"\u002Fcalculus\u002Ftechniques-of-integration\u002Ftrigonometric-integrals-and-substitution":16125,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fpartial-fractions-and-integration-strategy":16129,"\u002Fcalculus\u002Ftechniques-of-integration\u002Fapproximate-and-improper-integrals":16133,"\u002Fcalculus\u002Fparametric-and-polar\u002Fparametric-curves-and-their-calculus":16137,"\u002Fcalculus\u002Fparametric-and-polar\u002Fpolar-coordinates":16142,"\u002Fcalculus\u002Fparametric-and-polar\u002Fconic-sections":16146,"\u002Fcalculus\u002Fsequences-and-series\u002Fsequences":16150,"\u002Fcalculus\u002Fsequences-and-series\u002Fseries-and-the-integral-test":16155,"\u002Fcalculus\u002Fsequences-and-series\u002Fthe-convergence-tests":16159,"\u002Fcalculus\u002Fsequences-and-series\u002Fpower-series":16163,"\u002Fcalculus\u002Fsequences-and-series\u002Ftaylor-and-maclaurin-series":16167,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvectors-and-the-dot-product":16171,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fthe-cross-product-lines-and-planes":16176,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fcylinders-and-quadric-surfaces":16180,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Fvector-functions-and-space-curves":16184,"\u002Fcalculus\u002Fvectors-and-space-curves\u002Farc-length-curvature-and-motion":16188,"\u002Fcalculus\u002Fpartial-derivatives\u002Ffunctions-of-several-variables":16192,"\u002Fcalculus\u002Fpartial-derivatives\u002Fpartial-derivatives":16197,"\u002Fcalculus\u002Fpartial-derivatives\u002Ftangent-planes-and-the-chain-rule":16200,"\u002Fcalculus\u002Fpartial-derivatives\u002Fdirectional-derivatives-and-the-gradient":16204,"\u002Fcalculus\u002Fpartial-derivatives\u002Foptimization-and-lagrange-multipliers":16208,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fdouble-integrals":16212,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Ftriple-integrals-and-coordinate-systems":16217,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fvector-fields-and-line-integrals":16221,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fgreens-theorem-curl-and-divergence":16225,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fsurface-integrals":16229,"\u002Fcalculus\u002Fmultiple-integrals-and-vector-calculus\u002Fstokes-and-the-divergence-theorem":16233,"\u002Fcalculus":16237,"\u002Fmechanics\u002Ffoundations\u002Fmeasurement-and-dimensions":16240,"\u002Fmechanics\u002Ffoundations\u002Fvector-algebra":16244,"\u002Fmechanics\u002Fkinematics\u002Fone-dimensional-motion":16248,"\u002Fmechanics\u002Fkinematics\u002Fmotion-graphs":16253,"\u002Fmechanics\u002Fkinematics\u002Fprojectile-motion":16257,"\u002Fmechanics\u002Fkinematics\u002Frelative-motion":16261,"\u002Fmechanics\u002Fkinematics\u002Fcircular-motion":16265,"\u002Fmechanics\u002Fdynamics\u002Fnewtons-laws":16269,"\u002Fmechanics\u002Fdynamics\u002Ffree-body-diagrams":16274,"\u002Fmechanics\u002Fdynamics\u002Ffriction-and-curved-motion":16278,"\u002Fmechanics\u002Fdynamics\u002Fnumerical-dynamics":16282,"\u002Fmechanics\u002Fdynamics\u002Fcenter-of-mass-systems":16286,"\u002Fmechanics\u002Fenergy\u002Fwork-and-kinetic-energy":16290,"\u002Fmechanics\u002Fenergy\u002Fpotential-energy":16295,"\u002Fmechanics\u002Fenergy\u002Fmultiparticle-work":16299,"\u002Fmechanics\u002Fenergy\u002Fmass-energy-and-binding":16303,"\u002Fmechanics\u002Fenergy\u002Fphotons-and-quantization":16307,"\u002Fmechanics\u002Fmomentum\u002Fmomentum-and-collisions":16311,"\u002Fmechanics\u002Fmomentum\u002Fcenter-of-mass-collisions":16316,"\u002Fmechanics\u002Fmomentum\u002Frocket-propulsion":16320,"\u002Fmechanics\u002Frotation\u002Frotational-inertia":16324,"\u002Fmechanics\u002Frotation\u002Frotational-dynamics":16329,"\u002Fmechanics\u002Frotation\u002Frolling-motion":16333,"\u002Fmechanics\u002Frotation\u002Fangular-momentum":16337,"\u002Fmechanics\u002Frotation\u002Frolling-resistance":16341,"\u002Fmechanics\u002Frotation\u002Fgyroscopic-precession":16345,"\u002Fmechanics\u002Fgravity-and-matter\u002Fkeplerian-orbits":16349,"\u002Fmechanics\u002Fgravity-and-matter\u002Fgravitational-fields":16354,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstatic-equilibrium":16358,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-statics":16362,"\u002Fmechanics\u002Fgravity-and-matter\u002Ffluid-flow":16366,"\u002Fmechanics\u002Fgravity-and-matter\u002Forbital-motion":16370,"\u002Fmechanics\u002Fgravity-and-matter\u002Fstress-and-elasticity":16374,"\u002Fmechanics\u002Foscillations-waves\u002Fdamped-oscillators":16378,"\u002Fmechanics\u002Foscillations-waves\u002Ftravelling-waves":16383,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-superposition":16387,"\u002Fmechanics\u002Foscillations-waves\u002Fstanding-waves":16391,"\u002Fmechanics\u002Foscillations-waves\u002Fsound-waves":16395,"\u002Fmechanics\u002Foscillations-waves\u002Fdoppler-effect":16399,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-packets":16403,"\u002Fmechanics\u002Foscillations-waves\u002Fbeats-and-coupling":16407,"\u002Fmechanics\u002Foscillations-waves\u002Fsimple-harmonic-motion":16411,"\u002Fmechanics\u002Foscillations-waves\u002Fpendulum-motion":16415,"\u002Fmechanics\u002Foscillations-waves\u002Fdriven-oscillators":16419,"\u002Fmechanics\u002Foscillations-waves\u002Fwave-boundaries":16423,"\u002Fmechanics\u002Fthermodynamics\u002Fkinetic-theory-of-ideal-gases":16427,"\u002Fmechanics\u002Fthermodynamics\u002Ffirst-law-of-thermodynamics":16432,"\u002Fmechanics\u002Fthermodynamics\u002Fentropy-and-the-second-law":16436,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-processes":16440,"\u002Fmechanics\u002Fthermodynamics\u002Fphase-changes":16444,"\u002Fmechanics\u002Fthermodynamics\u002Fthermal-machines":16448,"\u002Fmechanics":16452,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcharge-and-conductors":16455,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Fcoulombs-law":16460,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-and-force":16464,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-field-maps":16468,"\u002Felectricity-and-magnetism\u002Felectric-fields\u002Felectric-dipoles":16472,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fcontinuous-charge-fields":16476,"\u002Felectricity-and-magnetism\u002Fcontinuous-charge-distributions\u002Fgauss-law-and-conductors":16481,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpoint-charge-potential":16485,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fpotential-gradients-and-equipotentials":16490,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Felectrostatic-energy-and-pressure":16494,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Flaplace-boundary-problems":16498,"\u002Felectricity-and-magnetism\u002Felectric-potential\u002Fcontinuous-charge-potentials":16502,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitance-fundamentals":16506,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-networks":16511,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fcapacitor-energy-and-force":16515,"\u002Felectricity-and-magnetism\u002Fcapacitance\u002Fdielectric-polarization-and-breakdown":16519,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fcurrent-and-resistance":16523,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Fkirchhoff-network-analysis":16528,"\u002Felectricity-and-magnetism\u002Fdirect-current-circuits\u002Frc-transients":16532,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-trajectories":16536,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fhall-effect":16541,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-force-on-conductors":16545,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmagnetic-dipoles":16549,"\u002Felectricity-and-magnetism\u002Fmagnetic-field\u002Fmass-spectrometry":16553,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmoving-charge-fields":16557,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fbiot-savart-law":16562,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fcircular-current-loops":16566,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Famperes-law":16570,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fgauss-law-for-magnetism":16574,"\u002Felectricity-and-magnetism\u002Fmagnetic-sources\u002Fmagnetic-materials":16578,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-flux":16582,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Ffaradays-law":16587,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Flenzs-law":16591,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmotional-emf":16595,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Feddy-currents":16599,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fself-inductance":16603,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Fmagnetic-energy":16607,"\u002Felectricity-and-magnetism\u002Felectromagnetic-induction\u002Frl-circuits":16611,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-fundamentals":16615,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Freactance":16620,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Frlc-resonance":16624,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Fac-power":16628,"\u002Felectricity-and-magnetism\u002Falternating-current\u002Ftransformers":16632,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdisplacement-current":16636,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-waves":16641,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Felectromagnetic-momentum":16645,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fdipole-radiation":16649,"\u002Felectricity-and-magnetism\u002Fmaxwell-electromagnetic-waves\u002Fpolarization":16653,"\u002Felectricity-and-magnetism\u002Foptics\u002Freflection-and-refraction":16657,"\u002Felectricity-and-magnetism\u002Foptics\u002Fthin-lenses":16662,"\u002Felectricity-and-magnetism\u002Foptics\u002Fspherical-mirrors":16666,"\u002Felectricity-and-magnetism":16670,"\u002Flinear-algebra\u002Flinear-systems\u002Fsystems-and-echelon-forms":16673,"\u002Flinear-algebra\u002Flinear-systems\u002Fvector-and-matrix-equations":16678,"\u002Flinear-algebra\u002Flinear-systems\u002Fsolution-sets-and-applications":16682,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-independence":16686,"\u002Flinear-algebra\u002Flinear-systems\u002Flinear-transformations":16690,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-operations":16694,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fmatrix-inverse-and-invertibility":16699,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fpartitioned-matrices-and-lu":16703,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fsubspaces-dimension-rank":16707,"\u002Flinear-algebra\u002Fmatrix-algebra\u002Fapplications-leontief-and-graphics":16711,"\u002Flinear-algebra\u002Fdeterminants\u002Fdeterminants-and-cofactors":16715,"\u002Flinear-algebra\u002Fdeterminants\u002Fproperties-of-determinants":16720,"\u002Flinear-algebra\u002Fdeterminants\u002Fcramer-volume-and-area":16724,"\u002Flinear-algebra\u002Fvector-spaces\u002Fvector-spaces-and-subspaces":16728,"\u002Flinear-algebra\u002Fvector-spaces\u002Fnull-and-column-spaces":16733,"\u002Flinear-algebra\u002Fvector-spaces\u002Fbases-and-independent-sets":16737,"\u002Flinear-algebra\u002Fvector-spaces\u002Fcoordinate-systems":16741,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdimension-and-rank":16745,"\u002Flinear-algebra\u002Fvector-spaces\u002Fchange-of-basis":16749,"\u002Flinear-algebra\u002Fvector-spaces\u002Fdifference-equations-and-markov":16753,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-eigenvalues":16757,"\u002Flinear-algebra\u002Feigenvalues\u002Fthe-characteristic-equation":16762,"\u002Flinear-algebra\u002Feigenvalues\u002Fdiagonalization":16766,"\u002Flinear-algebra\u002Feigenvalues\u002Feigenvectors-and-linear-transformations":16770,"\u002Flinear-algebra\u002Feigenvalues\u002Fcomplex-eigenvalues":16774,"\u002Flinear-algebra\u002Feigenvalues\u002Fdynamical-systems":16778,"\u002Flinear-algebra\u002Feigenvalues\u002Fpower-method":16782,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-length-orthogonality":16786,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Forthogonal-sets-and-projections":16791,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fgram-schmidt-and-qr":16795,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-problems":16799,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Fleast-squares-applications":16803,"\u002Flinear-algebra\u002Forthogonality-least-squares\u002Finner-product-spaces":16807,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fdiagonalizing-symmetric-matrices":16811,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fquadratic-forms":16816,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fconstrained-optimization":16820,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsingular-value-decomposition":16824,"\u002Flinear-algebra\u002Fsymmetric-quadratic-svd\u002Fsvd-applications-pca-imaging":16828,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-thinking-and-matrix-computation":16832,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Flu-and-cholesky":16837,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fconditioning-and-floating-point":16841,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fstability-and-error-analysis":16845,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fqr-and-numerical-least-squares":16849,"\u002Flinear-algebra\u002Fnumerical-linear-algebra\u002Fnumerical-eigenvalues-and-svd":16853,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-combinations":16857,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Faffine-independence-and-barycentric-coordinates":16862,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fconvex-combinations-and-convex-sets":16866,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fhyperplanes-and-polytopes":16870,"\u002Flinear-algebra\u002Fgeometry-of-vector-spaces\u002Fcurves-and-surfaces":16874,"\u002Flinear-algebra":16878,"\u002Ftheory-of-computation":16881,"\u002Fcomputer-architecture\u002Ffoundations\u002Fbits-bytes-and-words":16884,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-representation":16888,"\u002Fcomputer-architecture\u002Ffoundations\u002Finteger-arithmetic":16892,"\u002Fcomputer-architecture\u002Ffoundations\u002Ffloating-point":16896,"\u002Fcomputer-architecture\u002Ffoundations\u002Fboolean-algebra-and-bit-manipulation":16900,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fthe-machines-view":16904,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fdata-movement":16909,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farithmetic-and-logic":16913,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fcontrol-flow":16917,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fprocedures":16921,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Farrays-structs-and-alignment":16925,"\u002Fcomputer-architecture\u002Fmachine-level-x86-64\u002Fmemory-layout-and-buffer-overflows":16929,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fwhat-an-isa-is":16933,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Finstruction-formats-and-operands":16938,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Faddressing-modes":16942,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fthe-y86-64-instruction-set":16946,"\u002Fcomputer-architecture\u002Finstruction-set-architecture\u002Fy86-64-programming":16950,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Ftransistors-gates-and-boolean-functions":16954,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fcombinational-logic-and-hcl":16959,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmultiplexers-decoders-and-the-alu":16963,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fmemory-elements-latches-flip-flops-and-clocking":16967,"\u002Fcomputer-architecture\u002Fdigital-logic\u002Fregister-files-and-random-access-memory":16971,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-fetch-decode-execute-cycle":16975,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fthe-seq-stages":16980,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fcontrol-logic-and-sequencing":16984,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Fassembling-seq":16988,"\u002Fcomputer-architecture\u002Fprocessor-design\u002Ftracing-a-program":16992,"\u002Fcomputer-architecture\u002Fpipelining\u002Fpipelining-principles":16996,"\u002Fcomputer-architecture\u002Fpipelining\u002Ffrom-seq-to-pipe":17001,"\u002Fcomputer-architecture\u002Fpipelining\u002Fdata-hazards-stalling-and-forwarding":17005,"\u002Fcomputer-architecture\u002Fpipelining\u002Fcontrol-hazards-and-branch-prediction":17009,"\u002Fcomputer-architecture\u002Fpipelining\u002Fthe-complete-pipe-processor":17013,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fstorage-technologies-and-the-latency-gap":17017,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Flocality":17022,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-memories-direct-mapped":17026,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fset-associative-and-write-policies":17030,"\u002Fcomputer-architecture\u002Fmemory-hierarchy\u002Fcache-performance-and-cache-friendly-code":17034,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Faddress-spaces-and-translation":17038,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fpage-tables-and-page-faults":17043,"\u002Fcomputer-architecture\u002Fvirtual-memory\u002Fthe-tlb-and-multi-level-page-tables":17047,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Fexceptional-control-flow":17051,"\u002Fcomputer-architecture\u002Fexceptions-and-io\u002Finterrupts-and-the-kernel":17056,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fprocesses-threads-and-parallelism":17060,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fhardware-multithreading":17065,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fcache-coherence":17069,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmemory-consistency-and-synchronization":17073,"\u002Fcomputer-architecture\u002Fmultithreading-and-multicore\u002Fmulticore-organization":17077,"\u002Fcomputer-architecture\u002Fcapstone\u002Fthe-whole-machine":17081,"\u002Fcomputer-architecture\u002Fcapstone\u002Fassembling-a-complete-cpu":17086,"\u002Fcomputer-architecture":17090,"\u002Fdifferential-equations\u002Ffoundations\u002Fmodels-and-direction-fields":17093,"\u002Fdifferential-equations\u002Ffoundations\u002Fclassification-and-terminology":17097,"\u002Fdifferential-equations\u002Ffirst-order\u002Flinear-first-order-integrating-factors":17101,"\u002Fdifferential-equations\u002Ffirst-order\u002Fseparable-and-exact":17106,"\u002Fdifferential-equations\u002Ffirst-order\u002Fmodeling-first-order":17110,"\u002Fdifferential-equations\u002Ffirst-order\u002Fautonomous-and-population-dynamics":17114,"\u002Fdifferential-equations\u002Ffirst-order\u002Fexistence-uniqueness-euler":17118,"\u002Fdifferential-equations\u002Ffirst-order\u002Ffirst-order-difference-equations":17122,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhomogeneous-constant-coefficients":17126,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fcomplex-and-repeated-roots":17131,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fnonhomogeneous-undetermined-coefficients":17135,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fvariation-of-parameters":17139,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fmechanical-electrical-vibrations":17143,"\u002Fdifferential-equations\u002Fsecond-order-linear\u002Fhigher-order-linear":17147,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fpower-series-ordinary-points":17151,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fregular-singular-frobenius":17156,"\u002Fdifferential-equations\u002Fseries-solutions\u002Fbessel-and-special-functions":17160,"\u002Fdifferential-equations\u002Flaplace\u002Flaplace-definition-ivps":17164,"\u002Fdifferential-equations\u002Flaplace\u002Fstep-impulse-convolution":17169,"\u002Fdifferential-equations\u002Fsystems\u002Fmatrices-eigenvalues-review":17173,"\u002Fdifferential-equations\u002Fsystems\u002Fconstant-coefficient-systems-phase-portraits":17178,"\u002Fdifferential-equations\u002Fsystems\u002Frepeated-eigenvalues-fundamental-matrices":17182,"\u002Fdifferential-equations\u002Fnumerical\u002Feuler-and-runge-kutta":17186,"\u002Fdifferential-equations\u002Fnumerical\u002Fmultistep-systems-stability":17191,"\u002Fdifferential-equations\u002Fnonlinear\u002Fphase-plane-autonomous-stability":17195,"\u002Fdifferential-equations\u002Fnonlinear\u002Flocally-linear-and-liapunov":17200,"\u002Fdifferential-equations\u002Fnonlinear\u002Fcompeting-species-predator-prey-limit-cycles":17204,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Ffourier-series":17208,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fheat-wave-laplace-equations":17213,"\u002Fdifferential-equations\u002Fpdes-fourier-bvp\u002Fsturm-liouville":17217,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fcalculus-of-variations":17221,"\u002Fdifferential-equations\u002Fhistory-variations\u002Fhistorical-notes":17226,"\u002Fdifferential-equations":17230,"\u002Frelativity\u002Ffoundations\u002Fspecial-relativity-postulates":17233,"\u002Frelativity\u002Ffoundations\u002Florentz-transformation-spacetime":17238,"\u002Frelativity\u002Ffoundations\u002Ftime-dilation-length-contraction":17242,"\u002Frelativity\u002Ffoundations\u002Frelativistic-momentum-energy":17246,"\u002Frelativity\u002Ffoundations\u002Fgeneral-relativity":17250,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fminkowski-spacetime-and-the-interval":17254,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Ffour-vectors-and-index-notation":17259,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fthe-lorentz-group-and-rapidity":17263,"\u002Frelativity\u002Fspacetime-and-the-lorentz-group\u002Fdoppler-aberration-and-appearance":17267,"\u002Frelativity\u002Frelativistic-dynamics\u002Ffour-momentum-force-and-accelerated-motion":17271,"\u002Frelativity\u002Frelativistic-dynamics\u002Fparticle-decays-and-two-body-kinematics":17276,"\u002Frelativity\u002Frelativistic-dynamics\u002Fcollisions-thresholds-and-the-cm-frame":17280,"\u002Frelativity\u002Frelativistic-dynamics\u002Fmandelstam-variables-and-invariants":17284,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ffour-current-and-the-four-potential":17288,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fthe-electromagnetic-field-tensor":17293,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Ftransformation-of-electric-and-magnetic-fields":17297,"\u002Frelativity\u002Fcovariant-electrodynamics\u002Fcovariant-maxwell-and-the-stress-energy-tensor":17301,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-equivalence-principle-formalized":17305,"\u002Frelativity\u002Fcurved-spacetime\u002Fmanifolds-vectors-and-the-metric":17310,"\u002Frelativity\u002Fcurved-spacetime\u002Fcovariant-derivative-and-christoffel-symbols":17314,"\u002Frelativity\u002Fcurved-spacetime\u002Fgeodesics-and-the-geodesic-equation":17318,"\u002Frelativity\u002Fcurved-spacetime\u002Fcurvature-riemann-and-geodesic-deviation":17322,"\u002Frelativity\u002Fcurved-spacetime\u002Fthe-einstein-field-equations":17326,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fthe-schwarzschild-metric":17330,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Fgeodesics-and-orbits-in-schwarzschild":17335,"\u002Frelativity\u002Fthe-schwarzschild-solution\u002Flight-bending-and-null-geodesics":17339,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fperihelion-precession-of-mercury":17343,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fdeflection-of-light-and-gravitational-lensing":17348,"\u002Frelativity\u002Ftests-of-general-relativity\u002Fgravitational-redshift-and-shapiro-delay":17352,"\u002Frelativity\u002Ftests-of-general-relativity\u002Frelativity-in-technology-gps":17356,"\u002Frelativity\u002Fblack-holes\u002Fhorizons-and-coordinate-singularities":17360,"\u002Frelativity\u002Fblack-holes\u002Frotating-and-charged-black-holes":17365,"\u002Frelativity\u002Fblack-holes\u002Fblack-hole-thermodynamics":17369,"\u002Frelativity\u002Fgravitational-waves\u002Flinearized-gravity-and-wave-solutions":17373,"\u002Frelativity\u002Fgravitational-waves\u002Fgeneration-and-the-quadrupole-formula":17378,"\u002Frelativity\u002Fgravitational-waves\u002Fdetection-ligo-and-the-first-events":17382,"\u002Frelativity\u002Fcosmological-bridge\u002Fthe-cosmological-principle-and-flrw-metric":17386,"\u002Frelativity\u002Fcosmological-bridge\u002Ffriedmann-equations-and-cosmic-dynamics":17391,"\u002Frelativity":17395,"\u002Fphysical-computing":17398,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fblackbody-radiation-and-the-planck-quantum":17401,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-photoelectric-effect-and-the-photon":17406,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fx-rays-and-the-compton-effect":17410,"\u002Fquantum-mechanics\u002Fold-quantum-theory\u002Fthe-old-quantum-theory-bohr-and-sommerfeld":17414,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fde-broglie-waves-and-electron-diffraction":17418,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fwave-packets-and-the-probability-interpretation":17423,"\u002Fquantum-mechanics\u002Fmatter-waves\u002Fthe-uncertainty-principle":17427,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-schrodinger-equation-in-one-dimension":17431,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-free-particle-and-wave-packet-dynamics":17436,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fparticle-in-infinite-and-finite-square-wells":17440,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Foperators-expectation-values-and-the-harmonic-oscillator":17444,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fthe-dirac-delta-potential":17448,"\u002Fquantum-mechanics\u002Fwave-mechanics-1d\u002Fbarrier-penetration-and-quantum-tunneling":17452,"\u002Fquantum-mechanics\u002Fformalism\u002Fhilbert-space-and-dirac-notation":17456,"\u002Fquantum-mechanics\u002Fformalism\u002Fobservables-hermitian-operators-and-eigenvalues":17461,"\u002Fquantum-mechanics\u002Fformalism\u002Fthe-postulates-and-quantum-measurement":17465,"\u002Fquantum-mechanics\u002Fformalism\u002Fposition-momentum-and-continuous-spectra":17469,"\u002Fquantum-mechanics\u002Fformalism\u002Fcommutators-and-the-generalized-uncertainty-principle":17473,"\u002Fquantum-mechanics\u002Fformalism\u002Ftime-evolution-schrodinger-and-heisenberg-pictures":17477,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fladder-operators-and-the-number-states":17481,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fcoherent-and-squeezed-states":17486,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fsymmetries-generators-and-conservation-laws":17490,"\u002Fquantum-mechanics\u002Foscillator-and-symmetry\u002Fparity-time-reversal-and-discrete-symmetries":17494,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Forbital-angular-momentum-and-spherical-harmonics":17498,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Fthe-angular-momentum-algebra":17502,"\u002Fquantum-mechanics\u002Fangular-momentum\u002Faddition-of-angular-momenta-and-clebsch-gordan":17506,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-schrodinger-equation-in-three-dimensions":17510,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-hydrogen-atom":17515,"\u002Fquantum-mechanics\u002Fcentral-potentials\u002Fthe-isotropic-oscillator-and-hidden-symmetry":17519,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-half-pauli-matrices-and-stern-gerlach":17523,"\u002Fquantum-mechanics\u002Fspin\u002Fspin-in-a-magnetic-field-precession-and-resonance":17528,"\u002Fquantum-mechanics\u002Fspin\u002Ftwo-level-systems-and-the-bloch-sphere":17532,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fidentical-particles-and-exchange-symmetry":17536,"\u002Fquantum-mechanics\u002Fidentical-particles\u002Fthe-pauli-principle-atoms-and-the-periodic-table":17541,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ftime-independent-perturbation-theory":17545,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Ffine-structure-and-the-real-hydrogen-atom":17550,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-zeeman-and-stark-effects":17554,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-variational-method":17558,"\u002Fquantum-mechanics\u002Fapproximation-methods\u002Fthe-wkb-approximation":17562,"\u002Fquantum-mechanics":17566,"\u002Freal-analysis\u002Ffoundations\u002Fsets-logic-functions":17569,"\u002Freal-analysis\u002Ffoundations\u002Fordered-fields-completeness":17574,"\u002Freal-analysis\u002Ffoundations\u002Fabsolute-value-bounds":17578,"\u002Freal-analysis\u002Ffoundations\u002Fintervals-uncountability":17582,"\u002Freal-analysis\u002Fsequences-series\u002Fsequences-limits":17586,"\u002Freal-analysis\u002Fsequences-series\u002Flimit-laws-monotone":17591,"\u002Freal-analysis\u002Fsequences-series\u002Flimsup-bolzano-weierstrass":17595,"\u002Freal-analysis\u002Fsequences-series\u002Fcauchy-completeness":17599,"\u002Freal-analysis\u002Fsequences-series\u002Fseries-convergence":17603,"\u002Freal-analysis\u002Fsequences-series\u002Fabsolute-conditional-rearrangement":17607,"\u002Freal-analysis\u002Fmetric-spaces\u002Fmetric-spaces-norms":17611,"\u002Freal-analysis\u002Fmetric-spaces\u002Fopen-closed-sets":17616,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconvergence-completeness":17620,"\u002Freal-analysis\u002Fmetric-spaces\u002Fcompactness":17624,"\u002Freal-analysis\u002Fmetric-spaces\u002Fconnectedness":17628,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-of-functions":17632,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuous-functions":17636,"\u002Freal-analysis\u002Fcontinuity\u002Fevt-ivt":17640,"\u002Freal-analysis\u002Fcontinuity\u002Funiform-continuity":17644,"\u002Freal-analysis\u002Fcontinuity\u002Fcontinuity-metric-spaces":17648,"\u002Freal-analysis\u002Fcontinuity\u002Flimits-infinity-monotone":17652,"\u002Freal-analysis\u002Fdifferentiation\u002Fthe-derivative":17656,"\u002Freal-analysis\u002Fdifferentiation\u002Fmean-value-theorem":17661,"\u002Freal-analysis\u002Fdifferentiation\u002Ftaylors-theorem":17665,"\u002Freal-analysis\u002Fdifferentiation\u002Finverse-function-1d":17669,"\u002Freal-analysis\u002Friemann-integration\u002Fdarboux-integral":17673,"\u002Freal-analysis\u002Friemann-integration\u002Fintegrability-classes":17678,"\u002Freal-analysis\u002Friemann-integration\u002Fproperties-of-the-integral":17682,"\u002Freal-analysis\u002Friemann-integration\u002Ffundamental-theorem":17686,"\u002Freal-analysis\u002Friemann-integration\u002Flog-exp-improper":17689,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpointwise-uniform-convergence":17693,"\u002Freal-analysis\u002Ffunction-sequences\u002Finterchange-of-limits":17698,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpower-series-weierstrass":17702,"\u002Freal-analysis\u002Ffunction-sequences\u002Fpicard-ode":17706,"\u002Freal-analysis\u002Fseveral-variables\u002Fdifferentiability-rn":17710,"\u002Freal-analysis\u002Fseveral-variables\u002Fgradient-chain-rule":17715,"\u002Freal-analysis\u002Fseveral-variables\u002Fhigher-derivatives-taylor-extrema":17719,"\u002Freal-analysis\u002Fseveral-variables\u002Finverse-implicit-theorems":17723,"\u002Freal-analysis\u002Fseveral-variables\u002Fmultiple-integrals":17727,"\u002Freal-analysis":17731,"\u002Fabstract-algebra\u002Ffoundations\u002Fsets-functions-relations":17734,"\u002Fabstract-algebra\u002Ffoundations\u002Fintegers-and-modular-arithmetic":17738,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fgroup-axioms-and-first-examples":17742,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fdihedral-and-symmetric-groups":17747,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fmatrix-and-quaternion-groups":17751,"\u002Fabstract-algebra\u002Fgroups-and-symmetry\u002Fhomomorphisms-and-group-actions":17755,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fsubgroups-and-substructures":17759,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcyclic-groups":17764,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fgeneration-and-subgroup-lattices":17768,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcosets-lagrange-and-normal-subgroups":17772,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fisomorphism-theorems":17776,"\u002Fabstract-algebra\u002Fsubgroups-and-quotients\u002Fcomposition-series-and-the-alternating-group":17780,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Factions-and-cayleys-theorem":17784,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fconjugation-and-the-class-equation":17789,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fsylow-theorems":17793,"\u002Fabstract-algebra\u002Fgroup-actions-and-sylow\u002Fautomorphisms-and-simple-groups":17797,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fdirect-products-and-finite-abelian-groups":17801,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fsemidirect-products":17806,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fnilpotent-and-solvable-groups":17810,"\u002Fabstract-algebra\u002Fproducts-and-group-structure\u002Fclassifying-small-groups":17814,"\u002Fabstract-algebra\u002Fring-theory\u002Frings-definitions-and-examples":17818,"\u002Fabstract-algebra\u002Fring-theory\u002Fideals-quotients-and-homomorphisms":17823,"\u002Fabstract-algebra\u002Fring-theory\u002Ffractions-and-the-chinese-remainder-theorem":17827,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Feuclidean-domains-pids-ufds":17831,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fpolynomial-rings-over-fields":17836,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Fgauss-lemma-and-unique-factorization":17840,"\u002Fabstract-algebra\u002Ffactorization-and-polynomials\u002Firreducibility-criteria-and-groebner":17844,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fintroduction-to-modules":17848,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ffree-modules-and-direct-sums":17853,"\u002Fabstract-algebra\u002Fmodule-theory\u002Ftensor-products-and-exact-sequences":17857,"\u002Fabstract-algebra\u002Fmodule-theory\u002Fvector-spaces-and-linear-maps":17861,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fstructure-theorem-over-pids":17865,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Frational-canonical-form":17870,"\u002Fabstract-algebra\u002Fmodules-over-pids\u002Fjordan-canonical-form":17874,"\u002Fabstract-algebra\u002Ffield-theory\u002Ffield-extensions-and-algebraic-elements":17878,"\u002Fabstract-algebra\u002Ffield-theory\u002Fstraightedge-and-compass-constructions":17883,"\u002Fabstract-algebra\u002Ffield-theory\u002Fsplitting-fields-and-algebraic-closure":17887,"\u002Fabstract-algebra\u002Ffield-theory\u002Fseparable-and-cyclotomic-extensions":17891,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fthe-galois-correspondence":17895,"\u002Fabstract-algebra\u002Fgalois-theory\u002Ffinite-fields":17900,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fcyclotomic-and-abelian-extensions":17904,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fgalois-groups-of-polynomials":17908,"\u002Fabstract-algebra\u002Fgalois-theory\u002Fsolvability-by-radicals-and-the-quintic":17912,"\u002Fabstract-algebra\u002Fcapstone\u002Fcommutative-algebra-and-algebraic-geometry":17916,"\u002Fabstract-algebra\u002Fcapstone\u002Frepresentation-and-character-theory":17921,"\u002Fabstract-algebra":17925,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fatomic-spectra-rutherford":17928,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-model-hydrogen":17933,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fx-ray-spectra-franck-hertz":17937,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fbohr-sommerfeld-old-quantum-theory":17941,"\u002Fatomic-physics\u002Fearly-models-and-old-quantum-theory\u002Fold-quantum-theory-limits-wkb":17945,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fschrodinger-3d-hydrogen":17949,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fhydrogen-wave-functions":17954,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fradial-equation-in-full":17958,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fsymmetry-degeneracy-runge-lenz":17962,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fexpectation-values-virial":17966,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Fquantum-defects-alkali-spectra":17970,"\u002Fatomic-physics\u002Fquantum-hydrogen-atom\u002Frydberg-atoms":17974,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Frelativistic-kinetic-correction":17978,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fspin-orbit-thomas-precession":17983,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdarwin-term-fine-structure-formula":17987,"\u002Fatomic-physics\u002Ffine-structure-and-the-dirac-atom\u002Fdirac-equation-hydrogen":17991,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Flamb-shift-qed":17995,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fhyperfine-structure-21cm":18000,"\u002Fatomic-physics\u002Fqed-corrections-and-hyperfine-structure\u002Fnuclear-effects-isotope-shift":18004,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fperiodic-table-atomic-spectra":18008,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fcentral-field-self-consistent":18013,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fidentical-particles-hartree-fock":18017,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhelium-two-electron-atom":18021,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fls-jj-coupling-term-symbols":18025,"\u002Fatomic-physics\u002Fmany-electron-atoms\u002Fhund-rules-ground-terms":18029,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fzeeman-effect":18033,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fpaschen-back-intermediate":18038,"\u002Fatomic-physics\u002Fatoms-in-external-fields\u002Fstark-effect-polarizability":18042,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Ftime-dependent-perturbation-golden-rule":18046,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fdipole-approximation-einstein-coefficients":18051,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Fselection-rules-forbidden-transitions":18055,"\u002Fatomic-physics\u002Fradiative-transitions-and-line-shapes\u002Flifetimes-and-line-shapes":18059,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Flaser-principles":18063,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fspectroscopy-techniques":18068,"\u002Fatomic-physics\u002Flasers-and-spectroscopy\u002Fline-catalog-nist-asd":18072,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Flaser-cooling-doppler":18076,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fsub-doppler-trapping":18081,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Fbose-einstein-condensation":18085,"\u002Fatomic-physics\u002Fmodern-atomic-physics\u002Foptical-clocks-precision":18089,"\u002Fatomic-physics":18093,"\u002Fdatabases":18096,"\u002Fcategory-theory\u002Ffoundations\u002Fwhat-is-a-category":18099,"\u002Fcategory-theory\u002Ffoundations\u002Fexamples-of-categories":18103,"\u002Fcategory-theory\u002Ffoundations\u002Fspecial-morphisms":18107,"\u002Fcategory-theory\u002Ffoundations\u002Ffunctors":18111,"\u002Fcategory-theory\u002Ffoundations\u002Fnatural-transformations":18115,"\u002Fcategory-theory\u002Ffoundations\u002Fsize-and-set-theory":18119,"\u002Fcategory-theory\u002Funiversal-properties\u002Funiversal-properties":18123,"\u002Fcategory-theory\u002Funiversal-properties\u002Fproducts-and-coproducts":18128,"\u002Fcategory-theory\u002Funiversal-properties\u002Fconstructions-on-categories":18132,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Frepresentable-functors":18136,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-lemma":18141,"\u002Fcategory-theory\u002Frepresentables-yoneda\u002Fyoneda-consequences":18145,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits":18149,"\u002Fcategory-theory\u002Flimits-colimits\u002Fproducts-equalizers-pullbacks":18154,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcolimits":18158,"\u002Fcategory-theory\u002Flimits-colimits\u002Fcomputing-limits":18162,"\u002Fcategory-theory\u002Flimits-colimits\u002Flimits-and-functors":18166,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions":18170,"\u002Fcategory-theory\u002Fadjunctions\u002Funits-and-counits":18175,"\u002Fcategory-theory\u002Fadjunctions\u002Fadjunctions-via-universal-arrows":18179,"\u002Fcategory-theory\u002Fadjunctions\u002Ffree-forgetful-adjunctions":18183,"\u002Fcategory-theory\u002Fadjoints-limits\u002Flimits-via-adjoints":18187,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fpresheaf-limits-colimits":18192,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoints-preserve-limits":18196,"\u002Fcategory-theory\u002Fadjoints-limits\u002Fadjoint-functor-theorem":18200,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fmonads":18204,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-eilenberg-moore":18209,"\u002Fcategory-theory\u002Fmonads-algebras\u002Fkleisli-and-programming":18213,"\u002Fcategory-theory\u002Fmonads-algebras\u002Falgebras-for-endofunctors":18217,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Fcartesian-closed-categories":18221,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Flambda-calculus-correspondence":18226,"\u002Fcategory-theory\u002Fcartesian-closed-lambda\u002Ffixed-points-and-recursion":18230,"\u002Fcategory-theory":18234,"\u002Fdeep-learning\u002Fmathematical-background\u002Flinear-algebra-for-deep-learning":18237,"\u002Fdeep-learning\u002Fmathematical-background\u002Fprobability-and-information-theory":18241,"\u002Fdeep-learning\u002Fmathematical-background\u002Fnumerical-computation":18245,"\u002Fdeep-learning\u002Fmathematical-background\u002Fcalculus":18249,"\u002Fdeep-learning\u002Ffoundations\u002Fwhat-is-deep-learning":18252,"\u002Fdeep-learning\u002Ffoundations\u002Fmachine-learning-refresher":18256,"\u002Fdeep-learning\u002Ffoundations\u002Flinear-models-and-the-perceptron":18260,"\u002Fdeep-learning\u002Fneural-networks\u002Fthe-multilayer-perceptron":18264,"\u002Fdeep-learning\u002Fneural-networks\u002Factivation-functions":18269,"\u002Fdeep-learning\u002Fneural-networks\u002Funiversal-approximation":18273,"\u002Fdeep-learning\u002Fneural-networks\u002Fbackpropagation":18277,"\u002Fdeep-learning\u002Fneural-networks\u002Floss-functions-and-output-units":18281,"\u002Fdeep-learning\u002Foptimization\u002Fgradient-descent-and-sgd":18285,"\u002Fdeep-learning\u002Foptimization\u002Fmomentum-and-adaptive-methods":18290,"\u002Fdeep-learning\u002Foptimization\u002Finitialization":18294,"\u002Fdeep-learning\u002Foptimization\u002Fthe-optimization-landscape":18298,"\u002Fdeep-learning\u002Foptimization\u002Fsecond-order-and-approximate-methods":18302,"\u002Fdeep-learning\u002Fregularization\u002Fregularization-overview":18306,"\u002Fdeep-learning\u002Fregularization\u002Fdropout-and-data-augmentation":18311,"\u002Fdeep-learning\u002Fregularization\u002Fearly-stopping-and-parameter-sharing":18315,"\u002Fdeep-learning\u002Fregularization\u002Fnormalization":18319,"\u002Fdeep-learning\u002Farchitectures\u002Fconvolutional-networks":18323,"\u002Fdeep-learning\u002Farchitectures\u002Fcnn-architectures":18328,"\u002Fdeep-learning\u002Farchitectures\u002Frecurrent-networks":18332,"\u002Fdeep-learning\u002Farchitectures\u002Flstm-and-gru":18336,"\u002Fdeep-learning\u002Farchitectures\u002Fattention-and-transformers":18340,"\u002Fdeep-learning\u002Farchitectures\u002Fthe-transformer-architecture":18344,"\u002Fdeep-learning\u002Farchitectures\u002Ftransformers-in-practice":18348,"\u002Fdeep-learning\u002Farchitectures\u002Fgraph-neural-networks":18352,"\u002Fdeep-learning\u002Farchitectures\u002Fstate-space-models":18356,"\u002Fdeep-learning\u002Ftheory\u002Fgeneralization-theory":18360,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-robustness":18365,"\u002Fdeep-learning\u002Ftheory\u002Fadversarial-defenses":18369,"\u002Fdeep-learning\u002Ftheory\u002Fbayesian-and-ensemble-methods":18373,"\u002Fdeep-learning\u002Ftheory\u002Fdeep-equilibrium-models":18377,"\u002Fdeep-learning\u002Fgenerative-models\u002Flinear-factor-models":18381,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoencoders":18386,"\u002Fdeep-learning\u002Fgenerative-models\u002Fvariational-autoencoders":18390,"\u002Fdeep-learning\u002Fgenerative-models\u002Fgenerative-adversarial-networks":18394,"\u002Fdeep-learning\u002Fgenerative-models\u002Fautoregressive-and-normalizing-flows":18398,"\u002Fdeep-learning\u002Fgenerative-models\u002Fenergy-based-and-boltzmann-machines":18402,"\u002Fdeep-learning\u002Fgenerative-models\u002Fdiffusion-and-score-based-models":18406,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fstructured-probabilistic-models":18410,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fmonte-carlo-and-mcmc":18415,"\u002Fdeep-learning\u002Fprobabilistic-methods\u002Fapproximate-inference":18419,"\u002Fdeep-learning\u002Fpractical\u002Fpractical-methodology":18423,"\u002Fdeep-learning\u002Fpractical\u002Fhyperparameters-and-debugging":18428,"\u002Fdeep-learning\u002Fpractical\u002Frepresentation-learning":18432,"\u002Fdeep-learning\u002Fpractical\u002Ftransfer-learning":18436,"\u002Fdeep-learning\u002Fpractical\u002Fapplications":18440,"\u002Fdeep-learning\u002Fpractical\u002Fmodel-compression-and-distillation":18444,"\u002Fdeep-learning\u002Fpractical\u002Fmeta-learning-and-few-shot":18448,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Flarge-language-models":18452,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fscaling-inference-and-alignment":18456,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fseq2seq-pretraining-and-bart":18459,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ftext-to-text-transfer-and-conditional-generation":18463,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fspeech-and-audio-models":18467,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fself-supervised-speech-and-synthesis":18471,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fai-agents":18475,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fagent-memory-retrieval-and-orchestration":18478,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmixture-of-experts":18482,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Fmultimodal-models":18486,"\u002Fdeep-learning\u002Flarge-models-and-agents\u002Ffusion-and-vision-language-models":18490,"\u002Fdeep-learning\u002Freinforcement-learning\u002Ffoundations-of-reinforcement-learning":18494,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fmodel-free-prediction-and-control":18497,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fdeep-q-networks":18501,"\u002Fdeep-learning\u002Freinforcement-learning\u002Fpolicy-gradients-and-actor-critic":18505,"\u002Fdeep-learning\u002Freinforcement-learning\u002Frl-from-human-feedback":18508,"\u002Fdeep-learning":18509,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fequilibrium-state-variables-zeroth-law":18512,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Ffirst-law-heat-and-work":18516,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fsecond-law-entropy-and-the-carnot-bound":18520,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fthermodynamic-potentials-and-maxwell-relations":18524,"\u002Fstatistical-mechanics\u002Fthermodynamics\u002Fstability-response-functions-and-the-third-law":18528,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fclassical-statistics-and-equipartition":18532,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fphase-space-and-liouvilles-theorem":18537,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fensembles-and-the-equal-probability-postulate":18541,"\u002Fstatistical-mechanics\u002Ffoundations\u002Fstatistical-entropy-boltzmann-and-gibbs":18545,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fmicrocanonical-ensemble-and-entropy":18549,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fequilibrium-conditions-temperature-pressure-chemical-potential":18554,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Fideal-gas-phase-space-and-the-sackur-tetrode-entropy":18558,"\u002Fstatistical-mechanics\u002Fmicrocanonical\u002Ftwo-state-systems-paramagnets-and-negative-temperature":18562,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fcanonical-ensemble-and-the-boltzmann-distribution":18566,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fpartition-function-and-the-helmholtz-free-energy":18571,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fenergy-fluctuations-and-ensemble-equivalence":18575,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fthe-einstein-solid-and-harmonic-systems":18579,"\u002Fstatistical-mechanics\u002Fcanonical\u002Fparamagnetism-and-the-schottky-anomaly":18583,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fideal-gas-partition-function-and-the-gibbs-paradox":18587,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fequipartition-and-the-virial-theorem":18592,"\u002Fstatistical-mechanics\u002Fclassical-gas\u002Fmolecular-gases-rotation-and-vibration":18596,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fgrand-canonical-ensemble-and-the-grand-partition-function":18600,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fchemical-potential-fugacity-and-number-fluctuations":18605,"\u002Fstatistical-mechanics\u002Fgrand-canonical\u002Fensemble-summary-and-the-thermodynamic-web":18609,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fquantum-statistics-bose-einstein-and-fermi-dirac":18613,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fderiving-the-quantum-distributions":18618,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fthe-classical-limit-and-quantum-concentration":18622,"\u002Fstatistical-mechanics\u002Fquantum-statistics\u002Fideal-quantum-gases-general-framework":18626,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-and-the-fermion-gas":18630,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthe-photon-gas-and-plancks-radiation-law":18635,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fblackbody-thermodynamics-and-radiation-pressure":18639,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fphonons-and-the-debye-model":18643,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fbose-einstein-condensation-derived":18647,"\u002Fstatistical-mechanics\u002Fbose-systems\u002Fthermodynamics-of-the-bose-gas-and-superfluidity":18651,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fthe-ideal-fermi-gas-at-zero-temperature":18655,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fsommerfeld-expansion-and-electrons-in-metals":18660,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":18664,"\u002Fstatistical-mechanics\u002Ffermi-gas\u002Fneutron-stars-and-nuclear-matter":18668,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-cluster-expansion-and-virial-coefficients":18672,"\u002Fstatistical-mechanics\u002Finteractions\u002Fthe-van-der-waals-gas-and-liquid-gas-coexistence":18677,"\u002Fstatistical-mechanics\u002Finteractions\u002Fquantum-gases-with-interactions-and-exchange":18681,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fphases-coexistence-and-classification":18685,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-ising-model-and-exact-solutions":18690,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fmean-field-theory-and-the-weiss-model":18694,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fcritical-exponents-and-landau-theory":18698,"\u002Fstatistical-mechanics\u002Fphase-transitions\u002Fthe-renormalization-group-idea":18702,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fthermodynamic-fluctuations-and-response":18706,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Fbrownian-motion-and-the-langevin-equation":18711,"\u002Fstatistical-mechanics\u002Ffluctuations\u002Flinear-response-and-the-fluctuation-dissipation-theorem":18715,"\u002Fstatistical-mechanics":18719,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fbonding-mechanisms":18722,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fmolecular-orbitals-and-h2-plus":18727,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fhydrogen-molecule-and-exchange":18731,"\u002Fcondensed-matter\u002Fmolecules-and-bonding\u002Fvan-der-waals-forces":18735,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Frotational-vibrational-spectra":18739,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Fanharmonicity-and-rovibrational-structure":18744,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Framan-and-electronic-bands":18748,"\u002Fcondensed-matter\u002Fmolecular-spectra\u002Flasers-and-masers":18752,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fstructure-of-solids":18756,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fbravais-lattices-and-crystal-systems":18761,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Freciprocal-lattice-and-brillouin-zones":18765,"\u002Fcondensed-matter\u002Fcrystal-structure\u002Fdiffraction-and-structure-factors":18769,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonon-dispersion":18773,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fphonons-quantization-and-dos":18778,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fdebye-einstein-heat-capacity":18782,"\u002Fcondensed-matter\u002Flattice-dynamics\u002Fanharmonicity-and-thermal-transport":18786,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ffree-electron-gas-and-conduction":18790,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fsommerfeld-model-and-heat-capacity":18795,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Ftransport-and-the-hall-effect":18799,"\u002Fcondensed-matter\u002Ffree-electron-fermi-gas\u002Fscreening-and-plasmons":18803,"\u002Fcondensed-matter\u002Fband-theory\u002Fblochs-theorem-and-energy-bands":18807,"\u002Fcondensed-matter\u002Fband-theory\u002Fnearly-free-electron-model":18812,"\u002Fcondensed-matter\u002Fband-theory\u002Ftight-binding-method":18816,"\u002Fcondensed-matter\u002Fband-theory\u002Ffermi-surfaces-and-semiclassical-dynamics":18820,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fsemiconductor-bands-and-junctions":18824,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fintrinsic-and-extrinsic-semiconductors":18829,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fcarrier-transport-and-recombination":18833,"\u002Fcondensed-matter\u002Fsemiconductors\u002Fthe-pn-junction":18837,"\u002Fcondensed-matter\u002Fsemiconductors\u002Ftransistors-and-optoelectronics":18841,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fdielectrics-and-polarization":18845,"\u002Fcondensed-matter\u002Fdielectrics-and-ferroelectrics\u002Fferroelectrics-and-piezoelectrics":18850,"\u002Fcondensed-matter\u002Fmagnetism\u002Fdiamagnetism-and-paramagnetism":18854,"\u002Fcondensed-matter\u002Fmagnetism\u002Fexchange-and-ferromagnetism":18859,"\u002Fcondensed-matter\u002Fmagnetism\u002Fantiferromagnetism-and-domains":18863,"\u002Fcondensed-matter\u002Fmagnetism\u002Fspin-waves-and-magnons":18867,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fsuperconductivity-phenomenology":18871,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Flondon-theory-and-the-meissner-effect":18876,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fginzburg-landau-theory":18880,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fbcs-theory":18884,"\u002Fcondensed-matter\u002Fsuperconductivity\u002Fjosephson-and-high-tc":18888,"\u002Fcondensed-matter\u002Fnanostructures\u002Fquantum-wells-wires-and-dots":18892,"\u002Fcondensed-matter\u002Fnanostructures\u002Finteger-quantum-hall-effect":18897,"\u002Fcondensed-matter\u002Fnanostructures\u002Ffractional-quantum-hall-and-topology":18901,"\u002Fcondensed-matter\u002Fnanostructures\u002Fgraphene-and-dirac-materials":18905,"\u002Fcondensed-matter":18909,"\u002Flogic\u002Ffoundations\u002Flogic-as-a-mathematical-model":18912,"\u002Flogic\u002Fsentential-logic\u002Fformal-languages-and-well-formed-formulas":18916,"\u002Flogic\u002Fsentential-logic\u002Ftruth-assignments-and-tautologies":18921,"\u002Flogic\u002Fsentential-logic\u002Funique-readability-and-parsing":18925,"\u002Flogic\u002Fsentential-logic\u002Finduction-and-recursion":18929,"\u002Flogic\u002Fsentential-logic\u002Fexpressive-completeness-and-normal-forms":18933,"\u002Flogic\u002Fsentential-logic\u002Fboolean-circuits":18937,"\u002Flogic\u002Fsentential-logic\u002Fcompactness-and-effectiveness":18941,"\u002Flogic\u002Ffirst-order-languages\u002Ffirst-order-languages":18945,"\u002Flogic\u002Ffirst-order-languages\u002Fstructures-truth-and-satisfaction":18950,"\u002Flogic\u002Ffirst-order-languages\u002Fdefinability-and-elementary-equivalence":18954,"\u002Flogic\u002Ffirst-order-languages\u002Fterms-substitution-and-parsing":18958,"\u002Flogic\u002Fdeductive-calculus\u002Fa-deductive-calculus":18962,"\u002Flogic\u002Fdeductive-calculus\u002Fdeduction-theorem-and-derived-rules":18967,"\u002Flogic\u002Fdeductive-calculus\u002Fsoundness":18971,"\u002Flogic\u002Fdeductive-calculus\u002Fcompleteness-and-consistency":18975,"\u002Flogic\u002Fmodels-and-theories\u002Fcompactness-and-lowenheim-skolem":18979,"\u002Flogic\u002Fmodels-and-theories\u002Ftheories-elementary-classes-and-categoricity":18984,"\u002Flogic\u002Fmodels-and-theories\u002Finterpretations-between-theories":18988,"\u002Flogic\u002Fmodels-and-theories\u002Fnonstandard-analysis":18992,"\u002Flogic\u002Farithmetic-and-definability\u002Fdefinability-in-arithmetic":18996,"\u002Flogic\u002Farithmetic-and-definability\u002Fnatural-numbers-with-successor":19001,"\u002Flogic\u002Farithmetic-and-definability\u002Fpresburger-and-reducts":19005,"\u002Flogic\u002Farithmetic-and-definability\u002Fa-subtheory-and-representability":19009,"\u002Flogic\u002Fincompleteness\u002Farithmetization-of-syntax":19013,"\u002Flogic\u002Fincompleteness\u002Fincompleteness-and-undecidability":19018,"\u002Flogic\u002Fincompleteness\u002Fsecond-incompleteness-theorem":19022,"\u002Flogic\u002Fcomputability-and-representability\u002Frecursive-functions":19026,"\u002Flogic\u002Fcomputability-and-representability\u002Frepresenting-exponentiation":19031,"\u002Flogic\u002Fsecond-order-logic\u002Fsecond-order-languages":19035,"\u002Flogic\u002Fsecond-order-logic\u002Fskolem-functions-and-many-sorted-logic":19040,"\u002Flogic\u002Fsecond-order-logic\u002Fgeneral-structures":19044,"\u002Flogic":19048,"\u002Freinforcement-learning\u002Ffoundations\u002Fwhat-is-reinforcement-learning":19051,"\u002Freinforcement-learning\u002Ffoundations\u002Fa-brief-history-of-rl":19055,"\u002Freinforcement-learning\u002Ffoundations\u002Fmulti-armed-bandits":19059,"\u002Freinforcement-learning\u002Ffoundations\u002Fbandit-exploration-algorithms":19063,"\u002Freinforcement-learning\u002Ffoundations\u002Fmarkov-decision-processes":19067,"\u002Freinforcement-learning\u002Ffoundations\u002Fvalue-functions-and-optimality":19071,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdynamic-programming":19075,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdp-async-and-gpi":19079,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-methods":19083,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-off-policy":19087,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftemporal-difference-learning":19091,"\u002Freinforcement-learning\u002Ftabular-methods\u002Ftd-control-sarsa-and-q-learning":19095,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-bootstrapping":19099,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fn-step-off-policy-methods":19103,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-and-learning":19107,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fplanning-focusing-and-decision-time":19111,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fdecision-time-planning":19115,"\u002Freinforcement-learning\u002Ftabular-methods\u002Fmonte-carlo-tree-search":19119,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-prediction":19123,"\u002Freinforcement-learning\u002Fapproximation\u002Ffeature-construction-and-nonlinear":19128,"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-control":19132,"\u002Freinforcement-learning\u002Fapproximation\u002Faverage-reward-control":19136,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad":19140,"\u002Freinforcement-learning\u002Fapproximation\u002Fbellman-error-and-gradient-td":19144,"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces":19148,"\u002Freinforcement-learning\u002Fapproximation\u002Ftrue-online-and-sarsa-lambda":19152,"\u002Freinforcement-learning\u002Fapproximation\u002Fpolicy-gradient-methods":19156,"\u002Freinforcement-learning\u002Fapproximation\u002Factor-critic-and-continuous-actions":19160,"\u002Freinforcement-learning\u002Fapproximation\u002Fleast-squares-and-memory-based-methods":19164,"\u002Freinforcement-learning\u002Fapproximation\u002Fmemory-and-kernel-methods":19168,"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces":19172,"\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces":19176,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdeep-q-networks":19180,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fdqn-improvements":19184,"\u002Freinforcement-learning\u002Fdeep-rl\u002Factor-critic-and-ppo":19188,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fppo-and-continuous-control":19192,"\u002Freinforcement-learning\u002Fdeep-rl\u002Fcase-studies":19196,"\u002Freinforcement-learning\u002Fdeep-rl\u002Frl-beyond-games":19200,"\u002Freinforcement-learning\u002Fdeep-rl\u002Ffrontiers":19204,"\u002Freinforcement-learning\u002Fdeep-rl\u002Freward-design-and-open-problems":19208,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow":19212,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fdistributional-and-rainbow-part-2":19217,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control":19221,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fcontinuous-control-part-2":19225,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl":19229,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmodel-based-rl-part-2":19233,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration":19237,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fexploration-part-2":19241,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl":19245,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Foffline-rl-part-2":19249,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl":19253,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fimitation-and-inverse-rl-part-2":19257,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl":19261,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmulti-agent-rl-part-2":19265,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl":19269,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fhierarchical-rl-part-2":19273,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Frlhf-and-language-models":19277,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps":19281,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fpartial-observability-pomdps-part-2":19285,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl":19289,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fsafe-and-constrained-rl-part-2":19293,"\u002Freinforcement-learning\u002Fmodern-deep-rl\u002Fmeta-rl-and-generalization":19297,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fpsychology-of-reinforcement":19301,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Finstrumental-conditioning-and-control":19306,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-and-td-error":19310,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fdopamine-in-the-brain":19314,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fanimal-learning-and-cognition":19318,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fcognitive-maps-and-planning":19322,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fneuroscience-of-reinforcement":19326,"\u002Freinforcement-learning\u002Fminds-and-brains\u002Fseveral-learning-systems":19330,"\u002Freinforcement-learning":19334,"\u002Fartificial-intelligence\u002Ffoundations\u002Fwhat-is-ai":19336,"\u002Fartificial-intelligence\u002Ffoundations\u002Ffoundations-of-ai":19340,"\u002Fartificial-intelligence\u002Ffoundations\u002Fintelligent-agents":19344,"\u002Fartificial-intelligence\u002Ffoundations\u002Fagent-architectures":19348,"\u002Fartificial-intelligence\u002Fsearch\u002Funinformed-search":19352,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-strategies-compared":19357,"\u002Fartificial-intelligence\u002Fsearch\u002Finformed-search":19361,"\u002Fartificial-intelligence\u002Fsearch\u002Fheuristic-functions":19365,"\u002Fartificial-intelligence\u002Fsearch\u002Flocal-search":19369,"\u002Fartificial-intelligence\u002Fsearch\u002Fpopulation-and-continuous-search":19373,"\u002Fartificial-intelligence\u002Fsearch\u002Fadversarial-search":19377,"\u002Fartificial-intelligence\u002Fsearch\u002Fgames-of-chance-and-imperfect-information":19381,"\u002Fartificial-intelligence\u002Fsearch\u002Fconstraint-satisfaction":19385,"\u002Fartificial-intelligence\u002Fsearch\u002Fcsp-search-and-structure":19389,"\u002Fartificial-intelligence\u002Fsearch\u002Fsearch-under-uncertainty":19393,"\u002Fartificial-intelligence\u002Fsearch\u002Fbelief-state-and-online-search":19397,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-logic":19401,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fpropositional-inference":19406,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic":19410,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-logic-in-use":19414,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Finference-and-resolution":19418,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Ffirst-order-resolution":19422,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fclassical-planning":19426,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-graphs-and-graphplan":19430,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-in-the-real-world":19434,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fplanning-under-uncertainty":19438,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Fknowledge-representation":19442,"\u002Fartificial-intelligence\u002Flogic-and-planning\u002Freasoning-systems-and-defaults":19446,"\u002Fartificial-intelligence\u002Funcertainty\u002Fprobability-and-bayes":19450,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayes-rule-and-naive-bayes":19455,"\u002Fartificial-intelligence\u002Funcertainty\u002Fbayesian-networks":19459,"\u002Fartificial-intelligence\u002Funcertainty\u002Finference-in-bayesian-networks":19463,"\u002Fartificial-intelligence\u002Funcertainty\u002Freasoning-over-time":19467,"\u002Fartificial-intelligence\u002Funcertainty\u002Ftracking-and-data-association":19471,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmaking-decisions":19475,"\u002Fartificial-intelligence\u002Funcertainty\u002Fmarkov-decision-processes":19479,"\u002Fartificial-intelligence\u002Funcertainty\u002Fdecision-networks-and-game-theory":19482,"\u002Fartificial-intelligence\u002Funcertainty\u002Fgame-theory-and-mechanism-design":19486,"\u002Fartificial-intelligence\u002Flearning\u002Flearning-from-examples":19490,"\u002Fartificial-intelligence\u002Flearning\u002Ftheory-and-model-families":19495,"\u002Fartificial-intelligence\u002Flearning\u002Fprobabilistic-learning":19499,"\u002Fartificial-intelligence\u002Flearning\u002Fexpectation-maximization":19503,"\u002Fartificial-intelligence\u002Flearning\u002Freinforcement-learning":19507,"\u002Fartificial-intelligence\u002Flearning\u002Fgeneralization-and-policy-search":19510,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-in-learning":19514,"\u002Fartificial-intelligence\u002Flearning\u002Fknowledge-based-learning-methods":19518,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fvision-and-perception":19522,"\u002Fartificial-intelligence\u002Ffrontiers\u002Freconstructing-the-3d-world":19527,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobotics":19531,"\u002Fartificial-intelligence\u002Ffrontiers\u002Frobot-planning-and-control":19535,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnatural-language-in-ai":19539,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fnlp-grammar-translation-and-speech":19543,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fphilosophy-and-future":19547,"\u002Fartificial-intelligence\u002Ffrontiers\u002Fai-ethics-and-future":19551,"\u002Fartificial-intelligence":19555,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-constituents-nuclide-chart":19558,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-size-charge-distributions":19563,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-masses-binding-energy":19567,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fsemi-empirical-mass-formula":19571,"\u002Fnuclear-physics\u002Fnuclear-properties\u002Fnuclear-moments-multipoles":19575,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnuclear-force-shell-overview":19579,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fthe-deuteron":19584,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fnucleon-nucleon-scattering":19588,"\u002Fnuclear-physics\u002Fnuclear-force-deuteron\u002Fmeson-theory-isospin":19592,"\u002Fnuclear-physics\u002Fnuclear-models\u002Ffermi-gas-model":19596,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fliquid-drop-collective-coordinates":19601,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fshell-model-single-particle":19605,"\u002Fnuclear-physics\u002Fnuclear-models\u002Fcollective-model-rotations-vibrations":19609,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-law-modes":19613,"\u002Fnuclear-physics\u002Fradioactive-decay\u002Fdecay-kinetics-equilibrium":19618,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-decay-gamow-theory":19622,"\u002Fnuclear-physics\u002Falpha-decay\u002Falpha-fine-structure-hindrance":19627,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fbeta-decay-energetics-neutrino":19631,"\u002Fnuclear-physics\u002Fbeta-decay\u002Ffermi-theory-beta-decay":19636,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fweak-interaction-parity-violation":19640,"\u002Fnuclear-physics\u002Fbeta-decay\u002Fdouble-beta-decay-neutrino-mass":19644,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fgamma-multipole-radiation":19648,"\u002Fnuclear-physics\u002Fgamma-decay\u002Finternal-conversion-isomers":19653,"\u002Fnuclear-physics\u002Fgamma-decay\u002Fangular-correlations-mossbauer":19657,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Freaction-kinematics-cross-sections":19661,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fcompound-nucleus-resonances":19666,"\u002Fnuclear-physics\u002Fnuclear-reactions\u002Fdirect-reactions-optical-model":19670,"\u002Fnuclear-physics\u002Ffission\u002Ffission-barrier-dynamics":19674,"\u002Fnuclear-physics\u002Ffission\u002Fchain-reactions-reactor-physics":19679,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Ffusion-reactions-confinement":19683,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fstellar-nucleosynthesis":19688,"\u002Fnuclear-physics\u002Ffusion-nucleosynthesis\u002Fbig-bang-nucleosynthesis":19692,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fcharged-particle-stopping-power":19696,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fphoton-neutron-interactions":19701,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fradiation-detectors":19705,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fdosimetry-radiation-biology":19709,"\u002Fnuclear-physics\u002Fradiation-matter-applications\u002Fnuclear-applications-dating-medicine":19713,"\u002Fnuclear-physics":19717,"\u002Fnatural-language-processing\u002Ffoundations\u002Fwhat-is-nlp":19720,"\u002Fnatural-language-processing\u002Ffoundations\u002Fregex-and-text-normalization":19724,"\u002Fnatural-language-processing\u002Ffoundations\u002Fminimum-edit-distance":19728,"\u002Fnatural-language-processing\u002Ffoundations\u002Fn-gram-language-models":19732,"\u002Fnatural-language-processing\u002Ffoundations\u002Fsmoothing-and-backoff":19736,"\u002Fnatural-language-processing\u002Fclassification\u002Fnaive-bayes-and-sentiment":19740,"\u002Fnatural-language-processing\u002Fclassification\u002Fevaluating-classifiers":19745,"\u002Fnatural-language-processing\u002Fclassification\u002Flogistic-regression":19749,"\u002Fnatural-language-processing\u002Fclassification\u002Fsentiment-and-affect-lexicons":19753,"\u002Fnatural-language-processing\u002Fsemantics\u002Fvector-semantics-and-embeddings":19757,"\u002Fnatural-language-processing\u002Fsemantics\u002Fstatic-word-embeddings":19762,"\u002Fnatural-language-processing\u002Fsemantics\u002Fneural-language-models":19766,"\u002Fnatural-language-processing\u002Fsequences\u002Fsequence-labeling":19770,"\u002Fnatural-language-processing\u002Fsequences\u002Fcrfs-and-neural-taggers":19774,"\u002Fnatural-language-processing\u002Fsequences\u002Frnns-and-lstms":19778,"\u002Fnatural-language-processing\u002Ftransformers\u002Ftransformers-and-attention":19782,"\u002Fnatural-language-processing\u002Ftransformers\u002Fthe-transformer-architecture":19786,"\u002Fnatural-language-processing\u002Ftransformers\u002Flarge-language-models":19789,"\u002Fnatural-language-processing\u002Ftransformers\u002Fllm-pretraining-and-scaling":19792,"\u002Fnatural-language-processing\u002Ftransformers\u002Ffine-tuning-and-prompting":19796,"\u002Fnatural-language-processing\u002Ftransformers\u002Fprompting-and-alignment":19800,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-parsing":19804,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcky-scoring-and-evaluation":19809,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdependency-parsing":19813,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fgraph-based-and-neural-dependency-parsing":19817,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fword-senses-and-wsd":19821,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fwsd-in-practice-and-induction":19825,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-roles-and-information-extraction":19829,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Frelations-events-and-templates":19833,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoreference-and-discourse":19837,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcoherence-and-discourse-structure":19841,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Flogical-semantics":19845,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fcompositional-semantics-and-description-logics":19849,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fsemantic-parsing":19853,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fneural-semantic-parsing":19857,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Finformation-extraction":19861,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftimes-events-and-templates":19865,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fdiscourse-coherence":19869,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fentity-based-and-global-coherence":19873,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Fconstituency-grammars":19877,"\u002Fnatural-language-processing\u002Flinguistic-structure\u002Ftreebanks-and-lexicalized-grammars":19881,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation":19885,"\u002Fnatural-language-processing\u002Fapplications\u002Fmachine-translation-decoding-and-evaluation":19889,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering":19893,"\u002Fnatural-language-processing\u002Fapplications\u002Fquestion-answering-knowledge-and-llms":19897,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-and-chatbots":19901,"\u002Fnatural-language-processing\u002Fapplications\u002Fdialogue-systems-and-assistants":19905,"\u002Fnatural-language-processing\u002Fapplications\u002Ftext-summarization":19909,"\u002Fnatural-language-processing\u002Fapplications\u002Fabstractive-summarization-and-evaluation":19913,"\u002Fnatural-language-processing\u002Fspeech\u002Fphonetics":19917,"\u002Fnatural-language-processing\u002Fspeech\u002Facoustic-phonetics":19922,"\u002Fnatural-language-processing\u002Fspeech\u002Fautomatic-speech-recognition":19926,"\u002Fnatural-language-processing\u002Fspeech\u002Fasr-evaluation-and-applications":19930,"\u002Fnatural-language-processing":19934,"\u002Fparticle-physics\u002Ffoundations\u002Fhistorical-overview-particle-zoo":19937,"\u002Fparticle-physics\u002Ffoundations\u002Fparticle-physics-basic-concepts":19941,"\u002Fparticle-physics\u002Ffoundations\u002Ffundamental-interactions-force-carriers":19945,"\u002Fparticle-physics\u002Funits-kinematics\u002Fnatural-units-and-scales":19949,"\u002Fparticle-physics\u002Funits-kinematics\u002Ffour-vectors-invariant-mass":19954,"\u002Fparticle-physics\u002Funits-kinematics\u002Fdecay-scattering-kinematics-mandelstam":19958,"\u002Fparticle-physics\u002Funits-kinematics\u002Fcross-sections-golden-rule":19962,"\u002Fparticle-physics\u002Fsymmetries\u002Fconservation-laws-symmetries":19966,"\u002Fparticle-physics\u002Fsymmetries\u002Fdiscrete-symmetries-cpt":19971,"\u002Fparticle-physics\u002Fsymmetries\u002Fparity-violation-weak":19975,"\u002Fparticle-physics\u002Fsymmetries\u002Fsu2-su3-flavor-symmetry":19979,"\u002Fparticle-physics\u002Fquark-model\u002Feightfold-way-su3":19983,"\u002Fparticle-physics\u002Fquark-model\u002Fmeson-spectroscopy":19988,"\u002Fparticle-physics\u002Fquark-model\u002Fbaryon-spectroscopy":19992,"\u002Fparticle-physics\u002Fquark-model\u002Fcolor-confinement-exotics":19996,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fklein-gordon-equation":20000,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fdirac-equation-spinors":20005,"\u002Fparticle-physics\u002Frelativistic-wave-equations\u002Fantiparticles-hole-theory":20009,"\u002Fparticle-physics\u002Fqed\u002Ffeynman-rules-qed":20013,"\u002Fparticle-physics\u002Fqed\u002Fqed-tree-processes":20018,"\u002Fparticle-physics\u002Fqed\u002Frenormalization-running-coupling":20022,"\u002Fparticle-physics\u002Fqed\u002Felectron-g-2":20026,"\u002Fparticle-physics\u002Fweak-interaction\u002Fva-structure-weak":20030,"\u002Fparticle-physics\u002Fweak-interaction\u002Fw-z-bosons-decays":20035,"\u002Fparticle-physics\u002Fweak-interaction\u002Fckm-matrix":20039,"\u002Fparticle-physics\u002Fweak-interaction\u002Fcp-violation-kaons-b-mesons":20043,"\u002Fparticle-physics\u002Fqcd\u002Fcolor-su3-gluons":20047,"\u002Fparticle-physics\u002Fqcd\u002Fasymptotic-freedom-confinement":20052,"\u002Fparticle-physics\u002Fqcd\u002Fdeep-inelastic-scattering-partons":20056,"\u002Fparticle-physics\u002Fqcd\u002Fjets-hadronization":20060,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Felectroweak-su2-u1":20064,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fspontaneous-symmetry-breaking":20069,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-mechanism":20073,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fhiggs-boson-discovery":20077,"\u002Fparticle-physics\u002Felectroweak-higgs\u002Fstandard-model":20081,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-oscillations":20085,"\u002Fparticle-physics\u002Fneutrinos\u002Fneutrino-mass-pmns":20090,"\u002Fparticle-physics\u002Fneutrinos\u002Fdirac-majorana-experiments":20094,"\u002Fparticle-physics\u002Fexperiment\u002Faccelerators-luminosity":20098,"\u002Fparticle-physics\u002Fexperiment\u002Fdetectors-subsystems":20103,"\u002Fparticle-physics\u002Fexperiment\u002Fhow-discoveries-are-made":20107,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fbeyond-standard-model":20111,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fgrand-unified-theories":20115,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fsupersymmetry":20119,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fhierarchy-problem-naturalness":20123,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fdark-matter-candidates":20127,"\u002Fparticle-physics\u002Fbeyond-standard-model\u002Fmatter-antimatter-open-questions":20131,"\u002Fparticle-physics":20135,"\u002Fastrophysics-cosmology\u002Forientation\u002Fthe-sun-and-stars":20138,"\u002Fastrophysics-cosmology\u002Forientation\u002Fstellar-death-final-states":20143,"\u002Fastrophysics-cosmology\u002Forientation\u002Fgalaxies-and-cosmology":20147,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fmagnitudes-fluxes-and-the-distance-modulus":20151,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fstellar-spectra-and-spectral-classification":20156,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Ftelescopes-and-detectors-across-the-spectrum":20160,"\u002Fastrophysics-cosmology\u002Fobservational-foundations\u002Fthe-cosmic-distance-ladder":20164,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fblackbody-radiation-and-specific-intensity":20168,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fradiative-transfer-and-the-transfer-equation":20173,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fspectral-line-formation-and-broadening":20177,"\u002Fastrophysics-cosmology\u002Fradiation-and-matter\u002Fopacity-and-the-rosseland-mean":20181,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fhydrostatic-equilibrium-and-the-virial-theorem":20185,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equations-of-stellar-structure":20190,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-equation-of-state-and-polytropes":20194,"\u002Fastrophysics-cosmology\u002Fstellar-structure\u002Fthe-standard-solar-model":20198,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fthermonuclear-reaction-rates-and-the-gamow-peak":20202,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhydrogen-burning-pp-chains-and-cno":20207,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fhelium-burning-and-the-triple-alpha-process":20211,"\u002Fastrophysics-cosmology\u002Fnuclear-astrophysics\u002Fadvanced-burning-and-neutron-capture-nucleosynthesis":20215,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fphases-of-the-interstellar-medium":20219,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fmolecular-clouds-and-gravitational-collapse":20224,"\u002Fastrophysics-cosmology\u002Fism-and-star-formation\u002Fprotostars-and-the-pre-main-sequence":20228,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-main-sequence-and-its-structure":20232,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fpost-main-sequence-low-mass-evolution":20237,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fthe-evolution-of-massive-stars":20241,"\u002Fastrophysics-cosmology\u002Fstellar-evolution\u002Fstellar-pulsation-and-the-instability-strip":20245,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fwhite-dwarfs-and-the-chandrasekhar-limit":20249,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fcore-collapse-supernovae":20253,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fthermonuclear-supernovae-type-ia":20257,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fneutron-stars-and-pulsars":20261,"\u002Fastrophysics-cosmology\u002Fstellar-death-and-compact-remnants\u002Fblack-holes-schwarzschild-and-kerr":20265,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fbinary-systems-and-mass-transfer":20269,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Faccreting-compact-objects":20274,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fgravitational-waves-from-inspiraling-binaries":20278,"\u002Fastrophysics-cosmology\u002Fbinaries-and-gravitational-waves\u002Fmultimessenger-astronomy-and-gamma-ray-bursts":20282,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fthe-milky-way":20286,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-morphology-and-classification":20291,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-rotation-curves-and-dark-matter":20295,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Factive-galactic-nuclei-and-supermassive-black-holes":20299,"\u002Fastrophysics-cosmology\u002Fgalaxies\u002Fgalaxy-clusters-and-large-scale-structure":20303,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-expanding-universe-and-hubbles-law":20307,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-frw-metric-and-cosmological-redshift":20312,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fthe-friedmann-equations-and-cosmic-dynamics":20316,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fcosmological-models-and-distances":20319,"\u002Fastrophysics-cosmology\u002Fcosmology-expansion-and-dynamics\u002Fdark-energy-and-the-accelerating-universe":20323,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fthe-thermal-history-of-the-universe":20327,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fbig-bang-nucleosynthesis":20332,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Frecombination-and-the-cosmic-microwave-background":20336,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcmb-anisotropies-and-cosmological-parameters":20340,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fcosmic-inflation":20344,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fstructure-formation-and-the-growth-of-perturbations":20348,"\u002Fastrophysics-cosmology\u002Fthe-hot-big-bang\u002Fdark-matter-dark-energy-and-open-questions":20352,"\u002Fastrophysics-cosmology":20356,"\u002Fcolophon":20359,"\u002F":20362},{"path":15684,"title":15685,"module":15686,"summary":15687},"\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":15689,"title":15690,"module":15686,"summary":15691},"\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":15693,"title":15694,"module":15686,"summary":15695},"\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":15697,"title":15698,"module":15686,"summary":15699},"\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":15701,"title":15702,"module":15686,"summary":15703},"\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":15705,"title":15706,"module":15686,"summary":15707},"\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":15709,"title":15710,"module":15711,"summary":15712},"\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":15714,"title":15715,"module":15711,"summary":15716},"\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":15718,"title":15719,"module":15711,"summary":15720},"\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":15722,"title":15723,"module":15711,"summary":15724},"\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":15726,"title":15727,"module":15728,"summary":15729},"\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":15731,"title":15732,"module":15728,"summary":15733},"\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":15735,"title":15736,"module":15728,"summary":15737},"\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":15739,"title":15740,"module":15728,"summary":15741},"\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":15743,"title":15744,"module":15745,"summary":15746},"\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":15748,"title":15749,"module":15745,"summary":15750},"\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":15752,"title":15753,"module":15745,"summary":15754},"\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":15756,"title":15757,"module":15745,"summary":15758},"\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":15760,"title":15761,"module":15745,"summary":15762},"\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":15764,"title":15765,"module":15745,"summary":15766},"\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":15768,"title":15769,"module":15745,"summary":15770},"\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":15772,"title":15773,"module":15745,"summary":15774},"\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":15776,"title":15777,"module":15745,"summary":15778},"\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":15780,"title":15781,"module":15745,"summary":15782},"\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":15784,"title":15785,"module":15745,"summary":15786},"\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":15788,"title":15789,"module":15745,"summary":15790},"\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":15792,"title":15793,"module":15794,"summary":15795},"\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":15797,"title":15798,"module":15794,"summary":15799},"\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":15801,"title":15802,"module":15794,"summary":15803},"\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":15805,"title":15806,"module":15794,"summary":15807},"\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":15809,"title":15810,"module":15794,"summary":15811},"\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":15813,"title":15814,"module":15794,"summary":15815},"\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":15817,"title":15818,"module":15794,"summary":15819},"\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":15821,"title":15822,"module":15794,"summary":15823},"\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":15825,"title":15826,"module":15827,"summary":15828},"\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":15830,"title":15831,"module":15827,"summary":15832},"\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":15834,"title":15835,"module":15827,"summary":15836},"\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":15838,"title":15839,"module":15827,"summary":15840},"\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":15842,"title":15843,"module":15827,"summary":15844},"\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":15846,"title":15847,"module":15827,"summary":15848},"\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":15850,"title":15851,"module":15827,"summary":15852},"\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":15854,"title":15855,"module":15827,"summary":15856},"\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":15858,"title":15859,"module":15827,"summary":15860},"\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":15862,"title":15863,"module":15827,"summary":15864},"\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":15866,"title":15867,"module":15827,"summary":15868},"\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":15870,"title":15871,"module":15827,"summary":15872},"\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":15874,"title":15875,"module":15827,"summary":15876},"\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":15878,"title":15879,"module":15827,"summary":15880},"\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":15882,"title":15883,"module":15884,"summary":15885},"\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":15887,"title":15888,"module":15884,"summary":15889},"\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":15891,"title":15892,"module":15884,"summary":15893},"\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":15895,"title":15896,"module":15884,"summary":15897},"\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":15899,"title":15900,"module":15884,"summary":15901},"\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":15903,"title":15904,"module":15905,"summary":15906},"\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":15908,"title":15909,"module":15905,"summary":15910},"\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":15912,"title":15913,"module":15905,"summary":15914},"\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":15916,"title":15917,"module":15905,"summary":15918},"\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":15920,"title":15921,"module":15905,"summary":15922},"\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":15924,"title":15925,"module":15905,"summary":15926},"\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":15928,"title":15929,"module":15905,"summary":15930},"\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":15932,"title":15933,"module":15905,"summary":15934},"\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":15936,"title":15937,"module":15905,"summary":15938},"\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":15940,"title":15941,"module":15905,"summary":15942},"\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":15944,"title":15945,"module":15905,"summary":15946},"\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":15948,"title":15949,"module":15950,"summary":15951},"\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":15953,"title":15954,"module":15950,"summary":15955},"\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":15957,"title":15958,"module":15950,"summary":15959},"\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":15961,"title":15962,"module":15950,"summary":15963},"\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":15965,"title":15966,"module":15967,"summary":15968},"\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":15970,"title":15971,"module":15967,"summary":15972},"\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":15974,"title":15975,"module":15967,"summary":15976},"\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":15978,"title":15979,"module":15967,"summary":15980},"\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":15982,"title":15983,"module":15967,"summary":15984},"\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":15986,"title":15987,"module":15967,"summary":15988},"\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":15990,"title":15991,"module":15967,"summary":15992},"\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":15994,"title":15995,"module":15996,"summary":15997},"\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":15999,"title":16000,"module":15996,"summary":16001},"\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":16003,"title":16004,"module":15996,"summary":16005},"\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":16007,"title":16008,"module":15996,"summary":16009},"\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":16011,"title":16012,"module":16013,"summary":16014},"\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":16016,"title":16017,"module":16013,"summary":16018},"\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":16020,"title":16021,"module":16013,"summary":16022},"\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":16024,"title":16025,"module":16013,"summary":16026},"\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":16028,"title":16029,"module":6,"summary":6},"\u002Falgorithms","Algorithms",{"path":16031,"title":16032,"module":16033,"summary":16034},"\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":16036,"title":16037,"module":16033,"summary":16038},"\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":16040,"title":16041,"module":16033,"summary":16042},"\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":16044,"title":16045,"module":16033,"summary":16046},"\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":16048,"title":16049,"module":16050,"summary":16051},"\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":16053,"title":16054,"module":16050,"summary":16055},"\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":16057,"title":16058,"module":16050,"summary":16059},"\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":16061,"title":16062,"module":16050,"summary":16063},"\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":16065,"title":16066,"module":16067,"summary":16068},"\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":16070,"title":16071,"module":16067,"summary":16072},"\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":16074,"title":16075,"module":16067,"summary":16076},"\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":16078,"title":16079,"module":16067,"summary":16080},"\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":16082,"title":16083,"module":16084,"summary":16085},"\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":16087,"title":16088,"module":16084,"summary":16089},"\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":16091,"title":16092,"module":16084,"summary":16093},"\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":16095,"title":16096,"module":16097,"summary":16098},"\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":16100,"title":16101,"module":16097,"summary":16102},"\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":16104,"title":16105,"module":16097,"summary":16106},"\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":16108,"title":16109,"module":16110,"summary":16111},"\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":16113,"title":16114,"module":16110,"summary":16115},"\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":16117,"title":16118,"module":16110,"summary":16119},"\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":16121,"title":16122,"module":16123,"summary":16124},"\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":16126,"title":16127,"module":16123,"summary":16128},"\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":16130,"title":16131,"module":16123,"summary":16132},"\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":16134,"title":16135,"module":16123,"summary":16136},"\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":16138,"title":16139,"module":16140,"summary":16141},"\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":16143,"title":16144,"module":16140,"summary":16145},"\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":16147,"title":16148,"module":16140,"summary":16149},"\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":16151,"title":16152,"module":16153,"summary":16154},"\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":16156,"title":16157,"module":16153,"summary":16158},"\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":16160,"title":16161,"module":16153,"summary":16162},"\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":16164,"title":16165,"module":16153,"summary":16166},"\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":16168,"title":16169,"module":16153,"summary":16170},"\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":16172,"title":16173,"module":16174,"summary":16175},"\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":16177,"title":16178,"module":16174,"summary":16179},"\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":16181,"title":16182,"module":16174,"summary":16183},"\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":16185,"title":16186,"module":16174,"summary":16187},"\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":16189,"title":16190,"module":16174,"summary":16191},"\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":16193,"title":16194,"module":16195,"summary":16196},"\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":16198,"title":16195,"module":16195,"summary":16199},"\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":16201,"title":16202,"module":16195,"summary":16203},"\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":16205,"title":16206,"module":16195,"summary":16207},"\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":16209,"title":16210,"module":16195,"summary":16211},"\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":16213,"title":16214,"module":16215,"summary":16216},"\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":16218,"title":16219,"module":16215,"summary":16220},"\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":16222,"title":16223,"module":16215,"summary":16224},"\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":16226,"title":16227,"module":16215,"summary":16228},"\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":16230,"title":16231,"module":16215,"summary":16232},"\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":16234,"title":16235,"module":16215,"summary":16236},"\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":16238,"title":16239,"module":6,"summary":6},"\u002Fcalculus","Calculus",{"path":16241,"title":16242,"module":15686,"summary":16243},"\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":16245,"title":16246,"module":15686,"summary":16247},"\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":16249,"title":16250,"module":16251,"summary":16252},"\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":16254,"title":16255,"module":16251,"summary":16256},"\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":16258,"title":16259,"module":16251,"summary":16260},"\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":16262,"title":16263,"module":16251,"summary":16264},"\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":16266,"title":16267,"module":16251,"summary":16268},"\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":16270,"title":16271,"module":16272,"summary":16273},"\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":16275,"title":16276,"module":16272,"summary":16277},"\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":16279,"title":16280,"module":16272,"summary":16281},"\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":16283,"title":16284,"module":16272,"summary":16285},"\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":16287,"title":16288,"module":16272,"summary":16289},"\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":16291,"title":16292,"module":16293,"summary":16294},"\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":16296,"title":16297,"module":16293,"summary":16298},"\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":16300,"title":16301,"module":16293,"summary":16302},"\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":16304,"title":16305,"module":16293,"summary":16306},"\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":16308,"title":16309,"module":16293,"summary":16310},"\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":16312,"title":16313,"module":16314,"summary":16315},"\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":16317,"title":16318,"module":16314,"summary":16319},"\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":16321,"title":16322,"module":16314,"summary":16323},"\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":16325,"title":16326,"module":16327,"summary":16328},"\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":16330,"title":16331,"module":16327,"summary":16332},"\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":16334,"title":16335,"module":16327,"summary":16336},"\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":16338,"title":16339,"module":16327,"summary":16340},"\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":16342,"title":16343,"module":16327,"summary":16344},"\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":16346,"title":16347,"module":16327,"summary":16348},"\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":16350,"title":16351,"module":16352,"summary":16353},"\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":16355,"title":16356,"module":16352,"summary":16357},"\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":16359,"title":16360,"module":16352,"summary":16361},"\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":16363,"title":16364,"module":16352,"summary":16365},"\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":16367,"title":16368,"module":16352,"summary":16369},"\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":16371,"title":16372,"module":16352,"summary":16373},"\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":16375,"title":16376,"module":16352,"summary":16377},"\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":16379,"title":16380,"module":16381,"summary":16382},"\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":16384,"title":16385,"module":16381,"summary":16386},"\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":16388,"title":16389,"module":16381,"summary":16390},"\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":16392,"title":16393,"module":16381,"summary":16394},"\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":16396,"title":16397,"module":16381,"summary":16398},"\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":16400,"title":16401,"module":16381,"summary":16402},"\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":16404,"title":16405,"module":16381,"summary":16406},"\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":16408,"title":16409,"module":16381,"summary":16410},"\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":16412,"title":16413,"module":16381,"summary":16414},"\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":16416,"title":16417,"module":16381,"summary":16418},"\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":16420,"title":16421,"module":16381,"summary":16422},"\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":16424,"title":16425,"module":16381,"summary":16426},"\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":16428,"title":16429,"module":16430,"summary":16431},"\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":16433,"title":16434,"module":16430,"summary":16435},"\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":16437,"title":16438,"module":16430,"summary":16439},"\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":16441,"title":16442,"module":16430,"summary":16443},"\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":16445,"title":16446,"module":16430,"summary":16447},"\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":16449,"title":16450,"module":16430,"summary":16451},"\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":16453,"title":16454,"module":6,"summary":6},"\u002Fmechanics","Mechanics & Dynamics",{"path":16456,"title":16457,"module":16458,"summary":16459},"\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":16461,"title":16462,"module":16458,"summary":16463},"\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":16465,"title":16466,"module":16458,"summary":16467},"\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":16469,"title":16470,"module":16458,"summary":16471},"\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":16473,"title":16474,"module":16458,"summary":16475},"\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":16477,"title":16478,"module":16479,"summary":16480},"\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":16482,"title":16483,"module":16479,"summary":16484},"\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":16486,"title":16487,"module":16488,"summary":16489},"\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":16491,"title":16492,"module":16488,"summary":16493},"\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":16495,"title":16496,"module":16488,"summary":16497},"\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":16499,"title":16500,"module":16488,"summary":16501},"\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":16503,"title":16504,"module":16488,"summary":16505},"\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":16507,"title":16508,"module":16509,"summary":16510},"\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":16512,"title":16513,"module":16509,"summary":16514},"\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":16516,"title":16517,"module":16509,"summary":16518},"\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":16520,"title":16521,"module":16509,"summary":16522},"\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":16524,"title":16525,"module":16526,"summary":16527},"\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":16529,"title":16530,"module":16526,"summary":16531},"\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":16533,"title":16534,"module":16526,"summary":16535},"\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":16537,"title":16538,"module":16539,"summary":16540},"\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":16542,"title":16543,"module":16539,"summary":16544},"\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":16546,"title":16547,"module":16539,"summary":16548},"\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":16550,"title":16551,"module":16539,"summary":16552},"\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":16554,"title":16555,"module":16539,"summary":16556},"\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":16558,"title":16559,"module":16560,"summary":16561},"\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":16563,"title":16564,"module":16560,"summary":16565},"\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":16567,"title":16568,"module":16560,"summary":16569},"\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":16571,"title":16572,"module":16560,"summary":16573},"\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":16575,"title":16576,"module":16560,"summary":16577},"\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":16579,"title":16580,"module":16560,"summary":16581},"\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":16583,"title":16584,"module":16585,"summary":16586},"\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":16588,"title":16589,"module":16585,"summary":16590},"\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":16592,"title":16593,"module":16585,"summary":16594},"\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":16596,"title":16597,"module":16585,"summary":16598},"\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":16600,"title":16601,"module":16585,"summary":16602},"\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":16604,"title":16605,"module":16585,"summary":16606},"\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":16608,"title":16609,"module":16585,"summary":16610},"\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":16612,"title":16613,"module":16585,"summary":16614},"\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":16616,"title":16617,"module":16618,"summary":16619},"\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":16621,"title":16622,"module":16618,"summary":16623},"\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":16625,"title":16626,"module":16618,"summary":16627},"\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":16629,"title":16630,"module":16618,"summary":16631},"\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":16633,"title":16634,"module":16618,"summary":16635},"\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":16637,"title":16638,"module":16639,"summary":16640},"\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":16642,"title":16643,"module":16639,"summary":16644},"\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":16646,"title":16647,"module":16639,"summary":16648},"\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":16650,"title":16651,"module":16639,"summary":16652},"\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":16654,"title":16655,"module":16639,"summary":16656},"\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":16658,"title":16659,"module":16660,"summary":16661},"\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":16663,"title":16664,"module":16660,"summary":16665},"\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":16667,"title":16668,"module":16660,"summary":16669},"\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":16671,"title":16672,"module":6,"summary":6},"\u002Felectricity-and-magnetism","Electricity & Magnetism",{"path":16674,"title":16675,"module":16676,"summary":16677},"\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":16679,"title":16680,"module":16676,"summary":16681},"\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":16683,"title":16684,"module":16676,"summary":16685},"\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":16687,"title":16688,"module":16676,"summary":16689},"\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":16691,"title":16692,"module":16676,"summary":16693},"\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":16695,"title":16696,"module":16697,"summary":16698},"\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":16700,"title":16701,"module":16697,"summary":16702},"\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":16704,"title":16705,"module":16697,"summary":16706},"\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":16708,"title":16709,"module":16697,"summary":16710},"\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":16712,"title":16713,"module":16697,"summary":16714},"\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":16716,"title":16717,"module":16718,"summary":16719},"\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":16721,"title":16722,"module":16718,"summary":16723},"\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":16725,"title":16726,"module":16718,"summary":16727},"\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":16729,"title":16730,"module":16731,"summary":16732},"\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":16734,"title":16735,"module":16731,"summary":16736},"\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":16738,"title":16739,"module":16731,"summary":16740},"\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":16742,"title":16743,"module":16731,"summary":16744},"\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":16746,"title":16747,"module":16731,"summary":16748},"\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":16750,"title":16751,"module":16731,"summary":16752},"\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":16754,"title":16755,"module":16731,"summary":16756},"\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":16758,"title":16759,"module":16760,"summary":16761},"\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":16763,"title":16764,"module":16760,"summary":16765},"\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":16767,"title":16768,"module":16760,"summary":16769},"\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":16771,"title":16772,"module":16760,"summary":16773},"\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":16775,"title":16776,"module":16760,"summary":16777},"\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":16779,"title":16780,"module":16760,"summary":16781},"\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":16783,"title":16784,"module":16760,"summary":16785},"\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":16787,"title":16788,"module":16789,"summary":16790},"\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":16792,"title":16793,"module":16789,"summary":16794},"\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":16796,"title":16797,"module":16789,"summary":16798},"\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":16800,"title":16801,"module":16789,"summary":16802},"\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":16804,"title":16805,"module":16789,"summary":16806},"\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":16808,"title":16809,"module":16789,"summary":16810},"\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":16812,"title":16813,"module":16814,"summary":16815},"\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":16817,"title":16818,"module":16814,"summary":16819},"\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":16821,"title":16822,"module":16814,"summary":16823},"\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":16825,"title":16826,"module":16814,"summary":16827},"\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":16829,"title":16830,"module":16814,"summary":16831},"\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":16833,"title":16834,"module":16835,"summary":16836},"\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":16838,"title":16839,"module":16835,"summary":16840},"\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":16842,"title":16843,"module":16835,"summary":16844},"\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":16846,"title":16847,"module":16835,"summary":16848},"\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":16850,"title":16851,"module":16835,"summary":16852},"\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":16854,"title":16855,"module":16835,"summary":16856},"\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":16858,"title":16859,"module":16860,"summary":16861},"\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":16863,"title":16864,"module":16860,"summary":16865},"\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":16867,"title":16868,"module":16860,"summary":16869},"\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":16871,"title":16872,"module":16860,"summary":16873},"\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":16875,"title":16876,"module":16860,"summary":16877},"\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":16879,"title":16880,"module":6,"summary":6},"\u002Flinear-algebra","Linear Algebra",{"path":16882,"title":16883,"module":6,"summary":6},"\u002Ftheory-of-computation","Theory of Computation",{"path":16885,"title":16886,"module":15686,"summary":16887},"\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":16889,"title":16890,"module":15686,"summary":16891},"\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":16893,"title":16894,"module":15686,"summary":16895},"\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":16897,"title":16898,"module":15686,"summary":16899},"\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":16901,"title":16902,"module":15686,"summary":16903},"\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":16905,"title":16906,"module":16907,"summary":16908},"\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":16910,"title":16911,"module":16907,"summary":16912},"\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":16914,"title":16915,"module":16907,"summary":16916},"\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":16918,"title":16919,"module":16907,"summary":16920},"\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":16922,"title":16923,"module":16907,"summary":16924},"\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":16926,"title":16927,"module":16907,"summary":16928},"\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":16930,"title":16931,"module":16907,"summary":16932},"\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":16934,"title":16935,"module":16936,"summary":16937},"\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":16939,"title":16940,"module":16936,"summary":16941},"\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":16943,"title":16944,"module":16936,"summary":16945},"\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":16947,"title":16948,"module":16936,"summary":16949},"\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":16951,"title":16952,"module":16936,"summary":16953},"\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":16955,"title":16956,"module":16957,"summary":16958},"\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":16960,"title":16961,"module":16957,"summary":16962},"\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":16964,"title":16965,"module":16957,"summary":16966},"\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":16968,"title":16969,"module":16957,"summary":16970},"\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":16972,"title":16973,"module":16957,"summary":16974},"\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":16976,"title":16977,"module":16978,"summary":16979},"\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":16981,"title":16982,"module":16978,"summary":16983},"\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":16985,"title":16986,"module":16978,"summary":16987},"\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":16989,"title":16990,"module":16978,"summary":16991},"\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":16993,"title":16994,"module":16978,"summary":16995},"\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":16997,"title":16998,"module":16999,"summary":17000},"\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":17002,"title":17003,"module":16999,"summary":17004},"\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":17006,"title":17007,"module":16999,"summary":17008},"\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":17010,"title":17011,"module":16999,"summary":17012},"\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":17014,"title":17015,"module":16999,"summary":17016},"\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":17018,"title":17019,"module":17020,"summary":17021},"\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":17023,"title":17024,"module":17020,"summary":17025},"\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":17027,"title":17028,"module":17020,"summary":17029},"\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":17031,"title":17032,"module":17020,"summary":17033},"\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":17035,"title":17036,"module":17020,"summary":17037},"\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":17039,"title":17040,"module":17041,"summary":17042},"\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":17044,"title":17045,"module":17041,"summary":17046},"\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":17048,"title":17049,"module":17041,"summary":17050},"\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":17052,"title":17053,"module":17054,"summary":17055},"\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":17057,"title":17058,"module":17054,"summary":17059},"\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":17061,"title":17062,"module":17063,"summary":17064},"\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":17066,"title":17067,"module":17063,"summary":17068},"\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":17070,"title":17071,"module":17063,"summary":17072},"\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":17074,"title":17075,"module":17063,"summary":17076},"\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":17078,"title":17079,"module":17063,"summary":17080},"\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":17082,"title":17083,"module":17084,"summary":17085},"\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":17087,"title":17088,"module":17084,"summary":17089},"\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":17091,"title":17092,"module":6,"summary":6},"\u002Fcomputer-architecture","Computer Architecture",{"path":17094,"title":17095,"module":15686,"summary":17096},"\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":17098,"title":17099,"module":15686,"summary":17100},"\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":17102,"title":17103,"module":17104,"summary":17105},"\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":17107,"title":17108,"module":17104,"summary":17109},"\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":17111,"title":17112,"module":17104,"summary":17113},"\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":17115,"title":17116,"module":17104,"summary":17117},"\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":17119,"title":17120,"module":17104,"summary":17121},"\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":17123,"title":17124,"module":17104,"summary":17125},"\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":17127,"title":17128,"module":17129,"summary":17130},"\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":17132,"title":17133,"module":17129,"summary":17134},"\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":17136,"title":17137,"module":17129,"summary":17138},"\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":17140,"title":17141,"module":17129,"summary":17142},"\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":17144,"title":17145,"module":17129,"summary":17146},"\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":17148,"title":17149,"module":17129,"summary":17150},"\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":17152,"title":17153,"module":17154,"summary":17155},"\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":17157,"title":17158,"module":17154,"summary":17159},"\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":17161,"title":17162,"module":17154,"summary":17163},"\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":17165,"title":17166,"module":17167,"summary":17168},"\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":17170,"title":17171,"module":17167,"summary":17172},"\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":17174,"title":17175,"module":17176,"summary":17177},"\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":17179,"title":17180,"module":17176,"summary":17181},"\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":17183,"title":17184,"module":17176,"summary":17185},"\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":17187,"title":17188,"module":17189,"summary":17190},"\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":17192,"title":17193,"module":17189,"summary":17194},"\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":17196,"title":17197,"module":17198,"summary":17199},"\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":17201,"title":17202,"module":17198,"summary":17203},"\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":17205,"title":17206,"module":17198,"summary":17207},"\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":17209,"title":17210,"module":17211,"summary":17212},"\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":17214,"title":17215,"module":17211,"summary":17216},"\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":17218,"title":17219,"module":17211,"summary":17220},"\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":17222,"title":17223,"module":17224,"summary":17225},"\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":17227,"title":17228,"module":17224,"summary":17229},"\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":17231,"title":17232,"module":6,"summary":6},"\u002Fdifferential-equations","Differential Equations",{"path":17234,"title":17235,"module":17236,"summary":17237},"\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":17239,"title":17240,"module":17236,"summary":17241},"\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":17243,"title":17244,"module":17236,"summary":17245},"\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":17247,"title":17248,"module":17236,"summary":17249},"\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":17251,"title":17252,"module":17236,"summary":17253},"\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":17255,"title":17256,"module":17257,"summary":17258},"\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":17260,"title":17261,"module":17257,"summary":17262},"\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":17264,"title":17265,"module":17257,"summary":17266},"\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":17268,"title":17269,"module":17257,"summary":17270},"\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":17272,"title":17273,"module":17274,"summary":17275},"\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":17277,"title":17278,"module":17274,"summary":17279},"\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":17281,"title":17282,"module":17274,"summary":17283},"\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":17285,"title":17286,"module":17274,"summary":17287},"\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":17289,"title":17290,"module":17291,"summary":17292},"\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":17294,"title":17295,"module":17291,"summary":17296},"\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":17298,"title":17299,"module":17291,"summary":17300},"\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":17302,"title":17303,"module":17291,"summary":17304},"\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":17306,"title":17307,"module":17308,"summary":17309},"\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":17311,"title":17312,"module":17308,"summary":17313},"\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":17315,"title":17316,"module":17308,"summary":17317},"\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":17319,"title":17320,"module":17308,"summary":17321},"\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":17323,"title":17324,"module":17308,"summary":17325},"\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":17327,"title":17328,"module":17308,"summary":17329},"\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":17331,"title":17332,"module":17333,"summary":17334},"\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":17336,"title":17337,"module":17333,"summary":17338},"\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":17340,"title":17341,"module":17333,"summary":17342},"\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":17344,"title":17345,"module":17346,"summary":17347},"\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":17349,"title":17350,"module":17346,"summary":17351},"\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":17353,"title":17354,"module":17346,"summary":17355},"\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":17357,"title":17358,"module":17346,"summary":17359},"\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":17361,"title":17362,"module":17363,"summary":17364},"\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":17366,"title":17367,"module":17363,"summary":17368},"\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":17370,"title":17371,"module":17363,"summary":17372},"\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":17374,"title":17375,"module":17376,"summary":17377},"\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":17379,"title":17380,"module":17376,"summary":17381},"\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":17383,"title":17384,"module":17376,"summary":17385},"\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":17387,"title":17388,"module":17389,"summary":17390},"\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":17392,"title":17393,"module":17389,"summary":17394},"\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":17396,"title":17397,"module":6,"summary":6},"\u002Frelativity","Relativity",{"path":17399,"title":17400,"module":6,"summary":6},"\u002Fphysical-computing","Physical Computing",{"path":17402,"title":17403,"module":17404,"summary":17405},"\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":17407,"title":17408,"module":17404,"summary":17409},"\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":17411,"title":17412,"module":17404,"summary":17413},"\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":17415,"title":17416,"module":17404,"summary":17417},"\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":17419,"title":17420,"module":17421,"summary":17422},"\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":17424,"title":17425,"module":17421,"summary":17426},"\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":17428,"title":17429,"module":17421,"summary":17430},"\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":17432,"title":17433,"module":17434,"summary":17435},"\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":17437,"title":17438,"module":17434,"summary":17439},"\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":17441,"title":17442,"module":17434,"summary":17443},"\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":17445,"title":17446,"module":17434,"summary":17447},"\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":17449,"title":17450,"module":17434,"summary":17451},"\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":17453,"title":17454,"module":17434,"summary":17455},"\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":17457,"title":17458,"module":17459,"summary":17460},"\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":17462,"title":17463,"module":17459,"summary":17464},"\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":17466,"title":17467,"module":17459,"summary":17468},"\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":17470,"title":17471,"module":17459,"summary":17472},"\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":17474,"title":17475,"module":17459,"summary":17476},"\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":17478,"title":17479,"module":17459,"summary":17480},"\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":17482,"title":17483,"module":17484,"summary":17485},"\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":17487,"title":17488,"module":17484,"summary":17489},"\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":17491,"title":17492,"module":17484,"summary":17493},"\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":17495,"title":17496,"module":17484,"summary":17497},"\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":17499,"title":17500,"module":16339,"summary":17501},"\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":17503,"title":17504,"module":16339,"summary":17505},"\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":17507,"title":17508,"module":16339,"summary":17509},"\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":17511,"title":17512,"module":17513,"summary":17514},"\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":17516,"title":17517,"module":17513,"summary":17518},"\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":17520,"title":17521,"module":17513,"summary":17522},"\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":17524,"title":17525,"module":17526,"summary":17527},"\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":17529,"title":17530,"module":17526,"summary":17531},"\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":17533,"title":17534,"module":17526,"summary":17535},"\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":17537,"title":17538,"module":17539,"summary":17540},"\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":17542,"title":17543,"module":17539,"summary":17544},"\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":17546,"title":17547,"module":17548,"summary":17549},"\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":17551,"title":17552,"module":17548,"summary":17553},"\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":17555,"title":17556,"module":17548,"summary":17557},"\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":17559,"title":17560,"module":17548,"summary":17561},"\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":17563,"title":17564,"module":17548,"summary":17565},"\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":17567,"title":17568,"module":6,"summary":6},"\u002Fquantum-mechanics","Quantum Mechanics",{"path":17570,"title":17571,"module":17572,"summary":17573},"\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":17575,"title":17576,"module":17572,"summary":17577},"\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":17579,"title":17580,"module":17572,"summary":17581},"\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":17583,"title":17584,"module":17572,"summary":17585},"\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":17587,"title":17588,"module":17589,"summary":17590},"\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":17592,"title":17593,"module":17589,"summary":17594},"\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":17596,"title":17597,"module":17589,"summary":17598},"\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":17600,"title":17601,"module":17589,"summary":17602},"\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":17604,"title":17605,"module":17589,"summary":17606},"\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":17608,"title":17609,"module":17589,"summary":17610},"\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":17612,"title":17613,"module":17614,"summary":17615},"\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":17617,"title":17618,"module":17614,"summary":17619},"\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":17621,"title":17622,"module":17614,"summary":17623},"\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":17625,"title":17626,"module":17614,"summary":17627},"\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":17629,"title":17630,"module":17614,"summary":17631},"\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":17633,"title":17634,"module":16033,"summary":17635},"\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":17637,"title":17638,"module":16033,"summary":17639},"\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":17641,"title":17642,"module":16033,"summary":17643},"\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":17645,"title":17646,"module":16033,"summary":17647},"\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":17649,"title":17650,"module":16033,"summary":17651},"\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":17653,"title":17654,"module":16033,"summary":17655},"\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":17657,"title":17658,"module":17659,"summary":17660},"\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":17662,"title":17663,"module":17659,"summary":17664},"\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":17666,"title":17667,"module":17659,"summary":17668},"\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":17670,"title":17671,"module":17659,"summary":17672},"\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":17674,"title":17675,"module":17676,"summary":17677},"\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":17679,"title":17680,"module":17676,"summary":17681},"\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":17683,"title":17684,"module":17676,"summary":17685},"\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":17687,"title":16088,"module":17676,"summary":17688},"\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":17690,"title":17691,"module":17676,"summary":17692},"\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":17694,"title":17695,"module":17696,"summary":17697},"\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":17699,"title":17700,"module":17696,"summary":17701},"\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":17703,"title":17704,"module":17696,"summary":17705},"\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":17707,"title":17708,"module":17696,"summary":17709},"\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":17711,"title":17712,"module":17713,"summary":17714},"\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":17716,"title":17717,"module":17713,"summary":17718},"\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":17720,"title":17721,"module":17713,"summary":17722},"\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":17724,"title":17725,"module":17713,"summary":17726},"\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":17728,"title":17729,"module":17713,"summary":17730},"\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":17732,"title":17733,"module":6,"summary":6},"\u002Freal-analysis","Real Analysis",{"path":17735,"title":17736,"module":15686,"summary":17737},"\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":17739,"title":17740,"module":15686,"summary":17741},"\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":17743,"title":17744,"module":17745,"summary":17746},"\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":17748,"title":17749,"module":17745,"summary":17750},"\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":17752,"title":17753,"module":17745,"summary":17754},"\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":17756,"title":17757,"module":17745,"summary":17758},"\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":17760,"title":17761,"module":17762,"summary":17763},"\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":17765,"title":17766,"module":17762,"summary":17767},"\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":17769,"title":17770,"module":17762,"summary":17771},"\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":17773,"title":17774,"module":17762,"summary":17775},"\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":17777,"title":17778,"module":17762,"summary":17779},"\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":17781,"title":17782,"module":17762,"summary":17783},"\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":17785,"title":17786,"module":17787,"summary":17788},"\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":17790,"title":17791,"module":17787,"summary":17792},"\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":17794,"title":17795,"module":17787,"summary":17796},"\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":17798,"title":17799,"module":17787,"summary":17800},"\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":17802,"title":17803,"module":17804,"summary":17805},"\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":17807,"title":17808,"module":17804,"summary":17809},"\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":17811,"title":17812,"module":17804,"summary":17813},"\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":17815,"title":17816,"module":17804,"summary":17817},"\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":17819,"title":17820,"module":17821,"summary":17822},"\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":17824,"title":17825,"module":17821,"summary":17826},"\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":17828,"title":17829,"module":17821,"summary":17830},"\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":17832,"title":17833,"module":17834,"summary":17835},"\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":17837,"title":17838,"module":17834,"summary":17839},"\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":17841,"title":17842,"module":17834,"summary":17843},"\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":17845,"title":17846,"module":17834,"summary":17847},"\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":17849,"title":17850,"module":17851,"summary":17852},"\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":17854,"title":17855,"module":17851,"summary":17856},"\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":17858,"title":17859,"module":17851,"summary":17860},"\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":17862,"title":17863,"module":17851,"summary":17864},"\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":17866,"title":17867,"module":17868,"summary":17869},"\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":17871,"title":17872,"module":17868,"summary":17873},"\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":17875,"title":17876,"module":17868,"summary":17877},"\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":17879,"title":17880,"module":17881,"summary":17882},"\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":17884,"title":17885,"module":17881,"summary":17886},"\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":17888,"title":17889,"module":17881,"summary":17890},"\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":17892,"title":17893,"module":17881,"summary":17894},"\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":17896,"title":17897,"module":17898,"summary":17899},"\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":17901,"title":17902,"module":17898,"summary":17903},"\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":17905,"title":17906,"module":17898,"summary":17907},"\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":17909,"title":17910,"module":17898,"summary":17911},"\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":17913,"title":17914,"module":17898,"summary":17915},"\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":17917,"title":17918,"module":17919,"summary":17920},"\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":17922,"title":17923,"module":17919,"summary":17924},"\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":17926,"title":17927,"module":6,"summary":6},"\u002Fabstract-algebra","Abstract Algebra",{"path":17929,"title":17930,"module":17931,"summary":17932},"\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":17934,"title":17935,"module":17931,"summary":17936},"\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":17938,"title":17939,"module":17931,"summary":17940},"\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":17942,"title":17943,"module":17931,"summary":17944},"\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":17946,"title":17947,"module":17931,"summary":17948},"\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":17950,"title":17951,"module":17952,"summary":17953},"\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":17955,"title":17956,"module":17952,"summary":17957},"\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":17959,"title":17960,"module":17952,"summary":17961},"\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":17963,"title":17964,"module":17952,"summary":17965},"\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":17967,"title":17968,"module":17952,"summary":17969},"\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":17971,"title":17972,"module":17952,"summary":17973},"\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":17975,"title":17976,"module":17952,"summary":17977},"\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":17979,"title":17980,"module":17981,"summary":17982},"\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":17984,"title":17985,"module":17981,"summary":17986},"\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":17988,"title":17989,"module":17981,"summary":17990},"\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":17992,"title":17993,"module":17981,"summary":17994},"\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":17996,"title":17997,"module":17998,"summary":17999},"\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":18001,"title":18002,"module":17998,"summary":18003},"\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":18005,"title":18006,"module":17998,"summary":18007},"\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":18009,"title":18010,"module":18011,"summary":18012},"\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":18014,"title":18015,"module":18011,"summary":18016},"\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":18018,"title":18019,"module":18011,"summary":18020},"\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":18022,"title":18023,"module":18011,"summary":18024},"\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":18026,"title":18027,"module":18011,"summary":18028},"\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":18030,"title":18031,"module":18011,"summary":18032},"\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":18034,"title":18035,"module":18036,"summary":18037},"\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":18039,"title":18040,"module":18036,"summary":18041},"\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":18043,"title":18044,"module":18036,"summary":18045},"\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":18047,"title":18048,"module":18049,"summary":18050},"\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":18052,"title":18053,"module":18049,"summary":18054},"\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":18056,"title":18057,"module":18049,"summary":18058},"\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":18060,"title":18061,"module":18049,"summary":18062},"\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":18064,"title":18065,"module":18066,"summary":18067},"\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":18069,"title":18070,"module":18066,"summary":18071},"\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":18073,"title":18074,"module":18066,"summary":18075},"\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":18077,"title":18078,"module":18079,"summary":18080},"\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":18082,"title":18083,"module":18079,"summary":18084},"\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":18086,"title":18087,"module":18079,"summary":18088},"\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":18090,"title":18091,"module":18079,"summary":18092},"\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":18094,"title":18095,"module":6,"summary":6},"\u002Fatomic-physics","Atomic Physics",{"path":18097,"title":18098,"module":6,"summary":6},"\u002Fdatabases","Databases",{"path":18100,"title":18101,"module":15686,"summary":18102},"\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":18104,"title":18105,"module":15686,"summary":18106},"\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":18108,"title":18109,"module":15686,"summary":18110},"\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":18112,"title":18113,"module":15686,"summary":18114},"\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":18116,"title":18117,"module":15686,"summary":18118},"\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":18120,"title":18121,"module":15686,"summary":18122},"\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":18124,"title":18125,"module":18126,"summary":18127},"\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":18129,"title":18130,"module":18126,"summary":18131},"\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":18133,"title":18134,"module":18126,"summary":18135},"\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":18137,"title":18138,"module":18139,"summary":18140},"\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":18142,"title":18143,"module":18139,"summary":18144},"\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":18146,"title":18147,"module":18139,"summary":18148},"\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":18150,"title":18151,"module":18152,"summary":18153},"\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":18155,"title":18156,"module":18152,"summary":18157},"\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":18159,"title":18160,"module":18152,"summary":18161},"\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":18163,"title":18164,"module":18152,"summary":18165},"\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":18167,"title":18168,"module":18152,"summary":18169},"\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":18171,"title":18172,"module":18173,"summary":18174},"\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":18176,"title":18177,"module":18173,"summary":18178},"\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":18180,"title":18181,"module":18173,"summary":18182},"\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":18184,"title":18185,"module":18173,"summary":18186},"\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":18188,"title":18189,"module":18190,"summary":18191},"\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":18193,"title":18194,"module":18190,"summary":18195},"\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":18197,"title":18198,"module":18190,"summary":18199},"\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":18201,"title":18202,"module":18190,"summary":18203},"\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":18205,"title":18206,"module":18207,"summary":18208},"\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":18210,"title":18211,"module":18207,"summary":18212},"\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":18214,"title":18215,"module":18207,"summary":18216},"\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":18218,"title":18219,"module":18207,"summary":18220},"\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":18222,"title":18223,"module":18224,"summary":18225},"\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":18227,"title":18228,"module":18224,"summary":18229},"\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":18231,"title":18232,"module":18224,"summary":18233},"\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":18235,"title":18236,"module":6,"summary":6},"\u002Fcategory-theory","Category Theory",{"path":18238,"title":16880,"module":18239,"summary":18240},"\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":18242,"title":18243,"module":18239,"summary":18244},"\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":18246,"title":18247,"module":18239,"summary":18248},"\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":18250,"title":16239,"module":18239,"summary":18251},"\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":18253,"title":18254,"module":15686,"summary":18255},"\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":18257,"title":18258,"module":15686,"summary":18259},"\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":18261,"title":18262,"module":15686,"summary":18263},"\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":18265,"title":18266,"module":18267,"summary":18268},"\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":18270,"title":18271,"module":18267,"summary":18272},"\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":18274,"title":18275,"module":18267,"summary":18276},"\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":18278,"title":18279,"module":18267,"summary":18280},"\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":18282,"title":18283,"module":18267,"summary":18284},"\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":18286,"title":18287,"module":18288,"summary":18289},"\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":18291,"title":18292,"module":18288,"summary":18293},"\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":18295,"title":18296,"module":18288,"summary":18297},"\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":18299,"title":18300,"module":18288,"summary":18301},"\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":18303,"title":18304,"module":18288,"summary":18305},"\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":18307,"title":18308,"module":18309,"summary":18310},"\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":18312,"title":18313,"module":18309,"summary":18314},"\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":18316,"title":18317,"module":18309,"summary":18318},"\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":18320,"title":18321,"module":18309,"summary":18322},"\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":18324,"title":18325,"module":18326,"summary":18327},"\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":18329,"title":18330,"module":18326,"summary":18331},"\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":18333,"title":18334,"module":18326,"summary":18335},"\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":18337,"title":18338,"module":18326,"summary":18339},"\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":18341,"title":18342,"module":18326,"summary":18343},"\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":18345,"title":18346,"module":18326,"summary":18347},"\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":18349,"title":18350,"module":18326,"summary":18351},"\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":18353,"title":18354,"module":18326,"summary":18355},"\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":18357,"title":18358,"module":18326,"summary":18359},"\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":18361,"title":18362,"module":18363,"summary":18364},"\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":18366,"title":18367,"module":18363,"summary":18368},"\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":18370,"title":18371,"module":18363,"summary":18372},"\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":18374,"title":18375,"module":18363,"summary":18376},"\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":18378,"title":18379,"module":18363,"summary":18380},"\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":18382,"title":18383,"module":18384,"summary":18385},"\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":18387,"title":18388,"module":18384,"summary":18389},"\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":18391,"title":18392,"module":18384,"summary":18393},"\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":18395,"title":18396,"module":18384,"summary":18397},"\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":18399,"title":18400,"module":18384,"summary":18401},"\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":18403,"title":18404,"module":18384,"summary":18405},"\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":18407,"title":18408,"module":18384,"summary":18409},"\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":18411,"title":18412,"module":18413,"summary":18414},"\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":18416,"title":18417,"module":18413,"summary":18418},"\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":18420,"title":18421,"module":18413,"summary":18422},"\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":18424,"title":18425,"module":18426,"summary":18427},"\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":18429,"title":18430,"module":18426,"summary":18431},"\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":18433,"title":18434,"module":18426,"summary":18435},"\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":18437,"title":18438,"module":18426,"summary":18439},"\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":18441,"title":18442,"module":18426,"summary":18443},"\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":18445,"title":18446,"module":18426,"summary":18447},"\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":18449,"title":18450,"module":18426,"summary":18451},"\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":17,"title":18453,"module":18454,"summary":18455},"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":13823,"title":18457,"module":18454,"summary":18458},"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":18460,"title":18461,"module":18454,"summary":18462},"\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":18464,"title":18465,"module":18454,"summary":18466},"\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":18468,"title":18469,"module":18454,"summary":18470},"\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":18472,"title":18473,"module":18454,"summary":18474},"\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":13883,"title":18476,"module":18454,"summary":18477},"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":18479,"title":18480,"module":18454,"summary":18481},"\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":18483,"title":18484,"module":18454,"summary":18485},"\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":18487,"title":18488,"module":18454,"summary":18489},"\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":18491,"title":18492,"module":18454,"summary":18493},"\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":27,"title":18495,"module":14751,"summary":18496},"Foundations of 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":18498,"title":18499,"module":14751,"summary":18500},"\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":18502,"title":18503,"module":14751,"summary":18504},"\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":22,"title":18506,"module":14751,"summary":18507},"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":14753,"title":5,"module":14751,"summary":14768},{"path":18510,"title":18511,"module":6,"summary":6},"\u002Fdeep-learning","Deep Learning",{"path":18513,"title":18514,"module":16430,"summary":18515},"\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":18517,"title":18518,"module":16430,"summary":18519},"\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":18521,"title":18522,"module":16430,"summary":18523},"\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":18525,"title":18526,"module":16430,"summary":18527},"\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":18529,"title":18530,"module":16430,"summary":18531},"\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":18533,"title":18534,"module":18535,"summary":18536},"\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":18538,"title":18539,"module":18535,"summary":18540},"\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":18542,"title":18543,"module":18535,"summary":18544},"\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":18546,"title":18547,"module":18535,"summary":18548},"\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":18550,"title":18551,"module":18552,"summary":18553},"\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":18555,"title":18556,"module":18552,"summary":18557},"\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":18559,"title":18560,"module":18552,"summary":18561},"\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":18563,"title":18564,"module":18552,"summary":18565},"\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":18567,"title":18568,"module":18569,"summary":18570},"\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":18572,"title":18573,"module":18569,"summary":18574},"\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":18576,"title":18577,"module":18569,"summary":18578},"\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":18580,"title":18581,"module":18569,"summary":18582},"\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":18584,"title":18585,"module":18569,"summary":18586},"\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":18588,"title":18589,"module":18590,"summary":18591},"\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":18593,"title":18594,"module":18590,"summary":18595},"\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":18597,"title":18598,"module":18590,"summary":18599},"\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":18601,"title":18602,"module":18603,"summary":18604},"\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":18606,"title":18607,"module":18603,"summary":18608},"\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":18610,"title":18611,"module":18603,"summary":18612},"\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":18614,"title":18615,"module":18616,"summary":18617},"\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":18619,"title":18620,"module":18616,"summary":18621},"\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":18623,"title":18624,"module":18616,"summary":18625},"\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":18627,"title":18628,"module":18616,"summary":18629},"\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":18631,"title":18632,"module":18633,"summary":18634},"\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":18636,"title":18637,"module":18633,"summary":18638},"\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":18640,"title":18641,"module":18633,"summary":18642},"\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":18644,"title":18645,"module":18633,"summary":18646},"\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":18648,"title":18649,"module":18633,"summary":18650},"\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":18652,"title":18653,"module":18633,"summary":18654},"\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":18656,"title":18657,"module":18658,"summary":18659},"\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":18661,"title":18662,"module":18658,"summary":18663},"\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":18665,"title":18666,"module":18658,"summary":18667},"\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":18669,"title":18670,"module":18658,"summary":18671},"\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":18673,"title":18674,"module":18675,"summary":18676},"\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":18678,"title":18679,"module":18675,"summary":18680},"\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":18682,"title":18683,"module":18675,"summary":18684},"\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":18686,"title":18687,"module":18688,"summary":18689},"\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":18691,"title":18692,"module":18688,"summary":18693},"\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":18695,"title":18696,"module":18688,"summary":18697},"\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":18699,"title":18700,"module":18688,"summary":18701},"\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":18703,"title":18704,"module":18688,"summary":18705},"\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":18707,"title":18708,"module":18709,"summary":18710},"\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":18712,"title":18713,"module":18709,"summary":18714},"\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":18716,"title":18717,"module":18709,"summary":18718},"\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":18720,"title":18721,"module":6,"summary":6},"\u002Fstatistical-mechanics","Statistical Mechanics",{"path":18723,"title":18724,"module":18725,"summary":18726},"\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":18728,"title":18729,"module":18725,"summary":18730},"\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":18732,"title":18733,"module":18725,"summary":18734},"\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":18736,"title":18737,"module":18725,"summary":18738},"\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":18740,"title":18741,"module":18742,"summary":18743},"\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":18745,"title":18746,"module":18742,"summary":18747},"\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":18749,"title":18750,"module":18742,"summary":18751},"\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":18753,"title":18754,"module":18742,"summary":18755},"\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":18757,"title":18758,"module":18759,"summary":18760},"\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":18762,"title":18763,"module":18759,"summary":18764},"\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":18766,"title":18767,"module":18759,"summary":18768},"\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":18770,"title":18771,"module":18759,"summary":18772},"\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":18774,"title":18775,"module":18776,"summary":18777},"\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":18779,"title":18780,"module":18776,"summary":18781},"\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":18783,"title":18784,"module":18776,"summary":18785},"\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":18787,"title":18788,"module":18776,"summary":18789},"\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":18791,"title":18792,"module":18793,"summary":18794},"\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":18796,"title":18797,"module":18793,"summary":18798},"\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":18800,"title":18801,"module":18793,"summary":18802},"\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":18804,"title":18805,"module":18793,"summary":18806},"\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":18808,"title":18809,"module":18810,"summary":18811},"\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":18813,"title":18814,"module":18810,"summary":18815},"\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":18817,"title":18818,"module":18810,"summary":18819},"\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":18821,"title":18822,"module":18810,"summary":18823},"\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":18825,"title":18826,"module":18827,"summary":18828},"\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":18830,"title":18831,"module":18827,"summary":18832},"\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":18834,"title":18835,"module":18827,"summary":18836},"\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":18838,"title":18839,"module":18827,"summary":18840},"\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":18842,"title":18843,"module":18827,"summary":18844},"\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":18846,"title":18847,"module":18848,"summary":18849},"\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":18851,"title":18852,"module":18848,"summary":18853},"\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":18855,"title":18856,"module":18857,"summary":18858},"\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":18860,"title":18861,"module":18857,"summary":18862},"\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":18864,"title":18865,"module":18857,"summary":18866},"\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":18868,"title":18869,"module":18857,"summary":18870},"\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":18872,"title":18873,"module":18874,"summary":18875},"\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":18877,"title":18878,"module":18874,"summary":18879},"\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":18881,"title":18882,"module":18874,"summary":18883},"\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":18885,"title":18886,"module":18874,"summary":18887},"\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":18889,"title":18890,"module":18874,"summary":18891},"\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":18893,"title":18894,"module":18895,"summary":18896},"\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":18898,"title":18899,"module":18895,"summary":18900},"\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":18902,"title":18903,"module":18895,"summary":18904},"\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":18906,"title":18907,"module":18895,"summary":18908},"\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":18910,"title":18911,"module":6,"summary":6},"\u002Fcondensed-matter","Condensed Matter Physics",{"path":18913,"title":18914,"module":15686,"summary":18915},"\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":18917,"title":18918,"module":18919,"summary":18920},"\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":18922,"title":18923,"module":18919,"summary":18924},"\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":18926,"title":18927,"module":18919,"summary":18928},"\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":18930,"title":18931,"module":18919,"summary":18932},"\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":18934,"title":18935,"module":18919,"summary":18936},"\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":18938,"title":18939,"module":18919,"summary":18940},"\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":18942,"title":18943,"module":18919,"summary":18944},"\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":18946,"title":18947,"module":18948,"summary":18949},"\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":18951,"title":18952,"module":18948,"summary":18953},"\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":18955,"title":18956,"module":18948,"summary":18957},"\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":18959,"title":18960,"module":18948,"summary":18961},"\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":18963,"title":18964,"module":18965,"summary":18966},"\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":18968,"title":18969,"module":18965,"summary":18970},"\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":18972,"title":18973,"module":18965,"summary":18974},"\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":18976,"title":18977,"module":18965,"summary":18978},"\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":18980,"title":18981,"module":18982,"summary":18983},"\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":18985,"title":18986,"module":18982,"summary":18987},"\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":18989,"title":18990,"module":18982,"summary":18991},"\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":18993,"title":18994,"module":18982,"summary":18995},"\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":18997,"title":18998,"module":18999,"summary":19000},"\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":19002,"title":19003,"module":18999,"summary":19004},"\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":19006,"title":19007,"module":18999,"summary":19008},"\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":19010,"title":19011,"module":18999,"summary":19012},"\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":19014,"title":19015,"module":19016,"summary":19017},"\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":19019,"title":19020,"module":19016,"summary":19021},"\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":19023,"title":19024,"module":19016,"summary":19025},"\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":19027,"title":19028,"module":19029,"summary":19030},"\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":19032,"title":19033,"module":19029,"summary":19034},"\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":19036,"title":19037,"module":19038,"summary":19039},"\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":19041,"title":19042,"module":19038,"summary":19043},"\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":19045,"title":19046,"module":19038,"summary":19047},"\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":19049,"title":19050,"module":6,"summary":6},"\u002Flogic","Logic",{"path":19052,"title":19053,"module":15686,"summary":19054},"\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":19056,"title":19057,"module":15686,"summary":19058},"\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":19060,"title":19061,"module":15686,"summary":19062},"\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":19064,"title":19065,"module":15686,"summary":19066},"\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":19068,"title":19069,"module":15686,"summary":19070},"\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":19072,"title":19073,"module":15686,"summary":19074},"\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":19076,"title":15905,"module":19077,"summary":19078},"\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":19080,"title":19081,"module":19077,"summary":19082},"\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":19084,"title":19085,"module":19077,"summary":19086},"\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":19088,"title":19089,"module":19077,"summary":19090},"\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":19092,"title":19093,"module":19077,"summary":19094},"\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":19096,"title":19097,"module":19077,"summary":19098},"\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":19100,"title":19101,"module":19077,"summary":19102},"\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":19104,"title":19105,"module":19077,"summary":19106},"\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":19108,"title":19109,"module":19077,"summary":19110},"\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":19112,"title":19113,"module":19077,"summary":19114},"\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":19116,"title":19117,"module":19077,"summary":19118},"\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":19120,"title":19121,"module":19077,"summary":19122},"\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":19124,"title":19125,"module":19126,"summary":19127},"\u002Freinforcement-learning\u002Fapproximation\u002Fon-policy-prediction","On-Policy Prediction with Approximation","Approximate Solution Methods","Every tabular method so far stored one number per state, which fails once the state space is large or continuous. We replace the table with a parameterized value function $\\hat v(s,\\mathbf{w})$, define the mean squared value error it should minimize under the on-policy distribution, and derive stochastic- and semi-gradient learning rules — the semi-gradient TD(0) update that bootstraps and so is not a true gradient. Linear methods make the analysis clean and give the TD fixed point; feature construction (polynomials, Fourier basis, coarse and tile coding, RBFs) supplies the vectors $\\mathbf{x}(s)$, and neural networks are the nonlinear bridge to deep RL.\n",{"path":19129,"title":19130,"module":19126,"summary":19131},"\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":19133,"title":19134,"module":19126,"summary":19135},"\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":19137,"title":19138,"module":19126,"summary":19139},"\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":19141,"title":19142,"module":19126,"summary":19143},"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-and-the-deadly-triad","Off-Policy Methods and the Deadly Triad","Off-policy learning with function approximation is where the convergence guarantees of reinforcement learning fail. We extend the tabular off-policy updates to semi-gradient form with per-step importance sampling, show Baird's counterexample driving the weights to infinity, and identify the cause: the deadly triad of function approximation, bootstrapping, and off-policy training — any two are safe, all three can diverge. The divergence is not caused by sampling noise: a fully synchronous dynamic-programming update blows up just the same, which is what makes the triad a structural hazard rather than a fluke.\n",{"path":19145,"title":19146,"module":19126,"summary":19147},"\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":19149,"title":19150,"module":19126,"summary":19151},"\u002Freinforcement-learning\u002Fapproximation\u002Feligibility-traces","Eligibility Traces","n-step methods unify TD and Monte Carlo by storing the last n feature vectors; eligibility traces do the same job with a single short-term memory vector. The λ-return averages every n-step return under a geometric weighting; the forward view looks ahead to that average, and the backward view produces nearly the same updates online through a decaying trace vector. We build the λ-return, TD(λ) with its trace, the two ways λ recovers TD(0) and Monte Carlo, a note on the exact equivalence of true online TD(λ), and Sarsa(λ) for control.\n",{"path":19153,"title":19154,"module":19126,"summary":19155},"\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":19157,"title":19158,"module":19126,"summary":19159},"\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":19161,"title":19162,"module":19126,"summary":19163},"\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":19165,"title":19166,"module":19126,"summary":19167},"\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":19169,"title":19170,"module":19126,"summary":19171},"\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":19173,"title":19174,"module":19126,"summary":19175},"\u002Freinforcement-learning\u002Fapproximation\u002Foff-policy-eligibility-traces","Off-Policy Eligibility Traces","Eligibility traces meet off-policy learning and function approximation — the corner where stability gets hard. We first let the bootstrapping and discounting parameters vary with state, so a single generalized return covers episodic and continuing tasks and folds termination into the discount. Then we fold the per-decision importance ratio into the trace with a control-variate correction, and build Watkins's Q(λ) and its importance-sampling-free successor Tree-Backup(λ) — all correct in expectation, but still semi-gradient, so the deadly triad and its fixes wait for the next lesson.\n",{"path":19177,"title":19178,"module":19126,"summary":19179},"\u002Freinforcement-learning\u002Fapproximation\u002Fstable-off-policy-traces","Stable Off-Policy Methods with Traces","Off-policy traces get the expected target right, but with $\\lambda \u003C 1$ they bootstrap, so off-policy plus bootstrapping plus function approximation is the deadly triad and the weights can diverge. This lesson carries the two one-step fixes to traces: GTD(λ) and GQ(λ) add a second weight vector and a gradient correction for true gradient descent on the projected Bellman error, while Emphatic TD(λ) reweights updates through a followon trace and interest to recover the on-policy stability. It closes with the implementation reality that traces are cheap because they are sparse, and with Retrace and V-trace — the clipped-ratio descendants that make off-policy traces work at deep-RL scale.\n",{"path":19181,"title":18503,"module":19182,"summary":19183},"\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":19185,"title":19186,"module":19182,"summary":19187},"\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":19189,"title":19190,"module":19182,"summary":19191},"\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":19193,"title":19194,"module":19182,"summary":19195},"\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":19197,"title":19198,"module":19182,"summary":19199},"\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":19201,"title":19202,"module":19182,"summary":19203},"\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":19205,"title":19206,"module":19182,"summary":19207},"\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":19209,"title":19210,"module":19182,"summary":19211},"\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":19213,"title":19214,"module":19215,"summary":19216},"\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":19218,"title":19219,"module":19215,"summary":19220},"\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":19222,"title":19223,"module":19215,"summary":19224},"\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":19226,"title":19227,"module":19215,"summary":19228},"\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":19230,"title":19231,"module":19215,"summary":19232},"\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":19234,"title":19235,"module":19215,"summary":19236},"\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":19238,"title":19239,"module":19215,"summary":19240},"\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":19242,"title":19243,"module":19215,"summary":19244},"\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":19246,"title":19247,"module":19215,"summary":19248},"\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":19250,"title":19251,"module":19215,"summary":19252},"\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":19254,"title":19255,"module":19215,"summary":19256},"\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":19258,"title":19259,"module":19215,"summary":19260},"\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":19262,"title":19263,"module":19215,"summary":19264},"\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":19266,"title":19267,"module":19215,"summary":19268},"\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":19270,"title":19271,"module":19215,"summary":19272},"\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":19274,"title":19275,"module":19215,"summary":19276},"\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":19278,"title":19279,"module":19215,"summary":19280},"\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":19282,"title":19283,"module":19215,"summary":19284},"\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":19286,"title":19287,"module":19215,"summary":19288},"\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":19290,"title":19291,"module":19215,"summary":19292},"\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":19294,"title":19295,"module":19215,"summary":19296},"\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":19298,"title":19299,"module":19215,"summary":19300},"\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":19302,"title":19303,"module":19304,"summary":19305},"\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":19307,"title":19308,"module":19304,"summary":19309},"\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":19311,"title":19312,"module":19304,"summary":19313},"\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":19315,"title":19316,"module":19304,"summary":19317},"\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":19319,"title":19320,"module":19304,"summary":19321},"\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":19323,"title":19324,"module":19304,"summary":19325},"\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":19327,"title":19328,"module":19304,"summary":19329},"\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":19331,"title":19332,"module":19304,"summary":19333},"\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":19335,"title":14751,"module":6,"summary":6},"\u002Freinforcement-learning",{"path":19337,"title":19338,"module":15686,"summary":19339},"\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":19341,"title":19342,"module":15686,"summary":19343},"\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":19345,"title":19346,"module":15686,"summary":19347},"\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":19349,"title":19350,"module":15686,"summary":19351},"\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":19353,"title":19354,"module":19355,"summary":19356},"\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":19358,"title":19359,"module":19355,"summary":19360},"\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":19362,"title":19363,"module":19355,"summary":19364},"\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":19366,"title":19367,"module":19355,"summary":19368},"\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":19370,"title":19371,"module":19355,"summary":19372},"\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":19374,"title":19375,"module":19355,"summary":19376},"\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":19378,"title":19379,"module":19355,"summary":19380},"\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":19382,"title":19383,"module":19355,"summary":19384},"\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":19386,"title":19387,"module":19355,"summary":19388},"\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":19390,"title":19391,"module":19355,"summary":19392},"\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":19394,"title":19395,"module":19355,"summary":19396},"\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":19398,"title":19399,"module":19355,"summary":19400},"\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":19402,"title":19403,"module":19404,"summary":19405},"\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":19407,"title":19408,"module":19404,"summary":19409},"\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":19411,"title":19412,"module":19404,"summary":19413},"\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":19415,"title":19416,"module":19404,"summary":19417},"\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":19419,"title":19420,"module":19404,"summary":19421},"\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":19423,"title":19424,"module":19404,"summary":19425},"\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":19427,"title":19428,"module":19404,"summary":19429},"\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":19431,"title":19432,"module":19404,"summary":19433},"\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":19435,"title":19436,"module":19404,"summary":19437},"\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":19439,"title":19440,"module":19404,"summary":19441},"\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":19443,"title":19444,"module":19404,"summary":19445},"\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":19447,"title":19448,"module":19404,"summary":19449},"\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":19451,"title":19452,"module":19453,"summary":19454},"\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":19456,"title":19457,"module":19453,"summary":19458},"\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":19460,"title":19461,"module":19453,"summary":19462},"\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":19464,"title":19465,"module":19453,"summary":19466},"\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":19468,"title":19469,"module":19453,"summary":19470},"\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":19472,"title":19473,"module":19453,"summary":19474},"\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":19476,"title":19477,"module":19453,"summary":19478},"\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":19480,"title":19069,"module":19453,"summary":19481},"\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":19483,"title":19484,"module":19453,"summary":19485},"\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":19487,"title":19488,"module":19453,"summary":19489},"\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":19491,"title":19492,"module":19493,"summary":19494},"\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":19496,"title":19497,"module":19493,"summary":19498},"\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":19500,"title":19501,"module":19493,"summary":19502},"\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":19504,"title":19505,"module":19493,"summary":19506},"\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":19508,"title":14751,"module":19493,"summary":19509},"\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":19511,"title":19512,"module":19493,"summary":19513},"\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":19515,"title":19516,"module":19493,"summary":19517},"\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":19519,"title":19520,"module":19493,"summary":19521},"\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":19523,"title":19524,"module":19525,"summary":19526},"\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":19528,"title":19529,"module":19525,"summary":19530},"\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":19532,"title":19533,"module":19525,"summary":19534},"\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":19536,"title":19537,"module":19525,"summary":19538},"\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":19540,"title":19541,"module":19525,"summary":19542},"\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":19544,"title":19545,"module":19525,"summary":19546},"\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":19548,"title":19549,"module":19525,"summary":19550},"\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":19552,"title":19553,"module":19525,"summary":19554},"\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":19556,"title":19557,"module":6,"summary":6},"\u002Fartificial-intelligence","Artificial Intelligence",{"path":19559,"title":19560,"module":19561,"summary":19562},"\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":19564,"title":19565,"module":19561,"summary":19566},"\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":19568,"title":19569,"module":19561,"summary":19570},"\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":19572,"title":19573,"module":19561,"summary":19574},"\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":19576,"title":19577,"module":19561,"summary":19578},"\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":19580,"title":19581,"module":19582,"summary":19583},"\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":19585,"title":19586,"module":19582,"summary":19587},"\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":19589,"title":19590,"module":19582,"summary":19591},"\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":19593,"title":19594,"module":19582,"summary":19595},"\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":19597,"title":19598,"module":19599,"summary":19600},"\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":19602,"title":19603,"module":19599,"summary":19604},"\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":19606,"title":19607,"module":19599,"summary":19608},"\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":19610,"title":19611,"module":19599,"summary":19612},"\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":19614,"title":19615,"module":19616,"summary":19617},"\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":19619,"title":19620,"module":19616,"summary":19621},"\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":19623,"title":19624,"module":19625,"summary":19626},"\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":19628,"title":19629,"module":19625,"summary":19630},"\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":19632,"title":19633,"module":19634,"summary":19635},"\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":19637,"title":19638,"module":19634,"summary":19639},"\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":19641,"title":19642,"module":19634,"summary":19643},"\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":19645,"title":19646,"module":19634,"summary":19647},"\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":19649,"title":19650,"module":19651,"summary":19652},"\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":19654,"title":19655,"module":19651,"summary":19656},"\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":19658,"title":19659,"module":19651,"summary":19660},"\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":19662,"title":19663,"module":19664,"summary":19665},"\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":19667,"title":19668,"module":19664,"summary":19669},"\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":19671,"title":19672,"module":19664,"summary":19673},"\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":19675,"title":19676,"module":19677,"summary":19678},"\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":19680,"title":19681,"module":19677,"summary":19682},"\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":19684,"title":19685,"module":19686,"summary":19687},"\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":19689,"title":19690,"module":19686,"summary":19691},"\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":19693,"title":19694,"module":19686,"summary":19695},"\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":19697,"title":19698,"module":19699,"summary":19700},"\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":19702,"title":19703,"module":19699,"summary":19704},"\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":19706,"title":19707,"module":19699,"summary":19708},"\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":19710,"title":19711,"module":19699,"summary":19712},"\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":19714,"title":19715,"module":19699,"summary":19716},"\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":19718,"title":19719,"module":6,"summary":6},"\u002Fnuclear-physics","Nuclear Physics",{"path":19721,"title":19722,"module":15686,"summary":19723},"\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":19725,"title":19726,"module":15686,"summary":19727},"\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":19729,"title":19730,"module":15686,"summary":19731},"\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":19733,"title":19734,"module":15686,"summary":19735},"\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":19737,"title":19738,"module":15686,"summary":19739},"\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":19741,"title":19742,"module":19743,"summary":19744},"\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":19746,"title":19747,"module":19743,"summary":19748},"\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":19750,"title":19751,"module":19743,"summary":19752},"\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":19754,"title":19755,"module":19743,"summary":19756},"\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":19758,"title":19759,"module":19760,"summary":19761},"\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":19763,"title":19764,"module":19760,"summary":19765},"\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":19767,"title":19768,"module":19760,"summary":19769},"\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":19771,"title":19772,"module":16152,"summary":19773},"\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":19775,"title":19776,"module":16152,"summary":19777},"\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":19779,"title":19780,"module":16152,"summary":19781},"\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":19783,"title":19784,"module":16634,"summary":19785},"\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":19787,"title":18346,"module":16634,"summary":19788},"\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":19790,"title":18453,"module":16634,"summary":19791},"\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":19793,"title":19794,"module":16634,"summary":19795},"\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":19797,"title":19798,"module":16634,"summary":19799},"\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":19801,"title":19802,"module":16634,"summary":19803},"\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":19805,"title":19806,"module":19807,"summary":19808},"\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":19810,"title":19811,"module":19807,"summary":19812},"\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":19814,"title":19815,"module":19807,"summary":19816},"\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":19818,"title":19819,"module":19807,"summary":19820},"\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":19822,"title":19823,"module":19807,"summary":19824},"\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":19826,"title":19827,"module":19807,"summary":19828},"\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":19830,"title":19831,"module":19807,"summary":19832},"\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":19834,"title":19835,"module":19807,"summary":19836},"\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":19838,"title":19839,"module":19807,"summary":19840},"\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":19842,"title":19843,"module":19807,"summary":19844},"\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":19846,"title":19847,"module":19807,"summary":19848},"\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":19850,"title":19851,"module":19807,"summary":19852},"\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":19854,"title":19855,"module":19807,"summary":19856},"\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":19858,"title":19859,"module":19807,"summary":19860},"\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":19862,"title":19863,"module":19807,"summary":19864},"\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":19866,"title":19867,"module":19807,"summary":19868},"\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":19870,"title":19871,"module":19807,"summary":19872},"\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":19874,"title":19875,"module":19807,"summary":19876},"\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":19878,"title":19879,"module":19807,"summary":19880},"\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":19882,"title":19883,"module":19807,"summary":19884},"\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":19886,"title":19887,"module":18442,"summary":19888},"\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":19890,"title":19891,"module":18442,"summary":19892},"\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":19894,"title":19895,"module":18442,"summary":19896},"\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":19898,"title":19899,"module":18442,"summary":19900},"\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":19902,"title":19903,"module":18442,"summary":19904},"\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":19906,"title":19907,"module":18442,"summary":19908},"\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":19910,"title":19911,"module":18442,"summary":19912},"\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":19914,"title":19915,"module":18442,"summary":19916},"\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":19918,"title":19919,"module":19920,"summary":19921},"\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":19923,"title":19924,"module":19920,"summary":19925},"\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":19927,"title":19928,"module":19920,"summary":19929},"\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":19931,"title":19932,"module":19920,"summary":19933},"\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":19935,"title":19936,"module":6,"summary":6},"\u002Fnatural-language-processing","Natural Language Processing",{"path":19938,"title":19939,"module":15686,"summary":19940},"\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":19942,"title":19943,"module":15686,"summary":19944},"\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":19946,"title":19947,"module":15686,"summary":19948},"\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":19950,"title":19951,"module":19952,"summary":19953},"\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":19955,"title":19956,"module":19952,"summary":19957},"\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":19959,"title":19960,"module":19952,"summary":19961},"\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":19963,"title":19964,"module":19952,"summary":19965},"\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":19967,"title":19968,"module":19969,"summary":19970},"\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":19972,"title":19973,"module":19969,"summary":19974},"\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":19976,"title":19977,"module":19969,"summary":19978},"\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":19980,"title":19981,"module":19969,"summary":19982},"\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":19984,"title":19985,"module":19986,"summary":19987},"\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":19989,"title":19990,"module":19986,"summary":19991},"\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":19993,"title":19994,"module":19986,"summary":19995},"\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":19997,"title":19998,"module":19986,"summary":19999},"\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":20001,"title":20002,"module":20003,"summary":20004},"\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":20006,"title":20007,"module":20003,"summary":20008},"\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":20010,"title":20011,"module":20003,"summary":20012},"\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":20014,"title":20015,"module":20016,"summary":20017},"\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":20019,"title":20020,"module":20016,"summary":20021},"\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":20023,"title":20024,"module":20016,"summary":20025},"\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":20027,"title":20028,"module":20016,"summary":20029},"\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":20031,"title":20032,"module":20033,"summary":20034},"\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":20036,"title":20037,"module":20033,"summary":20038},"\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":20040,"title":20041,"module":20033,"summary":20042},"\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":20044,"title":20045,"module":20033,"summary":20046},"\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":20048,"title":20049,"module":20050,"summary":20051},"\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":20053,"title":20054,"module":20050,"summary":20055},"\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":20057,"title":20058,"module":20050,"summary":20059},"\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":20061,"title":20062,"module":20050,"summary":20063},"\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":20065,"title":20066,"module":20067,"summary":20068},"\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":20070,"title":20071,"module":20067,"summary":20072},"\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":20074,"title":20075,"module":20067,"summary":20076},"\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":20078,"title":20079,"module":20067,"summary":20080},"\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":20082,"title":20083,"module":20067,"summary":20084},"\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":20086,"title":20087,"module":20088,"summary":20089},"\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":20091,"title":20092,"module":20088,"summary":20093},"\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":20095,"title":20096,"module":20088,"summary":20097},"\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":20099,"title":20100,"module":20101,"summary":20102},"\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":20104,"title":20105,"module":20101,"summary":20106},"\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":20108,"title":20109,"module":20101,"summary":20110},"\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":20112,"title":20113,"module":20113,"summary":20114},"\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":20116,"title":20117,"module":20113,"summary":20118},"\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":20120,"title":20121,"module":20113,"summary":20122},"\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":20124,"title":20125,"module":20113,"summary":20126},"\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":20128,"title":20129,"module":20113,"summary":20130},"\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":20132,"title":20133,"module":20113,"summary":20134},"\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":20136,"title":20137,"module":6,"summary":6},"\u002Fparticle-physics","Particle Physics",{"path":20139,"title":20140,"module":20141,"summary":20142},"\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":20144,"title":20145,"module":20141,"summary":20146},"\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":20148,"title":20149,"module":20141,"summary":20150},"\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":20152,"title":20153,"module":20154,"summary":20155},"\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":20157,"title":20158,"module":20154,"summary":20159},"\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":20161,"title":20162,"module":20154,"summary":20163},"\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":20165,"title":20166,"module":20154,"summary":20167},"\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":20169,"title":20170,"module":20171,"summary":20172},"\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":20174,"title":20175,"module":20171,"summary":20176},"\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":20178,"title":20179,"module":20171,"summary":20180},"\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":20182,"title":20183,"module":20171,"summary":20184},"\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":20186,"title":20187,"module":20188,"summary":20189},"\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":20191,"title":20192,"module":20188,"summary":20193},"\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":20195,"title":20196,"module":20188,"summary":20197},"\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":20199,"title":20200,"module":20188,"summary":20201},"\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":20203,"title":20204,"module":20205,"summary":20206},"\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":20208,"title":20209,"module":20205,"summary":20210},"\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":20212,"title":20213,"module":20205,"summary":20214},"\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":20216,"title":20217,"module":20205,"summary":20218},"\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":20220,"title":20221,"module":20222,"summary":20223},"\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":20225,"title":20226,"module":20222,"summary":20227},"\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":20229,"title":20230,"module":20222,"summary":20231},"\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":20233,"title":20234,"module":20235,"summary":20236},"\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":20238,"title":20239,"module":20235,"summary":20240},"\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":20242,"title":20243,"module":20235,"summary":20244},"\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":20246,"title":20247,"module":20235,"summary":20248},"\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":20250,"title":18666,"module":20251,"summary":20252},"\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":20254,"title":20255,"module":20251,"summary":20256},"\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":20258,"title":20259,"module":20251,"summary":20260},"\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":20262,"title":20263,"module":20251,"summary":20264},"\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":20266,"title":20267,"module":20251,"summary":20268},"\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":20270,"title":20271,"module":20272,"summary":20273},"\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":20275,"title":20276,"module":20272,"summary":20277},"\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":20279,"title":20280,"module":20272,"summary":20281},"\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":20283,"title":20284,"module":20272,"summary":20285},"\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":20287,"title":20288,"module":20289,"summary":20290},"\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":20292,"title":20293,"module":20289,"summary":20294},"\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":20296,"title":20297,"module":20289,"summary":20298},"\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":20300,"title":20301,"module":20289,"summary":20302},"\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":20304,"title":20305,"module":20289,"summary":20306},"\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":20308,"title":20309,"module":20310,"summary":20311},"\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":20313,"title":20314,"module":20310,"summary":20315},"\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":20317,"title":17393,"module":20310,"summary":20318},"\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":20320,"title":20321,"module":20310,"summary":20322},"\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":20324,"title":20325,"module":20310,"summary":20326},"\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":20328,"title":20329,"module":20330,"summary":20331},"\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":20333,"title":20334,"module":20330,"summary":20335},"\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":20337,"title":20338,"module":20330,"summary":20339},"\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":20341,"title":20342,"module":20330,"summary":20343},"\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":20345,"title":20346,"module":20330,"summary":20347},"\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":20349,"title":20350,"module":20330,"summary":20351},"\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":20353,"title":20354,"module":20330,"summary":20355},"\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":20357,"title":20358,"module":6,"summary":6},"\u002Fastrophysics-cosmology","Astrophysics & Cosmology",{"path":20360,"title":20361,"module":6,"summary":6},"\u002Fcolophon","Colophon",{"path":10120,"title":20363,"module":6,"summary":6},"Study Notes",[20365,20376,20385,20398,20411,20422,20443,20456,20473,20482,20499,20524],{"module":18239,"moduleNumber":8271,"slug":20366,"lessons":20367},"mathematical-background",[20368,20370,20372,20374],{"title":16880,"path":18238,"lessonNumber":8271,"topics":20369,"summary":18240},[18239],{"title":18243,"path":18242,"lessonNumber":8277,"topics":20371,"summary":18244},[18239],{"title":18247,"path":18246,"lessonNumber":8283,"topics":20373,"summary":18248},[18239],{"title":16239,"path":18250,"lessonNumber":8289,"topics":20375,"summary":18251},[18239],{"module":15686,"moduleNumber":8277,"slug":20377,"lessons":20378},"foundations",[20379,20381,20383],{"title":18254,"path":18253,"lessonNumber":8271,"topics":20380,"summary":18255},[15686],{"title":18258,"path":18257,"lessonNumber":8277,"topics":20382,"summary":18259},[15686],{"title":18262,"path":18261,"lessonNumber":8283,"topics":20384,"summary":18263},[15686],{"module":18267,"moduleNumber":8283,"slug":20386,"lessons":20387},"neural-networks",[20388,20390,20392,20394,20396],{"title":18266,"path":18265,"lessonNumber":8271,"topics":20389,"summary":18268},[18267],{"title":18271,"path":18270,"lessonNumber":8277,"topics":20391,"summary":18272},[18267],{"title":18275,"path":18274,"lessonNumber":8283,"topics":20393,"summary":18276},[18267],{"title":18279,"path":18278,"lessonNumber":8289,"topics":20395,"summary":18280},[18267],{"title":18283,"path":18282,"lessonNumber":8295,"topics":20397,"summary":18284},[18267],{"module":18288,"moduleNumber":8289,"slug":20399,"lessons":20400},"optimization",[20401,20403,20405,20407,20409],{"title":18287,"path":18286,"lessonNumber":8271,"topics":20402,"summary":18289},[18288],{"title":18292,"path":18291,"lessonNumber":8277,"topics":20404,"summary":18293},[18288],{"title":18296,"path":18295,"lessonNumber":8283,"topics":20406,"summary":18297},[18288],{"title":18300,"path":18299,"lessonNumber":8289,"topics":20408,"summary":18301},[18288],{"title":18304,"path":18303,"lessonNumber":8295,"topics":20410,"summary":18305},[18288],{"module":18309,"moduleNumber":8295,"slug":20412,"lessons":20413},"regularization",[20414,20416,20418,20420],{"title":18308,"path":18307,"lessonNumber":8271,"topics":20415,"summary":18310},[18309],{"title":18313,"path":18312,"lessonNumber":8277,"topics":20417,"summary":18314},[18309],{"title":18317,"path":18316,"lessonNumber":8283,"topics":20419,"summary":18318},[18309],{"title":18321,"path":18320,"lessonNumber":8289,"topics":20421,"summary":18322},[18309],{"module":18326,"moduleNumber":8301,"slug":20423,"lessons":20424},"architectures",[20425,20427,20429,20431,20433,20435,20437,20439,20441],{"title":18325,"path":18324,"lessonNumber":8271,"topics":20426,"summary":18327},[18326],{"title":18330,"path":18329,"lessonNumber":8277,"topics":20428,"summary":18331},[18326],{"title":18334,"path":18333,"lessonNumber":8283,"topics":20430,"summary":18335},[18326],{"title":18338,"path":18337,"lessonNumber":8289,"topics":20432,"summary":18339},[18326],{"title":18342,"path":18341,"lessonNumber":8295,"topics":20434,"summary":18343},[18326],{"title":18346,"path":18345,"lessonNumber":8301,"topics":20436,"summary":18347},[18326],{"title":18350,"path":18349,"lessonNumber":8307,"topics":20438,"summary":18351},[18326],{"title":18354,"path":18353,"lessonNumber":8313,"topics":20440,"summary":18355},[18326],{"title":18358,"path":18357,"lessonNumber":8319,"topics":20442,"summary":18359},[18326],{"module":18363,"moduleNumber":8307,"slug":20444,"lessons":20445},"theory",[20446,20448,20450,20452,20454],{"title":18362,"path":18361,"lessonNumber":8271,"topics":20447,"summary":18364},[18363],{"title":18367,"path":18366,"lessonNumber":8277,"topics":20449,"summary":18368},[18363],{"title":18371,"path":18370,"lessonNumber":8283,"topics":20451,"summary":18372},[18363],{"title":18375,"path":18374,"lessonNumber":8289,"topics":20453,"summary":18376},[18363],{"title":18379,"path":18378,"lessonNumber":8295,"topics":20455,"summary":18380},[18363],{"module":18384,"moduleNumber":8313,"slug":20457,"lessons":20458},"generative-models",[20459,20461,20463,20465,20467,20469,20471],{"title":18383,"path":18382,"lessonNumber":8271,"topics":20460,"summary":18385},[18384],{"title":18388,"path":18387,"lessonNumber":8277,"topics":20462,"summary":18389},[18384],{"title":18392,"path":18391,"lessonNumber":8283,"topics":20464,"summary":18393},[18384],{"title":18396,"path":18395,"lessonNumber":8289,"topics":20466,"summary":18397},[18384],{"title":18400,"path":18399,"lessonNumber":8295,"topics":20468,"summary":18401},[18384],{"title":18404,"path":18403,"lessonNumber":8301,"topics":20470,"summary":18405},[18384],{"title":18408,"path":18407,"lessonNumber":8307,"topics":20472,"summary":18409},[18384],{"module":18413,"moduleNumber":8319,"slug":20474,"lessons":20475},"probabilistic-methods",[20476,20478,20480],{"title":18412,"path":18411,"lessonNumber":8271,"topics":20477,"summary":18414},[18413],{"title":18417,"path":18416,"lessonNumber":8277,"topics":20479,"summary":18418},[18413],{"title":18421,"path":18420,"lessonNumber":8283,"topics":20481,"summary":18422},[18413],{"module":18426,"moduleNumber":8325,"slug":20483,"lessons":20484},"practical",[20485,20487,20489,20491,20493,20495,20497],{"title":18425,"path":18424,"lessonNumber":8271,"topics":20486,"summary":18427},[18426],{"title":18430,"path":18429,"lessonNumber":8277,"topics":20488,"summary":18431},[18426],{"title":18434,"path":18433,"lessonNumber":8283,"topics":20490,"summary":18435},[18426],{"title":18438,"path":18437,"lessonNumber":8289,"topics":20492,"summary":18439},[18426],{"title":18442,"path":18441,"lessonNumber":8295,"topics":20494,"summary":18443},[18426],{"title":18446,"path":18445,"lessonNumber":8301,"topics":20496,"summary":18447},[18426],{"title":18450,"path":18449,"lessonNumber":8307,"topics":20498,"summary":18451},[18426],{"module":18454,"moduleNumber":8331,"slug":20500,"lessons":20501},"large-models-and-agents",[20502,20504,20506,20508,20510,20512,20514,20516,20518,20520,20522],{"title":18453,"path":17,"lessonNumber":8271,"topics":20503,"summary":18455},[18454],{"title":18457,"path":13823,"lessonNumber":8277,"topics":20505,"summary":18458},[18454],{"title":18461,"path":18460,"lessonNumber":8283,"topics":20507,"summary":18462},[18454],{"title":18465,"path":18464,"lessonNumber":8289,"topics":20509,"summary":18466},[18454],{"title":18469,"path":18468,"lessonNumber":8295,"topics":20511,"summary":18470},[18454],{"title":18473,"path":18472,"lessonNumber":8301,"topics":20513,"summary":18474},[18454],{"title":18476,"path":13883,"lessonNumber":8307,"topics":20515,"summary":18477},[18454],{"title":18480,"path":18479,"lessonNumber":8313,"topics":20517,"summary":18481},[18454],{"title":18484,"path":18483,"lessonNumber":8319,"topics":20519,"summary":18485},[18454],{"title":18488,"path":18487,"lessonNumber":8325,"topics":20521,"summary":18489},[18454],{"title":18492,"path":18491,"lessonNumber":8331,"topics":20523,"summary":18493},[18454],{"module":14751,"moduleNumber":8337,"slug":20525,"lessons":20526},"reinforcement-learning",[20527,20529,20531,20533,20535],{"title":18495,"path":27,"lessonNumber":8271,"topics":20528,"summary":18496},[14751],{"title":18499,"path":18498,"lessonNumber":8277,"topics":20530,"summary":18500},[14751],{"title":18503,"path":18502,"lessonNumber":8283,"topics":20532,"summary":18504},[14751],{"title":18506,"path":22,"lessonNumber":8289,"topics":20534,"summary":18507},[14751],{"title":5,"path":14753,"lessonNumber":8295,"topics":20536,"summary":14768},[14751],"\u003Csvg style=\"width:100%;max-width:624.947px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 468.711 176.240\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg fill=\"var(--tk-soft-neutral)\" style=\"stroke-width:.8\">\u003Cpath d=\"M-2.608 32.128H65.68V.83H-2.608Z\"\u002F>\u003Cg transform=\"translate(-26.654 -60.165)\">\u003Cpath d=\"M31.850 77.564Q31.850 77.226 31.991 76.935Q32.131 76.645 32.375 76.431Q32.619 76.218 32.924 76.103Q33.228 75.989 33.553 75.989Q33.823 75.989 34.086 76.088Q34.349 76.187 34.540 76.365L34.540 74.967Q34.540 74.697 34.433 74.635Q34.325 74.574 34.014 74.574L34.014 74.293L35.091 74.218L35.091 78.402Q35.091 78.590 35.145 78.673Q35.200 78.757 35.301 78.776Q35.402 78.795 35.617 78.795L35.617 79.075L34.510 79.143L34.510 78.726Q34.093 79.143 33.467 79.143Q33.036 79.143 32.664 78.931Q32.291 78.720 32.071 78.359Q31.850 77.998 31.850 77.564M33.525 78.921Q33.734 78.921 33.920 78.849Q34.106 78.778 34.260 78.641Q34.414 78.504 34.510 78.326L34.510 76.717Q34.424 76.570 34.279 76.450Q34.134 76.330 33.964 76.271Q33.795 76.211 33.614 76.211Q33.054 76.211 32.785 76.600Q32.517 76.990 32.517 77.571Q32.517 78.142 32.751 78.532Q32.985 78.921 33.525 78.921M36.225 77.540Q36.225 77.219 36.350 76.930Q36.475 76.641 36.701 76.418Q36.926 76.194 37.222 76.074Q37.517 75.954 37.835 75.954Q38.163 75.954 38.425 76.054Q38.686 76.153 38.862 76.335Q39.038 76.518 39.132 76.776Q39.226 77.034 39.226 77.366Q39.226 77.458 39.144 77.479L36.889 77.479L36.889 77.540Q36.889 78.128 37.172 78.511Q37.456 78.894 38.023 78.894Q38.345 78.894 38.613 78.701Q38.881 78.508 38.970 78.193Q38.977 78.152 39.052 78.138L39.144 78.138Q39.226 78.162 39.226 78.234Q39.226 78.241 39.220 78.268Q39.107 78.665 38.736 78.904Q38.365 79.143 37.941 79.143Q37.504 79.143 37.104 78.935Q36.704 78.726 36.465 78.359Q36.225 77.992 36.225 77.540M36.895 77.270L38.710 77.270Q38.710 76.993 38.613 76.741Q38.515 76.488 38.317 76.332Q38.119 76.177 37.835 76.177Q37.558 76.177 37.345 76.335Q37.131 76.494 37.013 76.749Q36.895 77.004 36.895 77.270M41.496 79.075L39.862 79.075L39.862 78.795Q40.091 78.795 40.240 78.761Q40.389 78.726 40.389 78.586L40.389 76.737Q40.389 76.467 40.281 76.406Q40.173 76.344 39.862 76.344L39.862 76.064L40.922 75.989L40.922 76.638Q41.093 76.330 41.397 76.159Q41.701 75.989 42.046 75.989Q42.446 75.989 42.723 76.129Q43 76.269 43.085 76.617Q43.253 76.324 43.552 76.156Q43.851 75.989 44.196 75.989Q44.702 75.989 44.986 76.212Q45.269 76.436 45.269 76.932L45.269 78.586Q45.269 78.723 45.418 78.759Q45.567 78.795 45.792 78.795L45.792 79.075L44.162 79.075L44.162 78.795Q44.388 78.795 44.538 78.759Q44.688 78.723 44.688 78.586L44.688 76.946Q44.688 76.611 44.569 76.411Q44.449 76.211 44.135 76.211Q43.865 76.211 43.630 76.347Q43.396 76.484 43.258 76.718Q43.119 76.952 43.119 77.226L43.119 78.586Q43.119 78.723 43.268 78.759Q43.417 78.795 43.642 78.795L43.642 79.075L42.012 79.075L42.012 78.795Q42.241 78.795 42.390 78.761Q42.538 78.726 42.538 78.586L42.538 76.946Q42.538 76.611 42.419 76.411Q42.299 76.211 41.985 76.211Q41.715 76.211 41.481 76.347Q41.246 76.484 41.108 76.718Q40.970 76.952 40.970 77.226L40.970 78.586Q40.970 78.723 41.120 78.759Q41.270 78.795 41.496 78.795L41.496 79.075M46.339 77.592Q46.339 77.250 46.474 76.951Q46.609 76.652 46.849 76.428Q47.088 76.204 47.406 76.079Q47.724 75.954 48.055 75.954Q48.499 75.954 48.899 76.170Q49.299 76.385 49.533 76.763Q49.767 77.140 49.767 77.592Q49.767 77.933 49.626 78.217Q49.484 78.501 49.239 78.708Q48.995 78.914 48.686 79.029Q48.376 79.143 48.055 79.143Q47.624 79.143 47.223 78.942Q46.821 78.740 46.580 78.388Q46.339 78.036 46.339 77.592M48.055 78.894Q48.657 78.894 48.880 78.516Q49.104 78.138 49.104 77.506Q49.104 76.894 48.870 76.535Q48.636 76.177 48.055 76.177Q47.002 76.177 47.002 77.506Q47.002 78.138 47.228 78.516Q47.453 78.894 48.055 78.894M52.044 79.075L50.410 79.075L50.410 78.795Q50.639 78.795 50.788 78.761Q50.936 78.726 50.936 78.586L50.936 76.737Q50.936 76.467 50.829 76.406Q50.721 76.344 50.410 76.344L50.410 76.064L51.470 75.989L51.470 76.638Q51.640 76.330 51.945 76.159Q52.249 75.989 52.594 75.989Q53.100 75.989 53.384 76.212Q53.667 76.436 53.667 76.932L53.667 78.586Q53.667 78.723 53.816 78.759Q53.965 78.795 54.190 78.795L54.190 79.075L52.560 79.075L52.560 78.795Q52.789 78.795 52.938 78.761Q53.086 78.726 53.086 78.586L53.086 76.946Q53.086 76.611 52.967 76.411Q52.847 76.211 52.533 76.211Q52.263 76.211 52.028 76.347Q51.794 76.484 51.656 76.718Q51.517 76.952 51.517 77.226L51.517 78.586Q51.517 78.723 51.668 78.759Q51.818 78.795 52.044 78.795L52.044 79.075M54.778 79.068L54.778 78.005Q54.778 77.981 54.806 77.954Q54.833 77.927 54.857 77.927L54.966 77.927Q55.031 77.927 55.045 77.985Q55.140 78.419 55.387 78.670Q55.633 78.921 56.046 78.921Q56.388 78.921 56.641 78.788Q56.894 78.655 56.894 78.347Q56.894 78.190 56.800 78.075Q56.706 77.961 56.567 77.892Q56.429 77.824 56.262 77.786L55.681 77.687Q55.325 77.619 55.052 77.398Q54.778 77.178 54.778 76.836Q54.778 76.587 54.889 76.412Q55 76.238 55.187 76.139Q55.373 76.040 55.588 75.997Q55.804 75.954 56.046 75.954Q56.460 75.954 56.740 76.136L56.955 75.961Q56.966 75.958 56.973 75.956Q56.979 75.954 56.990 75.954L57.041 75.954Q57.068 75.954 57.092 75.978Q57.116 76.002 57.116 76.030L57.116 76.877Q57.116 76.898 57.092 76.925Q57.068 76.952 57.041 76.952L56.928 76.952Q56.901 76.952 56.875 76.927Q56.849 76.901 56.849 76.877Q56.849 76.641 56.744 76.477Q56.638 76.313 56.455 76.231Q56.272 76.149 56.039 76.149Q55.711 76.149 55.455 76.252Q55.199 76.354 55.199 76.631Q55.199 76.826 55.381 76.935Q55.564 77.045 55.793 77.086L56.368 77.192Q56.614 77.240 56.827 77.368Q57.041 77.496 57.178 77.699Q57.314 77.903 57.314 78.152Q57.314 78.665 56.949 78.904Q56.583 79.143 56.046 79.143Q55.551 79.143 55.219 78.849L54.953 79.123Q54.932 79.143 54.905 79.143L54.857 79.143Q54.833 79.143 54.806 79.116Q54.778 79.089 54.778 79.068M58.470 78.234L58.470 76.337L57.830 76.337L57.830 76.115Q58.148 76.115 58.365 75.905Q58.582 75.695 58.683 75.385Q58.784 75.076 58.784 74.768L59.051 74.768L59.051 76.057L60.127 76.057L60.127 76.337L59.051 76.337L59.051 78.221Q59.051 78.497 59.155 78.696Q59.259 78.894 59.519 78.894Q59.676 78.894 59.782 78.790Q59.888 78.685 59.938 78.532Q59.987 78.378 59.987 78.221L59.987 77.807L60.254 77.807L60.254 78.234Q60.254 78.460 60.155 78.670Q60.056 78.880 59.871 79.012Q59.686 79.143 59.457 79.143Q59.020 79.143 58.745 78.906Q58.470 78.668 58.470 78.234M62.814 79.075L61.078 79.075L61.078 78.795Q61.307 78.795 61.455 78.761Q61.604 78.726 61.604 78.586L61.604 76.737Q61.604 76.467 61.496 76.406Q61.389 76.344 61.078 76.344L61.078 76.064L62.106 75.989L62.106 76.696Q62.236 76.388 62.479 76.189Q62.722 75.989 63.039 75.989Q63.258 75.989 63.429 76.113Q63.600 76.238 63.600 76.450Q63.600 76.587 63.501 76.686Q63.402 76.785 63.268 76.785Q63.132 76.785 63.033 76.686Q62.933 76.587 62.933 76.450Q62.933 76.310 63.033 76.211Q62.742 76.211 62.542 76.407Q62.342 76.604 62.250 76.898Q62.158 77.192 62.158 77.472L62.158 78.586Q62.158 78.795 62.814 78.795L62.814 79.075M64.243 78.347Q64.243 78.015 64.466 77.788Q64.690 77.561 65.034 77.433Q65.377 77.304 65.750 77.252Q66.122 77.199 66.427 77.199L66.427 76.946Q66.427 76.741 66.319 76.561Q66.211 76.382 66.030 76.279Q65.849 76.177 65.640 76.177Q65.234 76.177 64.998 76.269Q65.087 76.306 65.133 76.390Q65.179 76.474 65.179 76.576Q65.179 76.672 65.133 76.751Q65.087 76.829 65.006 76.874Q64.926 76.918 64.837 76.918Q64.687 76.918 64.586 76.821Q64.485 76.723 64.485 76.576Q64.485 75.954 65.640 75.954Q65.852 75.954 66.102 76.018Q66.351 76.081 66.553 76.200Q66.755 76.320 66.881 76.505Q67.008 76.689 67.008 76.932L67.008 78.508Q67.008 78.624 67.069 78.720Q67.131 78.815 67.244 78.815Q67.353 78.815 67.418 78.721Q67.483 78.627 67.483 78.508L67.483 78.060L67.749 78.060L67.749 78.508Q67.749 78.778 67.522 78.943Q67.295 79.109 67.015 79.109Q66.806 79.109 66.669 78.955Q66.533 78.802 66.509 78.586Q66.362 78.853 66.080 78.998Q65.798 79.143 65.473 79.143Q65.196 79.143 64.912 79.068Q64.629 78.993 64.436 78.814Q64.243 78.634 64.243 78.347M64.858 78.347Q64.858 78.521 64.959 78.651Q65.059 78.781 65.215 78.851Q65.370 78.921 65.535 78.921Q65.753 78.921 65.962 78.824Q66.170 78.726 66.298 78.545Q66.427 78.364 66.427 78.138L66.427 77.410Q66.102 77.410 65.736 77.501Q65.370 77.592 65.114 77.804Q64.858 78.015 64.858 78.347M68.693 78.234L68.693 76.337L68.054 76.337L68.054 76.115Q68.371 76.115 68.588 75.905Q68.806 75.695 68.906 75.385Q69.007 75.076 69.007 74.768L69.274 74.768L69.274 76.057L70.350 76.057L70.350 76.337L69.274 76.337L69.274 78.221Q69.274 78.497 69.378 78.696Q69.482 78.894 69.742 78.894Q69.899 78.894 70.005 78.790Q70.111 78.685 70.161 78.532Q70.210 78.378 70.210 78.221L70.210 77.807L70.477 77.807L70.477 78.234Q70.477 78.460 70.378 78.670Q70.279 78.880 70.094 79.012Q69.910 79.143 69.681 79.143Q69.243 79.143 68.968 78.906Q68.693 78.668 68.693 78.234M72.904 79.075L71.352 79.075L71.352 78.795Q71.578 78.795 71.726 78.761Q71.875 78.726 71.875 78.586L71.875 76.737Q71.875 76.549 71.827 76.465Q71.779 76.382 71.682 76.363Q71.584 76.344 71.372 76.344L71.372 76.064L72.429 75.989L72.429 78.586Q72.429 78.726 72.560 78.761Q72.692 78.795 72.904 78.795L72.904 79.075M71.632 74.768Q71.632 74.597 71.755 74.478Q71.878 74.358 72.049 74.358Q72.217 74.358 72.340 74.478Q72.463 74.597 72.463 74.768Q72.463 74.943 72.340 75.066Q72.217 75.189 72.049 75.189Q71.878 75.189 71.755 75.066Q71.632 74.943 71.632 74.768M73.509 77.592Q73.509 77.250 73.644 76.951Q73.779 76.652 74.018 76.428Q74.257 76.204 74.575 76.079Q74.893 75.954 75.224 75.954Q75.669 75.954 76.069 76.170Q76.469 76.385 76.703 76.763Q76.937 77.140 76.937 77.592Q76.937 77.933 76.795 78.217Q76.653 78.501 76.409 78.708Q76.164 78.914 75.855 79.029Q75.546 79.143 75.224 79.143Q74.794 79.143 74.392 78.942Q73.991 78.740 73.750 78.388Q73.509 78.036 73.509 77.592M75.224 78.894Q75.826 78.894 76.050 78.516Q76.274 78.138 76.274 77.506Q76.274 76.894 76.040 76.535Q75.806 76.177 75.224 76.177Q74.172 76.177 74.172 77.506Q74.172 78.138 74.397 78.516Q74.623 78.894 75.224 78.894M79.213 79.075L77.579 79.075L77.579 78.795Q77.808 78.795 77.957 78.761Q78.106 78.726 78.106 78.586L78.106 76.737Q78.106 76.467 77.998 76.406Q77.890 76.344 77.579 76.344L77.579 76.064L78.639 75.989L78.639 76.638Q78.810 76.330 79.114 76.159Q79.418 75.989 79.764 75.989Q80.269 75.989 80.553 76.212Q80.837 76.436 80.837 76.932L80.837 78.586Q80.837 78.723 80.985 78.759Q81.134 78.795 81.360 78.795L81.360 79.075L79.729 79.075L79.729 78.795Q79.958 78.795 80.107 78.761Q80.256 78.726 80.256 78.586L80.256 76.946Q80.256 76.611 80.136 76.411Q80.016 76.211 79.702 76.211Q79.432 76.211 79.198 76.347Q78.964 76.484 78.825 76.718Q78.687 76.952 78.687 77.226L78.687 78.586Q78.687 78.723 78.837 78.759Q78.988 78.795 79.213 78.795L79.213 79.075M81.948 79.068L81.948 78.005Q81.948 77.981 81.975 77.954Q82.002 77.927 82.026 77.927L82.136 77.927Q82.201 77.927 82.214 77.985Q82.310 78.419 82.556 78.670Q82.802 78.921 83.216 78.921Q83.557 78.921 83.810 78.788Q84.063 78.655 84.063 78.347Q84.063 78.190 83.969 78.075Q83.875 77.961 83.737 77.892Q83.599 77.824 83.431 77.786L82.850 77.687Q82.494 77.619 82.221 77.398Q81.948 77.178 81.948 76.836Q81.948 76.587 82.059 76.412Q82.170 76.238 82.356 76.139Q82.542 76.040 82.758 75.997Q82.973 75.954 83.216 75.954Q83.629 75.954 83.910 76.136L84.125 75.961Q84.135 75.958 84.142 75.956Q84.149 75.954 84.159 75.954L84.210 75.954Q84.238 75.954 84.262 75.978Q84.286 76.002 84.286 76.030L84.286 76.877Q84.286 76.898 84.262 76.925Q84.238 76.952 84.210 76.952L84.098 76.952Q84.070 76.952 84.045 76.927Q84.019 76.901 84.019 76.877Q84.019 76.641 83.913 76.477Q83.807 76.313 83.624 76.231Q83.441 76.149 83.209 76.149Q82.881 76.149 82.624 76.252Q82.368 76.354 82.368 76.631Q82.368 76.826 82.551 76.935Q82.734 77.045 82.963 77.086L83.537 77.192Q83.783 77.240 83.997 77.368Q84.210 77.496 84.347 77.699Q84.484 77.903 84.484 78.152Q84.484 78.665 84.118 78.904Q83.752 79.143 83.216 79.143Q82.720 79.143 82.389 78.849L82.122 79.123Q82.101 79.143 82.074 79.143L82.026 79.143Q82.002 79.143 81.975 79.116Q81.948 79.089 81.948 79.068\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-neutral)\" style=\"stroke-width:.8\">\u003Cpath d=\"M128.275 32.128h68.287V.83h-68.287Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(102.442 -56.165)\">\u003Cpath d=\"M33.532 71.075L31.898 71.075L31.898 70.795Q32.127 70.795 32.276 70.761Q32.425 70.726 32.425 70.586L32.425 66.967Q32.425 66.697 32.317 66.635Q32.209 66.574 31.898 66.574L31.898 66.293L32.978 66.218L32.978 68.604Q33.084 68.419 33.262 68.277Q33.440 68.136 33.648 68.062Q33.857 67.989 34.082 67.989Q34.588 67.989 34.872 68.212Q35.156 68.436 35.156 68.932L35.156 70.586Q35.156 70.723 35.304 70.759Q35.453 70.795 35.679 70.795L35.679 71.075L34.048 71.075L34.048 70.795Q34.277 70.795 34.426 70.761Q34.575 70.726 34.575 70.586L34.575 68.946Q34.575 68.611 34.455 68.411Q34.335 68.211 34.021 68.211Q33.751 68.211 33.517 68.347Q33.283 68.484 33.144 68.718Q33.006 68.952 33.006 69.226L33.006 70.586Q33.006 70.723 33.156 70.759Q33.307 70.795 33.532 70.795\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(102.442 -56.165)\">\u003Cpath d=\"M36.626 70.241L36.626 68.737Q36.626 68.467 36.518 68.406Q36.410 68.344 36.099 68.344L36.099 68.064L37.207 67.989L37.207 70.221L37.207 70.241Q37.207 70.521 37.258 70.665Q37.309 70.808 37.451 70.865Q37.593 70.921 37.880 70.921Q38.133 70.921 38.338 70.781Q38.543 70.641 38.659 70.415Q38.776 70.190 38.776 69.940L38.776 68.737Q38.776 68.467 38.668 68.406Q38.560 68.344 38.249 68.344L38.249 68.064L39.357 67.989L39.357 70.402Q39.357 70.593 39.410 70.675Q39.463 70.757 39.563 70.776Q39.664 70.795 39.880 70.795L39.880 71.075L38.803 71.143L38.803 70.579Q38.694 70.761 38.548 70.884Q38.403 71.007 38.217 71.075Q38.030 71.143 37.829 71.143Q36.626 71.143 36.626 70.241M42.149 71.075L40.515 71.075L40.515 70.795Q40.744 70.795 40.893 70.761Q41.042 70.726 41.042 70.586L41.042 68.737Q41.042 68.467 40.934 68.406Q40.826 68.344 40.515 68.344L40.515 68.064L41.575 67.989L41.575 68.638Q41.746 68.330 42.050 68.159Q42.354 67.989 42.699 67.989Q43.099 67.989 43.376 68.129Q43.653 68.269 43.738 68.617Q43.906 68.324 44.205 68.156Q44.504 67.989 44.849 67.989Q45.355 67.989 45.639 68.212Q45.923 68.436 45.923 68.932L45.923 70.586Q45.923 70.723 46.071 70.759Q46.220 70.795 46.445 70.795L46.445 71.075L44.815 71.075L44.815 70.795Q45.041 70.795 45.191 70.759Q45.341 70.723 45.341 70.586L45.341 68.946Q45.341 68.611 45.222 68.411Q45.102 68.211 44.788 68.211Q44.518 68.211 44.284 68.347Q44.050 68.484 43.911 68.718Q43.773 68.952 43.773 69.226L43.773 70.586Q43.773 70.723 43.921 70.759Q44.070 70.795 44.296 70.795L44.296 71.075L42.665 71.075L42.665 70.795Q42.894 70.795 43.043 70.761Q43.192 70.726 43.192 70.586L43.192 68.946Q43.192 68.611 43.072 68.411Q42.952 68.211 42.638 68.211Q42.368 68.211 42.134 68.347Q41.900 68.484 41.761 68.718Q41.623 68.952 41.623 69.226L41.623 70.586Q41.623 70.723 41.773 70.759Q41.924 70.795 42.149 70.795L42.149 71.075M47.091 70.347Q47.091 70.015 47.315 69.788Q47.539 69.561 47.883 69.433Q48.226 69.304 48.599 69.252Q48.971 69.199 49.276 69.199L49.276 68.946Q49.276 68.741 49.168 68.561Q49.060 68.382 48.879 68.279Q48.698 68.177 48.489 68.177Q48.083 68.177 47.847 68.269Q47.936 68.306 47.982 68.390Q48.028 68.474 48.028 68.576Q48.028 68.672 47.982 68.751Q47.936 68.829 47.855 68.874Q47.775 68.918 47.686 68.918Q47.536 68.918 47.435 68.821Q47.334 68.723 47.334 68.576Q47.334 67.954 48.489 67.954Q48.701 67.954 48.951 68.018Q49.200 68.081 49.402 68.200Q49.604 68.320 49.730 68.505Q49.857 68.689 49.857 68.932L49.857 70.508Q49.857 70.624 49.918 70.720Q49.980 70.815 50.092 70.815Q50.202 70.815 50.267 70.721Q50.332 70.627 50.332 70.508L50.332 70.060L50.598 70.060L50.598 70.508Q50.598 70.778 50.371 70.943Q50.144 71.109 49.863 71.109Q49.655 71.109 49.518 70.955Q49.382 70.802 49.358 70.586Q49.211 70.853 48.929 70.998Q48.647 71.143 48.322 71.143Q48.045 71.143 47.761 71.068Q47.478 70.993 47.285 70.814Q47.091 70.634 47.091 70.347M47.707 70.347Q47.707 70.521 47.808 70.651Q47.908 70.781 48.064 70.851Q48.219 70.921 48.383 70.921Q48.602 70.921 48.811 70.824Q49.019 70.726 49.147 70.545Q49.276 70.364 49.276 70.138L49.276 69.410Q48.951 69.410 48.585 69.501Q48.219 69.592 47.963 69.804Q47.707 70.015 47.707 70.347M52.697 71.075L51.063 71.075L51.063 70.795Q51.292 70.795 51.441 70.761Q51.590 70.726 51.590 70.586L51.590 68.737Q51.590 68.467 51.482 68.406Q51.374 68.344 51.063 68.344L51.063 68.064L52.123 67.989L52.123 68.638Q52.294 68.330 52.598 68.159Q52.902 67.989 53.247 67.989Q53.753 67.989 54.037 68.212Q54.320 68.436 54.320 68.932L54.320 70.586Q54.320 70.723 54.469 70.759Q54.618 70.795 54.843 70.795L54.843 71.075L53.213 71.075L53.213 70.795Q53.442 70.795 53.591 70.761Q53.739 70.726 53.739 70.586L53.739 68.946Q53.739 68.611 53.620 68.411Q53.500 68.211 53.186 68.211Q52.916 68.211 52.682 68.347Q52.447 68.484 52.309 68.718Q52.171 68.952 52.171 69.226L52.171 70.586Q52.171 70.723 52.321 70.759Q52.471 70.795 52.697 70.795\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(102.442 -56.165)\">\u003Cpath d=\"M58.135 69.564Q58.135 69.236 58.270 68.935Q58.405 68.635 58.641 68.414Q58.877 68.194 59.181 68.074Q59.486 67.954 59.810 67.954Q60.316 67.954 60.665 68.057Q61.013 68.159 61.013 68.535Q61.013 68.682 60.916 68.783Q60.819 68.884 60.672 68.884Q60.518 68.884 60.419 68.785Q60.320 68.686 60.320 68.535Q60.320 68.347 60.460 68.255Q60.258 68.204 59.817 68.204Q59.462 68.204 59.233 68.400Q59.004 68.597 58.903 68.906Q58.802 69.216 58.802 69.564Q58.802 69.913 58.928 70.219Q59.055 70.525 59.310 70.709Q59.564 70.894 59.920 70.894Q60.142 70.894 60.326 70.810Q60.511 70.726 60.646 70.571Q60.781 70.415 60.839 70.207Q60.853 70.152 60.907 70.152L61.020 70.152Q61.051 70.152 61.073 70.176Q61.095 70.200 61.095 70.234L61.095 70.255Q61.010 70.542 60.822 70.740Q60.634 70.938 60.369 71.041Q60.104 71.143 59.810 71.143Q59.380 71.143 58.992 70.937Q58.604 70.730 58.370 70.367Q58.135 70.005 58.135 69.564M61.642 69.592Q61.642 69.250 61.777 68.951Q61.912 68.652 62.152 68.428Q62.391 68.204 62.709 68.079Q63.027 67.954 63.358 67.954Q63.802 67.954 64.202 68.170Q64.602 68.385 64.836 68.763Q65.071 69.140 65.071 69.592Q65.071 69.933 64.929 70.217Q64.787 70.501 64.542 70.708Q64.298 70.914 63.989 71.029Q63.679 71.143 63.358 71.143Q62.927 71.143 62.526 70.942Q62.124 70.740 61.883 70.388Q61.642 70.036 61.642 69.592M63.358 70.894Q63.960 70.894 64.184 70.516Q64.407 70.138 64.407 69.506Q64.407 68.894 64.173 68.535Q63.939 68.177 63.358 68.177Q62.305 68.177 62.305 69.506Q62.305 70.138 62.531 70.516Q62.757 70.894 63.358 70.894M67.347 71.075L65.713 71.075L65.713 70.795Q65.942 70.795 66.091 70.761Q66.239 70.726 66.239 70.586L66.239 68.737Q66.239 68.467 66.132 68.406Q66.024 68.344 65.713 68.344L65.713 68.064L66.773 67.989L66.773 68.638Q66.944 68.330 67.248 68.159Q67.552 67.989 67.897 67.989Q68.297 67.989 68.574 68.129Q68.851 68.269 68.936 68.617Q69.104 68.324 69.403 68.156Q69.702 67.989 70.047 67.989Q70.553 67.989 70.837 68.212Q71.120 68.436 71.120 68.932L71.120 70.586Q71.120 70.723 71.269 70.759Q71.418 70.795 71.643 70.795L71.643 71.075L70.013 71.075L70.013 70.795Q70.238 70.795 70.389 70.759Q70.539 70.723 70.539 70.586L70.539 68.946Q70.539 68.611 70.420 68.411Q70.300 68.211 69.986 68.211Q69.716 68.211 69.481 68.347Q69.247 68.484 69.109 68.718Q68.970 68.952 68.970 69.226L68.970 70.586Q68.970 70.723 69.119 70.759Q69.268 70.795 69.493 70.795L69.493 71.075L67.863 71.075L67.863 70.795Q68.092 70.795 68.241 70.761Q68.389 70.726 68.389 70.586L68.389 68.946Q68.389 68.611 68.270 68.411Q68.150 68.211 67.836 68.211Q67.566 68.211 67.331 68.347Q67.097 68.484 66.959 68.718Q66.821 68.952 66.821 69.226L66.821 70.586Q66.821 70.723 66.971 70.759Q67.121 70.795 67.347 70.795L67.347 71.075M73.875 72.432L72.245 72.432L72.245 72.152Q72.474 72.152 72.623 72.117Q72.771 72.083 72.771 71.943L72.771 68.597Q72.771 68.426 72.634 68.385Q72.498 68.344 72.245 68.344L72.245 68.064L73.325 67.989L73.325 68.395Q73.547 68.194 73.834 68.091Q74.121 67.989 74.429 67.989Q74.856 67.989 75.220 68.202Q75.584 68.416 75.798 68.780Q76.011 69.144 76.011 69.564Q76.011 70.009 75.772 70.373Q75.533 70.737 75.140 70.940Q74.747 71.143 74.302 71.143Q74.036 71.143 73.788 71.043Q73.540 70.942 73.352 70.761L73.352 71.943Q73.352 72.080 73.501 72.116Q73.650 72.152 73.875 72.152L73.875 72.432M73.352 68.744L73.352 70.354Q73.486 70.607 73.728 70.764Q73.971 70.921 74.248 70.921Q74.576 70.921 74.829 70.720Q75.082 70.518 75.215 70.200Q75.348 69.882 75.348 69.564Q75.348 69.335 75.283 69.106Q75.218 68.877 75.090 68.679Q74.962 68.481 74.767 68.361Q74.572 68.242 74.340 68.242Q74.046 68.242 73.778 68.371Q73.509 68.501 73.352 68.744M76.705 70.347Q76.705 70.015 76.929 69.788Q77.153 69.561 77.497 69.433Q77.840 69.304 78.213 69.252Q78.585 69.199 78.889 69.199L78.889 68.946Q78.889 68.741 78.782 68.561Q78.674 68.382 78.493 68.279Q78.312 68.177 78.103 68.177Q77.696 68.177 77.461 68.269Q77.550 68.306 77.596 68.390Q77.642 68.474 77.642 68.576Q77.642 68.672 77.596 68.751Q77.550 68.829 77.469 68.874Q77.389 68.918 77.300 68.918Q77.150 68.918 77.049 68.821Q76.948 68.723 76.948 68.576Q76.948 67.954 78.103 67.954Q78.315 67.954 78.565 68.018Q78.814 68.081 79.016 68.200Q79.217 68.320 79.344 68.505Q79.470 68.689 79.470 68.932L79.470 70.508Q79.470 70.624 79.532 70.720Q79.593 70.815 79.706 70.815Q79.816 70.815 79.881 70.721Q79.946 70.627 79.946 70.508L79.946 70.060L80.212 70.060L80.212 70.508Q80.212 70.778 79.985 70.943Q79.758 71.109 79.477 71.109Q79.269 71.109 79.132 70.955Q78.995 70.802 78.971 70.586Q78.824 70.853 78.542 70.998Q78.260 71.143 77.936 71.143Q77.659 71.143 77.375 71.068Q77.092 70.993 76.898 70.814Q76.705 70.634 76.705 70.347M77.321 70.347Q77.321 70.521 77.421 70.651Q77.522 70.781 77.678 70.851Q77.833 70.921 77.997 70.921Q78.216 70.921 78.425 70.824Q78.633 70.726 78.761 70.545Q78.889 70.364 78.889 70.138L78.889 69.410Q78.565 69.410 78.199 69.501Q77.833 69.592 77.577 69.804Q77.321 70.015 77.321 70.347M82.379 71.075L80.643 71.075L80.643 70.795Q80.872 70.795 81.020 70.761Q81.169 70.726 81.169 70.586L81.169 68.737Q81.169 68.467 81.061 68.406Q80.954 68.344 80.643 68.344L80.643 68.064L81.672 67.989L81.672 68.696Q81.801 68.388 82.044 68.189Q82.287 67.989 82.605 67.989Q82.823 67.989 82.994 68.113Q83.165 68.238 83.165 68.450Q83.165 68.587 83.066 68.686Q82.967 68.785 82.834 68.785Q82.697 68.785 82.598 68.686Q82.499 68.587 82.499 68.450Q82.499 68.310 82.598 68.211Q82.307 68.211 82.107 68.407Q81.907 68.604 81.815 68.898Q81.723 69.192 81.723 69.472L81.723 70.586Q81.723 70.795 82.379 70.795L82.379 71.075M85.366 71.075L83.815 71.075L83.815 70.795Q84.040 70.795 84.189 70.761Q84.338 70.726 84.338 70.586L84.338 68.737Q84.338 68.549 84.290 68.465Q84.242 68.382 84.144 68.363Q84.047 68.344 83.835 68.344L83.835 68.064L84.891 67.989L84.891 70.586Q84.891 70.726 85.023 70.761Q85.154 70.795 85.366 70.795L85.366 71.075M84.095 66.768Q84.095 66.597 84.218 66.478Q84.341 66.358 84.512 66.358Q84.679 66.358 84.802 66.478Q84.925 66.597 84.925 66.768Q84.925 66.943 84.802 67.066Q84.679 67.189 84.512 67.189Q84.341 67.189 84.218 67.066Q84.095 66.943 84.095 66.768M87.896 69.821L85.838 69.821L85.838 69.318L87.896 69.318\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(102.442 -56.165)\">\u003Cpath d=\"M52.921 79.068L52.921 78.005Q52.921 77.981 52.949 77.954Q52.976 77.927 53 77.927L53.109 77.927Q53.174 77.927 53.188 77.985Q53.284 78.419 53.530 78.670Q53.776 78.921 54.190 78.921Q54.531 78.921 54.784 78.788Q55.037 78.655 55.037 78.347Q55.037 78.190 54.943 78.075Q54.849 77.961 54.711 77.892Q54.572 77.824 54.405 77.786L53.824 77.687Q53.468 77.619 53.195 77.398Q52.921 77.178 52.921 76.836Q52.921 76.587 53.033 76.412Q53.144 76.238 53.330 76.139Q53.516 76.040 53.732 75.997Q53.947 75.954 54.190 75.954Q54.603 75.954 54.883 76.136L55.099 75.961Q55.109 75.958 55.116 75.956Q55.123 75.954 55.133 75.954L55.184 75.954Q55.211 75.954 55.235 75.978Q55.259 76.002 55.259 76.030L55.259 76.877Q55.259 76.898 55.235 76.925Q55.211 76.952 55.184 76.952L55.071 76.952Q55.044 76.952 55.018 76.927Q54.993 76.901 54.993 76.877Q54.993 76.641 54.887 76.477Q54.781 76.313 54.598 76.231Q54.415 76.149 54.183 76.149Q53.855 76.149 53.598 76.252Q53.342 76.354 53.342 76.631Q53.342 76.826 53.525 76.935Q53.708 77.045 53.937 77.086L54.511 77.192Q54.757 77.240 54.971 77.368Q55.184 77.496 55.321 77.699Q55.458 77.903 55.458 78.152Q55.458 78.665 55.092 78.904Q54.726 79.143 54.190 79.143Q53.694 79.143 53.362 78.849L53.096 79.123Q53.075 79.143 53.048 79.143L53 79.143Q52.976 79.143 52.949 79.116Q52.921 79.089 52.921 79.068M56.045 77.592Q56.045 77.250 56.180 76.951Q56.315 76.652 56.555 76.428Q56.794 76.204 57.112 76.079Q57.430 75.954 57.761 75.954Q58.206 75.954 58.606 76.170Q59.005 76.385 59.240 76.763Q59.474 77.140 59.474 77.592Q59.474 77.933 59.332 78.217Q59.190 78.501 58.946 78.708Q58.701 78.914 58.392 79.029Q58.083 79.143 57.761 79.143Q57.331 79.143 56.929 78.942Q56.527 78.740 56.286 78.388Q56.045 78.036 56.045 77.592M57.761 78.894Q58.363 78.894 58.587 78.516Q58.811 78.138 58.811 77.506Q58.811 76.894 58.576 76.535Q58.342 76.177 57.761 76.177Q56.709 76.177 56.709 77.506Q56.709 78.138 56.934 78.516Q57.160 78.894 57.761 78.894M61.750 79.075L60.116 79.075L60.116 78.795Q60.345 78.795 60.494 78.761Q60.643 78.726 60.643 78.586L60.643 76.737Q60.643 76.467 60.535 76.406Q60.427 76.344 60.116 76.344L60.116 76.064L61.176 75.989L61.176 76.638Q61.347 76.330 61.651 76.159Q61.955 75.989 62.300 75.989Q62.806 75.989 63.090 76.212Q63.374 76.436 63.374 76.932L63.374 78.586Q63.374 78.723 63.522 78.759Q63.671 78.795 63.897 78.795L63.897 79.075L62.266 79.075L62.266 78.795Q62.495 78.795 62.644 78.761Q62.793 78.726 62.793 78.586L62.793 76.946Q62.793 76.611 62.673 76.411Q62.553 76.211 62.239 76.211Q61.969 76.211 61.735 76.347Q61.501 76.484 61.362 76.718Q61.224 76.952 61.224 77.226L61.224 78.586Q61.224 78.723 61.374 78.759Q61.524 78.795 61.750 78.795L61.750 79.075M64.484 79.068L64.484 78.005Q64.484 77.981 64.512 77.954Q64.539 77.927 64.563 77.927L64.672 77.927Q64.737 77.927 64.751 77.985Q64.847 78.419 65.093 78.670Q65.339 78.921 65.753 78.921Q66.094 78.921 66.347 78.788Q66.600 78.655 66.600 78.347Q66.600 78.190 66.506 78.075Q66.412 77.961 66.274 77.892Q66.135 77.824 65.968 77.786L65.387 77.687Q65.031 77.619 64.758 77.398Q64.484 77.178 64.484 76.836Q64.484 76.587 64.596 76.412Q64.707 76.238 64.893 76.139Q65.079 76.040 65.294 75.997Q65.510 75.954 65.753 75.954Q66.166 75.954 66.446 76.136L66.662 75.961Q66.672 75.958 66.679 75.956Q66.686 75.954 66.696 75.954L66.747 75.954Q66.774 75.954 66.798 75.978Q66.822 76.002 66.822 76.030L66.822 76.877Q66.822 76.898 66.798 76.925Q66.774 76.952 66.747 76.952L66.634 76.952Q66.607 76.952 66.581 76.927Q66.556 76.901 66.556 76.877Q66.556 76.641 66.450 76.477Q66.344 76.313 66.161 76.231Q65.978 76.149 65.746 76.149Q65.418 76.149 65.161 76.252Q64.905 76.354 64.905 76.631Q64.905 76.826 65.088 76.935Q65.271 77.045 65.500 77.086L66.074 77.192Q66.320 77.240 66.534 77.368Q66.747 77.496 66.884 77.699Q67.021 77.903 67.021 78.152Q67.021 78.665 66.655 78.904Q66.289 79.143 65.753 79.143Q65.257 79.143 64.925 78.849L64.659 79.123Q64.638 79.143 64.611 79.143L64.563 79.143Q64.539 79.143 64.512 79.116Q64.484 79.089 64.484 79.068\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-neutral)\" style=\"stroke-width:.8\">\u003Cpath d=\"M259.158 32.128h68.286V.83h-68.286Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(238.58 -60.846)\">\u003Cpath d=\"M33.494 80.432L31.864 80.432L31.864 80.152Q32.093 80.152 32.242 80.117Q32.390 80.083 32.390 79.943L32.390 76.597Q32.390 76.426 32.254 76.385Q32.117 76.344 31.864 76.344L31.864 76.064L32.944 75.989L32.944 76.395Q33.166 76.194 33.453 76.091Q33.741 75.989 34.048 75.989Q34.475 75.989 34.839 76.202Q35.203 76.416 35.417 76.780Q35.631 77.144 35.631 77.564Q35.631 78.009 35.391 78.373Q35.152 78.737 34.759 78.940Q34.366 79.143 33.922 79.143Q33.655 79.143 33.407 79.043Q33.160 78.942 32.972 78.761L32.972 79.943Q32.972 80.080 33.120 80.116Q33.269 80.152 33.494 80.152L33.494 80.432M32.972 76.744L32.972 78.354Q33.105 78.607 33.348 78.764Q33.590 78.921 33.867 78.921Q34.195 78.921 34.448 78.720Q34.701 78.518 34.834 78.200Q34.968 77.882 34.968 77.564Q34.968 77.335 34.903 77.106Q34.838 76.877 34.710 76.679Q34.581 76.481 34.387 76.361Q34.192 76.242 33.959 76.242Q33.665 76.242 33.397 76.371Q33.129 76.501 32.972 76.744M38.016 79.075L36.280 79.075L36.280 78.795Q36.509 78.795 36.658 78.761Q36.807 78.726 36.807 78.586L36.807 76.737Q36.807 76.467 36.699 76.406Q36.591 76.344 36.280 76.344L36.280 76.064L37.309 75.989L37.309 76.696Q37.439 76.388 37.682 76.189Q37.924 75.989 38.242 75.989Q38.461 75.989 38.632 76.113Q38.803 76.238 38.803 76.450Q38.803 76.587 38.703 76.686Q38.604 76.785 38.471 76.785Q38.334 76.785 38.235 76.686Q38.136 76.587 38.136 76.450Q38.136 76.310 38.235 76.211Q37.945 76.211 37.745 76.407Q37.545 76.604 37.453 76.898Q37.360 77.192 37.360 77.472L37.360 78.586Q37.360 78.795 38.016 78.795L38.016 79.075M39.346 77.592Q39.346 77.250 39.481 76.951Q39.616 76.652 39.855 76.428Q40.095 76.204 40.412 76.079Q40.730 75.954 41.062 75.954Q41.506 75.954 41.906 76.170Q42.306 76.385 42.540 76.763Q42.774 77.140 42.774 77.592Q42.774 77.933 42.632 78.217Q42.491 78.501 42.246 78.708Q42.002 78.914 41.692 79.029Q41.383 79.143 41.062 79.143Q40.631 79.143 40.230 78.942Q39.828 78.740 39.587 78.388Q39.346 78.036 39.346 77.592M41.062 78.894Q41.663 78.894 41.887 78.516Q42.111 78.138 42.111 77.506Q42.111 76.894 41.877 76.535Q41.643 76.177 41.062 76.177Q40.009 76.177 40.009 77.506Q40.009 78.138 40.235 78.516Q40.460 78.894 41.062 78.894M45.051 79.075L43.417 79.075L43.417 78.795Q43.646 78.795 43.795 78.761Q43.943 78.726 43.943 78.586L43.943 76.737Q43.943 76.467 43.836 76.406Q43.728 76.344 43.417 76.344L43.417 76.064L44.476 75.989L44.476 76.638Q44.647 76.330 44.952 76.159Q45.256 75.989 45.601 75.989Q46.001 75.989 46.278 76.129Q46.555 76.269 46.640 76.617Q46.807 76.324 47.107 76.156Q47.406 75.989 47.751 75.989Q48.257 75.989 48.540 76.212Q48.824 76.436 48.824 76.932L48.824 78.586Q48.824 78.723 48.973 78.759Q49.121 78.795 49.347 78.795L49.347 79.075L47.717 79.075L47.717 78.795Q47.942 78.795 48.093 78.759Q48.243 78.723 48.243 78.586L48.243 76.946Q48.243 76.611 48.123 76.411Q48.004 76.211 47.689 76.211Q47.419 76.211 47.185 76.347Q46.951 76.484 46.813 76.718Q46.674 76.952 46.674 77.226L46.674 78.586Q46.674 78.723 46.823 78.759Q46.972 78.795 47.197 78.795L47.197 79.075L45.567 79.075L45.567 78.795Q45.796 78.795 45.944 78.761Q46.093 78.726 46.093 78.586L46.093 76.946Q46.093 76.611 45.974 76.411Q45.854 76.211 45.539 76.211Q45.269 76.211 45.035 76.347Q44.801 76.484 44.663 76.718Q44.524 76.952 44.524 77.226L44.524 78.586Q44.524 78.723 44.675 78.759Q44.825 78.795 45.051 78.795L45.051 79.075M51.579 80.432L49.949 80.432L49.949 80.152Q50.178 80.152 50.326 80.117Q50.475 80.083 50.475 79.943L50.475 76.597Q50.475 76.426 50.338 76.385Q50.202 76.344 49.949 76.344L49.949 76.064L51.029 75.989L51.029 76.395Q51.251 76.194 51.538 76.091Q51.825 75.989 52.133 75.989Q52.560 75.989 52.924 76.202Q53.288 76.416 53.502 76.780Q53.715 77.144 53.715 77.564Q53.715 78.009 53.476 78.373Q53.237 78.737 52.844 78.940Q52.451 79.143 52.006 79.143Q51.740 79.143 51.492 79.043Q51.244 78.942 51.056 78.761L51.056 79.943Q51.056 80.080 51.205 80.116Q51.353 80.152 51.579 80.152L51.579 80.432M51.056 76.744L51.056 78.354Q51.189 78.607 51.432 78.764Q51.675 78.921 51.952 78.921Q52.280 78.921 52.533 78.720Q52.786 78.518 52.919 78.200Q53.052 77.882 53.052 77.564Q53.052 77.335 52.987 77.106Q52.922 76.877 52.794 76.679Q52.666 76.481 52.471 76.361Q52.276 76.242 52.044 76.242Q51.750 76.242 51.482 76.371Q51.213 76.501 51.056 76.744M54.877 78.234L54.877 76.337L54.238 76.337L54.238 76.115Q54.556 76.115 54.773 75.905Q54.990 75.695 55.091 75.385Q55.192 75.076 55.192 74.768L55.458 74.768L55.458 76.057L56.535 76.057L56.535 76.337L55.458 76.337L55.458 78.221Q55.458 78.497 55.563 78.696Q55.667 78.894 55.927 78.894Q56.084 78.894 56.190 78.790Q56.296 78.685 56.345 78.532Q56.395 78.378 56.395 78.221L56.395 77.807L56.661 77.807L56.661 78.234Q56.661 78.460 56.562 78.670Q56.463 78.880 56.279 79.012Q56.094 79.143 55.865 79.143Q55.428 79.143 55.152 78.906Q54.877 78.668 54.877 78.234M57.472 79.068L57.472 78.005Q57.472 77.981 57.499 77.954Q57.526 77.927 57.550 77.927L57.660 77.927Q57.724 77.927 57.738 77.985Q57.834 78.419 58.080 78.670Q58.326 78.921 58.740 78.921Q59.081 78.921 59.334 78.788Q59.587 78.655 59.587 78.347Q59.587 78.190 59.493 78.075Q59.399 77.961 59.261 77.892Q59.122 77.824 58.955 77.786L58.374 77.687Q58.018 77.619 57.745 77.398Q57.472 77.178 57.472 76.836Q57.472 76.587 57.583 76.412Q57.694 76.238 57.880 76.139Q58.066 76.040 58.282 75.997Q58.497 75.954 58.740 75.954Q59.153 75.954 59.433 76.136L59.649 75.961Q59.659 75.958 59.666 75.956Q59.673 75.954 59.683 75.954L59.734 75.954Q59.762 75.954 59.786 75.978Q59.809 76.002 59.809 76.030L59.809 76.877Q59.809 76.898 59.786 76.925Q59.762 76.952 59.734 76.952L59.621 76.952Q59.594 76.952 59.568 76.927Q59.543 76.901 59.543 76.877Q59.543 76.641 59.437 76.477Q59.331 76.313 59.148 76.231Q58.965 76.149 58.733 76.149Q58.405 76.149 58.148 76.252Q57.892 76.354 57.892 76.631Q57.892 76.826 58.075 76.935Q58.258 77.045 58.487 77.086L59.061 77.192Q59.307 77.240 59.521 77.368Q59.734 77.496 59.871 77.699Q60.008 77.903 60.008 78.152Q60.008 78.665 59.642 78.904Q59.276 79.143 58.740 79.143Q58.244 79.143 57.912 78.849L57.646 79.123Q57.625 79.143 57.598 79.143L57.550 79.143Q57.526 79.143 57.499 79.116Q57.472 79.089 57.472 79.068\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(238.58 -60.846)\">\u003Cpath d=\"M63.310 77.592Q63.310 77.250 63.445 76.951Q63.580 76.652 63.820 76.428Q64.059 76.204 64.377 76.079Q64.695 75.954 65.026 75.954Q65.471 75.954 65.870 76.170Q66.270 76.385 66.505 76.763Q66.739 77.140 66.739 77.592Q66.739 77.933 66.597 78.217Q66.455 78.501 66.211 78.708Q65.966 78.914 65.657 79.029Q65.348 79.143 65.026 79.143Q64.596 79.143 64.194 78.942Q63.792 78.740 63.551 78.388Q63.310 78.036 63.310 77.592M65.026 78.894Q65.628 78.894 65.852 78.516Q66.076 78.138 66.076 77.506Q66.076 76.894 65.841 76.535Q65.607 76.177 65.026 76.177Q63.974 76.177 63.974 77.506Q63.974 78.138 64.199 78.516Q64.425 78.894 65.026 78.894M69.015 79.075L67.381 79.075L67.381 78.795Q67.610 78.795 67.759 78.761Q67.908 78.726 67.908 78.586L67.908 76.737Q67.908 76.467 67.800 76.406Q67.692 76.344 67.381 76.344L67.381 76.064L68.441 75.989L68.441 76.638Q68.612 76.330 68.916 76.159Q69.220 75.989 69.565 75.989Q70.071 75.989 70.355 76.212Q70.639 76.436 70.639 76.932L70.639 78.586Q70.639 78.723 70.787 78.759Q70.936 78.795 71.162 78.795L71.162 79.075L69.531 79.075L69.531 78.795Q69.760 78.795 69.909 78.761Q70.058 78.726 70.058 78.586L70.058 76.946Q70.058 76.611 69.938 76.411Q69.818 76.211 69.504 76.211Q69.234 76.211 69 76.347Q68.766 76.484 68.627 76.718Q68.489 76.952 68.489 77.226L68.489 78.586Q68.489 78.723 68.639 78.759Q68.789 78.795 69.015 78.795L69.015 79.075M73.417 79.075L71.814 79.075L71.814 78.795Q72.040 78.795 72.189 78.761Q72.337 78.726 72.337 78.586L72.337 74.967Q72.337 74.697 72.230 74.635Q72.122 74.574 71.814 74.574L71.814 74.293L72.891 74.218L72.891 78.586Q72.891 78.723 73.041 78.759Q73.192 78.795 73.417 78.795L73.417 79.075M74.347 80.210Q74.477 80.278 74.614 80.278Q74.785 80.278 74.935 80.189Q75.085 80.100 75.196 79.955Q75.308 79.810 75.386 79.642L75.649 79.075L74.480 76.549Q74.405 76.402 74.275 76.370Q74.145 76.337 73.913 76.337L73.913 76.057L75.434 76.057L75.434 76.337Q75.085 76.337 75.085 76.484Q75.089 76.505 75.090 76.522Q75.092 76.539 75.092 76.549L75.950 78.408L76.723 76.737Q76.757 76.669 76.757 76.590Q76.757 76.477 76.673 76.407Q76.589 76.337 76.476 76.337L76.476 76.057L77.673 76.057L77.673 76.337Q77.454 76.337 77.281 76.441Q77.109 76.546 77.016 76.737L75.680 79.642Q75.509 80.012 75.239 80.258Q74.969 80.504 74.614 80.504Q74.344 80.504 74.125 80.338Q73.906 80.172 73.906 79.909Q73.906 79.772 73.998 79.683Q74.091 79.595 74.231 79.595Q74.368 79.595 74.456 79.683Q74.545 79.772 74.545 79.909Q74.545 80.012 74.492 80.090Q74.439 80.169 74.347 80.210\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath d=\"M-6.875 97.57h76.822V60.58H-6.875Z\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(-13.397 2.733)\">\u003Cpath d=\"M35.130 79.075L32.337 79.075L32.337 78.778Q33.399 78.778 33.399 78.516L33.399 74.348Q32.970 74.563 32.290 74.563L32.290 74.266Q33.309 74.266 33.825 73.755L33.970 73.755Q34.044 73.774 34.063 73.852L34.063 78.516Q34.063 78.778 35.130 78.778L35.130 79.075M36.501 78.610Q36.501 78.427 36.638 78.290Q36.774 78.153 36.966 78.153Q37.157 78.153 37.290 78.286Q37.423 78.419 37.423 78.610Q37.423 78.809 37.290 78.942Q37.157 79.075 36.966 79.075Q36.774 79.075 36.638 78.938Q36.501 78.802 36.501 78.610\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-13.397 2.733)\">\u003Cpath d=\"M42.398 79.153L42.398 77.348Q42.398 77.321 42.429 77.290Q42.460 77.259 42.484 77.259L42.589 77.259Q42.620 77.259 42.650 77.288Q42.679 77.317 42.679 77.348Q42.679 78.130 43.195 78.538Q43.710 78.946 44.519 78.946Q44.816 78.946 45.071 78.796Q45.327 78.645 45.478 78.389Q45.628 78.134 45.628 77.837Q45.628 77.438 45.382 77.134Q45.136 76.829 44.765 76.747L43.644 76.489Q43.304 76.415 43.017 76.194Q42.730 75.973 42.564 75.655Q42.398 75.337 42.398 74.985Q42.398 74.555 42.628 74.200Q42.859 73.845 43.239 73.643Q43.620 73.442 44.046 73.442Q44.296 73.442 44.542 73.501Q44.788 73.559 45.007 73.682Q45.226 73.805 45.390 73.985L45.718 73.489Q45.749 73.442 45.788 73.442L45.835 73.442Q45.862 73.442 45.894 73.473Q45.925 73.505 45.925 73.532L45.925 75.341Q45.925 75.364 45.894 75.395Q45.862 75.427 45.835 75.427L45.734 75.427Q45.702 75.427 45.673 75.397Q45.644 75.368 45.644 75.341Q45.644 75.208 45.601 75.022Q45.558 74.837 45.493 74.682Q45.429 74.528 45.329 74.370Q45.230 74.212 45.140 74.122Q44.710 73.716 44.046 73.716Q43.769 73.716 43.509 73.848Q43.249 73.981 43.091 74.216Q42.933 74.450 42.933 74.731Q42.933 75.087 43.173 75.358Q43.413 75.630 43.780 75.716L44.894 75.970Q45.171 76.036 45.404 76.190Q45.636 76.345 45.806 76.563Q45.976 76.782 46.070 77.040Q46.163 77.298 46.163 77.587Q46.163 77.915 46.038 78.218Q45.913 78.520 45.679 78.757Q45.445 78.993 45.152 79.118Q44.859 79.243 44.519 79.243Q43.503 79.243 42.933 78.700L42.605 79.196Q42.573 79.243 42.534 79.243L42.484 79.243Q42.460 79.243 42.429 79.212Q42.398 79.180 42.398 79.153M49.566 79.075L46.980 79.075L46.980 78.778Q47.300 78.778 47.544 78.731Q47.788 78.684 47.788 78.516L47.788 74.173Q47.788 74.001 47.544 73.954Q47.300 73.907 46.980 73.907L46.980 73.610L51.597 73.610L51.827 75.458L51.546 75.458Q51.460 74.774 51.298 74.454Q51.136 74.134 50.796 74.020Q50.456 73.907 49.765 73.907L48.956 73.907Q48.737 73.907 48.646 73.950Q48.554 73.993 48.554 74.173L48.554 76.196L49.163 76.196Q49.589 76.196 49.786 76.130Q49.984 76.063 50.066 75.870Q50.148 75.677 50.148 75.259L50.429 75.259L50.429 77.427L50.148 77.427Q50.148 77.009 50.066 76.815Q49.984 76.622 49.786 76.555Q49.589 76.489 49.163 76.489L48.554 76.489L48.554 78.516Q48.554 78.680 48.872 78.729Q49.191 78.778 49.566 78.778L49.566 79.075M56.773 79.075L53.730 79.075L53.730 78.778Q54.491 78.778 54.675 78.739Q54.718 78.727 54.767 78.694Q54.816 78.661 54.841 78.618Q54.866 78.575 54.866 78.516L54.866 74.173Q54.866 73.997 54.775 73.952Q54.683 73.907 54.468 73.907L54.073 73.907Q53.378 73.907 53.089 74.196Q52.941 74.345 52.878 74.665Q52.816 74.985 52.773 75.458L52.491 75.458L52.652 73.610L57.851 73.610L58.011 75.458L57.730 75.458Q57.687 74.950 57.624 74.647Q57.562 74.345 57.409 74.196Q57.124 73.907 56.425 73.907L56.034 73.907Q55.820 73.907 55.728 73.950Q55.636 73.993 55.636 74.173L55.636 78.516Q55.636 78.591 55.691 78.655Q55.745 78.720 55.827 78.739Q56.011 78.778 56.773 78.778\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath d=\"M124.007 97.57h76.823V60.58h-76.823Z\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(100.417 2.778)\">\u003Cpath d=\"M35.122 79.075L31.962 79.075L31.962 78.868Q31.962 78.841 31.985 78.809L33.337 77.411Q33.716 77.024 33.964 76.735Q34.212 76.446 34.386 76.089Q34.559 75.731 34.559 75.341Q34.559 74.993 34.427 74.700Q34.294 74.407 34.040 74.229Q33.786 74.052 33.431 74.052Q33.071 74.052 32.780 74.247Q32.489 74.442 32.345 74.770L32.399 74.770Q32.583 74.770 32.708 74.891Q32.833 75.013 32.833 75.204Q32.833 75.384 32.708 75.513Q32.583 75.641 32.399 75.641Q32.220 75.641 32.091 75.513Q31.962 75.384 31.962 75.204Q31.962 74.802 32.182 74.466Q32.403 74.130 32.768 73.942Q33.134 73.755 33.536 73.755Q34.016 73.755 34.432 73.942Q34.849 74.130 35.100 74.491Q35.352 74.852 35.352 75.341Q35.352 75.700 35.198 76.003Q35.044 76.305 34.792 76.565Q34.540 76.825 34.190 77.110Q33.841 77.395 33.673 77.548L32.743 78.388L33.458 78.388Q34.833 78.388 34.872 78.348Q34.942 78.270 34.985 78.085Q35.028 77.899 35.071 77.610L35.352 77.610L35.122 79.075M36.501 78.610Q36.501 78.427 36.638 78.290Q36.774 78.153 36.966 78.153Q37.157 78.153 37.290 78.286Q37.423 78.419 37.423 78.610Q37.423 78.809 37.290 78.942Q37.157 79.075 36.966 79.075Q36.774 79.075 36.638 78.938Q36.501 78.802 36.501 78.610\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(100.417 2.778)\">\u003Cpath d=\"M44.171 79.075L42.191 79.075L42.191 78.778Q42.460 78.778 42.628 78.733Q42.796 78.688 42.796 78.516L42.796 76.380Q42.796 76.165 42.734 76.069Q42.671 75.973 42.554 75.952Q42.437 75.930 42.191 75.930L42.191 75.634L43.359 75.548L43.359 76.333Q43.437 76.122 43.589 75.936Q43.741 75.751 43.941 75.649Q44.140 75.548 44.366 75.548Q44.612 75.548 44.804 75.692Q44.995 75.837 44.995 76.067Q44.995 76.223 44.890 76.333Q44.784 76.442 44.628 76.442Q44.472 76.442 44.362 76.333Q44.253 76.223 44.253 76.067Q44.253 75.907 44.359 75.802Q44.034 75.802 43.820 76.030Q43.605 76.259 43.509 76.598Q43.413 76.938 43.413 77.243L43.413 78.516Q43.413 78.684 43.640 78.731Q43.866 78.778 44.171 78.778L44.171 79.075M45.476 77.321Q45.476 76.841 45.708 76.425Q45.941 76.009 46.351 75.759Q46.761 75.509 47.237 75.509Q47.968 75.509 48.366 75.950Q48.765 76.391 48.765 77.122Q48.765 77.227 48.671 77.251L46.222 77.251L46.222 77.321Q46.222 77.731 46.343 78.087Q46.464 78.442 46.736 78.659Q47.007 78.876 47.437 78.876Q47.800 78.876 48.097 78.647Q48.394 78.419 48.495 78.067Q48.503 78.020 48.589 78.005L48.671 78.005Q48.765 78.032 48.765 78.114Q48.765 78.122 48.757 78.153Q48.695 78.380 48.556 78.563Q48.417 78.747 48.226 78.880Q48.034 79.013 47.816 79.083Q47.597 79.153 47.359 79.153Q46.987 79.153 46.650 79.016Q46.312 78.880 46.044 78.628Q45.777 78.376 45.626 78.036Q45.476 77.696 45.476 77.321M46.230 77.013L48.191 77.013Q48.191 76.708 48.089 76.417Q47.987 76.126 47.771 75.944Q47.554 75.763 47.237 75.763Q46.937 75.763 46.706 75.950Q46.476 76.138 46.353 76.429Q46.230 76.720 46.230 77.013M50.839 79.044L49.769 76.188Q49.702 76.009 49.571 75.966Q49.441 75.923 49.183 75.923L49.183 75.626L50.862 75.626L50.862 75.923Q50.413 75.923 50.413 76.122Q50.417 76.138 50.419 76.155Q50.421 76.173 50.421 76.188L51.214 78.282L51.925 76.372Q51.890 76.278 51.890 76.233Q51.890 76.188 51.855 76.188Q51.788 76.009 51.657 75.966Q51.527 75.923 51.273 75.923L51.273 75.626L52.862 75.626L52.862 75.923Q52.413 75.923 52.413 76.122Q52.417 76.141 52.419 76.159Q52.421 76.177 52.421 76.188L53.253 78.403L54.007 76.403Q54.030 76.345 54.030 76.274Q54.030 76.114 53.894 76.018Q53.757 75.923 53.589 75.923L53.589 75.626L54.976 75.626L54.976 75.923Q54.741 75.923 54.564 76.050Q54.386 76.177 54.304 76.403L53.320 79.044Q53.265 79.153 53.152 79.153L53.093 79.153Q52.980 79.153 52.937 79.044L52.077 76.770L51.222 79.044Q51.183 79.153 51.062 79.153L51.007 79.153Q50.894 79.153 50.839 79.044\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(100.417 2.778)\">\u003Cpath d=\"M55.254 78.243Q55.254 77.759 55.656 77.464Q56.059 77.169 56.609 77.050Q57.160 76.930 57.652 76.930L57.652 76.641Q57.652 76.415 57.537 76.208Q57.422 76.001 57.225 75.882Q57.027 75.763 56.797 75.763Q56.371 75.763 56.086 75.868Q56.156 75.895 56.203 75.950Q56.250 76.005 56.275 76.075Q56.301 76.145 56.301 76.220Q56.301 76.325 56.250 76.417Q56.199 76.509 56.107 76.559Q56.016 76.610 55.910 76.610Q55.805 76.610 55.713 76.559Q55.621 76.509 55.570 76.417Q55.520 76.325 55.520 76.220Q55.520 75.802 55.908 75.655Q56.297 75.509 56.797 75.509Q57.129 75.509 57.482 75.639Q57.836 75.770 58.064 76.024Q58.293 76.278 58.293 76.626L58.293 78.427Q58.293 78.559 58.365 78.669Q58.438 78.778 58.566 78.778Q58.691 78.778 58.760 78.673Q58.828 78.567 58.828 78.427L58.828 77.915L59.109 77.915L59.109 78.427Q59.109 78.630 58.992 78.788Q58.875 78.946 58.693 79.030Q58.512 79.114 58.309 79.114Q58.078 79.114 57.926 78.942Q57.773 78.770 57.742 78.540Q57.582 78.821 57.273 78.987Q56.965 79.153 56.613 79.153Q56.102 79.153 55.678 78.930Q55.254 78.708 55.254 78.243M55.941 78.243Q55.941 78.528 56.168 78.714Q56.395 78.899 56.688 78.899Q56.934 78.899 57.158 78.782Q57.383 78.665 57.518 78.462Q57.652 78.259 57.652 78.005L57.652 77.173Q57.387 77.173 57.102 77.227Q56.816 77.282 56.545 77.411Q56.273 77.540 56.107 77.747Q55.941 77.954 55.941 78.243M61.410 79.075L59.430 79.075L59.430 78.778Q59.699 78.778 59.867 78.733Q60.035 78.688 60.035 78.516L60.035 76.380Q60.035 76.165 59.973 76.069Q59.910 75.973 59.793 75.952Q59.676 75.930 59.430 75.930L59.430 75.634L60.598 75.548L60.598 76.333Q60.676 76.122 60.828 75.936Q60.980 75.751 61.180 75.649Q61.379 75.548 61.605 75.548Q61.852 75.548 62.043 75.692Q62.234 75.837 62.234 76.067Q62.234 76.223 62.129 76.333Q62.023 76.442 61.867 76.442Q61.711 76.442 61.602 76.333Q61.492 76.223 61.492 76.067Q61.492 75.907 61.598 75.802Q61.273 75.802 61.059 76.030Q60.844 76.259 60.748 76.598Q60.652 76.938 60.652 77.243L60.652 78.516Q60.652 78.684 60.879 78.731Q61.105 78.778 61.410 78.778L61.410 79.075M64.531 79.153Q64.051 79.153 63.643 78.909Q63.234 78.665 62.996 78.251Q62.758 77.837 62.758 77.348Q62.758 76.856 63.016 76.440Q63.273 76.024 63.705 75.786Q64.137 75.548 64.629 75.548Q65.250 75.548 65.699 75.985L65.699 74.356Q65.699 74.141 65.637 74.046Q65.574 73.950 65.457 73.929Q65.340 73.907 65.094 73.907L65.094 73.610L66.316 73.524L66.316 78.333Q66.316 78.544 66.379 78.639Q66.441 78.735 66.559 78.757Q66.676 78.778 66.926 78.778L66.926 79.075L65.676 79.153L65.676 78.669Q65.211 79.153 64.531 79.153M64.598 78.899Q64.938 78.899 65.231 78.708Q65.523 78.516 65.676 78.220L65.676 76.388Q65.527 76.114 65.266 75.958Q65.004 75.802 64.691 75.802Q64.066 75.802 63.783 76.249Q63.500 76.696 63.500 77.356Q63.500 78.001 63.752 78.450Q64.004 78.899 64.598 78.899\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(100.417 2.778)\">\u003Cpath d=\"M72.204 79.075L70.349 79.075L70.349 78.778Q70.622 78.778 70.790 78.731Q70.958 78.684 70.958 78.516L70.958 76.380Q70.958 76.165 70.895 76.069Q70.833 75.973 70.714 75.952Q70.595 75.930 70.349 75.930L70.349 75.634L71.540 75.548L71.540 76.282Q71.653 76.067 71.847 75.899Q72.040 75.731 72.278 75.639Q72.516 75.548 72.770 75.548Q73.731 75.548 73.907 76.259Q74.091 75.930 74.419 75.739Q74.747 75.548 75.126 75.548Q76.302 75.548 76.302 76.626L76.302 78.516Q76.302 78.684 76.470 78.731Q76.638 78.778 76.907 78.778L76.907 79.075L75.052 79.075L75.052 78.778Q75.325 78.778 75.493 78.733Q75.661 78.688 75.661 78.516L75.661 76.641Q75.661 76.255 75.536 76.028Q75.411 75.802 75.059 75.802Q74.755 75.802 74.499 75.964Q74.243 76.126 74.095 76.395Q73.946 76.665 73.946 76.962L73.946 78.516Q73.946 78.684 74.116 78.731Q74.286 78.778 74.556 78.778L74.556 79.075L72.700 79.075L72.700 78.778Q72.974 78.778 73.141 78.731Q73.309 78.684 73.309 78.516L73.309 76.641Q73.309 76.255 73.184 76.028Q73.059 75.802 72.708 75.802Q72.403 75.802 72.147 75.964Q71.891 76.126 71.743 76.395Q71.595 76.665 71.595 76.962L71.595 78.516Q71.595 78.684 71.765 78.731Q71.934 78.778 72.204 78.778L72.204 79.075M77.352 77.380Q77.352 76.876 77.608 76.444Q77.864 76.013 78.300 75.761Q78.735 75.509 79.235 75.509Q79.622 75.509 79.964 75.653Q80.306 75.798 80.567 76.059Q80.829 76.321 80.972 76.657Q81.114 76.993 81.114 77.380Q81.114 77.872 80.850 78.282Q80.587 78.692 80.157 78.923Q79.727 79.153 79.235 79.153Q78.743 79.153 78.309 78.921Q77.876 78.688 77.614 78.280Q77.352 77.872 77.352 77.380M79.235 78.876Q79.692 78.876 79.944 78.653Q80.196 78.430 80.284 78.079Q80.372 77.727 80.372 77.282Q80.372 76.852 80.278 76.514Q80.184 76.177 79.931 75.970Q79.677 75.763 79.235 75.763Q78.587 75.763 78.343 76.179Q78.099 76.595 78.099 77.282Q78.099 77.727 78.186 78.079Q78.274 78.430 78.526 78.653Q78.778 78.876 79.235 78.876\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(100.417 2.778)\">\u003Cpath d=\"M83.661 79.153Q83.180 79.153 82.772 78.909Q82.364 78.665 82.126 78.251Q81.887 77.837 81.887 77.348Q81.887 76.856 82.145 76.440Q82.403 76.024 82.835 75.786Q83.266 75.548 83.758 75.548Q84.379 75.548 84.829 75.985L84.829 74.356Q84.829 74.141 84.766 74.046Q84.704 73.950 84.586 73.929Q84.469 73.907 84.223 73.907L84.223 73.610L85.446 73.524L85.446 78.333Q85.446 78.544 85.508 78.639Q85.571 78.735 85.688 78.757Q85.805 78.778 86.055 78.778L86.055 79.075L84.805 79.153L84.805 78.669Q84.340 79.153 83.661 79.153M83.727 78.899Q84.067 78.899 84.360 78.708Q84.653 78.516 84.805 78.220L84.805 76.388Q84.657 76.114 84.395 75.958Q84.133 75.802 83.821 75.802Q83.196 75.802 82.913 76.249Q82.629 76.696 82.629 77.356Q82.629 78.001 82.881 78.450Q83.133 78.899 83.727 78.899M86.563 77.321Q86.563 76.841 86.795 76.425Q87.028 76.009 87.438 75.759Q87.848 75.509 88.325 75.509Q89.055 75.509 89.454 75.950Q89.852 76.391 89.852 77.122Q89.852 77.227 89.758 77.251L87.309 77.251L87.309 77.321Q87.309 77.731 87.430 78.087Q87.551 78.442 87.823 78.659Q88.094 78.876 88.524 78.876Q88.887 78.876 89.184 78.647Q89.481 78.419 89.583 78.067Q89.590 78.020 89.676 78.005L89.758 78.005Q89.852 78.032 89.852 78.114Q89.852 78.122 89.844 78.153Q89.782 78.380 89.643 78.563Q89.504 78.747 89.313 78.880Q89.122 79.013 88.903 79.083Q88.684 79.153 88.446 79.153Q88.075 79.153 87.737 79.016Q87.399 78.880 87.131 78.628Q86.864 78.376 86.713 78.036Q86.563 77.696 86.563 77.321M87.317 77.013L89.278 77.013Q89.278 76.708 89.176 76.417Q89.075 76.126 88.858 75.944Q88.641 75.763 88.325 75.763Q88.024 75.763 87.793 75.950Q87.563 76.138 87.440 76.429Q87.317 76.720 87.317 77.013M92.254 79.075L90.422 79.075L90.422 78.778Q90.696 78.778 90.864 78.731Q91.032 78.684 91.032 78.516L91.032 74.356Q91.032 74.141 90.969 74.046Q90.907 73.950 90.788 73.929Q90.668 73.907 90.422 73.907L90.422 73.610L91.645 73.524L91.645 78.516Q91.645 78.684 91.813 78.731Q91.981 78.778 92.254 78.778\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath d=\"M254.89 97.57h76.822V60.58H254.89Z\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(232.647 2.778)\">\u003Cpath d=\"M32.329 78.442Q32.520 78.716 32.876 78.843Q33.231 78.970 33.614 78.970Q33.950 78.970 34.159 78.784Q34.368 78.598 34.464 78.305Q34.559 78.013 34.559 77.700Q34.559 77.376 34.462 77.081Q34.364 76.786 34.151 76.602Q33.938 76.419 33.606 76.419L33.040 76.419Q33.009 76.419 32.979 76.389Q32.950 76.360 32.950 76.333L32.950 76.251Q32.950 76.216 32.979 76.190Q33.009 76.165 33.040 76.165L33.520 76.130Q33.806 76.130 34.003 75.925Q34.200 75.720 34.296 75.425Q34.391 75.130 34.391 74.852Q34.391 74.473 34.192 74.235Q33.993 73.997 33.614 73.997Q33.294 73.997 33.005 74.104Q32.716 74.212 32.552 74.434Q32.731 74.434 32.854 74.561Q32.977 74.688 32.977 74.860Q32.977 75.032 32.852 75.157Q32.727 75.282 32.552 75.282Q32.380 75.282 32.255 75.157Q32.130 75.032 32.130 74.860Q32.130 74.493 32.354 74.245Q32.579 73.997 32.919 73.876Q33.259 73.755 33.614 73.755Q33.962 73.755 34.325 73.876Q34.688 73.997 34.936 74.247Q35.184 74.497 35.184 74.852Q35.184 75.337 34.866 75.720Q34.548 76.102 34.071 76.274Q34.622 76.384 35.022 76.770Q35.423 77.157 35.423 77.692Q35.423 78.149 35.159 78.505Q34.895 78.860 34.474 79.052Q34.052 79.243 33.614 79.243Q33.204 79.243 32.811 79.108Q32.419 78.973 32.153 78.688Q31.888 78.403 31.888 77.985Q31.888 77.790 32.020 77.661Q32.153 77.532 32.345 77.532Q32.470 77.532 32.573 77.591Q32.677 77.649 32.739 77.755Q32.802 77.860 32.802 77.985Q32.802 78.180 32.667 78.311Q32.532 78.442 32.329 78.442M36.501 78.610Q36.501 78.427 36.638 78.290Q36.774 78.153 36.966 78.153Q37.157 78.153 37.290 78.286Q37.423 78.419 37.423 78.610Q37.423 78.809 37.290 78.942Q37.157 79.075 36.966 79.075Q36.774 79.075 36.638 78.938Q36.501 78.802 36.501 78.610\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(232.647 2.778)\">\u003Cpath d=\"M44.636 79.075L42.277 79.075L42.277 78.778Q42.601 78.778 42.843 78.731Q43.085 78.684 43.085 78.516L43.085 74.173Q43.085 74.001 42.843 73.954Q42.601 73.907 42.277 73.907L42.277 73.610L44.870 73.610Q45.202 73.610 45.589 73.696Q45.976 73.782 46.323 73.956Q46.671 74.130 46.890 74.411Q47.109 74.692 47.109 75.059Q47.109 75.384 46.907 75.649Q46.706 75.915 46.400 76.091Q46.093 76.266 45.765 76.356Q46.132 76.477 46.392 76.747Q46.652 77.016 46.702 77.380L46.796 78.075Q46.866 78.524 46.964 78.755Q47.062 78.985 47.359 78.985Q47.605 78.985 47.737 78.768Q47.870 78.552 47.870 78.290Q47.890 78.216 47.972 78.196L48.054 78.196Q48.148 78.220 48.148 78.313Q48.148 78.544 48.050 78.759Q47.952 78.973 47.775 79.108Q47.597 79.243 47.366 79.243Q46.769 79.243 46.351 78.956Q45.933 78.669 45.933 78.098L45.933 77.403Q45.933 77.122 45.780 76.907Q45.628 76.692 45.378 76.579Q45.128 76.466 44.855 76.466L43.827 76.466L43.827 78.516Q43.827 78.680 44.071 78.729Q44.316 78.778 44.636 78.778L44.636 79.075M43.827 74.173L43.827 76.212L44.757 76.212Q45.077 76.212 45.345 76.159Q45.612 76.106 45.812 75.979Q46.011 75.852 46.128 75.620Q46.245 75.388 46.245 75.059Q46.245 74.407 45.849 74.157Q45.452 73.907 44.757 73.907L44.230 73.907Q44.011 73.907 43.919 73.950Q43.827 73.993 43.827 74.173M52.890 79.075L48.507 79.075L48.507 78.778Q48.827 78.778 49.071 78.731Q49.316 78.684 49.316 78.516L49.316 74.173Q49.316 74.001 49.071 73.954Q48.827 73.907 48.507 73.907L48.507 73.610L51.093 73.610L51.093 73.907Q50.081 73.907 50.081 74.173L50.081 78.516Q50.081 78.688 50.173 78.733Q50.265 78.778 50.484 78.778L51.171 78.778Q51.644 78.778 51.952 78.653Q52.261 78.528 52.439 78.298Q52.616 78.067 52.708 77.741Q52.800 77.415 52.843 76.962L53.124 76.962\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(232.647 2.778)\">\u003Cpath d=\"M58.624 79.075L56.639 79.075L56.639 78.778Q56.913 78.778 57.081 78.731Q57.249 78.684 57.249 78.516L57.249 75.923L56.608 75.923L56.608 75.626L57.249 75.626L57.249 74.692Q57.249 74.427 57.366 74.190Q57.483 73.954 57.676 73.790Q57.870 73.626 58.118 73.534Q58.366 73.442 58.632 73.442Q58.917 73.442 59.141 73.600Q59.366 73.759 59.366 74.036Q59.366 74.192 59.260 74.302Q59.155 74.411 58.991 74.411Q58.835 74.411 58.725 74.302Q58.616 74.192 58.616 74.036Q58.616 73.829 58.776 73.723Q58.678 73.700 58.585 73.700Q58.354 73.700 58.182 73.856Q58.010 74.013 57.924 74.249Q57.839 74.485 57.839 74.708L57.839 75.626L58.807 75.626L58.807 75.923L57.862 75.923L57.862 78.516Q57.862 78.684 58.089 78.731Q58.315 78.778 58.624 78.778\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(232.647 2.778)\">\u003Cpath d=\"M61.647 79.075L59.869 79.075L59.869 78.778Q60.143 78.778 60.311 78.731Q60.479 78.684 60.479 78.516L60.479 76.380Q60.479 76.165 60.422 76.069Q60.365 75.973 60.252 75.952Q60.139 75.930 59.893 75.930L59.893 75.634L61.092 75.548L61.092 78.516Q61.092 78.684 61.238 78.731Q61.385 78.778 61.647 78.778L61.647 79.075M60.205 74.153Q60.205 73.962 60.340 73.831Q60.475 73.700 60.670 73.700Q60.791 73.700 60.895 73.763Q60.998 73.825 61.061 73.929Q61.123 74.032 61.123 74.153Q61.123 74.348 60.992 74.483Q60.861 74.618 60.670 74.618Q60.471 74.618 60.338 74.485Q60.205 74.352 60.205 74.153M64.076 79.075L62.221 79.075L62.221 78.778Q62.494 78.778 62.662 78.731Q62.830 78.684 62.830 78.516L62.830 76.380Q62.830 76.165 62.768 76.069Q62.705 75.973 62.586 75.952Q62.467 75.930 62.221 75.930L62.221 75.634L63.412 75.548L63.412 76.282Q63.526 76.067 63.719 75.899Q63.912 75.731 64.151 75.639Q64.389 75.548 64.643 75.548Q65.811 75.548 65.811 76.626L65.811 78.516Q65.811 78.684 65.981 78.731Q66.151 78.778 66.420 78.778L66.420 79.075L64.565 79.075L64.565 78.778Q64.838 78.778 65.006 78.731Q65.174 78.684 65.174 78.516L65.174 76.641Q65.174 76.259 65.053 76.030Q64.932 75.802 64.580 75.802Q64.268 75.802 64.014 75.964Q63.760 76.126 63.613 76.395Q63.467 76.665 63.467 76.962L63.467 78.516Q63.467 78.684 63.637 78.731Q63.807 78.778 64.076 78.778L64.076 79.075M66.865 77.321Q66.865 76.841 67.098 76.425Q67.330 76.009 67.740 75.759Q68.151 75.509 68.627 75.509Q69.358 75.509 69.756 75.950Q70.154 76.391 70.154 77.122Q70.154 77.227 70.061 77.251L67.612 77.251L67.612 77.321Q67.612 77.731 67.733 78.087Q67.854 78.442 68.125 78.659Q68.397 78.876 68.826 78.876Q69.190 78.876 69.487 78.647Q69.783 78.419 69.885 78.067Q69.893 78.020 69.979 78.005L70.061 78.005Q70.154 78.032 70.154 78.114Q70.154 78.122 70.147 78.153Q70.084 78.380 69.945 78.563Q69.807 78.747 69.615 78.880Q69.424 79.013 69.205 79.083Q68.987 79.153 68.748 79.153Q68.377 79.153 68.039 79.016Q67.701 78.880 67.434 78.628Q67.166 78.376 67.016 78.036Q66.865 77.696 66.865 77.321M67.619 77.013L69.580 77.013Q69.580 76.708 69.479 76.417Q69.377 76.126 69.160 75.944Q68.944 75.763 68.627 75.763Q68.326 75.763 68.096 75.950Q67.865 76.138 67.742 76.429Q67.619 76.720 67.619 77.013M72.756 77.626L70.502 77.626L70.502 77.075L72.756 77.075L72.756 77.626M74.100 78.114L74.100 75.923L73.397 75.923L73.397 75.669Q73.752 75.669 73.994 75.436Q74.237 75.204 74.348 74.856Q74.459 74.509 74.459 74.153L74.740 74.153L74.740 75.626L75.916 75.626L75.916 75.923L74.740 75.923L74.740 78.098Q74.740 78.419 74.860 78.647Q74.979 78.876 75.260 78.876Q75.440 78.876 75.557 78.753Q75.674 78.630 75.727 78.450Q75.779 78.270 75.779 78.098L75.779 77.626L76.061 77.626L76.061 78.114Q76.061 78.368 75.955 78.608Q75.850 78.848 75.653 79.001Q75.455 79.153 75.197 79.153Q74.881 79.153 74.629 79.030Q74.377 78.907 74.238 78.673Q74.100 78.438 74.100 78.114M77.463 78.122L77.463 76.380Q77.463 76.165 77.401 76.069Q77.338 75.973 77.219 75.952Q77.100 75.930 76.854 75.930L76.854 75.634L78.100 75.548L78.100 78.098L78.100 78.122Q78.100 78.434 78.154 78.596Q78.209 78.759 78.360 78.829Q78.510 78.899 78.830 78.899Q79.260 78.899 79.533 78.561Q79.807 78.223 79.807 77.778L79.807 76.380Q79.807 76.165 79.744 76.069Q79.682 75.973 79.563 75.952Q79.444 75.930 79.197 75.930L79.197 75.634L80.444 75.548L80.444 78.333Q80.444 78.544 80.506 78.639Q80.569 78.735 80.688 78.757Q80.807 78.778 81.053 78.778L81.053 79.075L79.830 79.153L79.830 78.532Q79.662 78.821 79.381 78.987Q79.100 79.153 78.779 79.153Q77.463 79.153 77.463 78.122M83.428 79.075L81.572 79.075L81.572 78.778Q81.846 78.778 82.014 78.731Q82.182 78.684 82.182 78.516L82.182 76.380Q82.182 76.165 82.119 76.069Q82.057 75.973 81.938 75.952Q81.819 75.930 81.572 75.930L81.572 75.634L82.764 75.548L82.764 76.282Q82.877 76.067 83.070 75.899Q83.264 75.731 83.502 75.639Q83.740 75.548 83.994 75.548Q85.162 75.548 85.162 76.626L85.162 78.516Q85.162 78.684 85.332 78.731Q85.502 78.778 85.772 78.778L85.772 79.075L83.916 79.075L83.916 78.778Q84.190 78.778 84.358 78.731Q84.526 78.684 84.526 78.516L84.526 76.641Q84.526 76.259 84.404 76.030Q84.283 75.802 83.932 75.802Q83.619 75.802 83.365 75.964Q83.112 76.126 82.965 76.395Q82.819 76.665 82.819 76.962L82.819 78.516Q82.819 78.684 82.988 78.731Q83.158 78.778 83.428 78.778L83.428 79.075M86.217 77.321Q86.217 76.841 86.449 76.425Q86.682 76.009 87.092 75.759Q87.502 75.509 87.979 75.509Q88.709 75.509 89.108 75.950Q89.506 76.391 89.506 77.122Q89.506 77.227 89.412 77.251L86.963 77.251L86.963 77.321Q86.963 77.731 87.084 78.087Q87.205 78.442 87.477 78.659Q87.748 78.876 88.178 78.876Q88.541 78.876 88.838 78.647Q89.135 78.419 89.237 78.067Q89.244 78.020 89.330 78.005L89.412 78.005Q89.506 78.032 89.506 78.114Q89.506 78.122 89.498 78.153Q89.436 78.380 89.297 78.563Q89.158 78.747 88.967 78.880Q88.776 79.013 88.557 79.083Q88.338 79.153 88.100 79.153Q87.729 79.153 87.391 79.016Q87.053 78.880 86.785 78.628Q86.518 78.376 86.367 78.036Q86.217 77.696 86.217 77.321M86.971 77.013L88.932 77.013Q88.932 76.708 88.830 76.417Q88.729 76.126 88.512 75.944Q88.295 75.763 87.979 75.763Q87.678 75.763 87.447 75.950Q87.217 76.138 87.094 76.429Q86.971 76.720 86.971 77.013\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M31.536 32.528v25.053\"\u002F>\u003Cpath stroke=\"none\" d=\"m31.536 60.181 2.08-4.16-2.08 1.56-2.08-1.56\"\u002F>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M162.419 32.528v25.053\"\u002F>\u003Cpath stroke=\"none\" d=\"m162.419 60.181 2.08-4.16-2.08 1.56-2.08-1.56\"\u002F>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M293.3 32.528v25.053\"\u002F>\u003Cpath stroke=\"none\" d=\"m293.3 60.181 2.08-4.16-2.08 1.56-2.079-1.56\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M70.347 79.075h50.66\"\u002F>\u003Cpath stroke=\"none\" d=\"m123.607 79.075-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003Cg transform=\"translate(60.862 8.594)\">\u003Cpath d=\"M33.600 79.075L31.864 79.075L31.864 78.795Q32.093 78.795 32.242 78.761Q32.390 78.726 32.390 78.586L32.390 76.737Q32.390 76.467 32.283 76.406Q32.175 76.344 31.864 76.344L31.864 76.064L32.893 75.989L32.893 76.696Q33.023 76.388 33.265 76.189Q33.508 75.989 33.826 75.989Q34.045 75.989 34.216 76.113Q34.387 76.238 34.387 76.450Q34.387 76.587 34.287 76.686Q34.188 76.785 34.055 76.785Q33.918 76.785 33.819 76.686Q33.720 76.587 33.720 76.450Q33.720 76.310 33.819 76.211Q33.529 76.211 33.329 76.407Q33.129 76.604 33.036 76.898Q32.944 77.192 32.944 77.472L32.944 78.586Q32.944 78.795 33.600 78.795L33.600 79.075M34.930 77.540Q34.930 77.219 35.055 76.930Q35.180 76.641 35.405 76.418Q35.631 76.194 35.926 76.074Q36.222 75.954 36.540 75.954Q36.868 75.954 37.130 76.054Q37.391 76.153 37.567 76.335Q37.743 76.518 37.837 76.776Q37.931 77.034 37.931 77.366Q37.931 77.458 37.849 77.479L35.593 77.479L35.593 77.540Q35.593 78.128 35.877 78.511Q36.161 78.894 36.728 78.894Q37.049 78.894 37.317 78.701Q37.586 78.508 37.675 78.193Q37.682 78.152 37.757 78.138L37.849 78.138Q37.931 78.162 37.931 78.234Q37.931 78.241 37.924 78.268Q37.811 78.665 37.441 78.904Q37.070 79.143 36.646 79.143Q36.208 79.143 35.808 78.935Q35.409 78.726 35.169 78.359Q34.930 77.992 34.930 77.540M35.600 77.270L37.415 77.270Q37.415 76.993 37.317 76.741Q37.220 76.488 37.022 76.332Q36.824 76.177 36.540 76.177Q36.263 76.177 36.049 76.335Q35.836 76.494 35.718 76.749Q35.600 77.004 35.600 77.270M40.317 79.075L38.584 79.075L38.584 78.795Q38.809 78.795 38.958 78.761Q39.107 78.726 39.107 78.586L39.107 76.337L38.519 76.337L38.519 76.057L39.107 76.057L39.107 75.240Q39.107 74.922 39.285 74.674Q39.462 74.427 39.753 74.286Q40.043 74.146 40.354 74.146Q40.611 74.146 40.814 74.288Q41.017 74.430 41.017 74.673Q41.017 74.809 40.918 74.908Q40.819 75.008 40.682 75.008Q40.546 75.008 40.447 74.908Q40.348 74.809 40.348 74.673Q40.348 74.492 40.488 74.399Q40.409 74.372 40.310 74.372Q40.101 74.372 39.948 74.505Q39.794 74.638 39.713 74.842Q39.633 75.045 39.633 75.254L39.633 76.057L40.522 76.057L40.522 76.337L39.661 76.337L39.661 78.586Q39.661 78.795 40.317 78.795\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M201.23 79.075h50.66\"\u002F>\u003Cpath stroke=\"none\" d=\"m254.49 79.075-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(184.473 8.594)\">\u003Cpath d=\"M33.600 79.075L31.864 79.075L31.864 78.795Q32.093 78.795 32.242 78.761Q32.390 78.726 32.390 78.586L32.390 76.737Q32.390 76.467 32.283 76.406Q32.175 76.344 31.864 76.344L31.864 76.064L32.893 75.989L32.893 76.696Q33.023 76.388 33.265 76.189Q33.508 75.989 33.826 75.989Q34.045 75.989 34.216 76.113Q34.387 76.238 34.387 76.450Q34.387 76.587 34.287 76.686Q34.188 76.785 34.055 76.785Q33.918 76.785 33.819 76.686Q33.720 76.587 33.720 76.450Q33.720 76.310 33.819 76.211Q33.529 76.211 33.329 76.407Q33.129 76.604 33.036 76.898Q32.944 77.192 32.944 77.472L32.944 78.586Q32.944 78.795 33.600 78.795L33.600 79.075M34.930 77.540Q34.930 77.219 35.055 76.930Q35.180 76.641 35.405 76.418Q35.631 76.194 35.926 76.074Q36.222 75.954 36.540 75.954Q36.868 75.954 37.130 76.054Q37.391 76.153 37.567 76.335Q37.743 76.518 37.837 76.776Q37.931 77.034 37.931 77.366Q37.931 77.458 37.849 77.479L35.593 77.479L35.593 77.540Q35.593 78.128 35.877 78.511Q36.161 78.894 36.728 78.894Q37.049 78.894 37.317 78.701Q37.586 78.508 37.675 78.193Q37.682 78.152 37.757 78.138L37.849 78.138Q37.931 78.162 37.931 78.234Q37.931 78.241 37.924 78.268Q37.811 78.665 37.441 78.904Q37.070 79.143 36.646 79.143Q36.208 79.143 35.808 78.935Q35.409 78.726 35.169 78.359Q34.930 77.992 34.930 77.540M35.600 77.270L37.415 77.270Q37.415 76.993 37.317 76.741Q37.220 76.488 37.022 76.332Q36.824 76.177 36.540 76.177Q36.263 76.177 36.049 76.335Q35.836 76.494 35.718 76.749Q35.600 77.004 35.600 77.270M39.907 79.048L38.926 76.549Q38.864 76.406 38.746 76.371Q38.628 76.337 38.413 76.337L38.413 76.057L39.893 76.057L39.893 76.337Q39.514 76.337 39.514 76.498Q39.514 76.508 39.527 76.549L40.242 78.381L40.915 76.676Q40.884 76.604 40.884 76.576Q40.884 76.549 40.857 76.549Q40.795 76.402 40.677 76.370Q40.559 76.337 40.348 76.337L40.348 76.057L41.745 76.057L41.745 76.337Q41.369 76.337 41.369 76.498Q41.369 76.529 41.376 76.549L42.132 78.487L42.819 76.737Q42.839 76.686 42.839 76.631Q42.839 76.491 42.726 76.414Q42.614 76.337 42.474 76.337L42.474 76.057L43.694 76.057L43.694 76.337Q43.489 76.337 43.333 76.443Q43.178 76.549 43.106 76.737L42.200 79.048Q42.166 79.143 42.053 79.143L41.985 79.143Q41.875 79.143 41.838 79.048L41.055 77.045L40.269 79.048Q40.235 79.143 40.122 79.143L40.054 79.143Q39.944 79.143 39.907 79.048\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(184.473 8.594)\">\u003Cpath d=\"M44.082 78.347Q44.082 78.015 44.305 77.788Q44.529 77.561 44.873 77.433Q45.216 77.304 45.589 77.252Q45.961 77.199 46.266 77.199L46.266 76.946Q46.266 76.741 46.158 76.561Q46.050 76.382 45.869 76.279Q45.688 76.177 45.480 76.177Q45.073 76.177 44.837 76.269Q44.926 76.306 44.972 76.390Q45.018 76.474 45.018 76.576Q45.018 76.672 44.972 76.751Q44.926 76.829 44.845 76.874Q44.765 76.918 44.676 76.918Q44.526 76.918 44.425 76.821Q44.324 76.723 44.324 76.576Q44.324 75.954 45.480 75.954Q45.691 75.954 45.941 76.018Q46.190 76.081 46.392 76.200Q46.594 76.320 46.720 76.505Q46.847 76.689 46.847 76.932L46.847 78.508Q46.847 78.624 46.908 78.720Q46.970 78.815 47.083 78.815Q47.192 78.815 47.257 78.721Q47.322 78.627 47.322 78.508L47.322 78.060L47.588 78.060L47.588 78.508Q47.588 78.778 47.361 78.943Q47.134 79.109 46.854 79.109Q46.645 79.109 46.508 78.955Q46.372 78.802 46.348 78.586Q46.201 78.853 45.919 78.998Q45.637 79.143 45.312 79.143Q45.035 79.143 44.751 79.068Q44.468 78.993 44.275 78.814Q44.082 78.634 44.082 78.347M44.697 78.347Q44.697 78.521 44.798 78.651Q44.898 78.781 45.054 78.851Q45.209 78.921 45.374 78.921Q45.592 78.921 45.801 78.824Q46.009 78.726 46.137 78.545Q46.266 78.364 46.266 78.138L46.266 77.410Q45.941 77.410 45.575 77.501Q45.209 77.592 44.953 77.804Q44.697 78.015 44.697 78.347M49.755 79.075L48.019 79.075L48.019 78.795Q48.248 78.795 48.397 78.761Q48.545 78.726 48.545 78.586L48.545 76.737Q48.545 76.467 48.438 76.406Q48.330 76.344 48.019 76.344L48.019 76.064L49.048 75.989L49.048 76.696Q49.178 76.388 49.420 76.189Q49.663 75.989 49.981 75.989Q50.200 75.989 50.371 76.113Q50.542 76.238 50.542 76.450Q50.542 76.587 50.442 76.686Q50.343 76.785 50.210 76.785Q50.073 76.785 49.974 76.686Q49.875 76.587 49.875 76.450Q49.875 76.310 49.974 76.211Q49.684 76.211 49.484 76.407Q49.284 76.604 49.191 76.898Q49.099 77.192 49.099 77.472L49.099 78.586Q49.099 78.795 49.755 78.795L49.755 79.075M51.126 77.564Q51.126 77.226 51.266 76.935Q51.406 76.645 51.651 76.431Q51.895 76.218 52.199 76.103Q52.503 75.989 52.828 75.989Q53.098 75.989 53.361 76.088Q53.625 76.187 53.816 76.365L53.816 74.967Q53.816 74.697 53.708 74.635Q53.601 74.574 53.290 74.574L53.290 74.293L54.366 74.218L54.366 78.402Q54.366 78.590 54.421 78.673Q54.476 78.757 54.576 78.776Q54.677 78.795 54.893 78.795L54.893 79.075L53.785 79.143L53.785 78.726Q53.368 79.143 52.743 79.143Q52.312 79.143 51.939 78.931Q51.567 78.720 51.346 78.359Q51.126 77.998 51.126 77.564M52.801 78.921Q53.009 78.921 53.196 78.849Q53.382 78.778 53.536 78.641Q53.689 78.504 53.785 78.326L53.785 76.717Q53.700 76.570 53.554 76.450Q53.409 76.330 53.240 76.271Q53.071 76.211 52.890 76.211Q52.329 76.211 52.061 76.600Q51.792 76.990 51.792 77.571Q51.792 78.142 52.027 78.532Q52.261 78.921 52.801 78.921\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg style=\"stroke-dasharray:3.0,3.0;stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M332.112 79.075C390.041 62.003 390.041-60.343 162.42-60.343S-65.203 62.003-9.77 78.34\"\u002F>\u003Cpath stroke=\"none\" d=\"m-7.275 79.075-3.402-3.17.908 2.435-2.084 1.554\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(93.597 -143.151)\">\u003Cpath d=\"M34.048 79.075L31.915 79.075L31.915 78.795Q32.637 78.795 32.637 78.586L32.637 74.785Q32.637 74.574 31.915 74.574L31.915 74.293L34.048 74.293L34.048 74.574Q33.327 74.574 33.327 74.785L33.327 77.011L35.696 74.973Q35.812 74.864 35.812 74.762Q35.812 74.669 35.735 74.621Q35.658 74.574 35.562 74.574L35.562 74.293L37.165 74.293L37.165 74.574Q36.878 74.574 36.601 74.679Q36.325 74.785 36.106 74.973L34.722 76.163L36.413 78.395Q36.550 78.573 36.660 78.658Q36.769 78.743 36.894 78.769Q37.018 78.795 37.268 78.795L37.268 79.075L35.385 79.075L35.385 78.795Q35.538 78.795 35.644 78.764Q35.750 78.733 35.750 78.620Q35.750 78.535 35.638 78.395L34.253 76.570L33.327 77.359L33.327 78.586Q33.327 78.795 34.048 78.795L34.048 79.075M42.002 79.075L38.034 79.075L38.034 78.795Q38.755 78.795 38.755 78.586L38.755 74.785Q38.755 74.574 38.034 74.574L38.034 74.293L40.348 74.293L40.348 74.574Q40.030 74.574 39.737 74.609Q39.445 74.645 39.445 74.785L39.445 78.586Q39.445 78.726 39.534 78.761Q39.623 78.795 39.811 78.795L40.433 78.795Q40.847 78.795 41.125 78.692Q41.404 78.590 41.569 78.388Q41.735 78.186 41.817 77.899Q41.899 77.612 41.944 77.205L42.210 77.205\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(93.597 -143.151)\">\u003Cpath d=\"M47.275 79.075L45.672 79.075L45.672 78.795Q45.898 78.795 46.047 78.761Q46.195 78.726 46.195 78.586L46.195 74.967Q46.195 74.697 46.088 74.635Q45.980 74.574 45.672 74.574L45.672 74.293L46.749 74.218L46.749 78.586Q46.749 78.723 46.899 78.759Q47.050 78.795 47.275 78.795L47.275 79.075M47.829 77.540Q47.829 77.219 47.954 76.930Q48.079 76.641 48.304 76.418Q48.530 76.194 48.825 76.074Q49.121 75.954 49.439 75.954Q49.767 75.954 50.029 76.054Q50.290 76.153 50.466 76.335Q50.642 76.518 50.736 76.776Q50.830 77.034 50.830 77.366Q50.830 77.458 50.748 77.479L48.492 77.479L48.492 77.540Q48.492 78.128 48.776 78.511Q49.060 78.894 49.627 78.894Q49.948 78.894 50.217 78.701Q50.485 78.508 50.574 78.193Q50.581 78.152 50.656 78.138L50.748 78.138Q50.830 78.162 50.830 78.234Q50.830 78.241 50.823 78.268Q50.710 78.665 50.340 78.904Q49.969 79.143 49.545 79.143Q49.107 79.143 48.708 78.935Q48.308 78.726 48.068 78.359Q47.829 77.992 47.829 77.540M48.499 77.270L50.314 77.270Q50.314 76.993 50.217 76.741Q50.119 76.488 49.921 76.332Q49.723 76.177 49.439 76.177Q49.162 76.177 48.949 76.335Q48.735 76.494 48.617 76.749Q48.499 77.004 48.499 77.270M51.476 78.347Q51.476 78.015 51.700 77.788Q51.924 77.561 52.267 77.433Q52.611 77.304 52.983 77.252Q53.356 77.199 53.660 77.199L53.660 76.946Q53.660 76.741 53.553 76.561Q53.445 76.382 53.264 76.279Q53.083 76.177 52.874 76.177Q52.467 76.177 52.231 76.269Q52.320 76.306 52.366 76.390Q52.413 76.474 52.413 76.576Q52.413 76.672 52.366 76.751Q52.320 76.829 52.240 76.874Q52.160 76.918 52.071 76.918Q51.920 76.918 51.820 76.821Q51.719 76.723 51.719 76.576Q51.719 75.954 52.874 75.954Q53.086 75.954 53.335 76.018Q53.585 76.081 53.787 76.200Q53.988 76.320 54.115 76.505Q54.241 76.689 54.241 76.932L54.241 78.508Q54.241 78.624 54.303 78.720Q54.364 78.815 54.477 78.815Q54.586 78.815 54.651 78.721Q54.716 78.627 54.716 78.508L54.716 78.060L54.983 78.060L54.983 78.508Q54.983 78.778 54.756 78.943Q54.528 79.109 54.248 79.109Q54.040 79.109 53.903 78.955Q53.766 78.802 53.742 78.586Q53.595 78.853 53.313 78.998Q53.031 79.143 52.707 79.143Q52.430 79.143 52.146 79.068Q51.862 78.993 51.669 78.814Q51.476 78.634 51.476 78.347M52.091 78.347Q52.091 78.521 52.192 78.651Q52.293 78.781 52.449 78.851Q52.604 78.921 52.768 78.921Q52.987 78.921 53.195 78.824Q53.404 78.726 53.532 78.545Q53.660 78.364 53.660 78.138L53.660 77.410Q53.335 77.410 52.970 77.501Q52.604 77.592 52.348 77.804Q52.091 78.015 52.091 78.347M55.400 79.068L55.400 78.005Q55.400 77.981 55.427 77.954Q55.455 77.927 55.479 77.927L55.588 77.927Q55.653 77.927 55.667 77.985Q55.762 78.419 56.008 78.670Q56.254 78.921 56.668 78.921Q57.010 78.921 57.263 78.788Q57.516 78.655 57.516 78.347Q57.516 78.190 57.422 78.075Q57.328 77.961 57.189 77.892Q57.051 77.824 56.883 77.786L56.302 77.687Q55.947 77.619 55.673 77.398Q55.400 77.178 55.400 76.836Q55.400 76.587 55.511 76.412Q55.622 76.238 55.808 76.139Q55.995 76.040 56.210 75.997Q56.425 75.954 56.668 75.954Q57.082 75.954 57.362 76.136L57.577 75.961Q57.587 75.958 57.594 75.956Q57.601 75.954 57.611 75.954L57.663 75.954Q57.690 75.954 57.714 75.978Q57.738 76.002 57.738 76.030L57.738 76.877Q57.738 76.898 57.714 76.925Q57.690 76.952 57.663 76.952L57.550 76.952Q57.522 76.952 57.497 76.927Q57.471 76.901 57.471 76.877Q57.471 76.641 57.365 76.477Q57.259 76.313 57.076 76.231Q56.894 76.149 56.661 76.149Q56.333 76.149 56.077 76.252Q55.820 76.354 55.820 76.631Q55.820 76.826 56.003 76.935Q56.186 77.045 56.415 77.086L56.989 77.192Q57.235 77.240 57.449 77.368Q57.663 77.496 57.799 77.699Q57.936 77.903 57.936 78.152Q57.936 78.665 57.570 78.904Q57.205 79.143 56.668 79.143Q56.172 79.143 55.841 78.849L55.574 79.123Q55.554 79.143 55.526 79.143L55.479 79.143Q55.455 79.143 55.427 79.116Q55.400 79.089 55.400 79.068M60.247 79.075L58.613 79.075L58.613 78.795Q58.842 78.795 58.991 78.761Q59.139 78.726 59.139 78.586L59.139 74.967Q59.139 74.697 59.032 74.635Q58.924 74.574 58.613 74.574L58.613 74.293L59.693 74.218L59.693 76.604Q59.799 76.419 59.977 76.277Q60.154 76.136 60.363 76.062Q60.571 75.989 60.797 75.989Q61.303 75.989 61.586 76.212Q61.870 76.436 61.870 76.932L61.870 78.586Q61.870 78.723 62.019 78.759Q62.168 78.795 62.393 78.795L62.393 79.075L60.763 79.075L60.763 78.795Q60.992 78.795 61.140 78.761Q61.289 78.726 61.289 78.586L61.289 76.946Q61.289 76.611 61.169 76.411Q61.050 76.211 60.735 76.211Q60.465 76.211 60.231 76.347Q59.997 76.484 59.859 76.718Q59.720 76.952 59.720 77.226L59.720 78.586Q59.720 78.723 59.871 78.759Q60.021 78.795 60.247 78.795\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(93.597 -143.151)\">\u003Cpath d=\"M66.219 78.234L66.219 76.337L65.580 76.337L65.580 76.115Q65.898 76.115 66.115 75.905Q66.332 75.695 66.432 75.385Q66.533 75.076 66.533 74.768L66.800 74.768L66.800 76.057L67.877 76.057L67.877 76.337L66.800 76.337L66.800 78.221Q66.800 78.497 66.904 78.696Q67.008 78.894 67.268 78.894Q67.425 78.894 67.531 78.790Q67.637 78.685 67.687 78.532Q67.736 78.378 67.736 78.221L67.736 77.807L68.003 77.807L68.003 78.234Q68.003 78.460 67.904 78.670Q67.805 78.880 67.620 79.012Q67.436 79.143 67.207 79.143Q66.769 79.143 66.494 78.906Q66.219 78.668 66.219 78.234M68.772 77.592Q68.772 77.250 68.907 76.951Q69.042 76.652 69.281 76.428Q69.521 76.204 69.838 76.079Q70.156 75.954 70.488 75.954Q70.932 75.954 71.332 76.170Q71.732 76.385 71.966 76.763Q72.200 77.140 72.200 77.592Q72.200 77.933 72.058 78.217Q71.917 78.501 71.672 78.708Q71.428 78.914 71.118 79.029Q70.809 79.143 70.488 79.143Q70.057 79.143 69.656 78.942Q69.254 78.740 69.013 78.388Q68.772 78.036 68.772 77.592M70.488 78.894Q71.089 78.894 71.313 78.516Q71.537 78.138 71.537 77.506Q71.537 76.894 71.303 76.535Q71.069 76.177 70.488 76.177Q69.435 76.177 69.435 77.506Q69.435 78.138 69.661 78.516Q69.886 78.894 70.488 78.894\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(93.597 -143.151)\">\u003Cpath d=\"M77.247 79.075L75.511 79.075L75.511 78.795Q75.740 78.795 75.889 78.761Q76.037 78.726 76.037 78.586L76.037 76.737Q76.037 76.467 75.930 76.406Q75.822 76.344 75.511 76.344L75.511 76.064L76.540 75.989L76.540 76.696Q76.670 76.388 76.912 76.189Q77.155 75.989 77.473 75.989Q77.692 75.989 77.863 76.113Q78.034 76.238 78.034 76.450Q78.034 76.587 77.934 76.686Q77.835 76.785 77.702 76.785Q77.565 76.785 77.466 76.686Q77.367 76.587 77.367 76.450Q77.367 76.310 77.466 76.211Q77.176 76.211 76.976 76.407Q76.776 76.604 76.683 76.898Q76.591 77.192 76.591 77.472L76.591 78.586Q76.591 78.795 77.247 78.795L77.247 79.075M78.577 77.540Q78.577 77.219 78.702 76.930Q78.827 76.641 79.052 76.418Q79.278 76.194 79.573 76.074Q79.869 75.954 80.187 75.954Q80.515 75.954 80.777 76.054Q81.038 76.153 81.214 76.335Q81.390 76.518 81.484 76.776Q81.578 77.034 81.578 77.366Q81.578 77.458 81.496 77.479L79.240 77.479L79.240 77.540Q79.240 78.128 79.524 78.511Q79.808 78.894 80.375 78.894Q80.696 78.894 80.964 78.701Q81.233 78.508 81.322 78.193Q81.329 78.152 81.404 78.138L81.496 78.138Q81.578 78.162 81.578 78.234Q81.578 78.241 81.571 78.268Q81.458 78.665 81.088 78.904Q80.717 79.143 80.293 79.143Q79.855 79.143 79.455 78.935Q79.056 78.726 78.816 78.359Q78.577 77.992 78.577 77.540M79.247 77.270L81.062 77.270Q81.062 76.993 80.964 76.741Q80.867 76.488 80.669 76.332Q80.471 76.177 80.187 76.177Q79.910 76.177 79.696 76.335Q79.483 76.494 79.365 76.749Q79.247 77.004 79.247 77.270M83.964 79.075L82.231 79.075L82.231 78.795Q82.456 78.795 82.605 78.761Q82.754 78.726 82.754 78.586L82.754 76.337L82.166 76.337L82.166 76.057L82.754 76.057L82.754 75.240Q82.754 74.922 82.932 74.674Q83.109 74.427 83.400 74.286Q83.690 74.146 84.001 74.146Q84.258 74.146 84.461 74.288Q84.664 74.430 84.664 74.673Q84.664 74.809 84.565 74.908Q84.466 75.008 84.329 75.008Q84.193 75.008 84.094 74.908Q83.995 74.809 83.995 74.673Q83.995 74.492 84.135 74.399Q84.056 74.372 83.957 74.372Q83.748 74.372 83.595 74.505Q83.441 74.638 83.360 74.842Q83.280 75.045 83.280 75.254L83.280 76.057L84.169 76.057L84.169 76.337L83.308 76.337L83.308 78.586Q83.308 78.795 83.964 78.795L83.964 79.075M84.603 77.540Q84.603 77.219 84.728 76.930Q84.852 76.641 85.078 76.418Q85.304 76.194 85.599 76.074Q85.895 75.954 86.213 75.954Q86.541 75.954 86.802 76.054Q87.064 76.153 87.240 76.335Q87.416 76.518 87.510 76.776Q87.604 77.034 87.604 77.366Q87.604 77.458 87.522 77.479L85.266 77.479L85.266 77.540Q85.266 78.128 85.550 78.511Q85.833 78.894 86.401 78.894Q86.722 78.894 86.990 78.701Q87.259 78.508 87.348 78.193Q87.354 78.152 87.430 78.138L87.522 78.138Q87.604 78.162 87.604 78.234Q87.604 78.241 87.597 78.268Q87.484 78.665 87.113 78.904Q86.743 79.143 86.319 79.143Q85.881 79.143 85.481 78.935Q85.081 78.726 84.842 78.359Q84.603 77.992 84.603 77.540M85.273 77.270L87.088 77.270Q87.088 76.993 86.990 76.741Q86.893 76.488 86.695 76.332Q86.496 76.177 86.213 76.177Q85.936 76.177 85.722 76.335Q85.509 76.494 85.391 76.749Q85.273 77.004 85.273 77.270M89.942 79.075L88.205 79.075L88.205 78.795Q88.434 78.795 88.583 78.761Q88.732 78.726 88.732 78.586L88.732 76.737Q88.732 76.467 88.624 76.406Q88.516 76.344 88.205 76.344L88.205 76.064L89.234 75.989L89.234 76.696Q89.364 76.388 89.607 76.189Q89.850 75.989 90.167 75.989Q90.386 75.989 90.557 76.113Q90.728 76.238 90.728 76.450Q90.728 76.587 90.629 76.686Q90.530 76.785 90.396 76.785Q90.260 76.785 90.161 76.686Q90.061 76.587 90.061 76.450Q90.061 76.310 90.161 76.211Q89.870 76.211 89.670 76.407Q89.470 76.604 89.378 76.898Q89.286 77.192 89.286 77.472L89.286 78.586Q89.286 78.795 89.942 78.795L89.942 79.075M91.271 77.540Q91.271 77.219 91.396 76.930Q91.521 76.641 91.746 76.418Q91.972 76.194 92.268 76.074Q92.563 75.954 92.881 75.954Q93.209 75.954 93.471 76.054Q93.732 76.153 93.908 76.335Q94.084 76.518 94.178 76.776Q94.272 77.034 94.272 77.366Q94.272 77.458 94.190 77.479L91.934 77.479L91.934 77.540Q91.934 78.128 92.218 78.511Q92.502 78.894 93.069 78.894Q93.391 78.894 93.659 78.701Q93.927 78.508 94.016 78.193Q94.023 78.152 94.098 78.138L94.190 78.138Q94.272 78.162 94.272 78.234Q94.272 78.241 94.266 78.268Q94.153 78.665 93.782 78.904Q93.411 79.143 92.987 79.143Q92.550 79.143 92.150 78.935Q91.750 78.726 91.511 78.359Q91.271 77.992 91.271 77.540M91.941 77.270L93.756 77.270Q93.756 76.993 93.659 76.741Q93.561 76.488 93.363 76.332Q93.165 76.177 92.881 76.177Q92.604 76.177 92.391 76.335Q92.177 76.494 92.059 76.749Q91.941 77.004 91.941 77.270M96.542 79.075L94.908 79.075L94.908 78.795Q95.137 78.795 95.286 78.761Q95.434 78.726 95.434 78.586L95.434 76.737Q95.434 76.467 95.327 76.406Q95.219 76.344 94.908 76.344L94.908 76.064L95.968 75.989L95.968 76.638Q96.139 76.330 96.443 76.159Q96.747 75.989 97.092 75.989Q97.598 75.989 97.882 76.212Q98.165 76.436 98.165 76.932L98.165 78.586Q98.165 78.723 98.314 78.759Q98.463 78.795 98.688 78.795L98.688 79.075L97.058 79.075L97.058 78.795Q97.287 78.795 97.436 78.761Q97.584 78.726 97.584 78.586L97.584 76.946Q97.584 76.611 97.465 76.411Q97.345 76.211 97.031 76.211Q96.761 76.211 96.527 76.347Q96.292 76.484 96.154 76.718Q96.016 76.952 96.016 77.226L96.016 78.586Q96.016 78.723 96.166 78.759Q96.316 78.795 96.542 78.795L96.542 79.075M99.276 77.564Q99.276 77.236 99.411 76.935Q99.546 76.635 99.782 76.414Q100.018 76.194 100.322 76.074Q100.626 75.954 100.951 75.954Q101.457 75.954 101.806 76.057Q102.154 76.159 102.154 76.535Q102.154 76.682 102.057 76.783Q101.959 76.884 101.812 76.884Q101.659 76.884 101.559 76.785Q101.460 76.686 101.460 76.535Q101.460 76.347 101.600 76.255Q101.399 76.204 100.958 76.204Q100.602 76.204 100.373 76.400Q100.144 76.597 100.044 76.906Q99.943 77.216 99.943 77.564Q99.943 77.913 100.069 78.219Q100.196 78.525 100.450 78.709Q100.705 78.894 101.060 78.894Q101.283 78.894 101.467 78.810Q101.652 78.726 101.787 78.571Q101.922 78.415 101.980 78.207Q101.994 78.152 102.048 78.152L102.161 78.152Q102.192 78.152 102.214 78.176Q102.236 78.200 102.236 78.234L102.236 78.255Q102.151 78.542 101.963 78.740Q101.775 78.938 101.510 79.041Q101.245 79.143 100.951 79.143Q100.520 79.143 100.132 78.937Q99.745 78.730 99.510 78.367Q99.276 78.005 99.276 77.564M102.783 77.540Q102.783 77.219 102.908 76.930Q103.033 76.641 103.258 76.418Q103.484 76.194 103.779 76.074Q104.075 75.954 104.393 75.954Q104.721 75.954 104.983 76.054Q105.244 76.153 105.420 76.335Q105.596 76.518 105.690 76.776Q105.784 77.034 105.784 77.366Q105.784 77.458 105.702 77.479L103.446 77.479L103.446 77.540Q103.446 78.128 103.730 78.511Q104.014 78.894 104.581 78.894Q104.902 78.894 105.171 78.701Q105.439 78.508 105.528 78.193Q105.535 78.152 105.610 78.138L105.702 78.138Q105.784 78.162 105.784 78.234Q105.784 78.241 105.777 78.268Q105.664 78.665 105.294 78.904Q104.923 79.143 104.499 79.143Q104.061 79.143 103.662 78.935Q103.262 78.726 103.022 78.359Q102.783 77.992 102.783 77.540M103.453 77.270L105.268 77.270Q105.268 76.993 105.171 76.741Q105.073 76.488 104.875 76.332Q104.677 76.177 104.393 76.177Q104.116 76.177 103.902 76.335Q103.689 76.494 103.571 76.749Q103.453 77.004 103.453 77.270\" fill=\"currentColor\" stroke=\"currentColor\" 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-stage RLHF pipeline. SFT yields the reference policy; the reward model learns from human comparisons; RL fine-tunes the policy under a KL leash.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:341.474px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 256.106 100.487\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-55.803 15.773h185.188\"\u002F>\u003Cpath stroke=\"none\" d=\"m131.985 15.773-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(100.472 1.75)\">\u003Cpath d=\"M37.310 15.773L35.574 15.773L35.574 15.493Q35.803 15.493 35.952 15.459Q36.100 15.424 36.100 15.284L36.100 13.435Q36.100 13.165 35.993 13.104Q35.885 13.042 35.574 13.042L35.574 12.762L36.603 12.687L36.603 13.394Q36.733 13.086 36.975 12.887Q37.218 12.687 37.536 12.687Q37.755 12.687 37.926 12.811Q38.097 12.936 38.097 13.148Q38.097 13.285 37.997 13.384Q37.898 13.483 37.765 13.483Q37.628 13.483 37.529 13.384Q37.430 13.285 37.430 13.148Q37.430 13.008 37.529 12.909Q37.239 12.909 37.039 13.105Q36.839 13.302 36.746 13.596Q36.654 13.890 36.654 14.170L36.654 15.284Q36.654 15.493 37.310 15.493L37.310 15.773M38.640 14.238Q38.640 13.917 38.765 13.628Q38.890 13.339 39.115 13.116Q39.341 12.892 39.636 12.772Q39.932 12.652 40.250 12.652Q40.578 12.652 40.840 12.752Q41.101 12.851 41.277 13.033Q41.453 13.216 41.547 13.474Q41.641 13.732 41.641 14.064Q41.641 14.156 41.559 14.177L39.303 14.177L39.303 14.238Q39.303 14.826 39.587 15.209Q39.871 15.592 40.438 15.592Q40.759 15.592 41.027 15.399Q41.296 15.206 41.385 14.891Q41.392 14.850 41.467 14.836L41.559 14.836Q41.641 14.860 41.641 14.932Q41.641 14.939 41.634 14.966Q41.521 15.363 41.151 15.602Q40.780 15.841 40.356 15.841Q39.918 15.841 39.518 15.633Q39.119 15.424 38.879 15.057Q38.640 14.690 38.640 14.238M39.310 13.968L41.125 13.968Q41.125 13.691 41.027 13.439Q40.930 13.186 40.732 13.030Q40.534 12.875 40.250 12.875Q39.973 12.875 39.759 13.033Q39.546 13.192 39.428 13.447Q39.310 13.702 39.310 13.968M43.617 15.746L42.636 13.247Q42.574 13.104 42.456 13.069Q42.338 13.035 42.123 13.035L42.123 12.755L43.603 12.755L43.603 13.035Q43.224 13.035 43.224 13.196Q43.224 13.206 43.237 13.247L43.952 15.079L44.625 13.374Q44.594 13.302 44.594 13.274Q44.594 13.247 44.567 13.247Q44.505 13.100 44.387 13.068Q44.269 13.035 44.058 13.035L44.058 12.755L45.455 12.755L45.455 13.035Q45.079 13.035 45.079 13.196Q45.079 13.227 45.086 13.247L45.842 15.185L46.529 13.435Q46.549 13.384 46.549 13.329Q46.549 13.189 46.436 13.112Q46.324 13.035 46.184 13.035L46.184 12.755L47.404 12.755L47.404 13.035Q47.199 13.035 47.043 13.141Q46.888 13.247 46.816 13.435L45.910 15.746Q45.876 15.841 45.763 15.841L45.695 15.841Q45.585 15.841 45.548 15.746L44.765 13.743L43.979 15.746Q43.945 15.841 43.832 15.841L43.764 15.841Q43.654 15.841 43.617 15.746\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(100.472 1.75)\">\u003Cpath d=\"M47.793 15.045Q47.793 14.713 48.016 14.486Q48.240 14.259 48.584 14.131Q48.927 14.002 49.300 13.950Q49.672 13.897 49.977 13.897L49.977 13.644Q49.977 13.439 49.869 13.259Q49.761 13.080 49.580 12.977Q49.399 12.875 49.191 12.875Q48.784 12.875 48.548 12.967Q48.637 13.004 48.683 13.088Q48.729 13.172 48.729 13.274Q48.729 13.370 48.683 13.449Q48.637 13.527 48.556 13.572Q48.476 13.616 48.387 13.616Q48.237 13.616 48.136 13.519Q48.035 13.421 48.035 13.274Q48.035 12.652 49.191 12.652Q49.402 12.652 49.652 12.716Q49.901 12.779 50.103 12.898Q50.305 13.018 50.431 13.203Q50.558 13.387 50.558 13.630L50.558 15.206Q50.558 15.322 50.619 15.418Q50.681 15.513 50.794 15.513Q50.903 15.513 50.968 15.419Q51.033 15.325 51.033 15.206L51.033 14.758L51.299 14.758L51.299 15.206Q51.299 15.476 51.072 15.641Q50.845 15.807 50.565 15.807Q50.356 15.807 50.219 15.653Q50.083 15.500 50.059 15.284Q49.912 15.551 49.630 15.696Q49.348 15.841 49.023 15.841Q48.746 15.841 48.462 15.766Q48.179 15.691 47.986 15.512Q47.793 15.332 47.793 15.045M48.408 15.045Q48.408 15.219 48.509 15.349Q48.609 15.479 48.765 15.549Q48.920 15.619 49.085 15.619Q49.303 15.619 49.512 15.522Q49.720 15.424 49.848 15.243Q49.977 15.062 49.977 14.836L49.977 14.108Q49.652 14.108 49.286 14.199Q48.920 14.290 48.664 14.502Q48.408 14.713 48.408 15.045M53.466 15.773L51.730 15.773L51.730 15.493Q51.959 15.493 52.108 15.459Q52.256 15.424 52.256 15.284L52.256 13.435Q52.256 13.165 52.149 13.104Q52.041 13.042 51.730 13.042L51.730 12.762L52.759 12.687L52.759 13.394Q52.889 13.086 53.131 12.887Q53.374 12.687 53.692 12.687Q53.911 12.687 54.082 12.811Q54.253 12.936 54.253 13.148Q54.253 13.285 54.153 13.384Q54.054 13.483 53.921 13.483Q53.784 13.483 53.685 13.384Q53.586 13.285 53.586 13.148Q53.586 13.008 53.685 12.909Q53.395 12.909 53.195 13.105Q52.995 13.302 52.902 13.596Q52.810 13.890 52.810 14.170L52.810 15.284Q52.810 15.493 53.466 15.493L53.466 15.773M54.837 14.262Q54.837 13.924 54.977 13.633Q55.117 13.343 55.362 13.129Q55.606 12.916 55.910 12.801Q56.214 12.687 56.539 12.687Q56.809 12.687 57.072 12.786Q57.336 12.885 57.527 13.063L57.527 11.665Q57.527 11.395 57.419 11.333Q57.312 11.272 57.001 11.272L57.001 10.991L58.077 10.916L58.077 15.100Q58.077 15.288 58.132 15.371Q58.187 15.455 58.287 15.474Q58.388 15.493 58.604 15.493L58.604 15.773L57.496 15.841L57.496 15.424Q57.079 15.841 56.454 15.841Q56.023 15.841 55.650 15.629Q55.278 15.418 55.057 15.057Q54.837 14.696 54.837 14.262M56.512 15.619Q56.720 15.619 56.907 15.547Q57.093 15.476 57.247 15.339Q57.400 15.202 57.496 15.024L57.496 13.415Q57.411 13.268 57.265 13.148Q57.120 13.028 56.951 12.969Q56.782 12.909 56.601 12.909Q56.040 12.909 55.772 13.298Q55.503 13.688 55.503 14.269Q55.503 14.840 55.738 15.230Q55.972 15.619 56.512 15.619\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(100.472 1.75)\">\u003Cpath d=\"M61.915 16.306Q61.915 16.060 62.112 15.876Q62.309 15.691 62.565 15.612Q62.428 15.500 62.356 15.339Q62.285 15.178 62.285 14.997Q62.285 14.676 62.496 14.430Q62.162 14.132 62.162 13.722Q62.162 13.261 62.551 12.974Q62.941 12.687 63.419 12.687Q63.891 12.687 64.226 12.933Q64.400 12.779 64.611 12.697Q64.821 12.615 65.050 12.615Q65.214 12.615 65.335 12.722Q65.456 12.830 65.456 12.994Q65.456 13.090 65.385 13.162Q65.313 13.233 65.221 13.233Q65.121 13.233 65.051 13.160Q64.981 13.086 64.981 12.987Q64.981 12.933 64.995 12.902L65.002 12.888Q65.009 12.868 65.017 12.857Q65.026 12.847 65.029 12.840Q64.674 12.840 64.387 13.063Q64.674 13.356 64.674 13.722Q64.674 14.037 64.489 14.269Q64.305 14.502 64.016 14.630Q63.727 14.758 63.419 14.758Q63.218 14.758 63.026 14.708Q62.835 14.659 62.657 14.549Q62.565 14.676 62.565 14.819Q62.565 15.001 62.693 15.136Q62.821 15.271 63.006 15.271L63.638 15.271Q64.086 15.271 64.455 15.342Q64.824 15.414 65.084 15.643Q65.344 15.872 65.344 16.306Q65.344 16.627 65.048 16.829Q64.752 17.031 64.349 17.120Q63.946 17.209 63.631 17.209Q63.313 17.209 62.910 17.120Q62.507 17.031 62.211 16.829Q61.915 16.627 61.915 16.306M62.370 16.306Q62.370 16.535 62.589 16.684Q62.808 16.833 63.100 16.901Q63.392 16.969 63.631 16.969Q63.795 16.969 64.004 16.933Q64.212 16.898 64.419 16.817Q64.626 16.737 64.757 16.609Q64.889 16.481 64.889 16.306Q64.889 15.954 64.508 15.860Q64.127 15.766 63.624 15.766L63.006 15.766Q62.767 15.766 62.568 15.917Q62.370 16.067 62.370 16.306M63.419 14.519Q64.086 14.519 64.086 13.722Q64.086 12.922 63.419 12.922Q62.749 12.922 62.749 13.722Q62.749 14.519 63.419 14.519M65.996 15.045Q65.996 14.713 66.220 14.486Q66.444 14.259 66.788 14.131Q67.131 14.002 67.504 13.950Q67.876 13.897 68.181 13.897L68.181 13.644Q68.181 13.439 68.073 13.259Q67.965 13.080 67.784 12.977Q67.603 12.875 67.394 12.875Q66.988 12.875 66.752 12.967Q66.841 13.004 66.887 13.088Q66.933 13.172 66.933 13.274Q66.933 13.370 66.887 13.449Q66.841 13.527 66.760 13.572Q66.680 13.616 66.591 13.616Q66.441 13.616 66.340 13.519Q66.239 13.421 66.239 13.274Q66.239 12.652 67.394 12.652Q67.606 12.652 67.856 12.716Q68.105 12.779 68.307 12.898Q68.509 13.018 68.635 13.203Q68.762 13.387 68.762 13.630L68.762 15.206Q68.762 15.322 68.823 15.418Q68.885 15.513 68.997 15.513Q69.107 15.513 69.172 15.419Q69.237 15.325 69.237 15.206L69.237 14.758L69.503 14.758L69.503 15.206Q69.503 15.476 69.276 15.641Q69.049 15.807 68.768 15.807Q68.560 15.807 68.423 15.653Q68.287 15.500 68.263 15.284Q68.116 15.551 67.834 15.696Q67.552 15.841 67.227 15.841Q66.950 15.841 66.666 15.766Q66.383 15.691 66.190 15.512Q65.996 15.332 65.996 15.045M66.612 15.045Q66.612 15.219 66.713 15.349Q66.813 15.479 66.969 15.549Q67.124 15.619 67.288 15.619Q67.507 15.619 67.716 15.522Q67.924 15.424 68.052 15.243Q68.181 15.062 68.181 14.836L68.181 14.108Q67.856 14.108 67.490 14.199Q67.124 14.290 66.868 14.502Q66.612 14.713 66.612 15.045M71.564 17.130L69.934 17.130L69.934 16.850Q70.163 16.850 70.312 16.815Q70.460 16.781 70.460 16.641L70.460 13.295Q70.460 13.124 70.324 13.083Q70.187 13.042 69.934 13.042L69.934 12.762L71.014 12.687L71.014 13.093Q71.236 12.892 71.523 12.789Q71.810 12.687 72.118 12.687Q72.545 12.687 72.909 12.900Q73.273 13.114 73.487 13.478Q73.701 13.842 73.701 14.262Q73.701 14.707 73.461 15.071Q73.222 15.435 72.829 15.638Q72.436 15.841 71.992 15.841Q71.725 15.841 71.477 15.741Q71.229 15.640 71.041 15.459L71.041 16.641Q71.041 16.778 71.190 16.814Q71.339 16.850 71.564 16.850L71.564 17.130M71.041 13.442L71.041 15.052Q71.175 15.305 71.417 15.462Q71.660 15.619 71.937 15.619Q72.265 15.619 72.518 15.418Q72.771 15.216 72.904 14.898Q73.038 14.580 73.038 14.262Q73.038 14.033 72.973 13.804Q72.908 13.575 72.779 13.377Q72.651 13.179 72.456 13.059Q72.262 12.940 72.029 12.940Q71.735 12.940 71.467 13.069Q71.199 13.199 71.041 13.442\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M35.246 21.464v-77.068\"\u002F>\u003Cpath stroke=\"none\" d=\"m35.246-58.204-2.08 4.16 2.08-1.56 2.08 1.56\"\u002F>\u003Cg transform=\"translate(-12.007 -79.46)\">\u003Cpath d=\"M37.772 15.773L35.639 15.773L35.639 15.493Q36.360 15.493 36.360 15.284L36.360 11.483Q36.360 11.272 35.639 11.272L35.639 10.991L38.305 10.991Q38.715 10.991 39.136 11.145Q39.556 11.299 39.840 11.603Q40.123 11.907 40.123 12.321Q40.123 12.639 39.956 12.885Q39.788 13.131 39.512 13.297Q39.235 13.462 38.913 13.546Q38.592 13.630 38.305 13.630L37.051 13.630L37.051 15.284Q37.051 15.493 37.772 15.493L37.772 15.773M37.023 11.483L37.023 13.380L38.110 13.380Q38.719 13.380 39.033 13.143Q39.348 12.905 39.348 12.321Q39.348 11.928 39.202 11.694Q39.057 11.460 38.785 11.366Q38.514 11.272 38.110 11.272L37.389 11.272Q37.201 11.272 37.112 11.306Q37.023 11.340 37.023 11.483M43.059 17.523Q42.509 17.123 42.138 16.568Q41.767 16.012 41.586 15.366Q41.405 14.720 41.405 14.023Q41.405 13.510 41.506 13.015Q41.607 12.519 41.812 12.068Q42.017 11.617 42.330 11.225Q42.642 10.834 43.059 10.530Q43.070 10.526 43.077 10.525Q43.083 10.523 43.094 10.523L43.162 10.523Q43.196 10.523 43.218 10.547Q43.241 10.571 43.241 10.608Q43.241 10.653 43.213 10.670Q42.865 10.971 42.612 11.355Q42.359 11.740 42.207 12.181Q42.055 12.622 41.983 13.078Q41.911 13.534 41.911 14.023Q41.911 15.024 42.220 15.911Q42.530 16.798 43.213 17.383Q43.241 17.400 43.241 17.444Q43.241 17.482 43.218 17.506Q43.196 17.530 43.162 17.530L43.094 17.530Q43.087 17.526 43.078 17.525Q43.070 17.523 43.059 17.523M45.438 15.746L44.457 13.247Q44.396 13.104 44.278 13.069Q44.160 13.035 43.945 13.035L43.945 12.755L45.425 12.755L45.425 13.035Q45.045 13.035 45.045 13.196Q45.045 13.206 45.059 13.247L45.773 15.079L46.447 13.374Q46.416 13.302 46.416 13.274Q46.416 13.247 46.389 13.247Q46.327 13.100 46.209 13.068Q46.091 13.035 45.879 13.035L45.879 12.755L47.277 12.755L47.277 13.035Q46.901 13.035 46.901 13.196Q46.901 13.227 46.908 13.247L47.663 15.185L48.350 13.435Q48.371 13.384 48.371 13.329Q48.371 13.189 48.258 13.112Q48.145 13.035 48.005 13.035L48.005 12.755L49.225 12.755L49.225 13.035Q49.020 13.035 48.865 13.141Q48.709 13.247 48.638 13.435L47.732 15.746Q47.698 15.841 47.585 15.841L47.517 15.841Q47.407 15.841 47.370 15.746L46.587 13.743L45.801 15.746Q45.767 15.841 45.654 15.841L45.585 15.841Q45.476 15.841 45.438 15.746M51.372 15.773L49.820 15.773L49.820 15.493Q50.046 15.493 50.194 15.459Q50.343 15.424 50.343 15.284L50.343 13.435Q50.343 13.247 50.295 13.163Q50.247 13.080 50.150 13.061Q50.053 13.042 49.841 13.042L49.841 12.762L50.897 12.687L50.897 15.284Q50.897 15.424 51.028 15.459Q51.160 15.493 51.372 15.493L51.372 15.773M50.100 11.466Q50.100 11.295 50.224 11.176Q50.347 11.056 50.517 11.056Q50.685 11.056 50.808 11.176Q50.931 11.295 50.931 11.466Q50.931 11.641 50.808 11.764Q50.685 11.887 50.517 11.887Q50.347 11.887 50.224 11.764Q50.100 11.641 50.100 11.466M53.700 15.773L52.066 15.773L52.066 15.493Q52.295 15.493 52.444 15.459Q52.592 15.424 52.592 15.284L52.592 13.435Q52.592 13.165 52.485 13.104Q52.377 13.042 52.066 13.042L52.066 12.762L53.125 12.687L53.125 13.336Q53.296 13.028 53.600 12.857Q53.905 12.687 54.250 12.687Q54.756 12.687 55.039 12.910Q55.323 13.134 55.323 13.630L55.323 15.284Q55.323 15.421 55.472 15.457Q55.621 15.493 55.846 15.493L55.846 15.773L54.216 15.773L54.216 15.493Q54.445 15.493 54.593 15.459Q54.742 15.424 54.742 15.284L54.742 13.644Q54.742 13.309 54.622 13.109Q54.503 12.909 54.188 12.909Q53.918 12.909 53.684 13.045Q53.450 13.182 53.312 13.416Q53.173 13.650 53.173 13.924L53.173 15.284Q53.173 15.421 53.324 15.457Q53.474 15.493 53.700 15.493L53.700 15.773M56.755 17.530L56.687 17.530Q56.653 17.530 56.631 17.504Q56.608 17.479 56.608 17.444Q56.608 17.400 56.639 17.383Q56.995 17.079 57.244 16.689Q57.494 16.299 57.646 15.867Q57.798 15.435 57.868 14.966Q57.938 14.498 57.938 14.023Q57.938 13.544 57.868 13.078Q57.798 12.611 57.644 12.176Q57.490 11.740 57.239 11.352Q56.988 10.964 56.639 10.670Q56.608 10.653 56.608 10.608Q56.608 10.574 56.631 10.549Q56.653 10.523 56.687 10.523L56.755 10.523Q56.766 10.523 56.774 10.525Q56.783 10.526 56.793 10.530Q57.336 10.930 57.709 11.483Q58.081 12.037 58.263 12.683Q58.444 13.329 58.444 14.023Q58.444 14.724 58.263 15.371Q58.081 16.019 57.707 16.573Q57.333 17.127 56.793 17.523Q56.783 17.523 56.774 17.525Q56.766 17.526 56.755 17.530\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-44.422-15.525h170.717\" style=\"stroke-dasharray:3.0,3.0\"\u002F>\u003Cg transform=\"translate(-100.85 -29.548)\">\u003Cpath d=\"M38.572 15.773L36.042 15.773L36.042 15.493Q37.010 15.493 37.010 15.284L37.010 11.665Q36.617 11.853 35.995 11.853L35.995 11.572Q36.412 11.572 36.776 11.471Q37.140 11.371 37.396 11.125L37.522 11.125Q37.587 11.142 37.604 11.210L37.604 15.284Q37.604 15.493 38.572 15.493L38.572 15.773M39.717 17.356Q39.717 17.338 39.730 17.291L42.386 10.629Q42.441 10.523 42.547 10.523Q42.612 10.523 42.663 10.574Q42.714 10.626 42.714 10.690Q42.714 10.714 42.713 10.726Q42.711 10.738 42.707 10.755L40.055 17.417Q39.983 17.523 39.894 17.523Q39.826 17.523 39.771 17.472Q39.717 17.420 39.717 17.356M46.536 15.773L43.651 15.773L43.651 15.571Q43.651 15.541 43.678 15.513L44.926 14.296Q44.997 14.221 45.040 14.179Q45.083 14.136 45.162 14.057Q45.575 13.644 45.806 13.286Q46.037 12.929 46.037 12.505Q46.037 12.273 45.958 12.070Q45.879 11.866 45.737 11.716Q45.596 11.565 45.401 11.485Q45.206 11.405 44.974 11.405Q44.663 11.405 44.404 11.564Q44.146 11.723 44.017 12L44.037 12Q44.204 12 44.312 12.111Q44.420 12.222 44.420 12.386Q44.420 12.543 44.310 12.656Q44.201 12.769 44.037 12.769Q43.876 12.769 43.764 12.656Q43.651 12.543 43.651 12.386Q43.651 12.010 43.859 11.723Q44.068 11.436 44.403 11.280Q44.738 11.125 45.093 11.125Q45.517 11.125 45.896 11.283Q46.276 11.442 46.510 11.759Q46.744 12.075 46.744 12.505Q46.744 12.816 46.604 13.085Q46.464 13.353 46.259 13.558Q46.054 13.763 45.691 14.045Q45.329 14.327 45.220 14.423L44.365 15.151L45.008 15.151Q45.271 15.151 45.560 15.149Q45.849 15.148 46.067 15.139Q46.286 15.130 46.303 15.113Q46.365 15.048 46.402 14.881Q46.440 14.713 46.477 14.471L46.744 14.471\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"M-44.422-46.823h170.717\" style=\"stroke-dasharray:3.0,3.0\"\u002F>\u003Cg transform=\"translate(-92.877 -60.34)\">\u003Cpath d=\"M38.572 15.773L36.042 15.773L36.042 15.493Q37.010 15.493 37.010 15.284L37.010 11.665Q36.617 11.853 35.995 11.853L35.995 11.572Q36.412 11.572 36.776 11.471Q37.140 11.371 37.396 11.125L37.522 11.125Q37.587 11.142 37.604 11.210L37.604 15.284Q37.604 15.493 38.572 15.493\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-accent)\" d=\"m-55.803 15.402 2.305-.052 2.305-.058 2.305-.066 2.305-.075 2.305-.085 2.305-.096 2.305-.11 2.305-.124 2.305-.141 2.304-.16 2.305-.18 2.305-.205 2.305-.23 2.305-.262 2.305-.294 2.305-.332 2.305-.372 2.305-.42 2.305-.47 2.305-.525 2.305-.587 2.305-.654 2.304-.727 2.305-.805 2.305-.888 2.305-.978 2.305-1.07 2.305-1.167L11.04 3l2.305-1.368L15.65.162l2.305-1.567 2.305-1.661 2.305-1.75 2.305-1.829 2.305-1.897 2.304-1.952 2.305-1.995 2.305-2.017 2.305-2.026L38.7-18.55l2.305-1.992 2.305-1.957 2.305-1.895 2.305-1.83 2.305-1.75 2.305-1.667 2.305-1.565 2.305-1.468 2.304-1.375 2.305-1.263 2.305-1.166 2.305-1.076 2.305-.976 2.305-.89 2.305-.803 2.305-.728 2.305-.657 2.305-.586 2.305-.527 2.305-.47 2.305-.418 2.305-.375 2.304-.333 2.305-.29 2.305-.262 2.305-.23 2.305-.208 2.305-.18 2.305-.162 2.305-.14 2.305-.119 2.305-.11 2.305-.1 2.305-.086 2.305-.07 2.304-.07 2.305-.056 2.305-.051\" style=\"stroke-width:1.2\"\u002F>\u003Cpath fill=\"var(--tk-accent)\" stroke=\"none\" d=\"M37.246-15.525a2 2 0 1 0-4 0 2 2 0 0 0 4 0m-2 0\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cg transform=\"translate(47.635 -18.636)\">\u003Cpath d=\"M37.228 15.773L35.625 15.773L35.625 15.493Q35.851 15.493 36 15.459Q36.148 15.424 36.148 15.284L36.148 11.665Q36.148 11.395 36.041 11.333Q35.933 11.272 35.625 11.272L35.625 10.991L36.702 10.916L36.702 15.284Q36.702 15.421 36.852 15.457Q37.003 15.493 37.228 15.493L37.228 15.773M37.782 14.290Q37.782 13.948 37.917 13.649Q38.052 13.350 38.291 13.126Q38.531 12.902 38.849 12.777Q39.166 12.652 39.498 12.652Q39.942 12.652 40.342 12.868Q40.742 13.083 40.976 13.461Q41.210 13.838 41.210 14.290Q41.210 14.631 41.069 14.915Q40.927 15.199 40.682 15.406Q40.438 15.612 40.129 15.727Q39.819 15.841 39.498 15.841Q39.067 15.841 38.666 15.640Q38.264 15.438 38.023 15.086Q37.782 14.734 37.782 14.290M39.498 15.592Q40.100 15.592 40.323 15.214Q40.547 14.836 40.547 14.204Q40.547 13.592 40.313 13.233Q40.079 12.875 39.498 12.875Q38.445 12.875 38.445 14.204Q38.445 14.836 38.671 15.214Q38.896 15.592 39.498 15.592M41.764 16.306Q41.764 16.060 41.961 15.876Q42.157 15.691 42.413 15.612Q42.277 15.500 42.205 15.339Q42.133 15.178 42.133 14.997Q42.133 14.676 42.345 14.430Q42.010 14.132 42.010 13.722Q42.010 13.261 42.400 12.974Q42.789 12.687 43.268 12.687Q43.740 12.687 44.075 12.933Q44.249 12.779 44.459 12.697Q44.669 12.615 44.898 12.615Q45.062 12.615 45.184 12.722Q45.305 12.830 45.305 12.994Q45.305 13.090 45.233 13.162Q45.162 13.233 45.069 13.233Q44.970 13.233 44.900 13.160Q44.830 13.086 44.830 12.987Q44.830 12.933 44.844 12.902L44.850 12.888Q44.857 12.868 44.866 12.857Q44.874 12.847 44.878 12.840Q44.522 12.840 44.235 13.063Q44.522 13.356 44.522 13.722Q44.522 14.037 44.338 14.269Q44.153 14.502 43.864 14.630Q43.576 14.758 43.268 14.758Q43.066 14.758 42.875 14.708Q42.684 14.659 42.506 14.549Q42.413 14.676 42.413 14.819Q42.413 15.001 42.542 15.136Q42.670 15.271 42.854 15.271L43.487 15.271Q43.934 15.271 44.304 15.342Q44.673 15.414 44.933 15.643Q45.192 15.872 45.192 16.306Q45.192 16.627 44.897 16.829Q44.601 17.031 44.198 17.120Q43.794 17.209 43.480 17.209Q43.162 17.209 42.759 17.120Q42.355 17.031 42.060 16.829Q41.764 16.627 41.764 16.306M42.219 16.306Q42.219 16.535 42.437 16.684Q42.656 16.833 42.948 16.901Q43.241 16.969 43.480 16.969Q43.644 16.969 43.852 16.933Q44.061 16.898 44.268 16.817Q44.475 16.737 44.606 16.609Q44.738 16.481 44.738 16.306Q44.738 15.954 44.357 15.860Q43.975 15.766 43.473 15.766L42.854 15.766Q42.615 15.766 42.417 15.917Q42.219 16.067 42.219 16.306M43.268 14.519Q43.934 14.519 43.934 13.722Q43.934 12.922 43.268 12.922Q42.598 12.922 42.598 13.722Q42.598 14.519 43.268 14.519M47.404 15.773L45.852 15.773L45.852 15.493Q46.078 15.493 46.226 15.459Q46.375 15.424 46.375 15.284L46.375 13.435Q46.375 13.247 46.327 13.163Q46.279 13.080 46.182 13.061Q46.084 13.042 45.872 13.042L45.872 12.762L46.929 12.687L46.929 15.284Q46.929 15.424 47.060 15.459Q47.192 15.493 47.404 15.493L47.404 15.773M46.132 11.466Q46.132 11.295 46.255 11.176Q46.378 11.056 46.549 11.056Q46.717 11.056 46.840 11.176Q46.963 11.295 46.963 11.466Q46.963 11.641 46.840 11.764Q46.717 11.887 46.549 11.887Q46.378 11.887 46.255 11.764Q46.132 11.641 46.132 11.466M48.050 15.766L48.050 14.703Q48.050 14.679 48.077 14.652Q48.104 14.625 48.128 14.625L48.238 14.625Q48.303 14.625 48.316 14.683Q48.412 15.117 48.658 15.368Q48.904 15.619 49.318 15.619Q49.660 15.619 49.913 15.486Q50.165 15.353 50.165 15.045Q50.165 14.888 50.071 14.773Q49.977 14.659 49.839 14.590Q49.701 14.522 49.533 14.484L48.952 14.385Q48.597 14.317 48.323 14.096Q48.050 13.876 48.050 13.534Q48.050 13.285 48.161 13.110Q48.272 12.936 48.458 12.837Q48.644 12.738 48.860 12.695Q49.075 12.652 49.318 12.652Q49.731 12.652 50.012 12.834L50.227 12.659Q50.237 12.656 50.244 12.654Q50.251 12.652 50.261 12.652L50.312 12.652Q50.340 12.652 50.364 12.676Q50.388 12.700 50.388 12.728L50.388 13.575Q50.388 13.596 50.364 13.623Q50.340 13.650 50.312 13.650L50.200 13.650Q50.172 13.650 50.147 13.625Q50.121 13.599 50.121 13.575Q50.121 13.339 50.015 13.175Q49.909 13.011 49.726 12.929Q49.543 12.847 49.311 12.847Q48.983 12.847 48.726 12.950Q48.470 13.052 48.470 13.329Q48.470 13.524 48.653 13.633Q48.836 13.743 49.065 13.784L49.639 13.890Q49.885 13.938 50.099 14.066Q50.312 14.194 50.449 14.397Q50.586 14.601 50.586 14.850Q50.586 15.363 50.220 15.602Q49.854 15.841 49.318 15.841Q48.822 15.841 48.491 15.547L48.224 15.821Q48.204 15.841 48.176 15.841L48.128 15.841Q48.104 15.841 48.077 15.814Q48.050 15.787 48.050 15.766M51.741 14.932L51.741 13.035L51.102 13.035L51.102 12.813Q51.420 12.813 51.637 12.603Q51.854 12.393 51.955 12.083Q52.056 11.774 52.056 11.466L52.322 11.466L52.322 12.755L53.399 12.755L53.399 13.035L52.322 13.035L52.322 14.919Q52.322 15.195 52.426 15.394Q52.531 15.592 52.790 15.592Q52.948 15.592 53.054 15.488Q53.160 15.383 53.209 15.230Q53.259 15.076 53.259 14.919L53.259 14.505L53.525 14.505L53.525 14.932Q53.525 15.158 53.426 15.368Q53.327 15.578 53.142 15.710Q52.958 15.841 52.729 15.841Q52.291 15.841 52.016 15.604Q51.741 15.366 51.741 14.932M55.952 15.773L54.400 15.773L54.400 15.493Q54.626 15.493 54.775 15.459Q54.923 15.424 54.923 15.284L54.923 13.435Q54.923 13.247 54.875 13.163Q54.828 13.080 54.730 13.061Q54.633 13.042 54.421 13.042L54.421 12.762L55.477 12.687L55.477 15.284Q55.477 15.424 55.609 15.459Q55.740 15.493 55.952 15.493L55.952 15.773M54.681 11.466Q54.681 11.295 54.804 11.176Q54.927 11.056 55.098 11.056Q55.265 11.056 55.388 11.176Q55.511 11.295 55.511 11.466Q55.511 11.641 55.388 11.764Q55.265 11.887 55.098 11.887Q54.927 11.887 54.804 11.764Q54.681 11.641 54.681 11.466M56.598 14.262Q56.598 13.934 56.733 13.633Q56.868 13.333 57.104 13.112Q57.340 12.892 57.644 12.772Q57.948 12.652 58.273 12.652Q58.779 12.652 59.127 12.755Q59.476 12.857 59.476 13.233Q59.476 13.380 59.379 13.481Q59.281 13.582 59.134 13.582Q58.980 13.582 58.881 13.483Q58.782 13.384 58.782 13.233Q58.782 13.045 58.922 12.953Q58.721 12.902 58.280 12.902Q57.924 12.902 57.695 13.098Q57.466 13.295 57.365 13.604Q57.265 13.914 57.265 14.262Q57.265 14.611 57.391 14.917Q57.517 15.223 57.772 15.407Q58.027 15.592 58.382 15.592Q58.604 15.592 58.789 15.508Q58.974 15.424 59.109 15.269Q59.244 15.113 59.302 14.905Q59.315 14.850 59.370 14.850L59.483 14.850Q59.514 14.850 59.536 14.874Q59.558 14.898 59.558 14.932L59.558 14.953Q59.473 15.240 59.285 15.438Q59.097 15.636 58.832 15.739Q58.567 15.841 58.273 15.841Q57.842 15.841 57.454 15.635Q57.066 15.428 56.832 15.065Q56.598 14.703 56.598 14.262\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fsvg>\u003Cfigcaption class=\"tikz-cap\">Bradley-Terry: preference probability is the logistic of the reward gap \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">Δ\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0269em;\">w\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:0.2222em;\">\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 mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\" style=\"margin-right:0.0197em;\">l\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>, passing through \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1.1901em;vertical-align:-0.345em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8451em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.394em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.345em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> at \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">Δ\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>.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:485.231px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 363.923 63.590\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg fill=\"var(--tk-soft-neutral)\" style=\"stroke-width:.8\">\u003Cpath d=\"M-61.87-37.727H.726V-71.87H-61.87Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-27.577 1.75)\">\u003Cpath d=\"M-28.508-54.798L-30.244-54.798L-30.244-55.078Q-30.015-55.078-29.866-55.112Q-29.718-55.147-29.718-55.287L-29.718-57.136Q-29.718-57.406-29.825-57.467Q-29.933-57.529-30.244-57.529L-30.244-57.809L-29.215-57.884L-29.215-57.177Q-29.085-57.485-28.843-57.684Q-28.600-57.884-28.282-57.884Q-28.063-57.884-27.892-57.760Q-27.721-57.635-27.721-57.423Q-27.721-57.286-27.821-57.187Q-27.920-57.088-28.053-57.088Q-28.190-57.088-28.289-57.187Q-28.388-57.286-28.388-57.423Q-28.388-57.563-28.289-57.662Q-28.579-57.662-28.779-57.466Q-28.979-57.269-29.072-56.975Q-29.164-56.681-29.164-56.401L-29.164-55.287Q-29.164-55.078-28.508-55.078L-28.508-54.798M-27.178-56.333Q-27.178-56.654-27.053-56.943Q-26.928-57.232-26.703-57.455Q-26.477-57.679-26.182-57.799Q-25.886-57.919-25.568-57.919Q-25.240-57.919-24.978-57.819Q-24.717-57.720-24.541-57.538Q-24.365-57.355-24.271-57.097Q-24.177-56.839-24.177-56.507Q-24.177-56.415-24.259-56.394L-26.515-56.394L-26.515-56.333Q-26.515-55.745-26.231-55.362Q-25.947-54.979-25.380-54.979Q-25.059-54.979-24.791-55.172Q-24.522-55.365-24.433-55.680Q-24.426-55.721-24.351-55.735L-24.259-55.735Q-24.177-55.711-24.177-55.639Q-24.177-55.632-24.184-55.605Q-24.297-55.208-24.667-54.969Q-25.038-54.730-25.462-54.730Q-25.900-54.730-26.300-54.938Q-26.699-55.147-26.939-55.514Q-27.178-55.881-27.178-56.333M-26.508-56.603L-24.693-56.603Q-24.693-56.880-24.791-57.132Q-24.888-57.385-25.086-57.541Q-25.284-57.696-25.568-57.696Q-25.845-57.696-26.059-57.538Q-26.272-57.379-26.390-57.124Q-26.508-56.869-26.508-56.603M-21.791-54.798L-23.524-54.798L-23.524-55.078Q-23.299-55.078-23.150-55.112Q-23.001-55.147-23.001-55.287L-23.001-57.536L-23.589-57.536L-23.589-57.816L-23.001-57.816L-23.001-58.633Q-23.001-58.951-22.823-59.199Q-22.646-59.446-22.355-59.587Q-22.065-59.727-21.754-59.727Q-21.497-59.727-21.294-59.585Q-21.091-59.443-21.091-59.200Q-21.091-59.064-21.190-58.965Q-21.289-58.865-21.426-58.865Q-21.562-58.865-21.661-58.965Q-21.760-59.064-21.760-59.200Q-21.760-59.381-21.620-59.474Q-21.699-59.501-21.798-59.501Q-22.007-59.501-22.160-59.368Q-22.314-59.235-22.395-59.031Q-22.475-58.828-22.475-58.619L-22.475-57.816L-21.586-57.816L-21.586-57.536L-22.447-57.536L-22.447-55.287Q-22.447-55.078-21.791-55.078L-21.791-54.798M-21.152-56.333Q-21.152-56.654-21.027-56.943Q-20.903-57.232-20.677-57.455Q-20.451-57.679-20.156-57.799Q-19.860-57.919-19.542-57.919Q-19.214-57.919-18.953-57.819Q-18.691-57.720-18.515-57.538Q-18.339-57.355-18.245-57.097Q-18.151-56.839-18.151-56.507Q-18.151-56.415-18.233-56.394L-20.489-56.394L-20.489-56.333Q-20.489-55.745-20.205-55.362Q-19.922-54.979-19.354-54.979Q-19.033-54.979-18.765-55.172Q-18.496-55.365-18.407-55.680Q-18.401-55.721-18.325-55.735L-18.233-55.735Q-18.151-55.711-18.151-55.639Q-18.151-55.632-18.158-55.605Q-18.271-55.208-18.642-54.969Q-19.012-54.730-19.436-54.730Q-19.874-54.730-20.274-54.938Q-20.674-55.147-20.913-55.514Q-21.152-55.881-21.152-56.333M-20.482-56.603L-18.667-56.603Q-18.667-56.880-18.765-57.132Q-18.862-57.385-19.060-57.541Q-19.259-57.696-19.542-57.696Q-19.819-57.696-20.033-57.538Q-20.246-57.379-20.364-57.124Q-20.482-56.869-20.482-56.603M-15.813-54.798L-17.550-54.798L-17.550-55.078Q-17.321-55.078-17.172-55.112Q-17.023-55.147-17.023-55.287L-17.023-57.136Q-17.023-57.406-17.131-57.467Q-17.239-57.529-17.550-57.529L-17.550-57.809L-16.521-57.884L-16.521-57.177Q-16.391-57.485-16.148-57.684Q-15.905-57.884-15.588-57.884Q-15.369-57.884-15.198-57.760Q-15.027-57.635-15.027-57.423Q-15.027-57.286-15.126-57.187Q-15.225-57.088-15.359-57.088Q-15.495-57.088-15.594-57.187Q-15.694-57.286-15.694-57.423Q-15.694-57.563-15.594-57.662Q-15.885-57.662-16.085-57.466Q-16.285-57.269-16.377-56.975Q-16.469-56.681-16.469-56.401L-16.469-55.287Q-16.469-55.078-15.813-55.078L-15.813-54.798M-14.484-56.333Q-14.484-56.654-14.359-56.943Q-14.234-57.232-14.009-57.455Q-13.783-57.679-13.487-57.799Q-13.192-57.919-12.874-57.919Q-12.546-57.919-12.284-57.819Q-12.023-57.720-11.847-57.538Q-11.671-57.355-11.577-57.097Q-11.483-56.839-11.483-56.507Q-11.483-56.415-11.565-56.394L-13.821-56.394L-13.821-56.333Q-13.821-55.745-13.537-55.362Q-13.253-54.979-12.686-54.979Q-12.364-54.979-12.096-55.172Q-11.828-55.365-11.739-55.680Q-11.732-55.721-11.657-55.735L-11.565-55.735Q-11.483-55.711-11.483-55.639Q-11.483-55.632-11.489-55.605Q-11.602-55.208-11.973-54.969Q-12.344-54.730-12.768-54.730Q-13.205-54.730-13.605-54.938Q-14.005-55.147-14.244-55.514Q-14.484-55.881-14.484-56.333M-13.814-56.603L-11.999-56.603Q-11.999-56.880-12.096-57.132Q-12.194-57.385-12.392-57.541Q-12.590-57.696-12.874-57.696Q-13.151-57.696-13.364-57.538Q-13.578-57.379-13.696-57.124Q-13.814-56.869-13.814-56.603M-9.213-54.798L-10.847-54.798L-10.847-55.078Q-10.618-55.078-10.469-55.112Q-10.321-55.147-10.321-55.287L-10.321-57.136Q-10.321-57.406-10.428-57.467Q-10.536-57.529-10.847-57.529L-10.847-57.809L-9.787-57.884L-9.787-57.235Q-9.616-57.543-9.312-57.714Q-9.008-57.884-8.663-57.884Q-8.157-57.884-7.873-57.661Q-7.590-57.437-7.590-56.941L-7.590-55.287Q-7.590-55.150-7.441-55.114Q-7.292-55.078-7.067-55.078L-7.067-54.798L-8.697-54.798L-8.697-55.078Q-8.468-55.078-8.319-55.112Q-8.171-55.147-8.171-55.287L-8.171-56.927Q-8.171-57.262-8.290-57.462Q-8.410-57.662-8.724-57.662Q-8.994-57.662-9.228-57.526Q-9.463-57.389-9.601-57.155Q-9.739-56.921-9.739-56.647L-9.739-55.287Q-9.739-55.150-9.589-55.114Q-9.439-55.078-9.213-55.078L-9.213-54.798M-6.479-56.309Q-6.479-56.637-6.344-56.938Q-6.209-57.238-5.973-57.459Q-5.737-57.679-5.433-57.799Q-5.129-57.919-4.804-57.919Q-4.298-57.919-3.949-57.816Q-3.601-57.714-3.601-57.338Q-3.601-57.191-3.698-57.090Q-3.796-56.989-3.943-56.989Q-4.096-56.989-4.196-57.088Q-4.295-57.187-4.295-57.338Q-4.295-57.526-4.155-57.618Q-4.356-57.669-4.797-57.669Q-5.153-57.669-5.382-57.473Q-5.611-57.276-5.711-56.967Q-5.812-56.657-5.812-56.309Q-5.812-55.960-5.686-55.654Q-5.559-55.348-5.305-55.164Q-5.050-54.979-4.695-54.979Q-4.472-54.979-4.288-55.063Q-4.103-55.147-3.968-55.302Q-3.833-55.458-3.775-55.666Q-3.761-55.721-3.707-55.721L-3.594-55.721Q-3.563-55.721-3.541-55.697Q-3.519-55.673-3.519-55.639L-3.519-55.618Q-3.604-55.331-3.792-55.133Q-3.980-54.935-4.245-54.832Q-4.510-54.730-4.804-54.730Q-5.235-54.730-5.623-54.936Q-6.010-55.143-6.245-55.506Q-6.479-55.868-6.479-56.309M-2.972-56.333Q-2.972-56.654-2.847-56.943Q-2.722-57.232-2.497-57.455Q-2.271-57.679-1.976-57.799Q-1.680-57.919-1.362-57.919Q-1.034-57.919-0.772-57.819Q-0.511-57.720-0.335-57.538Q-0.159-57.355-0.065-57.097Q0.029-56.839 0.029-56.507Q0.029-56.415-0.053-56.394L-2.309-56.394L-2.309-56.333Q-2.309-55.745-2.025-55.362Q-1.741-54.979-1.174-54.979Q-0.853-54.979-0.584-55.172Q-0.316-55.365-0.227-55.680Q-0.220-55.721-0.145-55.735L-0.053-55.735Q0.029-55.711 0.029-55.639Q0.029-55.632 0.022-55.605Q-0.091-55.208-0.461-54.969Q-0.832-54.730-1.256-54.730Q-1.694-54.730-2.093-54.938Q-2.493-55.147-2.733-55.514Q-2.972-55.881-2.972-56.333M-2.302-56.603L-0.487-56.603Q-0.487-56.880-0.584-57.132Q-0.682-57.385-0.880-57.541Q-1.078-57.696-1.362-57.696Q-1.639-57.696-1.853-57.538Q-2.066-57.379-2.184-57.124Q-2.302-56.869-2.302-56.603\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-27.577 1.75)\">\u003Cpath d=\"M5.490-53.048Q4.940-53.448 4.569-54.003Q4.198-54.559 4.017-55.205Q3.836-55.851 3.836-56.548Q3.836-57.061 3.936-57.556Q4.037-58.052 4.242-58.503Q4.447-58.954 4.760-59.346Q5.073-59.737 5.490-60.041Q5.500-60.045 5.507-60.046Q5.514-60.048 5.524-60.048L5.592-60.048Q5.627-60.048 5.649-60.024Q5.671-60 5.671-59.963Q5.671-59.918 5.644-59.901Q5.295-59.600 5.042-59.216Q4.789-58.831 4.637-58.390Q4.485-57.949 4.413-57.493Q4.341-57.037 4.341-56.548Q4.341-55.547 4.651-54.660Q4.960-53.773 5.644-53.188Q5.671-53.171 5.671-53.127Q5.671-53.089 5.649-53.065Q5.627-53.041 5.592-53.041L5.524-53.041Q5.517-53.045 5.509-53.046Q5.500-53.048 5.490-53.048M6.655-54.736L6.655-56.309Q6.655-56.336 6.681-56.362Q6.707-56.387 6.734-56.387L6.847-56.387Q6.874-56.387 6.898-56.360Q6.922-56.333 6.922-56.309Q6.922-55.964 7.054-55.700Q7.185-55.437 7.414-55.268Q7.643-55.099 7.946-55.018Q8.248-54.938 8.590-54.938Q8.857-54.938 9.092-55.066Q9.328-55.194 9.473-55.417Q9.619-55.639 9.619-55.905Q9.619-56.128 9.513-56.324Q9.407-56.521 9.226-56.656Q9.045-56.791 8.819-56.842L7.790-57.074Q7.479-57.146 7.219-57.332Q6.960-57.519 6.807-57.790Q6.655-58.062 6.655-58.377Q6.655-58.763 6.869-59.070Q7.083-59.378 7.430-59.549Q7.776-59.720 8.156-59.720Q8.385-59.720 8.614-59.667Q8.843-59.614 9.041-59.506Q9.239-59.399 9.393-59.235L9.687-59.675Q9.711-59.720 9.752-59.720L9.800-59.720Q9.831-59.720 9.853-59.694Q9.875-59.669 9.875-59.641L9.875-58.066Q9.875-58.045 9.851-58.018Q9.827-57.990 9.800-57.990L9.687-57.990Q9.626-57.990 9.612-58.066Q9.571-58.479 9.390-58.799Q9.209-59.118 8.898-59.293Q8.587-59.467 8.156-59.467Q7.906-59.467 7.667-59.356Q7.428-59.245 7.277-59.047Q7.127-58.848 7.127-58.585Q7.127-58.373 7.235-58.192Q7.342-58.011 7.518-57.891Q7.694-57.772 7.903-57.731L8.932-57.502Q9.250-57.430 9.516-57.225Q9.783-57.020 9.935-56.726Q10.087-56.432 10.087-56.100Q10.087-55.707 9.882-55.372Q9.677-55.037 9.332-54.848Q8.986-54.658 8.590-54.658Q8.170-54.658 7.790-54.771Q7.411-54.883 7.141-55.133L6.847-54.699Q6.819-54.658 6.782-54.658L6.734-54.658Q6.707-54.658 6.681-54.683Q6.655-54.709 6.655-54.736M13.276-54.798L10.962-54.798L10.962-55.078Q11.683-55.078 11.683-55.287L11.683-59.088Q11.683-59.299 10.962-59.299L10.962-59.580L15.146-59.580L15.358-57.943L15.091-57.943Q15.012-58.554 14.860-58.833Q14.708-59.111 14.404-59.205Q14.100-59.299 13.474-59.299L12.739-59.299Q12.551-59.299 12.463-59.265Q12.374-59.231 12.374-59.088L12.374-57.331L12.927-57.331Q13.296-57.331 13.481-57.385Q13.666-57.440 13.744-57.613Q13.823-57.785 13.823-58.151L14.089-58.151L14.089-56.234L13.823-56.234Q13.823-56.599 13.744-56.772Q13.666-56.944 13.481-56.997Q13.296-57.050 12.927-57.050L12.374-57.050L12.374-55.287Q12.374-55.078 13.276-55.078L13.276-54.798M19.962-54.798L17.224-54.798L17.224-55.078Q17.572-55.078 17.909-55.114Q18.246-55.150 18.246-55.287L18.246-59.088Q18.246-59.231 18.157-59.265Q18.068-59.299 17.883-59.299L17.525-59.299Q17.224-59.299 17.008-59.252Q16.793-59.204 16.636-59.047Q16.499-58.913 16.439-58.635Q16.379-58.356 16.342-57.943L16.075-57.943L16.222-59.580L20.956-59.580L21.103-57.943L20.837-57.943Q20.799-58.356 20.743-58.633Q20.686-58.910 20.543-59.047Q20.382-59.207 20.170-59.253Q19.958-59.299 19.654-59.299L19.302-59.299Q19.117-59.299 19.028-59.265Q18.940-59.231 18.940-59.088L18.940-55.287Q18.940-55.150 19.276-55.114Q19.613-55.078 19.962-55.078L19.962-54.798M22.081-53.041L22.012-53.041Q21.978-53.041 21.956-53.067Q21.934-53.092 21.934-53.127Q21.934-53.171 21.964-53.188Q22.320-53.492 22.569-53.882Q22.819-54.272 22.971-54.704Q23.123-55.136 23.193-55.605Q23.263-56.073 23.263-56.548Q23.263-57.027 23.193-57.493Q23.123-57.960 22.969-58.395Q22.816-58.831 22.564-59.219Q22.313-59.607 21.964-59.901Q21.934-59.918 21.934-59.963Q21.934-59.997 21.956-60.022Q21.978-60.048 22.012-60.048L22.081-60.048Q22.091-60.048 22.099-60.046Q22.108-60.045 22.118-60.041Q22.662-59.641 23.034-59.088Q23.407-58.534 23.588-57.888Q23.769-57.242 23.769-56.548Q23.769-55.847 23.588-55.200Q23.407-54.552 23.033-53.998Q22.658-53.444 22.118-53.048Q22.108-53.048 22.099-53.046Q22.091-53.045 22.081-53.041\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath d=\"M80.394-37.727h62.596V-71.87H80.394Z\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(131.812 1.75)\">\u003Cpath d=\"M-28.614-53.441L-30.244-53.441L-30.244-53.721Q-30.015-53.721-29.866-53.756Q-29.718-53.790-29.718-53.930L-29.718-57.276Q-29.718-57.447-29.854-57.488Q-29.991-57.529-30.244-57.529L-30.244-57.809L-29.164-57.884L-29.164-57.478Q-28.942-57.679-28.655-57.782Q-28.367-57.884-28.060-57.884Q-27.633-57.884-27.269-57.671Q-26.905-57.457-26.691-57.093Q-26.477-56.729-26.477-56.309Q-26.477-55.864-26.717-55.500Q-26.956-55.136-27.349-54.933Q-27.742-54.730-28.186-54.730Q-28.453-54.730-28.701-54.830Q-28.948-54.931-29.136-55.112L-29.136-53.930Q-29.136-53.793-28.988-53.757Q-28.839-53.721-28.614-53.721L-28.614-53.441M-29.136-57.129L-29.136-55.519Q-29.003-55.266-28.760-55.109Q-28.518-54.952-28.241-54.952Q-27.913-54.952-27.660-55.153Q-27.407-55.355-27.274-55.673Q-27.140-55.991-27.140-56.309Q-27.140-56.538-27.205-56.767Q-27.270-56.996-27.398-57.194Q-27.527-57.392-27.721-57.512Q-27.916-57.631-28.149-57.631Q-28.443-57.631-28.711-57.502Q-28.979-57.372-29.136-57.129\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(131.812 1.75)\">\u003Cpath d=\"M-25.667-56.281Q-25.667-56.623-25.532-56.922Q-25.397-57.221-25.157-57.445Q-24.918-57.669-24.600-57.794Q-24.282-57.919-23.951-57.919Q-23.506-57.919-23.107-57.703Q-22.707-57.488-22.472-57.110Q-22.238-56.733-22.238-56.281Q-22.238-55.940-22.380-55.656Q-22.522-55.372-22.766-55.165Q-23.011-54.959-23.320-54.844Q-23.629-54.730-23.951-54.730Q-24.381-54.730-24.783-54.931Q-25.185-55.133-25.426-55.485Q-25.667-55.837-25.667-56.281M-23.951-54.979Q-23.349-54.979-23.125-55.357Q-22.901-55.735-22.901-56.367Q-22.901-56.979-23.136-57.338Q-23.370-57.696-23.951-57.696Q-25.003-57.696-25.003-56.367Q-25.003-55.735-24.778-55.357Q-24.552-54.979-23.951-54.979M-19.976-54.798L-21.579-54.798L-21.579-55.078Q-21.353-55.078-21.204-55.112Q-21.056-55.147-21.056-55.287L-21.056-58.906Q-21.056-59.176-21.163-59.238Q-21.271-59.299-21.579-59.299L-21.579-59.580L-20.502-59.655L-20.502-55.287Q-20.502-55.150-20.352-55.114Q-20.201-55.078-19.976-55.078L-19.976-54.798M-17.764-54.798L-19.316-54.798L-19.316-55.078Q-19.090-55.078-18.942-55.112Q-18.793-55.147-18.793-55.287L-18.793-57.136Q-18.793-57.324-18.841-57.408Q-18.889-57.491-18.986-57.510Q-19.084-57.529-19.295-57.529L-19.295-57.809L-18.239-57.884L-18.239-55.287Q-18.239-55.147-18.108-55.112Q-17.976-55.078-17.764-55.078L-17.764-54.798M-19.036-59.105Q-19.036-59.276-18.913-59.395Q-18.790-59.515-18.619-59.515Q-18.451-59.515-18.328-59.395Q-18.205-59.276-18.205-59.105Q-18.205-58.930-18.328-58.807Q-18.451-58.684-18.619-58.684Q-18.790-58.684-18.913-58.807Q-19.036-58.930-19.036-59.105M-17.118-56.309Q-17.118-56.637-16.983-56.938Q-16.848-57.238-16.612-57.459Q-16.377-57.679-16.072-57.799Q-15.768-57.919-15.443-57.919Q-14.938-57.919-14.589-57.816Q-14.240-57.714-14.240-57.338Q-14.240-57.191-14.338-57.090Q-14.435-56.989-14.582-56.989Q-14.736-56.989-14.835-57.088Q-14.934-57.187-14.934-57.338Q-14.934-57.526-14.794-57.618Q-14.996-57.669-15.437-57.669Q-15.792-57.669-16.021-57.473Q-16.250-57.276-16.351-56.967Q-16.452-56.657-16.452-56.309Q-16.452-55.960-16.325-55.654Q-16.199-55.348-15.944-55.164Q-15.690-54.979-15.334-54.979Q-15.112-54.979-14.927-55.063Q-14.743-55.147-14.608-55.302Q-14.473-55.458-14.415-55.666Q-14.401-55.721-14.346-55.721L-14.233-55.721Q-14.203-55.721-14.180-55.697Q-14.158-55.673-14.158-55.639L-14.158-55.618Q-14.244-55.331-14.432-55.133Q-14.620-54.935-14.885-54.832Q-15.149-54.730-15.443-54.730Q-15.874-54.730-16.262-54.936Q-16.650-55.143-16.884-55.506Q-17.118-55.868-17.118-56.309M-13.235-53.663Q-13.106-53.595-12.969-53.595Q-12.798-53.595-12.648-53.684Q-12.497-53.773-12.386-53.918Q-12.275-54.063-12.196-54.231L-11.933-54.798L-13.102-57.324Q-13.177-57.471-13.307-57.503Q-13.437-57.536-13.669-57.536L-13.669-57.816L-12.148-57.816L-12.148-57.536Q-12.497-57.536-12.497-57.389Q-12.494-57.368-12.492-57.351Q-12.490-57.334-12.490-57.324L-11.632-55.465L-10.860-57.136Q-10.826-57.204-10.826-57.283Q-10.826-57.396-10.909-57.466Q-10.993-57.536-11.106-57.536L-11.106-57.816L-9.910-57.816L-9.910-57.536Q-10.128-57.536-10.301-57.432Q-10.474-57.327-10.566-57.136L-11.902-54.231Q-12.073-53.861-12.343-53.615Q-12.613-53.369-12.969-53.369Q-13.239-53.369-13.458-53.535Q-13.676-53.701-13.676-53.964Q-13.676-54.101-13.584-54.190Q-13.492-54.278-13.352-54.278Q-13.215-54.278-13.126-54.190Q-13.037-54.101-13.037-53.964Q-13.037-53.861-13.090-53.783Q-13.143-53.704-13.235-53.663\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-width:.8\">\u003Cpath d=\"M222.657-37.727h62.596V-71.87h-62.596Z\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(263.788 1.75)\">\u003Cpath d=\"M-28.576-54.798L-30.210-54.798L-30.210-55.078Q-29.981-55.078-29.832-55.112Q-29.683-55.147-29.683-55.287L-29.683-58.906Q-29.683-59.176-29.791-59.238Q-29.899-59.299-30.210-59.299L-30.210-59.580L-29.130-59.655L-29.130-57.269Q-29.024-57.454-28.846-57.596Q-28.668-57.737-28.460-57.811Q-28.251-57.884-28.026-57.884Q-27.520-57.884-27.236-57.661Q-26.952-57.437-26.952-56.941L-26.952-55.287Q-26.952-55.150-26.804-55.114Q-26.655-55.078-26.429-55.078L-26.429-54.798L-28.060-54.798L-28.060-55.078Q-27.831-55.078-27.682-55.112Q-27.533-55.147-27.533-55.287L-27.533-56.927Q-27.533-57.262-27.653-57.462Q-27.773-57.662-28.087-57.662Q-28.357-57.662-28.591-57.526Q-28.825-57.389-28.964-57.155Q-29.102-56.921-29.102-56.647L-29.102-55.287Q-29.102-55.150-28.952-55.114Q-28.801-55.078-28.576-55.078L-28.576-54.798M-24.225-54.798L-25.777-54.798L-25.777-55.078Q-25.551-55.078-25.402-55.112Q-25.254-55.147-25.254-55.287L-25.254-57.136Q-25.254-57.324-25.301-57.408Q-25.349-57.491-25.447-57.510Q-25.544-57.529-25.756-57.529L-25.756-57.809L-24.700-57.884L-24.700-55.287Q-24.700-55.147-24.568-55.112Q-24.437-55.078-24.225-55.078L-24.225-54.798M-25.496-59.105Q-25.496-59.276-25.373-59.395Q-25.250-59.515-25.079-59.515Q-24.912-59.515-24.789-59.395Q-24.666-59.276-24.666-59.105Q-24.666-58.930-24.789-58.807Q-24.912-58.684-25.079-58.684Q-25.250-58.684-25.373-58.807Q-25.496-58.930-25.496-59.105M-23.620-54.265Q-23.620-54.511-23.423-54.695Q-23.227-54.880-22.970-54.959Q-23.107-55.071-23.179-55.232Q-23.251-55.393-23.251-55.574Q-23.251-55.895-23.039-56.141Q-23.374-56.439-23.374-56.849Q-23.374-57.310-22.984-57.597Q-22.594-57.884-22.116-57.884Q-21.644-57.884-21.309-57.638Q-21.135-57.792-20.925-57.874Q-20.715-57.956-20.486-57.956Q-20.322-57.956-20.200-57.849Q-20.079-57.741-20.079-57.577Q-20.079-57.481-20.151-57.409Q-20.222-57.338-20.315-57.338Q-20.414-57.338-20.484-57.411Q-20.554-57.485-20.554-57.584Q-20.554-57.638-20.540-57.669L-20.533-57.683Q-20.527-57.703-20.518-57.714Q-20.509-57.724-20.506-57.731Q-20.862-57.731-21.149-57.508Q-20.862-57.215-20.862-56.849Q-20.862-56.534-21.046-56.302Q-21.231-56.069-21.520-55.941Q-21.808-55.813-22.116-55.813Q-22.318-55.813-22.509-55.863Q-22.700-55.912-22.878-56.022Q-22.970-55.895-22.970-55.752Q-22.970-55.570-22.842-55.435Q-22.714-55.300-22.530-55.300L-21.897-55.300Q-21.449-55.300-21.080-55.229Q-20.711-55.157-20.451-54.928Q-20.192-54.699-20.192-54.265Q-20.192-53.944-20.487-53.742Q-20.783-53.540-21.186-53.451Q-21.590-53.362-21.904-53.362Q-22.222-53.362-22.625-53.451Q-23.029-53.540-23.324-53.742Q-23.620-53.944-23.620-54.265M-23.165-54.265Q-23.165-54.036-22.947-53.887Q-22.728-53.738-22.436-53.670Q-22.143-53.602-21.904-53.602Q-21.740-53.602-21.531-53.638Q-21.323-53.673-21.116-53.754Q-20.909-53.834-20.778-53.962Q-20.646-54.090-20.646-54.265Q-20.646-54.617-21.027-54.711Q-21.408-54.805-21.911-54.805L-22.530-54.805Q-22.769-54.805-22.967-54.654Q-23.165-54.504-23.165-54.265M-22.116-56.052Q-21.449-56.052-21.449-56.849Q-21.449-57.649-22.116-57.649Q-22.786-57.649-22.786-56.849Q-22.786-56.052-22.116-56.052M-17.915-54.798L-19.549-54.798L-19.549-55.078Q-19.320-55.078-19.171-55.112Q-19.023-55.147-19.023-55.287L-19.023-58.906Q-19.023-59.176-19.130-59.238Q-19.238-59.299-19.549-59.299L-19.549-59.580L-18.469-59.655L-18.469-57.269Q-18.363-57.454-18.185-57.596Q-18.008-57.737-17.799-57.811Q-17.591-57.884-17.365-57.884Q-16.859-57.884-16.575-57.661Q-16.292-57.437-16.292-56.941L-16.292-55.287Q-16.292-55.150-16.143-55.114Q-15.994-55.078-15.769-55.078L-15.769-54.798L-17.399-54.798L-17.399-55.078Q-17.170-55.078-17.021-55.112Q-16.873-55.147-16.873-55.287L-16.873-56.927Q-16.873-57.262-16.992-57.462Q-17.112-57.662-17.426-57.662Q-17.697-57.662-17.931-57.526Q-18.165-57.389-18.303-57.155Q-18.442-56.921-18.442-56.647L-18.442-55.287Q-18.442-55.150-18.291-55.114Q-18.141-55.078-17.915-55.078\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(263.788 1.75)\">\u003Cpath d=\"M-10.730-54.798L-12.466-54.798L-12.466-55.078Q-12.237-55.078-12.088-55.112Q-11.940-55.147-11.940-55.287L-11.940-57.136Q-11.940-57.406-12.047-57.467Q-12.155-57.529-12.466-57.529L-12.466-57.809L-11.437-57.884L-11.437-57.177Q-11.307-57.485-11.065-57.684Q-10.822-57.884-10.504-57.884Q-10.285-57.884-10.114-57.760Q-9.943-57.635-9.943-57.423Q-9.943-57.286-10.043-57.187Q-10.142-57.088-10.275-57.088Q-10.412-57.088-10.511-57.187Q-10.610-57.286-10.610-57.423Q-10.610-57.563-10.511-57.662Q-10.801-57.662-11.001-57.466Q-11.201-57.269-11.294-56.975Q-11.386-56.681-11.386-56.401L-11.386-55.287Q-11.386-55.078-10.730-55.078L-10.730-54.798M-9.400-56.333Q-9.400-56.654-9.275-56.943Q-9.150-57.232-8.925-57.455Q-8.699-57.679-8.404-57.799Q-8.108-57.919-7.790-57.919Q-7.462-57.919-7.200-57.819Q-6.939-57.720-6.763-57.538Q-6.587-57.355-6.493-57.097Q-6.399-56.839-6.399-56.507Q-6.399-56.415-6.481-56.394L-8.737-56.394L-8.737-56.333Q-8.737-55.745-8.453-55.362Q-8.169-54.979-7.602-54.979Q-7.281-54.979-7.013-55.172Q-6.744-55.365-6.655-55.680Q-6.648-55.721-6.573-55.735L-6.481-55.735Q-6.399-55.711-6.399-55.639Q-6.399-55.632-6.406-55.605Q-6.519-55.208-6.889-54.969Q-7.260-54.730-7.684-54.730Q-8.122-54.730-8.522-54.938Q-8.921-55.147-9.161-55.514Q-9.400-55.881-9.400-56.333M-8.730-56.603L-6.915-56.603Q-6.915-56.880-7.013-57.132Q-7.110-57.385-7.308-57.541Q-7.506-57.696-7.790-57.696Q-8.067-57.696-8.281-57.538Q-8.494-57.379-8.612-57.124Q-8.730-56.869-8.730-56.603M-4.423-54.825L-5.404-57.324Q-5.466-57.467-5.584-57.502Q-5.702-57.536-5.917-57.536L-5.917-57.816L-4.437-57.816L-4.437-57.536Q-4.816-57.536-4.816-57.375Q-4.816-57.365-4.803-57.324L-4.088-55.492L-3.415-57.197Q-3.446-57.269-3.446-57.297Q-3.446-57.324-3.473-57.324Q-3.535-57.471-3.653-57.503Q-3.771-57.536-3.982-57.536L-3.982-57.816L-2.585-57.816L-2.585-57.536Q-2.961-57.536-2.961-57.375Q-2.961-57.344-2.954-57.324L-2.198-55.386L-1.511-57.136Q-1.491-57.187-1.491-57.242Q-1.491-57.382-1.604-57.459Q-1.716-57.536-1.857-57.536L-1.857-57.816L-0.636-57.816L-0.636-57.536Q-0.841-57.536-0.997-57.430Q-1.152-57.324-1.224-57.136L-2.130-54.825Q-2.164-54.730-2.277-54.730L-2.345-54.730Q-2.455-54.730-2.492-54.825L-3.275-56.828L-4.061-54.825Q-4.095-54.730-4.208-54.730L-4.276-54.730Q-4.386-54.730-4.423-54.825\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(263.788 1.75)\">\u003Cpath d=\"M-0.247-55.526Q-0.247-55.858-0.024-56.085Q0.200-56.312 0.544-56.440Q0.887-56.569 1.260-56.621Q1.632-56.674 1.937-56.674L1.937-56.927Q1.937-57.132 1.829-57.312Q1.721-57.491 1.540-57.594Q1.359-57.696 1.151-57.696Q0.744-57.696 0.508-57.604Q0.597-57.567 0.643-57.483Q0.689-57.399 0.689-57.297Q0.689-57.201 0.643-57.122Q0.597-57.044 0.516-56.999Q0.436-56.955 0.347-56.955Q0.197-56.955 0.096-57.052Q-0.005-57.150-0.005-57.297Q-0.005-57.919 1.151-57.919Q1.362-57.919 1.612-57.855Q1.861-57.792 2.063-57.673Q2.265-57.553 2.391-57.368Q2.518-57.184 2.518-56.941L2.518-55.365Q2.518-55.249 2.579-55.153Q2.641-55.058 2.754-55.058Q2.863-55.058 2.928-55.152Q2.993-55.246 2.993-55.365L2.993-55.813L3.259-55.813L3.259-55.365Q3.259-55.095 3.032-54.930Q2.805-54.764 2.525-54.764Q2.316-54.764 2.179-54.918Q2.043-55.071 2.019-55.287Q1.872-55.020 1.590-54.875Q1.308-54.730 0.983-54.730Q0.706-54.730 0.422-54.805Q0.139-54.880-0.054-55.059Q-0.247-55.239-0.247-55.526M0.368-55.526Q0.368-55.352 0.469-55.222Q0.569-55.092 0.725-55.022Q0.880-54.952 1.045-54.952Q1.263-54.952 1.472-55.049Q1.680-55.147 1.808-55.328Q1.937-55.509 1.937-55.735L1.937-56.463Q1.612-56.463 1.246-56.372Q0.880-56.281 0.624-56.069Q0.368-55.858 0.368-55.526M5.426-54.798L3.690-54.798L3.690-55.078Q3.919-55.078 4.068-55.112Q4.216-55.147 4.216-55.287L4.216-57.136Q4.216-57.406 4.109-57.467Q4.001-57.529 3.690-57.529L3.690-57.809L4.719-57.884L4.719-57.177Q4.849-57.485 5.091-57.684Q5.334-57.884 5.652-57.884Q5.871-57.884 6.042-57.760Q6.213-57.635 6.213-57.423Q6.213-57.286 6.113-57.187Q6.014-57.088 5.881-57.088Q5.744-57.088 5.645-57.187Q5.546-57.286 5.546-57.423Q5.546-57.563 5.645-57.662Q5.355-57.662 5.155-57.466Q4.955-57.269 4.862-56.975Q4.770-56.681 4.770-56.401L4.770-55.287Q4.770-55.078 5.426-55.078L5.426-54.798M6.797-56.309Q6.797-56.647 6.937-56.938Q7.077-57.228 7.322-57.442Q7.566-57.655 7.870-57.770Q8.174-57.884 8.499-57.884Q8.769-57.884 9.032-57.785Q9.296-57.686 9.487-57.508L9.487-58.906Q9.487-59.176 9.379-59.238Q9.272-59.299 8.961-59.299L8.961-59.580L10.037-59.655L10.037-55.471Q10.037-55.283 10.092-55.200Q10.147-55.116 10.247-55.097Q10.348-55.078 10.564-55.078L10.564-54.798L9.456-54.730L9.456-55.147Q9.039-54.730 8.414-54.730Q7.983-54.730 7.610-54.942Q7.238-55.153 7.017-55.514Q6.797-55.875 6.797-56.309M8.472-54.952Q8.680-54.952 8.867-55.024Q9.053-55.095 9.207-55.232Q9.360-55.369 9.456-55.547L9.456-57.156Q9.371-57.303 9.225-57.423Q9.080-57.543 8.911-57.602Q8.742-57.662 8.561-57.662Q8-57.662 7.732-57.273Q7.463-56.883 7.463-56.302Q7.463-55.731 7.698-55.341Q7.932-54.952 8.472-54.952\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"M79.994-54.798H4.326\"\u002F>\u003Cpath stroke=\"none\" d=\"m1.126-54.798 5.12 2.56-1.92-2.56 1.92-2.56\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(46.67 -5.294)\">\u003Cpath d=\"M-28.060-54.798L-30.193-54.798L-30.193-55.078Q-29.471-55.078-29.471-55.287L-29.471-59.088Q-29.471-59.299-30.193-59.299L-30.193-59.580L-28.060-59.580L-28.060-59.299Q-28.781-59.299-28.781-59.088L-28.781-56.862L-26.412-58.900Q-26.296-59.009-26.296-59.111Q-26.296-59.204-26.373-59.252Q-26.450-59.299-26.546-59.299L-26.546-59.580L-24.943-59.580L-24.943-59.299Q-25.230-59.299-25.507-59.194Q-25.783-59.088-26.002-58.900L-27.386-57.710L-25.695-55.478Q-25.558-55.300-25.448-55.215Q-25.339-55.130-25.214-55.104Q-25.090-55.078-24.840-55.078L-24.840-54.798L-26.723-54.798L-26.723-55.078Q-26.570-55.078-26.464-55.109Q-26.358-55.140-26.358-55.253Q-26.358-55.338-26.470-55.478L-27.855-57.303L-28.781-56.514L-28.781-55.287Q-28.781-55.078-28.060-55.078L-28.060-54.798M-20.106-54.798L-24.074-54.798L-24.074-55.078Q-23.353-55.078-23.353-55.287L-23.353-59.088Q-23.353-59.299-24.074-59.299L-24.074-59.580L-21.760-59.580L-21.760-59.299Q-22.078-59.299-22.371-59.264Q-22.663-59.228-22.663-59.088L-22.663-55.287Q-22.663-55.147-22.574-55.112Q-22.485-55.078-22.297-55.078L-21.675-55.078Q-21.261-55.078-20.983-55.181Q-20.704-55.283-20.539-55.485Q-20.373-55.687-20.291-55.974Q-20.209-56.261-20.164-56.668L-19.898-56.668\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(46.67 -5.294)\">\u003Cpath d=\"M-14.857-53.441L-16.487-53.441L-16.487-53.721Q-16.258-53.721-16.109-53.756Q-15.961-53.790-15.961-53.930L-15.961-57.276Q-15.961-57.447-16.097-57.488Q-16.234-57.529-16.487-57.529L-16.487-57.809L-15.407-57.884L-15.407-57.478Q-15.185-57.679-14.898-57.782Q-14.610-57.884-14.303-57.884Q-13.876-57.884-13.512-57.671Q-13.148-57.457-12.934-57.093Q-12.720-56.729-12.720-56.309Q-12.720-55.864-12.960-55.500Q-13.199-55.136-13.592-54.933Q-13.985-54.730-14.429-54.730Q-14.696-54.730-14.944-54.830Q-15.191-54.931-15.379-55.112L-15.379-53.930Q-15.379-53.793-15.231-53.757Q-15.082-53.721-14.857-53.721L-14.857-53.441M-15.379-57.129L-15.379-55.519Q-15.246-55.266-15.003-55.109Q-14.761-54.952-14.484-54.952Q-14.156-54.952-13.903-55.153Q-13.650-55.355-13.517-55.673Q-13.383-55.991-13.383-56.309Q-13.383-56.538-13.448-56.767Q-13.513-56.996-13.641-57.194Q-13.770-57.392-13.964-57.512Q-14.159-57.631-14.392-57.631Q-14.686-57.631-14.954-57.502Q-15.222-57.372-15.379-57.129M-11.510-55.632L-11.510-57.136Q-11.510-57.406-11.618-57.467Q-11.726-57.529-12.037-57.529L-12.037-57.809L-10.929-57.884L-10.929-55.652L-10.929-55.632Q-10.929-55.352-10.878-55.208Q-10.827-55.065-10.685-55.008Q-10.543-54.952-10.256-54.952Q-10.003-54.952-9.798-55.092Q-9.593-55.232-9.477-55.458Q-9.360-55.683-9.360-55.933L-9.360-57.136Q-9.360-57.406-9.468-57.467Q-9.576-57.529-9.887-57.529L-9.887-57.809L-8.779-57.884L-8.779-55.471Q-8.779-55.280-8.726-55.198Q-8.673-55.116-8.573-55.097Q-8.472-55.078-8.256-55.078L-8.256-54.798L-9.333-54.730L-9.333-55.294Q-9.442-55.112-9.588-54.989Q-9.733-54.866-9.919-54.798Q-10.106-54.730-10.307-54.730Q-11.510-54.730-11.510-55.632M-6.001-54.798L-7.604-54.798L-7.604-55.078Q-7.378-55.078-7.229-55.112Q-7.081-55.147-7.081-55.287L-7.081-58.906Q-7.081-59.176-7.188-59.238Q-7.296-59.299-7.604-59.299L-7.604-59.580L-6.527-59.655L-6.527-55.287Q-6.527-55.150-6.377-55.114Q-6.226-55.078-6.001-55.078L-6.001-54.798M-3.738-54.798L-5.341-54.798L-5.341-55.078Q-5.115-55.078-4.967-55.112Q-4.818-55.147-4.818-55.287L-4.818-58.906Q-4.818-59.176-4.926-59.238Q-5.033-59.299-5.341-59.299L-5.341-59.580L-4.264-59.655L-4.264-55.287Q-4.264-55.150-4.114-55.114Q-3.963-55.078-3.738-55.078L-3.738-54.798M-3.143-54.805L-3.143-55.868Q-3.143-55.892-3.116-55.919Q-3.088-55.946-3.065-55.946L-2.955-55.946Q-2.890-55.946-2.877-55.888Q-2.781-55.454-2.535-55.203Q-2.289-54.952-1.875-54.952Q-1.533-54.952-1.280-55.085Q-1.027-55.218-1.027-55.526Q-1.027-55.683-1.121-55.798Q-1.215-55.912-1.354-55.981Q-1.492-56.049-1.660-56.087L-2.241-56.186Q-2.596-56.254-2.870-56.475Q-3.143-56.695-3.143-57.037Q-3.143-57.286-3.032-57.461Q-2.921-57.635-2.735-57.734Q-2.548-57.833-2.333-57.876Q-2.118-57.919-1.875-57.919Q-1.461-57.919-1.181-57.737L-0.966-57.912Q-0.956-57.915-0.949-57.917Q-0.942-57.919-0.932-57.919L-0.880-57.919Q-0.853-57.919-0.829-57.895Q-0.805-57.871-0.805-57.843L-0.805-56.996Q-0.805-56.975-0.829-56.948Q-0.853-56.921-0.880-56.921L-0.993-56.921Q-1.021-56.921-1.046-56.946Q-1.072-56.972-1.072-56.996Q-1.072-57.232-1.178-57.396Q-1.284-57.560-1.467-57.642Q-1.649-57.724-1.882-57.724Q-2.210-57.724-2.466-57.621Q-2.723-57.519-2.723-57.242Q-2.723-57.047-2.540-56.938Q-2.357-56.828-2.128-56.787L-1.554-56.681Q-1.308-56.633-1.094-56.505Q-0.880-56.377-0.744-56.174Q-0.607-55.970-0.607-55.721Q-0.607-55.208-0.973-54.969Q-1.338-54.730-1.875-54.730Q-2.371-54.730-2.702-55.024L-2.969-54.750Q-2.989-54.730-3.017-54.730L-3.065-54.730Q-3.088-54.730-3.116-54.757Q-3.143-54.784-3.143-54.805\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(46.67 -5.294)\">\u003Cpath d=\"M3.530-54.798L3.263-54.798L3.263-58.906Q3.263-59.176 3.156-59.238Q3.048-59.299 2.737-59.299L2.737-59.580L3.817-59.655L3.817-57.485Q4.026-57.676 4.311-57.780Q4.596-57.884 4.894-57.884Q5.212-57.884 5.509-57.763Q5.806-57.642 6.029-57.426Q6.251-57.211 6.377-56.926Q6.504-56.640 6.504-56.309Q6.504-55.864 6.264-55.500Q6.025-55.136 5.632-54.933Q5.239-54.730 4.795-54.730Q4.600-54.730 4.410-54.786Q4.221-54.842 4.060-54.947Q3.899-55.051 3.759-55.212L3.530-54.798M3.845-57.143L3.845-55.526Q3.981-55.266 4.222-55.109Q4.463-54.952 4.740-54.952Q5.034-54.952 5.246-55.059Q5.458-55.167 5.591-55.359Q5.724-55.550 5.783-55.789Q5.841-56.028 5.841-56.309Q5.841-56.668 5.747-56.972Q5.653-57.276 5.425-57.469Q5.198-57.662 4.832-57.662Q4.532-57.662 4.265-57.526Q3.998-57.389 3.845-57.143M7.198-55.526Q7.198-55.858 7.421-56.085Q7.645-56.312 7.989-56.440Q8.332-56.569 8.705-56.621Q9.077-56.674 9.382-56.674L9.382-56.927Q9.382-57.132 9.274-57.312Q9.166-57.491 8.985-57.594Q8.804-57.696 8.596-57.696Q8.189-57.696 7.953-57.604Q8.042-57.567 8.088-57.483Q8.134-57.399 8.134-57.297Q8.134-57.201 8.088-57.122Q8.042-57.044 7.961-56.999Q7.881-56.955 7.792-56.955Q7.642-56.955 7.541-57.052Q7.440-57.150 7.440-57.297Q7.440-57.919 8.596-57.919Q8.807-57.919 9.057-57.855Q9.306-57.792 9.508-57.673Q9.710-57.553 9.836-57.368Q9.963-57.184 9.963-56.941L9.963-55.365Q9.963-55.249 10.024-55.153Q10.086-55.058 10.199-55.058Q10.308-55.058 10.373-55.152Q10.438-55.246 10.438-55.365L10.438-55.813L10.704-55.813L10.704-55.365Q10.704-55.095 10.477-54.930Q10.250-54.764 9.970-54.764Q9.761-54.764 9.624-54.918Q9.488-55.071 9.464-55.287Q9.317-55.020 9.035-54.875Q8.753-54.730 8.428-54.730Q8.151-54.730 7.867-54.805Q7.584-54.880 7.391-55.059Q7.198-55.239 7.198-55.526M7.813-55.526Q7.813-55.352 7.914-55.222Q8.014-55.092 8.170-55.022Q8.326-54.952 8.490-54.952Q8.708-54.952 8.917-55.049Q9.125-55.147 9.253-55.328Q9.382-55.509 9.382-55.735L9.382-56.463Q9.057-56.463 8.691-56.372Q8.326-56.281 8.069-56.069Q7.813-55.858 7.813-55.526M11.121-56.309Q11.121-56.637 11.256-56.938Q11.391-57.238 11.627-57.459Q11.863-57.679 12.167-57.799Q12.471-57.919 12.796-57.919Q13.302-57.919 13.651-57.816Q13.999-57.714 13.999-57.338Q13.999-57.191 13.902-57.090Q13.805-56.989 13.658-56.989Q13.504-56.989 13.405-57.088Q13.305-57.187 13.305-57.338Q13.305-57.526 13.446-57.618Q13.244-57.669 12.803-57.669Q12.448-57.669 12.219-57.473Q11.990-57.276 11.889-56.967Q11.788-56.657 11.788-56.309Q11.788-55.960 11.914-55.654Q12.041-55.348 12.295-55.164Q12.550-54.979 12.906-54.979Q13.128-54.979 13.312-55.063Q13.497-55.147 13.632-55.302Q13.767-55.458 13.825-55.666Q13.839-55.721 13.893-55.721L14.006-55.721Q14.037-55.721 14.059-55.697Q14.081-55.673 14.081-55.639L14.081-55.618Q13.996-55.331 13.808-55.133Q13.620-54.935 13.355-54.832Q13.090-54.730 12.796-54.730Q12.366-54.730 11.978-54.936Q11.590-55.143 11.356-55.506Q11.121-55.868 11.121-56.309\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(46.67 -5.294)\">\u003Cpath d=\"M16.063-54.798L14.480-54.798L14.480-55.078Q14.709-55.078 14.858-55.112Q15.006-55.147 15.006-55.287L15.006-58.906Q15.006-59.176 14.899-59.238Q14.791-59.299 14.480-59.299L14.480-59.580L15.560-59.655L15.560-56.367L16.545-57.136Q16.750-57.273 16.750-57.423Q16.750-57.467 16.709-57.502Q16.668-57.536 16.623-57.536L16.623-57.816L17.987-57.816L17.987-57.536Q17.498-57.536 16.979-57.136L16.422-56.702L17.399-55.478Q17.601-55.232 17.734-55.155Q17.867-55.078 18.154-55.078L18.154-54.798L16.722-54.798L16.722-55.078Q16.910-55.078 16.910-55.191Q16.910-55.287 16.756-55.478L16.022-56.387L15.540-56.008L15.540-55.287Q15.540-55.150 15.688-55.114Q15.837-55.078 16.063-55.078\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" style=\"stroke-width:1.2\">\u003Cpath fill=\"none\" d=\"M143.39-54.798h75.667\"\u002F>\u003Cpath stroke=\"none\" d=\"m222.257-54.798-5.12-2.56 1.92 2.56-1.92 2.56\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(181.968 -5.294)\">\u003Cpath d=\"M-28.508-54.798L-30.244-54.798L-30.244-55.078Q-30.015-55.078-29.866-55.112Q-29.718-55.147-29.718-55.287L-29.718-57.136Q-29.718-57.406-29.825-57.467Q-29.933-57.529-30.244-57.529L-30.244-57.809L-29.215-57.884L-29.215-57.177Q-29.085-57.485-28.843-57.684Q-28.600-57.884-28.282-57.884Q-28.063-57.884-27.892-57.760Q-27.721-57.635-27.721-57.423Q-27.721-57.286-27.821-57.187Q-27.920-57.088-28.053-57.088Q-28.190-57.088-28.289-57.187Q-28.388-57.286-28.388-57.423Q-28.388-57.563-28.289-57.662Q-28.579-57.662-28.779-57.466Q-28.979-57.269-29.072-56.975Q-29.164-56.681-29.164-56.401L-29.164-55.287Q-29.164-55.078-28.508-55.078L-28.508-54.798M-27.178-56.333Q-27.178-56.654-27.053-56.943Q-26.928-57.232-26.703-57.455Q-26.477-57.679-26.182-57.799Q-25.886-57.919-25.568-57.919Q-25.240-57.919-24.978-57.819Q-24.717-57.720-24.541-57.538Q-24.365-57.355-24.271-57.097Q-24.177-56.839-24.177-56.507Q-24.177-56.415-24.259-56.394L-26.515-56.394L-26.515-56.333Q-26.515-55.745-26.231-55.362Q-25.947-54.979-25.380-54.979Q-25.059-54.979-24.791-55.172Q-24.522-55.365-24.433-55.680Q-24.426-55.721-24.351-55.735L-24.259-55.735Q-24.177-55.711-24.177-55.639Q-24.177-55.632-24.184-55.605Q-24.297-55.208-24.667-54.969Q-25.038-54.730-25.462-54.730Q-25.900-54.730-26.300-54.938Q-26.699-55.147-26.939-55.514Q-27.178-55.881-27.178-56.333M-26.508-56.603L-24.693-56.603Q-24.693-56.880-24.791-57.132Q-24.888-57.385-25.086-57.541Q-25.284-57.696-25.568-57.696Q-25.845-57.696-26.059-57.538Q-26.272-57.379-26.390-57.124Q-26.508-56.869-26.508-56.603M-22.201-54.825L-23.182-57.324Q-23.244-57.467-23.362-57.502Q-23.480-57.536-23.695-57.536L-23.695-57.816L-22.215-57.816L-22.215-57.536Q-22.594-57.536-22.594-57.375Q-22.594-57.365-22.581-57.324L-21.866-55.492L-21.193-57.197Q-21.224-57.269-21.224-57.297Q-21.224-57.324-21.251-57.324Q-21.313-57.471-21.431-57.503Q-21.549-57.536-21.760-57.536L-21.760-57.816L-20.363-57.816L-20.363-57.536Q-20.739-57.536-20.739-57.375Q-20.739-57.344-20.732-57.324L-19.976-55.386L-19.289-57.136Q-19.269-57.187-19.269-57.242Q-19.269-57.382-19.382-57.459Q-19.494-57.536-19.634-57.536L-19.634-57.816L-18.414-57.816L-18.414-57.536Q-18.619-57.536-18.775-57.430Q-18.930-57.324-19.002-57.136L-19.908-54.825Q-19.942-54.730-20.055-54.730L-20.123-54.730Q-20.233-54.730-20.270-54.825L-21.053-56.828L-21.839-54.825Q-21.873-54.730-21.986-54.730L-22.054-54.730Q-22.164-54.730-22.201-54.825\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(181.968 -5.294)\">\u003Cpath d=\"M-18.025-55.526Q-18.025-55.858-17.802-56.085Q-17.578-56.312-17.234-56.440Q-16.891-56.569-16.518-56.621Q-16.146-56.674-15.841-56.674L-15.841-56.927Q-15.841-57.132-15.949-57.312Q-16.057-57.491-16.238-57.594Q-16.419-57.696-16.627-57.696Q-17.034-57.696-17.270-57.604Q-17.181-57.567-17.135-57.483Q-17.089-57.399-17.089-57.297Q-17.089-57.201-17.135-57.122Q-17.181-57.044-17.262-56.999Q-17.342-56.955-17.431-56.955Q-17.581-56.955-17.682-57.052Q-17.783-57.150-17.783-57.297Q-17.783-57.919-16.627-57.919Q-16.416-57.919-16.166-57.855Q-15.917-57.792-15.715-57.673Q-15.513-57.553-15.387-57.368Q-15.260-57.184-15.260-56.941L-15.260-55.365Q-15.260-55.249-15.199-55.153Q-15.137-55.058-15.024-55.058Q-14.915-55.058-14.850-55.152Q-14.785-55.246-14.785-55.365L-14.785-55.813L-14.519-55.813L-14.519-55.365Q-14.519-55.095-14.746-54.930Q-14.973-54.764-15.253-54.764Q-15.462-54.764-15.599-54.918Q-15.735-55.071-15.759-55.287Q-15.906-55.020-16.188-54.875Q-16.470-54.730-16.795-54.730Q-17.072-54.730-17.356-54.805Q-17.639-54.880-17.832-55.059Q-18.025-55.239-18.025-55.526M-17.410-55.526Q-17.410-55.352-17.309-55.222Q-17.209-55.092-17.053-55.022Q-16.898-54.952-16.733-54.952Q-16.515-54.952-16.306-55.049Q-16.098-55.147-15.970-55.328Q-15.841-55.509-15.841-55.735L-15.841-56.463Q-16.166-56.463-16.532-56.372Q-16.898-56.281-17.154-56.069Q-17.410-55.858-17.410-55.526M-12.352-54.798L-14.088-54.798L-14.088-55.078Q-13.859-55.078-13.710-55.112Q-13.562-55.147-13.562-55.287L-13.562-57.136Q-13.562-57.406-13.669-57.467Q-13.777-57.529-14.088-57.529L-14.088-57.809L-13.059-57.884L-13.059-57.177Q-12.929-57.485-12.687-57.684Q-12.444-57.884-12.126-57.884Q-11.907-57.884-11.736-57.760Q-11.565-57.635-11.565-57.423Q-11.565-57.286-11.665-57.187Q-11.764-57.088-11.897-57.088Q-12.034-57.088-12.133-57.187Q-12.232-57.286-12.232-57.423Q-12.232-57.563-12.133-57.662Q-12.423-57.662-12.623-57.466Q-12.823-57.269-12.916-56.975Q-13.008-56.681-13.008-56.401L-13.008-55.287Q-13.008-55.078-12.352-55.078L-12.352-54.798M-10.981-56.309Q-10.981-56.647-10.841-56.938Q-10.701-57.228-10.456-57.442Q-10.212-57.655-9.908-57.770Q-9.604-57.884-9.279-57.884Q-9.009-57.884-8.746-57.785Q-8.482-57.686-8.291-57.508L-8.291-58.906Q-8.291-59.176-8.399-59.238Q-8.506-59.299-8.817-59.299L-8.817-59.580L-7.741-59.655L-7.741-55.471Q-7.741-55.283-7.686-55.200Q-7.631-55.116-7.531-55.097Q-7.430-55.078-7.214-55.078L-7.214-54.798L-8.322-54.730L-8.322-55.147Q-8.739-54.730-9.364-54.730Q-9.795-54.730-10.168-54.942Q-10.540-55.153-10.761-55.514Q-10.981-55.875-10.981-56.309M-9.306-54.952Q-9.098-54.952-8.911-55.024Q-8.725-55.095-8.571-55.232Q-8.418-55.369-8.322-55.547L-8.322-57.156Q-8.407-57.303-8.553-57.423Q-8.698-57.543-8.867-57.602Q-9.036-57.662-9.217-57.662Q-9.778-57.662-10.046-57.273Q-10.315-56.883-10.315-56.302Q-10.315-55.731-10.080-55.341Q-9.846-54.952-9.306-54.952\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(181.968 -5.294)\">\u003Cpath d=\"M-2.218-53.441L-3.848-53.441L-3.848-53.721Q-3.619-53.721-3.470-53.756Q-3.322-53.790-3.322-53.930L-3.322-57.276Q-3.322-57.447-3.458-57.488Q-3.595-57.529-3.848-57.529L-3.848-57.809L-2.768-57.884L-2.768-57.478Q-2.546-57.679-2.259-57.782Q-1.971-57.884-1.664-57.884Q-1.237-57.884-0.873-57.671Q-0.509-57.457-0.295-57.093Q-0.081-56.729-0.081-56.309Q-0.081-55.864-0.321-55.500Q-0.560-55.136-0.953-54.933Q-1.346-54.730-1.790-54.730Q-2.057-54.730-2.305-54.830Q-2.552-54.931-2.740-55.112L-2.740-53.930Q-2.740-53.793-2.592-53.757Q-2.443-53.721-2.218-53.721L-2.218-53.441M-2.740-57.129L-2.740-55.519Q-2.607-55.266-2.364-55.109Q-2.122-54.952-1.845-54.952Q-1.517-54.952-1.264-55.153Q-1.011-55.355-0.878-55.673Q-0.744-55.991-0.744-56.309Q-0.744-56.538-0.809-56.767Q-0.874-56.996-1.002-57.194Q-1.131-57.392-1.325-57.512Q-1.520-57.631-1.753-57.631Q-2.047-57.631-2.315-57.502Q-2.583-57.372-2.740-57.129M1.129-55.632L1.129-57.136Q1.129-57.406 1.021-57.467Q0.913-57.529 0.602-57.529L0.602-57.809L1.710-57.884L1.710-55.652L1.710-55.632Q1.710-55.352 1.761-55.208Q1.812-55.065 1.954-55.008Q2.096-54.952 2.383-54.952Q2.636-54.952 2.841-55.092Q3.046-55.232 3.162-55.458Q3.279-55.683 3.279-55.933L3.279-57.136Q3.279-57.406 3.171-57.467Q3.063-57.529 2.752-57.529L2.752-57.809L3.860-57.884L3.860-55.471Q3.860-55.280 3.913-55.198Q3.966-55.116 4.066-55.097Q4.167-55.078 4.383-55.078L4.383-54.798L3.306-54.730L3.306-55.294Q3.197-55.112 3.051-54.989Q2.906-54.866 2.720-54.798Q2.533-54.730 2.332-54.730Q1.129-54.730 1.129-55.632M6.638-54.798L5.035-54.798L5.035-55.078Q5.261-55.078 5.410-55.112Q5.558-55.147 5.558-55.287L5.558-58.906Q5.558-59.176 5.451-59.238Q5.343-59.299 5.035-59.299L5.035-59.580L6.112-59.655L6.112-55.287Q6.112-55.150 6.262-55.114Q6.413-55.078 6.638-55.078L6.638-54.798M8.901-54.798L7.298-54.798L7.298-55.078Q7.524-55.078 7.672-55.112Q7.821-55.147 7.821-55.287L7.821-58.906Q7.821-59.176 7.713-59.238Q7.606-59.299 7.298-59.299L7.298-59.580L8.375-59.655L8.375-55.287Q8.375-55.150 8.525-55.114Q8.676-55.078 8.901-55.078L8.901-54.798M9.496-54.805L9.496-55.868Q9.496-55.892 9.523-55.919Q9.551-55.946 9.574-55.946L9.684-55.946Q9.749-55.946 9.762-55.888Q9.858-55.454 10.104-55.203Q10.350-54.952 10.764-54.952Q11.106-54.952 11.359-55.085Q11.612-55.218 11.612-55.526Q11.612-55.683 11.518-55.798Q11.424-55.912 11.285-55.981Q11.147-56.049 10.979-56.087L10.398-56.186Q10.043-56.254 9.769-56.475Q9.496-56.695 9.496-57.037Q9.496-57.286 9.607-57.461Q9.718-57.635 9.904-57.734Q10.091-57.833 10.306-57.876Q10.521-57.919 10.764-57.919Q11.178-57.919 11.458-57.737L11.673-57.912Q11.683-57.915 11.690-57.917Q11.697-57.919 11.707-57.919L11.759-57.919Q11.786-57.919 11.810-57.895Q11.834-57.871 11.834-57.843L11.834-56.996Q11.834-56.975 11.810-56.948Q11.786-56.921 11.759-56.921L11.646-56.921Q11.618-56.921 11.593-56.946Q11.567-56.972 11.567-56.996Q11.567-57.232 11.461-57.396Q11.355-57.560 11.172-57.642Q10.990-57.724 10.757-57.724Q10.429-57.724 10.173-57.621Q9.916-57.519 9.916-57.242Q9.916-57.047 10.099-56.938Q10.282-56.828 10.511-56.787L11.085-56.681Q11.331-56.633 11.545-56.505Q11.759-56.377 11.895-56.174Q12.032-55.970 12.032-55.721Q12.032-55.208 11.666-54.969Q11.301-54.730 10.764-54.730Q10.268-54.730 9.937-55.024L9.670-54.750Q9.650-54.730 9.622-54.730L9.574-54.730Q9.551-54.730 9.523-54.757Q9.496-54.784 9.496-54.805\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(181.968 -5.294)\">\u003Cpath d=\"M15.421-55.526Q15.421-55.858 15.644-56.085Q15.868-56.312 16.212-56.440Q16.555-56.569 16.928-56.621Q17.300-56.674 17.605-56.674L17.605-56.927Q17.605-57.132 17.497-57.312Q17.389-57.491 17.208-57.594Q17.027-57.696 16.819-57.696Q16.412-57.696 16.176-57.604Q16.265-57.567 16.311-57.483Q16.357-57.399 16.357-57.297Q16.357-57.201 16.311-57.122Q16.265-57.044 16.184-56.999Q16.104-56.955 16.015-56.955Q15.865-56.955 15.764-57.052Q15.663-57.150 15.663-57.297Q15.663-57.919 16.819-57.919Q17.030-57.919 17.280-57.855Q17.529-57.792 17.731-57.673Q17.933-57.553 18.059-57.368Q18.186-57.184 18.186-56.941L18.186-55.365Q18.186-55.249 18.247-55.153Q18.309-55.058 18.422-55.058Q18.531-55.058 18.596-55.152Q18.661-55.246 18.661-55.365L18.661-55.813L18.927-55.813L18.927-55.365Q18.927-55.095 18.700-54.930Q18.473-54.764 18.193-54.764Q17.984-54.764 17.847-54.918Q17.711-55.071 17.687-55.287Q17.540-55.020 17.258-54.875Q16.976-54.730 16.651-54.730Q16.374-54.730 16.090-54.805Q15.807-54.880 15.614-55.059Q15.421-55.239 15.421-55.526M16.036-55.526Q16.036-55.352 16.137-55.222Q16.237-55.092 16.393-55.022Q16.548-54.952 16.713-54.952Q16.931-54.952 17.140-55.049Q17.348-55.147 17.476-55.328Q17.605-55.509 17.605-55.735L17.605-56.463Q17.280-56.463 16.914-56.372Q16.548-56.281 16.292-56.069Q16.036-55.858 16.036-55.526\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(181.968 -5.294)\">\u003Cpath d=\"M20.521-54.825L19.540-57.324Q19.479-57.467 19.361-57.502Q19.243-57.536 19.027-57.536L19.027-57.816L20.507-57.816L20.507-57.536Q20.128-57.536 20.128-57.375Q20.128-57.365 20.142-57.324L20.856-55.492L21.529-57.197Q21.499-57.269 21.499-57.297Q21.499-57.324 21.471-57.324Q21.410-57.471 21.292-57.503Q21.174-57.536 20.962-57.536L20.962-57.816L22.360-57.816L22.360-57.536Q21.984-57.536 21.984-57.375Q21.984-57.344 21.991-57.324L22.746-55.386L23.433-57.136Q23.454-57.187 23.454-57.242Q23.454-57.382 23.341-57.459Q23.228-57.536 23.088-57.536L23.088-57.816L24.308-57.816L24.308-57.536Q24.103-57.536 23.948-57.430Q23.792-57.324 23.720-57.136L22.815-54.825Q22.780-54.730 22.668-54.730L22.599-54.730Q22.490-54.730 22.452-54.825L21.670-56.828L20.883-54.825Q20.849-54.730 20.736-54.730L20.668-54.730Q20.559-54.730 20.521-54.825\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(181.968 -5.294)\">\u003Cpath d=\"M24.685-55.526Q24.685-55.858 24.908-56.085Q25.132-56.312 25.476-56.440Q25.819-56.569 26.192-56.621Q26.564-56.674 26.869-56.674L26.869-56.927Q26.869-57.132 26.761-57.312Q26.653-57.491 26.472-57.594Q26.291-57.696 26.083-57.696Q25.676-57.696 25.440-57.604Q25.529-57.567 25.575-57.483Q25.621-57.399 25.621-57.297Q25.621-57.201 25.575-57.122Q25.529-57.044 25.448-56.999Q25.368-56.955 25.279-56.955Q25.129-56.955 25.028-57.052Q24.927-57.150 24.927-57.297Q24.927-57.919 26.083-57.919Q26.294-57.919 26.544-57.855Q26.793-57.792 26.995-57.673Q27.197-57.553 27.323-57.368Q27.450-57.184 27.450-56.941L27.450-55.365Q27.450-55.249 27.511-55.153Q27.573-55.058 27.686-55.058Q27.795-55.058 27.860-55.152Q27.925-55.246 27.925-55.365L27.925-55.813L28.191-55.813L28.191-55.365Q28.191-55.095 27.964-54.930Q27.737-54.764 27.457-54.764Q27.248-54.764 27.111-54.918Q26.975-55.071 26.951-55.287Q26.804-55.020 26.522-54.875Q26.240-54.730 25.915-54.730Q25.638-54.730 25.354-54.805Q25.071-54.880 24.878-55.059Q24.685-55.239 24.685-55.526M25.300-55.526Q25.300-55.352 25.401-55.222Q25.501-55.092 25.657-55.022Q25.812-54.952 25.977-54.952Q26.195-54.952 26.404-55.049Q26.612-55.147 26.740-55.328Q26.869-55.509 26.869-55.735L26.869-56.463Q26.544-56.463 26.178-56.372Q25.812-56.281 25.556-56.069Q25.300-55.858 25.300-55.526\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(181.968 -5.294)\">\u003Cpath d=\"M28.731-53.663Q28.861-53.595 28.998-53.595Q29.169-53.595 29.319-53.684Q29.470-53.773 29.581-53.918Q29.692-54.063 29.770-54.231L30.034-54.798L28.865-57.324Q28.790-57.471 28.660-57.503Q28.530-57.536 28.297-57.536L28.297-57.816L29.818-57.816L29.818-57.536Q29.470-57.536 29.470-57.389Q29.473-57.368 29.475-57.351Q29.477-57.334 29.477-57.324L30.334-55.465L31.107-57.136Q31.141-57.204 31.141-57.283Q31.141-57.396 31.057-57.466Q30.974-57.536 30.861-57.536L30.861-57.816L32.057-57.816L32.057-57.536Q31.838-57.536 31.666-57.432Q31.493-57.327 31.401-57.136L30.064-54.231Q29.894-53.861 29.624-53.615Q29.353-53.369 28.998-53.369Q28.728-53.369 28.509-53.535Q28.290-53.701 28.290-53.964Q28.290-54.101 28.383-54.190Q28.475-54.278 28.615-54.278Q28.752-54.278 28.841-54.190Q28.930-54.101 28.930-53.964Q28.930-53.861 28.877-53.783Q28.824-53.704 28.731-53.663\" 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(108.357 35.424)\">\u003Cpath d=\"M-29.451-54.798L-29.718-54.798L-29.718-58.906Q-29.718-59.176-29.825-59.238Q-29.933-59.299-30.244-59.299L-30.244-59.580L-29.164-59.655L-29.164-57.485Q-28.955-57.676-28.670-57.780Q-28.384-57.884-28.087-57.884Q-27.769-57.884-27.472-57.763Q-27.175-57.642-26.952-57.426Q-26.730-57.211-26.604-56.926Q-26.477-56.640-26.477-56.309Q-26.477-55.864-26.717-55.500Q-26.956-55.136-27.349-54.933Q-27.742-54.730-28.186-54.730Q-28.381-54.730-28.571-54.786Q-28.760-54.842-28.921-54.947Q-29.082-55.051-29.222-55.212L-29.451-54.798M-29.136-57.143L-29.136-55.526Q-29-55.266-28.759-55.109Q-28.518-54.952-28.241-54.952Q-27.947-54.952-27.735-55.059Q-27.523-55.167-27.390-55.359Q-27.257-55.550-27.198-55.789Q-27.140-56.028-27.140-56.309Q-27.140-56.668-27.234-56.972Q-27.328-57.276-27.556-57.469Q-27.783-57.662-28.149-57.662Q-28.449-57.662-28.716-57.526Q-28.983-57.389-29.136-57.143M-25.783-55.526Q-25.783-55.858-25.560-56.085Q-25.336-56.312-24.992-56.440Q-24.649-56.569-24.276-56.621Q-23.904-56.674-23.599-56.674L-23.599-56.927Q-23.599-57.132-23.707-57.312Q-23.815-57.491-23.996-57.594Q-24.177-57.696-24.385-57.696Q-24.792-57.696-25.028-57.604Q-24.939-57.567-24.893-57.483Q-24.847-57.399-24.847-57.297Q-24.847-57.201-24.893-57.122Q-24.939-57.044-25.020-56.999Q-25.100-56.955-25.189-56.955Q-25.339-56.955-25.440-57.052Q-25.541-57.150-25.541-57.297Q-25.541-57.919-24.385-57.919Q-24.174-57.919-23.924-57.855Q-23.675-57.792-23.473-57.673Q-23.271-57.553-23.145-57.368Q-23.018-57.184-23.018-56.941L-23.018-55.365Q-23.018-55.249-22.957-55.153Q-22.895-55.058-22.782-55.058Q-22.673-55.058-22.608-55.152Q-22.543-55.246-22.543-55.365L-22.543-55.813L-22.277-55.813L-22.277-55.365Q-22.277-55.095-22.504-54.930Q-22.731-54.764-23.011-54.764Q-23.220-54.764-23.357-54.918Q-23.493-55.071-23.517-55.287Q-23.664-55.020-23.946-54.875Q-24.228-54.730-24.553-54.730Q-24.830-54.730-25.114-54.805Q-25.397-54.880-25.590-55.059Q-25.783-55.239-25.783-55.526M-25.168-55.526Q-25.168-55.352-25.067-55.222Q-24.967-55.092-24.811-55.022Q-24.655-54.952-24.491-54.952Q-24.273-54.952-24.064-55.049Q-23.856-55.147-23.728-55.328Q-23.599-55.509-23.599-55.735L-23.599-56.463Q-23.924-56.463-24.290-56.372Q-24.655-56.281-24.912-56.069Q-25.168-55.858-25.168-55.526M-20.192-54.798L-21.795-54.798L-21.795-55.078Q-21.569-55.078-21.420-55.112Q-21.272-55.147-21.272-55.287L-21.272-58.906Q-21.272-59.176-21.379-59.238Q-21.487-59.299-21.795-59.299L-21.795-59.580L-20.718-59.655L-20.718-55.287Q-20.718-55.150-20.568-55.114Q-20.417-55.078-20.192-55.078L-20.192-54.798M-19.539-55.526Q-19.539-55.858-19.315-56.085Q-19.091-56.312-18.748-56.440Q-18.404-56.569-18.031-56.621Q-17.659-56.674-17.355-56.674L-17.355-56.927Q-17.355-57.132-17.462-57.312Q-17.570-57.491-17.751-57.594Q-17.932-57.696-18.141-57.696Q-18.548-57.696-18.783-57.604Q-18.695-57.567-18.648-57.483Q-18.602-57.399-18.602-57.297Q-18.602-57.201-18.648-57.122Q-18.695-57.044-18.775-56.999Q-18.855-56.955-18.944-56.955Q-19.094-56.955-19.195-57.052Q-19.296-57.150-19.296-57.297Q-19.296-57.919-18.141-57.919Q-17.929-57.919-17.679-57.855Q-17.430-57.792-17.228-57.673Q-17.027-57.553-16.900-57.368Q-16.774-57.184-16.774-56.941L-16.774-55.365Q-16.774-55.249-16.712-55.153Q-16.651-55.058-16.538-55.058Q-16.428-55.058-16.364-55.152Q-16.299-55.246-16.299-55.365L-16.299-55.813L-16.032-55.813L-16.032-55.365Q-16.032-55.095-16.259-54.930Q-16.487-54.764-16.767-54.764Q-16.975-54.764-17.112-54.918Q-17.249-55.071-17.273-55.287Q-17.420-55.020-17.702-54.875Q-17.984-54.730-18.308-54.730Q-18.585-54.730-18.869-54.805Q-19.153-54.880-19.346-55.059Q-19.539-55.239-19.539-55.526M-18.924-55.526Q-18.924-55.352-18.823-55.222Q-18.722-55.092-18.566-55.022Q-18.411-54.952-18.247-54.952Q-18.028-54.952-17.820-55.049Q-17.611-55.147-17.483-55.328Q-17.355-55.509-17.355-55.735L-17.355-56.463Q-17.679-56.463-18.045-56.372Q-18.411-56.281-18.667-56.069Q-18.924-55.858-18.924-55.526M-13.933-54.798L-15.567-54.798L-15.567-55.078Q-15.338-55.078-15.189-55.112Q-15.041-55.147-15.041-55.287L-15.041-57.136Q-15.041-57.406-15.148-57.467Q-15.256-57.529-15.567-57.529L-15.567-57.809L-14.508-57.884L-14.508-57.235Q-14.337-57.543-14.032-57.714Q-13.728-57.884-13.383-57.884Q-12.877-57.884-12.593-57.661Q-12.310-57.437-12.310-56.941L-12.310-55.287Q-12.310-55.150-12.161-55.114Q-12.012-55.078-11.787-55.078L-11.787-54.798L-13.417-54.798L-13.417-55.078Q-13.188-55.078-13.040-55.112Q-12.891-55.147-12.891-55.287L-12.891-56.927Q-12.891-57.262-13.010-57.462Q-13.130-57.662-13.445-57.662Q-13.715-57.662-13.949-57.526Q-14.183-57.389-14.321-57.155Q-14.460-56.921-14.460-56.647L-14.460-55.287Q-14.460-55.150-14.309-55.114Q-14.159-55.078-13.933-55.078L-13.933-54.798M-11.199-56.309Q-11.199-56.637-11.064-56.938Q-10.929-57.238-10.693-57.459Q-10.457-57.679-10.153-57.799Q-9.849-57.919-9.524-57.919Q-9.018-57.919-8.670-57.816Q-8.321-57.714-8.321-57.338Q-8.321-57.191-8.418-57.090Q-8.516-56.989-8.663-56.989Q-8.817-56.989-8.916-57.088Q-9.015-57.187-9.015-57.338Q-9.015-57.526-8.875-57.618Q-9.076-57.669-9.517-57.669Q-9.873-57.669-10.102-57.473Q-10.331-57.276-10.432-56.967Q-10.532-56.657-10.532-56.309Q-10.532-55.960-10.406-55.654Q-10.280-55.348-10.025-55.164Q-9.770-54.979-9.415-54.979Q-9.193-54.979-9.008-55.063Q-8.823-55.147-8.688-55.302Q-8.553-55.458-8.495-55.666Q-8.482-55.721-8.427-55.721L-8.314-55.721Q-8.283-55.721-8.261-55.697Q-8.239-55.673-8.239-55.639L-8.239-55.618Q-8.324-55.331-8.512-55.133Q-8.700-54.935-8.965-54.832Q-9.230-54.730-9.524-54.730Q-9.955-54.730-10.343-54.936Q-10.731-55.143-10.965-55.506Q-11.199-55.868-11.199-56.309M-7.692-56.333Q-7.692-56.654-7.567-56.943Q-7.443-57.232-7.217-57.455Q-6.991-57.679-6.696-57.799Q-6.400-57.919-6.082-57.919Q-5.754-57.919-5.493-57.819Q-5.231-57.720-5.055-57.538Q-4.879-57.355-4.785-57.097Q-4.691-56.839-4.691-56.507Q-4.691-56.415-4.773-56.394L-7.029-56.394L-7.029-56.333Q-7.029-55.745-6.745-55.362Q-6.462-54.979-5.894-54.979Q-5.573-54.979-5.305-55.172Q-5.036-55.365-4.947-55.680Q-4.941-55.721-4.865-55.735L-4.773-55.735Q-4.691-55.711-4.691-55.639Q-4.691-55.632-4.698-55.605Q-4.811-55.208-5.182-54.969Q-5.552-54.730-5.976-54.730Q-6.414-54.730-6.814-54.938Q-7.214-55.147-7.453-55.514Q-7.692-55.881-7.692-56.333M-7.022-56.603L-5.207-56.603Q-5.207-56.880-5.305-57.132Q-5.402-57.385-5.600-57.541Q-5.799-57.696-6.082-57.696Q-6.359-57.696-6.573-57.538Q-6.786-57.379-6.904-57.124Q-7.022-56.869-7.022-56.603\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(108.357 35.424)\">\u003Cpath d=\"M-1.383-54.805L-1.383-55.868Q-1.383-55.892-1.355-55.919Q-1.328-55.946-1.304-55.946L-1.195-55.946Q-1.130-55.946-1.116-55.888Q-1.020-55.454-0.774-55.203Q-0.528-54.952-0.114-54.952Q0.227-54.952 0.480-55.085Q0.733-55.218 0.733-55.526Q0.733-55.683 0.639-55.798Q0.545-55.912 0.407-55.981Q0.268-56.049 0.101-56.087L-0.480-56.186Q-0.836-56.254-1.109-56.475Q-1.383-56.695-1.383-57.037Q-1.383-57.286-1.271-57.461Q-1.160-57.635-0.974-57.734Q-0.788-57.833-0.572-57.876Q-0.357-57.919-0.114-57.919Q0.299-57.919 0.579-57.737L0.795-57.912Q0.805-57.915 0.812-57.917Q0.819-57.919 0.829-57.919L0.880-57.919Q0.907-57.919 0.931-57.895Q0.955-57.871 0.955-57.843L0.955-56.996Q0.955-56.975 0.931-56.948Q0.907-56.921 0.880-56.921L0.767-56.921Q0.740-56.921 0.714-56.946Q0.689-56.972 0.689-56.996Q0.689-57.232 0.583-57.396Q0.477-57.560 0.294-57.642Q0.111-57.724-0.121-57.724Q-0.449-57.724-0.706-57.621Q-0.962-57.519-0.962-57.242Q-0.962-57.047-0.779-56.938Q-0.596-56.828-0.367-56.787L0.207-56.681Q0.453-56.633 0.667-56.505Q0.880-56.377 1.017-56.174Q1.154-55.970 1.154-55.721Q1.154-55.208 0.788-54.969Q0.422-54.730-0.114-54.730Q-0.610-54.730-0.942-55.024L-1.208-54.750Q-1.229-54.730-1.256-54.730L-1.304-54.730Q-1.328-54.730-1.355-54.757Q-1.383-54.784-1.383-54.805M1.741-56.333Q1.741-56.654 1.866-56.943Q1.991-57.232 2.217-57.455Q2.442-57.679 2.738-57.799Q3.033-57.919 3.351-57.919Q3.679-57.919 3.941-57.819Q4.202-57.720 4.378-57.538Q4.554-57.355 4.648-57.097Q4.742-56.839 4.742-56.507Q4.742-56.415 4.660-56.394L2.405-56.394L2.405-56.333Q2.405-55.745 2.688-55.362Q2.972-54.979 3.539-54.979Q3.861-54.979 4.129-55.172Q4.397-55.365 4.486-55.680Q4.493-55.721 4.568-55.735L4.660-55.735Q4.742-55.711 4.742-55.639Q4.742-55.632 4.736-55.605Q4.623-55.208 4.252-54.969Q3.881-54.730 3.457-54.730Q3.020-54.730 2.620-54.938Q2.220-55.147 1.981-55.514Q1.741-55.881 1.741-56.333M2.411-56.603L4.226-56.603Q4.226-56.880 4.129-57.132Q4.032-57.385 3.833-57.541Q3.635-57.696 3.351-57.696Q3.074-57.696 2.861-57.538Q2.647-57.379 2.529-57.124Q2.411-56.869 2.411-56.603M5.857-55.639L5.857-57.536L5.218-57.536L5.218-57.758Q5.535-57.758 5.752-57.968Q5.970-58.178 6.070-58.488Q6.171-58.797 6.171-59.105L6.438-59.105L6.438-57.816L7.514-57.816L7.514-57.536L6.438-57.536L6.438-55.652Q6.438-55.376 6.542-55.177Q6.646-54.979 6.906-54.979Q7.063-54.979 7.169-55.083Q7.275-55.188 7.325-55.341Q7.374-55.495 7.374-55.652L7.374-56.066L7.641-56.066L7.641-55.639Q7.641-55.413 7.542-55.203Q7.443-54.993 7.258-54.861Q7.074-54.730 6.845-54.730Q6.407-54.730 6.132-54.967Q5.857-55.205 5.857-55.639\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(108.357 35.424)\">\u003Cpath d=\"M11.967-54.798L11.700-54.798L11.700-58.906Q11.700-59.176 11.593-59.238Q11.485-59.299 11.174-59.299L11.174-59.580L12.254-59.655L12.254-57.485Q12.463-57.676 12.748-57.780Q13.034-57.884 13.331-57.884Q13.649-57.884 13.946-57.763Q14.243-57.642 14.466-57.426Q14.688-57.211 14.814-56.926Q14.941-56.640 14.941-56.309Q14.941-55.864 14.701-55.500Q14.462-55.136 14.069-54.933Q13.676-54.730 13.232-54.730Q13.037-54.730 12.847-54.786Q12.658-54.842 12.497-54.947Q12.336-55.051 12.196-55.212L11.967-54.798M12.282-57.143L12.282-55.526Q12.418-55.266 12.659-55.109Q12.900-54.952 13.177-54.952Q13.471-54.952 13.683-55.059Q13.895-55.167 14.028-55.359Q14.161-55.550 14.220-55.789Q14.278-56.028 14.278-56.309Q14.278-56.668 14.184-56.972Q14.090-57.276 13.862-57.469Q13.635-57.662 13.269-57.662Q12.969-57.662 12.702-57.526Q12.435-57.389 12.282-57.143\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(108.357 35.424)\">\u003Cpath d=\"M15.697-53.663Q15.827-53.595 15.964-53.595Q16.135-53.595 16.285-53.684Q16.436-53.773 16.547-53.918Q16.658-54.063 16.736-54.231L17-54.798L15.831-57.324Q15.756-57.471 15.626-57.503Q15.496-57.536 15.263-57.536L15.263-57.816L16.784-57.816L16.784-57.536Q16.436-57.536 16.436-57.389Q16.439-57.368 16.441-57.351Q16.443-57.334 16.443-57.324L17.300-55.465L18.073-57.136Q18.107-57.204 18.107-57.283Q18.107-57.396 18.023-57.466Q17.940-57.536 17.827-57.536L17.827-57.816L19.023-57.816L19.023-57.536Q18.804-57.536 18.632-57.432Q18.459-57.327 18.367-57.136L17.030-54.231Q16.860-53.861 16.590-53.615Q16.319-53.369 15.964-53.369Q15.694-53.369 15.475-53.535Q15.256-53.701 15.256-53.964Q15.256-54.101 15.349-54.190Q15.441-54.278 15.581-54.278Q15.718-54.278 15.807-54.190Q15.896-54.101 15.896-53.964Q15.896-53.861 15.843-53.783Q15.790-53.704 15.697-53.663\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(108.357 35.424)\">\u003Cpath d=\"M23.065-54.798L22.798-54.798L22.798-58.906Q22.798-59.176 22.691-59.238Q22.583-59.299 22.272-59.299L22.272-59.580L23.352-59.655L23.352-57.485Q23.561-57.676 23.846-57.780Q24.131-57.884 24.429-57.884Q24.747-57.884 25.044-57.763Q25.341-57.642 25.564-57.426Q25.786-57.211 25.912-56.926Q26.039-56.640 26.039-56.309Q26.039-55.864 25.799-55.500Q25.560-55.136 25.167-54.933Q24.774-54.730 24.330-54.730Q24.135-54.730 23.945-54.786Q23.756-54.842 23.595-54.947Q23.434-55.051 23.294-55.212L23.065-54.798M23.380-57.143L23.380-55.526Q23.516-55.266 23.757-55.109Q23.998-54.952 24.275-54.952Q24.569-54.952 24.781-55.059Q24.993-55.167 25.126-55.359Q25.259-55.550 25.318-55.789Q25.376-56.028 25.376-56.309Q25.376-56.668 25.282-56.972Q25.188-57.276 24.960-57.469Q24.733-57.662 24.367-57.662Q24.067-57.662 23.800-57.526Q23.533-57.389 23.380-57.143\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(108.357 35.424)\">\u003Cpath d=\"M26.849-56.333Q26.849-56.654 26.974-56.943Q27.099-57.232 27.325-57.455Q27.550-57.679 27.846-57.799Q28.141-57.919 28.459-57.919Q28.787-57.919 29.049-57.819Q29.310-57.720 29.486-57.538Q29.662-57.355 29.756-57.097Q29.850-56.839 29.850-56.507Q29.850-56.415 29.768-56.394L27.513-56.394L27.513-56.333Q27.513-55.745 27.796-55.362Q28.080-54.979 28.647-54.979Q28.969-54.979 29.237-55.172Q29.505-55.365 29.594-55.680Q29.601-55.721 29.676-55.735L29.768-55.735Q29.850-55.711 29.850-55.639Q29.850-55.632 29.844-55.605Q29.731-55.208 29.360-54.969Q28.989-54.730 28.565-54.730Q28.128-54.730 27.728-54.938Q27.328-55.147 27.089-55.514Q26.849-55.881 26.849-56.333M27.519-56.603L29.334-56.603Q29.334-56.880 29.237-57.132Q29.139-57.385 28.941-57.541Q28.743-57.696 28.459-57.696Q28.182-57.696 27.969-57.538Q27.755-57.379 27.637-57.124Q27.519-56.869 27.519-56.603M30.965-55.639L30.965-57.536L30.326-57.536L30.326-57.758Q30.643-57.758 30.860-57.968Q31.077-58.178 31.178-58.488Q31.279-58.797 31.279-59.105L31.546-59.105L31.546-57.816L32.622-57.816L32.622-57.536L31.546-57.536L31.546-55.652Q31.546-55.376 31.650-55.177Q31.754-54.979 32.014-54.979Q32.171-54.979 32.277-55.083Q32.383-55.188 32.433-55.341Q32.482-55.495 32.482-55.652L32.482-56.066L32.749-56.066L32.749-55.639Q32.749-55.413 32.650-55.203Q32.551-54.993 32.366-54.861Q32.181-54.730 31.952-54.730Q31.515-54.730 31.240-54.967Q30.965-55.205 30.965-55.639M33.617-55.526Q33.617-55.858 33.841-56.085Q34.065-56.312 34.408-56.440Q34.752-56.569 35.124-56.621Q35.497-56.674 35.801-56.674L35.801-56.927Q35.801-57.132 35.693-57.312Q35.586-57.491 35.405-57.594Q35.223-57.696 35.015-57.696Q34.608-57.696 34.372-57.604Q34.461-57.567 34.507-57.483Q34.554-57.399 34.554-57.297Q34.554-57.201 34.507-57.122Q34.461-57.044 34.381-56.999Q34.301-56.955 34.212-56.955Q34.061-56.955 33.961-57.052Q33.860-57.150 33.860-57.297Q33.860-57.919 35.015-57.919Q35.227-57.919 35.476-57.855Q35.726-57.792 35.928-57.673Q36.129-57.553 36.256-57.368Q36.382-57.184 36.382-56.941L36.382-55.365Q36.382-55.249 36.444-55.153Q36.505-55.058 36.618-55.058Q36.727-55.058 36.792-55.152Q36.857-55.246 36.857-55.365L36.857-55.813L37.124-55.813L37.124-55.365Q37.124-55.095 36.897-54.930Q36.669-54.764 36.389-54.764Q36.180-54.764 36.044-54.918Q35.907-55.071 35.883-55.287Q35.736-55.020 35.454-54.875Q35.172-54.730 34.847-54.730Q34.571-54.730 34.287-54.805Q34.003-54.880 33.810-55.059Q33.617-55.239 33.617-55.526M34.232-55.526Q34.232-55.352 34.333-55.222Q34.434-55.092 34.589-55.022Q34.745-54.952 34.909-54.952Q35.128-54.952 35.336-55.049Q35.545-55.147 35.673-55.328Q35.801-55.509 35.801-55.735L35.801-56.463Q35.476-56.463 35.111-56.372Q34.745-56.281 34.489-56.069Q34.232-55.858 34.232-55.526\" 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 KL-anchored objective as a tug-of-war. Reward pulls the policy away from the reference toward high-scoring responses; the KL penalty pulls it back. The coefficient \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> sets the balance point.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:274.972px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 206.229 91.099\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg fill=\"var(--tk-soft-neutral)\" style=\"stroke-width:.8\">\u003Cpath d=\"M-65.203-16.462h28.452v-22.762h-28.452Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\">\u003Cg transform=\"translate(-4.184 .944)\">\u003Cpath d=\"M-50.497-26.905Q-50.497-27.124-50.364-27.288Q-50.231-27.452-50.016-27.452Q-49.883-27.452-49.789-27.368Q-49.696-27.284-49.696-27.148Q-49.696-26.968-49.823-26.825Q-49.950-26.683-50.129-26.683Q-49.946-26.468-49.563-26.468Q-49.012-26.468-48.661-26.960Q-48.309-27.452-48.153-28.089Q-48.504-27.765-48.946-27.765Q-49.270-27.765-49.520-27.874Q-49.770-27.984-49.909-28.208Q-50.047-28.433-50.047-28.765Q-50.047-29.015-49.973-29.302Q-49.899-29.589-49.762-29.941Q-49.625-30.292-49.563-30.460Q-49.473-30.683-49.473-30.866Q-49.473-31.116-49.641-31.116Q-49.957-31.116-50.166-30.810Q-50.375-30.503-50.481-30.116Q-50.493-30.042-50.563-30.042L-50.664-30.042Q-50.700-30.042-50.727-30.077Q-50.754-30.113-50.754-30.140L-50.754-30.171Q-50.629-30.632-50.332-31.001Q-50.036-31.370-49.625-31.370Q-49.430-31.370-49.256-31.286Q-49.082-31.202-48.981-31.050Q-48.879-30.898-48.879-30.691Q-48.879-30.538-48.938-30.402Q-49.028-30.171-49.129-29.905Q-49.231-29.640-49.288-29.458Q-49.344-29.277-49.389-29.062Q-49.434-28.847-49.434-28.652Q-49.434-28.378-49.311-28.198Q-49.188-28.019-48.930-28.019Q-48.411-28.019-48.008-28.659L-47.426-31.003Q-47.387-31.128-47.284-31.210Q-47.180-31.292-47.055-31.292Q-46.946-31.292-46.866-31.218Q-46.786-31.144-46.786-31.034Q-46.786-30.988-46.793-30.964L-47.579-27.812Q-47.688-27.390-47.989-27.019Q-48.289-26.648-48.709-26.429Q-49.129-26.210-49.571-26.210Q-49.930-26.210-50.213-26.388Q-50.497-26.566-50.497-26.905\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-4.184 .944)\">\u003Cpath d=\"M-43.735-26.732L-46.026-26.732L-46.026-26.990Q-45.150-26.990-45.150-27.163L-45.150-30.242Q-45.343-30.154-45.575-30.117Q-45.806-30.081-46.061-30.081L-46.061-30.338Q-45.683-30.338-45.362-30.423Q-45.042-30.508-44.813-30.722L-44.693-30.722Q-44.661-30.722-44.636-30.699Q-44.611-30.675-44.611-30.637L-44.611-27.163Q-44.611-26.990-43.735-26.990\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.180\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-neutral)\" style=\"stroke-width:.8\">\u003Cpath d=\"M-22.524-16.462H5.928v-22.762h-28.452Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\">\u003Cg transform=\"translate(38.495 .944)\">\u003Cpath d=\"M-50.497-26.905Q-50.497-27.124-50.364-27.288Q-50.231-27.452-50.016-27.452Q-49.883-27.452-49.789-27.368Q-49.696-27.284-49.696-27.148Q-49.696-26.968-49.823-26.825Q-49.950-26.683-50.129-26.683Q-49.946-26.468-49.563-26.468Q-49.012-26.468-48.661-26.960Q-48.309-27.452-48.153-28.089Q-48.504-27.765-48.946-27.765Q-49.270-27.765-49.520-27.874Q-49.770-27.984-49.909-28.208Q-50.047-28.433-50.047-28.765Q-50.047-29.015-49.973-29.302Q-49.899-29.589-49.762-29.941Q-49.625-30.292-49.563-30.460Q-49.473-30.683-49.473-30.866Q-49.473-31.116-49.641-31.116Q-49.957-31.116-50.166-30.810Q-50.375-30.503-50.481-30.116Q-50.493-30.042-50.563-30.042L-50.664-30.042Q-50.700-30.042-50.727-30.077Q-50.754-30.113-50.754-30.140L-50.754-30.171Q-50.629-30.632-50.332-31.001Q-50.036-31.370-49.625-31.370Q-49.430-31.370-49.256-31.286Q-49.082-31.202-48.981-31.050Q-48.879-30.898-48.879-30.691Q-48.879-30.538-48.938-30.402Q-49.028-30.171-49.129-29.905Q-49.231-29.640-49.288-29.458Q-49.344-29.277-49.389-29.062Q-49.434-28.847-49.434-28.652Q-49.434-28.378-49.311-28.198Q-49.188-28.019-48.930-28.019Q-48.411-28.019-48.008-28.659L-47.426-31.003Q-47.387-31.128-47.284-31.210Q-47.180-31.292-47.055-31.292Q-46.946-31.292-46.866-31.218Q-46.786-31.144-46.786-31.034Q-46.786-30.988-46.793-30.964L-47.579-27.812Q-47.688-27.390-47.989-27.019Q-48.289-26.648-48.709-26.429Q-49.129-26.210-49.571-26.210Q-49.930-26.210-50.213-26.388Q-50.497-26.566-50.497-26.905\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(38.495 .944)\">\u003Cpath d=\"M-43.735-26.732L-46.345-26.732L-46.345-26.917Q-46.339-26.940-46.319-26.966L-45.168-28.021Q-44.828-28.332-44.648-28.518Q-44.467-28.704-44.322-28.964Q-44.177-29.225-44.177-29.521Q-44.177-29.794-44.303-30.009Q-44.429-30.224-44.649-30.344Q-44.869-30.464-45.144-30.464Q-45.320-30.464-45.490-30.407Q-45.660-30.350-45.792-30.243Q-45.923-30.136-46.003-29.978Q-45.915-29.978-45.837-29.934Q-45.759-29.890-45.715-29.814Q-45.672-29.738-45.672-29.641Q-45.672-29.501-45.768-29.404Q-45.865-29.307-46.008-29.307Q-46.146-29.307-46.246-29.407Q-46.345-29.506-46.345-29.641Q-46.345-29.966-46.155-30.214Q-45.964-30.461-45.661-30.592Q-45.358-30.722-45.042-30.722Q-44.661-30.722-44.318-30.587Q-43.975-30.453-43.761-30.180Q-43.547-29.908-43.547-29.521Q-43.547-29.246-43.672-29.019Q-43.797-28.792-43.977-28.620Q-44.157-28.449-44.482-28.209Q-44.807-27.968-44.892-27.901L-45.648-27.297L-45.115-27.297Q-44.626-27.297-44.295-27.305Q-43.964-27.312-43.949-27.327Q-43.890-27.397-43.858-27.532Q-43.826-27.667-43.794-27.878L-43.547-27.878\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.180\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-neutral)\" style=\"stroke-width:.8\">\u003Cpath d=\"M20.155-16.462h28.453v-22.762H20.155Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\">\u003Cg transform=\"translate(81.174 .944)\">\u003Cpath d=\"M-50.497-26.905Q-50.497-27.124-50.364-27.288Q-50.231-27.452-50.016-27.452Q-49.883-27.452-49.789-27.368Q-49.696-27.284-49.696-27.148Q-49.696-26.968-49.823-26.825Q-49.950-26.683-50.129-26.683Q-49.946-26.468-49.563-26.468Q-49.012-26.468-48.661-26.960Q-48.309-27.452-48.153-28.089Q-48.504-27.765-48.946-27.765Q-49.270-27.765-49.520-27.874Q-49.770-27.984-49.909-28.208Q-50.047-28.433-50.047-28.765Q-50.047-29.015-49.973-29.302Q-49.899-29.589-49.762-29.941Q-49.625-30.292-49.563-30.460Q-49.473-30.683-49.473-30.866Q-49.473-31.116-49.641-31.116Q-49.957-31.116-50.166-30.810Q-50.375-30.503-50.481-30.116Q-50.493-30.042-50.563-30.042L-50.664-30.042Q-50.700-30.042-50.727-30.077Q-50.754-30.113-50.754-30.140L-50.754-30.171Q-50.629-30.632-50.332-31.001Q-50.036-31.370-49.625-31.370Q-49.430-31.370-49.256-31.286Q-49.082-31.202-48.981-31.050Q-48.879-30.898-48.879-30.691Q-48.879-30.538-48.938-30.402Q-49.028-30.171-49.129-29.905Q-49.231-29.640-49.288-29.458Q-49.344-29.277-49.389-29.062Q-49.434-28.847-49.434-28.652Q-49.434-28.378-49.311-28.198Q-49.188-28.019-48.930-28.019Q-48.411-28.019-48.008-28.659L-47.426-31.003Q-47.387-31.128-47.284-31.210Q-47.180-31.292-47.055-31.292Q-46.946-31.292-46.866-31.218Q-46.786-31.144-46.786-31.034Q-46.786-30.988-46.793-30.964L-47.579-27.812Q-47.688-27.390-47.989-27.019Q-48.289-26.648-48.709-26.429Q-49.129-26.210-49.571-26.210Q-49.930-26.210-50.213-26.388Q-50.497-26.566-50.497-26.905\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(81.174 .944)\">\u003Cpath d=\"M-46.003-27.183Q-45.707-26.846-44.977-26.846Q-44.719-26.846-44.539-26.974Q-44.359-27.101-44.271-27.309Q-44.183-27.517-44.183-27.775Q-44.183-28.170-44.390-28.441Q-44.596-28.712-44.983-28.712L-45.449-28.712Q-45.513-28.727-45.528-28.789L-45.528-28.856Q-45.513-28.912-45.449-28.929L-45.047-28.953Q-44.837-28.953-44.668-29.095Q-44.500-29.237-44.407-29.451Q-44.315-29.665-44.315-29.881Q-44.315-30.169-44.500-30.334Q-44.684-30.500-44.977-30.500Q-45.238-30.500-45.462-30.432Q-45.686-30.365-45.833-30.207Q-45.704-30.189-45.625-30.100Q-45.546-30.010-45.546-29.881Q-45.546-29.744-45.641-29.649Q-45.736-29.553-45.877-29.553Q-46.011-29.553-46.108-29.650Q-46.205-29.747-46.205-29.881Q-46.205-30.169-46.014-30.360Q-45.824-30.552-45.543-30.637Q-45.261-30.722-44.977-30.722Q-44.702-30.722-44.401-30.631Q-44.101-30.541-43.893-30.352Q-43.685-30.163-43.685-29.881Q-43.685-29.512-43.931-29.240Q-44.177-28.967-44.549-28.838Q-44.130-28.745-43.813-28.462Q-43.495-28.179-43.495-27.781Q-43.495-27.418-43.714-27.152Q-43.934-26.887-44.280-26.747Q-44.626-26.606-44.977-26.606Q-45.200-26.606-45.447-26.654Q-45.695-26.703-45.915-26.813Q-46.134-26.922-46.266-27.101Q-46.398-27.280-46.398-27.535Q-46.398-27.684-46.296-27.787Q-46.193-27.889-46.044-27.889Q-45.894-27.889-45.792-27.787Q-45.689-27.684-45.689-27.535Q-45.689-27.403-45.778-27.302Q-45.868-27.201-46.003-27.183\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.180\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr8\" font-size=\"8\">\u003Cg transform=\"translate(115.027 .444)\">\u003Cpath d=\"M-50.258-28.308Q-50.258-28.491-50.122-28.628Q-49.985-28.765-49.793-28.765Q-49.602-28.765-49.469-28.632Q-49.336-28.499-49.336-28.308Q-49.336-28.109-49.469-27.976Q-49.602-27.843-49.793-27.843Q-49.985-27.843-50.122-27.980Q-50.258-28.116-50.258-28.308\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(115.027 .444)\">\u003Cpath d=\"M-44.119-28.308Q-44.119-28.491-43.983-28.628Q-43.846-28.765-43.654-28.765Q-43.463-28.765-43.330-28.632Q-43.197-28.499-43.197-28.308Q-43.197-28.109-43.330-27.976Q-43.463-27.843-43.654-27.843Q-43.846-27.843-43.983-27.980Q-44.119-28.116-44.119-28.308\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(115.027 .444)\">\u003Cpath d=\"M-37.980-28.308Q-37.980-28.491-37.844-28.628Q-37.707-28.765-37.515-28.765Q-37.324-28.765-37.191-28.632Q-37.058-28.499-37.058-28.308Q-37.058-28.109-37.191-27.976Q-37.324-27.843-37.515-27.843Q-37.707-27.843-37.844-27.980Q-37.980-28.116-37.980-28.308\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath d=\"M94.132-16.462h28.453v-22.762H94.132Z\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\">\u003Cg transform=\"translate(154.305 .944)\">\u003Cpath d=\"M-50.497-26.905Q-50.497-27.124-50.364-27.288Q-50.231-27.452-50.016-27.452Q-49.883-27.452-49.789-27.368Q-49.696-27.284-49.696-27.148Q-49.696-26.968-49.823-26.825Q-49.950-26.683-50.129-26.683Q-49.946-26.468-49.563-26.468Q-49.012-26.468-48.661-26.960Q-48.309-27.452-48.153-28.089Q-48.504-27.765-48.946-27.765Q-49.270-27.765-49.520-27.874Q-49.770-27.984-49.909-28.208Q-50.047-28.433-50.047-28.765Q-50.047-29.015-49.973-29.302Q-49.899-29.589-49.762-29.941Q-49.625-30.292-49.563-30.460Q-49.473-30.683-49.473-30.866Q-49.473-31.116-49.641-31.116Q-49.957-31.116-50.166-30.810Q-50.375-30.503-50.481-30.116Q-50.493-30.042-50.563-30.042L-50.664-30.042Q-50.700-30.042-50.727-30.077Q-50.754-30.113-50.754-30.140L-50.754-30.171Q-50.629-30.632-50.332-31.001Q-50.036-31.370-49.625-31.370Q-49.430-31.370-49.256-31.286Q-49.082-31.202-48.981-31.050Q-48.879-30.898-48.879-30.691Q-48.879-30.538-48.938-30.402Q-49.028-30.171-49.129-29.905Q-49.231-29.640-49.288-29.458Q-49.344-29.277-49.389-29.062Q-49.434-28.847-49.434-28.652Q-49.434-28.378-49.311-28.198Q-49.188-28.019-48.930-28.019Q-48.411-28.019-48.008-28.659L-47.426-31.003Q-47.387-31.128-47.284-31.210Q-47.180-31.292-47.055-31.292Q-46.946-31.292-46.866-31.218Q-46.786-31.144-46.786-31.034Q-46.786-30.988-46.793-30.964L-47.579-27.812Q-47.688-27.390-47.989-27.019Q-48.289-26.648-48.709-26.429Q-49.129-26.210-49.571-26.210Q-49.930-26.210-50.213-26.388Q-50.497-26.566-50.497-26.905\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.240\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(154.305 .944)\">\u003Cpath d=\"M-46.038-26.601Q-46.017-26.683-46.004-26.715Q-45.991-26.747-45.935-26.756Q-45.420-26.756-45.241-26.785Q-45.091-26.806-45.047-26.937L-44.239-30.181Q-44.215-30.280-44.215-30.321Q-44.274-30.339-44.406-30.339L-44.863-30.339Q-45.214-30.339-45.424-30.277Q-45.633-30.216-45.761-30.091Q-45.888-29.967-45.970-29.791Q-46.052-29.615-46.193-29.234Q-46.214-29.173-46.272-29.173L-46.351-29.173Q-46.386-29.173-46.410-29.202Q-46.433-29.231-46.433-29.264Q-46.433-29.275-46.427-29.299L-45.985-30.541Q-45.964-30.597-45.906-30.597L-41.796-30.597Q-41.760-30.597-41.735-30.564Q-41.711-30.532-41.711-30.494L-41.892-29.252Q-41.904-29.173-41.977-29.173L-42.053-29.173Q-42.091-29.173-42.115-29.206Q-42.138-29.240-42.138-29.275Q-42.121-29.393-42.109-29.485Q-42.097-29.577-42.090-29.653Q-42.083-29.729-42.083-29.817Q-42.083-30.063-42.185-30.175Q-42.288-30.286-42.446-30.312Q-42.604-30.339-42.882-30.339L-43.331-30.339Q-43.445-30.339-43.493-30.330Q-43.542-30.321-43.577-30.277Q-43.612-30.233-43.632-30.157L-44.441-26.911Q-44.444-26.896-44.445-26.886Q-44.447-26.876-44.453-26.864Q-44.453-26.809-44.412-26.791Q-44.221-26.756-43.685-26.756Q-43.647-26.756-43.625-26.725Q-43.603-26.694-43.603-26.653Q-43.627-26.554-43.644-26.526Q-43.662-26.498-43.735-26.498L-45.953-26.498Q-45.991-26.498-46.014-26.532Q-46.038-26.565-46.038-26.601\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.180\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-36.35-27.843h10.826\"\u002F>\u003Cpath stroke=\"none\" d=\"m-22.924-27.843-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M6.328-27.843h10.827\"\u002F>\u003Cpath stroke=\"none\" d=\"m19.755-27.843-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M49.008-27.843h8.91\"\u002F>\u003Cpath stroke=\"none\" d=\"m60.517-27.843-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M82.222-27.843h8.91\"\u002F>\u003Cpath stroke=\"none\" d=\"m93.732-27.843-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-50.977-16.062v11.627\"\u002F>\u003Cpath stroke=\"none\" d=\"m-50.977-1.835 2.08-4.16-2.08 1.56-2.08-1.56\"\u002F>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-8.298-16.062v11.627\"\u002F>\u003Cpath stroke=\"none\" d=\"m-8.298-1.835 2.08-4.16-2.08 1.56-2.08-1.56\"\u002F>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M34.381-16.062v11.627\"\u002F>\u003Cpath stroke=\"none\" d=\"m34.381-1.835 2.08-4.16-2.08 1.56-2.08-1.56\"\u002F>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M108.359-16.062v11.627\"\u002F>\u003Cpath stroke=\"none\" d=\"m108.359-1.835 2.08-4.16-2.08 1.56-2.08-1.56\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-5.531 37.338)\">\u003Cpath d=\"M-48.465-27.843L-50.598-27.843L-50.598-28.123Q-49.876-28.123-49.876-28.332L-49.876-32.133Q-49.876-32.344-50.598-32.344L-50.598-32.625L-48.465-32.625L-48.465-32.344Q-49.186-32.344-49.186-32.133L-49.186-29.907L-46.817-31.945Q-46.701-32.054-46.701-32.156Q-46.701-32.249-46.778-32.297Q-46.855-32.344-46.951-32.344L-46.951-32.625L-45.348-32.625L-45.348-32.344Q-45.635-32.344-45.912-32.239Q-46.188-32.133-46.407-31.945L-47.791-30.755L-46.100-28.523Q-45.963-28.345-45.853-28.260Q-45.744-28.175-45.619-28.149Q-45.495-28.123-45.245-28.123L-45.245-27.843L-47.128-27.843L-47.128-28.123Q-46.975-28.123-46.869-28.154Q-46.763-28.185-46.763-28.298Q-46.763-28.383-46.875-28.523L-48.260-30.348L-49.186-29.559L-49.186-28.332Q-49.186-28.123-48.465-28.123L-48.465-27.843M-40.511-27.843L-44.479-27.843L-44.479-28.123Q-43.758-28.123-43.758-28.332L-43.758-32.133Q-43.758-32.344-44.479-32.344L-44.479-32.625L-42.165-32.625L-42.165-32.344Q-42.483-32.344-42.776-32.309Q-43.068-32.273-43.068-32.133L-43.068-28.332Q-43.068-28.192-42.979-28.157Q-42.890-28.123-42.702-28.123L-42.080-28.123Q-41.666-28.123-41.388-28.226Q-41.109-28.328-40.944-28.530Q-40.778-28.732-40.696-29.019Q-40.614-29.306-40.569-29.713L-40.303-29.713\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(37.148 37.338)\">\u003Cpath d=\"M-48.465-27.843L-50.598-27.843L-50.598-28.123Q-49.876-28.123-49.876-28.332L-49.876-32.133Q-49.876-32.344-50.598-32.344L-50.598-32.625L-48.465-32.625L-48.465-32.344Q-49.186-32.344-49.186-32.133L-49.186-29.907L-46.817-31.945Q-46.701-32.054-46.701-32.156Q-46.701-32.249-46.778-32.297Q-46.855-32.344-46.951-32.344L-46.951-32.625L-45.348-32.625L-45.348-32.344Q-45.635-32.344-45.912-32.239Q-46.188-32.133-46.407-31.945L-47.791-30.755L-46.100-28.523Q-45.963-28.345-45.853-28.260Q-45.744-28.175-45.619-28.149Q-45.495-28.123-45.245-28.123L-45.245-27.843L-47.128-27.843L-47.128-28.123Q-46.975-28.123-46.869-28.154Q-46.763-28.185-46.763-28.298Q-46.763-28.383-46.875-28.523L-48.260-30.348L-49.186-29.559L-49.186-28.332Q-49.186-28.123-48.465-28.123L-48.465-27.843M-40.511-27.843L-44.479-27.843L-44.479-28.123Q-43.758-28.123-43.758-28.332L-43.758-32.133Q-43.758-32.344-44.479-32.344L-44.479-32.625L-42.165-32.625L-42.165-32.344Q-42.483-32.344-42.776-32.309Q-43.068-32.273-43.068-32.133L-43.068-28.332Q-43.068-28.192-42.979-28.157Q-42.890-28.123-42.702-28.123L-42.080-28.123Q-41.666-28.123-41.388-28.226Q-41.109-28.328-40.944-28.530Q-40.778-28.732-40.696-29.019Q-40.614-29.306-40.569-29.713L-40.303-29.713\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(79.827 37.338)\">\u003Cpath d=\"M-48.465-27.843L-50.598-27.843L-50.598-28.123Q-49.876-28.123-49.876-28.332L-49.876-32.133Q-49.876-32.344-50.598-32.344L-50.598-32.625L-48.465-32.625L-48.465-32.344Q-49.186-32.344-49.186-32.133L-49.186-29.907L-46.817-31.945Q-46.701-32.054-46.701-32.156Q-46.701-32.249-46.778-32.297Q-46.855-32.344-46.951-32.344L-46.951-32.625L-45.348-32.625L-45.348-32.344Q-45.635-32.344-45.912-32.239Q-46.188-32.133-46.407-31.945L-47.791-30.755L-46.100-28.523Q-45.963-28.345-45.853-28.260Q-45.744-28.175-45.619-28.149Q-45.495-28.123-45.245-28.123L-45.245-27.843L-47.128-27.843L-47.128-28.123Q-46.975-28.123-46.869-28.154Q-46.763-28.185-46.763-28.298Q-46.763-28.383-46.875-28.523L-48.260-30.348L-49.186-29.559L-49.186-28.332Q-49.186-28.123-48.465-28.123L-48.465-27.843M-40.511-27.843L-44.479-27.843L-44.479-28.123Q-43.758-28.123-43.758-28.332L-43.758-32.133Q-43.758-32.344-44.479-32.344L-44.479-32.625L-42.165-32.625L-42.165-32.344Q-42.483-32.344-42.776-32.309Q-43.068-32.273-43.068-32.133L-43.068-28.332Q-43.068-28.192-42.979-28.157Q-42.890-28.123-42.702-28.123L-42.080-28.123Q-41.666-28.123-41.388-28.226Q-41.109-28.328-40.944-28.530Q-40.778-28.732-40.696-29.019Q-40.614-29.306-40.569-29.713L-40.303-29.713\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(153.804 37.338)\">\u003Cpath d=\"M-48.465-27.843L-50.598-27.843L-50.598-28.123Q-49.876-28.123-49.876-28.332L-49.876-32.133Q-49.876-32.344-50.598-32.344L-50.598-32.625L-48.465-32.625L-48.465-32.344Q-49.186-32.344-49.186-32.133L-49.186-29.907L-46.817-31.945Q-46.701-32.054-46.701-32.156Q-46.701-32.249-46.778-32.297Q-46.855-32.344-46.951-32.344L-46.951-32.625L-45.348-32.625L-45.348-32.344Q-45.635-32.344-45.912-32.239Q-46.188-32.133-46.407-31.945L-47.791-30.755L-46.100-28.523Q-45.963-28.345-45.853-28.260Q-45.744-28.175-45.619-28.149Q-45.495-28.123-45.245-28.123L-45.245-27.843L-47.128-27.843L-47.128-28.123Q-46.975-28.123-46.869-28.154Q-46.763-28.185-46.763-28.298Q-46.763-28.383-46.875-28.523L-48.260-30.348L-49.186-29.559L-49.186-28.332Q-49.186-28.123-48.465-28.123L-48.465-27.843M-40.511-27.843L-44.479-27.843L-44.479-28.123Q-43.758-28.123-43.758-28.332L-43.758-32.133Q-43.758-32.344-44.479-32.344L-44.479-32.625L-42.165-32.625L-42.165-32.344Q-42.483-32.344-42.776-32.309Q-43.068-32.273-43.068-32.133L-43.068-28.332Q-43.068-28.192-42.979-28.157Q-42.890-28.123-42.702-28.123L-42.080-28.123Q-41.666-28.123-41.388-28.226Q-41.109-28.328-40.944-28.530Q-40.778-28.732-40.696-29.019Q-40.614-29.306-40.569-29.713L-40.303-29.713\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M108.359-39.624v-13.049\"\u002F>\u003Cpath stroke=\"none\" d=\"m108.359-55.273-2.08 4.16 2.08-1.56 2.08 1.56\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(143.068 -36.233)\">\u003Cpath d=\"M-48.079-27.163L-48.079-29.419L-50.328-29.419Q-50.396-29.429-50.442-29.475Q-50.488-29.521-50.488-29.593Q-50.488-29.737-50.328-29.760L-48.079-29.760L-48.079-32.016Q-48.068-32.085-48.022-32.131Q-47.976-32.177-47.904-32.177Q-47.761-32.177-47.737-32.016L-47.737-29.760L-45.495-29.760Q-45.334-29.737-45.334-29.593Q-45.334-29.521-45.380-29.475Q-45.426-29.429-45.495-29.419L-47.737-29.419L-47.737-27.163Q-47.761-27.002-47.904-27.002Q-47.976-27.002-48.022-27.048Q-48.068-27.094-48.079-27.163\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(143.068 -36.233)\">\u003Cpath d=\"M-40.080-27.843L-41.816-27.843L-41.816-28.123Q-41.587-28.123-41.438-28.157Q-41.290-28.192-41.290-28.332L-41.290-30.181Q-41.290-30.451-41.397-30.512Q-41.505-30.574-41.816-30.574L-41.816-30.854L-40.787-30.929L-40.787-30.222Q-40.657-30.530-40.415-30.729Q-40.172-30.929-39.854-30.929Q-39.635-30.929-39.464-30.805Q-39.293-30.680-39.293-30.468Q-39.293-30.331-39.393-30.232Q-39.492-30.133-39.625-30.133Q-39.762-30.133-39.861-30.232Q-39.960-30.331-39.960-30.468Q-39.960-30.608-39.861-30.707Q-40.151-30.707-40.351-30.511Q-40.551-30.314-40.644-30.020Q-40.736-29.726-40.736-29.446L-40.736-28.332Q-40.736-28.123-40.080-28.123L-40.080-27.843M-38.750-29.378Q-38.750-29.699-38.625-29.988Q-38.500-30.277-38.275-30.500Q-38.049-30.724-37.754-30.844Q-37.458-30.964-37.140-30.964Q-36.812-30.964-36.550-30.864Q-36.289-30.765-36.113-30.583Q-35.937-30.400-35.843-30.142Q-35.749-29.884-35.749-29.552Q-35.749-29.460-35.831-29.439L-38.087-29.439L-38.087-29.378Q-38.087-28.790-37.803-28.407Q-37.519-28.024-36.952-28.024Q-36.631-28.024-36.363-28.217Q-36.094-28.410-36.005-28.725Q-35.998-28.766-35.923-28.780L-35.831-28.780Q-35.749-28.756-35.749-28.684Q-35.749-28.677-35.756-28.650Q-35.869-28.253-36.239-28.014Q-36.610-27.775-37.034-27.775Q-37.472-27.775-37.872-27.983Q-38.271-28.192-38.511-28.559Q-38.750-28.926-38.750-29.378M-38.080-29.648L-36.265-29.648Q-36.265-29.925-36.363-30.177Q-36.460-30.430-36.658-30.586Q-36.856-30.741-37.140-30.741Q-37.417-30.741-37.631-30.583Q-37.844-30.424-37.962-30.169Q-38.080-29.914-38.080-29.648M-33.773-27.870L-34.754-30.369Q-34.816-30.512-34.934-30.547Q-35.052-30.581-35.267-30.581L-35.267-30.861L-33.787-30.861L-33.787-30.581Q-34.166-30.581-34.166-30.420Q-34.166-30.410-34.153-30.369L-33.438-28.537L-32.765-30.242Q-32.796-30.314-32.796-30.342Q-32.796-30.369-32.823-30.369Q-32.885-30.516-33.003-30.548Q-33.121-30.581-33.332-30.581L-33.332-30.861L-31.935-30.861L-31.935-30.581Q-32.311-30.581-32.311-30.420Q-32.311-30.389-32.304-30.369L-31.548-28.431L-30.861-30.181Q-30.841-30.232-30.841-30.287Q-30.841-30.427-30.954-30.504Q-31.066-30.581-31.206-30.581L-31.206-30.861L-29.986-30.861L-29.986-30.581Q-30.191-30.581-30.347-30.475Q-30.502-30.369-30.574-30.181L-31.480-27.870Q-31.514-27.775-31.627-27.775L-31.695-27.775Q-31.805-27.775-31.842-27.870L-32.625-29.873L-33.411-27.870Q-33.445-27.775-33.558-27.775L-33.626-27.775Q-33.736-27.775-33.773-27.870\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(143.068 -36.233)\">\u003Cpath d=\"M-29.597-28.571Q-29.597-28.903-29.374-29.130Q-29.150-29.357-28.806-29.485Q-28.463-29.614-28.090-29.666Q-27.718-29.719-27.413-29.719L-27.413-29.972Q-27.413-30.177-27.521-30.357Q-27.629-30.536-27.810-30.639Q-27.991-30.741-28.199-30.741Q-28.606-30.741-28.842-30.649Q-28.753-30.612-28.707-30.528Q-28.661-30.444-28.661-30.342Q-28.661-30.246-28.707-30.167Q-28.753-30.089-28.834-30.044Q-28.914-30-29.003-30Q-29.153-30-29.254-30.097Q-29.355-30.195-29.355-30.342Q-29.355-30.964-28.199-30.964Q-27.988-30.964-27.738-30.900Q-27.489-30.837-27.287-30.718Q-27.085-30.598-26.959-30.413Q-26.832-30.229-26.832-29.986L-26.832-28.410Q-26.832-28.294-26.771-28.198Q-26.709-28.103-26.596-28.103Q-26.487-28.103-26.422-28.197Q-26.357-28.291-26.357-28.410L-26.357-28.858L-26.091-28.858L-26.091-28.410Q-26.091-28.140-26.318-27.975Q-26.545-27.809-26.825-27.809Q-27.034-27.809-27.171-27.963Q-27.307-28.116-27.331-28.332Q-27.478-28.065-27.760-27.920Q-28.042-27.775-28.367-27.775Q-28.644-27.775-28.928-27.850Q-29.211-27.925-29.404-28.104Q-29.597-28.284-29.597-28.571M-28.982-28.571Q-28.982-28.397-28.881-28.267Q-28.781-28.137-28.625-28.067Q-28.470-27.997-28.305-27.997Q-28.087-27.997-27.878-28.094Q-27.670-28.192-27.542-28.373Q-27.413-28.554-27.413-28.780L-27.413-29.508Q-27.738-29.508-28.104-29.417Q-28.470-29.326-28.726-29.114Q-28.982-28.903-28.982-28.571M-23.924-27.843L-25.660-27.843L-25.660-28.123Q-25.431-28.123-25.282-28.157Q-25.134-28.192-25.134-28.332L-25.134-30.181Q-25.134-30.451-25.241-30.512Q-25.349-30.574-25.660-30.574L-25.660-30.854L-24.631-30.929L-24.631-30.222Q-24.501-30.530-24.259-30.729Q-24.016-30.929-23.698-30.929Q-23.479-30.929-23.308-30.805Q-23.137-30.680-23.137-30.468Q-23.137-30.331-23.237-30.232Q-23.336-30.133-23.469-30.133Q-23.606-30.133-23.705-30.232Q-23.804-30.331-23.804-30.468Q-23.804-30.608-23.705-30.707Q-23.995-30.707-24.195-30.511Q-24.395-30.314-24.488-30.020Q-24.580-29.726-24.580-29.446L-24.580-28.332Q-24.580-28.123-23.924-28.123L-23.924-27.843M-22.553-29.354Q-22.553-29.692-22.413-29.983Q-22.273-30.273-22.028-30.487Q-21.784-30.700-21.480-30.815Q-21.176-30.929-20.851-30.929Q-20.581-30.929-20.318-30.830Q-20.054-30.731-19.863-30.553L-19.863-31.951Q-19.863-32.221-19.971-32.283Q-20.078-32.344-20.389-32.344L-20.389-32.625L-19.313-32.700L-19.313-28.516Q-19.313-28.328-19.258-28.245Q-19.203-28.161-19.103-28.142Q-19.002-28.123-18.786-28.123L-18.786-27.843L-19.894-27.775L-19.894-28.192Q-20.311-27.775-20.936-27.775Q-21.367-27.775-21.740-27.987Q-22.112-28.198-22.333-28.559Q-22.553-28.920-22.553-29.354M-20.878-27.997Q-20.670-27.997-20.483-28.069Q-20.297-28.140-20.143-28.277Q-19.990-28.414-19.894-28.592L-19.894-30.201Q-19.979-30.348-20.125-30.468Q-20.270-30.588-20.439-30.647Q-20.608-30.707-20.789-30.707Q-21.350-30.707-21.618-30.318Q-21.887-29.928-21.887-29.347Q-21.887-28.776-21.652-28.386Q-21.418-27.997-20.878-27.997\" 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\">Per-token reward shaping. Each generated token pays a small KL penalty (black) for drifting from the reference; the terminal token also collects the sequence reward (blue).\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:382.014px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 286.510 130.600\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-50.149 37.757H135.04\"\u002F>\u003Cpath stroke=\"none\" d=\"m137.64 37.757-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(191.521 2.43)\">\u003Cpath d=\"M-47.637 37.757L-49.770 37.757L-49.770 37.477Q-49.048 37.477-49.048 37.268L-49.048 33.467Q-49.048 33.256-49.770 33.256L-49.770 32.975L-47.637 32.975L-47.637 33.256Q-48.358 33.256-48.358 33.467L-48.358 35.693L-45.989 33.655Q-45.873 33.546-45.873 33.444Q-45.873 33.351-45.950 33.303Q-46.027 33.256-46.123 33.256L-46.123 32.975L-44.520 32.975L-44.520 33.256Q-44.807 33.256-45.084 33.361Q-45.360 33.467-45.579 33.655L-46.963 34.845L-45.272 37.077Q-45.135 37.255-45.025 37.340Q-44.916 37.425-44.791 37.451Q-44.667 37.477-44.417 37.477L-44.417 37.757L-46.300 37.757L-46.300 37.477Q-46.147 37.477-46.041 37.446Q-45.935 37.415-45.935 37.302Q-45.935 37.217-46.047 37.077L-47.432 35.252L-48.358 36.041L-48.358 37.268Q-48.358 37.477-47.637 37.477L-47.637 37.757M-39.683 37.757L-43.651 37.757L-43.651 37.477Q-42.930 37.477-42.930 37.268L-42.930 33.467Q-42.930 33.256-43.651 33.256L-43.651 32.975L-41.337 32.975L-41.337 33.256Q-41.655 33.256-41.948 33.291Q-42.240 33.327-42.240 33.467L-42.240 37.268Q-42.240 37.408-42.151 37.443Q-42.062 37.477-41.874 37.477L-41.252 37.477Q-40.838 37.477-40.560 37.374Q-40.281 37.272-40.116 37.070Q-39.950 36.868-39.868 36.581Q-39.786 36.294-39.741 35.887L-39.475 35.887\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(191.521 2.43)\">\u003Cpath d=\"M-34.280 37.757L-36.013 37.757L-36.013 37.477Q-35.787 37.477-35.638 37.443Q-35.490 37.408-35.490 37.268L-35.490 35.019L-36.078 35.019L-36.078 34.739L-35.490 34.739L-35.490 33.922Q-35.490 33.604-35.312 33.356Q-35.134 33.109-34.844 32.968Q-34.553 32.828-34.242 32.828Q-33.986 32.828-33.782 32.970Q-33.579 33.112-33.579 33.355Q-33.579 33.491-33.678 33.590Q-33.777 33.690-33.914 33.690Q-34.051 33.690-34.150 33.590Q-34.249 33.491-34.249 33.355Q-34.249 33.174-34.109 33.081Q-34.187 33.054-34.287 33.054Q-34.495 33.054-34.649 33.187Q-34.803 33.320-34.883 33.524Q-34.963 33.727-34.963 33.936L-34.963 34.739L-34.075 34.739L-34.075 35.019L-34.936 35.019L-34.936 37.268Q-34.936 37.477-34.280 37.477L-34.280 37.757M-31.850 37.757L-33.586 37.757L-33.586 37.477Q-33.357 37.477-33.208 37.443Q-33.059 37.408-33.059 37.268L-33.059 35.419Q-33.059 35.149-33.167 35.088Q-33.275 35.026-33.586 35.026L-33.586 34.746L-32.557 34.671L-32.557 35.378Q-32.427 35.070-32.184 34.871Q-31.942 34.671-31.624 34.671Q-31.405 34.671-31.234 34.795Q-31.063 34.920-31.063 35.132Q-31.063 35.269-31.163 35.368Q-31.262 35.467-31.395 35.467Q-31.532 35.467-31.631 35.368Q-31.730 35.269-31.730 35.132Q-31.730 34.992-31.631 34.893Q-31.921 34.893-32.121 35.089Q-32.321 35.286-32.413 35.580Q-32.506 35.874-32.506 36.154L-32.506 37.268Q-32.506 37.477-31.850 37.477L-31.850 37.757M-30.520 36.274Q-30.520 35.932-30.385 35.633Q-30.250 35.334-30.011 35.110Q-29.771 34.886-29.454 34.761Q-29.136 34.636-28.804 34.636Q-28.360 34.636-27.960 34.852Q-27.560 35.067-27.326 35.445Q-27.092 35.822-27.092 36.274Q-27.092 36.615-27.234 36.899Q-27.375 37.183-27.620 37.390Q-27.864 37.596-28.173 37.711Q-28.483 37.825-28.804 37.825Q-29.235 37.825-29.636 37.624Q-30.038 37.422-30.279 37.070Q-30.520 36.718-30.520 36.274M-28.804 37.576Q-28.203 37.576-27.979 37.198Q-27.755 36.820-27.755 36.188Q-27.755 35.576-27.989 35.217Q-28.223 34.859-28.804 34.859Q-29.857 34.859-29.857 36.188Q-29.857 36.820-29.631 37.198Q-29.406 37.576-28.804 37.576M-24.815 37.757L-26.449 37.757L-26.449 37.477Q-26.220 37.477-26.071 37.443Q-25.923 37.408-25.923 37.268L-25.923 35.419Q-25.923 35.149-26.030 35.088Q-26.138 35.026-26.449 35.026L-26.449 34.746L-25.390 34.671L-25.390 35.320Q-25.219 35.012-24.914 34.841Q-24.610 34.671-24.265 34.671Q-23.865 34.671-23.588 34.811Q-23.311 34.951-23.226 35.299Q-23.059 35.006-22.759 34.838Q-22.460 34.671-22.115 34.671Q-21.609 34.671-21.326 34.894Q-21.042 35.118-21.042 35.614L-21.042 37.268Q-21.042 37.405-20.893 37.441Q-20.745 37.477-20.519 37.477L-20.519 37.757L-22.149 37.757L-22.149 37.477Q-21.924 37.477-21.773 37.441Q-21.623 37.405-21.623 37.268L-21.623 35.628Q-21.623 35.293-21.743 35.093Q-21.862 34.893-22.177 34.893Q-22.447 34.893-22.681 35.029Q-22.915 35.166-23.053 35.400Q-23.192 35.634-23.192 35.908L-23.192 37.268Q-23.192 37.405-23.043 37.441Q-22.894 37.477-22.669 37.477L-22.669 37.757L-24.299 37.757L-24.299 37.477Q-24.070 37.477-23.922 37.443Q-23.773 37.408-23.773 37.268L-23.773 35.628Q-23.773 35.293-23.892 35.093Q-24.012 34.893-24.327 34.893Q-24.597 34.893-24.831 35.029Q-25.065 35.166-25.203 35.400Q-25.342 35.634-25.342 35.908L-25.342 37.268Q-25.342 37.405-25.191 37.441Q-25.041 37.477-24.815 37.477\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(191.521 2.43)\">\u003Cpath d=\"M-15.473 37.757L-17.209 37.757L-17.209 37.477Q-16.980 37.477-16.831 37.443Q-16.683 37.408-16.683 37.268L-16.683 35.419Q-16.683 35.149-16.790 35.088Q-16.898 35.026-17.209 35.026L-17.209 34.746L-16.180 34.671L-16.180 35.378Q-16.050 35.070-15.808 34.871Q-15.565 34.671-15.247 34.671Q-15.028 34.671-14.857 34.795Q-14.686 34.920-14.686 35.132Q-14.686 35.269-14.786 35.368Q-14.885 35.467-15.018 35.467Q-15.155 35.467-15.254 35.368Q-15.353 35.269-15.353 35.132Q-15.353 34.992-15.254 34.893Q-15.544 34.893-15.744 35.089Q-15.944 35.286-16.037 35.580Q-16.129 35.874-16.129 36.154L-16.129 37.268Q-16.129 37.477-15.473 37.477L-15.473 37.757M-14.143 36.222Q-14.143 35.901-14.018 35.612Q-13.893 35.323-13.668 35.100Q-13.442 34.876-13.147 34.756Q-12.851 34.636-12.533 34.636Q-12.205 34.636-11.943 34.736Q-11.682 34.835-11.506 35.017Q-11.330 35.200-11.236 35.458Q-11.142 35.716-11.142 36.048Q-11.142 36.140-11.224 36.161L-13.480 36.161L-13.480 36.222Q-13.480 36.810-13.196 37.193Q-12.912 37.576-12.345 37.576Q-12.024 37.576-11.756 37.383Q-11.487 37.190-11.398 36.875Q-11.391 36.834-11.316 36.820L-11.224 36.820Q-11.142 36.844-11.142 36.916Q-11.142 36.923-11.149 36.950Q-11.262 37.347-11.632 37.586Q-12.003 37.825-12.427 37.825Q-12.865 37.825-13.265 37.617Q-13.664 37.408-13.904 37.041Q-14.143 36.674-14.143 36.222M-13.473 35.952L-11.658 35.952Q-11.658 35.675-11.756 35.423Q-11.853 35.170-12.051 35.014Q-12.249 34.859-12.533 34.859Q-12.810 34.859-13.024 35.017Q-13.237 35.176-13.355 35.431Q-13.473 35.686-13.473 35.952M-8.756 37.757L-10.489 37.757L-10.489 37.477Q-10.264 37.477-10.115 37.443Q-9.966 37.408-9.966 37.268L-9.966 35.019L-10.554 35.019L-10.554 34.739L-9.966 34.739L-9.966 33.922Q-9.966 33.604-9.788 33.356Q-9.611 33.109-9.320 32.968Q-9.030 32.828-8.719 32.828Q-8.462 32.828-8.259 32.970Q-8.056 33.112-8.056 33.355Q-8.056 33.491-8.155 33.590Q-8.254 33.690-8.391 33.690Q-8.527 33.690-8.626 33.590Q-8.725 33.491-8.725 33.355Q-8.725 33.174-8.585 33.081Q-8.664 33.054-8.763 33.054Q-8.972 33.054-9.125 33.187Q-9.279 33.320-9.360 33.524Q-9.440 33.727-9.440 33.936L-9.440 34.739L-8.551 34.739L-8.551 35.019L-9.412 35.019L-9.412 37.268Q-9.412 37.477-8.756 37.477L-8.756 37.757M-8.117 36.222Q-8.117 35.901-7.992 35.612Q-7.868 35.323-7.642 35.100Q-7.416 34.876-7.121 34.756Q-6.825 34.636-6.507 34.636Q-6.179 34.636-5.918 34.736Q-5.656 34.835-5.480 35.017Q-5.304 35.200-5.210 35.458Q-5.116 35.716-5.116 36.048Q-5.116 36.140-5.198 36.161L-7.454 36.161L-7.454 36.222Q-7.454 36.810-7.170 37.193Q-6.887 37.576-6.319 37.576Q-5.998 37.576-5.730 37.383Q-5.461 37.190-5.372 36.875Q-5.366 36.834-5.290 36.820L-5.198 36.820Q-5.116 36.844-5.116 36.916Q-5.116 36.923-5.123 36.950Q-5.236 37.347-5.607 37.586Q-5.977 37.825-6.401 37.825Q-6.839 37.825-7.239 37.617Q-7.639 37.408-7.878 37.041Q-8.117 36.674-8.117 36.222M-7.447 35.952L-5.632 35.952Q-5.632 35.675-5.730 35.423Q-5.827 35.170-6.025 35.014Q-6.224 34.859-6.507 34.859Q-6.784 34.859-6.998 35.017Q-7.211 35.176-7.329 35.431Q-7.447 35.686-7.447 35.952M-2.778 37.757L-4.515 37.757L-4.515 37.477Q-4.286 37.477-4.137 37.443Q-3.988 37.408-3.988 37.268L-3.988 35.419Q-3.988 35.149-4.096 35.088Q-4.204 35.026-4.515 35.026L-4.515 34.746L-3.486 34.671L-3.486 35.378Q-3.356 35.070-3.113 34.871Q-2.870 34.671-2.553 34.671Q-2.334 34.671-2.163 34.795Q-1.992 34.920-1.992 35.132Q-1.992 35.269-2.091 35.368Q-2.190 35.467-2.324 35.467Q-2.460 35.467-2.559 35.368Q-2.659 35.269-2.659 35.132Q-2.659 34.992-2.559 34.893Q-2.850 34.893-3.050 35.089Q-3.250 35.286-3.342 35.580Q-3.434 35.874-3.434 36.154L-3.434 37.268Q-3.434 37.477-2.778 37.477L-2.778 37.757M-1.449 36.222Q-1.449 35.901-1.324 35.612Q-1.199 35.323-0.974 35.100Q-0.748 34.876-0.452 34.756Q-0.157 34.636 0.161 34.636Q0.489 34.636 0.751 34.736Q1.012 34.835 1.188 35.017Q1.364 35.200 1.458 35.458Q1.552 35.716 1.552 36.048Q1.552 36.140 1.470 36.161L-0.786 36.161L-0.786 36.222Q-0.786 36.810-0.502 37.193Q-0.218 37.576 0.349 37.576Q0.671 37.576 0.939 37.383Q1.207 37.190 1.296 36.875Q1.303 36.834 1.378 36.820L1.470 36.820Q1.552 36.844 1.552 36.916Q1.552 36.923 1.546 36.950Q1.433 37.347 1.062 37.586Q0.691 37.825 0.267 37.825Q-0.170 37.825-0.570 37.617Q-0.970 37.408-1.209 37.041Q-1.449 36.674-1.449 36.222M-0.779 35.952L1.036 35.952Q1.036 35.675 0.939 35.423Q0.841 35.170 0.643 35.014Q0.445 34.859 0.161 34.859Q-0.116 34.859-0.329 35.017Q-0.543 35.176-0.661 35.431Q-0.779 35.686-0.779 35.952M3.822 37.757L2.188 37.757L2.188 37.477Q2.417 37.477 2.566 37.443Q2.714 37.408 2.714 37.268L2.714 35.419Q2.714 35.149 2.607 35.088Q2.499 35.026 2.188 35.026L2.188 34.746L3.248 34.671L3.248 35.320Q3.419 35.012 3.723 34.841Q4.027 34.671 4.372 34.671Q4.878 34.671 5.162 34.894Q5.445 35.118 5.445 35.614L5.445 37.268Q5.445 37.405 5.594 37.441Q5.743 37.477 5.968 37.477L5.968 37.757L4.338 37.757L4.338 37.477Q4.567 37.477 4.716 37.443Q4.864 37.408 4.864 37.268L4.864 35.628Q4.864 35.293 4.745 35.093Q4.625 34.893 4.311 34.893Q4.041 34.893 3.807 35.029Q3.572 35.166 3.434 35.400Q3.296 35.634 3.296 35.908L3.296 37.268Q3.296 37.405 3.446 37.441Q3.596 37.477 3.822 37.477L3.822 37.757M6.556 36.246Q6.556 35.918 6.691 35.617Q6.826 35.317 7.062 35.096Q7.298 34.876 7.602 34.756Q7.906 34.636 8.231 34.636Q8.737 34.636 9.086 34.739Q9.434 34.841 9.434 35.217Q9.434 35.364 9.337 35.465Q9.239 35.566 9.092 35.566Q8.939 35.566 8.839 35.467Q8.740 35.368 8.740 35.217Q8.740 35.029 8.880 34.937Q8.679 34.886 8.238 34.886Q7.882 34.886 7.653 35.082Q7.424 35.279 7.324 35.588Q7.223 35.898 7.223 36.246Q7.223 36.595 7.349 36.901Q7.476 37.207 7.730 37.391Q7.985 37.576 8.340 37.576Q8.563 37.576 8.747 37.492Q8.932 37.408 9.067 37.253Q9.202 37.097 9.260 36.889Q9.274 36.834 9.328 36.834L9.441 36.834Q9.472 36.834 9.494 36.858Q9.516 36.882 9.516 36.916L9.516 36.937Q9.431 37.224 9.243 37.422Q9.055 37.620 8.790 37.723Q8.525 37.825 8.231 37.825Q7.800 37.825 7.412 37.619Q7.025 37.412 6.790 37.049Q6.556 36.687 6.556 36.246M10.063 36.222Q10.063 35.901 10.188 35.612Q10.313 35.323 10.538 35.100Q10.764 34.876 11.059 34.756Q11.355 34.636 11.673 34.636Q12.001 34.636 12.263 34.736Q12.524 34.835 12.700 35.017Q12.876 35.200 12.970 35.458Q13.064 35.716 13.064 36.048Q13.064 36.140 12.982 36.161L10.726 36.161L10.726 36.222Q10.726 36.810 11.010 37.193Q11.294 37.576 11.861 37.576Q12.182 37.576 12.451 37.383Q12.719 37.190 12.808 36.875Q12.815 36.834 12.890 36.820L12.982 36.820Q13.064 36.844 13.064 36.916Q13.064 36.923 13.057 36.950Q12.944 37.347 12.574 37.586Q12.203 37.825 11.779 37.825Q11.341 37.825 10.942 37.617Q10.542 37.408 10.302 37.041Q10.063 36.674 10.063 36.222M10.733 35.952L12.548 35.952Q12.548 35.675 12.451 35.423Q12.353 35.170 12.155 35.014Q11.957 34.859 11.673 34.859Q11.396 34.859 11.182 35.017Q10.969 35.176 10.851 35.431Q10.733 35.686 10.733 35.952\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-50.149 37.757v-94.139\"\u002F>\u003Cpath stroke=\"none\" d=\"m-50.149-58.982-2.08 4.16 2.08-1.56 2.08 1.56\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-12.122 -101.833)\">\u003Cpath d=\"M-48.211 37.825Q-48.642 37.825-49.018 37.613Q-49.394 37.402-49.614 37.041Q-49.835 36.680-49.835 36.246Q-49.835 35.802-49.597 35.441Q-49.359 35.081-48.973 34.876Q-48.587 34.671-48.146 34.671Q-47.828 34.671-47.545 34.826Q-47.261 34.982-47.076 35.258L-46.830 34.671L-46.594 34.671L-46.594 38.625Q-46.594 38.762-46.444 38.798Q-46.294 38.834-46.068 38.834L-46.068 39.114L-47.698 39.114L-47.698 38.834Q-47.473 38.834-47.324 38.799Q-47.175 38.765-47.175 38.625L-47.175 37.364Q-47.374 37.583-47.645 37.704Q-47.917 37.825-48.211 37.825M-48.160 37.603Q-47.828 37.603-47.556 37.398Q-47.285 37.193-47.145 36.868L-47.145 35.734Q-47.203 35.511-47.334 35.329Q-47.466 35.146-47.656 35.035Q-47.845 34.924-48.071 34.924Q-48.399 34.924-48.650 35.120Q-48.901 35.317-49.035 35.624Q-49.168 35.932-49.168 36.253Q-49.168 36.554-49.052 36.870Q-48.936 37.186-48.707 37.395Q-48.478 37.603-48.160 37.603M-45.060 36.923L-45.060 35.419Q-45.060 35.149-45.167 35.088Q-45.275 35.026-45.586 35.026L-45.586 34.746L-44.479 34.671L-44.479 36.903L-44.479 36.923Q-44.479 37.203-44.427 37.347Q-44.376 37.490-44.234 37.547Q-44.092 37.603-43.805 37.603Q-43.552 37.603-43.347 37.463Q-43.142 37.323-43.026 37.097Q-42.910 36.872-42.910 36.622L-42.910 35.419Q-42.910 35.149-43.017 35.088Q-43.125 35.026-43.436 35.026L-43.436 34.746L-42.329 34.671L-42.329 37.084Q-42.329 37.275-42.276 37.357Q-42.223 37.439-42.122 37.458Q-42.021 37.477-41.806 37.477L-41.806 37.757L-42.882 37.825L-42.882 37.261Q-42.992 37.443-43.137 37.566Q-43.282 37.689-43.469 37.757Q-43.655 37.825-43.857 37.825Q-45.060 37.825-45.060 36.923M-41.160 37.029Q-41.160 36.697-40.936 36.470Q-40.712 36.243-40.368 36.115Q-40.025 35.986-39.652 35.934Q-39.280 35.881-38.976 35.881L-38.976 35.628Q-38.976 35.423-39.083 35.243Q-39.191 35.064-39.372 34.961Q-39.553 34.859-39.762 34.859Q-40.169 34.859-40.404 34.951Q-40.316 34.988-40.269 35.072Q-40.223 35.156-40.223 35.258Q-40.223 35.354-40.269 35.433Q-40.316 35.511-40.396 35.556Q-40.476 35.600-40.565 35.600Q-40.715 35.600-40.816 35.503Q-40.917 35.405-40.917 35.258Q-40.917 34.636-39.762 34.636Q-39.550 34.636-39.300 34.700Q-39.051 34.763-38.849 34.882Q-38.648 35.002-38.521 35.187Q-38.395 35.371-38.395 35.614L-38.395 37.190Q-38.395 37.306-38.333 37.402Q-38.272 37.497-38.159 37.497Q-38.049 37.497-37.984 37.403Q-37.920 37.309-37.920 37.190L-37.920 36.742L-37.653 36.742L-37.653 37.190Q-37.653 37.460-37.880 37.625Q-38.107 37.791-38.388 37.791Q-38.596 37.791-38.733 37.637Q-38.870 37.484-38.894 37.268Q-39.041 37.535-39.323 37.680Q-39.605 37.825-39.929 37.825Q-40.206 37.825-40.490 37.750Q-40.774 37.675-40.967 37.496Q-41.160 37.316-41.160 37.029M-40.545 37.029Q-40.545 37.203-40.444 37.333Q-40.343 37.463-40.187 37.533Q-40.032 37.603-39.868 37.603Q-39.649 37.603-39.441 37.506Q-39.232 37.408-39.104 37.227Q-38.976 37.046-38.976 36.820L-38.976 36.092Q-39.300 36.092-39.666 36.183Q-40.032 36.274-40.288 36.486Q-40.545 36.697-40.545 37.029M-35.568 37.757L-37.171 37.757L-37.171 37.477Q-36.945 37.477-36.797 37.443Q-36.648 37.408-36.648 37.268L-36.648 33.649Q-36.648 33.379-36.756 33.317Q-36.863 33.256-37.171 33.256L-37.171 32.975L-36.094 32.900L-36.094 37.268Q-36.094 37.405-35.944 37.441Q-35.794 37.477-35.568 37.477L-35.568 37.757M-33.357 37.757L-34.908 37.757L-34.908 37.477Q-34.683 37.477-34.534 37.443Q-34.385 37.408-34.385 37.268L-34.385 35.419Q-34.385 35.231-34.433 35.147Q-34.481 35.064-34.578 35.045Q-34.676 35.026-34.888 35.026L-34.888 34.746L-33.832 34.671L-33.832 37.268Q-33.832 37.408-33.700 37.443Q-33.568 37.477-33.357 37.477L-33.357 37.757M-34.628 33.450Q-34.628 33.279-34.505 33.160Q-34.382 33.040-34.211 33.040Q-34.044 33.040-33.920 33.160Q-33.797 33.279-33.797 33.450Q-33.797 33.625-33.920 33.748Q-34.044 33.871-34.211 33.871Q-34.382 33.871-34.505 33.748Q-34.628 33.625-34.628 33.450M-32.184 36.916L-32.184 35.019L-32.823 35.019L-32.823 34.797Q-32.505 34.797-32.288 34.587Q-32.071 34.377-31.971 34.067Q-31.870 33.758-31.870 33.450L-31.603 33.450L-31.603 34.739L-30.526 34.739L-30.526 35.019L-31.603 35.019L-31.603 36.903Q-31.603 37.179-31.499 37.378Q-31.395 37.576-31.135 37.576Q-30.978 37.576-30.872 37.472Q-30.766 37.367-30.716 37.214Q-30.667 37.060-30.667 36.903L-30.667 36.489L-30.400 36.489L-30.400 36.916Q-30.400 37.142-30.499 37.352Q-30.598 37.562-30.783 37.694Q-30.967 37.825-31.196 37.825Q-31.634 37.825-31.909 37.588Q-32.184 37.350-32.184 36.916\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-12.122 -101.833)\">\u003Cpath d=\"M-29.458 38.892Q-29.328 38.960-29.191 38.960Q-29.020 38.960-28.870 38.871Q-28.719 38.782-28.608 38.637Q-28.497 38.492-28.419 38.324L-28.155 37.757L-29.324 35.231Q-29.399 35.084-29.529 35.052Q-29.659 35.019-29.892 35.019L-29.892 34.739L-28.371 34.739L-28.371 35.019Q-28.719 35.019-28.719 35.166Q-28.716 35.187-28.714 35.204Q-28.712 35.221-28.712 35.231L-27.855 37.090L-27.082 35.419Q-27.048 35.351-27.048 35.272Q-27.048 35.159-27.132 35.089Q-27.215 35.019-27.328 35.019L-27.328 34.739L-26.132 34.739L-26.132 35.019Q-26.351 35.019-26.523 35.123Q-26.696 35.228-26.788 35.419L-28.125 38.324Q-28.295 38.694-28.565 38.940Q-28.836 39.186-29.191 39.186Q-29.461 39.186-29.680 39.020Q-29.899 38.854-29.899 38.591Q-29.899 38.454-29.806 38.365Q-29.714 38.277-29.574 38.277Q-29.437 38.277-29.348 38.365Q-29.259 38.454-29.259 38.591Q-29.259 38.694-29.312 38.772Q-29.365 38.851-29.458 38.892\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-warn)\" d=\"m-47.303 33.21 2.196-3.364 2.197-3.22 2.197-3.081 2.197-2.963 2.197-2.834 2.197-2.716 2.196-2.604 2.197-2.502 2.197-2.39 2.197-2.288 2.197-2.203 2.196-2.1 2.197-2.023 2.197-1.93 2.197-1.853 2.197-1.776 2.197-1.708 2.196-1.63 2.197-1.571 2.197-1.494 2.197-1.434 2.197-1.383 2.197-1.316 2.196-1.272 2.197-1.212 2.197-1.161 2.197-1.11 2.197-1.067 2.196-1.025 2.197-.982 2.197-.939 2.197-.896 2.197-.871 2.197-.82 2.196-.794 2.197-.76 2.197-.734 2.197-.692 2.197-.674 2.196-.641 2.197-.614 2.197-.59 2.197-.562 2.197-.538 2.197-.521 2.196-.495 2.197-.48 2.197-.46 2.197-.436 2.197-.418 2.197-.4 2.196-.385 2.197-.376 2.197-.35 2.197-.342 2.197-.324 2.196-.308 2.197-.298 2.197-.29 2.197-.274 2.197-.264 2.197-.248 2.196-.24 2.197-.238 2.197-.223 2.197-.204 2.197-.205 2.196-.196 2.197-.188 2.197-.18 2.197-.17 2.197-.17 2.197-.154 2.196-.154 2.197-.145 2.197-.136 2.197-.137 2.197-.13 2.197-.118\" style=\"stroke-width:1.2\"\u002F>\u003Cg fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\">\u003Cg fill=\"var(--tk-warn)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(116.363 -46.15)\">\u003Cpath d=\"M-48.191 39.114L-49.821 39.114L-49.821 38.834Q-49.592 38.834-49.443 38.799Q-49.295 38.765-49.295 38.625L-49.295 35.279Q-49.295 35.108-49.431 35.067Q-49.568 35.026-49.821 35.026L-49.821 34.746L-48.741 34.671L-48.741 35.077Q-48.519 34.876-48.232 34.773Q-47.944 34.671-47.637 34.671Q-47.210 34.671-46.846 34.884Q-46.482 35.098-46.268 35.462Q-46.054 35.826-46.054 36.246Q-46.054 36.691-46.294 37.055Q-46.533 37.419-46.926 37.622Q-47.319 37.825-47.763 37.825Q-48.030 37.825-48.278 37.725Q-48.525 37.624-48.713 37.443L-48.713 38.625Q-48.713 38.762-48.565 38.798Q-48.416 38.834-48.191 38.834L-48.191 39.114M-48.713 35.426L-48.713 37.036Q-48.580 37.289-48.337 37.446Q-48.095 37.603-47.818 37.603Q-47.490 37.603-47.237 37.402Q-46.984 37.200-46.851 36.882Q-46.717 36.564-46.717 36.246Q-46.717 36.017-46.782 35.788Q-46.847 35.559-46.975 35.361Q-47.104 35.163-47.298 35.043Q-47.493 34.924-47.726 34.924Q-48.020 34.924-48.288 35.053Q-48.556 35.183-48.713 35.426M-43.669 37.757L-45.405 37.757L-45.405 37.477Q-45.176 37.477-45.027 37.443Q-44.878 37.408-44.878 37.268L-44.878 35.419Q-44.878 35.149-44.986 35.088Q-45.094 35.026-45.405 35.026L-45.405 34.746L-44.376 34.671L-44.376 35.378Q-44.246 35.070-44.003 34.871Q-43.761 34.671-43.443 34.671Q-43.224 34.671-43.053 34.795Q-42.882 34.920-42.882 35.132Q-42.882 35.269-42.982 35.368Q-43.081 35.467-43.214 35.467Q-43.351 35.467-43.450 35.368Q-43.549 35.269-43.549 35.132Q-43.549 34.992-43.450 34.893Q-43.740 34.893-43.940 35.089Q-44.140 35.286-44.232 35.580Q-44.325 35.874-44.325 36.154L-44.325 37.268Q-44.325 37.477-43.669 37.477L-43.669 37.757M-42.339 36.274Q-42.339 35.932-42.204 35.633Q-42.069 35.334-41.830 35.110Q-41.590 34.886-41.273 34.761Q-40.955 34.636-40.623 34.636Q-40.179 34.636-39.779 34.852Q-39.379 35.067-39.145 35.445Q-38.911 35.822-38.911 36.274Q-38.911 36.615-39.053 36.899Q-39.194 37.183-39.439 37.390Q-39.683 37.596-39.993 37.711Q-40.302 37.825-40.623 37.825Q-41.054 37.825-41.455 37.624Q-41.857 37.422-42.098 37.070Q-42.339 36.718-42.339 36.274M-40.623 37.576Q-40.022 37.576-39.798 37.198Q-39.574 36.820-39.574 36.188Q-39.574 35.576-39.808 35.217Q-40.042 34.859-40.623 34.859Q-41.676 34.859-41.676 36.188Q-41.676 36.820-41.450 37.198Q-41.225 37.576-40.623 37.576\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(116.363 -46.15)\">\u003Cpath d=\"M-37.339 37.757L-38.662 37.757L-38.662 37.477Q-38.101 37.477-37.722 37.077L-37.007 36.280L-37.920 35.231Q-38.057 35.084-38.205 35.052Q-38.354 35.019-38.621 35.019L-38.621 34.739L-37.120 34.739L-37.120 35.019Q-37.312 35.019-37.312 35.153Q-37.312 35.183-37.281 35.231L-36.686 35.915L-36.245 35.419Q-36.132 35.289-36.132 35.173Q-36.132 35.111-36.170 35.065Q-36.208 35.019-36.266 35.019L-36.266 34.739L-34.950 34.739L-34.950 35.019Q-35.510 35.019-35.890 35.419L-36.512 36.120L-35.517 37.268Q-35.418 37.367-35.317 37.412Q-35.216 37.456-35.105 37.466Q-34.994 37.477-34.816 37.477L-34.816 37.757L-36.310 37.757L-36.310 37.477Q-36.245 37.477-36.185 37.443Q-36.126 37.408-36.126 37.343Q-36.126 37.296-36.156 37.268L-36.833 36.482L-37.366 37.077Q-37.479 37.207-37.479 37.323Q-37.479 37.388-37.438 37.432Q-37.397 37.477-37.339 37.477L-37.339 37.757M-33.986 38.892Q-33.856 38.960-33.719 38.960Q-33.548 38.960-33.398 38.871Q-33.248 38.782-33.137 38.637Q-33.025 38.492-32.947 38.324L-32.684 37.757L-33.853 35.231Q-33.928 35.084-34.058 35.052Q-34.188 35.019-34.420 35.019L-34.420 34.739L-32.899 34.739L-32.899 35.019Q-33.248 35.019-33.248 35.166Q-33.244 35.187-33.242 35.204Q-33.241 35.221-33.241 35.231L-32.383 37.090L-31.610 35.419Q-31.576 35.351-31.576 35.272Q-31.576 35.159-31.660 35.089Q-31.744 35.019-31.857 35.019L-31.857 34.739L-30.660 34.739L-30.660 35.019Q-30.879 35.019-31.052 35.123Q-31.224 35.228-31.316 35.419L-32.653 38.324Q-32.824 38.694-33.094 38.940Q-33.364 39.186-33.719 39.186Q-33.989 39.186-34.208 39.020Q-34.427 38.854-34.427 38.591Q-34.427 38.454-34.335 38.365Q-34.242 38.277-34.102 38.277Q-33.965 38.277-33.877 38.365Q-33.788 38.454-33.788 38.591Q-33.788 38.694-33.841 38.772Q-33.894 38.851-33.986 38.892\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(116.363 -46.15)\">\u003Cpath d=\"M-25.675 37.757L-27.411 37.757L-27.411 37.477Q-27.182 37.477-27.033 37.443Q-26.885 37.408-26.885 37.268L-26.885 35.419Q-26.885 35.149-26.992 35.088Q-27.100 35.026-27.411 35.026L-27.411 34.746L-26.382 34.671L-26.382 35.378Q-26.252 35.070-26.010 34.871Q-25.767 34.671-25.449 34.671Q-25.230 34.671-25.059 34.795Q-24.888 34.920-24.888 35.132Q-24.888 35.269-24.988 35.368Q-25.087 35.467-25.220 35.467Q-25.357 35.467-25.456 35.368Q-25.555 35.269-25.555 35.132Q-25.555 34.992-25.456 34.893Q-25.746 34.893-25.946 35.089Q-26.146 35.286-26.239 35.580Q-26.331 35.874-26.331 36.154L-26.331 37.268Q-26.331 37.477-25.675 37.477L-25.675 37.757M-24.345 36.222Q-24.345 35.901-24.220 35.612Q-24.095 35.323-23.870 35.100Q-23.644 34.876-23.349 34.756Q-23.053 34.636-22.735 34.636Q-22.407 34.636-22.145 34.736Q-21.884 34.835-21.708 35.017Q-21.532 35.200-21.438 35.458Q-21.344 35.716-21.344 36.048Q-21.344 36.140-21.426 36.161L-23.682 36.161L-23.682 36.222Q-23.682 36.810-23.398 37.193Q-23.114 37.576-22.547 37.576Q-22.226 37.576-21.958 37.383Q-21.689 37.190-21.600 36.875Q-21.593 36.834-21.518 36.820L-21.426 36.820Q-21.344 36.844-21.344 36.916Q-21.344 36.923-21.351 36.950Q-21.464 37.347-21.834 37.586Q-22.205 37.825-22.629 37.825Q-23.067 37.825-23.467 37.617Q-23.866 37.408-24.106 37.041Q-24.345 36.674-24.345 36.222M-23.675 35.952L-21.860 35.952Q-21.860 35.675-21.958 35.423Q-22.055 35.170-22.253 35.014Q-22.451 34.859-22.735 34.859Q-23.012 34.859-23.226 35.017Q-23.439 35.176-23.557 35.431Q-23.675 35.686-23.675 35.952M-19.368 37.730L-20.349 35.231Q-20.411 35.088-20.529 35.053Q-20.647 35.019-20.862 35.019L-20.862 34.739L-19.382 34.739L-19.382 35.019Q-19.761 35.019-19.761 35.180Q-19.761 35.190-19.748 35.231L-19.033 37.063L-18.360 35.358Q-18.391 35.286-18.391 35.258Q-18.391 35.231-18.418 35.231Q-18.480 35.084-18.598 35.052Q-18.716 35.019-18.927 35.019L-18.927 34.739L-17.530 34.739L-17.530 35.019Q-17.906 35.019-17.906 35.180Q-17.906 35.211-17.899 35.231L-17.143 37.169L-16.456 35.419Q-16.436 35.368-16.436 35.313Q-16.436 35.173-16.549 35.096Q-16.661 35.019-16.802 35.019L-16.802 34.739L-15.581 34.739L-15.581 35.019Q-15.786 35.019-15.942 35.125Q-16.097 35.231-16.169 35.419L-17.075 37.730Q-17.109 37.825-17.222 37.825L-17.290 37.825Q-17.400 37.825-17.437 37.730L-18.220 35.727L-19.006 37.730Q-19.040 37.825-19.153 37.825L-19.221 37.825Q-19.331 37.825-19.368 37.730\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(116.363 -46.15)\">\u003Cpath d=\"M-15.192 37.029Q-15.192 36.697-14.969 36.470Q-14.745 36.243-14.401 36.115Q-14.058 35.986-13.685 35.934Q-13.313 35.881-13.008 35.881L-13.008 35.628Q-13.008 35.423-13.116 35.243Q-13.224 35.064-13.405 34.961Q-13.586 34.859-13.794 34.859Q-14.201 34.859-14.437 34.951Q-14.348 34.988-14.302 35.072Q-14.256 35.156-14.256 35.258Q-14.256 35.354-14.302 35.433Q-14.348 35.511-14.429 35.556Q-14.509 35.600-14.598 35.600Q-14.748 35.600-14.849 35.503Q-14.950 35.405-14.950 35.258Q-14.950 34.636-13.794 34.636Q-13.583 34.636-13.333 34.700Q-13.084 34.763-12.882 34.882Q-12.680 35.002-12.554 35.187Q-12.427 35.371-12.427 35.614L-12.427 37.190Q-12.427 37.306-12.366 37.402Q-12.304 37.497-12.191 37.497Q-12.082 37.497-12.017 37.403Q-11.952 37.309-11.952 37.190L-11.952 36.742L-11.686 36.742L-11.686 37.190Q-11.686 37.460-11.913 37.625Q-12.140 37.791-12.420 37.791Q-12.629 37.791-12.766 37.637Q-12.902 37.484-12.926 37.268Q-13.073 37.535-13.355 37.680Q-13.637 37.825-13.962 37.825Q-14.239 37.825-14.523 37.750Q-14.806 37.675-14.999 37.496Q-15.192 37.316-15.192 37.029M-14.577 37.029Q-14.577 37.203-14.476 37.333Q-14.376 37.463-14.220 37.533Q-14.065 37.603-13.900 37.603Q-13.682 37.603-13.473 37.506Q-13.265 37.408-13.137 37.227Q-13.008 37.046-13.008 36.820L-13.008 36.092Q-13.333 36.092-13.699 36.183Q-14.065 36.274-14.321 36.486Q-14.577 36.697-14.577 37.029M-9.519 37.757L-11.255 37.757L-11.255 37.477Q-11.026 37.477-10.877 37.443Q-10.729 37.408-10.729 37.268L-10.729 35.419Q-10.729 35.149-10.836 35.088Q-10.944 35.026-11.255 35.026L-11.255 34.746L-10.226 34.671L-10.226 35.378Q-10.096 35.070-9.854 34.871Q-9.611 34.671-9.293 34.671Q-9.074 34.671-8.903 34.795Q-8.732 34.920-8.732 35.132Q-8.732 35.269-8.832 35.368Q-8.931 35.467-9.064 35.467Q-9.201 35.467-9.300 35.368Q-9.399 35.269-9.399 35.132Q-9.399 34.992-9.300 34.893Q-9.590 34.893-9.790 35.089Q-9.990 35.286-10.083 35.580Q-10.175 35.874-10.175 36.154L-10.175 37.268Q-10.175 37.477-9.519 37.477L-9.519 37.757M-8.148 36.246Q-8.148 35.908-8.008 35.617Q-7.868 35.327-7.623 35.113Q-7.379 34.900-7.075 34.785Q-6.771 34.671-6.446 34.671Q-6.176 34.671-5.913 34.770Q-5.649 34.869-5.458 35.047L-5.458 33.649Q-5.458 33.379-5.566 33.317Q-5.673 33.256-5.984 33.256L-5.984 32.975L-4.908 32.900L-4.908 37.084Q-4.908 37.272-4.853 37.355Q-4.798 37.439-4.698 37.458Q-4.597 37.477-4.381 37.477L-4.381 37.757L-5.489 37.825L-5.489 37.408Q-5.906 37.825-6.531 37.825Q-6.962 37.825-7.335 37.613Q-7.707 37.402-7.928 37.041Q-8.148 36.680-8.148 36.246M-6.473 37.603Q-6.265 37.603-6.078 37.531Q-5.892 37.460-5.738 37.323Q-5.585 37.186-5.489 37.008L-5.489 35.399Q-5.574 35.252-5.720 35.132Q-5.865 35.012-6.034 34.953Q-6.203 34.893-6.384 34.893Q-6.945 34.893-7.213 35.282Q-7.482 35.672-7.482 36.253Q-7.482 36.824-7.247 37.214Q-7.013 37.603-6.473 37.603\" fill=\"var(--tk-warn)\" stroke=\"var(--tk-warn)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-accent)\" d=\"m-47.303 16.258 2.196-1.975 2.197-2.075 2.197-2.166 2.197-2.267 2.197-2.337 2.197-2.42 2.196-2.482L-29.729-2l2.197-2.574 2.197-2.604 2.197-2.605 2.196-2.606 2.197-2.58 2.197-2.545 2.197-2.474 2.197-2.405 2.197-2.305 2.196-2.198 2.197-2.051 2.197-1.913 2.197-1.745 2.197-1.56 2.197-1.368 2.196-1.152 2.197-.938 2.197-.715 2.197-.476 2.197-.246 2.196.008 2.197.238 2.197.484 2.197.715 2.197.937 2.197 1.16 2.196 1.376 2.197 1.56 2.197 1.744 2.197 1.906 2.197 2.066 2.196 2.191 2.197 2.313 2.197 2.397 2.197 2.49 2.197 2.535 2.197 2.583 2.196 2.605 2.197 2.613 2.197 2.596 2.197 2.574L62.538.56l2.197 2.49 2.196 2.413 2.197 2.343 2.197 2.26 2.197 2.174 2.197 2.075 2.196 1.967 2.197 1.868 2.197 1.766 2.197 1.653 2.197 1.544 2.197 1.445 2.196 1.337 2.197 1.237 2.197 1.13 2.197 1.045 2.197.953 2.196.86 2.197.783 2.197.708 2.197.638 2.197.568 2.197.507 2.196.453 2.197.4 2.197.355 2.197.313 2.197.277 2.197.24\" style=\"stroke-width:1.2\"\u002F>\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(44.861 -85.985)\">\u003Cpath d=\"M-49.308 36.916L-49.308 35.019L-49.947 35.019L-49.947 34.797Q-49.629 34.797-49.412 34.587Q-49.195 34.377-49.095 34.067Q-48.994 33.758-48.994 33.450L-48.727 33.450L-48.727 34.739L-47.650 34.739L-47.650 35.019L-48.727 35.019L-48.727 36.903Q-48.727 37.179-48.623 37.378Q-48.519 37.576-48.259 37.576Q-48.102 37.576-47.996 37.472Q-47.890 37.367-47.840 37.214Q-47.791 37.060-47.791 36.903L-47.791 36.489L-47.524 36.489L-47.524 36.916Q-47.524 37.142-47.623 37.352Q-47.722 37.562-47.907 37.694Q-48.091 37.825-48.320 37.825Q-48.758 37.825-49.033 37.588Q-49.308 37.350-49.308 36.916M-44.964 37.757L-46.700 37.757L-46.700 37.477Q-46.471 37.477-46.323 37.443Q-46.174 37.408-46.174 37.268L-46.174 35.419Q-46.174 35.149-46.282 35.088Q-46.389 35.026-46.700 35.026L-46.700 34.746L-45.671 34.671L-45.671 35.378Q-45.542 35.070-45.299 34.871Q-45.056 34.671-44.738 34.671Q-44.520 34.671-44.349 34.795Q-44.178 34.920-44.178 35.132Q-44.178 35.269-44.277 35.368Q-44.376 35.467-44.509 35.467Q-44.646 35.467-44.745 35.368Q-44.844 35.269-44.844 35.132Q-44.844 34.992-44.745 34.893Q-45.036 34.893-45.236 35.089Q-45.436 35.286-45.528 35.580Q-45.620 35.874-45.620 36.154L-45.620 37.268Q-45.620 37.477-44.964 37.477L-44.964 37.757M-43.019 36.923L-43.019 35.419Q-43.019 35.149-43.127 35.088Q-43.234 35.026-43.545 35.026L-43.545 34.746L-42.438 34.671L-42.438 36.903L-42.438 36.923Q-42.438 37.203-42.387 37.347Q-42.336 37.490-42.194 37.547Q-42.052 37.603-41.765 37.603Q-41.512 37.603-41.307 37.463Q-41.102 37.323-40.985 37.097Q-40.869 36.872-40.869 36.622L-40.869 35.419Q-40.869 35.149-40.977 35.088Q-41.085 35.026-41.396 35.026L-41.396 34.746L-40.288 34.671L-40.288 37.084Q-40.288 37.275-40.235 37.357Q-40.182 37.439-40.081 37.458Q-39.981 37.477-39.765 37.477L-39.765 37.757L-40.842 37.825L-40.842 37.261Q-40.951 37.443-41.097 37.566Q-41.242 37.689-41.428 37.757Q-41.614 37.825-41.816 37.825Q-43.019 37.825-43.019 36.923M-39.218 36.222Q-39.218 35.901-39.094 35.612Q-38.969 35.323-38.743 35.100Q-38.518 34.876-38.222 34.756Q-37.926 34.636-37.608 34.636Q-37.280 34.636-37.019 34.736Q-36.757 34.835-36.581 35.017Q-36.405 35.200-36.311 35.458Q-36.217 35.716-36.217 36.048Q-36.217 36.140-36.299 36.161L-38.555 36.161L-38.555 36.222Q-38.555 36.810-38.272 37.193Q-37.988 37.576-37.420 37.576Q-37.099 37.576-36.831 37.383Q-36.563 37.190-36.474 36.875Q-36.467 36.834-36.392 36.820L-36.299 36.820Q-36.217 36.844-36.217 36.916Q-36.217 36.923-36.224 36.950Q-36.337 37.347-36.708 37.586Q-37.079 37.825-37.503 37.825Q-37.940 37.825-38.340 37.617Q-38.740 37.408-38.979 37.041Q-39.218 36.674-39.218 36.222M-38.548 35.952L-36.733 35.952Q-36.733 35.675-36.831 35.423Q-36.928 35.170-37.127 35.014Q-37.325 34.859-37.608 34.859Q-37.885 34.859-38.099 35.017Q-38.313 35.176-38.430 35.431Q-38.548 35.686-38.548 35.952\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(44.861 -85.985)\">\u003Cpath d=\"M-31.294 37.825Q-31.725 37.825-32.101 37.613Q-32.477 37.402-32.697 37.041Q-32.918 36.680-32.918 36.246Q-32.918 35.802-32.680 35.441Q-32.442 35.081-32.056 34.876Q-31.670 34.671-31.229 34.671Q-30.911 34.671-30.628 34.826Q-30.344 34.982-30.159 35.258L-29.913 34.671L-29.677 34.671L-29.677 38.625Q-29.677 38.762-29.527 38.798Q-29.377 38.834-29.151 38.834L-29.151 39.114L-30.781 39.114L-30.781 38.834Q-30.556 38.834-30.407 38.799Q-30.258 38.765-30.258 38.625L-30.258 37.364Q-30.457 37.583-30.728 37.704Q-31 37.825-31.294 37.825M-31.243 37.603Q-30.911 37.603-30.639 37.398Q-30.368 37.193-30.228 36.868L-30.228 35.734Q-30.286 35.511-30.417 35.329Q-30.549 35.146-30.739 35.035Q-30.928 34.924-31.154 34.924Q-31.482 34.924-31.733 35.120Q-31.984 35.317-32.118 35.624Q-32.251 35.932-32.251 36.253Q-32.251 36.554-32.135 36.870Q-32.019 37.186-31.790 37.395Q-31.561 37.603-31.243 37.603M-28.143 36.923L-28.143 35.419Q-28.143 35.149-28.250 35.088Q-28.358 35.026-28.669 35.026L-28.669 34.746L-27.562 34.671L-27.562 36.903L-27.562 36.923Q-27.562 37.203-27.510 37.347Q-27.459 37.490-27.317 37.547Q-27.175 37.603-26.888 37.603Q-26.635 37.603-26.430 37.463Q-26.225 37.323-26.109 37.097Q-25.993 36.872-25.993 36.622L-25.993 35.419Q-25.993 35.149-26.100 35.088Q-26.208 35.026-26.519 35.026L-26.519 34.746L-25.412 34.671L-25.412 37.084Q-25.412 37.275-25.359 37.357Q-25.306 37.439-25.205 37.458Q-25.104 37.477-24.889 37.477L-24.889 37.757L-25.965 37.825L-25.965 37.261Q-26.075 37.443-26.220 37.566Q-26.365 37.689-26.552 37.757Q-26.738 37.825-26.940 37.825Q-28.143 37.825-28.143 36.923M-24.243 37.029Q-24.243 36.697-24.019 36.470Q-23.795 36.243-23.451 36.115Q-23.108 35.986-22.735 35.934Q-22.363 35.881-22.059 35.881L-22.059 35.628Q-22.059 35.423-22.166 35.243Q-22.274 35.064-22.455 34.961Q-22.636 34.859-22.845 34.859Q-23.252 34.859-23.487 34.951Q-23.399 34.988-23.352 35.072Q-23.306 35.156-23.306 35.258Q-23.306 35.354-23.352 35.433Q-23.399 35.511-23.479 35.556Q-23.559 35.600-23.648 35.600Q-23.798 35.600-23.899 35.503Q-24 35.405-24 35.258Q-24 34.636-22.845 34.636Q-22.633 34.636-22.383 34.700Q-22.134 34.763-21.932 34.882Q-21.731 35.002-21.604 35.187Q-21.478 35.371-21.478 35.614L-21.478 37.190Q-21.478 37.306-21.416 37.402Q-21.355 37.497-21.242 37.497Q-21.132 37.497-21.067 37.403Q-21.003 37.309-21.003 37.190L-21.003 36.742L-20.736 36.742L-20.736 37.190Q-20.736 37.460-20.963 37.625Q-21.190 37.791-21.471 37.791Q-21.679 37.791-21.816 37.637Q-21.953 37.484-21.977 37.268Q-22.124 37.535-22.406 37.680Q-22.688 37.825-23.012 37.825Q-23.289 37.825-23.573 37.750Q-23.857 37.675-24.050 37.496Q-24.243 37.316-24.243 37.029M-23.628 37.029Q-23.628 37.203-23.527 37.333Q-23.426 37.463-23.270 37.533Q-23.115 37.603-22.951 37.603Q-22.732 37.603-22.524 37.506Q-22.315 37.408-22.187 37.227Q-22.059 37.046-22.059 36.820L-22.059 36.092Q-22.383 36.092-22.749 36.183Q-23.115 36.274-23.371 36.486Q-23.628 36.697-23.628 37.029M-18.651 37.757L-20.254 37.757L-20.254 37.477Q-20.028 37.477-19.880 37.443Q-19.731 37.408-19.731 37.268L-19.731 33.649Q-19.731 33.379-19.839 33.317Q-19.946 33.256-20.254 33.256L-20.254 32.975L-19.177 32.900L-19.177 37.268Q-19.177 37.405-19.027 37.441Q-18.877 37.477-18.651 37.477L-18.651 37.757M-16.440 37.757L-17.991 37.757L-17.991 37.477Q-17.766 37.477-17.617 37.443Q-17.468 37.408-17.468 37.268L-17.468 35.419Q-17.468 35.231-17.516 35.147Q-17.564 35.064-17.661 35.045Q-17.759 35.026-17.971 35.026L-17.971 34.746L-16.915 34.671L-16.915 37.268Q-16.915 37.408-16.783 37.443Q-16.651 37.477-16.440 37.477L-16.440 37.757M-17.711 33.450Q-17.711 33.279-17.588 33.160Q-17.465 33.040-17.294 33.040Q-17.127 33.040-17.003 33.160Q-16.880 33.279-16.880 33.450Q-16.880 33.625-17.003 33.748Q-17.127 33.871-17.294 33.871Q-17.465 33.871-17.588 33.748Q-17.711 33.625-17.711 33.450M-15.267 36.916L-15.267 35.019L-15.906 35.019L-15.906 34.797Q-15.588 34.797-15.371 34.587Q-15.154 34.377-15.054 34.067Q-14.953 33.758-14.953 33.450L-14.686 33.450L-14.686 34.739L-13.609 34.739L-13.609 35.019L-14.686 35.019L-14.686 36.903Q-14.686 37.179-14.582 37.378Q-14.478 37.576-14.218 37.576Q-14.061 37.576-13.955 37.472Q-13.849 37.367-13.799 37.214Q-13.750 37.060-13.750 36.903L-13.750 36.489L-13.483 36.489L-13.483 36.916Q-13.483 37.142-13.582 37.352Q-13.681 37.562-13.866 37.694Q-14.050 37.825-14.279 37.825Q-14.717 37.825-14.992 37.588Q-15.267 37.350-15.267 36.916\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(44.861 -85.985)\">\u003Cpath d=\"M-12.541 38.892Q-12.411 38.960-12.274 38.960Q-12.103 38.960-11.953 38.871Q-11.802 38.782-11.691 38.637Q-11.580 38.492-11.502 38.324L-11.238 37.757L-12.407 35.231Q-12.482 35.084-12.612 35.052Q-12.742 35.019-12.975 35.019L-12.975 34.739L-11.454 34.739L-11.454 35.019Q-11.802 35.019-11.802 35.166Q-11.799 35.187-11.797 35.204Q-11.795 35.221-11.795 35.231L-10.938 37.090L-10.165 35.419Q-10.131 35.351-10.131 35.272Q-10.131 35.159-10.215 35.089Q-10.298 35.019-10.411 35.019L-10.411 34.739L-9.215 34.739L-9.215 35.019Q-9.434 35.019-9.606 35.123Q-9.779 35.228-9.871 35.419L-11.208 38.324Q-11.378 38.694-11.648 38.940Q-11.919 39.186-12.274 39.186Q-12.544 39.186-12.763 39.020Q-12.982 38.854-12.982 38.591Q-12.982 38.454-12.889 38.365Q-12.797 38.277-12.657 38.277Q-12.520 38.277-12.431 38.365Q-12.342 38.454-12.342 38.591Q-12.342 38.694-12.395 38.772Q-12.448 38.851-12.541 38.892\" 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=\"M15.293 37.757v-76.822\" style=\"stroke-dasharray:3.0,3.0\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(51.323 11.24)\">\u003Cpath d=\"M-49.028 37.757L-49.295 37.757L-49.295 33.649Q-49.295 33.379-49.402 33.317Q-49.510 33.256-49.821 33.256L-49.821 32.975L-48.741 32.900L-48.741 35.070Q-48.532 34.879-48.247 34.775Q-47.962 34.671-47.664 34.671Q-47.346 34.671-47.049 34.792Q-46.752 34.913-46.529 35.129Q-46.307 35.344-46.181 35.629Q-46.054 35.915-46.054 36.246Q-46.054 36.691-46.294 37.055Q-46.533 37.419-46.926 37.622Q-47.319 37.825-47.763 37.825Q-47.958 37.825-48.148 37.769Q-48.337 37.713-48.498 37.608Q-48.659 37.504-48.799 37.343L-49.028 37.757M-48.713 35.412L-48.713 37.029Q-48.577 37.289-48.336 37.446Q-48.095 37.603-47.818 37.603Q-47.524 37.603-47.312 37.496Q-47.100 37.388-46.967 37.196Q-46.834 37.005-46.775 36.766Q-46.717 36.527-46.717 36.246Q-46.717 35.887-46.811 35.583Q-46.905 35.279-47.133 35.086Q-47.360 34.893-47.726 34.893Q-48.026 34.893-48.293 35.029Q-48.560 35.166-48.713 35.412\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(51.323 11.24)\">\u003Cpath d=\"M-45.244 36.222Q-45.244 35.901-45.119 35.612Q-44.994 35.323-44.768 35.100Q-44.543 34.876-44.247 34.756Q-43.952 34.636-43.634 34.636Q-43.306 34.636-43.044 34.736Q-42.783 34.835-42.607 35.017Q-42.431 35.200-42.337 35.458Q-42.243 35.716-42.243 36.048Q-42.243 36.140-42.325 36.161L-44.580 36.161L-44.580 36.222Q-44.580 36.810-44.297 37.193Q-44.013 37.576-43.446 37.576Q-43.124 37.576-42.856 37.383Q-42.588 37.190-42.499 36.875Q-42.492 36.834-42.417 36.820L-42.325 36.820Q-42.243 36.844-42.243 36.916Q-42.243 36.923-42.249 36.950Q-42.362 37.347-42.733 37.586Q-43.104 37.825-43.528 37.825Q-43.965 37.825-44.365 37.617Q-44.765 37.408-45.004 37.041Q-45.244 36.674-45.244 36.222M-44.574 35.952L-42.759 35.952Q-42.759 35.675-42.856 35.423Q-42.954 35.170-43.152 35.014Q-43.350 34.859-43.634 34.859Q-43.911 34.859-44.124 35.017Q-44.338 35.176-44.456 35.431Q-44.574 35.686-44.574 35.952M-41.655 37.750L-41.655 36.687Q-41.655 36.663-41.627 36.636Q-41.600 36.609-41.576 36.609L-41.467 36.609Q-41.402 36.609-41.388 36.667Q-41.292 37.101-41.046 37.352Q-40.800 37.603-40.387 37.603Q-40.045 37.603-39.792 37.470Q-39.539 37.337-39.539 37.029Q-39.539 36.872-39.633 36.757Q-39.727 36.643-39.865 36.574Q-40.004 36.506-40.171 36.468L-40.752 36.369Q-41.108 36.301-41.381 36.080Q-41.655 35.860-41.655 35.518Q-41.655 35.269-41.544 35.094Q-41.433 34.920-41.246 34.821Q-41.060 34.722-40.845 34.679Q-40.629 34.636-40.387 34.636Q-39.973 34.636-39.693 34.818L-39.477 34.643Q-39.467 34.640-39.460 34.638Q-39.454 34.636-39.443 34.636L-39.392 34.636Q-39.365 34.636-39.341 34.660Q-39.317 34.684-39.317 34.712L-39.317 35.559Q-39.317 35.580-39.341 35.607Q-39.365 35.634-39.392 35.634L-39.505 35.634Q-39.532 35.634-39.558 35.609Q-39.583 35.583-39.583 35.559Q-39.583 35.323-39.689 35.159Q-39.795 34.995-39.978 34.913Q-40.161 34.831-40.393 34.831Q-40.722 34.831-40.978 34.934Q-41.234 35.036-41.234 35.313Q-41.234 35.508-41.051 35.617Q-40.869 35.727-40.640 35.768L-40.065 35.874Q-39.819 35.922-39.606 36.050Q-39.392 36.178-39.255 36.381Q-39.119 36.585-39.119 36.834Q-39.119 37.347-39.484 37.586Q-39.850 37.825-40.387 37.825Q-40.882 37.825-41.214 37.531L-41.480 37.805Q-41.501 37.825-41.528 37.825L-41.576 37.825Q-41.600 37.825-41.627 37.798Q-41.655 37.771-41.655 37.750M-37.963 36.916L-37.963 35.019L-38.602 35.019L-38.602 34.797Q-38.285 34.797-38.068 34.587Q-37.850 34.377-37.750 34.067Q-37.649 33.758-37.649 33.450L-37.382 33.450L-37.382 34.739L-36.306 34.739L-36.306 35.019L-37.382 35.019L-37.382 36.903Q-37.382 37.179-37.278 37.378Q-37.174 37.576-36.914 37.576Q-36.757 37.576-36.651 37.472Q-36.545 37.367-36.495 37.214Q-36.446 37.060-36.446 36.903L-36.446 36.489L-36.179 36.489L-36.179 36.916Q-36.179 37.142-36.278 37.352Q-36.377 37.562-36.562 37.694Q-36.746 37.825-36.975 37.825Q-37.413 37.825-37.688 37.588Q-37.963 37.350-37.963 36.916\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(51.323 11.24)\">\u003Cpath d=\"M-30.462 37.757L-32.595 37.757L-32.595 37.477Q-31.873 37.477-31.873 37.268L-31.873 33.467Q-31.873 33.256-32.595 33.256L-32.595 32.975L-30.462 32.975L-30.462 33.256Q-31.183 33.256-31.183 33.467L-31.183 35.693L-28.814 33.655Q-28.698 33.546-28.698 33.444Q-28.698 33.351-28.775 33.303Q-28.852 33.256-28.948 33.256L-28.948 32.975L-27.345 32.975L-27.345 33.256Q-27.632 33.256-27.909 33.361Q-28.185 33.467-28.404 33.655L-29.788 34.845L-28.097 37.077Q-27.960 37.255-27.850 37.340Q-27.741 37.425-27.616 37.451Q-27.492 37.477-27.242 37.477L-27.242 37.757L-29.125 37.757L-29.125 37.477Q-28.972 37.477-28.866 37.446Q-28.760 37.415-28.760 37.302Q-28.760 37.217-28.872 37.077L-30.257 35.252L-31.183 36.041L-31.183 37.268Q-31.183 37.477-30.462 37.477L-30.462 37.757M-22.508 37.757L-26.476 37.757L-26.476 37.477Q-25.755 37.477-25.755 37.268L-25.755 33.467Q-25.755 33.256-26.476 33.256L-26.476 32.975L-24.162 32.975L-24.162 33.256Q-24.480 33.256-24.773 33.291Q-25.065 33.327-25.065 33.467L-25.065 37.268Q-25.065 37.408-24.976 37.443Q-24.887 37.477-24.699 37.477L-24.077 37.477Q-23.663 37.477-23.385 37.374Q-23.106 37.272-22.941 37.070Q-22.775 36.868-22.693 36.581Q-22.611 36.294-22.566 35.887L-22.300 35.887\" 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\">Reward over-optimization. True quality (blue) rises then falls as the policy travels in KL from the reference, while the proxy reward (red) keeps climbing.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:404.913px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 303.685 109.020\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg fill=\"var(--tk-soft-neutral)\" style=\"stroke-width:.8\">\u003Cpath d=\"M-65.203.58h91.048V-42.1h-91.048Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(-23.701 6.43)\">\u003Cpath d=\"M-16.452-28.760L-18.188-28.760L-18.188-29.040Q-17.959-29.040-17.810-29.074Q-17.662-29.109-17.662-29.249L-17.662-31.098Q-17.662-31.368-17.769-31.429Q-17.877-31.491-18.188-31.491L-18.188-31.771L-17.159-31.846L-17.159-31.139Q-17.029-31.447-16.787-31.646Q-16.544-31.846-16.226-31.846Q-16.007-31.846-15.836-31.722Q-15.665-31.597-15.665-31.385Q-15.665-31.248-15.765-31.149Q-15.864-31.050-15.997-31.050Q-16.134-31.050-16.233-31.149Q-16.332-31.248-16.332-31.385Q-16.332-31.525-16.233-31.624Q-16.523-31.624-16.723-31.428Q-16.923-31.231-17.016-30.937Q-17.108-30.643-17.108-30.363L-17.108-29.249Q-17.108-29.040-16.452-29.040L-16.452-28.760M-15.122-30.295Q-15.122-30.616-14.997-30.905Q-14.872-31.194-14.647-31.417Q-14.421-31.641-14.126-31.761Q-13.830-31.881-13.512-31.881Q-13.184-31.881-12.922-31.781Q-12.661-31.682-12.485-31.500Q-12.309-31.317-12.215-31.059Q-12.121-30.801-12.121-30.469Q-12.121-30.377-12.203-30.356L-14.459-30.356L-14.459-30.295Q-14.459-29.707-14.175-29.324Q-13.891-28.941-13.324-28.941Q-13.003-28.941-12.735-29.134Q-12.466-29.327-12.377-29.642Q-12.370-29.683-12.295-29.697L-12.203-29.697Q-12.121-29.673-12.121-29.601Q-12.121-29.594-12.128-29.567Q-12.241-29.170-12.611-28.931Q-12.982-28.692-13.406-28.692Q-13.844-28.692-14.244-28.900Q-14.643-29.109-14.883-29.476Q-15.122-29.843-15.122-30.295M-14.452-30.565L-12.637-30.565Q-12.637-30.842-12.735-31.094Q-12.832-31.347-13.030-31.503Q-13.228-31.658-13.512-31.658Q-13.789-31.658-14.003-31.500Q-14.216-31.341-14.334-31.086Q-14.452-30.831-14.452-30.565M-10.145-28.787L-11.126-31.286Q-11.188-31.429-11.306-31.464Q-11.424-31.498-11.639-31.498L-11.639-31.778L-10.159-31.778L-10.159-31.498Q-10.538-31.498-10.538-31.337Q-10.538-31.327-10.525-31.286L-9.810-29.454L-9.137-31.159Q-9.168-31.231-9.168-31.259Q-9.168-31.286-9.195-31.286Q-9.257-31.433-9.375-31.465Q-9.493-31.498-9.704-31.498L-9.704-31.778L-8.307-31.778L-8.307-31.498Q-8.683-31.498-8.683-31.337Q-8.683-31.306-8.676-31.286L-7.920-29.348L-7.233-31.098Q-7.213-31.149-7.213-31.204Q-7.213-31.344-7.326-31.421Q-7.438-31.498-7.578-31.498L-7.578-31.778L-6.358-31.778L-6.358-31.498Q-6.563-31.498-6.719-31.392Q-6.874-31.286-6.946-31.098L-7.852-28.787Q-7.886-28.692-7.999-28.692L-8.067-28.692Q-8.177-28.692-8.214-28.787L-8.997-30.790L-9.783-28.787Q-9.817-28.692-9.930-28.692L-9.998-28.692Q-10.108-28.692-10.145-28.787\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-23.701 6.43)\">\u003Cpath d=\"M-5.970-29.488Q-5.970-29.820-5.747-30.047Q-5.523-30.274-5.179-30.402Q-4.836-30.531-4.463-30.583Q-4.091-30.636-3.786-30.636L-3.786-30.889Q-3.786-31.094-3.894-31.274Q-4.002-31.453-4.183-31.556Q-4.364-31.658-4.572-31.658Q-4.979-31.658-5.215-31.566Q-5.126-31.529-5.080-31.445Q-5.034-31.361-5.034-31.259Q-5.034-31.163-5.080-31.084Q-5.126-31.006-5.207-30.961Q-5.287-30.917-5.376-30.917Q-5.526-30.917-5.627-31.014Q-5.728-31.112-5.728-31.259Q-5.728-31.881-4.572-31.881Q-4.361-31.881-4.111-31.817Q-3.862-31.754-3.660-31.635Q-3.458-31.515-3.332-31.330Q-3.205-31.146-3.205-30.903L-3.205-29.327Q-3.205-29.211-3.144-29.115Q-3.082-29.020-2.969-29.020Q-2.860-29.020-2.795-29.114Q-2.730-29.208-2.730-29.327L-2.730-29.775L-2.464-29.775L-2.464-29.327Q-2.464-29.057-2.691-28.892Q-2.918-28.726-3.198-28.726Q-3.407-28.726-3.544-28.880Q-3.680-29.033-3.704-29.249Q-3.851-28.982-4.133-28.837Q-4.415-28.692-4.740-28.692Q-5.017-28.692-5.301-28.767Q-5.584-28.842-5.777-29.021Q-5.970-29.201-5.970-29.488M-5.355-29.488Q-5.355-29.314-5.254-29.184Q-5.154-29.054-4.998-28.984Q-4.843-28.914-4.678-28.914Q-4.460-28.914-4.251-29.011Q-4.043-29.109-3.915-29.290Q-3.786-29.471-3.786-29.697L-3.786-30.425Q-4.111-30.425-4.477-30.334Q-4.843-30.243-5.099-30.031Q-5.355-29.820-5.355-29.488M-0.297-28.760L-2.033-28.760L-2.033-29.040Q-1.804-29.040-1.655-29.074Q-1.507-29.109-1.507-29.249L-1.507-31.098Q-1.507-31.368-1.614-31.429Q-1.722-31.491-2.033-31.491L-2.033-31.771L-1.004-31.846L-1.004-31.139Q-0.874-31.447-0.632-31.646Q-0.389-31.846-0.071-31.846Q0.148-31.846 0.319-31.722Q0.490-31.597 0.490-31.385Q0.490-31.248 0.390-31.149Q0.291-31.050 0.158-31.050Q0.021-31.050-0.078-31.149Q-0.177-31.248-0.177-31.385Q-0.177-31.525-0.078-31.624Q-0.368-31.624-0.568-31.428Q-0.768-31.231-0.861-30.937Q-0.953-30.643-0.953-30.363L-0.953-29.249Q-0.953-29.040-0.297-29.040L-0.297-28.760M1.074-30.271Q1.074-30.609 1.214-30.900Q1.354-31.190 1.599-31.404Q1.843-31.617 2.147-31.732Q2.451-31.846 2.776-31.846Q3.046-31.846 3.309-31.747Q3.573-31.648 3.764-31.470L3.764-32.868Q3.764-33.138 3.656-33.200Q3.549-33.261 3.238-33.261L3.238-33.542L4.314-33.617L4.314-29.433Q4.314-29.245 4.369-29.162Q4.424-29.078 4.524-29.059Q4.625-29.040 4.841-29.040L4.841-28.760L3.733-28.692L3.733-29.109Q3.316-28.692 2.691-28.692Q2.260-28.692 1.887-28.904Q1.515-29.115 1.294-29.476Q1.074-29.837 1.074-30.271M2.749-28.914Q2.957-28.914 3.144-28.986Q3.330-29.057 3.484-29.194Q3.637-29.331 3.733-29.509L3.733-31.118Q3.648-31.265 3.502-31.385Q3.357-31.505 3.188-31.564Q3.019-31.624 2.838-31.624Q2.277-31.624 2.009-31.235Q1.740-30.845 1.740-30.264Q1.740-29.693 1.975-29.303Q2.209-28.914 2.749-28.914\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-23.701 6.43)\">\u003Cpath d=\"M8.194-28.767L8.194-29.830Q8.194-29.854 8.222-29.881Q8.249-29.908 8.273-29.908L8.382-29.908Q8.447-29.908 8.461-29.850Q8.557-29.416 8.803-29.165Q9.049-28.914 9.463-28.914Q9.804-28.914 10.057-29.047Q10.310-29.180 10.310-29.488Q10.310-29.645 10.216-29.760Q10.122-29.874 9.984-29.943Q9.845-30.011 9.678-30.049L9.097-30.148Q8.741-30.216 8.468-30.437Q8.194-30.657 8.194-30.999Q8.194-31.248 8.306-31.423Q8.417-31.597 8.603-31.696Q8.789-31.795 9.005-31.838Q9.220-31.881 9.463-31.881Q9.876-31.881 10.156-31.699L10.372-31.874Q10.382-31.877 10.389-31.879Q10.396-31.881 10.406-31.881L10.457-31.881Q10.484-31.881 10.508-31.857Q10.532-31.833 10.532-31.805L10.532-30.958Q10.532-30.937 10.508-30.910Q10.484-30.883 10.457-30.883L10.344-30.883Q10.317-30.883 10.291-30.908Q10.266-30.934 10.266-30.958Q10.266-31.194 10.160-31.358Q10.054-31.522 9.871-31.604Q9.688-31.686 9.456-31.686Q9.128-31.686 8.871-31.583Q8.615-31.481 8.615-31.204Q8.615-31.009 8.798-30.900Q8.981-30.790 9.210-30.749L9.784-30.643Q10.030-30.595 10.244-30.467Q10.457-30.339 10.594-30.136Q10.731-29.932 10.731-29.683Q10.731-29.170 10.365-28.931Q9.999-28.692 9.463-28.692Q8.967-28.692 8.635-28.986L8.369-28.712Q8.348-28.692 8.321-28.692L8.273-28.692Q8.249-28.692 8.222-28.719Q8.194-28.746 8.194-28.767M13.004-27.403L11.373-27.403L11.373-27.683Q11.602-27.683 11.751-27.718Q11.900-27.752 11.900-27.892L11.900-31.238Q11.900-31.409 11.763-31.450Q11.626-31.491 11.373-31.491L11.373-31.771L12.453-31.846L12.453-31.440Q12.675-31.641 12.963-31.744Q13.250-31.846 13.557-31.846Q13.984-31.846 14.349-31.633Q14.713-31.419 14.926-31.055Q15.140-30.691 15.140-30.271Q15.140-29.826 14.901-29.462Q14.661-29.098 14.268-28.895Q13.875-28.692 13.431-28.692Q13.164-28.692 12.916-28.792Q12.669-28.893 12.481-29.074L12.481-27.892Q12.481-27.755 12.629-27.719Q12.778-27.683 13.004-27.683L13.004-27.403M12.481-31.091L12.481-29.481Q12.614-29.228 12.857-29.071Q13.099-28.914 13.376-28.914Q13.704-28.914 13.957-29.115Q14.210-29.317 14.343-29.635Q14.477-29.953 14.477-30.271Q14.477-30.500 14.412-30.729Q14.347-30.958 14.219-31.156Q14.090-31.354 13.896-31.474Q13.701-31.593 13.468-31.593Q13.174-31.593 12.906-31.464Q12.638-31.334 12.481-31.091M15.834-29.488Q15.834-29.820 16.057-30.047Q16.281-30.274 16.625-30.402Q16.968-30.531 17.341-30.583Q17.713-30.636 18.018-30.636L18.018-30.889Q18.018-31.094 17.910-31.274Q17.802-31.453 17.621-31.556Q17.440-31.658 17.232-31.658Q16.825-31.658 16.589-31.566Q16.678-31.529 16.724-31.445Q16.770-31.361 16.770-31.259Q16.770-31.163 16.724-31.084Q16.678-31.006 16.598-30.961Q16.517-30.917 16.428-30.917Q16.278-30.917 16.177-31.014Q16.076-31.112 16.076-31.259Q16.076-31.881 17.232-31.881Q17.443-31.881 17.693-31.817Q17.942-31.754 18.144-31.635Q18.346-31.515 18.472-31.330Q18.599-31.146 18.599-30.903L18.599-29.327Q18.599-29.211 18.660-29.115Q18.722-29.020 18.835-29.020Q18.944-29.020 19.009-29.114Q19.074-29.208 19.074-29.327L19.074-29.775L19.340-29.775L19.340-29.327Q19.340-29.057 19.113-28.892Q18.886-28.726 18.606-28.726Q18.397-28.726 18.260-28.880Q18.124-29.033 18.100-29.249Q17.953-28.982 17.671-28.837Q17.389-28.692 17.064-28.692Q16.787-28.692 16.504-28.767Q16.220-28.842 16.027-29.021Q15.834-29.201 15.834-29.488M16.449-29.488Q16.449-29.314 16.550-29.184Q16.651-29.054 16.806-28.984Q16.962-28.914 17.126-28.914Q17.344-28.914 17.553-29.011Q17.761-29.109 17.890-29.290Q18.018-29.471 18.018-29.697L18.018-30.425Q17.693-30.425 17.327-30.334Q16.962-30.243 16.705-30.031Q16.449-29.820 16.449-29.488M19.757-30.271Q19.757-30.599 19.892-30.900Q20.027-31.200 20.263-31.421Q20.499-31.641 20.803-31.761Q21.108-31.881 21.432-31.881Q21.938-31.881 22.287-31.778Q22.635-31.676 22.635-31.300Q22.635-31.153 22.538-31.052Q22.441-30.951 22.294-30.951Q22.140-30.951 22.041-31.050Q21.942-31.149 21.942-31.300Q21.942-31.488 22.082-31.580Q21.880-31.631 21.439-31.631Q21.084-31.631 20.855-31.435Q20.626-31.238 20.525-30.929Q20.424-30.619 20.424-30.271Q20.424-29.922 20.550-29.616Q20.677-29.310 20.932-29.126Q21.186-28.941 21.542-28.941Q21.764-28.941 21.948-29.025Q22.133-29.109 22.268-29.264Q22.403-29.420 22.461-29.628Q22.475-29.683 22.529-29.683L22.642-29.683Q22.673-29.683 22.695-29.659Q22.717-29.635 22.717-29.601L22.717-29.580Q22.632-29.293 22.444-29.095Q22.256-28.897 21.991-28.794Q21.726-28.692 21.432-28.692Q21.002-28.692 20.614-28.898Q20.226-29.105 19.992-29.468Q19.757-29.830 19.757-30.271M23.264-30.295Q23.264-30.616 23.389-30.905Q23.514-31.194 23.739-31.417Q23.965-31.641 24.261-31.761Q24.556-31.881 24.874-31.881Q25.202-31.881 25.464-31.781Q25.725-31.682 25.901-31.500Q26.077-31.317 26.171-31.059Q26.265-30.801 26.265-30.469Q26.265-30.377 26.183-30.356L23.927-30.356L23.927-30.295Q23.927-29.707 24.211-29.324Q24.495-28.941 25.062-28.941Q25.383-28.941 25.652-29.134Q25.920-29.327 26.009-29.642Q26.016-29.683 26.091-29.697L26.183-29.697Q26.265-29.673 26.265-29.601Q26.265-29.594 26.258-29.567Q26.146-29.170 25.775-28.931Q25.404-28.692 24.980-28.692Q24.543-28.692 24.143-28.900Q23.743-29.109 23.504-29.476Q23.264-29.843 23.264-30.295M23.934-30.565L25.749-30.565Q25.749-30.842 25.652-31.094Q25.554-31.347 25.356-31.503Q25.158-31.658 24.874-31.658Q24.597-31.658 24.384-31.500Q24.170-31.341 24.052-31.086Q23.934-30.831 23.934-30.565\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-23.701 6.43)\">\u003Cpath d=\"M-17.615-20.760L-19.351-20.760L-19.351-21.040Q-19.122-21.040-18.973-21.074Q-18.825-21.109-18.825-21.249L-18.825-23.098Q-18.825-23.368-18.932-23.429Q-19.040-23.491-19.351-23.491L-19.351-23.771L-18.322-23.846L-18.322-23.139Q-18.192-23.447-17.950-23.646Q-17.707-23.846-17.389-23.846Q-17.170-23.846-16.999-23.722Q-16.828-23.597-16.828-23.385Q-16.828-23.248-16.928-23.149Q-17.027-23.050-17.160-23.050Q-17.297-23.050-17.396-23.149Q-17.495-23.248-17.495-23.385Q-17.495-23.525-17.396-23.624Q-17.686-23.624-17.886-23.428Q-18.086-23.231-18.179-22.937Q-18.271-22.643-18.271-22.363L-18.271-21.249Q-18.271-21.040-17.615-21.040L-17.615-20.760M-16.285-22.295Q-16.285-22.616-16.160-22.905Q-16.035-23.194-15.810-23.417Q-15.584-23.641-15.289-23.761Q-14.993-23.881-14.675-23.881Q-14.347-23.881-14.085-23.781Q-13.824-23.682-13.648-23.500Q-13.472-23.317-13.378-23.059Q-13.284-22.801-13.284-22.469Q-13.284-22.377-13.366-22.356L-15.622-22.356L-15.622-22.295Q-15.622-21.707-15.338-21.324Q-15.054-20.941-14.487-20.941Q-14.166-20.941-13.898-21.134Q-13.629-21.327-13.540-21.642Q-13.533-21.683-13.458-21.697L-13.366-21.697Q-13.284-21.673-13.284-21.601Q-13.284-21.594-13.291-21.567Q-13.404-21.170-13.774-20.931Q-14.145-20.692-14.569-20.692Q-15.007-20.692-15.407-20.900Q-15.806-21.109-16.046-21.476Q-16.285-21.843-16.285-22.295M-15.615-22.565L-13.800-22.565Q-13.800-22.842-13.898-23.094Q-13.995-23.347-14.193-23.503Q-14.391-23.658-14.675-23.658Q-14.952-23.658-15.166-23.500Q-15.379-23.341-15.497-23.086Q-15.615-22.831-15.615-22.565M-11.308-20.787L-12.289-23.286Q-12.351-23.429-12.469-23.464Q-12.587-23.498-12.802-23.498L-12.802-23.778L-11.322-23.778L-11.322-23.498Q-11.701-23.498-11.701-23.337Q-11.701-23.327-11.688-23.286L-10.973-21.454L-10.300-23.159Q-10.331-23.231-10.331-23.259Q-10.331-23.286-10.358-23.286Q-10.420-23.433-10.538-23.465Q-10.656-23.498-10.867-23.498L-10.867-23.778L-9.470-23.778L-9.470-23.498Q-9.846-23.498-9.846-23.337Q-9.846-23.306-9.839-23.286L-9.083-21.348L-8.396-23.098Q-8.376-23.149-8.376-23.204Q-8.376-23.344-8.489-23.421Q-8.601-23.498-8.741-23.498L-8.741-23.778L-7.521-23.778L-7.521-23.498Q-7.726-23.498-7.882-23.392Q-8.037-23.286-8.109-23.098L-9.015-20.787Q-9.049-20.692-9.162-20.692L-9.230-20.692Q-9.340-20.692-9.377-20.787L-10.160-22.790L-10.946-20.787Q-10.980-20.692-11.093-20.692L-11.161-20.692Q-11.271-20.692-11.308-20.787\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-23.701 6.43)\">\u003Cpath d=\"M-7.132-21.488Q-7.132-21.820-6.909-22.047Q-6.685-22.274-6.341-22.402Q-5.998-22.531-5.625-22.583Q-5.253-22.636-4.948-22.636L-4.948-22.889Q-4.948-23.094-5.056-23.274Q-5.164-23.453-5.345-23.556Q-5.526-23.658-5.734-23.658Q-6.141-23.658-6.377-23.566Q-6.288-23.529-6.242-23.445Q-6.196-23.361-6.196-23.259Q-6.196-23.163-6.242-23.084Q-6.288-23.006-6.369-22.961Q-6.449-22.917-6.538-22.917Q-6.688-22.917-6.789-23.014Q-6.890-23.112-6.890-23.259Q-6.890-23.881-5.734-23.881Q-5.523-23.881-5.273-23.817Q-5.024-23.754-4.822-23.635Q-4.620-23.515-4.494-23.330Q-4.367-23.146-4.367-22.903L-4.367-21.327Q-4.367-21.211-4.306-21.115Q-4.244-21.020-4.131-21.020Q-4.022-21.020-3.957-21.114Q-3.892-21.208-3.892-21.327L-3.892-21.775L-3.626-21.775L-3.626-21.327Q-3.626-21.057-3.853-20.892Q-4.080-20.726-4.360-20.726Q-4.569-20.726-4.706-20.880Q-4.842-21.033-4.866-21.249Q-5.013-20.982-5.295-20.837Q-5.577-20.692-5.902-20.692Q-6.179-20.692-6.463-20.767Q-6.746-20.842-6.939-21.021Q-7.132-21.201-7.132-21.488M-6.517-21.488Q-6.517-21.314-6.416-21.184Q-6.316-21.054-6.160-20.984Q-6.005-20.914-5.840-20.914Q-5.622-20.914-5.413-21.011Q-5.205-21.109-5.077-21.290Q-4.948-21.471-4.948-21.697L-4.948-22.425Q-5.273-22.425-5.639-22.334Q-6.005-22.243-6.261-22.031Q-6.517-21.820-6.517-21.488M-1.459-20.760L-3.195-20.760L-3.195-21.040Q-2.966-21.040-2.817-21.074Q-2.669-21.109-2.669-21.249L-2.669-23.098Q-2.669-23.368-2.776-23.429Q-2.884-23.491-3.195-23.491L-3.195-23.771L-2.166-23.846L-2.166-23.139Q-2.036-23.447-1.794-23.646Q-1.551-23.846-1.233-23.846Q-1.014-23.846-0.843-23.722Q-0.672-23.597-0.672-23.385Q-0.672-23.248-0.772-23.149Q-0.871-23.050-1.004-23.050Q-1.141-23.050-1.240-23.149Q-1.339-23.248-1.339-23.385Q-1.339-23.525-1.240-23.624Q-1.530-23.624-1.730-23.428Q-1.930-23.231-2.023-22.937Q-2.115-22.643-2.115-22.363L-2.115-21.249Q-2.115-21.040-1.459-21.040L-1.459-20.760M-0.088-22.271Q-0.088-22.609 0.052-22.900Q0.192-23.190 0.437-23.404Q0.681-23.617 0.985-23.732Q1.289-23.846 1.614-23.846Q1.884-23.846 2.147-23.747Q2.411-23.648 2.602-23.470L2.602-24.868Q2.602-25.138 2.494-25.200Q2.387-25.261 2.076-25.261L2.076-25.542L3.152-25.617L3.152-21.433Q3.152-21.245 3.207-21.162Q3.262-21.078 3.362-21.059Q3.463-21.040 3.679-21.040L3.679-20.760L2.571-20.692L2.571-21.109Q2.154-20.692 1.529-20.692Q1.098-20.692 0.725-20.904Q0.353-21.115 0.132-21.476Q-0.088-21.837-0.088-22.271M1.587-20.914Q1.795-20.914 1.982-20.986Q2.168-21.057 2.322-21.194Q2.475-21.331 2.571-21.509L2.571-23.118Q2.486-23.265 2.340-23.385Q2.195-23.505 2.026-23.564Q1.857-23.624 1.676-23.624Q1.115-23.624 0.847-23.235Q0.578-22.845 0.578-22.264Q0.578-21.693 0.813-21.303Q1.047-20.914 1.587-20.914\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-23.701 6.43)\">\u003Cpath d=\"M8.713-20.760L7.079-20.760L7.079-21.040Q7.308-21.040 7.457-21.074Q7.606-21.109 7.606-21.249L7.606-23.098Q7.606-23.368 7.498-23.429Q7.390-23.491 7.079-23.491L7.079-23.771L8.139-23.846L8.139-23.197Q8.310-23.505 8.614-23.676Q8.918-23.846 9.263-23.846Q9.663-23.846 9.940-23.706Q10.217-23.566 10.302-23.218Q10.470-23.511 10.769-23.679Q11.068-23.846 11.413-23.846Q11.919-23.846 12.203-23.623Q12.487-23.399 12.487-22.903L12.487-21.249Q12.487-21.112 12.635-21.076Q12.784-21.040 13.009-21.040L13.009-20.760L11.379-20.760L11.379-21.040Q11.605-21.040 11.755-21.076Q11.905-21.112 11.905-21.249L11.905-22.889Q11.905-23.224 11.786-23.424Q11.666-23.624 11.352-23.624Q11.082-23.624 10.848-23.488Q10.613-23.351 10.475-23.117Q10.337-22.883 10.337-22.609L10.337-21.249Q10.337-21.112 10.485-21.076Q10.634-21.040 10.860-21.040L10.860-20.760L9.229-20.760L9.229-21.040Q9.458-21.040 9.607-21.074Q9.756-21.109 9.756-21.249L9.756-22.889Q9.756-23.224 9.636-23.424Q9.516-23.624 9.202-23.624Q8.932-23.624 8.698-23.488Q8.464-23.351 8.325-23.117Q8.187-22.883 8.187-22.609L8.187-21.249Q8.187-21.112 8.337-21.076Q8.488-21.040 8.713-21.040L8.713-20.760M13.556-22.243Q13.556-22.585 13.691-22.884Q13.826-23.183 14.066-23.407Q14.305-23.631 14.623-23.756Q14.941-23.881 15.272-23.881Q15.717-23.881 16.116-23.665Q16.516-23.450 16.750-23.072Q16.985-22.695 16.985-22.243Q16.985-21.902 16.843-21.618Q16.701-21.334 16.457-21.127Q16.212-20.921 15.903-20.806Q15.593-20.692 15.272-20.692Q14.842-20.692 14.440-20.893Q14.038-21.095 13.797-21.447Q13.556-21.799 13.556-22.243M15.272-20.941Q15.874-20.941 16.098-21.319Q16.321-21.697 16.321-22.329Q16.321-22.941 16.087-23.300Q15.853-23.658 15.272-23.658Q14.219-23.658 14.219-22.329Q14.219-21.697 14.445-21.319Q14.671-20.941 15.272-20.941\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-23.701 6.43)\">\u003Cpath d=\"M17.802-22.271Q17.802-22.609 17.943-22.900Q18.083-23.190 18.327-23.404Q18.571-23.617 18.876-23.732Q19.180-23.846 19.505-23.846Q19.775-23.846 20.038-23.747Q20.301-23.648 20.492-23.470L20.492-24.868Q20.492-25.138 20.385-25.200Q20.277-25.261 19.966-25.261L19.966-25.542L21.043-25.617L21.043-21.433Q21.043-21.245 21.097-21.162Q21.152-21.078 21.253-21.059Q21.354-21.040 21.569-21.040L21.569-20.760L20.462-20.692L20.462-21.109Q20.045-20.692 19.419-20.692Q18.988-20.692 18.616-20.904Q18.243-21.115 18.023-21.476Q17.802-21.837 17.802-22.271M19.477-20.914Q19.686-20.914 19.872-20.986Q20.058-21.057 20.212-21.194Q20.366-21.331 20.462-21.509L20.462-23.118Q20.376-23.265 20.231-23.385Q20.086-23.505 19.916-23.564Q19.747-23.624 19.566-23.624Q19.006-23.624 18.737-23.235Q18.469-22.845 18.469-22.264Q18.469-21.693 18.703-21.303Q18.937-20.914 19.477-20.914M22.177-22.295Q22.177-22.616 22.302-22.905Q22.427-23.194 22.653-23.417Q22.878-23.641 23.174-23.761Q23.469-23.881 23.787-23.881Q24.115-23.881 24.377-23.781Q24.638-23.682 24.814-23.500Q24.990-23.317 25.084-23.059Q25.178-22.801 25.178-22.469Q25.178-22.377 25.096-22.356L22.841-22.356L22.841-22.295Q22.841-21.707 23.124-21.324Q23.408-20.941 23.975-20.941Q24.297-20.941 24.565-21.134Q24.833-21.327 24.922-21.642Q24.929-21.683 25.004-21.697L25.096-21.697Q25.178-21.673 25.178-21.601Q25.178-21.594 25.172-21.567Q25.059-21.170 24.688-20.931Q24.317-20.692 23.893-20.692Q23.456-20.692 23.056-20.900Q22.656-21.109 22.417-21.476Q22.177-21.843 22.177-22.295M22.847-22.565L24.662-22.565Q24.662-22.842 24.565-23.094Q24.467-23.347 24.269-23.503Q24.071-23.658 23.787-23.658Q23.510-23.658 23.297-23.500Q23.083-23.341 22.965-23.086Q22.847-22.831 22.847-22.565M27.434-20.760L25.831-20.760L25.831-21.040Q26.057-21.040 26.206-21.074Q26.354-21.109 26.354-21.249L26.354-24.868Q26.354-25.138 26.247-25.200Q26.139-25.261 25.831-25.261L25.831-25.542L26.908-25.617L26.908-21.249Q26.908-21.112 27.058-21.076Q27.209-21.040 27.434-21.040\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath d=\"M133.966.58h91.049V-42.1h-91.05Z\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(166.877 5.75)\">\u003Cpath d=\"M-6.568-27.403L-8.198-27.403L-8.198-27.683Q-7.969-27.683-7.820-27.718Q-7.672-27.752-7.672-27.892L-7.672-31.238Q-7.672-31.409-7.808-31.450Q-7.945-31.491-8.198-31.491L-8.198-31.771L-7.118-31.846L-7.118-31.440Q-6.896-31.641-6.609-31.744Q-6.321-31.846-6.014-31.846Q-5.587-31.846-5.223-31.633Q-4.859-31.419-4.645-31.055Q-4.431-30.691-4.431-30.271Q-4.431-29.826-4.671-29.462Q-4.910-29.098-5.303-28.895Q-5.696-28.692-6.140-28.692Q-6.407-28.692-6.655-28.792Q-6.902-28.893-7.090-29.074L-7.090-27.892Q-7.090-27.755-6.942-27.719Q-6.793-27.683-6.568-27.683L-6.568-27.403M-7.090-31.091L-7.090-29.481Q-6.957-29.228-6.714-29.071Q-6.472-28.914-6.195-28.914Q-5.867-28.914-5.614-29.115Q-5.361-29.317-5.228-29.635Q-5.094-29.953-5.094-30.271Q-5.094-30.500-5.159-30.729Q-5.224-30.958-5.352-31.156Q-5.481-31.354-5.675-31.474Q-5.870-31.593-6.103-31.593Q-6.397-31.593-6.665-31.464Q-6.933-31.334-7.090-31.091\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(166.877 5.75)\">\u003Cpath d=\"M-3.621-30.243Q-3.621-30.585-3.486-30.884Q-3.351-31.183-3.111-31.407Q-2.872-31.631-2.554-31.756Q-2.236-31.881-1.905-31.881Q-1.460-31.881-1.061-31.665Q-0.661-31.450-0.426-31.072Q-0.192-30.695-0.192-30.243Q-0.192-29.902-0.334-29.618Q-0.476-29.334-0.720-29.127Q-0.965-28.921-1.274-28.806Q-1.583-28.692-1.905-28.692Q-2.335-28.692-2.737-28.893Q-3.139-29.095-3.380-29.447Q-3.621-29.799-3.621-30.243M-1.905-28.941Q-1.303-28.941-1.079-29.319Q-0.855-29.697-0.855-30.329Q-0.855-30.941-1.090-31.300Q-1.324-31.658-1.905-31.658Q-2.957-31.658-2.957-30.329Q-2.957-29.697-2.732-29.319Q-2.506-28.941-1.905-28.941M2.070-28.760L0.467-28.760L0.467-29.040Q0.693-29.040 0.842-29.074Q0.990-29.109 0.990-29.249L0.990-32.868Q0.990-33.138 0.883-33.200Q0.775-33.261 0.467-33.261L0.467-33.542L1.544-33.617L1.544-29.249Q1.544-29.112 1.694-29.076Q1.845-29.040 2.070-29.040L2.070-28.760M4.282-28.760L2.730-28.760L2.730-29.040Q2.956-29.040 3.104-29.074Q3.253-29.109 3.253-29.249L3.253-31.098Q3.253-31.286 3.205-31.370Q3.157-31.453 3.060-31.472Q2.962-31.491 2.751-31.491L2.751-31.771L3.807-31.846L3.807-29.249Q3.807-29.109 3.938-29.074Q4.070-29.040 4.282-29.040L4.282-28.760M3.010-33.067Q3.010-33.238 3.133-33.357Q3.256-33.477 3.427-33.477Q3.595-33.477 3.718-33.357Q3.841-33.238 3.841-33.067Q3.841-32.892 3.718-32.769Q3.595-32.646 3.427-32.646Q3.256-32.646 3.133-32.769Q3.010-32.892 3.010-33.067M4.928-30.271Q4.928-30.599 5.063-30.900Q5.198-31.200 5.434-31.421Q5.669-31.641 5.974-31.761Q6.278-31.881 6.603-31.881Q7.108-31.881 7.457-31.778Q7.806-31.676 7.806-31.300Q7.806-31.153 7.708-31.052Q7.611-30.951 7.464-30.951Q7.310-30.951 7.211-31.050Q7.112-31.149 7.112-31.300Q7.112-31.488 7.252-31.580Q7.050-31.631 6.609-31.631Q6.254-31.631 6.025-31.435Q5.796-31.238 5.695-30.929Q5.594-30.619 5.594-30.271Q5.594-29.922 5.721-29.616Q5.847-29.310 6.102-29.126Q6.356-28.941 6.712-28.941Q6.934-28.941 7.119-29.025Q7.303-29.109 7.438-29.264Q7.573-29.420 7.631-29.628Q7.645-29.683 7.700-29.683L7.813-29.683Q7.843-29.683 7.866-29.659Q7.888-29.635 7.888-29.601L7.888-29.580Q7.802-29.293 7.614-29.095Q7.426-28.897 7.161-28.794Q6.897-28.692 6.603-28.692Q6.172-28.692 5.784-28.898Q5.396-29.105 5.162-29.468Q4.928-29.830 4.928-30.271M8.811-27.625Q8.940-27.557 9.077-27.557Q9.248-27.557 9.398-27.646Q9.549-27.735 9.660-27.880Q9.771-28.025 9.850-28.193L10.113-28.760L8.944-31.286Q8.869-31.433 8.739-31.465Q8.609-31.498 8.377-31.498L8.377-31.778L9.898-31.778L9.898-31.498Q9.549-31.498 9.549-31.351Q9.552-31.330 9.554-31.313Q9.556-31.296 9.556-31.286L10.414-29.427L11.186-31.098Q11.220-31.166 11.220-31.245Q11.220-31.358 11.137-31.428Q11.053-31.498 10.940-31.498L10.940-31.778L12.136-31.778L12.136-31.498Q11.918-31.498 11.745-31.394Q11.572-31.289 11.480-31.098L10.144-28.193Q9.973-27.823 9.703-27.577Q9.433-27.331 9.077-27.331Q8.807-27.331 8.588-27.497Q8.370-27.663 8.370-27.926Q8.370-28.063 8.462-28.152Q8.554-28.240 8.694-28.240Q8.831-28.240 8.920-28.152Q9.009-28.063 9.009-27.926Q9.009-27.823 8.956-27.745Q8.903-27.666 8.811-27.625\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(166.877 5.75)\">\u003Cpath d=\"M15.385-28.767L15.385-29.830Q15.385-29.854 15.413-29.881Q15.440-29.908 15.464-29.908L15.573-29.908Q15.638-29.908 15.652-29.850Q15.748-29.416 15.994-29.165Q16.240-28.914 16.654-28.914Q16.995-28.914 17.248-29.047Q17.501-29.180 17.501-29.488Q17.501-29.645 17.407-29.760Q17.313-29.874 17.175-29.943Q17.036-30.011 16.869-30.049L16.288-30.148Q15.932-30.216 15.659-30.437Q15.385-30.657 15.385-30.999Q15.385-31.248 15.497-31.423Q15.608-31.597 15.794-31.696Q15.980-31.795 16.196-31.838Q16.411-31.881 16.654-31.881Q17.067-31.881 17.347-31.699L17.563-31.874Q17.573-31.877 17.580-31.879Q17.587-31.881 17.597-31.881L17.648-31.881Q17.675-31.881 17.699-31.857Q17.723-31.833 17.723-31.805L17.723-30.958Q17.723-30.937 17.699-30.910Q17.675-30.883 17.648-30.883L17.535-30.883Q17.508-30.883 17.482-30.908Q17.457-30.934 17.457-30.958Q17.457-31.194 17.351-31.358Q17.245-31.522 17.062-31.604Q16.879-31.686 16.647-31.686Q16.319-31.686 16.062-31.583Q15.806-31.481 15.806-31.204Q15.806-31.009 15.989-30.900Q16.172-30.790 16.401-30.749L16.975-30.643Q17.221-30.595 17.435-30.467Q17.648-30.339 17.785-30.136Q17.922-29.932 17.922-29.683Q17.922-29.170 17.556-28.931Q17.190-28.692 16.654-28.692Q16.158-28.692 15.826-28.986L15.560-28.712Q15.539-28.692 15.512-28.692L15.464-28.692Q15.440-28.692 15.413-28.719Q15.385-28.746 15.385-28.767M20.195-27.403L18.564-27.403L18.564-27.683Q18.793-27.683 18.942-27.718Q19.091-27.752 19.091-27.892L19.091-31.238Q19.091-31.409 18.954-31.450Q18.817-31.491 18.564-31.491L18.564-31.771L19.644-31.846L19.644-31.440Q19.866-31.641 20.154-31.744Q20.441-31.846 20.748-31.846Q21.175-31.846 21.540-31.633Q21.904-31.419 22.117-31.055Q22.331-30.691 22.331-30.271Q22.331-29.826 22.092-29.462Q21.852-29.098 21.459-28.895Q21.066-28.692 20.622-28.692Q20.355-28.692 20.107-28.792Q19.860-28.893 19.672-29.074L19.672-27.892Q19.672-27.755 19.820-27.719Q19.969-27.683 20.195-27.683L20.195-27.403M19.672-31.091L19.672-29.481Q19.805-29.228 20.048-29.071Q20.290-28.914 20.567-28.914Q20.895-28.914 21.148-29.115Q21.401-29.317 21.534-29.635Q21.668-29.953 21.668-30.271Q21.668-30.500 21.603-30.729Q21.538-30.958 21.410-31.156Q21.281-31.354 21.087-31.474Q20.892-31.593 20.659-31.593Q20.365-31.593 20.097-31.464Q19.829-31.334 19.672-31.091M23.025-29.488Q23.025-29.820 23.248-30.047Q23.472-30.274 23.816-30.402Q24.159-30.531 24.532-30.583Q24.904-30.636 25.209-30.636L25.209-30.889Q25.209-31.094 25.101-31.274Q24.993-31.453 24.812-31.556Q24.631-31.658 24.423-31.658Q24.016-31.658 23.780-31.566Q23.869-31.529 23.915-31.445Q23.961-31.361 23.961-31.259Q23.961-31.163 23.915-31.084Q23.869-31.006 23.789-30.961Q23.708-30.917 23.619-30.917Q23.469-30.917 23.368-31.014Q23.267-31.112 23.267-31.259Q23.267-31.881 24.423-31.881Q24.634-31.881 24.884-31.817Q25.133-31.754 25.335-31.635Q25.537-31.515 25.663-31.330Q25.790-31.146 25.790-30.903L25.790-29.327Q25.790-29.211 25.851-29.115Q25.913-29.020 26.026-29.020Q26.135-29.020 26.200-29.114Q26.265-29.208 26.265-29.327L26.265-29.775L26.531-29.775L26.531-29.327Q26.531-29.057 26.304-28.892Q26.077-28.726 25.797-28.726Q25.588-28.726 25.451-28.880Q25.315-29.033 25.291-29.249Q25.144-28.982 24.862-28.837Q24.580-28.692 24.255-28.692Q23.978-28.692 23.695-28.767Q23.411-28.842 23.218-29.021Q23.025-29.201 23.025-29.488M23.640-29.488Q23.640-29.314 23.741-29.184Q23.842-29.054 23.997-28.984Q24.153-28.914 24.317-28.914Q24.535-28.914 24.744-29.011Q24.952-29.109 25.081-29.290Q25.209-29.471 25.209-29.697L25.209-30.425Q24.884-30.425 24.518-30.334Q24.153-30.243 23.896-30.031Q23.640-29.820 23.640-29.488M26.948-30.271Q26.948-30.599 27.083-30.900Q27.218-31.200 27.454-31.421Q27.690-31.641 27.994-31.761Q28.299-31.881 28.623-31.881Q29.129-31.881 29.478-31.778Q29.826-31.676 29.826-31.300Q29.826-31.153 29.729-31.052Q29.632-30.951 29.485-30.951Q29.331-30.951 29.232-31.050Q29.133-31.149 29.133-31.300Q29.133-31.488 29.273-31.580Q29.071-31.631 28.630-31.631Q28.275-31.631 28.046-31.435Q27.817-31.238 27.716-30.929Q27.615-30.619 27.615-30.271Q27.615-29.922 27.741-29.616Q27.868-29.310 28.123-29.126Q28.377-28.941 28.733-28.941Q28.955-28.941 29.139-29.025Q29.324-29.109 29.459-29.264Q29.594-29.420 29.652-29.628Q29.666-29.683 29.720-29.683L29.833-29.683Q29.864-29.683 29.886-29.659Q29.908-29.635 29.908-29.601L29.908-29.580Q29.823-29.293 29.635-29.095Q29.447-28.897 29.182-28.794Q28.917-28.692 28.623-28.692Q28.193-28.692 27.805-28.898Q27.417-29.105 27.183-29.468Q26.948-29.830 26.948-30.271M30.455-30.295Q30.455-30.616 30.580-30.905Q30.705-31.194 30.930-31.417Q31.156-31.641 31.452-31.761Q31.747-31.881 32.065-31.881Q32.393-31.881 32.655-31.781Q32.916-31.682 33.092-31.500Q33.268-31.317 33.362-31.059Q33.456-30.801 33.456-30.469Q33.456-30.377 33.374-30.356L31.118-30.356L31.118-30.295Q31.118-29.707 31.402-29.324Q31.686-28.941 32.253-28.941Q32.574-28.941 32.843-29.134Q33.111-29.327 33.200-29.642Q33.207-29.683 33.282-29.697L33.374-29.697Q33.456-29.673 33.456-29.601Q33.456-29.594 33.449-29.567Q33.337-29.170 32.966-28.931Q32.595-28.692 32.171-28.692Q31.734-28.692 31.334-28.900Q30.934-29.109 30.695-29.476Q30.455-29.843 30.455-30.295M31.125-30.565L32.940-30.565Q32.940-30.842 32.843-31.094Q32.745-31.347 32.547-31.503Q32.349-31.658 32.065-31.658Q31.788-31.658 31.575-31.500Q31.361-31.341 31.243-31.086Q31.125-30.831 31.125-30.565\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(166.877 5.75)\">\u003Cpath d=\"M-17.721-19.403L-19.351-19.403L-19.351-19.683Q-19.122-19.683-18.973-19.718Q-18.825-19.752-18.825-19.892L-18.825-23.238Q-18.825-23.409-18.961-23.450Q-19.098-23.491-19.351-23.491L-19.351-23.771L-18.271-23.846L-18.271-23.440Q-18.049-23.641-17.762-23.744Q-17.474-23.846-17.167-23.846Q-16.740-23.846-16.376-23.633Q-16.012-23.419-15.798-23.055Q-15.584-22.691-15.584-22.271Q-15.584-21.826-15.824-21.462Q-16.063-21.098-16.456-20.895Q-16.849-20.692-17.293-20.692Q-17.560-20.692-17.808-20.792Q-18.055-20.893-18.243-21.074L-18.243-19.892Q-18.243-19.755-18.095-19.719Q-17.946-19.683-17.721-19.683L-17.721-19.403M-18.243-23.091L-18.243-21.481Q-18.110-21.228-17.867-21.071Q-17.625-20.914-17.348-20.914Q-17.020-20.914-16.767-21.115Q-16.514-21.317-16.381-21.635Q-16.247-21.953-16.247-22.271Q-16.247-22.500-16.312-22.729Q-16.377-22.958-16.505-23.156Q-16.634-23.354-16.828-23.474Q-17.023-23.593-17.256-23.593Q-17.550-23.593-17.818-23.464Q-18.086-23.334-18.243-23.091\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(166.877 5.75)\">\u003Cpath d=\"M-14.774-22.243Q-14.774-22.585-14.639-22.884Q-14.504-23.183-14.264-23.407Q-14.025-23.631-13.707-23.756Q-13.389-23.881-13.058-23.881Q-12.613-23.881-12.214-23.665Q-11.814-23.450-11.579-23.072Q-11.345-22.695-11.345-22.243Q-11.345-21.902-11.487-21.618Q-11.629-21.334-11.873-21.127Q-12.118-20.921-12.427-20.806Q-12.736-20.692-13.058-20.692Q-13.488-20.692-13.890-20.893Q-14.292-21.095-14.533-21.447Q-14.774-21.799-14.774-22.243M-13.058-20.941Q-12.456-20.941-12.232-21.319Q-12.008-21.697-12.008-22.329Q-12.008-22.941-12.243-23.300Q-12.477-23.658-13.058-23.658Q-14.110-23.658-14.110-22.329Q-14.110-21.697-13.885-21.319Q-13.659-20.941-13.058-20.941M-9.083-20.760L-10.686-20.760L-10.686-21.040Q-10.460-21.040-10.311-21.074Q-10.163-21.109-10.163-21.249L-10.163-24.868Q-10.163-25.138-10.270-25.200Q-10.378-25.261-10.686-25.261L-10.686-25.542L-9.609-25.617L-9.609-21.249Q-9.609-21.112-9.459-21.076Q-9.308-21.040-9.083-21.040L-9.083-20.760M-6.871-20.760L-8.423-20.760L-8.423-21.040Q-8.197-21.040-8.049-21.074Q-7.900-21.109-7.900-21.249L-7.900-23.098Q-7.900-23.286-7.948-23.370Q-7.996-23.453-8.093-23.472Q-8.191-23.491-8.402-23.491L-8.402-23.771L-7.346-23.846L-7.346-21.249Q-7.346-21.109-7.215-21.074Q-7.083-21.040-6.871-21.040L-6.871-20.760M-8.143-25.067Q-8.143-25.238-8.020-25.357Q-7.897-25.477-7.726-25.477Q-7.558-25.477-7.435-25.357Q-7.312-25.238-7.312-25.067Q-7.312-24.892-7.435-24.769Q-7.558-24.646-7.726-24.646Q-7.897-24.646-8.020-24.769Q-8.143-24.892-8.143-25.067M-6.225-22.271Q-6.225-22.599-6.090-22.900Q-5.955-23.200-5.719-23.421Q-5.484-23.641-5.179-23.761Q-4.875-23.881-4.550-23.881Q-4.045-23.881-3.696-23.778Q-3.347-23.676-3.347-23.300Q-3.347-23.153-3.445-23.052Q-3.542-22.951-3.689-22.951Q-3.843-22.951-3.942-23.050Q-4.041-23.149-4.041-23.300Q-4.041-23.488-3.901-23.580Q-4.103-23.631-4.544-23.631Q-4.899-23.631-5.128-23.435Q-5.357-23.238-5.458-22.929Q-5.559-22.619-5.559-22.271Q-5.559-21.922-5.432-21.616Q-5.306-21.310-5.051-21.126Q-4.797-20.941-4.441-20.941Q-4.219-20.941-4.034-21.025Q-3.850-21.109-3.715-21.264Q-3.580-21.420-3.522-21.628Q-3.508-21.683-3.453-21.683L-3.340-21.683Q-3.310-21.683-3.287-21.659Q-3.265-21.635-3.265-21.601L-3.265-21.580Q-3.351-21.293-3.539-21.095Q-3.727-20.897-3.992-20.794Q-4.256-20.692-4.550-20.692Q-4.981-20.692-5.369-20.898Q-5.757-21.105-5.991-21.468Q-6.225-21.830-6.225-22.271M-2.342-19.625Q-2.213-19.557-2.076-19.557Q-1.905-19.557-1.755-19.646Q-1.604-19.735-1.493-19.880Q-1.382-20.025-1.303-20.193L-1.040-20.760L-2.209-23.286Q-2.284-23.433-2.414-23.465Q-2.544-23.498-2.776-23.498L-2.776-23.778L-1.255-23.778L-1.255-23.498Q-1.604-23.498-1.604-23.351Q-1.601-23.330-1.599-23.313Q-1.597-23.296-1.597-23.286L-0.739-21.427L0.033-23.098Q0.067-23.166 0.067-23.245Q0.067-23.358-0.016-23.428Q-0.100-23.498-0.213-23.498L-0.213-23.778L0.983-23.778L0.983-23.498Q0.765-23.498 0.592-23.394Q0.419-23.289 0.327-23.098L-1.009-20.193Q-1.180-19.823-1.450-19.577Q-1.720-19.331-2.076-19.331Q-2.346-19.331-2.565-19.497Q-2.783-19.663-2.783-19.926Q-2.783-20.063-2.691-20.152Q-2.599-20.240-2.459-20.240Q-2.322-20.240-2.233-20.152Q-2.144-20.063-2.144-19.926Q-2.144-19.823-2.197-19.745Q-2.250-19.666-2.342-19.625\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(166.877 5.75)\">\u003Cpath d=\"M5.822-20.787L4.694-23.286Q4.622-23.433 4.492-23.465Q4.362-23.498 4.133-23.498L4.133-23.778L5.647-23.778L5.647-23.498Q5.295-23.498 5.295-23.351Q5.295-23.306 5.306-23.286L6.170-21.368L6.950-23.098Q6.984-23.166 6.984-23.245Q6.984-23.358 6.900-23.428Q6.816-23.498 6.697-23.498L6.697-23.778L7.893-23.778L7.893-23.498Q7.674-23.498 7.503-23.395Q7.333-23.293 7.244-23.098L6.208-20.787Q6.160-20.692 6.054-20.692L5.976-20.692Q5.870-20.692 5.822-20.787M8.433-20.767L8.433-21.830Q8.433-21.854 8.460-21.881Q8.488-21.908 8.512-21.908L8.621-21.908Q8.686-21.908 8.700-21.850Q8.795-21.416 9.042-21.165Q9.288-20.914 9.701-20.914Q10.043-20.914 10.296-21.047Q10.549-21.180 10.549-21.488Q10.549-21.645 10.455-21.760Q10.361-21.874 10.222-21.943Q10.084-22.011 9.917-22.049L9.335-22.148Q8.980-22.216 8.707-22.437Q8.433-22.657 8.433-22.999Q8.433-23.248 8.544-23.423Q8.655-23.597 8.842-23.696Q9.028-23.795 9.243-23.838Q9.459-23.881 9.701-23.881Q10.115-23.881 10.395-23.699L10.610-23.874Q10.621-23.877 10.627-23.879Q10.634-23.881 10.645-23.881L10.696-23.881Q10.723-23.881 10.747-23.857Q10.771-23.833 10.771-23.805L10.771-22.958Q10.771-22.937 10.747-22.910Q10.723-22.883 10.696-22.883L10.583-22.883Q10.556-22.883 10.530-22.908Q10.504-22.934 10.504-22.958Q10.504-23.194 10.398-23.358Q10.293-23.522 10.110-23.604Q9.927-23.686 9.694-23.686Q9.366-23.686 9.110-23.583Q8.854-23.481 8.854-23.204Q8.854-23.009 9.036-22.900Q9.219-22.790 9.448-22.749L10.022-22.643Q10.269-22.595 10.482-22.467Q10.696-22.339 10.833-22.136Q10.969-21.932 10.969-21.683Q10.969-21.170 10.604-20.931Q10.238-20.692 9.701-20.692Q9.206-20.692 8.874-20.986L8.607-20.712Q8.587-20.692 8.560-20.692L8.512-20.692Q8.488-20.692 8.460-20.719Q8.433-20.746 8.433-20.767\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(166.877 5.75)\">\u003Cpath d=\"M16.046-20.760L14.310-20.760L14.310-21.040Q14.539-21.040 14.688-21.074Q14.836-21.109 14.836-21.249L14.836-23.098Q14.836-23.368 14.729-23.429Q14.621-23.491 14.310-23.491L14.310-23.771L15.339-23.846L15.339-23.139Q15.469-23.447 15.711-23.646Q15.954-23.846 16.272-23.846Q16.491-23.846 16.662-23.722Q16.833-23.597 16.833-23.385Q16.833-23.248 16.733-23.149Q16.634-23.050 16.501-23.050Q16.364-23.050 16.265-23.149Q16.166-23.248 16.166-23.385Q16.166-23.525 16.265-23.624Q15.975-23.624 15.775-23.428Q15.575-23.231 15.482-22.937Q15.390-22.643 15.390-22.363L15.390-21.249Q15.390-21.040 16.046-21.040L16.046-20.760M17.376-22.295Q17.376-22.616 17.501-22.905Q17.626-23.194 17.851-23.417Q18.077-23.641 18.372-23.761Q18.668-23.881 18.986-23.881Q19.314-23.881 19.576-23.781Q19.837-23.682 20.013-23.500Q20.189-23.317 20.283-23.059Q20.377-22.801 20.377-22.469Q20.377-22.377 20.295-22.356L18.039-22.356L18.039-22.295Q18.039-21.707 18.323-21.324Q18.607-20.941 19.174-20.941Q19.495-20.941 19.763-21.134Q20.032-21.327 20.121-21.642Q20.128-21.683 20.203-21.697L20.295-21.697Q20.377-21.673 20.377-21.601Q20.377-21.594 20.370-21.567Q20.257-21.170 19.887-20.931Q19.516-20.692 19.092-20.692Q18.654-20.692 18.254-20.900Q17.855-21.109 17.615-21.476Q17.376-21.843 17.376-22.295M18.046-22.565L19.861-22.565Q19.861-22.842 19.763-23.094Q19.666-23.347 19.468-23.503Q19.270-23.658 18.986-23.658Q18.709-23.658 18.495-23.500Q18.282-23.341 18.164-23.086Q18.046-22.831 18.046-22.565M22.763-20.760L21.030-20.760L21.030-21.040Q21.255-21.040 21.404-21.074Q21.553-21.109 21.553-21.249L21.553-23.498L20.965-23.498L20.965-23.778L21.553-23.778L21.553-24.595Q21.553-24.913 21.731-25.161Q21.908-25.408 22.199-25.549Q22.489-25.689 22.800-25.689Q23.057-25.689 23.260-25.547Q23.463-25.405 23.463-25.162Q23.463-25.026 23.364-24.927Q23.265-24.827 23.128-24.827Q22.992-24.827 22.893-24.927Q22.794-25.026 22.794-25.162Q22.794-25.343 22.934-25.436Q22.855-25.463 22.756-25.463Q22.547-25.463 22.394-25.330Q22.240-25.197 22.159-24.993Q22.079-24.790 22.079-24.581L22.079-23.778L22.968-23.778L22.968-23.498L22.107-23.498L22.107-21.249Q22.107-21.040 22.763-21.040L22.763-20.760M23.402-22.295Q23.402-22.616 23.527-22.905Q23.651-23.194 23.877-23.417Q24.103-23.641 24.398-23.761Q24.694-23.881 25.012-23.881Q25.340-23.881 25.601-23.781Q25.863-23.682 26.039-23.500Q26.215-23.317 26.309-23.059Q26.403-22.801 26.403-22.469Q26.403-22.377 26.321-22.356L24.065-22.356L24.065-22.295Q24.065-21.707 24.349-21.324Q24.632-20.941 25.200-20.941Q25.521-20.941 25.789-21.134Q26.058-21.327 26.147-21.642Q26.153-21.683 26.229-21.697L26.321-21.697Q26.403-21.673 26.403-21.601Q26.403-21.594 26.396-21.567Q26.283-21.170 25.912-20.931Q25.542-20.692 25.118-20.692Q24.680-20.692 24.280-20.900Q23.880-21.109 23.641-21.476Q23.402-21.843 23.402-22.295M24.072-22.565L25.887-22.565Q25.887-22.842 25.789-23.094Q25.692-23.347 25.494-23.503Q25.295-23.658 25.012-23.658Q24.735-23.658 24.521-23.500Q24.308-23.341 24.190-23.086Q24.072-22.831 24.072-22.565M28.741-20.760L27.004-20.760L27.004-21.040Q27.233-21.040 27.382-21.074Q27.531-21.109 27.531-21.249L27.531-23.098Q27.531-23.368 27.423-23.429Q27.315-23.491 27.004-23.491L27.004-23.771L28.033-23.846L28.033-23.139Q28.163-23.447 28.406-23.646Q28.649-23.846 28.966-23.846Q29.185-23.846 29.356-23.722Q29.527-23.597 29.527-23.385Q29.527-23.248 29.428-23.149Q29.329-23.050 29.195-23.050Q29.059-23.050 28.960-23.149Q28.860-23.248 28.860-23.385Q28.860-23.525 28.960-23.624Q28.669-23.624 28.469-23.428Q28.269-23.231 28.177-22.937Q28.085-22.643 28.085-22.363L28.085-21.249Q28.085-21.040 28.741-21.040L28.741-20.760M30.070-22.295Q30.070-22.616 30.195-22.905Q30.320-23.194 30.545-23.417Q30.771-23.641 31.067-23.761Q31.362-23.881 31.680-23.881Q32.008-23.881 32.270-23.781Q32.531-23.682 32.707-23.500Q32.883-23.317 32.977-23.059Q33.071-22.801 33.071-22.469Q33.071-22.377 32.989-22.356L30.733-22.356L30.733-22.295Q30.733-21.707 31.017-21.324Q31.301-20.941 31.868-20.941Q32.190-20.941 32.458-21.134Q32.726-21.327 32.815-21.642Q32.822-21.683 32.897-21.697L32.989-21.697Q33.071-21.673 33.071-21.601Q33.071-21.594 33.065-21.567Q32.952-21.170 32.581-20.931Q32.210-20.692 31.786-20.692Q31.349-20.692 30.949-20.900Q30.549-21.109 30.310-21.476Q30.070-21.843 30.070-22.295M30.740-22.565L32.555-22.565Q32.555-22.842 32.458-23.094Q32.360-23.347 32.162-23.503Q31.964-23.658 31.680-23.658Q31.403-23.658 31.190-23.500Q30.976-23.341 30.858-23.086Q30.740-22.831 30.740-22.565M35.341-20.760L33.707-20.760L33.707-21.040Q33.936-21.040 34.085-21.074Q34.233-21.109 34.233-21.249L34.233-23.098Q34.233-23.368 34.126-23.429Q34.018-23.491 33.707-23.491L33.707-23.771L34.767-23.846L34.767-23.197Q34.938-23.505 35.242-23.676Q35.546-23.846 35.891-23.846Q36.397-23.846 36.681-23.623Q36.964-23.399 36.964-22.903L36.964-21.249Q36.964-21.112 37.113-21.076Q37.262-21.040 37.487-21.040L37.487-20.760L35.857-20.760L35.857-21.040Q36.086-21.040 36.235-21.074Q36.383-21.109 36.383-21.249L36.383-22.889Q36.383-23.224 36.264-23.424Q36.144-23.624 35.830-23.624Q35.560-23.624 35.326-23.488Q35.091-23.351 34.953-23.117Q34.815-22.883 34.815-22.609L34.815-21.249Q34.815-21.112 34.965-21.076Q35.115-21.040 35.341-21.040L35.341-20.760M38.075-22.271Q38.075-22.599 38.210-22.900Q38.345-23.200 38.581-23.421Q38.817-23.641 39.121-23.761Q39.425-23.881 39.750-23.881Q40.256-23.881 40.605-23.778Q40.953-23.676 40.953-23.300Q40.953-23.153 40.856-23.052Q40.758-22.951 40.611-22.951Q40.458-22.951 40.358-23.050Q40.259-23.149 40.259-23.300Q40.259-23.488 40.399-23.580Q40.198-23.631 39.757-23.631Q39.401-23.631 39.172-23.435Q38.943-23.238 38.843-22.929Q38.742-22.619 38.742-22.271Q38.742-21.922 38.868-21.616Q38.995-21.310 39.249-21.126Q39.504-20.941 39.859-20.941Q40.082-20.941 40.266-21.025Q40.451-21.109 40.586-21.264Q40.721-21.420 40.779-21.628Q40.793-21.683 40.847-21.683L40.960-21.683Q40.991-21.683 41.013-21.659Q41.035-21.635 41.035-21.601L41.035-21.580Q40.950-21.293 40.762-21.095Q40.574-20.897 40.309-20.794Q40.044-20.692 39.750-20.692Q39.319-20.692 38.931-20.898Q38.544-21.105 38.309-21.468Q38.075-21.830 38.075-22.271M41.582-22.295Q41.582-22.616 41.707-22.905Q41.832-23.194 42.057-23.417Q42.283-23.641 42.578-23.761Q42.874-23.881 43.192-23.881Q43.520-23.881 43.782-23.781Q44.043-23.682 44.219-23.500Q44.395-23.317 44.489-23.059Q44.583-22.801 44.583-22.469Q44.583-22.377 44.501-22.356L42.245-22.356L42.245-22.295Q42.245-21.707 42.529-21.324Q42.813-20.941 43.380-20.941Q43.701-20.941 43.970-21.134Q44.238-21.327 44.327-21.642Q44.334-21.683 44.409-21.697L44.501-21.697Q44.583-21.673 44.583-21.601Q44.583-21.594 44.576-21.567Q44.463-21.170 44.093-20.931Q43.722-20.692 43.298-20.692Q42.860-20.692 42.461-20.900Q42.061-21.109 41.821-21.476Q41.582-21.843 41.582-22.295M42.252-22.565L44.067-22.565Q44.067-22.842 43.970-23.094Q43.872-23.347 43.674-23.503Q43.476-23.658 43.192-23.658Q42.915-23.658 42.701-23.500Q42.488-23.341 42.370-23.086Q42.252-22.831 42.252-22.565\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M26.245-42.5c53.66-20.939 53.66-20.939 104.899-.944\"\u002F>\u003Cpath stroke=\"none\" d=\"m133.566-42.5-3.12-3.45.698 2.506-2.21 1.37\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(26.16 -42.927)\">\u003Cpath d=\"M-19.406-22.243Q-19.406-22.585-19.271-22.884Q-19.136-23.183-18.896-23.407Q-18.657-23.631-18.339-23.756Q-18.021-23.881-17.690-23.881Q-17.245-23.881-16.846-23.665Q-16.446-23.450-16.211-23.072Q-15.977-22.695-15.977-22.243Q-15.977-21.902-16.119-21.618Q-16.261-21.334-16.505-21.127Q-16.750-20.921-17.059-20.806Q-17.368-20.692-17.690-20.692Q-18.120-20.692-18.522-20.893Q-18.924-21.095-19.165-21.447Q-19.406-21.799-19.406-22.243M-17.690-20.941Q-17.088-20.941-16.864-21.319Q-16.640-21.697-16.640-22.329Q-16.640-22.941-16.875-23.300Q-17.109-23.658-17.690-23.658Q-18.742-23.658-18.742-22.329Q-18.742-21.697-18.517-21.319Q-18.291-20.941-17.690-20.941M-13.739-19.403L-15.369-19.403L-15.369-19.683Q-15.140-19.683-14.991-19.718Q-14.843-19.752-14.843-19.892L-14.843-23.238Q-14.843-23.409-14.979-23.450Q-15.116-23.491-15.369-23.491L-15.369-23.771L-14.289-23.846L-14.289-23.440Q-14.067-23.641-13.780-23.744Q-13.492-23.846-13.185-23.846Q-12.758-23.846-12.394-23.633Q-12.030-23.419-11.816-23.055Q-11.602-22.691-11.602-22.271Q-11.602-21.826-11.842-21.462Q-12.081-21.098-12.474-20.895Q-12.867-20.692-13.311-20.692Q-13.578-20.692-13.826-20.792Q-14.074-20.893-14.262-21.074L-14.262-19.892Q-14.262-19.755-14.113-19.719Q-13.964-19.683-13.739-19.683L-13.739-19.403M-14.262-23.091L-14.262-21.481Q-14.128-21.228-13.886-21.071Q-13.643-20.914-13.366-20.914Q-13.038-20.914-12.785-21.115Q-12.532-21.317-12.399-21.635Q-12.265-21.953-12.265-22.271Q-12.265-22.500-12.330-22.729Q-12.395-22.958-12.523-23.156Q-12.652-23.354-12.846-23.474Q-13.041-23.593-13.274-23.593Q-13.568-23.593-13.836-23.464Q-14.104-23.334-14.262-23.091M-10.440-21.601L-10.440-23.498L-11.079-23.498L-11.079-23.720Q-10.762-23.720-10.544-23.930Q-10.327-24.140-10.227-24.450Q-10.126-24.759-10.126-25.067L-9.859-25.067L-9.859-23.778L-8.783-23.778L-8.783-23.498L-9.859-23.498L-9.859-21.614Q-9.859-21.338-9.755-21.139Q-9.651-20.941-9.391-20.941Q-9.234-20.941-9.128-21.045Q-9.022-21.150-8.972-21.303Q-8.923-21.457-8.923-21.614L-8.923-22.028L-8.656-22.028L-8.656-21.601Q-8.656-21.375-8.755-21.165Q-8.854-20.955-9.039-20.823Q-9.223-20.692-9.452-20.692Q-9.890-20.692-10.165-20.929Q-10.440-21.167-10.440-21.601M-6.229-20.760L-7.781-20.760L-7.781-21.040Q-7.555-21.040-7.407-21.074Q-7.258-21.109-7.258-21.249L-7.258-23.098Q-7.258-23.286-7.306-23.370Q-7.354-23.453-7.451-23.472Q-7.549-23.491-7.761-23.491L-7.761-23.771L-6.704-23.846L-6.704-21.249Q-6.704-21.109-6.573-21.074Q-6.441-21.040-6.229-21.040L-6.229-20.760M-7.501-25.067Q-7.501-25.238-7.378-25.357Q-7.255-25.477-7.084-25.477Q-6.916-25.477-6.793-25.357Q-6.670-25.238-6.670-25.067Q-6.670-24.892-6.793-24.769Q-6.916-24.646-7.084-24.646Q-7.255-24.646-7.378-24.769Q-7.501-24.892-7.501-25.067M-3.902-20.760L-5.535-20.760L-5.535-21.040Q-5.306-21.040-5.158-21.074Q-5.009-21.109-5.009-21.249L-5.009-23.098Q-5.009-23.368-5.117-23.429Q-5.224-23.491-5.535-23.491L-5.535-23.771L-4.476-23.846L-4.476-23.197Q-4.305-23.505-4.001-23.676Q-3.697-23.846-3.351-23.846Q-2.951-23.846-2.675-23.706Q-2.398-23.566-2.312-23.218Q-2.145-23.511-1.846-23.679Q-1.547-23.846-1.201-23.846Q-0.696-23.846-0.412-23.623Q-0.128-23.399-0.128-22.903L-0.128-21.249Q-0.128-21.112 0.020-21.076Q0.169-21.040 0.395-21.040L0.395-20.760L-1.236-20.760L-1.236-21.040Q-1.010-21.040-0.860-21.076Q-0.709-21.112-0.709-21.249L-0.709-22.889Q-0.709-23.224-0.829-23.424Q-0.949-23.624-1.263-23.624Q-1.533-23.624-1.767-23.488Q-2.001-23.351-2.140-23.117Q-2.278-22.883-2.278-22.609L-2.278-21.249Q-2.278-21.112-2.129-21.076Q-1.981-21.040-1.755-21.040L-1.755-20.760L-3.386-20.760L-3.386-21.040Q-3.157-21.040-3.008-21.074Q-2.859-21.109-2.859-21.249L-2.859-22.889Q-2.859-23.224-2.979-23.424Q-3.098-23.624-3.413-23.624Q-3.683-23.624-3.917-23.488Q-4.151-23.351-4.290-23.117Q-4.428-22.883-4.428-22.609L-4.428-21.249Q-4.428-21.112-4.278-21.076Q-4.127-21.040-3.902-21.040\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(26.16 -42.927)\">\u003Cpath d=\"M1.356-21.594L1.356-23.098Q1.356-23.368 1.248-23.429Q1.140-23.491 0.829-23.491L0.829-23.771L1.937-23.846L1.937-21.614L1.937-21.594Q1.937-21.314 1.988-21.170Q2.039-21.027 2.181-20.970Q2.323-20.914 2.610-20.914Q2.863-20.914 3.068-21.054Q3.273-21.194 3.389-21.420Q3.506-21.645 3.506-21.895L3.506-23.098Q3.506-23.368 3.398-23.429Q3.290-23.491 2.979-23.491L2.979-23.771L4.087-23.846L4.087-21.433Q4.087-21.242 4.140-21.160Q4.193-21.078 4.293-21.059Q4.394-21.040 4.610-21.040L4.610-20.760L3.533-20.692L3.533-21.256Q3.424-21.074 3.278-20.951Q3.133-20.828 2.947-20.760Q2.760-20.692 2.559-20.692Q1.356-20.692 1.356-21.594M6.879-20.760L5.245-20.760L5.245-21.040Q5.474-21.040 5.623-21.074Q5.772-21.109 5.772-21.249L5.772-23.098Q5.772-23.368 5.664-23.429Q5.556-23.491 5.245-23.491L5.245-23.771L6.305-23.846L6.305-23.197Q6.476-23.505 6.780-23.676Q7.084-23.846 7.429-23.846Q7.829-23.846 8.106-23.706Q8.383-23.566 8.468-23.218Q8.636-23.511 8.935-23.679Q9.234-23.846 9.579-23.846Q10.085-23.846 10.369-23.623Q10.653-23.399 10.653-22.903L10.653-21.249Q10.653-21.112 10.801-21.076Q10.950-21.040 11.175-21.040L11.175-20.760L9.545-20.760L9.545-21.040Q9.771-21.040 9.921-21.076Q10.071-21.112 10.071-21.249L10.071-22.889Q10.071-23.224 9.952-23.424Q9.832-23.624 9.518-23.624Q9.248-23.624 9.014-23.488Q8.779-23.351 8.641-23.117Q8.503-22.883 8.503-22.609L8.503-21.249Q8.503-21.112 8.651-21.076Q8.800-21.040 9.026-21.040L9.026-20.760L7.395-20.760L7.395-21.040Q7.624-21.040 7.773-21.074Q7.922-21.109 7.922-21.249L7.922-22.889Q7.922-23.224 7.802-23.424Q7.682-23.624 7.368-23.624Q7.098-23.624 6.864-23.488Q6.630-23.351 6.491-23.117Q6.353-22.883 6.353-22.609L6.353-21.249Q6.353-21.112 6.503-21.076Q6.654-21.040 6.879-21.040L6.879-20.760M12.163-21.180Q12.163-21.348 12.286-21.471Q12.409-21.594 12.584-21.594Q12.751-21.594 12.874-21.471Q12.997-21.348 12.997-21.180Q12.997-21.006 12.874-20.883Q12.751-20.760 12.584-20.760Q12.409-20.760 12.286-20.883Q12.163-21.006 12.163-21.180M12.163-23.364Q12.163-23.532 12.286-23.655Q12.409-23.778 12.584-23.778Q12.751-23.778 12.874-23.655Q12.997-23.532 12.997-23.364Q12.997-23.190 12.874-23.067Q12.751-22.944 12.584-22.944Q12.409-22.944 12.286-23.067Q12.163-23.190 12.163-23.364\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(26.16 -42.927)\">\u003Cpath d=\"M19.337-20.760L17.601-20.760L17.601-21.040Q17.830-21.040 17.979-21.074Q18.127-21.109 18.127-21.249L18.127-23.098Q18.127-23.368 18.020-23.429Q17.912-23.491 17.601-23.491L17.601-23.771L18.630-23.846L18.630-23.139Q18.760-23.447 19.002-23.646Q19.245-23.846 19.563-23.846Q19.782-23.846 19.953-23.722Q20.124-23.597 20.124-23.385Q20.124-23.248 20.024-23.149Q19.925-23.050 19.792-23.050Q19.655-23.050 19.556-23.149Q19.457-23.248 19.457-23.385Q19.457-23.525 19.556-23.624Q19.266-23.624 19.066-23.428Q18.866-23.231 18.773-22.937Q18.681-22.643 18.681-22.363L18.681-21.249Q18.681-21.040 19.337-21.040L19.337-20.760M20.667-22.295Q20.667-22.616 20.792-22.905Q20.917-23.194 21.142-23.417Q21.368-23.641 21.663-23.761Q21.959-23.881 22.277-23.881Q22.605-23.881 22.867-23.781Q23.128-23.682 23.304-23.500Q23.480-23.317 23.574-23.059Q23.668-22.801 23.668-22.469Q23.668-22.377 23.586-22.356L21.330-22.356L21.330-22.295Q21.330-21.707 21.614-21.324Q21.898-20.941 22.465-20.941Q22.786-20.941 23.054-21.134Q23.323-21.327 23.412-21.642Q23.419-21.683 23.494-21.697L23.586-21.697Q23.668-21.673 23.668-21.601Q23.668-21.594 23.661-21.567Q23.548-21.170 23.178-20.931Q22.807-20.692 22.383-20.692Q21.945-20.692 21.545-20.900Q21.146-21.109 20.906-21.476Q20.667-21.843 20.667-22.295M21.337-22.565L23.152-22.565Q23.152-22.842 23.054-23.094Q22.957-23.347 22.759-23.503Q22.561-23.658 22.277-23.658Q22-23.658 21.786-23.500Q21.573-23.341 21.455-23.086Q21.337-22.831 21.337-22.565M25.644-20.787L24.663-23.286Q24.601-23.429 24.483-23.464Q24.365-23.498 24.150-23.498L24.150-23.778L25.630-23.778L25.630-23.498Q25.251-23.498 25.251-23.337Q25.251-23.327 25.264-23.286L25.979-21.454L26.652-23.159Q26.621-23.231 26.621-23.259Q26.621-23.286 26.594-23.286Q26.532-23.433 26.414-23.465Q26.296-23.498 26.085-23.498L26.085-23.778L27.482-23.778L27.482-23.498Q27.106-23.498 27.106-23.337Q27.106-23.306 27.113-23.286L27.869-21.348L28.556-23.098Q28.576-23.149 28.576-23.204Q28.576-23.344 28.463-23.421Q28.351-23.498 28.210-23.498L28.210-23.778L29.431-23.778L29.431-23.498Q29.226-23.498 29.070-23.392Q28.915-23.286 28.843-23.098L27.937-20.787Q27.903-20.692 27.790-20.692L27.722-20.692Q27.612-20.692 27.575-20.787L26.792-22.790L26.006-20.787Q25.972-20.692 25.859-20.692L25.791-20.692Q25.681-20.692 25.644-20.787\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(26.16 -42.927)\">\u003Cpath d=\"M29.719-22.295Q29.719-22.616 29.844-22.905Q29.969-23.194 30.195-23.417Q30.420-23.641 30.716-23.761Q31.011-23.881 31.329-23.881Q31.657-23.881 31.919-23.781Q32.180-23.682 32.356-23.500Q32.532-23.317 32.626-23.059Q32.720-22.801 32.720-22.469Q32.720-22.377 32.638-22.356L30.383-22.356L30.383-22.295Q30.383-21.707 30.666-21.324Q30.950-20.941 31.517-20.941Q31.839-20.941 32.107-21.134Q32.375-21.327 32.464-21.642Q32.471-21.683 32.546-21.697L32.638-21.697Q32.720-21.673 32.720-21.601Q32.720-21.594 32.714-21.567Q32.601-21.170 32.230-20.931Q31.859-20.692 31.435-20.692Q30.998-20.692 30.598-20.900Q30.198-21.109 29.959-21.476Q29.719-21.843 29.719-22.295M30.389-22.565L32.204-22.565Q32.204-22.842 32.107-23.094Q32.009-23.347 31.811-23.503Q31.613-23.658 31.329-23.658Q31.052-23.658 30.839-23.500Q30.625-23.341 30.507-23.086Q30.389-22.831 30.389-22.565M34.925-20.760L33.373-20.760L33.373-21.040Q33.599-21.040 33.748-21.074Q33.896-21.109 33.896-21.249L33.896-23.098Q33.896-23.286 33.848-23.370Q33.800-23.453 33.703-23.472Q33.606-23.491 33.394-23.491L33.394-23.771L34.450-23.846L34.450-21.249Q34.450-21.109 34.581-21.074Q34.713-21.040 34.925-21.040L34.925-20.760M33.654-25.067Q33.654-25.238 33.777-25.357Q33.900-25.477 34.071-25.477Q34.238-25.477 34.361-25.357Q34.484-25.238 34.484-25.067Q34.484-24.892 34.361-24.769Q34.238-24.646 34.071-24.646Q33.900-24.646 33.777-24.769Q33.654-24.892 33.654-25.067M35.530-20.227Q35.530-20.473 35.727-20.657Q35.923-20.842 36.179-20.921Q36.043-21.033 35.971-21.194Q35.899-21.355 35.899-21.536Q35.899-21.857 36.111-22.103Q35.776-22.401 35.776-22.811Q35.776-23.272 36.166-23.559Q36.555-23.846 37.034-23.846Q37.506-23.846 37.841-23.600Q38.015-23.754 38.225-23.836Q38.435-23.918 38.664-23.918Q38.828-23.918 38.950-23.811Q39.071-23.703 39.071-23.539Q39.071-23.443 38.999-23.371Q38.927-23.300 38.835-23.300Q38.736-23.300 38.666-23.373Q38.596-23.447 38.596-23.546Q38.596-23.600 38.610-23.631L38.616-23.645Q38.623-23.665 38.632-23.676Q38.640-23.686 38.644-23.693Q38.288-23.693 38.001-23.470Q38.288-23.177 38.288-22.811Q38.288-22.496 38.104-22.264Q37.919-22.031 37.630-21.903Q37.342-21.775 37.034-21.775Q36.832-21.775 36.641-21.825Q36.449-21.874 36.272-21.984Q36.179-21.857 36.179-21.714Q36.179-21.532 36.308-21.397Q36.436-21.262 36.620-21.262L37.253-21.262Q37.700-21.262 38.070-21.191Q38.439-21.119 38.698-20.890Q38.958-20.661 38.958-20.227Q38.958-19.906 38.663-19.704Q38.367-19.502 37.964-19.413Q37.560-19.324 37.246-19.324Q36.928-19.324 36.525-19.413Q36.121-19.502 35.826-19.704Q35.530-19.906 35.530-20.227M35.985-20.227Q35.985-19.998 36.203-19.849Q36.422-19.700 36.714-19.632Q37.007-19.564 37.246-19.564Q37.410-19.564 37.618-19.600Q37.827-19.635 38.034-19.716Q38.240-19.796 38.372-19.924Q38.504-20.052 38.504-20.227Q38.504-20.579 38.123-20.673Q37.741-20.767 37.239-20.767L36.620-20.767Q36.381-20.767 36.183-20.616Q35.985-20.466 35.985-20.227M37.034-22.014Q37.700-22.014 37.700-22.811Q37.700-23.611 37.034-23.611Q36.364-23.611 36.364-22.811Q36.364-22.014 37.034-22.014M41.235-20.760L39.601-20.760L39.601-21.040Q39.830-21.040 39.978-21.074Q40.127-21.109 40.127-21.249L40.127-24.868Q40.127-25.138 40.019-25.200Q39.912-25.261 39.601-25.261L39.601-25.542L40.681-25.617L40.681-23.231Q40.787-23.416 40.965-23.558Q41.142-23.699 41.351-23.773Q41.559-23.846 41.785-23.846Q42.291-23.846 42.574-23.623Q42.858-23.399 42.858-22.903L42.858-21.249Q42.858-21.112 43.007-21.076Q43.155-21.040 43.381-21.040L43.381-20.760L41.751-20.760L41.751-21.040Q41.980-21.040 42.128-21.074Q42.277-21.109 42.277-21.249L42.277-22.889Q42.277-23.224 42.157-23.424Q42.038-23.624 41.723-23.624Q41.453-23.624 41.219-23.488Q40.985-23.351 40.847-23.117Q40.708-22.883 40.708-22.609L40.708-21.249Q40.708-21.112 40.859-21.076Q41.009-21.040 41.235-21.040\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(26.16 -42.927)\">\u003Cpath d=\"M44.294-21.601L44.294-23.498L43.655-23.498L43.655-23.720Q43.973-23.720 44.190-23.930Q44.407-24.140 44.507-24.450Q44.608-24.759 44.608-25.067L44.875-25.067L44.875-23.778L45.952-23.778L45.952-23.498L44.875-23.498L44.875-21.614Q44.875-21.338 44.979-21.139Q45.083-20.941 45.343-20.941Q45.500-20.941 45.606-21.045Q45.712-21.150 45.762-21.303Q45.811-21.457 45.811-21.614L45.811-22.028L46.078-22.028L46.078-21.601Q46.078-21.375 45.979-21.165Q45.880-20.955 45.695-20.823Q45.511-20.692 45.282-20.692Q44.844-20.692 44.569-20.929Q44.294-21.167 44.294-21.601\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(26.16 -42.927)\">\u003Cpath d=\"M50.394-20.760L50.127-20.760L50.127-24.868Q50.127-25.138 50.020-25.200Q49.912-25.261 49.601-25.261L49.601-25.542L50.681-25.617L50.681-23.447Q50.890-23.638 51.175-23.742Q51.461-23.846 51.758-23.846Q52.076-23.846 52.373-23.725Q52.670-23.604 52.893-23.388Q53.115-23.173 53.241-22.888Q53.368-22.602 53.368-22.271Q53.368-21.826 53.128-21.462Q52.889-21.098 52.496-20.895Q52.103-20.692 51.659-20.692Q51.464-20.692 51.274-20.748Q51.085-20.804 50.924-20.909Q50.763-21.013 50.623-21.174L50.394-20.760M50.709-23.105L50.709-21.488Q50.845-21.228 51.086-21.071Q51.327-20.914 51.604-20.914Q51.898-20.914 52.110-21.021Q52.322-21.129 52.455-21.321Q52.588-21.512 52.647-21.751Q52.705-21.990 52.705-22.271Q52.705-22.630 52.611-22.934Q52.517-23.238 52.289-23.431Q52.062-23.624 51.696-23.624Q51.396-23.624 51.129-23.488Q50.862-23.351 50.709-23.105\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(26.16 -42.927)\">\u003Cpath d=\"M54.123-19.625Q54.253-19.557 54.390-19.557Q54.561-19.557 54.711-19.646Q54.862-19.735 54.973-19.880Q55.084-20.025 55.162-20.193L55.426-20.760L54.257-23.286Q54.182-23.433 54.052-23.465Q53.922-23.498 53.689-23.498L53.689-23.778L55.210-23.778L55.210-23.498Q54.862-23.498 54.862-23.351Q54.865-23.330 54.867-23.313Q54.869-23.296 54.869-23.286L55.726-21.427L56.499-23.098Q56.533-23.166 56.533-23.245Q56.533-23.358 56.449-23.428Q56.366-23.498 56.253-23.498L56.253-23.778L57.449-23.778L57.449-23.498Q57.230-23.498 57.058-23.394Q56.885-23.289 56.793-23.098L55.456-20.193Q55.286-19.823 55.016-19.577Q54.745-19.331 54.390-19.331Q54.120-19.331 53.901-19.497Q53.682-19.663 53.682-19.926Q53.682-20.063 53.775-20.152Q53.867-20.240 54.007-20.240Q54.144-20.240 54.233-20.152Q54.322-20.063 54.322-19.926Q54.322-19.823 54.269-19.745Q54.216-19.666 54.123-19.625\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(26.16 -42.927)\">\u003Cpath d=\"M60.643-22.295Q60.643-22.616 60.768-22.905Q60.893-23.194 61.119-23.417Q61.344-23.641 61.640-23.761Q61.935-23.881 62.253-23.881Q62.581-23.881 62.843-23.781Q63.104-23.682 63.280-23.500Q63.456-23.317 63.550-23.059Q63.644-22.801 63.644-22.469Q63.644-22.377 63.562-22.356L61.307-22.356L61.307-22.295Q61.307-21.707 61.590-21.324Q61.874-20.941 62.441-20.941Q62.763-20.941 63.031-21.134Q63.299-21.327 63.388-21.642Q63.395-21.683 63.470-21.697L63.562-21.697Q63.644-21.673 63.644-21.601Q63.644-21.594 63.638-21.567Q63.525-21.170 63.154-20.931Q62.783-20.692 62.359-20.692Q61.922-20.692 61.522-20.900Q61.122-21.109 60.883-21.476Q60.643-21.843 60.643-22.295M61.313-22.565L63.128-22.565Q63.128-22.842 63.031-23.094Q62.933-23.347 62.735-23.503Q62.537-23.658 62.253-23.658Q61.976-23.658 61.763-23.500Q61.549-23.341 61.431-23.086Q61.313-22.831 61.313-22.565M65.415-20.760L64.092-20.760L64.092-21.040Q64.653-21.040 65.032-21.440L65.746-22.237L64.834-23.286Q64.697-23.433 64.548-23.465Q64.400-23.498 64.133-23.498L64.133-23.778L65.634-23.778L65.634-23.498Q65.442-23.498 65.442-23.364Q65.442-23.334 65.473-23.286L66.068-22.602L66.509-23.098Q66.621-23.228 66.621-23.344Q66.621-23.406 66.584-23.452Q66.546-23.498 66.488-23.498L66.488-23.778L67.804-23.778L67.804-23.498Q67.244-23.498 66.864-23.098L66.242-22.397L67.237-21.249Q67.336-21.150 67.437-21.105Q67.537-21.061 67.649-21.051Q67.760-21.040 67.937-21.040L67.937-20.760L66.444-20.760L66.444-21.040Q66.509-21.040 66.568-21.074Q66.628-21.109 66.628-21.174Q66.628-21.221 66.598-21.249L65.921-22.035L65.388-21.440Q65.275-21.310 65.275-21.194Q65.275-21.129 65.316-21.085Q65.357-21.040 65.415-21.040L65.415-20.760M70.077-19.403L68.447-19.403L68.447-19.683Q68.676-19.683 68.824-19.718Q68.973-19.752 68.973-19.892L68.973-23.238Q68.973-23.409 68.836-23.450Q68.700-23.491 68.447-23.491L68.447-23.771L69.527-23.846L69.527-23.440Q69.749-23.641 70.036-23.744Q70.323-23.846 70.631-23.846Q71.058-23.846 71.422-23.633Q71.786-23.419 72-23.055Q72.213-22.691 72.213-22.271Q72.213-21.826 71.974-21.462Q71.735-21.098 71.342-20.895Q70.949-20.692 70.504-20.692Q70.238-20.692 69.990-20.792Q69.742-20.893 69.554-21.074L69.554-19.892Q69.554-19.755 69.703-19.719Q69.851-19.683 70.077-19.683L70.077-19.403M69.554-23.091L69.554-21.481Q69.687-21.228 69.930-21.071Q70.173-20.914 70.450-20.914Q70.778-20.914 71.031-21.115Q71.284-21.317 71.417-21.635Q71.550-21.953 71.550-22.271Q71.550-22.500 71.485-22.729Q71.420-22.958 71.292-23.156Q71.164-23.354 70.969-23.474Q70.774-23.593 70.542-23.593Q70.248-23.593 69.980-23.464Q69.711-23.334 69.554-23.091M74.978-19.010Q74.428-19.410 74.057-19.965Q73.686-20.521 73.505-21.167Q73.324-21.813 73.324-22.510Q73.324-23.023 73.425-23.518Q73.526-24.014 73.731-24.465Q73.936-24.916 74.249-25.308Q74.561-25.699 74.978-26.003Q74.989-26.007 74.995-26.008Q75.002-26.010 75.013-26.010L75.081-26.010Q75.115-26.010 75.137-25.986Q75.160-25.962 75.160-25.925Q75.160-25.880 75.132-25.863Q74.784-25.562 74.531-25.178Q74.278-24.793 74.126-24.352Q73.974-23.911 73.902-23.455Q73.830-22.999 73.830-22.510Q73.830-21.509 74.139-20.622Q74.449-19.735 75.132-19.150Q75.160-19.133 75.160-19.089Q75.160-19.051 75.137-19.027Q75.115-19.003 75.081-19.003L75.013-19.003Q75.006-19.007 74.997-19.008Q74.989-19.010 74.978-19.010M77.720-20.760L75.983-20.760L75.983-21.040Q76.212-21.040 76.361-21.074Q76.510-21.109 76.510-21.249L76.510-23.098Q76.510-23.368 76.402-23.429Q76.294-23.491 75.983-23.491L75.983-23.771L77.012-23.846L77.012-23.139Q77.142-23.447 77.385-23.646Q77.627-23.846 77.945-23.846Q78.164-23.846 78.335-23.722Q78.506-23.597 78.506-23.385Q78.506-23.248 78.407-23.149Q78.308-23.050 78.174-23.050Q78.037-23.050 77.938-23.149Q77.839-23.248 77.839-23.385Q77.839-23.525 77.938-23.624Q77.648-23.624 77.448-23.428Q77.248-23.231 77.156-22.937Q77.063-22.643 77.063-22.363L77.063-21.249Q77.063-21.040 77.720-21.040L77.720-20.760M79.049-22.295Q79.049-22.616 79.174-22.905Q79.299-23.194 79.524-23.417Q79.750-23.641 80.046-23.761Q80.341-23.881 80.659-23.881Q80.987-23.881 81.249-23.781Q81.510-23.682 81.686-23.500Q81.862-23.317 81.956-23.059Q82.050-22.801 82.050-22.469Q82.050-22.377 81.968-22.356L79.712-22.356L79.712-22.295Q79.712-21.707 79.996-21.324Q80.280-20.941 80.847-20.941Q81.168-20.941 81.437-21.134Q81.705-21.327 81.794-21.642Q81.801-21.683 81.876-21.697L81.968-21.697Q82.050-21.673 82.050-21.601Q82.050-21.594 82.043-21.567Q81.931-21.170 81.560-20.931Q81.189-20.692 80.765-20.692Q80.328-20.692 79.928-20.900Q79.528-21.109 79.288-21.476Q79.049-21.843 79.049-22.295M79.719-22.565L81.534-22.565Q81.534-22.842 81.437-23.094Q81.339-23.347 81.141-23.503Q80.943-23.658 80.659-23.658Q80.382-23.658 80.169-23.500Q79.955-23.341 79.837-23.086Q79.719-22.831 79.719-22.565M84.026-20.787L83.045-23.286Q82.983-23.429 82.865-23.464Q82.747-23.498 82.532-23.498L82.532-23.778L84.012-23.778L84.012-23.498Q83.633-23.498 83.633-23.337Q83.633-23.327 83.646-23.286L84.361-21.454L85.034-23.159Q85.003-23.231 85.003-23.259Q85.003-23.286 84.976-23.286Q84.914-23.433 84.797-23.465Q84.679-23.498 84.467-23.498L84.467-23.778L85.865-23.778L85.865-23.498Q85.489-23.498 85.489-23.337Q85.489-23.306 85.495-23.286L86.251-21.348L86.938-23.098Q86.958-23.149 86.958-23.204Q86.958-23.344 86.846-23.421Q86.733-23.498 86.593-23.498L86.593-23.778L87.813-23.778L87.813-23.498Q87.608-23.498 87.452-23.392Q87.297-23.286 87.225-23.098L86.319-20.787Q86.285-20.692 86.172-20.692L86.104-20.692Q85.995-20.692 85.957-20.787L85.174-22.790L84.388-20.787Q84.354-20.692 84.241-20.692L84.173-20.692Q84.063-20.692 84.026-20.787\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(26.16 -42.927)\">\u003Cpath d=\"M88.215-21.488Q88.215-21.820 88.438-22.047Q88.662-22.274 89.006-22.402Q89.349-22.531 89.722-22.583Q90.094-22.636 90.399-22.636L90.399-22.889Q90.399-23.094 90.291-23.274Q90.183-23.453 90.002-23.556Q89.821-23.658 89.613-23.658Q89.206-23.658 88.970-23.566Q89.059-23.529 89.105-23.445Q89.151-23.361 89.151-23.259Q89.151-23.163 89.105-23.084Q89.059-23.006 88.978-22.961Q88.898-22.917 88.809-22.917Q88.659-22.917 88.558-23.014Q88.457-23.112 88.457-23.259Q88.457-23.881 89.613-23.881Q89.824-23.881 90.074-23.817Q90.323-23.754 90.525-23.635Q90.727-23.515 90.853-23.330Q90.980-23.146 90.980-22.903L90.980-21.327Q90.980-21.211 91.041-21.115Q91.103-21.020 91.216-21.020Q91.325-21.020 91.390-21.114Q91.455-21.208 91.455-21.327L91.455-21.775L91.721-21.775L91.721-21.327Q91.721-21.057 91.494-20.892Q91.267-20.726 90.987-20.726Q90.778-20.726 90.641-20.880Q90.505-21.033 90.481-21.249Q90.334-20.982 90.052-20.837Q89.770-20.692 89.445-20.692Q89.168-20.692 88.884-20.767Q88.601-20.842 88.408-21.021Q88.215-21.201 88.215-21.488M88.830-21.488Q88.830-21.314 88.931-21.184Q89.031-21.054 89.187-20.984Q89.342-20.914 89.507-20.914Q89.725-20.914 89.934-21.011Q90.142-21.109 90.270-21.290Q90.399-21.471 90.399-21.697L90.399-22.425Q90.074-22.425 89.708-22.334Q89.342-22.243 89.086-22.031Q88.830-21.820 88.830-21.488M93.888-20.760L92.152-20.760L92.152-21.040Q92.381-21.040 92.530-21.074Q92.678-21.109 92.678-21.249L92.678-23.098Q92.678-23.368 92.571-23.429Q92.463-23.491 92.152-23.491L92.152-23.771L93.181-23.846L93.181-23.139Q93.311-23.447 93.553-23.646Q93.796-23.846 94.114-23.846Q94.333-23.846 94.504-23.722Q94.675-23.597 94.675-23.385Q94.675-23.248 94.575-23.149Q94.476-23.050 94.343-23.050Q94.206-23.050 94.107-23.149Q94.008-23.248 94.008-23.385Q94.008-23.525 94.107-23.624Q93.817-23.624 93.617-23.428Q93.417-23.231 93.324-22.937Q93.232-22.643 93.232-22.363L93.232-21.249Q93.232-21.040 93.888-21.040L93.888-20.760M95.259-22.271Q95.259-22.609 95.399-22.900Q95.539-23.190 95.784-23.404Q96.028-23.617 96.332-23.732Q96.636-23.846 96.961-23.846Q97.231-23.846 97.494-23.747Q97.758-23.648 97.949-23.470L97.949-24.868Q97.949-25.138 97.841-25.200Q97.734-25.261 97.423-25.261L97.423-25.542L98.499-25.617L98.499-21.433Q98.499-21.245 98.554-21.162Q98.609-21.078 98.709-21.059Q98.810-21.040 99.026-21.040L99.026-20.760L97.918-20.692L97.918-21.109Q97.501-20.692 96.876-20.692Q96.445-20.692 96.072-20.904Q95.700-21.115 95.479-21.476Q95.259-21.837 95.259-22.271M96.934-20.914Q97.142-20.914 97.329-20.986Q97.515-21.057 97.669-21.194Q97.822-21.331 97.918-21.509L97.918-23.118Q97.833-23.265 97.687-23.385Q97.542-23.505 97.373-23.564Q97.204-23.624 97.023-23.624Q96.462-23.624 96.194-23.235Q95.925-22.845 95.925-22.264Q95.925-21.693 96.160-21.303Q96.394-20.914 96.934-20.914\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(26.16 -42.927)\">\u003Cpath d=\"M102.554-19.177Q102.554-19.195 102.567-19.242L105.223-25.904Q105.278-26.010 105.384-26.010Q105.449-26.010 105.500-25.959Q105.551-25.907 105.551-25.843Q105.551-25.819 105.550-25.807Q105.548-25.795 105.544-25.778L102.892-19.116Q102.820-19.010 102.732-19.010Q102.663-19.010 102.608-19.061Q102.554-19.113 102.554-19.177\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(26.16 -42.927)\">\u003Cpath d=\"M109.866-20.760L109.599-20.760L109.599-24.868Q109.599-25.138 109.492-25.200Q109.384-25.261 109.073-25.261L109.073-25.542L110.153-25.617L110.153-23.447Q110.362-23.638 110.647-23.742Q110.933-23.846 111.230-23.846Q111.548-23.846 111.845-23.725Q112.142-23.604 112.365-23.388Q112.587-23.173 112.713-22.888Q112.840-22.602 112.840-22.271Q112.840-21.826 112.600-21.462Q112.361-21.098 111.968-20.895Q111.575-20.692 111.131-20.692Q110.936-20.692 110.746-20.748Q110.557-20.804 110.396-20.909Q110.235-21.013 110.095-21.174L109.866-20.760M110.181-23.105L110.181-21.488Q110.317-21.228 110.558-21.071Q110.799-20.914 111.076-20.914Q111.370-20.914 111.582-21.021Q111.794-21.129 111.927-21.321Q112.060-21.512 112.119-21.751Q112.177-21.990 112.177-22.271Q112.177-22.630 112.083-22.934Q111.989-23.238 111.761-23.431Q111.534-23.624 111.168-23.624Q110.868-23.624 110.601-23.488Q110.334-23.351 110.181-23.105\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(26.16 -42.927)\">\u003Cpath d=\"M113.650-22.295Q113.650-22.616 113.775-22.905Q113.900-23.194 114.126-23.417Q114.351-23.641 114.647-23.761Q114.942-23.881 115.260-23.881Q115.588-23.881 115.850-23.781Q116.111-23.682 116.287-23.500Q116.463-23.317 116.557-23.059Q116.651-22.801 116.651-22.469Q116.651-22.377 116.569-22.356L114.314-22.356L114.314-22.295Q114.314-21.707 114.597-21.324Q114.881-20.941 115.448-20.941Q115.770-20.941 116.038-21.134Q116.306-21.327 116.395-21.642Q116.402-21.683 116.477-21.697L116.569-21.697Q116.651-21.673 116.651-21.601Q116.651-21.594 116.645-21.567Q116.532-21.170 116.161-20.931Q115.790-20.692 115.366-20.692Q114.929-20.692 114.529-20.900Q114.129-21.109 113.890-21.476Q113.650-21.843 113.650-22.295M114.320-22.565L116.135-22.565Q116.135-22.842 116.038-23.094Q115.940-23.347 115.742-23.503Q115.544-23.658 115.260-23.658Q114.983-23.658 114.770-23.500Q114.556-23.341 114.438-23.086Q114.320-22.831 114.320-22.565M117.766-21.601L117.766-23.498L117.127-23.498L117.127-23.720Q117.444-23.720 117.661-23.930Q117.878-24.140 117.979-24.450Q118.080-24.759 118.080-25.067L118.347-25.067L118.347-23.778L119.423-23.778L119.423-23.498L118.347-23.498L118.347-21.614Q118.347-21.338 118.451-21.139Q118.555-20.941 118.815-20.941Q118.972-20.941 119.078-21.045Q119.184-21.150 119.234-21.303Q119.283-21.457 119.283-21.614L119.283-22.028L119.550-22.028L119.550-21.601Q119.550-21.375 119.451-21.165Q119.352-20.955 119.167-20.823Q118.982-20.692 118.753-20.692Q118.316-20.692 118.041-20.929Q117.766-21.167 117.766-21.601M120.418-21.488Q120.418-21.820 120.642-22.047Q120.866-22.274 121.209-22.402Q121.553-22.531 121.925-22.583Q122.298-22.636 122.602-22.636L122.602-22.889Q122.602-23.094 122.494-23.274Q122.387-23.453 122.206-23.556Q122.024-23.658 121.816-23.658Q121.409-23.658 121.173-23.566Q121.262-23.529 121.308-23.445Q121.355-23.361 121.355-23.259Q121.355-23.163 121.308-23.084Q121.262-23.006 121.182-22.961Q121.102-22.917 121.013-22.917Q120.862-22.917 120.762-23.014Q120.661-23.112 120.661-23.259Q120.661-23.881 121.816-23.881Q122.028-23.881 122.277-23.817Q122.527-23.754 122.729-23.635Q122.930-23.515 123.057-23.330Q123.183-23.146 123.183-22.903L123.183-21.327Q123.183-21.211 123.245-21.115Q123.306-21.020 123.419-21.020Q123.528-21.020 123.593-21.114Q123.658-21.208 123.658-21.327L123.658-21.775L123.925-21.775L123.925-21.327Q123.925-21.057 123.698-20.892Q123.470-20.726 123.190-20.726Q122.981-20.726 122.845-20.880Q122.708-21.033 122.684-21.249Q122.537-20.982 122.255-20.837Q121.973-20.692 121.648-20.692Q121.372-20.692 121.088-20.767Q120.804-20.842 120.611-21.021Q120.418-21.201 120.418-21.488M121.033-21.488Q121.033-21.314 121.134-21.184Q121.235-21.054 121.390-20.984Q121.546-20.914 121.710-20.914Q121.929-20.914 122.137-21.011Q122.346-21.109 122.474-21.290Q122.602-21.471 122.602-21.697L122.602-22.425Q122.277-22.425 121.912-22.334Q121.546-22.243 121.290-22.031Q121.033-21.820 121.033-21.488M124.663-19.003L124.595-19.003Q124.561-19.003 124.538-19.029Q124.516-19.054 124.516-19.089Q124.516-19.133 124.547-19.150Q124.902-19.454 125.152-19.844Q125.401-20.234 125.554-20.666Q125.706-21.098 125.776-21.567Q125.846-22.035 125.846-22.510Q125.846-22.989 125.776-23.455Q125.706-23.922 125.552-24.357Q125.398-24.793 125.147-25.181Q124.896-25.569 124.547-25.863Q124.516-25.880 124.516-25.925Q124.516-25.959 124.538-25.984Q124.561-26.010 124.595-26.010L124.663-26.010Q124.673-26.010 124.682-26.008Q124.690-26.007 124.701-26.003Q125.244-25.603 125.617-25.050Q125.989-24.496 126.170-23.850Q126.352-23.204 126.352-22.510Q126.352-21.809 126.170-21.162Q125.989-20.514 125.615-19.960Q125.241-19.406 124.701-19.010Q124.690-19.010 124.682-19.008Q124.673-19.007 124.663-19.003\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M133.566.98c-53.66 20.94-53.66 20.94-104.899.945\"\u002F>\u003Cpath stroke=\"none\" d=\"m26.245.98 3.12 3.45-.698-2.505 2.21-1.37\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(31.694 46.427)\">\u003Cpath d=\"M-17.748-20.760L-19.300-20.760L-19.300-21.040Q-19.074-21.040-18.925-21.074Q-18.777-21.109-18.777-21.249L-18.777-23.098Q-18.777-23.286-18.825-23.370Q-18.872-23.453-18.970-23.472Q-19.067-23.491-19.279-23.491L-19.279-23.771L-18.223-23.846L-18.223-21.249Q-18.223-21.109-18.091-21.074Q-17.960-21.040-17.748-21.040L-17.748-20.760M-19.019-25.067Q-19.019-25.238-18.896-25.357Q-18.773-25.477-18.602-25.477Q-18.435-25.477-18.312-25.357Q-18.189-25.238-18.189-25.067Q-18.189-24.892-18.312-24.769Q-18.435-24.646-18.602-24.646Q-18.773-24.646-18.896-24.769Q-19.019-24.892-19.019-25.067M-15.420-20.760L-17.054-20.760L-17.054-21.040Q-16.825-21.040-16.676-21.074Q-16.528-21.109-16.528-21.249L-16.528-23.098Q-16.528-23.368-16.635-23.429Q-16.743-23.491-17.054-23.491L-17.054-23.771L-15.994-23.846L-15.994-23.197Q-15.824-23.505-15.519-23.676Q-15.215-23.846-14.870-23.846Q-14.364-23.846-14.080-23.623Q-13.797-23.399-13.797-22.903L-13.797-21.249Q-13.797-21.112-13.648-21.076Q-13.499-21.040-13.274-21.040L-13.274-20.760L-14.904-20.760L-14.904-21.040Q-14.675-21.040-14.526-21.074Q-14.378-21.109-14.378-21.249L-14.378-22.889Q-14.378-23.224-14.497-23.424Q-14.617-23.624-14.931-23.624Q-15.201-23.624-15.436-23.488Q-15.670-23.351-15.808-23.117Q-15.947-22.883-15.947-22.609L-15.947-21.249Q-15.947-21.112-15.796-21.076Q-15.646-21.040-15.420-21.040\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(31.694 46.427)\">\u003Cpath d=\"M-11.310-20.787L-12.438-23.286Q-12.510-23.433-12.640-23.465Q-12.770-23.498-12.999-23.498L-12.999-23.778L-11.485-23.778L-11.485-23.498Q-11.837-23.498-11.837-23.351Q-11.837-23.306-11.826-23.286L-10.962-21.368L-10.182-23.098Q-10.148-23.166-10.148-23.245Q-10.148-23.358-10.232-23.428Q-10.316-23.498-10.435-23.498L-10.435-23.778L-9.239-23.778L-9.239-23.498Q-9.458-23.498-9.629-23.395Q-9.799-23.293-9.888-23.098L-10.924-20.787Q-10.972-20.692-11.078-20.692L-11.156-20.692Q-11.262-20.692-11.310-20.787\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(31.694 46.427)\">\u003Cpath d=\"M-8.955-22.295Q-8.955-22.616-8.830-22.905Q-8.705-23.194-8.479-23.417Q-8.254-23.641-7.958-23.761Q-7.663-23.881-7.345-23.881Q-7.017-23.881-6.755-23.781Q-6.494-23.682-6.318-23.500Q-6.142-23.317-6.048-23.059Q-5.954-22.801-5.954-22.469Q-5.954-22.377-6.036-22.356L-8.291-22.356L-8.291-22.295Q-8.291-21.707-8.008-21.324Q-7.724-20.941-7.157-20.941Q-6.835-20.941-6.567-21.134Q-6.299-21.327-6.210-21.642Q-6.203-21.683-6.128-21.697L-6.036-21.697Q-5.954-21.673-5.954-21.601Q-5.954-21.594-5.960-21.567Q-6.073-21.170-6.444-20.931Q-6.815-20.692-7.239-20.692Q-7.676-20.692-8.076-20.900Q-8.476-21.109-8.715-21.476Q-8.955-21.843-8.955-22.295M-8.285-22.565L-6.470-22.565Q-6.470-22.842-6.567-23.094Q-6.665-23.347-6.863-23.503Q-7.061-23.658-7.345-23.658Q-7.622-23.658-7.835-23.500Q-8.049-23.341-8.167-23.086Q-8.285-22.831-8.285-22.565M-3.616-20.760L-5.352-20.760L-5.352-21.040Q-5.123-21.040-4.974-21.074Q-4.826-21.109-4.826-21.249L-4.826-23.098Q-4.826-23.368-4.933-23.429Q-5.041-23.491-5.352-23.491L-5.352-23.771L-4.323-23.846L-4.323-23.139Q-4.193-23.447-3.951-23.646Q-3.708-23.846-3.390-23.846Q-3.171-23.846-3-23.722Q-2.830-23.597-2.830-23.385Q-2.830-23.248-2.929-23.149Q-3.028-23.050-3.161-23.050Q-3.298-23.050-3.397-23.149Q-3.496-23.248-3.496-23.385Q-3.496-23.525-3.397-23.624Q-3.687-23.624-3.887-23.428Q-4.087-23.231-4.180-22.937Q-4.272-22.643-4.272-22.363L-4.272-21.249Q-4.272-21.040-3.616-21.040L-3.616-20.760M-1.719-21.601L-1.719-23.498L-2.358-23.498L-2.358-23.720Q-2.040-23.720-1.823-23.930Q-1.606-24.140-1.505-24.450Q-1.404-24.759-1.404-25.067L-1.138-25.067L-1.138-23.778L-0.061-23.778L-0.061-23.498L-1.138-23.498L-1.138-21.614Q-1.138-21.338-1.033-21.139Q-0.929-20.941-0.669-20.941Q-0.512-20.941-0.406-21.045Q-0.300-21.150-0.251-21.303Q-0.201-21.457-0.201-21.614L-0.201-22.028L0.065-22.028L0.065-21.601Q0.065-21.375-0.034-21.165Q-0.133-20.955-0.317-20.823Q-0.502-20.692-0.731-20.692Q-1.168-20.692-1.444-20.929Q-1.719-21.167-1.719-21.601M1.275-21.180Q1.275-21.348 1.398-21.471Q1.522-21.594 1.696-21.594Q1.863-21.594 1.986-21.471Q2.109-21.348 2.109-21.180Q2.109-21.006 1.986-20.883Q1.863-20.760 1.696-20.760Q1.522-20.760 1.398-20.883Q1.275-21.006 1.275-21.180M1.275-23.364Q1.275-23.532 1.398-23.655Q1.522-23.778 1.696-23.778Q1.863-23.778 1.986-23.655Q2.109-23.532 2.109-23.364Q2.109-23.190 1.986-23.067Q1.863-22.944 1.696-22.944Q1.522-22.944 1.398-23.067Q1.275-23.190 1.275-23.364\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(31.694 46.427)\">\u003Cpath d=\"M8.462-20.760L6.726-20.760L6.726-21.040Q6.955-21.040 7.104-21.074Q7.252-21.109 7.252-21.249L7.252-23.098Q7.252-23.368 7.145-23.429Q7.037-23.491 6.726-23.491L6.726-23.771L7.755-23.846L7.755-23.139Q7.885-23.447 8.127-23.646Q8.370-23.846 8.688-23.846Q8.907-23.846 9.078-23.722Q9.249-23.597 9.249-23.385Q9.249-23.248 9.149-23.149Q9.050-23.050 8.917-23.050Q8.780-23.050 8.681-23.149Q8.582-23.248 8.582-23.385Q8.582-23.525 8.681-23.624Q8.391-23.624 8.191-23.428Q7.991-23.231 7.898-22.937Q7.806-22.643 7.806-22.363L7.806-21.249Q7.806-21.040 8.462-21.040L8.462-20.760M9.792-22.295Q9.792-22.616 9.917-22.905Q10.042-23.194 10.267-23.417Q10.493-23.641 10.788-23.761Q11.084-23.881 11.402-23.881Q11.730-23.881 11.992-23.781Q12.253-23.682 12.429-23.500Q12.605-23.317 12.699-23.059Q12.793-22.801 12.793-22.469Q12.793-22.377 12.711-22.356L10.455-22.356L10.455-22.295Q10.455-21.707 10.739-21.324Q11.023-20.941 11.590-20.941Q11.911-20.941 12.179-21.134Q12.448-21.327 12.537-21.642Q12.544-21.683 12.619-21.697L12.711-21.697Q12.793-21.673 12.793-21.601Q12.793-21.594 12.786-21.567Q12.673-21.170 12.303-20.931Q11.932-20.692 11.508-20.692Q11.070-20.692 10.670-20.900Q10.271-21.109 10.031-21.476Q9.792-21.843 9.792-22.295M10.462-22.565L12.277-22.565Q12.277-22.842 12.179-23.094Q12.082-23.347 11.884-23.503Q11.686-23.658 11.402-23.658Q11.125-23.658 10.911-23.500Q10.698-23.341 10.580-23.086Q10.462-22.831 10.462-22.565M14.769-20.787L13.788-23.286Q13.726-23.429 13.608-23.464Q13.490-23.498 13.275-23.498L13.275-23.778L14.755-23.778L14.755-23.498Q14.376-23.498 14.376-23.337Q14.376-23.327 14.389-23.286L15.104-21.454L15.777-23.159Q15.746-23.231 15.746-23.259Q15.746-23.286 15.719-23.286Q15.657-23.433 15.539-23.465Q15.421-23.498 15.210-23.498L15.210-23.778L16.607-23.778L16.607-23.498Q16.231-23.498 16.231-23.337Q16.231-23.306 16.238-23.286L16.994-21.348L17.681-23.098Q17.701-23.149 17.701-23.204Q17.701-23.344 17.588-23.421Q17.476-23.498 17.335-23.498L17.335-23.778L18.556-23.778L18.556-23.498Q18.351-23.498 18.195-23.392Q18.040-23.286 17.968-23.098L17.062-20.787Q17.028-20.692 16.915-20.692L16.847-20.692Q16.737-20.692 16.700-20.787L15.917-22.790L15.131-20.787Q15.097-20.692 14.984-20.692L14.916-20.692Q14.806-20.692 14.769-20.787\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(31.694 46.427)\">\u003Cpath d=\"M18.944-21.488Q18.944-21.820 19.167-22.047Q19.391-22.274 19.735-22.402Q20.078-22.531 20.451-22.583Q20.823-22.636 21.128-22.636L21.128-22.889Q21.128-23.094 21.020-23.274Q20.912-23.453 20.731-23.556Q20.550-23.658 20.342-23.658Q19.935-23.658 19.699-23.566Q19.788-23.529 19.834-23.445Q19.880-23.361 19.880-23.259Q19.880-23.163 19.834-23.084Q19.788-23.006 19.707-22.961Q19.627-22.917 19.538-22.917Q19.388-22.917 19.287-23.014Q19.186-23.112 19.186-23.259Q19.186-23.881 20.342-23.881Q20.553-23.881 20.803-23.817Q21.052-23.754 21.254-23.635Q21.456-23.515 21.582-23.330Q21.709-23.146 21.709-22.903L21.709-21.327Q21.709-21.211 21.770-21.115Q21.832-21.020 21.945-21.020Q22.054-21.020 22.119-21.114Q22.184-21.208 22.184-21.327L22.184-21.775L22.450-21.775L22.450-21.327Q22.450-21.057 22.223-20.892Q21.996-20.726 21.716-20.726Q21.507-20.726 21.370-20.880Q21.234-21.033 21.210-21.249Q21.063-20.982 20.781-20.837Q20.499-20.692 20.174-20.692Q19.897-20.692 19.613-20.767Q19.330-20.842 19.137-21.021Q18.944-21.201 18.944-21.488M19.559-21.488Q19.559-21.314 19.660-21.184Q19.760-21.054 19.916-20.984Q20.071-20.914 20.236-20.914Q20.454-20.914 20.663-21.011Q20.871-21.109 20.999-21.290Q21.128-21.471 21.128-21.697L21.128-22.425Q20.803-22.425 20.437-22.334Q20.071-22.243 19.815-22.031Q19.559-21.820 19.559-21.488M24.617-20.760L22.881-20.760L22.881-21.040Q23.110-21.040 23.259-21.074Q23.407-21.109 23.407-21.249L23.407-23.098Q23.407-23.368 23.300-23.429Q23.192-23.491 22.881-23.491L22.881-23.771L23.910-23.846L23.910-23.139Q24.040-23.447 24.282-23.646Q24.525-23.846 24.843-23.846Q25.062-23.846 25.233-23.722Q25.404-23.597 25.404-23.385Q25.404-23.248 25.304-23.149Q25.205-23.050 25.072-23.050Q24.935-23.050 24.836-23.149Q24.737-23.248 24.737-23.385Q24.737-23.525 24.836-23.624Q24.546-23.624 24.346-23.428Q24.146-23.231 24.053-22.937Q23.961-22.643 23.961-22.363L23.961-21.249Q23.961-21.040 24.617-21.040L24.617-20.760M25.988-22.271Q25.988-22.609 26.128-22.900Q26.268-23.190 26.513-23.404Q26.757-23.617 27.061-23.732Q27.365-23.846 27.690-23.846Q27.960-23.846 28.223-23.747Q28.487-23.648 28.678-23.470L28.678-24.868Q28.678-25.138 28.570-25.200Q28.463-25.261 28.152-25.261L28.152-25.542L29.228-25.617L29.228-21.433Q29.228-21.245 29.283-21.162Q29.338-21.078 29.438-21.059Q29.539-21.040 29.755-21.040L29.755-20.760L28.647-20.692L28.647-21.109Q28.230-20.692 27.605-20.692Q27.174-20.692 26.801-20.904Q26.429-21.115 26.208-21.476Q25.988-21.837 25.988-22.271M27.663-20.914Q27.871-20.914 28.058-20.986Q28.244-21.057 28.398-21.194Q28.551-21.331 28.647-21.509L28.647-23.118Q28.562-23.265 28.416-23.385Q28.271-23.505 28.102-23.564Q27.933-23.624 27.752-23.624Q27.191-23.624 26.923-23.235Q26.654-22.845 26.654-22.264Q26.654-21.693 26.889-21.303Q27.123-20.914 27.663-20.914\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(31.694 46.427)\">\u003Cpath d=\"M38.276-21.567L33.443-21.567Q33.375-21.577 33.329-21.623Q33.283-21.669 33.283-21.741Q33.283-21.806 33.329-21.852Q33.375-21.898 33.443-21.908L38.276-21.908Q38.345-21.898 38.391-21.852Q38.437-21.806 38.437-21.741Q38.437-21.669 38.391-21.623Q38.345-21.577 38.276-21.567M38.276-23.105L33.443-23.105Q33.375-23.115 33.329-23.161Q33.283-23.207 33.283-23.279Q33.283-23.423 33.443-23.447L38.276-23.447Q38.437-23.423 38.437-23.279Q38.437-23.207 38.391-23.161Q38.345-23.115 38.276-23.105\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(31.694 46.427)\">\u003Cpath d=\"M42.748-20.760L42.481-20.760L42.481-24.868Q42.481-25.138 42.374-25.200Q42.266-25.261 41.955-25.261L41.955-25.542L43.035-25.617L43.035-23.447Q43.244-23.638 43.529-23.742Q43.815-23.846 44.112-23.846Q44.430-23.846 44.727-23.725Q45.024-23.604 45.247-23.388Q45.469-23.173 45.595-22.888Q45.722-22.602 45.722-22.271Q45.722-21.826 45.482-21.462Q45.243-21.098 44.850-20.895Q44.457-20.692 44.013-20.692Q43.818-20.692 43.628-20.748Q43.439-20.804 43.278-20.909Q43.117-21.013 42.977-21.174L42.748-20.760M43.063-23.105L43.063-21.488Q43.199-21.228 43.440-21.071Q43.681-20.914 43.958-20.914Q44.252-20.914 44.464-21.021Q44.676-21.129 44.809-21.321Q44.942-21.512 45.001-21.751Q45.059-21.990 45.059-22.271Q45.059-22.630 44.965-22.934Q44.871-23.238 44.643-23.431Q44.416-23.624 44.050-23.624Q43.750-23.624 43.483-23.488Q43.216-23.351 43.063-23.105\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(31.694 46.427)\">\u003Cpath d=\"M46.532-22.295Q46.532-22.616 46.657-22.905Q46.782-23.194 47.008-23.417Q47.233-23.641 47.529-23.761Q47.824-23.881 48.142-23.881Q48.470-23.881 48.732-23.781Q48.993-23.682 49.169-23.500Q49.345-23.317 49.439-23.059Q49.533-22.801 49.533-22.469Q49.533-22.377 49.451-22.356L47.196-22.356L47.196-22.295Q47.196-21.707 47.479-21.324Q47.763-20.941 48.330-20.941Q48.652-20.941 48.920-21.134Q49.188-21.327 49.277-21.642Q49.284-21.683 49.359-21.697L49.451-21.697Q49.533-21.673 49.533-21.601Q49.533-21.594 49.527-21.567Q49.414-21.170 49.043-20.931Q48.672-20.692 48.248-20.692Q47.811-20.692 47.411-20.900Q47.011-21.109 46.772-21.476Q46.532-21.843 46.532-22.295M47.202-22.565L49.017-22.565Q49.017-22.842 48.920-23.094Q48.822-23.347 48.624-23.503Q48.426-23.658 48.142-23.658Q47.865-23.658 47.652-23.500Q47.438-23.341 47.320-23.086Q47.202-22.831 47.202-22.565M50.648-21.601L50.648-23.498L50.009-23.498L50.009-23.720Q50.326-23.720 50.543-23.930Q50.760-24.140 50.861-24.450Q50.962-24.759 50.962-25.067L51.229-25.067L51.229-23.778L52.305-23.778L52.305-23.498L51.229-23.498L51.229-21.614Q51.229-21.338 51.333-21.139Q51.437-20.941 51.697-20.941Q51.854-20.941 51.960-21.045Q52.066-21.150 52.116-21.303Q52.165-21.457 52.165-21.614L52.165-22.028L52.432-22.028L52.432-21.601Q52.432-21.375 52.333-21.165Q52.234-20.955 52.049-20.823Q51.864-20.692 51.635-20.692Q51.198-20.692 50.923-20.929Q50.648-21.167 50.648-21.601M53.300-21.488Q53.300-21.820 53.524-22.047Q53.748-22.274 54.091-22.402Q54.435-22.531 54.807-22.583Q55.180-22.636 55.484-22.636L55.484-22.889Q55.484-23.094 55.376-23.274Q55.269-23.453 55.088-23.556Q54.906-23.658 54.698-23.658Q54.291-23.658 54.055-23.566Q54.144-23.529 54.190-23.445Q54.237-23.361 54.237-23.259Q54.237-23.163 54.190-23.084Q54.144-23.006 54.064-22.961Q53.984-22.917 53.895-22.917Q53.744-22.917 53.644-23.014Q53.543-23.112 53.543-23.259Q53.543-23.881 54.698-23.881Q54.910-23.881 55.159-23.817Q55.409-23.754 55.611-23.635Q55.812-23.515 55.939-23.330Q56.065-23.146 56.065-22.903L56.065-21.327Q56.065-21.211 56.127-21.115Q56.188-21.020 56.301-21.020Q56.410-21.020 56.475-21.114Q56.540-21.208 56.540-21.327L56.540-21.775L56.807-21.775L56.807-21.327Q56.807-21.057 56.580-20.892Q56.352-20.726 56.072-20.726Q55.863-20.726 55.727-20.880Q55.590-21.033 55.566-21.249Q55.419-20.982 55.137-20.837Q54.855-20.692 54.530-20.692Q54.254-20.692 53.970-20.767Q53.686-20.842 53.493-21.021Q53.300-21.201 53.300-21.488M53.915-21.488Q53.915-21.314 54.016-21.184Q54.117-21.054 54.272-20.984Q54.428-20.914 54.592-20.914Q54.811-20.914 55.019-21.011Q55.228-21.109 55.356-21.290Q55.484-21.471 55.484-21.697L55.484-22.425Q55.159-22.425 54.794-22.334Q54.428-22.243 54.172-22.031Q53.915-21.820 53.915-21.488\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(31.694 46.427)\">\u003Cpath d=\"M61.602-20.760L59.999-20.760L59.999-21.040Q60.225-21.040 60.374-21.074Q60.522-21.109 60.522-21.249L60.522-24.868Q60.522-25.138 60.415-25.200Q60.307-25.261 59.999-25.261L59.999-25.542L61.076-25.617L61.076-21.249Q61.076-21.112 61.226-21.076Q61.377-21.040 61.602-21.040L61.602-20.760M62.156-22.243Q62.156-22.585 62.291-22.884Q62.426-23.183 62.665-23.407Q62.905-23.631 63.223-23.756Q63.540-23.881 63.872-23.881Q64.316-23.881 64.716-23.665Q65.116-23.450 65.350-23.072Q65.584-22.695 65.584-22.243Q65.584-21.902 65.443-21.618Q65.301-21.334 65.056-21.127Q64.812-20.921 64.503-20.806Q64.193-20.692 63.872-20.692Q63.441-20.692 63.040-20.893Q62.638-21.095 62.397-21.447Q62.156-21.799 62.156-22.243M63.872-20.941Q64.474-20.941 64.697-21.319Q64.921-21.697 64.921-22.329Q64.921-22.941 64.687-23.300Q64.453-23.658 63.872-23.658Q62.819-23.658 62.819-22.329Q62.819-21.697 63.045-21.319Q63.270-20.941 63.872-20.941M66.138-20.227Q66.138-20.473 66.335-20.657Q66.531-20.842 66.787-20.921Q66.651-21.033 66.579-21.194Q66.507-21.355 66.507-21.536Q66.507-21.857 66.719-22.103Q66.384-22.401 66.384-22.811Q66.384-23.272 66.774-23.559Q67.163-23.846 67.642-23.846Q68.114-23.846 68.449-23.600Q68.623-23.754 68.833-23.836Q69.043-23.918 69.272-23.918Q69.436-23.918 69.558-23.811Q69.679-23.703 69.679-23.539Q69.679-23.443 69.607-23.371Q69.536-23.300 69.443-23.300Q69.344-23.300 69.274-23.373Q69.204-23.447 69.204-23.546Q69.204-23.600 69.218-23.631L69.224-23.645Q69.231-23.665 69.240-23.676Q69.248-23.686 69.252-23.693Q68.896-23.693 68.609-23.470Q68.896-23.177 68.896-22.811Q68.896-22.496 68.712-22.264Q68.527-22.031 68.238-21.903Q67.950-21.775 67.642-21.775Q67.440-21.775 67.249-21.825Q67.058-21.874 66.880-21.984Q66.787-21.857 66.787-21.714Q66.787-21.532 66.916-21.397Q67.044-21.262 67.228-21.262L67.861-21.262Q68.308-21.262 68.678-21.191Q69.047-21.119 69.307-20.890Q69.566-20.661 69.566-20.227Q69.566-19.906 69.271-19.704Q68.975-19.502 68.572-19.413Q68.168-19.324 67.854-19.324Q67.536-19.324 67.133-19.413Q66.729-19.502 66.434-19.704Q66.138-19.906 66.138-20.227M66.593-20.227Q66.593-19.998 66.811-19.849Q67.030-19.700 67.322-19.632Q67.615-19.564 67.854-19.564Q68.018-19.564 68.226-19.600Q68.435-19.635 68.642-19.716Q68.849-19.796 68.980-19.924Q69.112-20.052 69.112-20.227Q69.112-20.579 68.731-20.673Q68.349-20.767 67.847-20.767L67.228-20.767Q66.989-20.767 66.791-20.616Q66.593-20.466 66.593-20.227M67.642-22.014Q68.308-22.014 68.308-22.811Q68.308-23.611 67.642-23.611Q66.972-23.611 66.972-22.811Q66.972-22.014 67.642-22.014M72.290-19.010Q71.740-19.410 71.369-19.965Q70.998-20.521 70.817-21.167Q70.636-21.813 70.636-22.510Q70.636-23.023 70.737-23.518Q70.838-24.014 71.043-24.465Q71.248-24.916 71.561-25.308Q71.873-25.699 72.290-26.003Q72.301-26.007 72.308-26.008Q72.314-26.010 72.325-26.010L72.393-26.010Q72.427-26.010 72.449-25.986Q72.472-25.962 72.472-25.925Q72.472-25.880 72.444-25.863Q72.096-25.562 71.843-25.178Q71.590-24.793 71.438-24.352Q71.286-23.911 71.214-23.455Q71.142-22.999 71.142-22.510Q71.142-21.509 71.451-20.622Q71.761-19.735 72.444-19.150Q72.472-19.133 72.472-19.089Q72.472-19.051 72.449-19.027Q72.427-19.003 72.393-19.003L72.325-19.003Q72.318-19.007 72.309-19.008Q72.301-19.010 72.290-19.010M74.926-19.403L73.295-19.403L73.295-19.683Q73.524-19.683 73.673-19.718Q73.822-19.752 73.822-19.892L73.822-23.238Q73.822-23.409 73.685-23.450Q73.548-23.491 73.295-23.491L73.295-23.771L74.375-23.846L74.375-23.440Q74.598-23.641 74.885-23.744Q75.172-23.846 75.479-23.846Q75.907-23.846 76.271-23.633Q76.635-23.419 76.848-23.055Q77.062-22.691 77.062-22.271Q77.062-21.826 76.823-21.462Q76.583-21.098 76.190-20.895Q75.797-20.692 75.353-20.692Q75.086-20.692 74.839-20.792Q74.591-20.893 74.403-21.074L74.403-19.892Q74.403-19.755 74.551-19.719Q74.700-19.683 74.926-19.683L74.926-19.403M74.403-23.091L74.403-21.481Q74.536-21.228 74.779-21.071Q75.021-20.914 75.298-20.914Q75.626-20.914 75.879-21.115Q76.132-21.317 76.266-21.635Q76.399-21.953 76.399-22.271Q76.399-22.500 76.334-22.729Q76.269-22.958 76.141-23.156Q76.013-23.354 75.818-23.474Q75.623-23.593 75.391-23.593Q75.097-23.593 74.828-23.464Q74.560-23.334 74.403-23.091\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(31.694 46.427)\">\u003Cpath d=\"M77.886-22.243Q77.886-22.585 78.021-22.884Q78.156-23.183 78.396-23.407Q78.635-23.631 78.953-23.756Q79.271-23.881 79.602-23.881Q80.047-23.881 80.446-23.665Q80.846-23.450 81.081-23.072Q81.315-22.695 81.315-22.243Q81.315-21.902 81.173-21.618Q81.031-21.334 80.787-21.127Q80.542-20.921 80.233-20.806Q79.924-20.692 79.602-20.692Q79.172-20.692 78.770-20.893Q78.368-21.095 78.127-21.447Q77.886-21.799 77.886-22.243M79.602-20.941Q80.204-20.941 80.428-21.319Q80.652-21.697 80.652-22.329Q80.652-22.941 80.417-23.300Q80.183-23.658 79.602-23.658Q78.550-23.658 78.550-22.329Q78.550-21.697 78.775-21.319Q79.001-20.941 79.602-20.941M83.577-20.760L81.974-20.760L81.974-21.040Q82.200-21.040 82.349-21.074Q82.497-21.109 82.497-21.249L82.497-24.868Q82.497-25.138 82.390-25.200Q82.282-25.261 81.974-25.261L81.974-25.542L83.051-25.617L83.051-21.249Q83.051-21.112 83.201-21.076Q83.352-21.040 83.577-21.040L83.577-20.760M85.789-20.760L84.237-20.760L84.237-21.040Q84.463-21.040 84.611-21.074Q84.760-21.109 84.760-21.249L84.760-23.098Q84.760-23.286 84.712-23.370Q84.664-23.453 84.567-23.472Q84.469-23.491 84.258-23.491L84.258-23.771L85.314-23.846L85.314-21.249Q85.314-21.109 85.445-21.074Q85.577-21.040 85.789-21.040L85.789-20.760M84.517-25.067Q84.517-25.238 84.640-25.357Q84.763-25.477 84.934-25.477Q85.102-25.477 85.225-25.357Q85.348-25.238 85.348-25.067Q85.348-24.892 85.225-24.769Q85.102-24.646 84.934-24.646Q84.763-24.646 84.640-24.769Q84.517-24.892 84.517-25.067M86.435-22.271Q86.435-22.599 86.570-22.900Q86.705-23.200 86.941-23.421Q87.176-23.641 87.481-23.761Q87.785-23.881 88.110-23.881Q88.615-23.881 88.964-23.778Q89.313-23.676 89.313-23.300Q89.313-23.153 89.215-23.052Q89.118-22.951 88.971-22.951Q88.817-22.951 88.718-23.050Q88.619-23.149 88.619-23.300Q88.619-23.488 88.759-23.580Q88.557-23.631 88.116-23.631Q87.761-23.631 87.532-23.435Q87.303-23.238 87.202-22.929Q87.101-22.619 87.101-22.271Q87.101-21.922 87.228-21.616Q87.354-21.310 87.609-21.126Q87.863-20.941 88.219-20.941Q88.441-20.941 88.626-21.025Q88.810-21.109 88.945-21.264Q89.080-21.420 89.138-21.628Q89.152-21.683 89.207-21.683L89.320-21.683Q89.350-21.683 89.373-21.659Q89.395-21.635 89.395-21.601L89.395-21.580Q89.309-21.293 89.121-21.095Q88.933-20.897 88.668-20.794Q88.404-20.692 88.110-20.692Q87.679-20.692 87.291-20.898Q86.903-21.105 86.669-21.468Q86.435-21.830 86.435-22.271M90.318-19.625Q90.447-19.557 90.584-19.557Q90.755-19.557 90.905-19.646Q91.056-19.735 91.167-19.880Q91.278-20.025 91.357-20.193L91.620-20.760L90.451-23.286Q90.376-23.433 90.246-23.465Q90.116-23.498 89.884-23.498L89.884-23.778L91.405-23.778L91.405-23.498Q91.056-23.498 91.056-23.351Q91.059-23.330 91.061-23.313Q91.063-23.296 91.063-23.286L91.921-21.427L92.693-23.098Q92.727-23.166 92.727-23.245Q92.727-23.358 92.644-23.428Q92.560-23.498 92.447-23.498L92.447-23.778L93.643-23.778L93.643-23.498Q93.425-23.498 93.252-23.394Q93.079-23.289 92.987-23.098L91.651-20.193Q91.480-19.823 91.210-19.577Q90.940-19.331 90.584-19.331Q90.314-19.331 90.095-19.497Q89.877-19.663 89.877-19.926Q89.877-20.063 89.969-20.152Q90.061-20.240 90.201-20.240Q90.338-20.240 90.427-20.152Q90.516-20.063 90.516-19.926Q90.516-19.823 90.463-19.745Q90.410-19.666 90.318-19.625\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(31.694 46.427)\">\u003Cpath d=\"M97.068-19.177Q97.068-19.195 97.081-19.242L99.737-25.904Q99.792-26.010 99.898-26.010Q99.963-26.010 100.014-25.959Q100.065-25.907 100.065-25.843Q100.065-25.819 100.064-25.807Q100.062-25.795 100.058-25.778L97.406-19.116Q97.334-19.010 97.246-19.010Q97.177-19.010 97.122-19.061Q97.068-19.113 97.068-19.177\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(31.694 46.427)\">\u003Cpath d=\"M105.323-20.760L103.587-20.760L103.587-21.040Q103.816-21.040 103.965-21.074Q104.113-21.109 104.113-21.249L104.113-23.098Q104.113-23.368 104.006-23.429Q103.898-23.491 103.587-23.491L103.587-23.771L104.616-23.846L104.616-23.139Q104.746-23.447 104.988-23.646Q105.231-23.846 105.549-23.846Q105.768-23.846 105.939-23.722Q106.110-23.597 106.110-23.385Q106.110-23.248 106.010-23.149Q105.911-23.050 105.778-23.050Q105.641-23.050 105.542-23.149Q105.443-23.248 105.443-23.385Q105.443-23.525 105.542-23.624Q105.252-23.624 105.052-23.428Q104.852-23.231 104.759-22.937Q104.667-22.643 104.667-22.363L104.667-21.249Q104.667-21.040 105.323-21.040L105.323-20.760M106.653-22.295Q106.653-22.616 106.778-22.905Q106.903-23.194 107.128-23.417Q107.354-23.641 107.649-23.761Q107.945-23.881 108.263-23.881Q108.591-23.881 108.853-23.781Q109.114-23.682 109.290-23.500Q109.466-23.317 109.560-23.059Q109.654-22.801 109.654-22.469Q109.654-22.377 109.572-22.356L107.316-22.356L107.316-22.295Q107.316-21.707 107.600-21.324Q107.884-20.941 108.451-20.941Q108.772-20.941 109.040-21.134Q109.309-21.327 109.398-21.642Q109.405-21.683 109.480-21.697L109.572-21.697Q109.654-21.673 109.654-21.601Q109.654-21.594 109.647-21.567Q109.534-21.170 109.164-20.931Q108.793-20.692 108.369-20.692Q107.931-20.692 107.531-20.900Q107.132-21.109 106.892-21.476Q106.653-21.843 106.653-22.295M107.323-22.565L109.138-22.565Q109.138-22.842 109.040-23.094Q108.943-23.347 108.745-23.503Q108.547-23.658 108.263-23.658Q107.986-23.658 107.772-23.500Q107.559-23.341 107.441-23.086Q107.323-22.831 107.323-22.565M112.040-20.760L110.307-20.760L110.307-21.040Q110.532-21.040 110.681-21.074Q110.830-21.109 110.830-21.249L110.830-23.498L110.242-23.498L110.242-23.778L110.830-23.778L110.830-24.595Q110.830-24.913 111.008-25.161Q111.185-25.408 111.476-25.549Q111.766-25.689 112.077-25.689Q112.334-25.689 112.537-25.547Q112.740-25.405 112.740-25.162Q112.740-25.026 112.641-24.927Q112.542-24.827 112.405-24.827Q112.269-24.827 112.170-24.927Q112.071-25.026 112.071-25.162Q112.071-25.343 112.211-25.436Q112.132-25.463 112.033-25.463Q111.824-25.463 111.671-25.330Q111.517-25.197 111.436-24.993Q111.356-24.790 111.356-24.581L111.356-23.778L112.245-23.778L112.245-23.498L111.384-23.498L111.384-21.249Q111.384-21.040 112.040-21.040\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(31.694 46.427)\">\u003Cpath d=\"M113.614-19.003L113.545-19.003Q113.511-19.003 113.489-19.029Q113.467-19.054 113.467-19.089Q113.467-19.133 113.498-19.150Q113.853-19.454 114.103-19.844Q114.352-20.234 114.504-20.666Q114.656-21.098 114.726-21.567Q114.796-22.035 114.796-22.510Q114.796-22.989 114.726-23.455Q114.656-23.922 114.502-24.357Q114.349-24.793 114.097-25.181Q113.846-25.569 113.498-25.863Q113.467-25.880 113.467-25.925Q113.467-25.959 113.489-25.984Q113.511-26.010 113.545-26.010L113.614-26.010Q113.624-26.010 113.633-26.008Q113.641-26.007 113.651-26.003Q114.195-25.603 114.567-25.050Q114.940-24.496 115.121-23.850Q115.302-23.204 115.302-22.510Q115.302-21.809 115.121-21.162Q114.940-20.514 114.566-19.960Q114.191-19.406 113.651-19.010Q113.641-19.010 113.633-19.008Q113.624-19.007 113.614-19.003\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(170.339 41.115)\">\u003Cpath d=\"M-16.514-20.760L-19.286-20.760L-19.286-21.040Q-18.565-21.040-18.565-21.249L-18.565-25.050Q-18.565-25.261-19.286-25.261L-19.286-25.542L-16.514-25.542Q-16.029-25.542-15.593-25.347Q-15.157-25.152-14.834-24.810Q-14.511-24.468-14.333-24.028Q-14.156-23.587-14.156-23.105Q-14.156-22.619-14.340-22.196Q-14.525-21.772-14.848-21.450Q-15.171-21.129-15.603-20.945Q-16.035-20.760-16.514-20.760M-17.902-25.050L-17.902-21.249Q-17.902-21.109-17.813-21.074Q-17.724-21.040-17.536-21.040L-16.712-21.040Q-16.087-21.040-15.683-21.302Q-15.280-21.563-15.092-22.030Q-14.904-22.496-14.904-23.105Q-14.904-23.583-14.996-23.976Q-15.089-24.369-15.338-24.667Q-15.577-24.954-15.941-25.108Q-16.305-25.261-16.712-25.261L-17.536-25.261Q-17.724-25.261-17.813-25.227Q-17.902-25.193-17.902-25.050M-11.134-20.760L-13.267-20.760L-13.267-21.040Q-12.546-21.040-12.546-21.249L-12.546-25.050Q-12.546-25.261-13.267-25.261L-13.267-25.542L-10.601-25.542Q-10.191-25.542-9.770-25.388Q-9.350-25.234-9.066-24.930Q-8.783-24.626-8.783-24.212Q-8.783-23.894-8.950-23.648Q-9.117-23.402-9.394-23.236Q-9.671-23.071-9.992-22.987Q-10.314-22.903-10.601-22.903L-11.855-22.903L-11.855-21.249Q-11.855-21.040-11.134-21.040L-11.134-20.760M-11.883-25.050L-11.883-23.153L-10.796-23.153Q-10.187-23.153-9.873-23.390Q-9.558-23.628-9.558-24.212Q-9.558-24.605-9.704-24.839Q-9.849-25.073-10.121-25.167Q-10.392-25.261-10.796-25.261L-11.517-25.261Q-11.705-25.261-11.794-25.227Q-11.883-25.193-11.883-25.050M-7.802-23.125Q-7.802-23.812-7.453-24.402Q-7.104-24.991-6.510-25.337Q-5.915-25.682-5.224-25.682Q-4.534-25.682-3.939-25.337Q-3.345-24.991-2.996-24.402Q-2.647-23.812-2.647-23.125Q-2.647-22.438-2.999-21.864Q-3.351-21.290-3.944-20.955Q-4.537-20.620-5.224-20.620Q-5.737-20.620-6.205-20.813Q-6.674-21.006-7.033-21.348Q-7.391-21.690-7.596-22.151Q-7.802-22.613-7.802-23.125M-5.224-20.880Q-4.619-20.880-4.213-21.225Q-3.806-21.570-3.616-22.108Q-3.427-22.647-3.427-23.245Q-3.427-23.679-3.538-24.067Q-3.649-24.455-3.874-24.762Q-4.100-25.070-4.442-25.250Q-4.783-25.429-5.224-25.429Q-5.662-25.429-6-25.253Q-6.339-25.077-6.568-24.762Q-6.797-24.448-6.909-24.055Q-7.022-23.662-7.022-23.245Q-7.022-22.657-6.831-22.114Q-6.639-21.570-6.233-21.225Q-5.826-20.880-5.224-20.880\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(170.339 41.115)\">\u003Cpath d=\"M1.391-21.601L1.391-23.498L0.752-23.498L0.752-23.720Q1.070-23.720 1.287-23.930Q1.504-24.140 1.604-24.450Q1.705-24.759 1.705-25.067L1.972-25.067L1.972-23.778L3.049-23.778L3.049-23.498L1.972-23.498L1.972-21.614Q1.972-21.338 2.076-21.139Q2.180-20.941 2.440-20.941Q2.597-20.941 2.703-21.045Q2.809-21.150 2.859-21.303Q2.908-21.457 2.908-21.614L2.908-22.028L3.175-22.028L3.175-21.601Q3.175-21.375 3.076-21.165Q2.977-20.955 2.792-20.823Q2.608-20.692 2.379-20.692Q1.941-20.692 1.666-20.929Q1.391-21.167 1.391-21.601M5.735-20.760L3.999-20.760L3.999-21.040Q4.228-21.040 4.376-21.074Q4.525-21.109 4.525-21.249L4.525-23.098Q4.525-23.368 4.417-23.429Q4.310-23.491 3.999-23.491L3.999-23.771L5.028-23.846L5.028-23.139Q5.157-23.447 5.400-23.646Q5.643-23.846 5.961-23.846Q6.179-23.846 6.350-23.722Q6.521-23.597 6.521-23.385Q6.521-23.248 6.422-23.149Q6.323-23.050 6.190-23.050Q6.053-23.050 5.954-23.149Q5.855-23.248 5.855-23.385Q5.855-23.525 5.954-23.624Q5.663-23.624 5.463-23.428Q5.263-23.231 5.171-22.937Q5.079-22.643 5.079-22.363L5.079-21.249Q5.079-21.040 5.735-21.040L5.735-20.760M7.164-21.488Q7.164-21.820 7.388-22.047Q7.612-22.274 7.955-22.402Q8.299-22.531 8.671-22.583Q9.044-22.636 9.348-22.636L9.348-22.889Q9.348-23.094 9.240-23.274Q9.133-23.453 8.951-23.556Q8.770-23.658 8.562-23.658Q8.155-23.658 7.919-23.566Q8.008-23.529 8.054-23.445Q8.100-23.361 8.100-23.259Q8.100-23.163 8.054-23.084Q8.008-23.006 7.928-22.961Q7.847-22.917 7.758-22.917Q7.608-22.917 7.507-23.014Q7.406-23.112 7.406-23.259Q7.406-23.881 8.562-23.881Q8.774-23.881 9.023-23.817Q9.273-23.754 9.474-23.635Q9.676-23.515 9.802-23.330Q9.929-23.146 9.929-22.903L9.929-21.327Q9.929-21.211 9.990-21.115Q10.052-21.020 10.165-21.020Q10.274-21.020 10.339-21.114Q10.404-21.208 10.404-21.327L10.404-21.775L10.671-21.775L10.671-21.327Q10.671-21.057 10.443-20.892Q10.216-20.726 9.936-20.726Q9.727-20.726 9.591-20.880Q9.454-21.033 9.430-21.249Q9.283-20.982 9.001-20.837Q8.719-20.692 8.394-20.692Q8.117-20.692 7.834-20.767Q7.550-20.842 7.357-21.021Q7.164-21.201 7.164-21.488M7.779-21.488Q7.779-21.314 7.880-21.184Q7.981-21.054 8.136-20.984Q8.292-20.914 8.456-20.914Q8.675-20.914 8.883-21.011Q9.092-21.109 9.220-21.290Q9.348-21.471 9.348-21.697L9.348-22.425Q9.023-22.425 8.657-22.334Q8.292-22.243 8.035-22.031Q7.779-21.820 7.779-21.488M12.704-20.760L11.153-20.760L11.153-21.040Q11.378-21.040 11.527-21.074Q11.675-21.109 11.675-21.249L11.675-23.098Q11.675-23.286 11.628-23.370Q11.580-23.453 11.482-23.472Q11.385-23.491 11.173-23.491L11.173-23.771L12.229-23.846L12.229-21.249Q12.229-21.109 12.361-21.074Q12.492-21.040 12.704-21.040L12.704-20.760M11.433-25.067Q11.433-25.238 11.556-25.357Q11.679-25.477 11.850-25.477Q12.017-25.477 12.140-25.357Q12.263-25.238 12.263-25.067Q12.263-24.892 12.140-24.769Q12.017-24.646 11.850-24.646Q11.679-24.646 11.556-24.769Q11.433-24.892 11.433-25.067M15.032-20.760L13.398-20.760L13.398-21.040Q13.627-21.040 13.776-21.074Q13.925-21.109 13.925-21.249L13.925-23.098Q13.925-23.368 13.817-23.429Q13.709-23.491 13.398-23.491L13.398-23.771L14.458-23.846L14.458-23.197Q14.629-23.505 14.933-23.676Q15.237-23.846 15.582-23.846Q16.088-23.846 16.372-23.623Q16.655-23.399 16.655-22.903L16.655-21.249Q16.655-21.112 16.804-21.076Q16.953-21.040 17.178-21.040L17.178-20.760L15.548-20.760L15.548-21.040Q15.777-21.040 15.926-21.074Q16.074-21.109 16.074-21.249L16.074-22.889Q16.074-23.224 15.955-23.424Q15.835-23.624 15.521-23.624Q15.251-23.624 15.017-23.488Q14.782-23.351 14.644-23.117Q14.506-22.883 14.506-22.609L14.506-21.249Q14.506-21.112 14.656-21.076Q14.806-21.040 15.032-21.040L15.032-20.760M17.766-20.767L17.766-21.830Q17.766-21.854 17.794-21.881Q17.821-21.908 17.845-21.908L17.954-21.908Q18.019-21.908 18.033-21.850Q18.129-21.416 18.375-21.165Q18.621-20.914 19.034-20.914Q19.376-20.914 19.629-21.047Q19.882-21.180 19.882-21.488Q19.882-21.645 19.788-21.760Q19.694-21.874 19.556-21.943Q19.417-22.011 19.250-22.049L18.669-22.148Q18.313-22.216 18.040-22.437Q17.766-22.657 17.766-22.999Q17.766-23.248 17.877-23.423Q17.988-23.597 18.175-23.696Q18.361-23.795 18.576-23.838Q18.792-23.881 19.034-23.881Q19.448-23.881 19.728-23.699L19.944-23.874Q19.954-23.877 19.961-23.879Q19.967-23.881 19.978-23.881L20.029-23.881Q20.056-23.881 20.080-23.857Q20.104-23.833 20.104-23.805L20.104-22.958Q20.104-22.937 20.080-22.910Q20.056-22.883 20.029-22.883L19.916-22.883Q19.889-22.883 19.863-22.908Q19.838-22.934 19.838-22.958Q19.838-23.194 19.732-23.358Q19.626-23.522 19.443-23.604Q19.260-23.686 19.028-23.686Q18.699-23.686 18.443-23.583Q18.187-23.481 18.187-23.204Q18.187-23.009 18.370-22.900Q18.552-22.790 18.781-22.749L19.356-22.643Q19.602-22.595 19.815-22.467Q20.029-22.339 20.166-22.136Q20.302-21.932 20.302-21.683Q20.302-21.170 19.937-20.931Q19.571-20.692 19.034-20.692Q18.539-20.692 18.207-20.986L17.941-20.712Q17.920-20.692 17.893-20.692L17.845-20.692Q17.821-20.692 17.794-20.719Q17.766-20.746 17.766-20.767\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(170.339 41.115)\">\u003Cpath d=\"M25.326-20.760L23.692-20.760L23.692-21.040Q23.921-21.040 24.070-21.074Q24.219-21.109 24.219-21.249L24.219-24.868Q24.219-25.138 24.111-25.200Q24.003-25.261 23.692-25.261L23.692-25.542L24.772-25.617L24.772-23.231Q24.878-23.416 25.056-23.558Q25.234-23.699 25.442-23.773Q25.651-23.846 25.876-23.846Q26.382-23.846 26.666-23.623Q26.950-23.399 26.950-22.903L26.950-21.249Q26.950-21.112 27.098-21.076Q27.247-21.040 27.473-21.040L27.473-20.760L25.842-20.760L25.842-21.040Q26.071-21.040 26.220-21.074Q26.369-21.109 26.369-21.249L26.369-22.889Q26.369-23.224 26.249-23.424Q26.129-23.624 25.815-23.624Q25.545-23.624 25.311-23.488Q25.077-23.351 24.938-23.117Q24.800-22.883 24.800-22.609L24.800-21.249Q24.800-21.112 24.950-21.076Q25.101-21.040 25.326-21.040L25.326-20.760M28.019-22.295Q28.019-22.616 28.144-22.905Q28.269-23.194 28.495-23.417Q28.720-23.641 29.016-23.761Q29.311-23.881 29.629-23.881Q29.957-23.881 30.219-23.781Q30.480-23.682 30.656-23.500Q30.832-23.317 30.926-23.059Q31.020-22.801 31.020-22.469Q31.020-22.377 30.938-22.356L28.683-22.356L28.683-22.295Q28.683-21.707 28.966-21.324Q29.250-20.941 29.817-20.941Q30.139-20.941 30.407-21.134Q30.675-21.327 30.764-21.642Q30.771-21.683 30.846-21.697L30.938-21.697Q31.020-21.673 31.020-21.601Q31.020-21.594 31.014-21.567Q30.901-21.170 30.530-20.931Q30.159-20.692 29.735-20.692Q29.298-20.692 28.898-20.900Q28.498-21.109 28.259-21.476Q28.019-21.843 28.019-22.295M28.689-22.565L30.504-22.565Q30.504-22.842 30.407-23.094Q30.309-23.347 30.111-23.503Q29.913-23.658 29.629-23.658Q29.352-23.658 29.139-23.500Q28.925-23.341 28.807-23.086Q28.689-22.831 28.689-22.565M33.358-20.760L31.622-20.760L31.622-21.040Q31.851-21.040 32-21.074Q32.148-21.109 32.148-21.249L32.148-23.098Q32.148-23.368 32.041-23.429Q31.933-23.491 31.622-23.491L31.622-23.771L32.651-23.846L32.651-23.139Q32.781-23.447 33.023-23.646Q33.266-23.846 33.584-23.846Q33.803-23.846 33.974-23.722Q34.144-23.597 34.144-23.385Q34.144-23.248 34.045-23.149Q33.946-23.050 33.813-23.050Q33.676-23.050 33.577-23.149Q33.478-23.248 33.478-23.385Q33.478-23.525 33.577-23.624Q33.287-23.624 33.087-23.428Q32.887-23.231 32.794-22.937Q32.702-22.643 32.702-22.363L32.702-21.249Q32.702-21.040 33.358-21.040L33.358-20.760M34.688-22.295Q34.688-22.616 34.813-22.905Q34.937-23.194 35.163-23.417Q35.389-23.641 35.684-23.761Q35.980-23.881 36.298-23.881Q36.626-23.881 36.887-23.781Q37.149-23.682 37.325-23.500Q37.501-23.317 37.595-23.059Q37.689-22.801 37.689-22.469Q37.689-22.377 37.607-22.356L35.351-22.356L35.351-22.295Q35.351-21.707 35.635-21.324Q35.918-20.941 36.486-20.941Q36.807-20.941 37.075-21.134Q37.344-21.327 37.433-21.642Q37.439-21.683 37.515-21.697L37.607-21.697Q37.689-21.673 37.689-21.601Q37.689-21.594 37.682-21.567Q37.569-21.170 37.198-20.931Q36.828-20.692 36.404-20.692Q35.966-20.692 35.566-20.900Q35.166-21.109 34.927-21.476Q34.688-21.843 34.688-22.295M35.358-22.565L37.173-22.565Q37.173-22.842 37.075-23.094Q36.978-23.347 36.780-23.503Q36.581-23.658 36.298-23.658Q36.021-23.658 35.807-23.500Q35.594-23.341 35.476-23.086Q35.358-22.831 35.358-22.565\" 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 DPO change of variable. The closed-form optimum maps a reward to a policy (\u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.6887em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6887em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">∗\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∝\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord text mtight\">\u003Cspan class=\"mord mtight\">ref\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">exp\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"mord\">\u002F\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>); taking logs inverts it, mapping the policy back to a reward (\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:0.0278em;\">r\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:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mord mathrm\" style=\"margin-right:0.0139em;\">log\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"mord\">\u002F\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">π\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3361em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord text mtight\">\u003Cspan class=\"mord mtight\">ref\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\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:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mord mathrm\" style=\"margin-right:0.0139em;\">log\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0715em;\">Z\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>). DPO fits the policy side, so the reward is never instantiated.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:439.469px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 329.602 108.634\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-56.655 23.53h190.879\"\u002F>\u003Cpath stroke=\"none\" d=\"m136.823 23.53-4.16-2.079 1.56 2.08-1.56 2.08\"\u002F>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(191.521 1.75)\">\u003Cpath d=\"M-48.901 23.531L-50.637 23.531L-50.637 23.251Q-50.408 23.251-50.259 23.217Q-50.111 23.182-50.111 23.042L-50.111 21.193Q-50.111 20.923-50.218 20.862Q-50.326 20.800-50.637 20.800L-50.637 20.520L-49.608 20.445L-49.608 21.152Q-49.478 20.844-49.236 20.645Q-48.993 20.445-48.675 20.445Q-48.456 20.445-48.285 20.569Q-48.114 20.694-48.114 20.906Q-48.114 21.043-48.214 21.142Q-48.313 21.241-48.446 21.241Q-48.583 21.241-48.682 21.142Q-48.781 21.043-48.781 20.906Q-48.781 20.766-48.682 20.667Q-48.972 20.667-49.172 20.863Q-49.372 21.060-49.465 21.354Q-49.557 21.648-49.557 21.928L-49.557 23.042Q-49.557 23.251-48.901 23.251L-48.901 23.531M-47.571 21.996Q-47.571 21.675-47.446 21.386Q-47.321 21.097-47.096 20.874Q-46.870 20.650-46.575 20.530Q-46.279 20.410-45.961 20.410Q-45.633 20.410-45.371 20.510Q-45.110 20.609-44.934 20.791Q-44.758 20.974-44.664 21.232Q-44.570 21.490-44.570 21.822Q-44.570 21.914-44.652 21.935L-46.908 21.935L-46.908 21.996Q-46.908 22.584-46.624 22.967Q-46.340 23.350-45.773 23.350Q-45.452 23.350-45.184 23.157Q-44.915 22.964-44.826 22.649Q-44.819 22.608-44.744 22.594L-44.652 22.594Q-44.570 22.618-44.570 22.690Q-44.570 22.697-44.577 22.724Q-44.690 23.121-45.060 23.360Q-45.431 23.599-45.855 23.599Q-46.293 23.599-46.693 23.391Q-47.092 23.182-47.332 22.815Q-47.571 22.448-47.571 21.996M-46.901 21.726L-45.086 21.726Q-45.086 21.449-45.184 21.197Q-45.281 20.944-45.479 20.788Q-45.677 20.633-45.961 20.633Q-46.238 20.633-46.452 20.791Q-46.665 20.950-46.783 21.205Q-46.901 21.460-46.901 21.726M-43.982 23.524L-43.982 22.461Q-43.982 22.437-43.955 22.410Q-43.927 22.383-43.903 22.383L-43.794 22.383Q-43.729 22.383-43.715 22.441Q-43.620 22.875-43.374 23.126Q-43.128 23.377-42.714 23.377Q-42.372 23.377-42.119 23.244Q-41.866 23.111-41.866 22.803Q-41.866 22.646-41.960 22.531Q-42.054 22.417-42.193 22.348Q-42.331 22.280-42.499 22.242L-43.080 22.143Q-43.435 22.075-43.709 21.854Q-43.982 21.634-43.982 21.292Q-43.982 21.043-43.871 20.868Q-43.760 20.694-43.574 20.595Q-43.387 20.496-43.172 20.453Q-42.957 20.410-42.714 20.410Q-42.300 20.410-42.020 20.592L-41.805 20.417Q-41.795 20.414-41.788 20.412Q-41.781 20.410-41.771 20.410L-41.719 20.410Q-41.692 20.410-41.668 20.434Q-41.644 20.458-41.644 20.486L-41.644 21.333Q-41.644 21.354-41.668 21.381Q-41.692 21.408-41.719 21.408L-41.832 21.408Q-41.860 21.408-41.885 21.383Q-41.911 21.357-41.911 21.333Q-41.911 21.097-42.017 20.933Q-42.123 20.769-42.306 20.687Q-42.488 20.605-42.721 20.605Q-43.049 20.605-43.305 20.708Q-43.562 20.810-43.562 21.087Q-43.562 21.282-43.379 21.391Q-43.196 21.501-42.967 21.542L-42.393 21.648Q-42.147 21.696-41.933 21.824Q-41.719 21.952-41.583 22.155Q-41.446 22.359-41.446 22.608Q-41.446 23.121-41.812 23.360Q-42.177 23.599-42.714 23.599Q-43.210 23.599-43.541 23.305L-43.808 23.579Q-43.828 23.599-43.856 23.599L-43.903 23.599Q-43.927 23.599-43.955 23.572Q-43.982 23.545-43.982 23.524M-39.173 24.888L-40.803 24.888L-40.803 24.608Q-40.574 24.608-40.426 24.573Q-40.277 24.539-40.277 24.399L-40.277 21.053Q-40.277 20.882-40.414 20.841Q-40.550 20.800-40.803 20.800L-40.803 20.520L-39.723 20.445L-39.723 20.851Q-39.501 20.650-39.214 20.547Q-38.927 20.445-38.619 20.445Q-38.192 20.445-37.828 20.658Q-37.464 20.872-37.250 21.236Q-37.037 21.600-37.037 22.020Q-37.037 22.465-37.276 22.829Q-37.515 23.193-37.908 23.396Q-38.301 23.599-38.746 23.599Q-39.012 23.599-39.260 23.499Q-39.508 23.398-39.696 23.217L-39.696 24.399Q-39.696 24.536-39.547 24.572Q-39.399 24.608-39.173 24.608L-39.173 24.888M-39.696 21.200L-39.696 22.810Q-39.563 23.063-39.320 23.220Q-39.077 23.377-38.800 23.377Q-38.472 23.377-38.219 23.176Q-37.966 22.974-37.833 22.656Q-37.700 22.338-37.700 22.020Q-37.700 21.791-37.765 21.562Q-37.830 21.333-37.958 21.135Q-38.086 20.937-38.281 20.817Q-38.476 20.698-38.708 20.698Q-39.002 20.698-39.270 20.827Q-39.539 20.957-39.696 21.200\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(191.521 1.75)\">\u003Cpath d=\"M-36.211 22.048Q-36.211 21.706-36.076 21.407Q-35.941 21.108-35.701 20.884Q-35.462 20.660-35.144 20.535Q-34.826 20.410-34.495 20.410Q-34.050 20.410-33.651 20.626Q-33.251 20.841-33.016 21.219Q-32.782 21.596-32.782 22.048Q-32.782 22.389-32.924 22.673Q-33.066 22.957-33.310 23.164Q-33.555 23.370-33.864 23.485Q-34.173 23.599-34.495 23.599Q-34.925 23.599-35.327 23.398Q-35.729 23.196-35.970 22.844Q-36.211 22.492-36.211 22.048M-34.495 23.350Q-33.893 23.350-33.669 22.972Q-33.445 22.594-33.445 21.962Q-33.445 21.350-33.680 20.991Q-33.914 20.633-34.495 20.633Q-35.547 20.633-35.547 21.962Q-35.547 22.594-35.322 22.972Q-35.096 23.350-34.495 23.350M-30.506 23.531L-32.140 23.531L-32.140 23.251Q-31.911 23.251-31.762 23.217Q-31.613 23.182-31.613 23.042L-31.613 21.193Q-31.613 20.923-31.721 20.862Q-31.829 20.800-32.140 20.800L-32.140 20.520L-31.080 20.445L-31.080 21.094Q-30.909 20.786-30.605 20.615Q-30.301 20.445-29.956 20.445Q-29.450 20.445-29.166 20.668Q-28.882 20.892-28.882 21.388L-28.882 23.042Q-28.882 23.179-28.734 23.215Q-28.585 23.251-28.359 23.251L-28.359 23.531L-29.990 23.531L-29.990 23.251Q-29.761 23.251-29.612 23.217Q-29.463 23.182-29.463 23.042L-29.463 21.402Q-29.463 21.067-29.583 20.867Q-29.703 20.667-30.017 20.667Q-30.287 20.667-30.521 20.803Q-30.755 20.940-30.894 21.174Q-31.032 21.408-31.032 21.682L-31.032 23.042Q-31.032 23.179-30.882 23.215Q-30.732 23.251-30.506 23.251L-30.506 23.531M-27.772 23.524L-27.772 22.461Q-27.772 22.437-27.744 22.410Q-27.717 22.383-27.693 22.383L-27.584 22.383Q-27.519 22.383-27.505 22.441Q-27.409 22.875-27.163 23.126Q-26.917 23.377-26.504 23.377Q-26.162 23.377-25.909 23.244Q-25.656 23.111-25.656 22.803Q-25.656 22.646-25.750 22.531Q-25.844 22.417-25.982 22.348Q-26.121 22.280-26.288 22.242L-26.869 22.143Q-27.225 22.075-27.498 21.854Q-27.772 21.634-27.772 21.292Q-27.772 21.043-27.661 20.868Q-27.549 20.694-27.363 20.595Q-27.177 20.496-26.962 20.453Q-26.746 20.410-26.504 20.410Q-26.090 20.410-25.810 20.592L-25.594 20.417Q-25.584 20.414-25.577 20.412Q-25.570 20.410-25.560 20.410L-25.509 20.410Q-25.482 20.410-25.458 20.434Q-25.434 20.458-25.434 20.486L-25.434 21.333Q-25.434 21.354-25.458 21.381Q-25.482 21.408-25.509 21.408L-25.622 21.408Q-25.649 21.408-25.675 21.383Q-25.700 21.357-25.700 21.333Q-25.700 21.097-25.806 20.933Q-25.912 20.769-26.095 20.687Q-26.278 20.605-26.510 20.605Q-26.838 20.605-27.095 20.708Q-27.351 20.810-27.351 21.087Q-27.351 21.282-27.168 21.391Q-26.985 21.501-26.756 21.542L-26.182 21.648Q-25.936 21.696-25.723 21.824Q-25.509 21.952-25.372 22.155Q-25.235 22.359-25.235 22.608Q-25.235 23.121-25.601 23.360Q-25.967 23.599-26.504 23.599Q-26.999 23.599-27.331 23.305L-27.597 23.579Q-27.618 23.599-27.645 23.599L-27.693 23.599Q-27.717 23.599-27.744 23.572Q-27.772 23.545-27.772 23.524M-24.648 21.996Q-24.648 21.675-24.523 21.386Q-24.398 21.097-24.172 20.874Q-23.947 20.650-23.651 20.530Q-23.356 20.410-23.038 20.410Q-22.710 20.410-22.448 20.510Q-22.187 20.609-22.011 20.791Q-21.835 20.974-21.741 21.232Q-21.647 21.490-21.647 21.822Q-21.647 21.914-21.729 21.935L-23.984 21.935L-23.984 21.996Q-23.984 22.584-23.701 22.967Q-23.417 23.350-22.850 23.350Q-22.528 23.350-22.260 23.157Q-21.992 22.964-21.903 22.649Q-21.896 22.608-21.821 22.594L-21.729 22.594Q-21.647 22.618-21.647 22.690Q-21.647 22.697-21.653 22.724Q-21.766 23.121-22.137 23.360Q-22.508 23.599-22.932 23.599Q-23.369 23.599-23.769 23.391Q-24.169 23.182-24.408 22.815Q-24.648 22.448-24.648 21.996M-23.978 21.726L-22.163 21.726Q-22.163 21.449-22.260 21.197Q-22.358 20.944-22.556 20.788Q-22.754 20.633-23.038 20.633Q-23.315 20.633-23.528 20.791Q-23.742 20.950-23.860 21.205Q-23.978 21.460-23.978 21.726M-21.059 23.524L-21.059 22.461Q-21.059 22.437-21.031 22.410Q-21.004 22.383-20.980 22.383L-20.871 22.383Q-20.806 22.383-20.792 22.441Q-20.696 22.875-20.450 23.126Q-20.204 23.377-19.791 23.377Q-19.449 23.377-19.196 23.244Q-18.943 23.111-18.943 22.803Q-18.943 22.646-19.037 22.531Q-19.131 22.417-19.269 22.348Q-19.408 22.280-19.575 22.242L-20.156 22.143Q-20.512 22.075-20.785 21.854Q-21.059 21.634-21.059 21.292Q-21.059 21.043-20.948 20.868Q-20.837 20.694-20.650 20.595Q-20.464 20.496-20.249 20.453Q-20.033 20.410-19.791 20.410Q-19.377 20.410-19.097 20.592L-18.881 20.417Q-18.871 20.414-18.864 20.412Q-18.858 20.410-18.847 20.410L-18.796 20.410Q-18.769 20.410-18.745 20.434Q-18.721 20.458-18.721 20.486L-18.721 21.333Q-18.721 21.354-18.745 21.381Q-18.769 21.408-18.796 21.408L-18.909 21.408Q-18.936 21.408-18.962 21.383Q-18.987 21.357-18.987 21.333Q-18.987 21.097-19.093 20.933Q-19.199 20.769-19.382 20.687Q-19.565 20.605-19.797 20.605Q-20.126 20.605-20.382 20.708Q-20.638 20.810-20.638 21.087Q-20.638 21.282-20.455 21.391Q-20.273 21.501-20.044 21.542L-19.469 21.648Q-19.223 21.696-19.010 21.824Q-18.796 21.952-18.659 22.155Q-18.523 22.359-18.523 22.608Q-18.523 23.121-18.888 23.360Q-19.254 23.599-19.791 23.599Q-20.286 23.599-20.618 23.305L-20.884 23.579Q-20.905 23.599-20.932 23.599L-20.980 23.599Q-21.004 23.599-21.031 23.572Q-21.059 23.545-21.059 23.524\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(191.521 1.75)\">\u003Cpath d=\"M-13.051 25.281Q-13.601 24.881-13.972 24.326Q-14.343 23.770-14.524 23.124Q-14.705 22.478-14.705 21.781Q-14.705 21.268-14.605 20.773Q-14.504 20.277-14.299 19.826Q-14.094 19.375-13.781 18.983Q-13.468 18.592-13.051 18.288Q-13.041 18.284-13.034 18.283Q-13.027 18.281-13.017 18.281L-12.949 18.281Q-12.914 18.281-12.892 18.305Q-12.870 18.329-12.870 18.366Q-12.870 18.411-12.897 18.428Q-13.246 18.729-13.499 19.113Q-13.752 19.498-13.904 19.939Q-14.056 20.380-14.128 20.836Q-14.200 21.292-14.200 21.781Q-14.200 22.782-13.890 23.669Q-13.581 24.556-12.897 25.141Q-12.870 25.158-12.870 25.202Q-12.870 25.240-12.892 25.264Q-12.914 25.288-12.949 25.288L-13.017 25.288Q-13.024 25.284-13.032 25.283Q-13.041 25.281-13.051 25.281M-10.392 23.531L-11.995 23.531L-11.995 23.251Q-11.769 23.251-11.621 23.217Q-11.472 23.182-11.472 23.042L-11.472 19.423Q-11.472 19.153-11.580 19.091Q-11.687 19.030-11.995 19.030L-11.995 18.749L-10.918 18.674L-10.918 23.042Q-10.918 23.179-10.768 23.215Q-10.618 23.251-10.392 23.251L-10.392 23.531M-9.838 22.048Q-9.838 21.706-9.703 21.407Q-9.568 21.108-9.329 20.884Q-9.090 20.660-8.772 20.535Q-8.454 20.410-8.122 20.410Q-7.678 20.410-7.278 20.626Q-6.878 20.841-6.644 21.219Q-6.410 21.596-6.410 22.048Q-6.410 22.389-6.552 22.673Q-6.694 22.957-6.938 23.164Q-7.182 23.370-7.492 23.485Q-7.801 23.599-8.122 23.599Q-8.553 23.599-8.955 23.398Q-9.356 23.196-9.597 22.844Q-9.838 22.492-9.838 22.048M-8.122 23.350Q-7.521 23.350-7.297 22.972Q-7.073 22.594-7.073 21.962Q-7.073 21.350-7.307 20.991Q-7.541 20.633-8.122 20.633Q-9.175 20.633-9.175 21.962Q-9.175 22.594-8.950 22.972Q-8.724 23.350-8.122 23.350\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(191.521 1.75)\">\u003Cpath d=\"M-4.633 23.504L-5.614 21.005Q-5.675 20.862-5.793 20.827Q-5.911 20.793-6.127 20.793L-6.127 20.513L-4.647 20.513L-4.647 20.793Q-5.026 20.793-5.026 20.954Q-5.026 20.964-5.012 21.005L-4.298 22.837L-3.625 21.132Q-3.655 21.060-3.655 21.032Q-3.655 21.005-3.683 21.005Q-3.744 20.858-3.862 20.826Q-3.980 20.793-4.192 20.793L-4.192 20.513L-2.794 20.513L-2.794 20.793Q-3.170 20.793-3.170 20.954Q-3.170 20.985-3.163 21.005L-2.408 22.943L-1.721 21.193Q-1.700 21.142-1.700 21.087Q-1.700 20.947-1.813 20.870Q-1.926 20.793-2.066 20.793L-2.066 20.513L-0.846 20.513L-0.846 20.793Q-1.051 20.793-1.206 20.899Q-1.362 21.005-1.434 21.193L-2.339 23.504Q-2.374 23.599-2.486 23.599L-2.555 23.599Q-2.664 23.599-2.702 23.504L-3.484 21.501L-4.271 23.504Q-4.305 23.599-4.418 23.599L-4.486 23.599Q-4.595 23.599-4.633 23.504\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(191.521 1.75)\">\u003Cpath d=\"M2.908 22.690L2.908 20.793L2.269 20.793L2.269 20.571Q2.587 20.571 2.804 20.361Q3.021 20.151 3.121 19.841Q3.222 19.532 3.222 19.224L3.489 19.224L3.489 20.513L4.566 20.513L4.566 20.793L3.489 20.793L3.489 22.677Q3.489 22.953 3.593 23.152Q3.697 23.350 3.957 23.350Q4.114 23.350 4.220 23.246Q4.326 23.141 4.376 22.988Q4.425 22.834 4.425 22.677L4.425 22.263L4.692 22.263L4.692 22.690Q4.692 22.916 4.593 23.126Q4.494 23.336 4.309 23.468Q4.125 23.599 3.896 23.599Q3.458 23.599 3.183 23.362Q2.908 23.124 2.908 22.690M5.461 22.048Q5.461 21.706 5.596 21.407Q5.731 21.108 5.970 20.884Q6.210 20.660 6.527 20.535Q6.845 20.410 7.177 20.410Q7.621 20.410 8.021 20.626Q8.421 20.841 8.655 21.219Q8.889 21.596 8.889 22.048Q8.889 22.389 8.747 22.673Q8.606 22.957 8.361 23.164Q8.117 23.370 7.807 23.485Q7.498 23.599 7.177 23.599Q6.746 23.599 6.345 23.398Q5.943 23.196 5.702 22.844Q5.461 22.492 5.461 22.048M7.177 23.350Q7.778 23.350 8.002 22.972Q8.226 22.594 8.226 21.962Q8.226 21.350 7.992 20.991Q7.758 20.633 7.177 20.633Q6.124 20.633 6.124 21.962Q6.124 22.594 6.350 22.972Q6.575 23.350 7.177 23.350\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(191.521 1.75)\">\u003Cpath d=\"M13.869 23.531L12.235 23.531L12.235 23.251Q12.464 23.251 12.613 23.217Q12.762 23.182 12.762 23.042L12.762 19.423Q12.762 19.153 12.654 19.091Q12.546 19.030 12.235 19.030L12.235 18.749L13.315 18.674L13.315 21.060Q13.421 20.875 13.599 20.733Q13.777 20.592 13.985 20.518Q14.194 20.445 14.419 20.445Q14.925 20.445 15.209 20.668Q15.493 20.892 15.493 21.388L15.493 23.042Q15.493 23.179 15.641 23.215Q15.790 23.251 16.016 23.251L16.016 23.531L14.385 23.531L14.385 23.251Q14.614 23.251 14.763 23.217Q14.912 23.182 14.912 23.042L14.912 21.402Q14.912 21.067 14.792 20.867Q14.672 20.667 14.358 20.667Q14.088 20.667 13.854 20.803Q13.620 20.940 13.481 21.174Q13.343 21.408 13.343 21.682L13.343 23.042Q13.343 23.179 13.493 23.215Q13.644 23.251 13.869 23.251L13.869 23.531M18.220 23.531L16.668 23.531L16.668 23.251Q16.894 23.251 17.043 23.217Q17.191 23.182 17.191 23.042L17.191 21.193Q17.191 21.005 17.144 20.921Q17.096 20.838 16.998 20.819Q16.901 20.800 16.689 20.800L16.689 20.520L17.745 20.445L17.745 23.042Q17.745 23.182 17.877 23.217Q18.008 23.251 18.220 23.251L18.220 23.531M16.949 19.224Q16.949 19.053 17.072 18.934Q17.195 18.814 17.366 18.814Q17.533 18.814 17.656 18.934Q17.779 19.053 17.779 19.224Q17.779 19.399 17.656 19.522Q17.533 19.645 17.366 19.645Q17.195 19.645 17.072 19.522Q16.949 19.399 16.949 19.224M18.825 24.064Q18.825 23.818 19.022 23.634Q19.218 23.449 19.475 23.370Q19.338 23.258 19.266 23.097Q19.194 22.936 19.194 22.755Q19.194 22.434 19.406 22.188Q19.071 21.890 19.071 21.480Q19.071 21.019 19.461 20.732Q19.851 20.445 20.329 20.445Q20.801 20.445 21.136 20.691Q21.310 20.537 21.520 20.455Q21.730 20.373 21.959 20.373Q22.123 20.373 22.245 20.480Q22.366 20.588 22.366 20.752Q22.366 20.848 22.294 20.920Q22.223 20.991 22.130 20.991Q22.031 20.991 21.961 20.918Q21.891 20.844 21.891 20.745Q21.891 20.691 21.905 20.660L21.912 20.646Q21.918 20.626 21.927 20.615Q21.935 20.605 21.939 20.598Q21.583 20.598 21.296 20.821Q21.583 21.114 21.583 21.480Q21.583 21.795 21.399 22.027Q21.214 22.260 20.925 22.388Q20.637 22.516 20.329 22.516Q20.127 22.516 19.936 22.466Q19.745 22.417 19.567 22.307Q19.475 22.434 19.475 22.577Q19.475 22.759 19.603 22.894Q19.731 23.029 19.915 23.029L20.548 23.029Q20.996 23.029 21.365 23.100Q21.734 23.172 21.994 23.401Q22.253 23.630 22.253 24.064Q22.253 24.385 21.958 24.587Q21.662 24.789 21.259 24.878Q20.855 24.967 20.541 24.967Q20.223 24.967 19.820 24.878Q19.416 24.789 19.121 24.587Q18.825 24.385 18.825 24.064M19.280 24.064Q19.280 24.293 19.498 24.442Q19.717 24.591 20.009 24.659Q20.302 24.727 20.541 24.727Q20.705 24.727 20.914 24.691Q21.122 24.656 21.329 24.575Q21.536 24.495 21.667 24.367Q21.799 24.239 21.799 24.064Q21.799 23.712 21.418 23.618Q21.037 23.524 20.534 23.524L19.915 23.524Q19.676 23.524 19.478 23.675Q19.280 23.825 19.280 24.064M20.329 22.277Q20.996 22.277 20.996 21.480Q20.996 20.680 20.329 20.680Q19.659 20.680 19.659 21.480Q19.659 22.277 20.329 22.277M24.530 23.531L22.896 23.531L22.896 23.251Q23.125 23.251 23.274 23.217Q23.422 23.182 23.422 23.042L23.422 19.423Q23.422 19.153 23.315 19.091Q23.207 19.030 22.896 19.030L22.896 18.749L23.976 18.674L23.976 21.060Q24.082 20.875 24.260 20.733Q24.437 20.592 24.646 20.518Q24.854 20.445 25.080 20.445Q25.586 20.445 25.870 20.668Q26.153 20.892 26.153 21.388L26.153 23.042Q26.153 23.179 26.302 23.215Q26.451 23.251 26.676 23.251L26.676 23.531L25.046 23.531L25.046 23.251Q25.275 23.251 25.424 23.217Q25.572 23.182 25.572 23.042L25.572 21.402Q25.572 21.067 25.453 20.867Q25.333 20.667 25.019 20.667Q24.748 20.667 24.514 20.803Q24.280 20.940 24.142 21.174Q24.003 21.408 24.003 21.682L24.003 23.042Q24.003 23.179 24.154 23.215Q24.304 23.251 24.530 23.251\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(191.521 1.75)\">\u003Cpath d=\"M31.715 23.531L29.979 23.531L29.979 23.251Q30.208 23.251 30.357 23.217Q30.505 23.182 30.505 23.042L30.505 21.193Q30.505 20.923 30.398 20.862Q30.290 20.800 29.979 20.800L29.979 20.520L31.008 20.445L31.008 21.152Q31.138 20.844 31.380 20.645Q31.623 20.445 31.941 20.445Q32.160 20.445 32.331 20.569Q32.502 20.694 32.502 20.906Q32.502 21.043 32.402 21.142Q32.303 21.241 32.170 21.241Q32.033 21.241 31.934 21.142Q31.835 21.043 31.835 20.906Q31.835 20.766 31.934 20.667Q31.644 20.667 31.444 20.863Q31.244 21.060 31.151 21.354Q31.059 21.648 31.059 21.928L31.059 23.042Q31.059 23.251 31.715 23.251L31.715 23.531M33.045 21.996Q33.045 21.675 33.170 21.386Q33.295 21.097 33.520 20.874Q33.746 20.650 34.041 20.530Q34.337 20.410 34.655 20.410Q34.983 20.410 35.245 20.510Q35.506 20.609 35.682 20.791Q35.858 20.974 35.952 21.232Q36.046 21.490 36.046 21.822Q36.046 21.914 35.964 21.935L33.708 21.935L33.708 21.996Q33.708 22.584 33.992 22.967Q34.276 23.350 34.843 23.350Q35.164 23.350 35.432 23.157Q35.701 22.964 35.790 22.649Q35.797 22.608 35.872 22.594L35.964 22.594Q36.046 22.618 36.046 22.690Q36.046 22.697 36.039 22.724Q35.926 23.121 35.556 23.360Q35.185 23.599 34.761 23.599Q34.323 23.599 33.923 23.391Q33.524 23.182 33.284 22.815Q33.045 22.448 33.045 21.996M33.715 21.726L35.530 21.726Q35.530 21.449 35.432 21.197Q35.335 20.944 35.137 20.788Q34.939 20.633 34.655 20.633Q34.378 20.633 34.164 20.791Q33.951 20.950 33.833 21.205Q33.715 21.460 33.715 21.726M38.022 23.504L37.041 21.005Q36.979 20.862 36.861 20.827Q36.743 20.793 36.528 20.793L36.528 20.513L38.008 20.513L38.008 20.793Q37.629 20.793 37.629 20.954Q37.629 20.964 37.642 21.005L38.357 22.837L39.030 21.132Q38.999 21.060 38.999 21.032Q38.999 21.005 38.972 21.005Q38.910 20.858 38.792 20.826Q38.674 20.793 38.463 20.793L38.463 20.513L39.860 20.513L39.860 20.793Q39.484 20.793 39.484 20.954Q39.484 20.985 39.491 21.005L40.247 22.943L40.934 21.193Q40.954 21.142 40.954 21.087Q40.954 20.947 40.841 20.870Q40.729 20.793 40.588 20.793L40.588 20.513L41.809 20.513L41.809 20.793Q41.604 20.793 41.448 20.899Q41.293 21.005 41.221 21.193L40.315 23.504Q40.281 23.599 40.168 23.599L40.100 23.599Q39.990 23.599 39.953 23.504L39.170 21.501L38.384 23.504Q38.350 23.599 38.237 23.599L38.169 23.599Q38.059 23.599 38.022 23.504\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(191.521 1.75)\">\u003Cpath d=\"M42.198 22.803Q42.198 22.471 42.421 22.244Q42.645 22.017 42.989 21.889Q43.332 21.760 43.705 21.708Q44.077 21.655 44.382 21.655L44.382 21.402Q44.382 21.197 44.274 21.017Q44.166 20.838 43.985 20.735Q43.804 20.633 43.596 20.633Q43.189 20.633 42.953 20.725Q43.042 20.762 43.088 20.846Q43.134 20.930 43.134 21.032Q43.134 21.128 43.088 21.207Q43.042 21.285 42.961 21.330Q42.881 21.374 42.792 21.374Q42.642 21.374 42.541 21.277Q42.440 21.179 42.440 21.032Q42.440 20.410 43.596 20.410Q43.807 20.410 44.057 20.474Q44.306 20.537 44.508 20.656Q44.710 20.776 44.836 20.961Q44.963 21.145 44.963 21.388L44.963 22.964Q44.963 23.080 45.024 23.176Q45.086 23.271 45.199 23.271Q45.308 23.271 45.373 23.177Q45.438 23.083 45.438 22.964L45.438 22.516L45.704 22.516L45.704 22.964Q45.704 23.234 45.477 23.399Q45.250 23.565 44.970 23.565Q44.761 23.565 44.624 23.411Q44.488 23.258 44.464 23.042Q44.317 23.309 44.035 23.454Q43.753 23.599 43.428 23.599Q43.151 23.599 42.867 23.524Q42.584 23.449 42.391 23.270Q42.198 23.090 42.198 22.803M42.813 22.803Q42.813 22.977 42.914 23.107Q43.014 23.237 43.170 23.307Q43.325 23.377 43.490 23.377Q43.708 23.377 43.917 23.280Q44.125 23.182 44.253 23.001Q44.382 22.820 44.382 22.594L44.382 21.866Q44.057 21.866 43.691 21.957Q43.325 22.048 43.069 22.260Q42.813 22.471 42.813 22.803M47.871 23.531L46.135 23.531L46.135 23.251Q46.364 23.251 46.513 23.217Q46.661 23.182 46.661 23.042L46.661 21.193Q46.661 20.923 46.554 20.862Q46.446 20.800 46.135 20.800L46.135 20.520L47.164 20.445L47.164 21.152Q47.294 20.844 47.536 20.645Q47.779 20.445 48.097 20.445Q48.316 20.445 48.487 20.569Q48.658 20.694 48.658 20.906Q48.658 21.043 48.558 21.142Q48.459 21.241 48.326 21.241Q48.189 21.241 48.090 21.142Q47.991 21.043 47.991 20.906Q47.991 20.766 48.090 20.667Q47.800 20.667 47.600 20.863Q47.400 21.060 47.307 21.354Q47.215 21.648 47.215 21.928L47.215 23.042Q47.215 23.251 47.871 23.251L47.871 23.531M49.242 22.020Q49.242 21.682 49.382 21.391Q49.522 21.101 49.767 20.887Q50.011 20.674 50.315 20.559Q50.619 20.445 50.944 20.445Q51.214 20.445 51.477 20.544Q51.741 20.643 51.932 20.821L51.932 19.423Q51.932 19.153 51.824 19.091Q51.717 19.030 51.406 19.030L51.406 18.749L52.482 18.674L52.482 22.858Q52.482 23.046 52.537 23.129Q52.592 23.213 52.692 23.232Q52.793 23.251 53.009 23.251L53.009 23.531L51.901 23.599L51.901 23.182Q51.484 23.599 50.859 23.599Q50.428 23.599 50.055 23.387Q49.683 23.176 49.462 22.815Q49.242 22.454 49.242 22.020M50.917 23.377Q51.125 23.377 51.312 23.305Q51.498 23.234 51.652 23.097Q51.805 22.960 51.901 22.782L51.901 21.173Q51.816 21.026 51.670 20.906Q51.525 20.786 51.356 20.727Q51.187 20.667 51.006 20.667Q50.445 20.667 50.177 21.056Q49.908 21.446 49.908 22.027Q49.908 22.598 50.143 22.988Q50.377 23.377 50.917 23.377M53.979 25.288L53.911 25.288Q53.877 25.288 53.855 25.262Q53.832 25.237 53.832 25.202Q53.832 25.158 53.863 25.141Q54.219 24.837 54.468 24.447Q54.718 24.057 54.870 23.625Q55.022 23.193 55.092 22.724Q55.162 22.256 55.162 21.781Q55.162 21.302 55.092 20.836Q55.022 20.369 54.868 19.934Q54.714 19.498 54.463 19.110Q54.212 18.722 53.863 18.428Q53.832 18.411 53.832 18.366Q53.832 18.332 53.855 18.307Q53.877 18.281 53.911 18.281L53.979 18.281Q53.990 18.281 53.998 18.283Q54.007 18.284 54.017 18.288Q54.560 18.688 54.933 19.241Q55.305 19.795 55.487 20.441Q55.668 21.087 55.668 21.781Q55.668 22.482 55.487 23.129Q55.305 23.777 54.931 24.331Q54.557 24.885 54.017 25.281Q54.007 25.281 53.998 25.283Q53.990 25.284 53.979 25.288\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-50.965 29.221v-85.603\"\u002F>\u003Cpath stroke=\"none\" d=\"m-50.965-58.982-2.08 4.16 2.08-1.56 2.08 1.56\"\u002F>\u003Cg transform=\"translate(-7.972 -87.607)\">\u003Cpath d=\"M-49.007 24.888L-50.637 24.888L-50.637 24.608Q-50.408 24.608-50.259 24.573Q-50.111 24.539-50.111 24.399L-50.111 21.053Q-50.111 20.882-50.247 20.841Q-50.384 20.800-50.637 20.800L-50.637 20.520L-49.557 20.445L-49.557 20.851Q-49.335 20.650-49.048 20.547Q-48.760 20.445-48.453 20.445Q-48.026 20.445-47.662 20.658Q-47.298 20.872-47.084 21.236Q-46.870 21.600-46.870 22.020Q-46.870 22.465-47.110 22.829Q-47.349 23.193-47.742 23.396Q-48.135 23.599-48.579 23.599Q-48.846 23.599-49.094 23.499Q-49.341 23.398-49.529 23.217L-49.529 24.399Q-49.529 24.536-49.381 24.572Q-49.232 24.608-49.007 24.608L-49.007 24.888M-49.529 21.200L-49.529 22.810Q-49.396 23.063-49.153 23.220Q-48.911 23.377-48.634 23.377Q-48.306 23.377-48.053 23.176Q-47.800 22.974-47.667 22.656Q-47.533 22.338-47.533 22.020Q-47.533 21.791-47.598 21.562Q-47.663 21.333-47.791 21.135Q-47.920 20.937-48.114 20.817Q-48.309 20.698-48.542 20.698Q-48.836 20.698-49.104 20.827Q-49.372 20.957-49.529 21.200M-44.485 23.531L-46.221 23.531L-46.221 23.251Q-45.992 23.251-45.843 23.217Q-45.694 23.182-45.694 23.042L-45.694 21.193Q-45.694 20.923-45.802 20.862Q-45.910 20.800-46.221 20.800L-46.221 20.520L-45.192 20.445L-45.192 21.152Q-45.062 20.844-44.819 20.645Q-44.577 20.445-44.259 20.445Q-44.040 20.445-43.869 20.569Q-43.698 20.694-43.698 20.906Q-43.698 21.043-43.798 21.142Q-43.897 21.241-44.030 21.241Q-44.167 21.241-44.266 21.142Q-44.365 21.043-44.365 20.906Q-44.365 20.766-44.266 20.667Q-44.556 20.667-44.756 20.863Q-44.956 21.060-45.048 21.354Q-45.141 21.648-45.141 21.928L-45.141 23.042Q-45.141 23.251-44.485 23.251L-44.485 23.531M-43.155 22.048Q-43.155 21.706-43.020 21.407Q-42.885 21.108-42.646 20.884Q-42.406 20.660-42.089 20.535Q-41.771 20.410-41.439 20.410Q-40.995 20.410-40.595 20.626Q-40.195 20.841-39.961 21.219Q-39.727 21.596-39.727 22.048Q-39.727 22.389-39.869 22.673Q-40.010 22.957-40.255 23.164Q-40.499 23.370-40.809 23.485Q-41.118 23.599-41.439 23.599Q-41.870 23.599-42.271 23.398Q-42.673 23.196-42.914 22.844Q-43.155 22.492-43.155 22.048M-41.439 23.350Q-40.838 23.350-40.614 22.972Q-40.390 22.594-40.390 21.962Q-40.390 21.350-40.624 20.991Q-40.858 20.633-41.439 20.633Q-42.492 20.633-42.492 21.962Q-42.492 22.594-42.266 22.972Q-42.041 23.350-41.439 23.350M-38.325 23.531L-38.592 23.531L-38.592 19.423Q-38.592 19.153-38.700 19.091Q-38.807 19.030-39.118 19.030L-39.118 18.749L-38.038 18.674L-38.038 20.844Q-37.830 20.653-37.544 20.549Q-37.259 20.445-36.962 20.445Q-36.644 20.445-36.346 20.566Q-36.049 20.687-35.827 20.903Q-35.605 21.118-35.478 21.403Q-35.352 21.689-35.352 22.020Q-35.352 22.465-35.591 22.829Q-35.830 23.193-36.223 23.396Q-36.616 23.599-37.061 23.599Q-37.256 23.599-37.445 23.543Q-37.635 23.487-37.796 23.382Q-37.956 23.278-38.096 23.117L-38.325 23.531M-38.011 21.186L-38.011 22.803Q-37.874 23.063-37.633 23.220Q-37.392 23.377-37.115 23.377Q-36.821 23.377-36.610 23.270Q-36.398 23.162-36.264 22.970Q-36.131 22.779-36.073 22.540Q-36.015 22.301-36.015 22.020Q-36.015 21.661-36.109 21.357Q-36.203 21.053-36.430 20.860Q-36.657 20.667-37.023 20.667Q-37.324 20.667-37.590 20.803Q-37.857 20.940-38.011 21.186\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cpath fill=\"none\" d=\"m-45.274 21.92 2.16-.335 2.161-.399 2.161-.467 2.16-.535 2.162-.62 2.16-.712 2.161-.808 2.161-.91 2.16-1.03 2.162-1.145 2.16-1.275 2.161-1.4 2.16-1.531 2.162-1.668 2.16-1.798 2.161-1.925 2.161-2.049 2.16-2.162 2.162-2.26 2.16-2.35L.102-5.88l2.161-2.47 2.16-2.493 2.162-2.493 2.16-2.476 2.161-2.413 2.16-2.34 2.162-2.215 2.16-2.083 2.161-1.9 2.161-1.708 2.16-1.486 2.162-1.24 2.16-.974 2.161-.7 2.161-.399 2.16-.106 2.162.175 2.16.49 2.161.768 2.16 1.041 2.162 1.31 2.16 1.542L49.8-26.29l2.161 1.958 2.16 2.117 2.162 2.254 2.16 2.357 2.161 2.436 2.161 2.476 2.16 2.499L67.088-7.7l2.16 2.453 2.161 2.402 2.16 2.333 2.162 2.238 2.16 2.134 2.161 2.015 2.161 1.895 2.16 1.765 2.162 1.628 2.16 1.497 2.161 1.372 2.161 1.234 2.16 1.116 2.162.996 2.16.888 2.161.78 2.16.689 2.162.598 2.16.517 2.161.444 2.161.382 2.16.324 2.162.274 2.16.233 2.161.194 2.161.159 2.16.13\" style=\"stroke-width:1.2\"\u002F>\u003Cg transform=\"translate(39.52 -59.016)\">\u003Cpath d=\"M-48.901 23.531L-50.637 23.531L-50.637 23.251Q-50.408 23.251-50.259 23.217Q-50.111 23.182-50.111 23.042L-50.111 21.193Q-50.111 20.923-50.218 20.862Q-50.326 20.800-50.637 20.800L-50.637 20.520L-49.608 20.445L-49.608 21.152Q-49.478 20.844-49.236 20.645Q-48.993 20.445-48.675 20.445Q-48.456 20.445-48.285 20.569Q-48.114 20.694-48.114 20.906Q-48.114 21.043-48.214 21.142Q-48.313 21.241-48.446 21.241Q-48.583 21.241-48.682 21.142Q-48.781 21.043-48.781 20.906Q-48.781 20.766-48.682 20.667Q-48.972 20.667-49.172 20.863Q-49.372 21.060-49.465 21.354Q-49.557 21.648-49.557 21.928L-49.557 23.042Q-49.557 23.251-48.901 23.251L-48.901 23.531M-47.571 21.996Q-47.571 21.675-47.446 21.386Q-47.321 21.097-47.096 20.874Q-46.870 20.650-46.575 20.530Q-46.279 20.410-45.961 20.410Q-45.633 20.410-45.371 20.510Q-45.110 20.609-44.934 20.791Q-44.758 20.974-44.664 21.232Q-44.570 21.490-44.570 21.822Q-44.570 21.914-44.652 21.935L-46.908 21.935L-46.908 21.996Q-46.908 22.584-46.624 22.967Q-46.340 23.350-45.773 23.350Q-45.452 23.350-45.184 23.157Q-44.915 22.964-44.826 22.649Q-44.819 22.608-44.744 22.594L-44.652 22.594Q-44.570 22.618-44.570 22.690Q-44.570 22.697-44.577 22.724Q-44.690 23.121-45.060 23.360Q-45.431 23.599-45.855 23.599Q-46.293 23.599-46.693 23.391Q-47.092 23.182-47.332 22.815Q-47.571 22.448-47.571 21.996M-46.901 21.726L-45.086 21.726Q-45.086 21.449-45.184 21.197Q-45.281 20.944-45.479 20.788Q-45.677 20.633-45.961 20.633Q-46.238 20.633-46.452 20.791Q-46.665 20.950-46.783 21.205Q-46.901 21.460-46.901 21.726M-42.184 23.531L-43.917 23.531L-43.917 23.251Q-43.692 23.251-43.543 23.217Q-43.394 23.182-43.394 23.042L-43.394 20.793L-43.982 20.793L-43.982 20.513L-43.394 20.513L-43.394 19.696Q-43.394 19.378-43.216 19.130Q-43.039 18.883-42.748 18.742Q-42.458 18.602-42.147 18.602Q-41.890 18.602-41.687 18.744Q-41.484 18.886-41.484 19.129Q-41.484 19.265-41.583 19.364Q-41.682 19.464-41.819 19.464Q-41.955 19.464-42.054 19.364Q-42.153 19.265-42.153 19.129Q-42.153 18.948-42.013 18.855Q-42.092 18.828-42.191 18.828Q-42.400 18.828-42.553 18.961Q-42.707 19.094-42.788 19.298Q-42.868 19.501-42.868 19.710L-42.868 20.513L-41.979 20.513L-41.979 20.793L-42.840 20.793L-42.840 23.042Q-42.840 23.251-42.184 23.251L-42.184 23.531M-41.545 21.996Q-41.545 21.675-41.420 21.386Q-41.296 21.097-41.070 20.874Q-40.844 20.650-40.549 20.530Q-40.253 20.410-39.935 20.410Q-39.607 20.410-39.346 20.510Q-39.084 20.609-38.908 20.791Q-38.732 20.974-38.638 21.232Q-38.544 21.490-38.544 21.822Q-38.544 21.914-38.626 21.935L-40.882 21.935L-40.882 21.996Q-40.882 22.584-40.598 22.967Q-40.315 23.350-39.747 23.350Q-39.426 23.350-39.158 23.157Q-38.889 22.964-38.800 22.649Q-38.794 22.608-38.718 22.594L-38.626 22.594Q-38.544 22.618-38.544 22.690Q-38.544 22.697-38.551 22.724Q-38.664 23.121-39.035 23.360Q-39.405 23.599-39.829 23.599Q-40.267 23.599-40.667 23.391Q-41.067 23.182-41.306 22.815Q-41.545 22.448-41.545 21.996M-40.875 21.726L-39.060 21.726Q-39.060 21.449-39.158 21.197Q-39.255 20.944-39.453 20.788Q-39.652 20.633-39.935 20.633Q-40.212 20.633-40.426 20.791Q-40.639 20.950-40.757 21.205Q-40.875 21.460-40.875 21.726M-36.206 23.531L-37.943 23.531L-37.943 23.251Q-37.714 23.251-37.565 23.217Q-37.416 23.182-37.416 23.042L-37.416 21.193Q-37.416 20.923-37.524 20.862Q-37.632 20.800-37.943 20.800L-37.943 20.520L-36.914 20.445L-36.914 21.152Q-36.784 20.844-36.541 20.645Q-36.298 20.445-35.981 20.445Q-35.762 20.445-35.591 20.569Q-35.420 20.694-35.420 20.906Q-35.420 21.043-35.519 21.142Q-35.618 21.241-35.752 21.241Q-35.888 21.241-35.987 21.142Q-36.087 21.043-36.087 20.906Q-36.087 20.766-35.987 20.667Q-36.278 20.667-36.478 20.863Q-36.678 21.060-36.770 21.354Q-36.862 21.648-36.862 21.928L-36.862 23.042Q-36.862 23.251-36.206 23.251L-36.206 23.531M-34.877 21.996Q-34.877 21.675-34.752 21.386Q-34.627 21.097-34.402 20.874Q-34.176 20.650-33.880 20.530Q-33.585 20.410-33.267 20.410Q-32.939 20.410-32.677 20.510Q-32.416 20.609-32.240 20.791Q-32.064 20.974-31.970 21.232Q-31.876 21.490-31.876 21.822Q-31.876 21.914-31.958 21.935L-34.214 21.935L-34.214 21.996Q-34.214 22.584-33.930 22.967Q-33.646 23.350-33.079 23.350Q-32.757 23.350-32.489 23.157Q-32.221 22.964-32.132 22.649Q-32.125 22.608-32.050 22.594L-31.958 22.594Q-31.876 22.618-31.876 22.690Q-31.876 22.697-31.882 22.724Q-31.995 23.121-32.366 23.360Q-32.737 23.599-33.161 23.599Q-33.598 23.599-33.998 23.391Q-34.398 23.182-34.637 22.815Q-34.877 22.448-34.877 21.996M-34.207 21.726L-32.392 21.726Q-32.392 21.449-32.489 21.197Q-32.587 20.944-32.785 20.788Q-32.983 20.633-33.267 20.633Q-33.544 20.633-33.757 20.791Q-33.971 20.950-34.089 21.205Q-34.207 21.460-34.207 21.726M-29.606 23.531L-31.240 23.531L-31.240 23.251Q-31.011 23.251-30.862 23.217Q-30.714 23.182-30.714 23.042L-30.714 21.193Q-30.714 20.923-30.821 20.862Q-30.929 20.800-31.240 20.800L-31.240 20.520L-30.180 20.445L-30.180 21.094Q-30.009 20.786-29.705 20.615Q-29.401 20.445-29.056 20.445Q-28.550 20.445-28.266 20.668Q-27.983 20.892-27.983 21.388L-27.983 23.042Q-27.983 23.179-27.834 23.215Q-27.685 23.251-27.460 23.251L-27.460 23.531L-29.090 23.531L-29.090 23.251Q-28.861 23.251-28.712 23.217Q-28.564 23.182-28.564 23.042L-28.564 21.402Q-28.564 21.067-28.683 20.867Q-28.803 20.667-29.117 20.667Q-29.387 20.667-29.621 20.803Q-29.856 20.940-29.994 21.174Q-30.132 21.408-30.132 21.682L-30.132 23.042Q-30.132 23.179-29.982 23.215Q-29.832 23.251-29.606 23.251L-29.606 23.531M-26.872 22.020Q-26.872 21.692-26.737 21.391Q-26.602 21.091-26.366 20.870Q-26.130 20.650-25.826 20.530Q-25.522 20.410-25.197 20.410Q-24.691 20.410-24.342 20.513Q-23.994 20.615-23.994 20.991Q-23.994 21.138-24.091 21.239Q-24.189 21.340-24.336 21.340Q-24.489 21.340-24.589 21.241Q-24.688 21.142-24.688 20.991Q-24.688 20.803-24.548 20.711Q-24.749 20.660-25.190 20.660Q-25.546 20.660-25.775 20.856Q-26.004 21.053-26.104 21.362Q-26.205 21.672-26.205 22.020Q-26.205 22.369-26.079 22.675Q-25.952 22.981-25.698 23.165Q-25.443 23.350-25.088 23.350Q-24.865 23.350-24.681 23.266Q-24.496 23.182-24.361 23.027Q-24.226 22.871-24.168 22.663Q-24.154 22.608-24.100 22.608L-23.987 22.608Q-23.956 22.608-23.934 22.632Q-23.912 22.656-23.912 22.690L-23.912 22.711Q-23.997 22.998-24.185 23.196Q-24.373 23.394-24.638 23.497Q-24.903 23.599-25.197 23.599Q-25.628 23.599-26.016 23.393Q-26.403 23.186-26.638 22.823Q-26.872 22.461-26.872 22.020M-23.365 21.996Q-23.365 21.675-23.240 21.386Q-23.115 21.097-22.890 20.874Q-22.664 20.650-22.369 20.530Q-22.073 20.410-21.755 20.410Q-21.427 20.410-21.165 20.510Q-20.904 20.609-20.728 20.791Q-20.552 20.974-20.458 21.232Q-20.364 21.490-20.364 21.822Q-20.364 21.914-20.446 21.935L-22.702 21.935L-22.702 21.996Q-22.702 22.584-22.418 22.967Q-22.134 23.350-21.567 23.350Q-21.246 23.350-20.977 23.157Q-20.709 22.964-20.620 22.649Q-20.613 22.608-20.538 22.594L-20.446 22.594Q-20.364 22.618-20.364 22.690Q-20.364 22.697-20.371 22.724Q-20.484 23.121-20.854 23.360Q-21.225 23.599-21.649 23.599Q-22.087 23.599-22.486 23.391Q-22.886 23.182-23.126 22.815Q-23.365 22.448-23.365 21.996M-22.695 21.726L-20.880 21.726Q-20.880 21.449-20.977 21.197Q-21.075 20.944-21.273 20.788Q-21.471 20.633-21.755 20.633Q-22.032 20.633-22.246 20.791Q-22.459 20.950-22.577 21.205Q-22.695 21.460-22.695 21.726\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cpath fill=\"none\" stroke=\"var(--tk-good)\" d=\"M-45.274 23.53h6.482l2.16-.01 2.162-.005 2.16-.007 2.161-.01 2.161-.015 2.16-.019 2.162-.026 2.16-.038 2.161-.048 2.16-.066 2.162-.085 2.16-.112 2.161-.147 2.161-.184 2.16-.237 2.162-.307 2.16-.376 2.161-.468 2.161-.572 2.16-.697 2.162-.836 2.16-.998 2.161-1.185 2.16-1.38 2.162-1.597 2.16-1.834 2.161-2.07 2.161-2.33 2.16-2.586 2.162-2.838 2.16-3.082 2.161-3.298 2.161-3.494 2.16-3.653 2.162-3.773 2.16-3.828 2.161-3.827 2.16-3.766L45.48-29.9l2.16-3.423 2.161-3.152 2.161-2.795 2.16-2.393 2.162-1.93 2.16-1.416 2.161-.871 2.161-.298 2.16.249 2.162.836 2.16 1.374 2.161 1.897 2.16 2.356 2.162 2.775 2.16 3.117 2.161 3.41 2.161 3.612 2.16 3.758 2.162 3.828 2.16 3.827 2.161 3.773 2.161 3.668 2.16 3.506 2.162 3.32 2.16 3.095 2.161 2.852 2.16 2.608 2.162 2.35 2.16 2.091 2.161 1.848 2.161 1.618 2.16 1.394 2.162 1.2 2.16 1.01 2.161.852 2.161.704 2.16.586\" style=\"stroke-width:1.2\"\u002F>\u003Cg fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\">\u003Cg fill=\"var(--tk-good)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(138.684 -63.222)\">\u003Cpath d=\"M-50.692 22.048Q-50.692 21.706-50.557 21.407Q-50.422 21.108-50.182 20.884Q-49.943 20.660-49.625 20.535Q-49.307 20.410-48.976 20.410Q-48.531 20.410-48.132 20.626Q-47.732 20.841-47.497 21.219Q-47.263 21.596-47.263 22.048Q-47.263 22.389-47.405 22.673Q-47.547 22.957-47.791 23.164Q-48.036 23.370-48.345 23.485Q-48.654 23.599-48.976 23.599Q-49.406 23.599-49.808 23.398Q-50.210 23.196-50.451 22.844Q-50.692 22.492-50.692 22.048M-48.976 23.350Q-48.374 23.350-48.150 22.972Q-47.926 22.594-47.926 21.962Q-47.926 21.350-48.161 20.991Q-48.395 20.633-48.976 20.633Q-50.028 20.633-50.028 21.962Q-50.028 22.594-49.803 22.972Q-49.577 23.350-48.976 23.350M-45.025 24.888L-46.655 24.888L-46.655 24.608Q-46.426 24.608-46.277 24.573Q-46.129 24.539-46.129 24.399L-46.129 21.053Q-46.129 20.882-46.265 20.841Q-46.402 20.800-46.655 20.800L-46.655 20.520L-45.575 20.445L-45.575 20.851Q-45.353 20.650-45.066 20.547Q-44.778 20.445-44.471 20.445Q-44.044 20.445-43.680 20.658Q-43.316 20.872-43.102 21.236Q-42.888 21.600-42.888 22.020Q-42.888 22.465-43.128 22.829Q-43.367 23.193-43.760 23.396Q-44.153 23.599-44.597 23.599Q-44.864 23.599-45.112 23.499Q-45.360 23.398-45.548 23.217L-45.548 24.399Q-45.548 24.536-45.399 24.572Q-45.250 24.608-45.025 24.608L-45.025 24.888M-45.548 21.200L-45.548 22.810Q-45.414 23.063-45.172 23.220Q-44.929 23.377-44.652 23.377Q-44.324 23.377-44.071 23.176Q-43.818 22.974-43.685 22.656Q-43.551 22.338-43.551 22.020Q-43.551 21.791-43.616 21.562Q-43.681 21.333-43.809 21.135Q-43.938 20.937-44.132 20.817Q-44.327 20.698-44.560 20.698Q-44.854 20.698-45.122 20.827Q-45.390 20.957-45.548 21.200M-41.726 22.690L-41.726 20.793L-42.365 20.793L-42.365 20.571Q-42.048 20.571-41.830 20.361Q-41.613 20.151-41.513 19.841Q-41.412 19.532-41.412 19.224L-41.145 19.224L-41.145 20.513L-40.069 20.513L-40.069 20.793L-41.145 20.793L-41.145 22.677Q-41.145 22.953-41.041 23.152Q-40.937 23.350-40.677 23.350Q-40.520 23.350-40.414 23.246Q-40.308 23.141-40.258 22.988Q-40.209 22.834-40.209 22.677L-40.209 22.263L-39.942 22.263L-39.942 22.690Q-39.942 22.916-40.041 23.126Q-40.140 23.336-40.325 23.468Q-40.509 23.599-40.738 23.599Q-41.176 23.599-41.451 23.362Q-41.726 23.124-41.726 22.690M-37.515 23.531L-39.067 23.531L-39.067 23.251Q-38.841 23.251-38.693 23.217Q-38.544 23.182-38.544 23.042L-38.544 21.193Q-38.544 21.005-38.592 20.921Q-38.640 20.838-38.737 20.819Q-38.835 20.800-39.047 20.800L-39.047 20.520L-37.990 20.445L-37.990 23.042Q-37.990 23.182-37.859 23.217Q-37.727 23.251-37.515 23.251L-37.515 23.531M-38.787 19.224Q-38.787 19.053-38.664 18.934Q-38.541 18.814-38.370 18.814Q-38.202 18.814-38.079 18.934Q-37.956 19.053-37.956 19.224Q-37.956 19.399-38.079 19.522Q-38.202 19.645-38.370 19.645Q-38.541 19.645-38.664 19.522Q-38.787 19.399-38.787 19.224M-35.188 23.531L-36.821 23.531L-36.821 23.251Q-36.592 23.251-36.444 23.217Q-36.295 23.182-36.295 23.042L-36.295 21.193Q-36.295 20.923-36.403 20.862Q-36.510 20.800-36.821 20.800L-36.821 20.520L-35.762 20.445L-35.762 21.094Q-35.591 20.786-35.287 20.615Q-34.983 20.445-34.637 20.445Q-34.237 20.445-33.961 20.585Q-33.684 20.725-33.598 21.073Q-33.431 20.780-33.132 20.612Q-32.833 20.445-32.487 20.445Q-31.982 20.445-31.698 20.668Q-31.414 20.892-31.414 21.388L-31.414 23.042Q-31.414 23.179-31.266 23.215Q-31.117 23.251-30.891 23.251L-30.891 23.531L-32.522 23.531L-32.522 23.251Q-32.296 23.251-32.146 23.215Q-31.995 23.179-31.995 23.042L-31.995 21.402Q-31.995 21.067-32.115 20.867Q-32.235 20.667-32.549 20.667Q-32.819 20.667-33.053 20.803Q-33.287 20.940-33.426 21.174Q-33.564 21.408-33.564 21.682L-33.564 23.042Q-33.564 23.179-33.415 23.215Q-33.267 23.251-33.041 23.251L-33.041 23.531L-34.672 23.531L-34.672 23.251Q-34.443 23.251-34.294 23.217Q-34.145 23.182-34.145 23.042L-34.145 21.402Q-34.145 21.067-34.265 20.867Q-34.384 20.667-34.699 20.667Q-34.969 20.667-35.203 20.803Q-35.437 20.940-35.576 21.174Q-35.714 21.408-35.714 21.682L-35.714 23.042Q-35.714 23.179-35.564 23.215Q-35.413 23.251-35.188 23.251L-35.188 23.531M-30.245 22.803Q-30.245 22.471-30.021 22.244Q-29.798 22.017-29.454 21.889Q-29.111 21.760-28.738 21.708Q-28.365 21.655-28.061 21.655L-28.061 21.402Q-28.061 21.197-28.169 21.017Q-28.277 20.838-28.458 20.735Q-28.639 20.633-28.847 20.633Q-29.254 20.633-29.490 20.725Q-29.401 20.762-29.355 20.846Q-29.309 20.930-29.309 21.032Q-29.309 21.128-29.355 21.207Q-29.401 21.285-29.481 21.330Q-29.562 21.374-29.651 21.374Q-29.801 21.374-29.902 21.277Q-30.003 21.179-30.003 21.032Q-30.003 20.410-28.847 20.410Q-28.635 20.410-28.386 20.474Q-28.136 20.537-27.935 20.656Q-27.733 20.776-27.607 20.961Q-27.480 21.145-27.480 21.388L-27.480 22.964Q-27.480 23.080-27.419 23.176Q-27.357 23.271-27.244 23.271Q-27.135 23.271-27.070 23.177Q-27.005 23.083-27.005 22.964L-27.005 22.516L-26.738 22.516L-26.738 22.964Q-26.738 23.234-26.966 23.399Q-27.193 23.565-27.473 23.565Q-27.682 23.565-27.819 23.411Q-27.955 23.258-27.979 23.042Q-28.126 23.309-28.408 23.454Q-28.690 23.599-29.015 23.599Q-29.292 23.599-29.575 23.524Q-29.859 23.449-30.052 23.270Q-30.245 23.090-30.245 22.803M-29.630 22.803Q-29.630 22.977-29.529 23.107Q-29.428 23.237-29.273 23.307Q-29.117 23.377-28.953 23.377Q-28.735 23.377-28.526 23.280Q-28.318 23.182-28.189 23.001Q-28.061 22.820-28.061 22.594L-28.061 21.866Q-28.386 21.866-28.752 21.957Q-29.117 22.048-29.374 22.260Q-29.630 22.471-29.630 22.803M-24.653 23.531L-26.257 23.531L-26.257 23.251Q-26.031 23.251-25.882 23.217Q-25.734 23.182-25.734 23.042L-25.734 19.423Q-25.734 19.153-25.841 19.091Q-25.949 19.030-26.257 19.030L-26.257 18.749L-25.180 18.674L-25.180 23.042Q-25.180 23.179-25.029 23.215Q-24.879 23.251-24.653 23.251\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(138.684 -63.222)\">\u003Cpath d=\"M-19.701 24.888L-21.331 24.888L-21.331 24.608Q-21.102 24.608-20.953 24.573Q-20.805 24.539-20.805 24.399L-20.805 21.053Q-20.805 20.882-20.941 20.841Q-21.078 20.800-21.331 20.800L-21.331 20.520L-20.251 20.445L-20.251 20.851Q-20.029 20.650-19.742 20.547Q-19.454 20.445-19.147 20.445Q-18.720 20.445-18.356 20.658Q-17.992 20.872-17.778 21.236Q-17.564 21.600-17.564 22.020Q-17.564 22.465-17.804 22.829Q-18.043 23.193-18.436 23.396Q-18.829 23.599-19.273 23.599Q-19.540 23.599-19.788 23.499Q-20.035 23.398-20.223 23.217L-20.223 24.399Q-20.223 24.536-20.075 24.572Q-19.926 24.608-19.701 24.608L-19.701 24.888M-20.223 21.200L-20.223 22.810Q-20.090 23.063-19.847 23.220Q-19.605 23.377-19.328 23.377Q-19 23.377-18.747 23.176Q-18.494 22.974-18.361 22.656Q-18.227 22.338-18.227 22.020Q-18.227 21.791-18.292 21.562Q-18.357 21.333-18.485 21.135Q-18.614 20.937-18.808 20.817Q-19.003 20.698-19.236 20.698Q-19.530 20.698-19.798 20.827Q-20.066 20.957-20.223 21.200\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(138.684 -63.222)\">\u003Cpath d=\"M-16.754 22.048Q-16.754 21.706-16.619 21.407Q-16.484 21.108-16.244 20.884Q-16.005 20.660-15.687 20.535Q-15.369 20.410-15.038 20.410Q-14.593 20.410-14.194 20.626Q-13.794 20.841-13.559 21.219Q-13.325 21.596-13.325 22.048Q-13.325 22.389-13.467 22.673Q-13.609 22.957-13.853 23.164Q-14.098 23.370-14.407 23.485Q-14.716 23.599-15.038 23.599Q-15.468 23.599-15.870 23.398Q-16.272 23.196-16.513 22.844Q-16.754 22.492-16.754 22.048M-15.038 23.350Q-14.436 23.350-14.212 22.972Q-13.988 22.594-13.988 21.962Q-13.988 21.350-14.223 20.991Q-14.457 20.633-15.038 20.633Q-16.090 20.633-16.090 21.962Q-16.090 22.594-15.865 22.972Q-15.639 23.350-15.038 23.350M-11.063 23.531L-12.666 23.531L-12.666 23.251Q-12.440 23.251-12.291 23.217Q-12.143 23.182-12.143 23.042L-12.143 19.423Q-12.143 19.153-12.250 19.091Q-12.358 19.030-12.666 19.030L-12.666 18.749L-11.589 18.674L-11.589 23.042Q-11.589 23.179-11.439 23.215Q-11.288 23.251-11.063 23.251L-11.063 23.531M-8.851 23.531L-10.403 23.531L-10.403 23.251Q-10.177 23.251-10.029 23.217Q-9.880 23.182-9.880 23.042L-9.880 21.193Q-9.880 21.005-9.928 20.921Q-9.976 20.838-10.073 20.819Q-10.171 20.800-10.382 20.800L-10.382 20.520L-9.326 20.445L-9.326 23.042Q-9.326 23.182-9.195 23.217Q-9.063 23.251-8.851 23.251L-8.851 23.531M-10.123 19.224Q-10.123 19.053-10 18.934Q-9.877 18.814-9.706 18.814Q-9.538 18.814-9.415 18.934Q-9.292 19.053-9.292 19.224Q-9.292 19.399-9.415 19.522Q-9.538 19.645-9.706 19.645Q-9.877 19.645-10 19.522Q-10.123 19.399-10.123 19.224M-8.205 22.020Q-8.205 21.692-8.070 21.391Q-7.935 21.091-7.699 20.870Q-7.464 20.650-7.159 20.530Q-6.855 20.410-6.530 20.410Q-6.025 20.410-5.676 20.513Q-5.327 20.615-5.327 20.991Q-5.327 21.138-5.425 21.239Q-5.522 21.340-5.669 21.340Q-5.823 21.340-5.922 21.241Q-6.021 21.142-6.021 20.991Q-6.021 20.803-5.881 20.711Q-6.083 20.660-6.524 20.660Q-6.879 20.660-7.108 20.856Q-7.337 21.053-7.438 21.362Q-7.539 21.672-7.539 22.020Q-7.539 22.369-7.412 22.675Q-7.286 22.981-7.031 23.165Q-6.777 23.350-6.421 23.350Q-6.199 23.350-6.014 23.266Q-5.830 23.182-5.695 23.027Q-5.560 22.871-5.502 22.663Q-5.488 22.608-5.433 22.608L-5.320 22.608Q-5.290 22.608-5.267 22.632Q-5.245 22.656-5.245 22.690L-5.245 22.711Q-5.331 22.998-5.519 23.196Q-5.707 23.394-5.972 23.497Q-6.236 23.599-6.530 23.599Q-6.961 23.599-7.349 23.393Q-7.737 23.186-7.971 22.823Q-8.205 22.461-8.205 22.020M-4.322 24.666Q-4.193 24.734-4.056 24.734Q-3.885 24.734-3.735 24.645Q-3.584 24.556-3.473 24.411Q-3.362 24.266-3.283 24.098L-3.020 23.531L-4.189 21.005Q-4.264 20.858-4.394 20.826Q-4.524 20.793-4.756 20.793L-4.756 20.513L-3.235 20.513L-3.235 20.793Q-3.584 20.793-3.584 20.940Q-3.581 20.961-3.579 20.978Q-3.577 20.995-3.577 21.005L-2.719 22.864L-1.947 21.193Q-1.913 21.125-1.913 21.046Q-1.913 20.933-1.996 20.863Q-2.080 20.793-2.193 20.793L-2.193 20.513L-0.997 20.513L-0.997 20.793Q-1.215 20.793-1.388 20.897Q-1.561 21.002-1.653 21.193L-2.989 24.098Q-3.160 24.468-3.430 24.714Q-3.700 24.960-4.056 24.960Q-4.326 24.960-4.545 24.794Q-4.763 24.628-4.763 24.365Q-4.763 24.228-4.671 24.139Q-4.579 24.051-4.439 24.051Q-4.302 24.051-4.213 24.139Q-4.124 24.228-4.124 24.365Q-4.124 24.468-4.177 24.546Q-4.230 24.625-4.322 24.666\" fill=\"var(--tk-good)\" stroke=\"var(--tk-good)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M34.393-46.178c14.227-7.113 14.227-7.113 25.963-3.593\"\u002F>\u003Cpath stroke=\"none\" d=\"M62.846-49.024 59.46-52.21l.896 2.44-2.092 1.544\"\u002F>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\">\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(53.9 -82.105)\">\u003Cpath d=\"M-50.124 22.690L-50.124 20.793L-50.763 20.793L-50.763 20.571Q-50.445 20.571-50.228 20.361Q-50.011 20.151-49.911 19.841Q-49.810 19.532-49.810 19.224L-49.543 19.224L-49.543 20.513L-48.466 20.513L-48.466 20.793L-49.543 20.793L-49.543 22.677Q-49.543 22.953-49.439 23.152Q-49.335 23.350-49.075 23.350Q-48.918 23.350-48.812 23.246Q-48.706 23.141-48.656 22.988Q-48.607 22.834-48.607 22.677L-48.607 22.263L-48.340 22.263L-48.340 22.690Q-48.340 22.916-48.439 23.126Q-48.538 23.336-48.723 23.468Q-48.907 23.599-49.136 23.599Q-49.574 23.599-49.849 23.362Q-50.124 23.124-50.124 22.690M-45.913 23.531L-47.465 23.531L-47.465 23.251Q-47.239 23.251-47.091 23.217Q-46.942 23.182-46.942 23.042L-46.942 21.193Q-46.942 21.005-46.990 20.921Q-47.038 20.838-47.135 20.819Q-47.233 20.800-47.444 20.800L-47.444 20.520L-46.388 20.445L-46.388 23.042Q-46.388 23.182-46.257 23.217Q-46.125 23.251-45.913 23.251L-45.913 23.531M-47.185 19.224Q-47.185 19.053-47.062 18.934Q-46.939 18.814-46.768 18.814Q-46.600 18.814-46.477 18.934Q-46.354 19.053-46.354 19.224Q-46.354 19.399-46.477 19.522Q-46.600 19.645-46.768 19.645Q-46.939 19.645-47.062 19.522Q-47.185 19.399-47.185 19.224M-43.599 23.531L-45.202 23.531L-45.202 23.251Q-44.977 23.251-44.828 23.217Q-44.679 23.182-44.679 23.042L-44.679 19.423Q-44.679 19.153-44.787 19.091Q-44.895 19.030-45.202 19.030L-45.202 18.749L-44.126 18.674L-44.126 23.042Q-44.126 23.179-43.975 23.215Q-43.825 23.251-43.599 23.251L-43.599 23.531M-42.478 22.690L-42.478 20.793L-43.117 20.793L-43.117 20.571Q-42.799 20.571-42.582 20.361Q-42.365 20.151-42.265 19.841Q-42.164 19.532-42.164 19.224L-41.897 19.224L-41.897 20.513L-40.820 20.513L-40.820 20.793L-41.897 20.793L-41.897 22.677Q-41.897 22.953-41.793 23.152Q-41.689 23.350-41.429 23.350Q-41.272 23.350-41.166 23.246Q-41.060 23.141-41.010 22.988Q-40.961 22.834-40.961 22.677L-40.961 22.263L-40.694 22.263L-40.694 22.690Q-40.694 22.916-40.793 23.126Q-40.892 23.336-41.077 23.468Q-41.261 23.599-41.490 23.599Q-41.928 23.599-42.203 23.362Q-42.478 23.124-42.478 22.690\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(53.9 -82.105)\">\u003Cpath d=\"M-36.371 23.531L-36.638 23.531L-36.638 19.423Q-36.638 19.153-36.745 19.091Q-36.853 19.030-37.164 19.030L-37.164 18.749L-36.084 18.674L-36.084 20.844Q-35.875 20.653-35.590 20.549Q-35.304 20.445-35.007 20.445Q-34.689 20.445-34.392 20.566Q-34.095 20.687-33.872 20.903Q-33.650 21.118-33.524 21.403Q-33.397 21.689-33.397 22.020Q-33.397 22.465-33.637 22.829Q-33.876 23.193-34.269 23.396Q-34.662 23.599-35.106 23.599Q-35.301 23.599-35.491 23.543Q-35.680 23.487-35.841 23.382Q-36.002 23.278-36.142 23.117L-36.371 23.531M-36.056 21.186L-36.056 22.803Q-35.920 23.063-35.679 23.220Q-35.438 23.377-35.161 23.377Q-34.867 23.377-34.655 23.270Q-34.443 23.162-34.310 22.970Q-34.177 22.779-34.118 22.540Q-34.060 22.301-34.060 22.020Q-34.060 21.661-34.154 21.357Q-34.248 21.053-34.476 20.860Q-34.703 20.667-35.069 20.667Q-35.369 20.667-35.636 20.803Q-35.903 20.940-36.056 21.186\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(53.9 -82.105)\">\u003Cpath d=\"M-32.642 24.666Q-32.512 24.734-32.375 24.734Q-32.204 24.734-32.054 24.645Q-31.903 24.556-31.792 24.411Q-31.681 24.266-31.603 24.098L-31.339 23.531L-32.508 21.005Q-32.583 20.858-32.713 20.826Q-32.843 20.793-33.076 20.793L-33.076 20.513L-31.555 20.513L-31.555 20.793Q-31.903 20.793-31.903 20.940Q-31.900 20.961-31.898 20.978Q-31.896 20.995-31.896 21.005L-31.039 22.864L-30.266 21.193Q-30.232 21.125-30.232 21.046Q-30.232 20.933-30.316 20.863Q-30.399 20.793-30.512 20.793L-30.512 20.513L-29.316 20.513L-29.316 20.793Q-29.535 20.793-29.707 20.897Q-29.880 21.002-29.972 21.193L-31.309 24.098Q-31.479 24.468-31.749 24.714Q-32.020 24.960-32.375 24.960Q-32.645 24.960-32.864 24.794Q-33.083 24.628-33.083 24.365Q-33.083 24.228-32.990 24.139Q-32.898 24.051-32.758 24.051Q-32.621 24.051-32.532 24.139Q-32.443 24.228-32.443 24.365Q-32.443 24.468-32.496 24.546Q-32.549 24.625-32.642 24.666\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(53.9 -82.105)\">\u003Cpath d=\"M-26.122 21.996Q-26.122 21.675-25.997 21.386Q-25.872 21.097-25.646 20.874Q-25.421 20.650-25.125 20.530Q-24.830 20.410-24.512 20.410Q-24.184 20.410-23.922 20.510Q-23.661 20.609-23.485 20.791Q-23.309 20.974-23.215 21.232Q-23.121 21.490-23.121 21.822Q-23.121 21.914-23.203 21.935L-25.458 21.935L-25.458 21.996Q-25.458 22.584-25.175 22.967Q-24.891 23.350-24.324 23.350Q-24.002 23.350-23.734 23.157Q-23.466 22.964-23.377 22.649Q-23.370 22.608-23.295 22.594L-23.203 22.594Q-23.121 22.618-23.121 22.690Q-23.121 22.697-23.127 22.724Q-23.240 23.121-23.611 23.360Q-23.982 23.599-24.406 23.599Q-24.843 23.599-25.243 23.391Q-25.643 23.182-25.882 22.815Q-26.122 22.448-26.122 21.996M-25.452 21.726L-23.637 21.726Q-23.637 21.449-23.734 21.197Q-23.832 20.944-24.030 20.788Q-24.228 20.633-24.512 20.633Q-24.789 20.633-25.002 20.791Q-25.216 20.950-25.334 21.205Q-25.452 21.460-25.452 21.726M-21.350 23.531L-22.673 23.531L-22.673 23.251Q-22.112 23.251-21.733 22.851L-21.019 22.054L-21.931 21.005Q-22.068 20.858-22.217 20.826Q-22.365 20.793-22.632 20.793L-22.632 20.513L-21.131 20.513L-21.131 20.793Q-21.323 20.793-21.323 20.927Q-21.323 20.957-21.292 21.005L-20.697 21.689L-20.256 21.193Q-20.144 21.063-20.144 20.947Q-20.144 20.885-20.181 20.839Q-20.219 20.793-20.277 20.793L-20.277 20.513L-18.961 20.513L-18.961 20.793Q-19.521 20.793-19.901 21.193L-20.523 21.894L-19.528 23.042Q-19.429 23.141-19.328 23.186Q-19.228 23.230-19.116 23.240Q-19.005 23.251-18.828 23.251L-18.828 23.531L-20.321 23.531L-20.321 23.251Q-20.256 23.251-20.197 23.217Q-20.137 23.182-20.137 23.117Q-20.137 23.070-20.167 23.042L-20.844 22.256L-21.377 22.851Q-21.490 22.981-21.490 23.097Q-21.490 23.162-21.449 23.206Q-21.408 23.251-21.350 23.251L-21.350 23.531M-16.688 24.888L-18.318 24.888L-18.318 24.608Q-18.089 24.608-17.941 24.573Q-17.792 24.539-17.792 24.399L-17.792 21.053Q-17.792 20.882-17.929 20.841Q-18.065 20.800-18.318 20.800L-18.318 20.520L-17.238 20.445L-17.238 20.851Q-17.016 20.650-16.729 20.547Q-16.442 20.445-16.134 20.445Q-15.707 20.445-15.343 20.658Q-14.979 20.872-14.765 21.236Q-14.552 21.600-14.552 22.020Q-14.552 22.465-14.791 22.829Q-15.030 23.193-15.423 23.396Q-15.816 23.599-16.261 23.599Q-16.527 23.599-16.775 23.499Q-17.023 23.398-17.211 23.217L-17.211 24.399Q-17.211 24.536-17.062 24.572Q-16.914 24.608-16.688 24.608L-16.688 24.888M-17.211 21.200L-17.211 22.810Q-17.078 23.063-16.835 23.220Q-16.592 23.377-16.315 23.377Q-15.987 23.377-15.734 23.176Q-15.481 22.974-15.348 22.656Q-15.215 22.338-15.215 22.020Q-15.215 21.791-15.280 21.562Q-15.345 21.333-15.473 21.135Q-15.601 20.937-15.796 20.817Q-15.991 20.698-16.223 20.698Q-16.517 20.698-16.785 20.827Q-17.054 20.957-17.211 21.200M-11.787 25.281Q-12.337 24.881-12.708 24.326Q-13.079 23.770-13.260 23.124Q-13.441 22.478-13.441 21.781Q-13.441 21.268-13.340 20.773Q-13.239 20.277-13.034 19.826Q-12.829 19.375-12.516 18.983Q-12.204 18.592-11.787 18.288Q-11.776 18.284-11.770 18.283Q-11.763 18.281-11.752 18.281L-11.684 18.281Q-11.650 18.281-11.628 18.305Q-11.605 18.329-11.605 18.366Q-11.605 18.411-11.633 18.428Q-11.981 18.729-12.234 19.113Q-12.487 19.498-12.639 19.939Q-12.791 20.380-12.863 20.836Q-12.935 21.292-12.935 21.781Q-12.935 22.782-12.626 23.669Q-12.316 24.556-11.633 25.141Q-11.605 25.158-11.605 25.202Q-11.605 25.240-11.628 25.264Q-11.650 25.288-11.684 25.288L-11.752 25.288Q-11.759 25.284-11.768 25.283Q-11.776 25.281-11.787 25.281M-9.045 23.531L-10.782 23.531L-10.782 23.251Q-10.553 23.251-10.404 23.217Q-10.255 23.182-10.255 23.042L-10.255 21.193Q-10.255 20.923-10.363 20.862Q-10.471 20.800-10.782 20.800L-10.782 20.520L-9.753 20.445L-9.753 21.152Q-9.623 20.844-9.380 20.645Q-9.138 20.445-8.820 20.445Q-8.601 20.445-8.430 20.569Q-8.259 20.694-8.259 20.906Q-8.259 21.043-8.358 21.142Q-8.457 21.241-8.591 21.241Q-8.728 21.241-8.827 21.142Q-8.926 21.043-8.926 20.906Q-8.926 20.766-8.827 20.667Q-9.117 20.667-9.317 20.863Q-9.517 21.060-9.609 21.354Q-9.702 21.648-9.702 21.928L-9.702 23.042Q-9.702 23.251-9.045 23.251L-9.045 23.531M-7.716 21.996Q-7.716 21.675-7.591 21.386Q-7.466 21.097-7.241 20.874Q-7.015 20.650-6.719 20.530Q-6.424 20.410-6.106 20.410Q-5.778 20.410-5.516 20.510Q-5.255 20.609-5.079 20.791Q-4.903 20.974-4.809 21.232Q-4.715 21.490-4.715 21.822Q-4.715 21.914-4.797 21.935L-7.053 21.935L-7.053 21.996Q-7.053 22.584-6.769 22.967Q-6.485 23.350-5.918 23.350Q-5.597 23.350-5.328 23.157Q-5.060 22.964-4.971 22.649Q-4.964 22.608-4.889 22.594L-4.797 22.594Q-4.715 22.618-4.715 22.690Q-4.715 22.697-4.722 22.724Q-4.834 23.121-5.205 23.360Q-5.576 23.599-6 23.599Q-6.437 23.599-6.837 23.391Q-7.237 23.182-7.477 22.815Q-7.716 22.448-7.716 21.996M-7.046 21.726L-5.231 21.726Q-5.231 21.449-5.328 21.197Q-5.426 20.944-5.624 20.788Q-5.822 20.633-6.106 20.633Q-6.383 20.633-6.596 20.791Q-6.810 20.950-6.928 21.205Q-7.046 21.460-7.046 21.726M-2.739 23.504L-3.720 21.005Q-3.782 20.862-3.900 20.827Q-4.018 20.793-4.233 20.793L-4.233 20.513L-2.753 20.513L-2.753 20.793Q-3.132 20.793-3.132 20.954Q-3.132 20.964-3.119 21.005L-2.404 22.837L-1.731 21.132Q-1.762 21.060-1.762 21.032Q-1.762 21.005-1.789 21.005Q-1.851 20.858-1.968 20.826Q-2.086 20.793-2.298 20.793L-2.298 20.513L-0.900 20.513L-0.900 20.793Q-1.276 20.793-1.276 20.954Q-1.276 20.985-1.270 21.005L-0.514 22.943L0.173 21.193Q0.193 21.142 0.193 21.087Q0.193 20.947 0.081 20.870Q-0.032 20.793-0.172 20.793L-0.172 20.513L1.048 20.513L1.048 20.793Q0.843 20.793 0.687 20.899Q0.532 21.005 0.460 21.193L-0.446 23.504Q-0.480 23.599-0.593 23.599L-0.661 23.599Q-0.770 23.599-0.808 23.504L-1.591 21.501L-2.377 23.504Q-2.411 23.599-2.524 23.599L-2.592 23.599Q-2.702 23.599-2.739 23.504\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(53.9 -82.105)\">\u003Cpath d=\"M1.450 22.803Q1.450 22.471 1.673 22.244Q1.897 22.017 2.241 21.889Q2.584 21.760 2.957 21.708Q3.329 21.655 3.634 21.655L3.634 21.402Q3.634 21.197 3.526 21.017Q3.418 20.838 3.237 20.735Q3.056 20.633 2.848 20.633Q2.441 20.633 2.205 20.725Q2.294 20.762 2.340 20.846Q2.386 20.930 2.386 21.032Q2.386 21.128 2.340 21.207Q2.294 21.285 2.213 21.330Q2.133 21.374 2.044 21.374Q1.894 21.374 1.793 21.277Q1.692 21.179 1.692 21.032Q1.692 20.410 2.848 20.410Q3.059 20.410 3.309 20.474Q3.558 20.537 3.760 20.656Q3.962 20.776 4.088 20.961Q4.215 21.145 4.215 21.388L4.215 22.964Q4.215 23.080 4.276 23.176Q4.338 23.271 4.451 23.271Q4.560 23.271 4.625 23.177Q4.690 23.083 4.690 22.964L4.690 22.516L4.956 22.516L4.956 22.964Q4.956 23.234 4.729 23.399Q4.502 23.565 4.222 23.565Q4.013 23.565 3.876 23.411Q3.740 23.258 3.716 23.042Q3.569 23.309 3.287 23.454Q3.005 23.599 2.680 23.599Q2.403 23.599 2.119 23.524Q1.836 23.449 1.643 23.270Q1.450 23.090 1.450 22.803M2.065 22.803Q2.065 22.977 2.166 23.107Q2.266 23.237 2.422 23.307Q2.577 23.377 2.742 23.377Q2.960 23.377 3.169 23.280Q3.377 23.182 3.505 23.001Q3.634 22.820 3.634 22.594L3.634 21.866Q3.309 21.866 2.943 21.957Q2.577 22.048 2.321 22.260Q2.065 22.471 2.065 22.803M7.123 23.531L5.387 23.531L5.387 23.251Q5.616 23.251 5.765 23.217Q5.913 23.182 5.913 23.042L5.913 21.193Q5.913 20.923 5.806 20.862Q5.698 20.800 5.387 20.800L5.387 20.520L6.416 20.445L6.416 21.152Q6.546 20.844 6.788 20.645Q7.031 20.445 7.349 20.445Q7.568 20.445 7.739 20.569Q7.910 20.694 7.910 20.906Q7.910 21.043 7.810 21.142Q7.711 21.241 7.578 21.241Q7.441 21.241 7.342 21.142Q7.243 21.043 7.243 20.906Q7.243 20.766 7.342 20.667Q7.052 20.667 6.852 20.863Q6.652 21.060 6.559 21.354Q6.467 21.648 6.467 21.928L6.467 23.042Q6.467 23.251 7.123 23.251L7.123 23.531M8.494 22.020Q8.494 21.682 8.634 21.391Q8.774 21.101 9.019 20.887Q9.263 20.674 9.567 20.559Q9.871 20.445 10.196 20.445Q10.466 20.445 10.729 20.544Q10.993 20.643 11.184 20.821L11.184 19.423Q11.184 19.153 11.076 19.091Q10.969 19.030 10.658 19.030L10.658 18.749L11.734 18.674L11.734 22.858Q11.734 23.046 11.789 23.129Q11.844 23.213 11.944 23.232Q12.045 23.251 12.261 23.251L12.261 23.531L11.153 23.599L11.153 23.182Q10.736 23.599 10.111 23.599Q9.680 23.599 9.307 23.387Q8.935 23.176 8.714 22.815Q8.494 22.454 8.494 22.020M10.169 23.377Q10.377 23.377 10.564 23.305Q10.750 23.234 10.904 23.097Q11.057 22.960 11.153 22.782L11.153 21.173Q11.068 21.026 10.922 20.906Q10.777 20.786 10.608 20.727Q10.439 20.667 10.258 20.667Q9.697 20.667 9.429 21.056Q9.160 21.446 9.160 22.027Q9.160 22.598 9.395 22.988Q9.629 23.377 10.169 23.377\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(53.9 -82.105)\">\u003Cpath d=\"M15.788 25.114Q15.788 25.096 15.801 25.049L18.457 18.387Q18.512 18.281 18.618 18.281Q18.683 18.281 18.734 18.332Q18.785 18.384 18.785 18.448Q18.785 18.472 18.784 18.484Q18.782 18.496 18.778 18.513L16.126 25.175Q16.054 25.281 15.966 25.281Q15.897 25.281 15.842 25.230Q15.788 25.178 15.788 25.114\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(53.9 -82.105)\">\u003Cpath d=\"M23.101 23.531L22.834 23.531L22.834 19.423Q22.834 19.153 22.727 19.091Q22.619 19.030 22.308 19.030L22.308 18.749L23.388 18.674L23.388 20.844Q23.597 20.653 23.882 20.549Q24.168 20.445 24.465 20.445Q24.783 20.445 25.080 20.566Q25.377 20.687 25.600 20.903Q25.822 21.118 25.948 21.403Q26.075 21.689 26.075 22.020Q26.075 22.465 25.835 22.829Q25.596 23.193 25.203 23.396Q24.810 23.599 24.366 23.599Q24.171 23.599 23.981 23.543Q23.792 23.487 23.631 23.382Q23.470 23.278 23.330 23.117L23.101 23.531M23.416 21.186L23.416 22.803Q23.552 23.063 23.793 23.220Q24.034 23.377 24.311 23.377Q24.605 23.377 24.817 23.270Q25.029 23.162 25.162 22.970Q25.295 22.779 25.354 22.540Q25.412 22.301 25.412 22.020Q25.412 21.661 25.318 21.357Q25.224 21.053 24.996 20.860Q24.769 20.667 24.403 20.667Q24.103 20.667 23.836 20.803Q23.569 20.940 23.416 21.186\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(53.9 -82.105)\">\u003Cpath d=\"M26.885 21.996Q26.885 21.675 27.010 21.386Q27.135 21.097 27.361 20.874Q27.586 20.650 27.882 20.530Q28.177 20.410 28.495 20.410Q28.823 20.410 29.085 20.510Q29.346 20.609 29.522 20.791Q29.698 20.974 29.792 21.232Q29.886 21.490 29.886 21.822Q29.886 21.914 29.804 21.935L27.549 21.935L27.549 21.996Q27.549 22.584 27.832 22.967Q28.116 23.350 28.683 23.350Q29.005 23.350 29.273 23.157Q29.541 22.964 29.630 22.649Q29.637 22.608 29.712 22.594L29.804 22.594Q29.886 22.618 29.886 22.690Q29.886 22.697 29.880 22.724Q29.767 23.121 29.396 23.360Q29.025 23.599 28.601 23.599Q28.164 23.599 27.764 23.391Q27.364 23.182 27.125 22.815Q26.885 22.448 26.885 21.996M27.555 21.726L29.370 21.726Q29.370 21.449 29.273 21.197Q29.175 20.944 28.977 20.788Q28.779 20.633 28.495 20.633Q28.218 20.633 28.005 20.791Q27.791 20.950 27.673 21.205Q27.555 21.460 27.555 21.726M31.001 22.690L31.001 20.793L30.362 20.793L30.362 20.571Q30.679 20.571 30.896 20.361Q31.113 20.151 31.214 19.841Q31.315 19.532 31.315 19.224L31.582 19.224L31.582 20.513L32.658 20.513L32.658 20.793L31.582 20.793L31.582 22.677Q31.582 22.953 31.686 23.152Q31.790 23.350 32.050 23.350Q32.207 23.350 32.313 23.246Q32.419 23.141 32.469 22.988Q32.518 22.834 32.518 22.677L32.518 22.263L32.785 22.263L32.785 22.690Q32.785 22.916 32.686 23.126Q32.587 23.336 32.402 23.468Q32.217 23.599 31.988 23.599Q31.551 23.599 31.276 23.362Q31.001 23.124 31.001 22.690M33.653 22.803Q33.653 22.471 33.877 22.244Q34.101 22.017 34.444 21.889Q34.788 21.760 35.160 21.708Q35.533 21.655 35.837 21.655L35.837 21.402Q35.837 21.197 35.729 21.017Q35.622 20.838 35.441 20.735Q35.259 20.633 35.051 20.633Q34.644 20.633 34.408 20.725Q34.497 20.762 34.543 20.846Q34.590 20.930 34.590 21.032Q34.590 21.128 34.543 21.207Q34.497 21.285 34.417 21.330Q34.337 21.374 34.248 21.374Q34.097 21.374 33.997 21.277Q33.896 21.179 33.896 21.032Q33.896 20.410 35.051 20.410Q35.263 20.410 35.512 20.474Q35.762 20.537 35.964 20.656Q36.165 20.776 36.292 20.961Q36.418 21.145 36.418 21.388L36.418 22.964Q36.418 23.080 36.480 23.176Q36.541 23.271 36.654 23.271Q36.763 23.271 36.828 23.177Q36.893 23.083 36.893 22.964L36.893 22.516L37.160 22.516L37.160 22.964Q37.160 23.234 36.933 23.399Q36.705 23.565 36.425 23.565Q36.216 23.565 36.080 23.411Q35.943 23.258 35.919 23.042Q35.772 23.309 35.490 23.454Q35.208 23.599 34.883 23.599Q34.607 23.599 34.323 23.524Q34.039 23.449 33.846 23.270Q33.653 23.090 33.653 22.803M34.268 22.803Q34.268 22.977 34.369 23.107Q34.470 23.237 34.625 23.307Q34.781 23.377 34.945 23.377Q35.164 23.377 35.372 23.280Q35.581 23.182 35.709 23.001Q35.837 22.820 35.837 22.594L35.837 21.866Q35.512 21.866 35.147 21.957Q34.781 22.048 34.525 22.260Q34.268 22.471 34.268 22.803M37.898 25.288L37.830 25.288Q37.796 25.288 37.773 25.262Q37.751 25.237 37.751 25.202Q37.751 25.158 37.782 25.141Q38.137 24.837 38.387 24.447Q38.636 24.057 38.789 23.625Q38.941 23.193 39.011 22.724Q39.081 22.256 39.081 21.781Q39.081 21.302 39.011 20.836Q38.941 20.369 38.787 19.934Q38.633 19.498 38.382 19.110Q38.131 18.722 37.782 18.428Q37.751 18.411 37.751 18.366Q37.751 18.332 37.773 18.307Q37.796 18.281 37.830 18.281L37.898 18.281Q37.908 18.281 37.917 18.283Q37.925 18.284 37.936 18.288Q38.479 18.688 38.852 19.241Q39.224 19.795 39.405 20.441Q39.587 21.087 39.587 21.781Q39.587 22.482 39.405 23.129Q39.224 23.777 38.850 24.331Q38.476 24.885 37.936 25.281Q37.925 25.281 37.917 25.283Q37.908 25.284 37.898 25.288\" 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 closed-form optimum reweights the reference 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=\"mop\">exp\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"mord\">\u002F\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>, tilting mass from low-reward (left) toward high-reward (right) responses; smaller \u003Cspan class=\"katex\">\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"base\">\u003Cspan class=\"strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0528em;\">β\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> tilts harder.\u003C\u002Ffigcaption>","\u003Csvg style=\"width:100%;max-width:579.423px;height:auto\" xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" viewBox=\"-75 -75 434.567 147.995\">\u003Cg stroke=\"currentColor\" style=\"stroke-miterlimit:10;stroke-width:.4\">\u003Cg fill=\"var(--tk-soft-neutral)\" style=\"stroke-width:.8\">\u003Cpath d=\"M-65.203 23.8H3.083V-7.498h-68.286Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\">\u003Cg transform=\"translate(-19.254 5.556)\">\u003Cpath d=\"M-29.102 1.508L-30.732 1.508L-30.732 1.228Q-30.503 1.228-30.354 1.193Q-30.206 1.159-30.206 1.019L-30.206-2.327Q-30.206-2.498-30.342-2.539Q-30.479-2.580-30.732-2.580L-30.732-2.860L-29.652-2.935L-29.652-2.529Q-29.430-2.730-29.143-2.833Q-28.855-2.935-28.548-2.935Q-28.121-2.935-27.757-2.722Q-27.393-2.508-27.179-2.144Q-26.965-1.780-26.965-1.360Q-26.965-0.915-27.205-0.551Q-27.444-0.187-27.837 0.016Q-28.230 0.219-28.674 0.219Q-28.941 0.219-29.189 0.119Q-29.436 0.018-29.624-0.163L-29.624 1.019Q-29.624 1.156-29.476 1.192Q-29.327 1.228-29.102 1.228L-29.102 1.508M-29.624-2.180L-29.624-0.570Q-29.491-0.317-29.248-0.160Q-29.006-0.003-28.729-0.003Q-28.401-0.003-28.148-0.204Q-27.895-0.406-27.762-0.724Q-27.628-1.042-27.628-1.360Q-27.628-1.589-27.693-1.818Q-27.758-2.047-27.886-2.245Q-28.015-2.443-28.209-2.563Q-28.404-2.682-28.637-2.682Q-28.931-2.682-29.199-2.553Q-29.467-2.423-29.624-2.180M-24.580 0.151L-26.316 0.151L-26.316-0.129Q-26.087-0.129-25.938-0.163Q-25.789-0.198-25.789-0.338L-25.789-2.187Q-25.789-2.457-25.897-2.518Q-26.005-2.580-26.316-2.580L-26.316-2.860L-25.287-2.935L-25.287-2.228Q-25.157-2.536-24.914-2.735Q-24.672-2.935-24.354-2.935Q-24.135-2.935-23.964-2.811Q-23.793-2.686-23.793-2.474Q-23.793-2.337-23.893-2.238Q-23.992-2.139-24.125-2.139Q-24.262-2.139-24.361-2.238Q-24.460-2.337-24.460-2.474Q-24.460-2.614-24.361-2.713Q-24.651-2.713-24.851-2.517Q-25.051-2.320-25.143-2.026Q-25.236-1.732-25.236-1.452L-25.236-0.338Q-25.236-0.129-24.580-0.129L-24.580 0.151M-23.250-1.384Q-23.250-1.705-23.125-1.994Q-23-2.283-22.775-2.506Q-22.549-2.730-22.254-2.850Q-21.958-2.970-21.640-2.970Q-21.312-2.970-21.050-2.870Q-20.789-2.771-20.613-2.589Q-20.437-2.406-20.343-2.148Q-20.249-1.890-20.249-1.558Q-20.249-1.466-20.331-1.445L-22.587-1.445L-22.587-1.384Q-22.587-0.796-22.303-0.413Q-22.019-0.030-21.452-0.030Q-21.131-0.030-20.862-0.223Q-20.594-0.416-20.505-0.731Q-20.498-0.772-20.423-0.786L-20.331-0.786Q-20.249-0.762-20.249-0.690Q-20.249-0.683-20.256-0.656Q-20.369-0.259-20.739-0.020Q-21.110 0.219-21.534 0.219Q-21.972 0.219-22.372 0.011Q-22.771-0.198-23.011-0.565Q-23.250-0.932-23.250-1.384M-22.580-1.654L-20.765-1.654Q-20.765-1.931-20.862-2.183Q-20.960-2.436-21.158-2.592Q-21.356-2.747-21.640-2.747Q-21.917-2.747-22.131-2.589Q-22.344-2.430-22.462-2.175Q-22.580-1.920-22.580-1.654M-17.863 0.151L-19.596 0.151L-19.596-0.129Q-19.371-0.129-19.222-0.163Q-19.073-0.198-19.073-0.338L-19.073-2.587L-19.661-2.587L-19.661-2.867L-19.073-2.867L-19.073-3.684Q-19.073-4.002-18.895-4.250Q-18.718-4.497-18.427-4.638Q-18.137-4.778-17.826-4.778Q-17.569-4.778-17.366-4.636Q-17.163-4.494-17.163-4.251Q-17.163-4.115-17.262-4.016Q-17.361-3.916-17.497-3.916Q-17.634-3.916-17.733-4.016Q-17.832-4.115-17.832-4.251Q-17.832-4.432-17.692-4.525Q-17.771-4.552-17.870-4.552Q-18.079-4.552-18.232-4.419Q-18.386-4.286-18.466-4.082Q-18.547-3.879-18.547-3.670L-18.547-2.867L-17.658-2.867L-17.658-2.587L-18.519-2.587L-18.519-0.338Q-18.519-0.129-17.863-0.129L-17.863 0.151M-17.224-1.384Q-17.224-1.705-17.099-1.994Q-16.975-2.283-16.749-2.506Q-16.523-2.730-16.228-2.850Q-15.932-2.970-15.614-2.970Q-15.286-2.970-15.025-2.870Q-14.763-2.771-14.587-2.589Q-14.411-2.406-14.317-2.148Q-14.223-1.890-14.223-1.558Q-14.223-1.466-14.305-1.445L-16.561-1.445L-16.561-1.384Q-16.561-0.796-16.277-0.413Q-15.994-0.030-15.426-0.030Q-15.105-0.030-14.837-0.223Q-14.568-0.416-14.479-0.731Q-14.473-0.772-14.397-0.786L-14.305-0.786Q-14.223-0.762-14.223-0.690Q-14.223-0.683-14.230-0.656Q-14.343-0.259-14.714-0.020Q-15.084 0.219-15.508 0.219Q-15.946 0.219-16.346 0.011Q-16.746-0.198-16.985-0.565Q-17.224-0.932-17.224-1.384M-16.554-1.654L-14.739-1.654Q-14.739-1.931-14.837-2.183Q-14.934-2.436-15.132-2.592Q-15.331-2.747-15.614-2.747Q-15.891-2.747-16.105-2.589Q-16.318-2.430-16.436-2.175Q-16.554-1.920-16.554-1.654M-11.885 0.151L-13.622 0.151L-13.622-0.129Q-13.393-0.129-13.244-0.163Q-13.095-0.198-13.095-0.338L-13.095-2.187Q-13.095-2.457-13.203-2.518Q-13.310-2.580-13.622-2.580L-13.622-2.860L-12.593-2.935L-12.593-2.228Q-12.463-2.536-12.220-2.735Q-11.977-2.935-11.660-2.935Q-11.441-2.935-11.270-2.811Q-11.099-2.686-11.099-2.474Q-11.099-2.337-11.198-2.238Q-11.297-2.139-11.431-2.139Q-11.567-2.139-11.666-2.238Q-11.766-2.337-11.766-2.474Q-11.766-2.614-11.666-2.713Q-11.957-2.713-12.157-2.517Q-12.357-2.320-12.449-2.026Q-12.541-1.732-12.541-1.452L-12.541-0.338Q-12.541-0.129-11.885-0.129L-11.885 0.151M-10.556-1.384Q-10.556-1.705-10.431-1.994Q-10.306-2.283-10.081-2.506Q-9.855-2.730-9.559-2.850Q-9.264-2.970-8.946-2.970Q-8.618-2.970-8.356-2.870Q-8.095-2.771-7.919-2.589Q-7.743-2.406-7.649-2.148Q-7.555-1.890-7.555-1.558Q-7.555-1.466-7.637-1.445L-9.893-1.445L-9.893-1.384Q-9.893-0.796-9.609-0.413Q-9.325-0.030-8.758-0.030Q-8.436-0.030-8.168-0.223Q-7.900-0.416-7.811-0.731Q-7.804-0.772-7.729-0.786L-7.637-0.786Q-7.555-0.762-7.555-0.690Q-7.555-0.683-7.561-0.656Q-7.674-0.259-8.045-0.020Q-8.416 0.219-8.840 0.219Q-9.277 0.219-9.677 0.011Q-10.077-0.198-10.316-0.565Q-10.556-0.932-10.556-1.384M-9.886-1.654L-8.071-1.654Q-8.071-1.931-8.168-2.183Q-8.266-2.436-8.464-2.592Q-8.662-2.747-8.946-2.747Q-9.223-2.747-9.436-2.589Q-9.650-2.430-9.768-2.175Q-9.886-1.920-9.886-1.654M-5.285 0.151L-6.919 0.151L-6.919-0.129Q-6.690-0.129-6.541-0.163Q-6.393-0.198-6.393-0.338L-6.393-2.187Q-6.393-2.457-6.500-2.518Q-6.608-2.580-6.919-2.580L-6.919-2.860L-5.859-2.935L-5.859-2.286Q-5.688-2.594-5.384-2.765Q-5.080-2.935-4.735-2.935Q-4.229-2.935-3.945-2.712Q-3.662-2.488-3.662-1.992L-3.662-0.338Q-3.662-0.201-3.513-0.165Q-3.364-0.129-3.139-0.129L-3.139 0.151L-4.769 0.151L-4.769-0.129Q-4.540-0.129-4.391-0.163Q-4.243-0.198-4.243-0.338L-4.243-1.978Q-4.243-2.313-4.362-2.513Q-4.482-2.713-4.796-2.713Q-5.066-2.713-5.300-2.577Q-5.535-2.440-5.673-2.206Q-5.811-1.972-5.811-1.698L-5.811-0.338Q-5.811-0.201-5.661-0.165Q-5.511-0.129-5.285-0.129L-5.285 0.151M-2.551-1.360Q-2.551-1.688-2.416-1.989Q-2.281-2.289-2.045-2.510Q-1.809-2.730-1.505-2.850Q-1.201-2.970-0.876-2.970Q-0.370-2.970-0.021-2.867Q0.327-2.765 0.327-2.389Q0.327-2.242 0.230-2.141Q0.132-2.040-0.015-2.040Q-0.168-2.040-0.268-2.139Q-0.367-2.238-0.367-2.389Q-0.367-2.577-0.227-2.669Q-0.428-2.720-0.869-2.720Q-1.225-2.720-1.454-2.524Q-1.683-2.327-1.783-2.018Q-1.884-1.708-1.884-1.360Q-1.884-1.011-1.758-0.705Q-1.631-0.399-1.377-0.215Q-1.122-0.030-0.767-0.030Q-0.544-0.030-0.360-0.114Q-0.175-0.198-0.040-0.353Q0.095-0.509 0.153-0.717Q0.167-0.772 0.221-0.772L0.334-0.772Q0.365-0.772 0.387-0.748Q0.409-0.724 0.409-0.690L0.409-0.669Q0.324-0.382 0.136-0.184Q-0.052 0.014-0.317 0.117Q-0.582 0.219-0.876 0.219Q-1.307 0.219-1.695 0.013Q-2.082-0.194-2.317-0.557Q-2.551-0.919-2.551-1.360M0.956-1.384Q0.956-1.705 1.081-1.994Q1.206-2.283 1.431-2.506Q1.657-2.730 1.952-2.850Q2.248-2.970 2.566-2.970Q2.894-2.970 3.156-2.870Q3.417-2.771 3.593-2.589Q3.769-2.406 3.863-2.148Q3.957-1.890 3.957-1.558Q3.957-1.466 3.875-1.445L1.619-1.445L1.619-1.384Q1.619-0.796 1.903-0.413Q2.187-0.030 2.754-0.030Q3.075-0.030 3.344-0.223Q3.612-0.416 3.701-0.731Q3.708-0.772 3.783-0.786L3.875-0.786Q3.957-0.762 3.957-0.690Q3.957-0.683 3.950-0.656Q3.837-0.259 3.467-0.020Q3.096 0.219 2.672 0.219Q2.234 0.219 1.835 0.011Q1.435-0.198 1.195-0.565Q0.956-0.932 0.956-1.384M1.626-1.654L3.441-1.654Q3.441-1.931 3.344-2.183Q3.246-2.436 3.048-2.592Q2.850-2.747 2.566-2.747Q2.289-2.747 2.075-2.589Q1.862-2.430 1.744-2.175Q1.626-1.920 1.626-1.654M4.545 0.144L4.545-0.919Q4.545-0.943 4.572-0.970Q4.600-0.997 4.624-0.997L4.733-0.997Q4.798-0.997 4.812-0.939Q4.907-0.505 5.153-0.254Q5.399-0.003 5.813-0.003Q6.155-0.003 6.408-0.136Q6.661-0.269 6.661-0.577Q6.661-0.734 6.567-0.849Q6.473-0.963 6.334-1.032Q6.196-1.100 6.028-1.138L5.447-1.237Q5.092-1.305 4.818-1.526Q4.545-1.746 4.545-2.088Q4.545-2.337 4.656-2.512Q4.767-2.686 4.953-2.785Q5.140-2.884 5.355-2.927Q5.570-2.970 5.813-2.970Q6.227-2.970 6.507-2.788L6.722-2.963Q6.732-2.966 6.739-2.968Q6.746-2.970 6.756-2.970L6.808-2.970Q6.835-2.970 6.859-2.946Q6.883-2.922 6.883-2.894L6.883-2.047Q6.883-2.026 6.859-1.999Q6.835-1.972 6.808-1.972L6.695-1.972Q6.668-1.972 6.642-1.997Q6.616-2.023 6.616-2.047Q6.616-2.283 6.510-2.447Q6.404-2.611 6.221-2.693Q6.039-2.775 5.806-2.775Q5.478-2.775 5.222-2.672Q4.965-2.570 4.965-2.293Q4.965-2.098 5.148-1.989Q5.331-1.879 5.560-1.838L6.134-1.732Q6.380-1.684 6.594-1.556Q6.808-1.428 6.944-1.225Q7.081-1.021 7.081-0.772Q7.081-0.259 6.715-0.020Q6.350 0.219 5.813 0.219Q5.317 0.219 4.986-0.075L4.719 0.199Q4.699 0.219 4.671 0.219L4.624 0.219Q4.600 0.219 4.572 0.192Q4.545 0.165 4.545 0.144\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-19.254 5.556)\">\u003Cpath d=\"M-26.841 9.901Q-27.391 9.501-27.762 8.946Q-28.133 8.390-28.314 7.744Q-28.495 7.098-28.495 6.401Q-28.495 5.888-28.395 5.393Q-28.294 4.897-28.089 4.446Q-27.884 3.995-27.571 3.603Q-27.258 3.212-26.841 2.908Q-26.831 2.904-26.824 2.903Q-26.817 2.901-26.807 2.901L-26.739 2.901Q-26.704 2.901-26.682 2.925Q-26.660 2.949-26.660 2.986Q-26.660 3.031-26.687 3.048Q-27.036 3.349-27.289 3.733Q-27.542 4.118-27.694 4.559Q-27.846 5-27.918 5.456Q-27.990 5.912-27.990 6.401Q-27.990 7.402-27.680 8.289Q-27.371 9.176-26.687 9.761Q-26.660 9.778-26.660 9.822Q-26.660 9.860-26.682 9.884Q-26.704 9.908-26.739 9.908L-26.807 9.908Q-26.814 9.904-26.822 9.903Q-26.831 9.901-26.841 9.901\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-19.254 5.556)\">\u003Cpath d=\"M-25.425 7.891Q-25.282 7.997-25.053 7.997Q-24.827 7.997-24.653 7.801Q-24.478 7.604-24.424 7.375L-24.109 6.114Q-24.037 5.847-24.037 5.714Q-24.037 5.523-24.149 5.405Q-24.260 5.287-24.451 5.287Q-24.680 5.287-24.878 5.411Q-25.077 5.536-25.218 5.740Q-25.360 5.943-25.418 6.162Q-25.429 6.227-25.487 6.227L-25.599 6.227Q-25.630 6.227-25.654 6.196Q-25.678 6.165-25.678 6.141L-25.678 6.114Q-25.565 5.687-25.212 5.376Q-24.858 5.065-24.437 5.065Q-24.266 5.065-24.102 5.118Q-23.938 5.171-23.805 5.275Q-23.672 5.379-23.597 5.533Q-23.456 5.328-23.255 5.196Q-23.053 5.065-22.834 5.065Q-22.544 5.065-22.315 5.200Q-22.086 5.335-22.086 5.605Q-22.086 5.724-22.139 5.828Q-22.192 5.933-22.289 5.996Q-22.387 6.059-22.506 6.059Q-22.622 6.059-22.704 5.986Q-22.786 5.912-22.786 5.793Q-22.786 5.652-22.696 5.538Q-22.605 5.423-22.472 5.393Q-22.626 5.287-22.848 5.287Q-23.002 5.287-23.132 5.381Q-23.262 5.475-23.350 5.611Q-23.439 5.748-23.487 5.912L-23.802 7.170Q-23.863 7.454-23.863 7.570Q-23.863 7.761-23.752 7.879Q-23.641 7.997-23.450 7.997Q-23.275 7.997-23.116 7.922Q-22.957 7.847-22.827 7.717Q-22.698 7.587-22.609 7.430Q-22.520 7.273-22.486 7.122Q-22.462 7.061-22.407 7.061L-22.298 7.061Q-22.264 7.061-22.241 7.086Q-22.219 7.112-22.219 7.143Q-22.219 7.156-22.226 7.170Q-22.294 7.450-22.479 7.690Q-22.663 7.929-22.923 8.074Q-23.183 8.219-23.467 8.219Q-23.733 8.219-23.962 8.100Q-24.191 7.980-24.304 7.751Q-24.434 7.949-24.637 8.084Q-24.841 8.219-25.070 8.219Q-25.350 8.219-25.581 8.084Q-25.811 7.949-25.811 7.683Q-25.811 7.502-25.690 7.365Q-25.569 7.228-25.391 7.228Q-25.271 7.228-25.191 7.300Q-25.111 7.372-25.111 7.491Q-25.111 7.631-25.198 7.746Q-25.285 7.860-25.425 7.891M-20.763 9.381Q-20.763 9.347-20.736 9.320Q-20.469 9.098-20.319 8.771Q-20.168 8.445-20.168 8.089L-20.168 8.052Q-20.271 8.151-20.442 8.151Q-20.619 8.151-20.741 8.030Q-20.862 7.908-20.862 7.731Q-20.862 7.560-20.741 7.438Q-20.619 7.317-20.442 7.317Q-20.182 7.317-20.062 7.556Q-19.943 7.796-19.943 8.089Q-19.943 8.493-20.114 8.862Q-20.285 9.231-20.582 9.487Q-20.613 9.508-20.637 9.508Q-20.681 9.508-20.722 9.467Q-20.763 9.426-20.763 9.381\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-19.254 5.556)\">\u003Cpath d=\"M-17.283 8.964Q-17.283 8.773-17.167 8.633Q-17.051 8.493-16.863 8.493Q-16.743 8.493-16.660 8.568Q-16.576 8.643-16.576 8.759Q-16.576 8.913-16.682 9.029Q-16.788 9.146-16.942 9.166Q-16.781 9.354-16.436 9.354Q-15.947 9.354-15.554 8.838Q-15.410 8.647-15.322 8.431Q-15.233 8.216-15.161 7.919Q-15.496 8.219-15.896 8.219Q-16.343 8.219-16.620 7.995Q-16.897 7.772-16.897 7.331Q-16.897 7.020-16.790 6.700Q-16.682 6.380-16.470 5.861Q-16.395 5.659-16.395 5.519Q-16.395 5.420-16.434 5.353Q-16.473 5.287-16.562 5.287Q-16.843 5.287-17.032 5.558Q-17.222 5.830-17.311 6.162Q-17.321 6.227-17.383 6.227L-17.492 6.227Q-17.523 6.227-17.547 6.196Q-17.571 6.165-17.571 6.141L-17.571 6.114Q-17.502 5.854-17.362 5.617Q-17.222 5.379-17.012 5.222Q-16.802 5.065-16.549 5.065Q-16.367 5.065-16.212 5.136Q-16.056 5.208-15.959 5.343Q-15.862 5.478-15.862 5.659Q-15.862 5.776-15.909 5.912Q-16.125 6.442-16.238 6.784Q-16.350 7.126-16.350 7.437Q-16.350 7.679-16.236 7.838Q-16.121 7.997-15.882 7.997Q-15.400 7.997-15.034 7.402L-14.532 5.393Q-14.501 5.280-14.409 5.206Q-14.317 5.133-14.204 5.133Q-14.101 5.133-14.030 5.196Q-13.958 5.259-13.958 5.359Q-13.958 5.382-13.959 5.396Q-13.961 5.410-13.965 5.427L-14.652 8.178Q-14.747 8.561-15.022 8.884Q-15.298 9.207-15.677 9.393Q-16.056 9.580-16.449 9.580Q-16.767 9.580-17.025 9.422Q-17.283 9.265-17.283 8.964\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-19.254 5.556)\">\u003Cpath d=\"M-12.794 8.506Q-12.794 8.399-12.768 8.286Q-12.743 8.172-12.690 8.021Q-12.638 7.869-12.564 7.692Q-12.491 7.515-12.489 7.505Q-12.445 7.401-12.445 7.305Q-12.445 7.127-12.584 7.127Q-12.791 7.127-12.944 7.315Q-13.097 7.503-13.160 7.740Q-13.177 7.772-13.219 7.786L-13.324 7.786Q-13.385 7.769-13.385 7.715Q-13.385 7.711-13.380 7.686Q-13.341 7.545-13.270 7.415Q-13.199 7.286-13.089 7.176Q-12.979 7.066-12.848 7.004Q-12.716 6.942-12.569 6.942Q-12.350 6.942-12.180 7.055Q-12.010 7.169-12.010 7.381Q-12.010 7.471-12.049 7.562Q-12.091 7.659-12.147 7.795Q-12.203 7.930-12.251 8.065Q-12.298 8.199-12.326 8.322Q-12.354 8.445-12.354 8.555Q-12.354 8.790-12.213 8.906Q-12.071 9.022-11.829 9.022Q-11.519 9.022-11.324 8.602Q-11.324 8.572-11.325 8.554Q-11.327 8.536-11.329 8.516Q-11.329 8.409-11.295 8.272L-11.024 7.191Q-11.004 7.105-10.930 7.050Q-10.855 6.995-10.770 6.995Q-10.689 6.995-10.632 7.047Q-10.575 7.098-10.575 7.176Q-10.575 7.193-10.576 7.204Q-10.577 7.215-10.579 7.227L-10.843 8.287Q-10.882 8.431-10.890 8.575Q-10.890 8.790-10.760 8.906Q-10.631 9.022-10.409 9.022Q-10.089 9.022-9.845 8.621Q-9.747 8.460-9.648 8.196Q-9.549 7.933-9.549 7.781Q-9.549 7.637-9.593 7.579Q-9.637 7.520-9.738 7.421Q-9.840 7.322-9.840 7.242Q-9.840 7.127-9.747 7.032Q-9.654 6.937-9.539 6.937Q-9.388 6.937-9.314 7.071Q-9.239 7.205-9.239 7.371Q-9.239 7.542-9.295 7.781Q-9.351 8.021-9.442 8.259Q-9.532 8.497-9.620 8.641Q-9.954 9.207-10.413 9.207Q-11.048 9.207-11.244 8.856Q-11.497 9.207-11.839 9.207Q-12.088 9.207-12.306 9.142Q-12.523 9.078-12.658 8.920Q-12.794 8.763-12.794 8.506\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.150\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-19.254 5.556)\">\u003Cpath d=\"M-7.281 9.381Q-7.281 9.347-7.253 9.320Q-6.987 9.098-6.836 8.771Q-6.686 8.445-6.686 8.089L-6.686 8.052Q-6.789 8.151-6.960 8.151Q-7.137 8.151-7.259 8.030Q-7.380 7.908-7.380 7.731Q-7.380 7.560-7.259 7.438Q-7.137 7.317-6.960 7.317Q-6.700 7.317-6.580 7.556Q-6.461 7.796-6.461 8.089Q-6.461 8.493-6.631 8.862Q-6.802 9.231-7.100 9.487Q-7.130 9.508-7.154 9.508Q-7.199 9.508-7.240 9.467Q-7.281 9.426-7.281 9.381\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-19.254 5.556)\">\u003Cpath d=\"M-3.807 8.964Q-3.807 8.773-3.691 8.633Q-3.575 8.493-3.387 8.493Q-3.267 8.493-3.184 8.568Q-3.100 8.643-3.100 8.759Q-3.100 8.913-3.206 9.029Q-3.312 9.146-3.466 9.166Q-3.305 9.354-2.960 9.354Q-2.471 9.354-2.078 8.838Q-1.934 8.647-1.846 8.431Q-1.757 8.216-1.685 7.919Q-2.020 8.219-2.420 8.219Q-2.867 8.219-3.144 7.995Q-3.421 7.772-3.421 7.331Q-3.421 7.020-3.314 6.700Q-3.206 6.380-2.994 5.861Q-2.919 5.659-2.919 5.519Q-2.919 5.420-2.958 5.353Q-2.997 5.287-3.086 5.287Q-3.367 5.287-3.556 5.558Q-3.746 5.830-3.835 6.162Q-3.845 6.227-3.907 6.227L-4.016 6.227Q-4.047 6.227-4.071 6.196Q-4.095 6.165-4.095 6.141L-4.095 6.114Q-4.026 5.854-3.886 5.617Q-3.746 5.379-3.536 5.222Q-3.326 5.065-3.073 5.065Q-2.891 5.065-2.736 5.136Q-2.580 5.208-2.483 5.343Q-2.386 5.478-2.386 5.659Q-2.386 5.776-2.433 5.912Q-2.649 6.442-2.762 6.784Q-2.874 7.126-2.874 7.437Q-2.874 7.679-2.760 7.838Q-2.645 7.997-2.406 7.997Q-1.924 7.997-1.558 7.402L-1.056 5.393Q-1.025 5.280-0.933 5.206Q-0.841 5.133-0.728 5.133Q-0.625 5.133-0.554 5.196Q-0.482 5.259-0.482 5.359Q-0.482 5.382-0.483 5.396Q-0.485 5.410-0.489 5.427L-1.176 8.178Q-1.271 8.561-1.546 8.884Q-1.822 9.207-2.201 9.393Q-2.580 9.580-2.973 9.580Q-3.291 9.580-3.549 9.422Q-3.807 9.265-3.807 8.964\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-19.254 5.556)\">\u003Cpath d=\"M0.167 8.782Q0.167 8.719 0.182 8.653L0.792 6.212Q0.812 6.143 0.812 6.102Q0.812 6.038 0.462 6.038Q0.436 6.038 0.411 6.009Q0.387 5.980 0.387 5.953Q0.387 5.816 0.482 5.796L1.251 5.743Q1.290 5.743 1.316 5.764Q1.341 5.784 1.341 5.821L1.341 5.843L0.626 8.692Q0.611 8.758 0.611 8.826Q0.611 8.931 0.660 9.007Q0.709 9.083 0.807 9.083Q0.919 9.083 0.998 8.980Q1.078 8.878 1.119 8.760Q1.161 8.643 1.207 8.472Q1.212 8.445 1.263 8.421L1.368 8.421Q1.432 8.443 1.432 8.492Q1.432 8.497 1.427 8.521Q1.388 8.687 1.305 8.860Q1.222 9.034 1.096 9.151Q0.970 9.268 0.797 9.268Q0.646 9.268 0.500 9.214Q0.355 9.161 0.261 9.052Q0.167 8.943 0.167 8.782\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.150\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(-19.254 5.556)\">\u003Cpath d=\"M3.185 9.908L3.116 9.908Q3.082 9.908 3.060 9.882Q3.038 9.857 3.038 9.822Q3.038 9.778 3.069 9.761Q3.424 9.457 3.674 9.067Q3.923 8.677 4.075 8.245Q4.227 7.813 4.297 7.344Q4.367 6.876 4.367 6.401Q4.367 5.922 4.297 5.456Q4.227 4.989 4.073 4.554Q3.920 4.118 3.668 3.730Q3.417 3.342 3.069 3.048Q3.038 3.031 3.038 2.986Q3.038 2.952 3.060 2.927Q3.082 2.901 3.116 2.901L3.185 2.901Q3.195 2.901 3.204 2.903Q3.212 2.904 3.222 2.908Q3.766 3.308 4.138 3.861Q4.511 4.415 4.692 5.061Q4.873 5.707 4.873 6.401Q4.873 7.102 4.692 7.749Q4.511 8.397 4.137 8.951Q3.762 9.505 3.222 9.901Q3.212 9.901 3.204 9.903Q3.195 9.904 3.185 9.908\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-neutral)\" style=\"stroke-width:.8\">\u003Cpath d=\"M54.298-21.724h68.287v-31.298H54.298Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(95.8 -43.094)\">\u003Cpath d=\"M-28.996 8.151L-30.732 8.151L-30.732 7.871Q-30.503 7.871-30.354 7.837Q-30.206 7.802-30.206 7.662L-30.206 5.813Q-30.206 5.543-30.313 5.482Q-30.421 5.420-30.732 5.420L-30.732 5.140L-29.703 5.065L-29.703 5.772Q-29.573 5.464-29.331 5.265Q-29.088 5.065-28.770 5.065Q-28.551 5.065-28.380 5.189Q-28.209 5.314-28.209 5.526Q-28.209 5.663-28.309 5.762Q-28.408 5.861-28.541 5.861Q-28.678 5.861-28.777 5.762Q-28.876 5.663-28.876 5.526Q-28.876 5.386-28.777 5.287Q-29.067 5.287-29.267 5.483Q-29.467 5.680-29.560 5.974Q-29.652 6.268-29.652 6.548L-29.652 7.662Q-29.652 7.871-28.996 7.871L-28.996 8.151M-27.666 6.616Q-27.666 6.295-27.541 6.006Q-27.416 5.717-27.191 5.494Q-26.965 5.270-26.670 5.150Q-26.374 5.030-26.056 5.030Q-25.728 5.030-25.466 5.130Q-25.205 5.229-25.029 5.411Q-24.853 5.594-24.759 5.852Q-24.665 6.110-24.665 6.442Q-24.665 6.534-24.747 6.555L-27.003 6.555L-27.003 6.616Q-27.003 7.204-26.719 7.587Q-26.435 7.970-25.868 7.970Q-25.547 7.970-25.279 7.777Q-25.010 7.584-24.921 7.269Q-24.914 7.228-24.839 7.214L-24.747 7.214Q-24.665 7.238-24.665 7.310Q-24.665 7.317-24.672 7.344Q-24.785 7.741-25.155 7.980Q-25.526 8.219-25.950 8.219Q-26.388 8.219-26.788 8.011Q-27.187 7.802-27.427 7.435Q-27.666 7.068-27.666 6.616M-26.996 6.346L-25.181 6.346Q-25.181 6.069-25.279 5.817Q-25.376 5.564-25.574 5.408Q-25.772 5.253-26.056 5.253Q-26.333 5.253-26.547 5.411Q-26.760 5.570-26.878 5.825Q-26.996 6.080-26.996 6.346M-22.689 8.124L-23.670 5.625Q-23.732 5.482-23.850 5.447Q-23.968 5.413-24.183 5.413L-24.183 5.133L-22.703 5.133L-22.703 5.413Q-23.082 5.413-23.082 5.574Q-23.082 5.584-23.069 5.625L-22.354 7.457L-21.681 5.752Q-21.712 5.680-21.712 5.652Q-21.712 5.625-21.739 5.625Q-21.801 5.478-21.919 5.446Q-22.037 5.413-22.248 5.413L-22.248 5.133L-20.851 5.133L-20.851 5.413Q-21.227 5.413-21.227 5.574Q-21.227 5.605-21.220 5.625L-20.464 7.563L-19.777 5.813Q-19.757 5.762-19.757 5.707Q-19.757 5.567-19.870 5.490Q-19.982 5.413-20.122 5.413L-20.122 5.133L-18.902 5.133L-18.902 5.413Q-19.107 5.413-19.263 5.519Q-19.418 5.625-19.490 5.813L-20.396 8.124Q-20.430 8.219-20.543 8.219L-20.611 8.219Q-20.721 8.219-20.758 8.124L-21.541 6.121L-22.327 8.124Q-22.361 8.219-22.474 8.219L-22.542 8.219Q-22.652 8.219-22.689 8.124\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(95.8 -43.094)\">\u003Cpath d=\"M-18.513 7.423Q-18.513 7.091-18.290 6.864Q-18.066 6.637-17.722 6.509Q-17.379 6.380-17.006 6.328Q-16.634 6.275-16.329 6.275L-16.329 6.022Q-16.329 5.817-16.437 5.637Q-16.545 5.458-16.726 5.355Q-16.907 5.253-17.115 5.253Q-17.522 5.253-17.758 5.345Q-17.669 5.382-17.623 5.466Q-17.577 5.550-17.577 5.652Q-17.577 5.748-17.623 5.827Q-17.669 5.905-17.750 5.950Q-17.830 5.994-17.919 5.994Q-18.069 5.994-18.170 5.897Q-18.271 5.799-18.271 5.652Q-18.271 5.030-17.115 5.030Q-16.904 5.030-16.654 5.094Q-16.405 5.157-16.203 5.276Q-16.001 5.396-15.875 5.581Q-15.748 5.765-15.748 6.008L-15.748 7.584Q-15.748 7.700-15.687 7.796Q-15.625 7.891-15.512 7.891Q-15.403 7.891-15.338 7.797Q-15.273 7.703-15.273 7.584L-15.273 7.136L-15.007 7.136L-15.007 7.584Q-15.007 7.854-15.234 8.019Q-15.461 8.185-15.741 8.185Q-15.950 8.185-16.087 8.031Q-16.223 7.878-16.247 7.662Q-16.394 7.929-16.676 8.074Q-16.958 8.219-17.283 8.219Q-17.560 8.219-17.844 8.144Q-18.127 8.069-18.320 7.890Q-18.513 7.710-18.513 7.423M-17.898 7.423Q-17.898 7.597-17.797 7.727Q-17.697 7.857-17.541 7.927Q-17.386 7.997-17.221 7.997Q-17.003 7.997-16.794 7.900Q-16.586 7.802-16.458 7.621Q-16.329 7.440-16.329 7.214L-16.329 6.486Q-16.654 6.486-17.020 6.577Q-17.386 6.668-17.642 6.880Q-17.898 7.091-17.898 7.423M-12.840 8.151L-14.576 8.151L-14.576 7.871Q-14.347 7.871-14.198 7.837Q-14.050 7.802-14.050 7.662L-14.050 5.813Q-14.050 5.543-14.157 5.482Q-14.265 5.420-14.576 5.420L-14.576 5.140L-13.547 5.065L-13.547 5.772Q-13.417 5.464-13.175 5.265Q-12.932 5.065-12.614 5.065Q-12.395 5.065-12.224 5.189Q-12.053 5.314-12.053 5.526Q-12.053 5.663-12.153 5.762Q-12.252 5.861-12.385 5.861Q-12.522 5.861-12.621 5.762Q-12.720 5.663-12.720 5.526Q-12.720 5.386-12.621 5.287Q-12.911 5.287-13.111 5.483Q-13.311 5.680-13.404 5.974Q-13.496 6.268-13.496 6.548L-13.496 7.662Q-13.496 7.871-12.840 7.871L-12.840 8.151M-11.469 6.640Q-11.469 6.302-11.329 6.011Q-11.189 5.721-10.944 5.507Q-10.700 5.294-10.396 5.179Q-10.092 5.065-9.767 5.065Q-9.497 5.065-9.234 5.164Q-8.970 5.263-8.779 5.441L-8.779 4.043Q-8.779 3.773-8.887 3.711Q-8.994 3.650-9.305 3.650L-9.305 3.369L-8.229 3.294L-8.229 7.478Q-8.229 7.666-8.174 7.749Q-8.119 7.833-8.019 7.852Q-7.918 7.871-7.702 7.871L-7.702 8.151L-8.810 8.219L-8.810 7.802Q-9.227 8.219-9.852 8.219Q-10.283 8.219-10.656 8.007Q-11.028 7.796-11.249 7.435Q-11.469 7.074-11.469 6.640M-9.794 7.997Q-9.586 7.997-9.399 7.925Q-9.213 7.854-9.059 7.717Q-8.906 7.580-8.810 7.402L-8.810 5.793Q-8.895 5.646-9.041 5.526Q-9.186 5.406-9.355 5.347Q-9.524 5.287-9.705 5.287Q-10.266 5.287-10.534 5.676Q-10.803 6.066-10.803 6.647Q-10.803 7.218-10.568 7.608Q-10.334 7.997-9.794 7.997\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(95.8 -43.094)\">\u003Cpath d=\"M-2.668 8.151L-4.302 8.151L-4.302 7.871Q-4.073 7.871-3.924 7.837Q-3.775 7.802-3.775 7.662L-3.775 5.813Q-3.775 5.543-3.883 5.482Q-3.991 5.420-4.302 5.420L-4.302 5.140L-3.242 5.065L-3.242 5.714Q-3.071 5.406-2.767 5.235Q-2.463 5.065-2.118 5.065Q-1.718 5.065-1.441 5.205Q-1.164 5.345-1.079 5.693Q-0.911 5.400-0.612 5.232Q-0.313 5.065 0.032 5.065Q0.538 5.065 0.822 5.288Q1.106 5.512 1.106 6.008L1.106 7.662Q1.106 7.799 1.254 7.835Q1.403 7.871 1.628 7.871L1.628 8.151L-0.002 8.151L-0.002 7.871Q0.224 7.871 0.374 7.835Q0.524 7.799 0.524 7.662L0.524 6.022Q0.524 5.687 0.405 5.487Q0.285 5.287-0.029 5.287Q-0.299 5.287-0.533 5.423Q-0.768 5.560-0.906 5.794Q-1.044 6.028-1.044 6.302L-1.044 7.662Q-1.044 7.799-0.896 7.835Q-0.747 7.871-0.521 7.871L-0.521 8.151L-2.152 8.151L-2.152 7.871Q-1.923 7.871-1.774 7.837Q-1.625 7.802-1.625 7.662L-1.625 6.022Q-1.625 5.687-1.745 5.487Q-1.865 5.287-2.179 5.287Q-2.449 5.287-2.683 5.423Q-2.917 5.560-3.056 5.794Q-3.194 6.028-3.194 6.302L-3.194 7.662Q-3.194 7.799-3.044 7.835Q-2.893 7.871-2.668 7.871L-2.668 8.151M2.175 6.668Q2.175 6.326 2.310 6.027Q2.445 5.728 2.685 5.504Q2.924 5.280 3.242 5.155Q3.560 5.030 3.891 5.030Q4.336 5.030 4.735 5.246Q5.135 5.461 5.369 5.839Q5.604 6.216 5.604 6.668Q5.604 7.009 5.462 7.293Q5.320 7.577 5.076 7.784Q4.831 7.990 4.522 8.105Q4.212 8.219 3.891 8.219Q3.461 8.219 3.059 8.018Q2.657 7.816 2.416 7.464Q2.175 7.112 2.175 6.668M3.891 7.970Q4.493 7.970 4.717 7.592Q4.940 7.214 4.940 6.582Q4.940 5.970 4.706 5.611Q4.472 5.253 3.891 5.253Q2.838 5.253 2.838 6.582Q2.838 7.214 3.064 7.592Q3.290 7.970 3.891 7.970\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(95.8 -43.094)\">\u003Cpath d=\"M6.421 6.640Q6.421 6.302 6.562 6.011Q6.702 5.721 6.946 5.507Q7.190 5.294 7.495 5.179Q7.799 5.065 8.124 5.065Q8.394 5.065 8.657 5.164Q8.920 5.263 9.111 5.441L9.111 4.043Q9.111 3.773 9.004 3.711Q8.896 3.650 8.585 3.650L8.585 3.369L9.662 3.294L9.662 7.478Q9.662 7.666 9.716 7.749Q9.771 7.833 9.872 7.852Q9.973 7.871 10.188 7.871L10.188 8.151L9.081 8.219L9.081 7.802Q8.664 8.219 8.038 8.219Q7.607 8.219 7.235 8.007Q6.862 7.796 6.642 7.435Q6.421 7.074 6.421 6.640M8.096 7.997Q8.305 7.997 8.491 7.925Q8.677 7.854 8.831 7.717Q8.985 7.580 9.081 7.402L9.081 5.793Q8.995 5.646 8.850 5.526Q8.705 5.406 8.535 5.347Q8.366 5.287 8.185 5.287Q7.625 5.287 7.356 5.676Q7.088 6.066 7.088 6.647Q7.088 7.218 7.322 7.608Q7.556 7.997 8.096 7.997M10.796 6.616Q10.796 6.295 10.921 6.006Q11.046 5.717 11.272 5.494Q11.497 5.270 11.793 5.150Q12.088 5.030 12.406 5.030Q12.734 5.030 12.996 5.130Q13.257 5.229 13.433 5.411Q13.609 5.594 13.703 5.852Q13.797 6.110 13.797 6.442Q13.797 6.534 13.715 6.555L11.460 6.555L11.460 6.616Q11.460 7.204 11.743 7.587Q12.027 7.970 12.594 7.970Q12.916 7.970 13.184 7.777Q13.452 7.584 13.541 7.269Q13.548 7.228 13.623 7.214L13.715 7.214Q13.797 7.238 13.797 7.310Q13.797 7.317 13.791 7.344Q13.678 7.741 13.307 7.980Q12.936 8.219 12.512 8.219Q12.075 8.219 11.675 8.011Q11.275 7.802 11.036 7.435Q10.796 7.068 10.796 6.616M11.466 6.346L13.281 6.346Q13.281 6.069 13.184 5.817Q13.086 5.564 12.888 5.408Q12.690 5.253 12.406 5.253Q12.129 5.253 11.916 5.411Q11.702 5.570 11.584 5.825Q11.466 6.080 11.466 6.346M16.053 8.151L14.450 8.151L14.450 7.871Q14.676 7.871 14.825 7.837Q14.973 7.802 14.973 7.662L14.973 4.043Q14.973 3.773 14.866 3.711Q14.758 3.650 14.450 3.650L14.450 3.369L15.527 3.294L15.527 7.662Q15.527 7.799 15.677 7.835Q15.828 7.871 16.053 7.871\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-neutral)\" style=\"stroke-width:.8\">\u003Cpath d=\"M173.8-21.724h68.286v-31.298H173.8Z\"\u002F>\u003Cg fill=\"currentColor\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(219.012 -39.316)\">\u003Cpath d=\"M-26.862 0.151L-28.968 0.151L-28.968-0.129Q-28.247-0.129-28.247-0.338L-28.247-4.139Q-28.247-4.350-28.968-4.350L-28.968-4.631L-26.630-4.631Q-26.319-4.631-25.976-4.557Q-25.632-4.484-25.311-4.328Q-24.989-4.173-24.788-3.927Q-24.586-3.681-24.586-3.356Q-24.586-3.069-24.774-2.838Q-24.962-2.607-25.239-2.459Q-25.516-2.310-25.806-2.228Q-25.471-2.125-25.237-1.893Q-25.003-1.661-24.959-1.332L-24.873-0.724Q-24.846-0.512-24.800-0.343Q-24.754-0.174-24.649-0.054Q-24.545 0.066-24.357 0.066Q-24.142 0.066-24.026-0.119Q-23.909-0.304-23.909-0.536Q-23.892-0.601-23.817-0.618L-23.725-0.618Q-23.643-0.598-23.643-0.516Q-23.643-0.310-23.733-0.124Q-23.824 0.062-23.990 0.177Q-24.155 0.291-24.357 0.291Q-24.695 0.291-24.989 0.190Q-25.283 0.089-25.468-0.133Q-25.653-0.355-25.653-0.703L-25.653-1.312Q-25.653-1.561-25.796-1.748Q-25.940-1.934-26.163-2.033Q-26.387-2.132-26.637-2.132L-27.584-2.132L-27.584-0.338Q-27.584-0.129-26.862-0.129L-26.862 0.151M-27.584-4.139L-27.584-2.354L-26.729-2.354Q-26.442-2.354-26.203-2.397Q-25.964-2.440-25.776-2.549Q-25.588-2.659-25.476-2.857Q-25.365-3.055-25.365-3.356Q-25.365-3.933-25.733-4.142Q-26.100-4.350-26.729-4.350L-27.218-4.350Q-27.406-4.350-27.495-4.316Q-27.584-4.282-27.584-4.139M-19.210 0.151L-23.178 0.151L-23.178-0.129Q-22.457-0.129-22.457-0.338L-22.457-4.139Q-22.457-4.350-23.178-4.350L-23.178-4.631L-20.864-4.631L-20.864-4.350Q-21.182-4.350-21.474-4.315Q-21.766-4.279-21.766-4.139L-21.766-0.338Q-21.766-0.198-21.677-0.163Q-21.589-0.129-21.401-0.129L-20.778-0.129Q-20.365-0.129-20.086-0.232Q-19.808-0.334-19.642-0.536Q-19.476-0.738-19.394-1.025Q-19.312-1.312-19.268-1.719L-19.001-1.719\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(219.012 -39.316)\">\u003Cpath d=\"M-13.473 1.901Q-14.023 1.501-14.394 0.946Q-14.765 0.390-14.946-0.256Q-15.127-0.902-15.127-1.599Q-15.127-2.112-15.027-2.607Q-14.926-3.103-14.721-3.554Q-14.516-4.005-14.203-4.397Q-13.890-4.788-13.473-5.092Q-13.463-5.096-13.456-5.097Q-13.449-5.099-13.439-5.099L-13.371-5.099Q-13.336-5.099-13.314-5.075Q-13.292-5.051-13.292-5.014Q-13.292-4.969-13.319-4.952Q-13.668-4.651-13.921-4.267Q-14.174-3.882-14.326-3.441Q-14.478-3-14.550-2.544Q-14.622-2.088-14.622-1.599Q-14.622-0.598-14.312 0.289Q-14.003 1.176-13.319 1.761Q-13.292 1.778-13.292 1.822Q-13.292 1.860-13.314 1.884Q-13.336 1.908-13.371 1.908L-13.439 1.908Q-13.446 1.904-13.454 1.903Q-13.463 1.901-13.473 1.901M-10.271 0.151L-12.403 0.151L-12.403-0.129Q-11.682-0.129-11.682-0.338L-11.682-4.139Q-11.682-4.350-12.403-4.350L-12.403-4.631L-9.737-4.631Q-9.327-4.631-8.907-4.477Q-8.486-4.323-8.203-4.019Q-7.919-3.715-7.919-3.301Q-7.919-2.983-8.086-2.737Q-8.254-2.491-8.531-2.325Q-8.808-2.160-9.129-2.076Q-9.450-1.992-9.737-1.992L-10.992-1.992L-10.992-0.338Q-10.992-0.129-10.271-0.129L-10.271 0.151M-11.019-4.139L-11.019-2.242L-9.932-2.242Q-9.324-2.242-9.009-2.479Q-8.695-2.717-8.695-3.301Q-8.695-3.694-8.840-3.928Q-8.985-4.162-9.257-4.256Q-9.529-4.350-9.932-4.350L-10.653-4.350Q-10.841-4.350-10.930-4.316Q-11.019-4.282-11.019-4.139M-4.901 0.151L-7.034 0.151L-7.034-0.129Q-6.313-0.129-6.313-0.338L-6.313-4.139Q-6.313-4.350-7.034-4.350L-7.034-4.631L-4.368-4.631Q-3.958-4.631-3.537-4.477Q-3.117-4.323-2.833-4.019Q-2.549-3.715-2.549-3.301Q-2.549-2.983-2.717-2.737Q-2.884-2.491-3.161-2.325Q-3.438-2.160-3.759-2.076Q-4.081-1.992-4.368-1.992L-5.622-1.992L-5.622-0.338Q-5.622-0.129-4.901-0.129L-4.901 0.151M-5.649-4.139L-5.649-2.242L-4.563-2.242Q-3.954-2.242-3.640-2.479Q-3.325-2.717-3.325-3.301Q-3.325-3.694-3.470-3.928Q-3.616-4.162-3.887-4.256Q-4.159-4.350-4.563-4.350L-5.284-4.350Q-5.472-4.350-5.561-4.316Q-5.649-4.282-5.649-4.139M-1.568-2.214Q-1.568-2.901-1.220-3.491Q-0.871-4.080-0.276-4.426Q0.318-4.771 1.009-4.771Q1.699-4.771 2.294-4.426Q2.889-4.080 3.237-3.491Q3.586-2.901 3.586-2.214Q3.586-1.527 3.234-0.953Q2.882-0.379 2.289-0.044Q1.696 0.291 1.009 0.291Q0.496 0.291 0.028 0.098Q-0.440-0.095-0.799-0.437Q-1.158-0.779-1.363-1.240Q-1.568-1.702-1.568-2.214M1.009 0.031Q1.614 0.031 2.021-0.314Q2.427-0.659 2.617-1.197Q2.807-1.736 2.807-2.334Q2.807-2.768 2.696-3.156Q2.584-3.544 2.359-3.851Q2.133-4.159 1.791-4.339Q1.450-4.518 1.009-4.518Q0.571-4.518 0.233-4.342Q-0.105-4.166-0.334-3.851Q-0.563-3.537-0.676-3.144Q-0.789-2.751-0.789-2.334Q-0.789-1.746-0.598-1.203Q-0.406-0.659 0-0.314Q0.407 0.031 1.009 0.031M4.717 1.908L4.649 1.908Q4.615 1.908 4.593 1.882Q4.570 1.857 4.570 1.822Q4.570 1.778 4.601 1.761Q4.957 1.457 5.206 1.067Q5.456 0.677 5.608 0.245Q5.760-0.187 5.830-0.656Q5.900-1.124 5.900-1.599Q5.900-2.078 5.830-2.544Q5.760-3.011 5.606-3.446Q5.452-3.882 5.201-4.270Q4.950-4.658 4.601-4.952Q4.570-4.969 4.570-5.014Q4.570-5.048 4.593-5.073Q4.615-5.099 4.649-5.099L4.717-5.099Q4.728-5.099 4.736-5.097Q4.745-5.096 4.755-5.092Q5.298-4.692 5.671-4.139Q6.043-3.585 6.225-2.939Q6.406-2.293 6.406-1.599Q6.406-0.898 6.225-0.251Q6.043 0.397 5.669 0.951Q5.295 1.505 4.755 1.901Q4.745 1.901 4.736 1.903Q4.728 1.904 4.717 1.908\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(219.012 -39.316)\">\u003Cpath d=\"M-28.162 8.831L-28.162 6.575L-30.411 6.575Q-30.479 6.565-30.525 6.519Q-30.571 6.473-30.571 6.401Q-30.571 6.257-30.411 6.234L-28.162 6.234L-28.162 3.978Q-28.151 3.909-28.105 3.863Q-28.059 3.817-27.987 3.817Q-27.844 3.817-27.820 3.978L-27.820 6.234L-25.578 6.234Q-25.417 6.257-25.417 6.401Q-25.417 6.473-25.463 6.519Q-25.509 6.565-25.578 6.575L-27.820 6.575L-27.820 8.831Q-27.844 8.992-27.987 8.992Q-28.059 8.992-28.105 8.946Q-28.151 8.900-28.162 8.831\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(219.012 -39.316)\">\u003Cpath d=\"M-19.715 8.151L-21.848 8.151L-21.848 7.871Q-21.126 7.871-21.126 7.662L-21.126 3.861Q-21.126 3.650-21.848 3.650L-21.848 3.369L-19.715 3.369L-19.715 3.650Q-20.436 3.650-20.436 3.861L-20.436 6.087L-18.067 4.049Q-17.951 3.940-17.951 3.838Q-17.951 3.745-18.028 3.697Q-18.105 3.650-18.201 3.650L-18.201 3.369L-16.598 3.369L-16.598 3.650Q-16.885 3.650-17.162 3.755Q-17.438 3.861-17.657 4.049L-19.041 5.239L-17.350 7.471Q-17.213 7.649-17.103 7.734Q-16.994 7.819-16.869 7.845Q-16.745 7.871-16.495 7.871L-16.495 8.151L-18.378 8.151L-18.378 7.871Q-18.225 7.871-18.119 7.840Q-18.013 7.809-18.013 7.696Q-18.013 7.611-18.125 7.471L-19.510 5.646L-20.436 6.435L-20.436 7.662Q-20.436 7.871-19.715 7.871L-19.715 8.151M-11.761 8.151L-15.729 8.151L-15.729 7.871Q-15.008 7.871-15.008 7.662L-15.008 3.861Q-15.008 3.650-15.729 3.650L-15.729 3.369L-13.415 3.369L-13.415 3.650Q-13.733 3.650-14.026 3.685Q-14.318 3.721-14.318 3.861L-14.318 7.662Q-14.318 7.802-14.229 7.837Q-14.140 7.871-13.952 7.871L-13.330 7.871Q-12.916 7.871-12.638 7.768Q-12.359 7.666-12.194 7.464Q-12.028 7.262-11.946 6.975Q-11.864 6.688-11.819 6.281L-11.553 6.281\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(219.012 -39.316)\">\u003Cpath d=\"M-6.488 8.151L-8.091 8.151L-8.091 7.871Q-7.865 7.871-7.716 7.837Q-7.568 7.802-7.568 7.662L-7.568 4.043Q-7.568 3.773-7.675 3.711Q-7.783 3.650-8.091 3.650L-8.091 3.369L-7.014 3.294L-7.014 7.662Q-7.014 7.799-6.864 7.835Q-6.713 7.871-6.488 7.871L-6.488 8.151M-5.934 6.616Q-5.934 6.295-5.809 6.006Q-5.684 5.717-5.459 5.494Q-5.233 5.270-4.938 5.150Q-4.642 5.030-4.324 5.030Q-3.996 5.030-3.734 5.130Q-3.473 5.229-3.297 5.411Q-3.121 5.594-3.027 5.852Q-2.933 6.110-2.933 6.442Q-2.933 6.534-3.015 6.555L-5.271 6.555L-5.271 6.616Q-5.271 7.204-4.987 7.587Q-4.703 7.970-4.136 7.970Q-3.815 7.970-3.546 7.777Q-3.278 7.584-3.189 7.269Q-3.182 7.228-3.107 7.214L-3.015 7.214Q-2.933 7.238-2.933 7.310Q-2.933 7.317-2.940 7.344Q-3.053 7.741-3.423 7.980Q-3.794 8.219-4.218 8.219Q-4.656 8.219-5.055 8.011Q-5.455 7.802-5.695 7.435Q-5.934 7.068-5.934 6.616M-5.264 6.346L-3.449 6.346Q-3.449 6.069-3.546 5.817Q-3.644 5.564-3.842 5.408Q-4.040 5.253-4.324 5.253Q-4.601 5.253-4.814 5.411Q-5.028 5.570-5.146 5.825Q-5.264 6.080-5.264 6.346M-2.287 7.423Q-2.287 7.091-2.063 6.864Q-1.839 6.637-1.496 6.509Q-1.152 6.380-0.780 6.328Q-0.407 6.275-0.103 6.275L-0.103 6.022Q-0.103 5.817-0.210 5.637Q-0.318 5.458-0.499 5.355Q-0.680 5.253-0.889 5.253Q-1.296 5.253-1.532 5.345Q-1.443 5.382-1.397 5.466Q-1.350 5.550-1.350 5.652Q-1.350 5.748-1.397 5.827Q-1.443 5.905-1.523 5.950Q-1.603 5.994-1.692 5.994Q-1.843 5.994-1.943 5.897Q-2.044 5.799-2.044 5.652Q-2.044 5.030-0.889 5.030Q-0.677 5.030-0.428 5.094Q-0.178 5.157 0.024 5.276Q0.225 5.396 0.352 5.581Q0.478 5.765 0.478 6.008L0.478 7.584Q0.478 7.700 0.540 7.796Q0.601 7.891 0.714 7.891Q0.823 7.891 0.888 7.797Q0.953 7.703 0.953 7.584L0.953 7.136L1.220 7.136L1.220 7.584Q1.220 7.854 0.993 8.019Q0.765 8.185 0.485 8.185Q0.277 8.185 0.140 8.031Q0.003 7.878-0.021 7.662Q-0.168 7.929-0.450 8.074Q-0.732 8.219-1.056 8.219Q-1.333 8.219-1.617 8.144Q-1.901 8.069-2.094 7.890Q-2.287 7.710-2.287 7.423M-1.672 7.423Q-1.672 7.597-1.571 7.727Q-1.470 7.857-1.314 7.927Q-1.159 7.997-0.995 7.997Q-0.776 7.997-0.568 7.900Q-0.359 7.802-0.231 7.621Q-0.103 7.440-0.103 7.214L-0.103 6.486Q-0.428 6.486-0.793 6.577Q-1.159 6.668-1.415 6.880Q-1.672 7.091-1.672 7.423M1.637 8.144L1.637 7.081Q1.637 7.057 1.664 7.030Q1.692 7.003 1.716 7.003L1.825 7.003Q1.890 7.003 1.904 7.061Q1.999 7.495 2.245 7.746Q2.491 7.997 2.905 7.997Q3.247 7.997 3.500 7.864Q3.753 7.731 3.753 7.423Q3.753 7.266 3.659 7.151Q3.565 7.037 3.426 6.968Q3.288 6.900 3.120 6.862L2.539 6.763Q2.184 6.695 1.910 6.474Q1.637 6.254 1.637 5.912Q1.637 5.663 1.748 5.488Q1.859 5.314 2.045 5.215Q2.232 5.116 2.447 5.073Q2.662 5.030 2.905 5.030Q3.319 5.030 3.599 5.212L3.814 5.037Q3.824 5.034 3.831 5.032Q3.838 5.030 3.848 5.030L3.900 5.030Q3.927 5.030 3.951 5.054Q3.975 5.078 3.975 5.106L3.975 5.953Q3.975 5.974 3.951 6.001Q3.927 6.028 3.900 6.028L3.787 6.028Q3.759 6.028 3.734 6.003Q3.708 5.977 3.708 5.953Q3.708 5.717 3.602 5.553Q3.496 5.389 3.313 5.307Q3.131 5.225 2.898 5.225Q2.570 5.225 2.314 5.328Q2.057 5.430 2.057 5.707Q2.057 5.902 2.240 6.011Q2.423 6.121 2.652 6.162L3.226 6.268Q3.472 6.316 3.686 6.444Q3.900 6.572 4.036 6.775Q4.173 6.979 4.173 7.228Q4.173 7.741 3.807 7.980Q3.442 8.219 2.905 8.219Q2.409 8.219 2.078 7.925L1.811 8.199Q1.791 8.219 1.763 8.219L1.716 8.219Q1.692 8.219 1.664 8.192Q1.637 8.165 1.637 8.144M6.484 8.151L4.850 8.151L4.850 7.871Q5.079 7.871 5.228 7.837Q5.376 7.802 5.376 7.662L5.376 4.043Q5.376 3.773 5.269 3.711Q5.161 3.650 4.850 3.650L4.850 3.369L5.930 3.294L5.930 5.680Q6.036 5.495 6.214 5.353Q6.391 5.212 6.600 5.138Q6.808 5.065 7.034 5.065Q7.540 5.065 7.823 5.288Q8.107 5.512 8.107 6.008L8.107 7.662Q8.107 7.799 8.256 7.835Q8.405 7.871 8.630 7.871L8.630 8.151L7 8.151L7 7.871Q7.229 7.871 7.377 7.837Q7.526 7.802 7.526 7.662L7.526 6.022Q7.526 5.687 7.406 5.487Q7.287 5.287 6.972 5.287Q6.702 5.287 6.468 5.423Q6.234 5.560 6.096 5.794Q5.957 6.028 5.957 6.302L5.957 7.662Q5.957 7.799 6.108 7.835Q6.258 7.871 6.484 7.871\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath d=\"M287.61-21.724h68.287v-31.298H287.61Z\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(342.362 -43.775)\">\u003Cpath d=\"M-29.102 9.508L-30.732 9.508L-30.732 9.228Q-30.503 9.228-30.354 9.193Q-30.206 9.159-30.206 9.019L-30.206 5.673Q-30.206 5.502-30.342 5.461Q-30.479 5.420-30.732 5.420L-30.732 5.140L-29.652 5.065L-29.652 5.471Q-29.430 5.270-29.143 5.167Q-28.855 5.065-28.548 5.065Q-28.121 5.065-27.757 5.278Q-27.393 5.492-27.179 5.856Q-26.965 6.220-26.965 6.640Q-26.965 7.085-27.205 7.449Q-27.444 7.813-27.837 8.016Q-28.230 8.219-28.674 8.219Q-28.941 8.219-29.189 8.119Q-29.436 8.018-29.624 7.837L-29.624 9.019Q-29.624 9.156-29.476 9.192Q-29.327 9.228-29.102 9.228L-29.102 9.508M-29.624 5.820L-29.624 7.430Q-29.491 7.683-29.248 7.840Q-29.006 7.997-28.729 7.997Q-28.401 7.997-28.148 7.796Q-27.895 7.594-27.762 7.276Q-27.628 6.958-27.628 6.640Q-27.628 6.411-27.693 6.182Q-27.758 5.953-27.886 5.755Q-28.015 5.557-28.209 5.437Q-28.404 5.318-28.637 5.318Q-28.931 5.318-29.199 5.447Q-29.467 5.577-29.624 5.820\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(342.362 -43.775)\">\u003Cpath d=\"M-26.155 6.668Q-26.155 6.326-26.020 6.027Q-25.885 5.728-25.645 5.504Q-25.406 5.280-25.088 5.155Q-24.770 5.030-24.439 5.030Q-23.994 5.030-23.595 5.246Q-23.195 5.461-22.960 5.839Q-22.726 6.216-22.726 6.668Q-22.726 7.009-22.868 7.293Q-23.010 7.577-23.254 7.784Q-23.499 7.990-23.808 8.105Q-24.117 8.219-24.439 8.219Q-24.869 8.219-25.271 8.018Q-25.673 7.816-25.914 7.464Q-26.155 7.112-26.155 6.668M-24.439 7.970Q-23.837 7.970-23.613 7.592Q-23.389 7.214-23.389 6.582Q-23.389 5.970-23.624 5.611Q-23.858 5.253-24.439 5.253Q-25.491 5.253-25.491 6.582Q-25.491 7.214-25.266 7.592Q-25.040 7.970-24.439 7.970M-20.464 8.151L-22.067 8.151L-22.067 7.871Q-21.841 7.871-21.692 7.837Q-21.544 7.802-21.544 7.662L-21.544 4.043Q-21.544 3.773-21.651 3.711Q-21.759 3.650-22.067 3.650L-22.067 3.369L-20.990 3.294L-20.990 7.662Q-20.990 7.799-20.840 7.835Q-20.689 7.871-20.464 7.871L-20.464 8.151M-18.252 8.151L-19.804 8.151L-19.804 7.871Q-19.578 7.871-19.430 7.837Q-19.281 7.802-19.281 7.662L-19.281 5.813Q-19.281 5.625-19.329 5.541Q-19.377 5.458-19.474 5.439Q-19.572 5.420-19.783 5.420L-19.783 5.140L-18.727 5.065L-18.727 7.662Q-18.727 7.802-18.596 7.837Q-18.464 7.871-18.252 7.871L-18.252 8.151M-19.524 3.844Q-19.524 3.673-19.401 3.554Q-19.278 3.434-19.107 3.434Q-18.939 3.434-18.816 3.554Q-18.693 3.673-18.693 3.844Q-18.693 4.019-18.816 4.142Q-18.939 4.265-19.107 4.265Q-19.278 4.265-19.401 4.142Q-19.524 4.019-19.524 3.844M-17.606 6.640Q-17.606 6.312-17.471 6.011Q-17.336 5.711-17.100 5.490Q-16.865 5.270-16.560 5.150Q-16.256 5.030-15.931 5.030Q-15.426 5.030-15.077 5.133Q-14.728 5.235-14.728 5.611Q-14.728 5.758-14.826 5.859Q-14.923 5.960-15.070 5.960Q-15.224 5.960-15.323 5.861Q-15.422 5.762-15.422 5.611Q-15.422 5.423-15.282 5.331Q-15.484 5.280-15.925 5.280Q-16.280 5.280-16.509 5.476Q-16.738 5.673-16.839 5.982Q-16.940 6.292-16.940 6.640Q-16.940 6.989-16.813 7.295Q-16.687 7.601-16.432 7.785Q-16.178 7.970-15.822 7.970Q-15.600 7.970-15.415 7.886Q-15.231 7.802-15.096 7.647Q-14.961 7.491-14.903 7.283Q-14.889 7.228-14.834 7.228L-14.721 7.228Q-14.691 7.228-14.668 7.252Q-14.646 7.276-14.646 7.310L-14.646 7.331Q-14.732 7.618-14.920 7.816Q-15.108 8.014-15.373 8.117Q-15.637 8.219-15.931 8.219Q-16.362 8.219-16.750 8.013Q-17.138 7.806-17.372 7.443Q-17.606 7.081-17.606 6.640M-13.723 9.286Q-13.594 9.354-13.457 9.354Q-13.286 9.354-13.136 9.265Q-12.985 9.176-12.874 9.031Q-12.763 8.886-12.684 8.718L-12.421 8.151L-13.590 5.625Q-13.665 5.478-13.795 5.446Q-13.925 5.413-14.157 5.413L-14.157 5.133L-12.636 5.133L-12.636 5.413Q-12.985 5.413-12.985 5.560Q-12.982 5.581-12.980 5.598Q-12.978 5.615-12.978 5.625L-12.120 7.484L-11.348 5.813Q-11.314 5.745-11.314 5.666Q-11.314 5.553-11.397 5.483Q-11.481 5.413-11.594 5.413L-11.594 5.133L-10.398 5.133L-10.398 5.413Q-10.616 5.413-10.789 5.517Q-10.962 5.622-11.054 5.813L-12.390 8.718Q-12.561 9.088-12.831 9.334Q-13.101 9.580-13.457 9.580Q-13.727 9.580-13.946 9.414Q-14.164 9.248-14.164 8.985Q-14.164 8.848-14.072 8.759Q-13.980 8.671-13.840 8.671Q-13.703 8.671-13.614 8.759Q-13.525 8.848-13.525 8.985Q-13.525 9.088-13.578 9.166Q-13.631 9.245-13.723 9.286\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-31.06-7.898c51.215-29.475 51.215-29.475 82.358-29.475\"\u002F>\u003Cpath stroke=\"none\" d=\"m53.898-37.373-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M122.985-37.373H170.8\"\u002F>\u003Cpath stroke=\"none\" d=\"m173.4-37.373-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M242.486-37.373h42.125\"\u002F>\u003Cpath stroke=\"none\" d=\"m287.21-37.373-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003C\u002Fg>\u003Cg stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(129.816 -72.227)\">\u003Cpath d=\"M-28.561 8.151L-30.667 8.151L-30.667 7.871Q-29.946 7.871-29.946 7.662L-29.946 3.861Q-29.946 3.650-30.667 3.650L-30.667 3.369L-28.329 3.369Q-28.018 3.369-27.675 3.443Q-27.331 3.516-27.010 3.672Q-26.688 3.827-26.487 4.073Q-26.285 4.319-26.285 4.644Q-26.285 4.931-26.473 5.162Q-26.661 5.393-26.938 5.541Q-27.215 5.690-27.505 5.772Q-27.170 5.875-26.936 6.107Q-26.702 6.339-26.658 6.668L-26.572 7.276Q-26.545 7.488-26.499 7.657Q-26.453 7.826-26.348 7.946Q-26.244 8.066-26.056 8.066Q-25.841 8.066-25.725 7.881Q-25.608 7.696-25.608 7.464Q-25.591 7.399-25.516 7.382L-25.424 7.382Q-25.342 7.402-25.342 7.484Q-25.342 7.690-25.432 7.876Q-25.523 8.062-25.689 8.177Q-25.854 8.291-26.056 8.291Q-26.394 8.291-26.688 8.190Q-26.982 8.089-27.167 7.867Q-27.352 7.645-27.352 7.297L-27.352 6.688Q-27.352 6.439-27.495 6.252Q-27.639 6.066-27.862 5.967Q-28.086 5.868-28.336 5.868L-29.283 5.868L-29.283 7.662Q-29.283 7.871-28.561 7.871L-28.561 8.151M-29.283 3.861L-29.283 5.646L-28.428 5.646Q-28.141 5.646-27.902 5.603Q-27.663 5.560-27.475 5.451Q-27.287 5.341-27.175 5.143Q-27.064 4.945-27.064 4.644Q-27.064 4.067-27.432 3.858Q-27.799 3.650-28.428 3.650L-28.917 3.650Q-29.105 3.650-29.194 3.684Q-29.283 3.718-29.283 3.861M-20.909 8.151L-24.877 8.151L-24.877 7.871Q-24.156 7.871-24.156 7.662L-24.156 3.861Q-24.156 3.650-24.877 3.650L-24.877 3.369L-22.563 3.369L-22.563 3.650Q-22.881 3.650-23.173 3.685Q-23.465 3.721-23.465 3.861L-23.465 7.662Q-23.465 7.802-23.376 7.837Q-23.288 7.871-23.100 7.871L-22.477 7.871Q-22.064 7.871-21.785 7.768Q-21.507 7.666-21.341 7.464Q-21.175 7.262-21.093 6.975Q-21.011 6.688-20.967 6.281L-20.700 6.281L-20.909 8.151M-17.802 8.151L-19.935 8.151L-19.935 7.871Q-19.213 7.871-19.213 7.662L-19.213 3.861Q-19.213 3.650-19.935 3.650L-19.935 3.369L-17.802 3.369L-17.802 3.650Q-18.523 3.650-18.523 3.861L-18.523 5.526L-16.212 5.526L-16.212 3.861Q-16.212 3.650-16.934 3.650L-16.934 3.369L-14.797 3.369L-14.797 3.650Q-15.518 3.650-15.518 3.861L-15.518 7.662Q-15.518 7.871-14.797 7.871L-14.797 8.151L-16.934 8.151L-16.934 7.871Q-16.212 7.871-16.212 7.662L-16.212 5.806L-18.523 5.806L-18.523 7.662Q-18.523 7.871-17.802 7.871L-17.802 8.151M-11.721 8.151L-14.035 8.151L-14.035 7.871Q-13.314 7.871-13.314 7.662L-13.314 3.861Q-13.314 3.650-14.035 3.650L-14.035 3.369L-9.852 3.369L-9.640 5.006L-9.906 5.006Q-9.985 4.395-10.137 4.116Q-10.289 3.838-10.593 3.744Q-10.897 3.650-11.523 3.650L-12.258 3.650Q-12.446 3.650-12.535 3.684Q-12.623 3.718-12.623 3.861L-12.623 5.618L-12.070 5.618Q-11.701 5.618-11.516 5.564Q-11.331 5.509-11.253 5.336Q-11.174 5.164-11.174 4.798L-10.908 4.798L-10.908 6.715L-11.174 6.715Q-11.174 6.350-11.253 6.177Q-11.331 6.005-11.516 5.952Q-11.701 5.899-12.070 5.899L-12.623 5.899L-12.623 7.662Q-12.623 7.871-11.721 7.871L-11.721 8.151M-8.542 7.731Q-8.542 7.563-8.419 7.440Q-8.296 7.317-8.122 7.317Q-7.955 7.317-7.831 7.440Q-7.708 7.563-7.708 7.731Q-7.708 7.905-7.831 8.028Q-7.955 8.151-8.122 8.151Q-8.296 8.151-8.419 8.028Q-8.542 7.905-8.542 7.731M-8.542 5.547Q-8.542 5.379-8.419 5.256Q-8.296 5.133-8.122 5.133Q-7.955 5.133-7.831 5.256Q-7.708 5.379-7.708 5.547Q-7.708 5.721-7.831 5.844Q-7.955 5.967-8.122 5.967Q-8.296 5.967-8.419 5.844Q-8.542 5.721-8.542 5.547\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(129.816 -72.227)\">\u003Cpath d=\"M-3.448 7.310L-3.448 5.413L-4.087 5.413L-4.087 5.191Q-3.769 5.191-3.552 4.981Q-3.335 4.771-3.235 4.461Q-3.134 4.152-3.134 3.844L-2.867 3.844L-2.867 5.133L-1.790 5.133L-1.790 5.413L-2.867 5.413L-2.867 7.297Q-2.867 7.573-2.763 7.772Q-2.659 7.970-2.399 7.970Q-2.242 7.970-2.136 7.866Q-2.030 7.761-1.980 7.608Q-1.931 7.454-1.931 7.297L-1.931 6.883L-1.664 6.883L-1.664 7.310Q-1.664 7.536-1.763 7.746Q-1.862 7.956-2.047 8.088Q-2.231 8.219-2.460 8.219Q-2.898 8.219-3.173 7.982Q-3.448 7.744-3.448 7.310\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(129.816 -72.227)\">\u003Cpath d=\"M0.323 8.124L-0.658 5.625Q-0.719 5.482-0.837 5.447Q-0.955 5.413-1.171 5.413L-1.171 5.133L0.309 5.133L0.309 5.413Q-0.070 5.413-0.070 5.574Q-0.070 5.584-0.056 5.625L0.658 7.457L1.331 5.752Q1.301 5.680 1.301 5.652Q1.301 5.625 1.273 5.625Q1.212 5.478 1.094 5.446Q0.976 5.413 0.764 5.413L0.764 5.133L2.162 5.133L2.162 5.413Q1.786 5.413 1.786 5.574Q1.786 5.605 1.793 5.625L2.548 7.563L3.235 5.813Q3.256 5.762 3.256 5.707Q3.256 5.567 3.143 5.490Q3.030 5.413 2.890 5.413L2.890 5.133L4.110 5.133L4.110 5.413Q3.905 5.413 3.750 5.519Q3.594 5.625 3.522 5.813L2.617 8.124Q2.582 8.219 2.470 8.219L2.401 8.219Q2.292 8.219 2.254 8.124L1.472 6.121L0.685 8.124Q0.651 8.219 0.538 8.219L0.470 8.219Q0.361 8.219 0.323 8.124\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(129.816 -72.227)\">\u003Cpath d=\"M4.387 6.668Q4.387 6.326 4.522 6.027Q4.657 5.728 4.897 5.504Q5.136 5.280 5.454 5.155Q5.772 5.030 6.103 5.030Q6.548 5.030 6.947 5.246Q7.347 5.461 7.582 5.839Q7.816 6.216 7.816 6.668Q7.816 7.009 7.674 7.293Q7.532 7.577 7.288 7.784Q7.043 7.990 6.734 8.105Q6.425 8.219 6.103 8.219Q5.673 8.219 5.271 8.018Q4.869 7.816 4.628 7.464Q4.387 7.112 4.387 6.668M6.103 7.970Q6.705 7.970 6.929 7.592Q7.153 7.214 7.153 6.582Q7.153 5.970 6.918 5.611Q6.684 5.253 6.103 5.253Q5.051 5.253 5.051 6.582Q5.051 7.214 5.276 7.592Q5.502 7.970 6.103 7.970\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(129.816 -72.227)\">\u003Cpath d=\"M11.108 8.144L11.108 7.081Q11.108 7.057 11.136 7.030Q11.163 7.003 11.187 7.003L11.296 7.003Q11.361 7.003 11.375 7.061Q11.471 7.495 11.717 7.746Q11.963 7.997 12.377 7.997Q12.718 7.997 12.971 7.864Q13.224 7.731 13.224 7.423Q13.224 7.266 13.130 7.151Q13.036 7.037 12.898 6.968Q12.759 6.900 12.592 6.862L12.011 6.763Q11.655 6.695 11.382 6.474Q11.108 6.254 11.108 5.912Q11.108 5.663 11.220 5.488Q11.331 5.314 11.517 5.215Q11.703 5.116 11.919 5.073Q12.134 5.030 12.377 5.030Q12.790 5.030 13.070 5.212L13.286 5.037Q13.296 5.034 13.303 5.032Q13.310 5.030 13.320 5.030L13.371 5.030Q13.398 5.030 13.422 5.054Q13.446 5.078 13.446 5.106L13.446 5.953Q13.446 5.974 13.422 6.001Q13.398 6.028 13.371 6.028L13.258 6.028Q13.231 6.028 13.205 6.003Q13.180 5.977 13.180 5.953Q13.180 5.717 13.074 5.553Q12.968 5.389 12.785 5.307Q12.602 5.225 12.370 5.225Q12.042 5.225 11.785 5.328Q11.529 5.430 11.529 5.707Q11.529 5.902 11.712 6.011Q11.895 6.121 12.124 6.162L12.698 6.268Q12.944 6.316 13.158 6.444Q13.371 6.572 13.508 6.775Q13.645 6.979 13.645 7.228Q13.645 7.741 13.279 7.980Q12.913 8.219 12.377 8.219Q11.881 8.219 11.549 7.925L11.283 8.199Q11.262 8.219 11.235 8.219L11.187 8.219Q11.163 8.219 11.136 8.192Q11.108 8.165 11.108 8.144M14.800 7.310L14.800 5.413L14.161 5.413L14.161 5.191Q14.479 5.191 14.696 4.981Q14.913 4.771 15.013 4.461Q15.114 4.152 15.114 3.844L15.381 3.844L15.381 5.133L16.458 5.133L16.458 5.413L15.381 5.413L15.381 7.297Q15.381 7.573 15.485 7.772Q15.589 7.970 15.849 7.970Q16.006 7.970 16.112 7.866Q16.218 7.761 16.268 7.608Q16.317 7.454 16.317 7.297L16.317 6.883L16.584 6.883L16.584 7.310Q16.584 7.536 16.485 7.746Q16.386 7.956 16.201 8.088Q16.017 8.219 15.788 8.219Q15.350 8.219 15.075 7.982Q14.800 7.744 14.800 7.310M17.452 7.423Q17.452 7.091 17.676 6.864Q17.900 6.637 18.243 6.509Q18.587 6.380 18.960 6.328Q19.332 6.275 19.636 6.275L19.636 6.022Q19.636 5.817 19.529 5.637Q19.421 5.458 19.240 5.355Q19.059 5.253 18.850 5.253Q18.443 5.253 18.208 5.345Q18.296 5.382 18.343 5.466Q18.389 5.550 18.389 5.652Q18.389 5.748 18.343 5.827Q18.296 5.905 18.216 5.950Q18.136 5.994 18.047 5.994Q17.897 5.994 17.796 5.897Q17.695 5.799 17.695 5.652Q17.695 5.030 18.850 5.030Q19.062 5.030 19.312 5.094Q19.561 5.157 19.763 5.276Q19.964 5.396 20.091 5.581Q20.217 5.765 20.217 6.008L20.217 7.584Q20.217 7.700 20.279 7.796Q20.340 7.891 20.453 7.891Q20.563 7.891 20.627 7.797Q20.692 7.703 20.692 7.584L20.692 7.136L20.959 7.136L20.959 7.584Q20.959 7.854 20.732 8.019Q20.504 8.185 20.224 8.185Q20.016 8.185 19.879 8.031Q19.742 7.878 19.718 7.662Q19.571 7.929 19.289 8.074Q19.007 8.219 18.683 8.219Q18.406 8.219 18.122 8.144Q17.838 8.069 17.645 7.890Q17.452 7.710 17.452 7.423M18.067 7.423Q18.067 7.597 18.168 7.727Q18.269 7.857 18.425 7.927Q18.580 7.997 18.744 7.997Q18.963 7.997 19.171 7.900Q19.380 7.802 19.508 7.621Q19.636 7.440 19.636 7.214L19.636 6.486Q19.312 6.486 18.946 6.577Q18.580 6.668 18.324 6.880Q18.067 7.091 18.067 7.423M21.335 8.684Q21.335 8.438 21.532 8.254Q21.728 8.069 21.984 7.990Q21.848 7.878 21.776 7.717Q21.704 7.556 21.704 7.375Q21.704 7.054 21.916 6.808Q21.581 6.510 21.581 6.100Q21.581 5.639 21.971 5.352Q22.360 5.065 22.839 5.065Q23.311 5.065 23.646 5.311Q23.820 5.157 24.030 5.075Q24.240 4.993 24.469 4.993Q24.633 4.993 24.755 5.100Q24.876 5.208 24.876 5.372Q24.876 5.468 24.804 5.540Q24.732 5.611 24.640 5.611Q24.541 5.611 24.471 5.538Q24.401 5.464 24.401 5.365Q24.401 5.311 24.415 5.280L24.421 5.266Q24.428 5.246 24.437 5.235Q24.445 5.225 24.449 5.218Q24.093 5.218 23.806 5.441Q24.093 5.734 24.093 6.100Q24.093 6.415 23.909 6.647Q23.724 6.880 23.435 7.008Q23.147 7.136 22.839 7.136Q22.637 7.136 22.446 7.086Q22.254 7.037 22.077 6.927Q21.984 7.054 21.984 7.197Q21.984 7.379 22.113 7.514Q22.241 7.649 22.425 7.649L23.058 7.649Q23.505 7.649 23.875 7.720Q24.244 7.792 24.503 8.021Q24.763 8.250 24.763 8.684Q24.763 9.005 24.468 9.207Q24.172 9.409 23.769 9.498Q23.365 9.587 23.051 9.587Q22.733 9.587 22.330 9.498Q21.926 9.409 21.631 9.207Q21.335 9.005 21.335 8.684M21.790 8.684Q21.790 8.913 22.008 9.062Q22.227 9.211 22.519 9.279Q22.812 9.347 23.051 9.347Q23.215 9.347 23.423 9.311Q23.632 9.276 23.839 9.195Q24.045 9.115 24.177 8.987Q24.309 8.859 24.309 8.684Q24.309 8.332 23.928 8.238Q23.546 8.144 23.044 8.144L22.425 8.144Q22.186 8.144 21.988 8.295Q21.790 8.445 21.790 8.684M22.839 6.897Q23.505 6.897 23.505 6.100Q23.505 5.300 22.839 5.300Q22.169 5.300 22.169 6.100Q22.169 6.897 22.839 6.897M25.317 6.616Q25.317 6.295 25.442 6.006Q25.566 5.717 25.792 5.494Q26.018 5.270 26.313 5.150Q26.609 5.030 26.927 5.030Q27.255 5.030 27.516 5.130Q27.778 5.229 27.954 5.411Q28.130 5.594 28.224 5.852Q28.318 6.110 28.318 6.442Q28.318 6.534 28.236 6.555L25.980 6.555L25.980 6.616Q25.980 7.204 26.264 7.587Q26.547 7.970 27.115 7.970Q27.436 7.970 27.704 7.777Q27.973 7.584 28.062 7.269Q28.068 7.228 28.144 7.214L28.236 7.214Q28.318 7.238 28.318 7.310Q28.318 7.317 28.311 7.344Q28.198 7.741 27.827 7.980Q27.457 8.219 27.033 8.219Q26.595 8.219 26.195 8.011Q25.795 7.802 25.556 7.435Q25.317 7.068 25.317 6.616M25.987 6.346L27.802 6.346Q27.802 6.069 27.704 5.817Q27.607 5.564 27.409 5.408Q27.211 5.253 26.927 5.253Q26.650 5.253 26.436 5.411Q26.223 5.570 26.105 5.825Q25.987 6.080 25.987 6.346M28.906 8.144L28.906 7.081Q28.906 7.057 28.933 7.030Q28.961 7.003 28.984 7.003L29.094 7.003Q29.159 7.003 29.172 7.061Q29.268 7.495 29.514 7.746Q29.760 7.997 30.174 7.997Q30.516 7.997 30.769 7.864Q31.022 7.731 31.022 7.423Q31.022 7.266 30.928 7.151Q30.834 7.037 30.695 6.968Q30.557 6.900 30.389 6.862L29.808 6.763Q29.453 6.695 29.179 6.474Q28.906 6.254 28.906 5.912Q28.906 5.663 29.017 5.488Q29.128 5.314 29.314 5.215Q29.501 5.116 29.716 5.073Q29.931 5.030 30.174 5.030Q30.587 5.030 30.868 5.212L31.083 5.037Q31.093 5.034 31.100 5.032Q31.107 5.030 31.117 5.030L31.169 5.030Q31.196 5.030 31.220 5.054Q31.244 5.078 31.244 5.106L31.244 5.953Q31.244 5.974 31.220 6.001Q31.196 6.028 31.169 6.028L31.056 6.028Q31.028 6.028 31.003 6.003Q30.977 5.977 30.977 5.953Q30.977 5.717 30.871 5.553Q30.765 5.389 30.582 5.307Q30.399 5.225 30.167 5.225Q29.839 5.225 29.583 5.328Q29.326 5.430 29.326 5.707Q29.326 5.902 29.509 6.011Q29.692 6.121 29.921 6.162L30.495 6.268Q30.741 6.316 30.955 6.444Q31.169 6.572 31.305 6.775Q31.442 6.979 31.442 7.228Q31.442 7.741 31.076 7.980Q30.711 8.219 30.174 8.219Q29.678 8.219 29.347 7.925L29.080 8.199Q29.060 8.219 29.032 8.219L28.984 8.219Q28.961 8.219 28.933 8.192Q28.906 8.165 28.906 8.144M32.570 9.381Q32.570 9.347 32.597 9.320Q32.867 9.091 33.016 8.768Q33.165 8.445 33.165 8.089L33.165 8.052Q33.055 8.151 32.891 8.151Q32.710 8.151 32.590 8.031Q32.471 7.912 32.471 7.731Q32.471 7.556 32.590 7.437Q32.710 7.317 32.891 7.317Q33.148 7.317 33.267 7.556Q33.387 7.796 33.387 8.089Q33.387 8.489 33.218 8.860Q33.048 9.231 32.751 9.487Q32.720 9.508 32.693 9.508Q32.652 9.508 32.611 9.467Q32.570 9.426 32.570 9.381\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(129.816 -72.227)\">\u003Cpath d=\"M37.056 8.144L37.056 7.081Q37.056 7.057 37.084 7.030Q37.111 7.003 37.135 7.003L37.244 7.003Q37.309 7.003 37.323 7.061Q37.419 7.495 37.665 7.746Q37.911 7.997 38.325 7.997Q38.666 7.997 38.919 7.864Q39.172 7.731 39.172 7.423Q39.172 7.266 39.078 7.151Q38.984 7.037 38.846 6.968Q38.707 6.900 38.540 6.862L37.959 6.763Q37.603 6.695 37.330 6.474Q37.056 6.254 37.056 5.912Q37.056 5.663 37.168 5.488Q37.279 5.314 37.465 5.215Q37.651 5.116 37.867 5.073Q38.082 5.030 38.325 5.030Q38.738 5.030 39.018 5.212L39.234 5.037Q39.244 5.034 39.251 5.032Q39.258 5.030 39.268 5.030L39.319 5.030Q39.346 5.030 39.370 5.054Q39.394 5.078 39.394 5.106L39.394 5.953Q39.394 5.974 39.370 6.001Q39.346 6.028 39.319 6.028L39.206 6.028Q39.179 6.028 39.153 6.003Q39.128 5.977 39.128 5.953Q39.128 5.717 39.022 5.553Q38.916 5.389 38.733 5.307Q38.550 5.225 38.318 5.225Q37.990 5.225 37.733 5.328Q37.477 5.430 37.477 5.707Q37.477 5.902 37.660 6.011Q37.843 6.121 38.072 6.162L38.646 6.268Q38.892 6.316 39.106 6.444Q39.319 6.572 39.456 6.775Q39.593 6.979 39.593 7.228Q39.593 7.741 39.227 7.980Q38.861 8.219 38.325 8.219Q37.829 8.219 37.497 7.925L37.231 8.199Q37.210 8.219 37.183 8.219L37.135 8.219Q37.111 8.219 37.084 8.192Q37.056 8.165 37.056 8.144M40.280 7.423Q40.280 7.091 40.503 6.864Q40.727 6.637 41.071 6.509Q41.414 6.380 41.787 6.328Q42.159 6.275 42.464 6.275L42.464 6.022Q42.464 5.817 42.356 5.637Q42.248 5.458 42.067 5.355Q41.886 5.253 41.678 5.253Q41.271 5.253 41.035 5.345Q41.124 5.382 41.170 5.466Q41.216 5.550 41.216 5.652Q41.216 5.748 41.170 5.827Q41.124 5.905 41.044 5.950Q40.963 5.994 40.874 5.994Q40.724 5.994 40.623 5.897Q40.522 5.799 40.522 5.652Q40.522 5.030 41.678 5.030Q41.889 5.030 42.139 5.094Q42.388 5.157 42.590 5.276Q42.792 5.396 42.918 5.581Q43.045 5.765 43.045 6.008L43.045 7.584Q43.045 7.700 43.106 7.796Q43.168 7.891 43.281 7.891Q43.390 7.891 43.455 7.797Q43.520 7.703 43.520 7.584L43.520 7.136L43.786 7.136L43.786 7.584Q43.786 7.854 43.559 8.019Q43.332 8.185 43.052 8.185Q42.843 8.185 42.706 8.031Q42.570 7.878 42.546 7.662Q42.399 7.929 42.117 8.074Q41.835 8.219 41.510 8.219Q41.233 8.219 40.950 8.144Q40.666 8.069 40.473 7.890Q40.280 7.710 40.280 7.423M40.895 7.423Q40.895 7.597 40.996 7.727Q41.096 7.857 41.252 7.927Q41.408 7.997 41.572 7.997Q41.790 7.997 41.999 7.900Q42.207 7.802 42.336 7.621Q42.464 7.440 42.464 7.214L42.464 6.486Q42.139 6.486 41.773 6.577Q41.408 6.668 41.151 6.880Q40.895 7.091 40.895 7.423M45.885 8.151L44.251 8.151L44.251 7.871Q44.480 7.871 44.629 7.837Q44.778 7.802 44.778 7.662L44.778 5.813Q44.778 5.543 44.670 5.482Q44.562 5.420 44.251 5.420L44.251 5.140L45.311 5.065L45.311 5.714Q45.482 5.406 45.786 5.235Q46.090 5.065 46.435 5.065Q46.835 5.065 47.112 5.205Q47.389 5.345 47.474 5.693Q47.642 5.400 47.941 5.232Q48.240 5.065 48.585 5.065Q49.091 5.065 49.375 5.288Q49.659 5.512 49.659 6.008L49.659 7.662Q49.659 7.799 49.807 7.835Q49.956 7.871 50.181 7.871L50.181 8.151L48.551 8.151L48.551 7.871Q48.777 7.871 48.927 7.835Q49.077 7.799 49.077 7.662L49.077 6.022Q49.077 5.687 48.958 5.487Q48.838 5.287 48.524 5.287Q48.254 5.287 48.020 5.423Q47.785 5.560 47.647 5.794Q47.509 6.028 47.509 6.302L47.509 7.662Q47.509 7.799 47.657 7.835Q47.806 7.871 48.032 7.871L48.032 8.151L46.401 8.151L46.401 7.871Q46.630 7.871 46.779 7.837Q46.928 7.802 46.928 7.662L46.928 6.022Q46.928 5.687 46.808 5.487Q46.688 5.287 46.374 5.287Q46.104 5.287 45.870 5.423Q45.636 5.560 45.497 5.794Q45.359 6.028 45.359 6.302L45.359 7.662Q45.359 7.799 45.509 7.835Q45.659 7.871 45.885 7.871L45.885 8.151M52.413 9.508L50.783 9.508L50.783 9.228Q51.012 9.228 51.161 9.193Q51.309 9.159 51.309 9.019L51.309 5.673Q51.309 5.502 51.173 5.461Q51.036 5.420 50.783 5.420L50.783 5.140L51.863 5.065L51.863 5.471Q52.085 5.270 52.372 5.167Q52.659 5.065 52.967 5.065Q53.394 5.065 53.758 5.278Q54.122 5.492 54.336 5.856Q54.550 6.220 54.550 6.640Q54.550 7.085 54.310 7.449Q54.071 7.813 53.678 8.016Q53.285 8.219 52.841 8.219Q52.574 8.219 52.326 8.119Q52.078 8.018 51.890 7.837L51.890 9.019Q51.890 9.156 52.039 9.192Q52.188 9.228 52.413 9.228L52.413 9.508M51.890 5.820L51.890 7.430Q52.024 7.683 52.266 7.840Q52.509 7.997 52.786 7.997Q53.114 7.997 53.367 7.796Q53.620 7.594 53.753 7.276Q53.887 6.958 53.887 6.640Q53.887 6.411 53.822 6.182Q53.757 5.953 53.628 5.755Q53.500 5.557 53.305 5.437Q53.111 5.318 52.878 5.318Q52.584 5.318 52.316 5.447Q52.048 5.577 51.890 5.820M56.853 8.151L55.250 8.151L55.250 7.871Q55.476 7.871 55.625 7.837Q55.773 7.802 55.773 7.662L55.773 4.043Q55.773 3.773 55.666 3.711Q55.558 3.650 55.250 3.650L55.250 3.369L56.327 3.294L56.327 7.662Q56.327 7.799 56.477 7.835Q56.628 7.871 56.853 7.871L56.853 8.151M59.065 8.151L57.513 8.151L57.513 7.871Q57.739 7.871 57.887 7.837Q58.036 7.802 58.036 7.662L58.036 5.813Q58.036 5.625 57.988 5.541Q57.940 5.458 57.843 5.439Q57.745 5.420 57.534 5.420L57.534 5.140L58.590 5.065L58.590 7.662Q58.590 7.802 58.721 7.837Q58.853 7.871 59.065 7.871L59.065 8.151M57.793 3.844Q57.793 3.673 57.916 3.554Q58.039 3.434 58.210 3.434Q58.378 3.434 58.501 3.554Q58.624 3.673 58.624 3.844Q58.624 4.019 58.501 4.142Q58.378 4.265 58.210 4.265Q58.039 4.265 57.916 4.142Q57.793 4.019 57.793 3.844M61.392 8.151L59.759 8.151L59.759 7.871Q59.988 7.871 60.136 7.837Q60.285 7.802 60.285 7.662L60.285 5.813Q60.285 5.543 60.177 5.482Q60.070 5.420 59.759 5.420L59.759 5.140L60.818 5.065L60.818 5.714Q60.989 5.406 61.293 5.235Q61.597 5.065 61.943 5.065Q62.449 5.065 62.732 5.288Q63.016 5.512 63.016 6.008L63.016 7.662Q63.016 7.799 63.165 7.835Q63.313 7.871 63.539 7.871L63.539 8.151L61.909 8.151L61.909 7.871Q62.138 7.871 62.286 7.837Q62.435 7.802 62.435 7.662L62.435 6.022Q62.435 5.687 62.315 5.487Q62.196 5.287 61.881 5.287Q61.611 5.287 61.377 5.423Q61.143 5.560 61.004 5.794Q60.866 6.028 60.866 6.302L60.866 7.662Q60.866 7.799 61.016 7.835Q61.167 7.871 61.392 7.871L61.392 8.151M64.086 8.684Q64.086 8.438 64.282 8.254Q64.479 8.069 64.735 7.990Q64.598 7.878 64.527 7.717Q64.455 7.556 64.455 7.375Q64.455 7.054 64.667 6.808Q64.332 6.510 64.332 6.100Q64.332 5.639 64.721 5.352Q65.111 5.065 65.590 5.065Q66.061 5.065 66.396 5.311Q66.571 5.157 66.781 5.075Q66.991 4.993 67.220 4.993Q67.384 4.993 67.505 5.100Q67.627 5.208 67.627 5.372Q67.627 5.468 67.555 5.540Q67.483 5.611 67.391 5.611Q67.292 5.611 67.222 5.538Q67.152 5.464 67.152 5.365Q67.152 5.311 67.165 5.280L67.172 5.266Q67.179 5.246 67.188 5.235Q67.196 5.225 67.200 5.218Q66.844 5.218 66.557 5.441Q66.844 5.734 66.844 6.100Q66.844 6.415 66.659 6.647Q66.475 6.880 66.186 7.008Q65.897 7.136 65.590 7.136Q65.388 7.136 65.197 7.086Q65.005 7.037 64.827 6.927Q64.735 7.054 64.735 7.197Q64.735 7.379 64.863 7.514Q64.992 7.649 65.176 7.649L65.808 7.649Q66.256 7.649 66.625 7.720Q66.994 7.792 67.254 8.021Q67.514 8.250 67.514 8.684Q67.514 9.005 67.218 9.207Q66.923 9.409 66.519 9.498Q66.116 9.587 65.802 9.587Q65.484 9.587 65.080 9.498Q64.677 9.409 64.381 9.207Q64.086 9.005 64.086 8.684M64.540 8.684Q64.540 8.913 64.759 9.062Q64.978 9.211 65.270 9.279Q65.562 9.347 65.802 9.347Q65.966 9.347 66.174 9.311Q66.383 9.276 66.589 9.195Q66.796 9.115 66.928 8.987Q67.059 8.859 67.059 8.684Q67.059 8.332 66.678 8.238Q66.297 8.144 65.795 8.144L65.176 8.144Q64.937 8.144 64.739 8.295Q64.540 8.445 64.540 8.684M65.590 6.897Q66.256 6.897 66.256 6.100Q66.256 5.300 65.590 5.300Q64.920 5.300 64.920 6.100Q64.920 6.897 65.590 6.897\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-soft-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath d=\"M287.61 69.325h68.287V38.027H287.61Z\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(342.362 47.275)\">\u003Cpath d=\"M-29.102 9.508L-30.732 9.508L-30.732 9.228Q-30.503 9.228-30.354 9.193Q-30.206 9.159-30.206 9.019L-30.206 5.673Q-30.206 5.502-30.342 5.461Q-30.479 5.420-30.732 5.420L-30.732 5.140L-29.652 5.065L-29.652 5.471Q-29.430 5.270-29.143 5.167Q-28.855 5.065-28.548 5.065Q-28.121 5.065-27.757 5.278Q-27.393 5.492-27.179 5.856Q-26.965 6.220-26.965 6.640Q-26.965 7.085-27.205 7.449Q-27.444 7.813-27.837 8.016Q-28.230 8.219-28.674 8.219Q-28.941 8.219-29.189 8.119Q-29.436 8.018-29.624 7.837L-29.624 9.019Q-29.624 9.156-29.476 9.192Q-29.327 9.228-29.102 9.228L-29.102 9.508M-29.624 5.820L-29.624 7.430Q-29.491 7.683-29.248 7.840Q-29.006 7.997-28.729 7.997Q-28.401 7.997-28.148 7.796Q-27.895 7.594-27.762 7.276Q-27.628 6.958-27.628 6.640Q-27.628 6.411-27.693 6.182Q-27.758 5.953-27.886 5.755Q-28.015 5.557-28.209 5.437Q-28.404 5.318-28.637 5.318Q-28.931 5.318-29.199 5.447Q-29.467 5.577-29.624 5.820\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(342.362 47.275)\">\u003Cpath d=\"M-26.155 6.668Q-26.155 6.326-26.020 6.027Q-25.885 5.728-25.645 5.504Q-25.406 5.280-25.088 5.155Q-24.770 5.030-24.439 5.030Q-23.994 5.030-23.595 5.246Q-23.195 5.461-22.960 5.839Q-22.726 6.216-22.726 6.668Q-22.726 7.009-22.868 7.293Q-23.010 7.577-23.254 7.784Q-23.499 7.990-23.808 8.105Q-24.117 8.219-24.439 8.219Q-24.869 8.219-25.271 8.018Q-25.673 7.816-25.914 7.464Q-26.155 7.112-26.155 6.668M-24.439 7.970Q-23.837 7.970-23.613 7.592Q-23.389 7.214-23.389 6.582Q-23.389 5.970-23.624 5.611Q-23.858 5.253-24.439 5.253Q-25.491 5.253-25.491 6.582Q-25.491 7.214-25.266 7.592Q-25.040 7.970-24.439 7.970M-20.464 8.151L-22.067 8.151L-22.067 7.871Q-21.841 7.871-21.692 7.837Q-21.544 7.802-21.544 7.662L-21.544 4.043Q-21.544 3.773-21.651 3.711Q-21.759 3.650-22.067 3.650L-22.067 3.369L-20.990 3.294L-20.990 7.662Q-20.990 7.799-20.840 7.835Q-20.689 7.871-20.464 7.871L-20.464 8.151M-18.252 8.151L-19.804 8.151L-19.804 7.871Q-19.578 7.871-19.430 7.837Q-19.281 7.802-19.281 7.662L-19.281 5.813Q-19.281 5.625-19.329 5.541Q-19.377 5.458-19.474 5.439Q-19.572 5.420-19.783 5.420L-19.783 5.140L-18.727 5.065L-18.727 7.662Q-18.727 7.802-18.596 7.837Q-18.464 7.871-18.252 7.871L-18.252 8.151M-19.524 3.844Q-19.524 3.673-19.401 3.554Q-19.278 3.434-19.107 3.434Q-18.939 3.434-18.816 3.554Q-18.693 3.673-18.693 3.844Q-18.693 4.019-18.816 4.142Q-18.939 4.265-19.107 4.265Q-19.278 4.265-19.401 4.142Q-19.524 4.019-19.524 3.844M-17.606 6.640Q-17.606 6.312-17.471 6.011Q-17.336 5.711-17.100 5.490Q-16.865 5.270-16.560 5.150Q-16.256 5.030-15.931 5.030Q-15.426 5.030-15.077 5.133Q-14.728 5.235-14.728 5.611Q-14.728 5.758-14.826 5.859Q-14.923 5.960-15.070 5.960Q-15.224 5.960-15.323 5.861Q-15.422 5.762-15.422 5.611Q-15.422 5.423-15.282 5.331Q-15.484 5.280-15.925 5.280Q-16.280 5.280-16.509 5.476Q-16.738 5.673-16.839 5.982Q-16.940 6.292-16.940 6.640Q-16.940 6.989-16.813 7.295Q-16.687 7.601-16.432 7.785Q-16.178 7.970-15.822 7.970Q-15.600 7.970-15.415 7.886Q-15.231 7.802-15.096 7.647Q-14.961 7.491-14.903 7.283Q-14.889 7.228-14.834 7.228L-14.721 7.228Q-14.691 7.228-14.668 7.252Q-14.646 7.276-14.646 7.310L-14.646 7.331Q-14.732 7.618-14.920 7.816Q-15.108 8.014-15.373 8.117Q-15.637 8.219-15.931 8.219Q-16.362 8.219-16.750 8.013Q-17.138 7.806-17.372 7.443Q-17.606 7.081-17.606 6.640M-13.723 9.286Q-13.594 9.354-13.457 9.354Q-13.286 9.354-13.136 9.265Q-12.985 9.176-12.874 9.031Q-12.763 8.886-12.684 8.718L-12.421 8.151L-13.590 5.625Q-13.665 5.478-13.795 5.446Q-13.925 5.413-14.157 5.413L-14.157 5.133L-12.636 5.133L-12.636 5.413Q-12.985 5.413-12.985 5.560Q-12.982 5.581-12.980 5.598Q-12.978 5.615-12.978 5.625L-12.120 7.484L-11.348 5.813Q-11.314 5.745-11.314 5.666Q-11.314 5.553-11.397 5.483Q-11.481 5.413-11.594 5.413L-11.594 5.133L-10.398 5.133L-10.398 5.413Q-10.616 5.413-10.789 5.517Q-10.962 5.622-11.054 5.813L-12.390 8.718Q-12.561 9.088-12.831 9.334Q-13.101 9.580-13.457 9.580Q-13.727 9.580-13.946 9.414Q-14.164 9.248-14.164 8.985Q-14.164 8.848-14.072 8.759Q-13.980 8.671-13.840 8.671Q-13.703 8.671-13.614 8.759Q-13.525 8.848-13.525 8.985Q-13.525 9.088-13.578 9.166Q-13.631 9.245-13.723 9.286\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cg fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" style=\"stroke-width:.8\">\u003Cpath fill=\"none\" d=\"M-31.06 24.2c51.215 29.476 239.003 29.476 315.67 29.476\"\u002F>\u003Cpath stroke=\"none\" d=\"m287.21 53.676-4.16-2.08 1.56 2.08-1.56 2.08\"\u002F>\u003Cg fill=\"var(--tk-accent)\" stroke=\"none\" font-family=\"cmr7\" font-size=\"7\">\u003Cg transform=\"translate(83.011 50.434)\">\u003Cpath d=\"M-27.895 8.151L-30.667 8.151L-30.667 7.871Q-29.946 7.871-29.946 7.662L-29.946 3.861Q-29.946 3.650-30.667 3.650L-30.667 3.369L-27.895 3.369Q-27.410 3.369-26.974 3.564Q-26.538 3.759-26.215 4.101Q-25.892 4.443-25.714 4.883Q-25.537 5.324-25.537 5.806Q-25.537 6.292-25.721 6.715Q-25.906 7.139-26.229 7.461Q-26.552 7.782-26.984 7.966Q-27.416 8.151-27.895 8.151M-29.283 3.861L-29.283 7.662Q-29.283 7.802-29.194 7.837Q-29.105 7.871-28.917 7.871L-28.093 7.871Q-27.468 7.871-27.064 7.609Q-26.661 7.348-26.473 6.881Q-26.285 6.415-26.285 5.806Q-26.285 5.328-26.377 4.935Q-26.470 4.542-26.719 4.244Q-26.958 3.957-27.322 3.803Q-27.686 3.650-28.093 3.650L-28.917 3.650Q-29.105 3.650-29.194 3.684Q-29.283 3.718-29.283 3.861M-22.515 8.151L-24.648 8.151L-24.648 7.871Q-23.927 7.871-23.927 7.662L-23.927 3.861Q-23.927 3.650-24.648 3.650L-24.648 3.369L-21.982 3.369Q-21.572 3.369-21.151 3.523Q-20.731 3.677-20.447 3.981Q-20.164 4.285-20.164 4.699Q-20.164 5.017-20.331 5.263Q-20.498 5.509-20.775 5.675Q-21.052 5.840-21.373 5.924Q-21.695 6.008-21.982 6.008L-23.236 6.008L-23.236 7.662Q-23.236 7.871-22.515 7.871L-22.515 8.151M-23.264 3.861L-23.264 5.758L-22.177 5.758Q-21.568 5.758-21.254 5.521Q-20.939 5.283-20.939 4.699Q-20.939 4.306-21.085 4.072Q-21.230 3.838-21.502 3.744Q-21.773 3.650-22.177 3.650L-22.898 3.650Q-23.086 3.650-23.175 3.684Q-23.264 3.718-23.264 3.861M-19.183 5.786Q-19.183 5.099-18.834 4.509Q-18.485 3.920-17.891 3.574Q-17.296 3.229-16.605 3.229Q-15.915 3.229-15.320 3.574Q-14.726 3.920-14.377 4.509Q-14.028 5.099-14.028 5.786Q-14.028 6.473-14.380 7.047Q-14.732 7.621-15.325 7.956Q-15.918 8.291-16.605 8.291Q-17.118 8.291-17.586 8.098Q-18.055 7.905-18.414 7.563Q-18.772 7.221-18.977 6.760Q-19.183 6.298-19.183 5.786M-16.605 8.031Q-16 8.031-15.594 7.686Q-15.187 7.341-14.997 6.803Q-14.808 6.264-14.808 5.666Q-14.808 5.232-14.919 4.844Q-15.030 4.456-15.255 4.149Q-15.481 3.841-15.823 3.661Q-16.164 3.482-16.605 3.482Q-17.043 3.482-17.381 3.658Q-17.720 3.834-17.949 4.149Q-18.178 4.463-18.290 4.856Q-18.403 5.249-18.403 5.666Q-18.403 6.254-18.212 6.797Q-18.020 7.341-17.614 7.686Q-17.207 8.031-16.605 8.031M-12.818 7.731Q-12.818 7.563-12.695 7.440Q-12.572 7.317-12.398 7.317Q-12.230 7.317-12.107 7.440Q-11.984 7.563-11.984 7.731Q-11.984 7.905-12.107 8.028Q-12.230 8.151-12.398 8.151Q-12.572 8.151-12.695 8.028Q-12.818 7.905-12.818 7.731M-12.818 5.547Q-12.818 5.379-12.695 5.256Q-12.572 5.133-12.398 5.133Q-12.230 5.133-12.107 5.256Q-11.984 5.379-11.984 5.547Q-11.984 5.721-12.107 5.844Q-12.230 5.967-12.398 5.967Q-12.572 5.967-12.695 5.844Q-12.818 5.721-12.818 5.547\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(83.011 50.434)\">\u003Cpath d=\"M-8.294 6.668Q-8.294 6.326-8.159 6.027Q-8.024 5.728-7.784 5.504Q-7.545 5.280-7.227 5.155Q-6.909 5.030-6.578 5.030Q-6.133 5.030-5.734 5.246Q-5.334 5.461-5.099 5.839Q-4.865 6.216-4.865 6.668Q-4.865 7.009-5.007 7.293Q-5.149 7.577-5.393 7.784Q-5.638 7.990-5.947 8.105Q-6.256 8.219-6.578 8.219Q-7.008 8.219-7.410 8.018Q-7.812 7.816-8.053 7.464Q-8.294 7.112-8.294 6.668M-6.578 7.970Q-5.976 7.970-5.752 7.592Q-5.528 7.214-5.528 6.582Q-5.528 5.970-5.763 5.611Q-5.997 5.253-6.578 5.253Q-7.630 5.253-7.630 6.582Q-7.630 7.214-7.405 7.592Q-7.179 7.970-6.578 7.970M-2.589 8.151L-4.223 8.151L-4.223 7.871Q-3.994 7.871-3.845 7.837Q-3.696 7.802-3.696 7.662L-3.696 5.813Q-3.696 5.543-3.804 5.482Q-3.912 5.420-4.223 5.420L-4.223 5.140L-3.163 5.065L-3.163 5.714Q-2.992 5.406-2.688 5.235Q-2.384 5.065-2.039 5.065Q-1.533 5.065-1.249 5.288Q-0.965 5.512-0.965 6.008L-0.965 7.662Q-0.965 7.799-0.817 7.835Q-0.668 7.871-0.442 7.871L-0.442 8.151L-2.073 8.151L-2.073 7.871Q-1.844 7.871-1.695 7.837Q-1.546 7.802-1.546 7.662L-1.546 6.022Q-1.546 5.687-1.666 5.487Q-1.786 5.287-2.100 5.287Q-2.370 5.287-2.604 5.423Q-2.838 5.560-2.977 5.794Q-3.115 6.028-3.115 6.302L-3.115 7.662Q-3.115 7.799-2.965 7.835Q-2.815 7.871-2.589 7.871L-2.589 8.151M0.104 6.616Q0.104 6.295 0.229 6.006Q0.354 5.717 0.579 5.494Q0.805 5.270 1.101 5.150Q1.396 5.030 1.714 5.030Q2.042 5.030 2.304 5.130Q2.565 5.229 2.741 5.411Q2.917 5.594 3.011 5.852Q3.105 6.110 3.105 6.442Q3.105 6.534 3.023 6.555L0.767 6.555L0.767 6.616Q0.767 7.204 1.051 7.587Q1.335 7.970 1.902 7.970Q2.224 7.970 2.492 7.777Q2.760 7.584 2.849 7.269Q2.856 7.228 2.931 7.214L3.023 7.214Q3.105 7.238 3.105 7.310Q3.105 7.317 3.099 7.344Q2.986 7.741 2.615 7.980Q2.244 8.219 1.820 8.219Q1.383 8.219 0.983 8.011Q0.583 7.802 0.344 7.435Q0.104 7.068 0.104 6.616M0.774 6.346L2.589 6.346Q2.589 6.069 2.492 5.817Q2.394 5.564 2.196 5.408Q1.998 5.253 1.714 5.253Q1.437 5.253 1.224 5.411Q1.010 5.570 0.892 5.825Q0.774 6.080 0.774 6.346\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(83.011 50.434)\">\u003Cpath d=\"M6.400 8.144L6.400 7.081Q6.400 7.057 6.428 7.030Q6.455 7.003 6.479 7.003L6.588 7.003Q6.653 7.003 6.667 7.061Q6.763 7.495 7.009 7.746Q7.255 7.997 7.669 7.997Q8.010 7.997 8.263 7.864Q8.516 7.731 8.516 7.423Q8.516 7.266 8.422 7.151Q8.328 7.037 8.190 6.968Q8.051 6.900 7.884 6.862L7.303 6.763Q6.947 6.695 6.674 6.474Q6.400 6.254 6.400 5.912Q6.400 5.663 6.512 5.488Q6.623 5.314 6.809 5.215Q6.995 5.116 7.211 5.073Q7.426 5.030 7.669 5.030Q8.082 5.030 8.362 5.212L8.578 5.037Q8.588 5.034 8.595 5.032Q8.602 5.030 8.612 5.030L8.663 5.030Q8.690 5.030 8.714 5.054Q8.738 5.078 8.738 5.106L8.738 5.953Q8.738 5.974 8.714 6.001Q8.690 6.028 8.663 6.028L8.550 6.028Q8.523 6.028 8.497 6.003Q8.472 5.977 8.472 5.953Q8.472 5.717 8.366 5.553Q8.260 5.389 8.077 5.307Q7.894 5.225 7.662 5.225Q7.334 5.225 7.077 5.328Q6.821 5.430 6.821 5.707Q6.821 5.902 7.004 6.011Q7.187 6.121 7.416 6.162L7.990 6.268Q8.236 6.316 8.450 6.444Q8.663 6.572 8.800 6.775Q8.937 6.979 8.937 7.228Q8.937 7.741 8.571 7.980Q8.205 8.219 7.669 8.219Q7.173 8.219 6.841 7.925L6.575 8.199Q6.554 8.219 6.527 8.219L6.479 8.219Q6.455 8.219 6.428 8.192Q6.400 8.165 6.400 8.144M10.140 7.317L10.140 5.813Q10.140 5.543 10.032 5.482Q9.924 5.420 9.613 5.420L9.613 5.140L10.721 5.065L10.721 7.297L10.721 7.317Q10.721 7.597 10.772 7.741Q10.823 7.884 10.965 7.941Q11.107 7.997 11.394 7.997Q11.647 7.997 11.852 7.857Q12.057 7.717 12.173 7.491Q12.290 7.266 12.290 7.016L12.290 5.813Q12.290 5.543 12.182 5.482Q12.074 5.420 11.763 5.420L11.763 5.140L12.871 5.065L12.871 7.478Q12.871 7.669 12.924 7.751Q12.977 7.833 13.077 7.852Q13.178 7.871 13.394 7.871L13.394 8.151L12.317 8.219L12.317 7.655Q12.208 7.837 12.062 7.960Q11.917 8.083 11.731 8.151Q11.544 8.219 11.343 8.219Q10.140 8.219 10.140 7.317M15.626 9.508L13.995 9.508L13.995 9.228Q14.224 9.228 14.373 9.193Q14.522 9.159 14.522 9.019L14.522 5.673Q14.522 5.502 14.385 5.461Q14.248 5.420 13.995 5.420L13.995 5.140L15.075 5.065L15.075 5.471Q15.297 5.270 15.585 5.167Q15.872 5.065 16.179 5.065Q16.607 5.065 16.971 5.278Q17.335 5.492 17.548 5.856Q17.762 6.220 17.762 6.640Q17.762 7.085 17.523 7.449Q17.283 7.813 16.890 8.016Q16.497 8.219 16.053 8.219Q15.786 8.219 15.538 8.119Q15.291 8.018 15.103 7.837L15.103 9.019Q15.103 9.156 15.251 9.192Q15.400 9.228 15.626 9.228L15.626 9.508M15.103 5.820L15.103 7.430Q15.236 7.683 15.479 7.840Q15.721 7.997 15.998 7.997Q16.326 7.997 16.579 7.796Q16.832 7.594 16.965 7.276Q17.099 6.958 17.099 6.640Q17.099 6.411 17.034 6.182Q16.969 5.953 16.841 5.755Q16.712 5.557 16.518 5.437Q16.323 5.318 16.090 5.318Q15.796 5.318 15.528 5.447Q15.260 5.577 15.103 5.820\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(83.011 50.434)\">\u003Cpath d=\"M18.576 6.616Q18.576 6.295 18.701 6.006Q18.826 5.717 19.052 5.494Q19.277 5.270 19.573 5.150Q19.868 5.030 20.186 5.030Q20.514 5.030 20.776 5.130Q21.037 5.229 21.213 5.411Q21.389 5.594 21.483 5.852Q21.577 6.110 21.577 6.442Q21.577 6.534 21.495 6.555L19.240 6.555L19.240 6.616Q19.240 7.204 19.523 7.587Q19.807 7.970 20.374 7.970Q20.696 7.970 20.964 7.777Q21.232 7.584 21.321 7.269Q21.328 7.228 21.403 7.214L21.495 7.214Q21.577 7.238 21.577 7.310Q21.577 7.317 21.571 7.344Q21.458 7.741 21.087 7.980Q20.716 8.219 20.292 8.219Q19.855 8.219 19.455 8.011Q19.055 7.802 18.816 7.435Q18.576 7.068 18.576 6.616M19.246 6.346L21.061 6.346Q21.061 6.069 20.964 5.817Q20.866 5.564 20.668 5.408Q20.470 5.253 20.186 5.253Q19.909 5.253 19.696 5.411Q19.482 5.570 19.364 5.825Q19.246 6.080 19.246 6.346M23.915 8.151L22.179 8.151L22.179 7.871Q22.408 7.871 22.557 7.837Q22.705 7.802 22.705 7.662L22.705 5.813Q22.705 5.543 22.598 5.482Q22.490 5.420 22.179 5.420L22.179 5.140L23.208 5.065L23.208 5.772Q23.338 5.464 23.580 5.265Q23.823 5.065 24.141 5.065Q24.360 5.065 24.531 5.189Q24.701 5.314 24.701 5.526Q24.701 5.663 24.602 5.762Q24.503 5.861 24.370 5.861Q24.233 5.861 24.134 5.762Q24.035 5.663 24.035 5.526Q24.035 5.386 24.134 5.287Q23.844 5.287 23.644 5.483Q23.444 5.680 23.351 5.974Q23.259 6.268 23.259 6.548L23.259 7.662Q23.259 7.871 23.915 7.871L23.915 8.151M26.875 8.124L25.747 5.625Q25.676 5.478 25.546 5.446Q25.416 5.413 25.187 5.413L25.187 5.133L26.701 5.133L26.701 5.413Q26.349 5.413 26.349 5.560Q26.349 5.605 26.359 5.625L27.224 7.543L28.003 5.813Q28.037 5.745 28.037 5.666Q28.037 5.553 27.954 5.483Q27.870 5.413 27.750 5.413L27.750 5.133L28.947 5.133L28.947 5.413Q28.728 5.413 28.557 5.516Q28.386 5.618 28.297 5.813L27.261 8.124Q27.214 8.219 27.108 8.219L27.029 8.219Q26.923 8.219 26.875 8.124M31.103 8.151L29.552 8.151L29.552 7.871Q29.777 7.871 29.926 7.837Q30.074 7.802 30.074 7.662L30.074 5.813Q30.074 5.625 30.027 5.541Q29.979 5.458 29.881 5.439Q29.784 5.420 29.572 5.420L29.572 5.140L30.628 5.065L30.628 7.662Q30.628 7.802 30.760 7.837Q30.891 7.871 31.103 7.871L31.103 8.151M29.832 3.844Q29.832 3.673 29.955 3.554Q30.078 3.434 30.249 3.434Q30.416 3.434 30.539 3.554Q30.662 3.673 30.662 3.844Q30.662 4.019 30.539 4.142Q30.416 4.265 30.249 4.265Q30.078 4.265 29.955 4.142Q29.832 4.019 29.832 3.844M31.749 8.144L31.749 7.081Q31.749 7.057 31.777 7.030Q31.804 7.003 31.828 7.003L31.937 7.003Q32.002 7.003 32.016 7.061Q32.112 7.495 32.358 7.746Q32.604 7.997 33.017 7.997Q33.359 7.997 33.612 7.864Q33.865 7.731 33.865 7.423Q33.865 7.266 33.771 7.151Q33.677 7.037 33.539 6.968Q33.400 6.900 33.233 6.862L32.652 6.763Q32.296 6.695 32.023 6.474Q31.749 6.254 31.749 5.912Q31.749 5.663 31.860 5.488Q31.971 5.314 32.158 5.215Q32.344 5.116 32.559 5.073Q32.775 5.030 33.017 5.030Q33.431 5.030 33.711 5.212L33.927 5.037Q33.937 5.034 33.944 5.032Q33.950 5.030 33.961 5.030L34.012 5.030Q34.039 5.030 34.063 5.054Q34.087 5.078 34.087 5.106L34.087 5.953Q34.087 5.974 34.063 6.001Q34.039 6.028 34.012 6.028L33.899 6.028Q33.872 6.028 33.846 6.003Q33.821 5.977 33.821 5.953Q33.821 5.717 33.715 5.553Q33.609 5.389 33.426 5.307Q33.243 5.225 33.011 5.225Q32.682 5.225 32.426 5.328Q32.170 5.430 32.170 5.707Q32.170 5.902 32.353 6.011Q32.535 6.121 32.764 6.162L33.339 6.268Q33.585 6.316 33.798 6.444Q34.012 6.572 34.149 6.775Q34.285 6.979 34.285 7.228Q34.285 7.741 33.920 7.980Q33.554 8.219 33.017 8.219Q32.522 8.219 32.190 7.925L31.924 8.199Q31.903 8.219 31.876 8.219L31.828 8.219Q31.804 8.219 31.777 8.192Q31.749 8.165 31.749 8.144M34.873 6.616Q34.873 6.295 34.998 6.006Q35.123 5.717 35.348 5.494Q35.574 5.270 35.870 5.150Q36.165 5.030 36.483 5.030Q36.811 5.030 37.073 5.130Q37.334 5.229 37.510 5.411Q37.686 5.594 37.780 5.852Q37.874 6.110 37.874 6.442Q37.874 6.534 37.792 6.555L35.536 6.555L35.536 6.616Q35.536 7.204 35.820 7.587Q36.104 7.970 36.671 7.970Q36.992 7.970 37.261 7.777Q37.529 7.584 37.618 7.269Q37.625 7.228 37.700 7.214L37.792 7.214Q37.874 7.238 37.874 7.310Q37.874 7.317 37.867 7.344Q37.755 7.741 37.384 7.980Q37.013 8.219 36.589 8.219Q36.152 8.219 35.752 8.011Q35.352 7.802 35.113 7.435Q34.873 7.068 34.873 6.616M35.543 6.346L37.358 6.346Q37.358 6.069 37.261 5.817Q37.163 5.564 36.965 5.408Q36.767 5.253 36.483 5.253Q36.206 5.253 35.993 5.411Q35.779 5.570 35.661 5.825Q35.543 6.080 35.543 6.346M38.462 6.640Q38.462 6.302 38.602 6.011Q38.742 5.721 38.987 5.507Q39.231 5.294 39.535 5.179Q39.840 5.065 40.164 5.065Q40.434 5.065 40.698 5.164Q40.961 5.263 41.152 5.441L41.152 4.043Q41.152 3.773 41.044 3.711Q40.937 3.650 40.626 3.650L40.626 3.369L41.702 3.294L41.702 7.478Q41.702 7.666 41.757 7.749Q41.812 7.833 41.913 7.852Q42.013 7.871 42.229 7.871L42.229 8.151L41.121 8.219L41.121 7.802Q40.704 8.219 40.079 8.219Q39.648 8.219 39.276 8.007Q38.903 7.796 38.683 7.435Q38.462 7.074 38.462 6.640M40.137 7.997Q40.345 7.997 40.532 7.925Q40.718 7.854 40.872 7.717Q41.026 7.580 41.121 7.402L41.121 5.793Q41.036 5.646 40.891 5.526Q40.745 5.406 40.576 5.347Q40.407 5.287 40.226 5.287Q39.665 5.287 39.397 5.676Q39.129 6.066 39.129 6.647Q39.129 7.218 39.363 7.608Q39.597 7.997 40.137 7.997\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(83.011 50.434)\">\u003Cpath d=\"M47.265 8.151L45.662 8.151L45.662 7.871Q45.888 7.871 46.037 7.837Q46.185 7.802 46.185 7.662L46.185 4.043Q46.185 3.773 46.078 3.711Q45.970 3.650 45.662 3.650L45.662 3.369L46.739 3.294L46.739 7.662Q46.739 7.799 46.889 7.835Q47.040 7.871 47.265 7.871L47.265 8.151M47.819 6.668Q47.819 6.326 47.954 6.027Q48.089 5.728 48.328 5.504Q48.568 5.280 48.886 5.155Q49.203 5.030 49.535 5.030Q49.979 5.030 50.379 5.246Q50.779 5.461 51.013 5.839Q51.247 6.216 51.247 6.668Q51.247 7.009 51.106 7.293Q50.964 7.577 50.719 7.784Q50.475 7.990 50.166 8.105Q49.856 8.219 49.535 8.219Q49.104 8.219 48.703 8.018Q48.301 7.816 48.060 7.464Q47.819 7.112 47.819 6.668M49.535 7.970Q50.137 7.970 50.360 7.592Q50.584 7.214 50.584 6.582Q50.584 5.970 50.350 5.611Q50.116 5.253 49.535 5.253Q48.482 5.253 48.482 6.582Q48.482 7.214 48.708 7.592Q48.933 7.970 49.535 7.970M51.801 8.684Q51.801 8.438 51.998 8.254Q52.194 8.069 52.450 7.990Q52.314 7.878 52.242 7.717Q52.170 7.556 52.170 7.375Q52.170 7.054 52.382 6.808Q52.047 6.510 52.047 6.100Q52.047 5.639 52.437 5.352Q52.826 5.065 53.305 5.065Q53.777 5.065 54.112 5.311Q54.286 5.157 54.496 5.075Q54.706 4.993 54.935 4.993Q55.099 4.993 55.221 5.100Q55.342 5.208 55.342 5.372Q55.342 5.468 55.270 5.540Q55.199 5.611 55.106 5.611Q55.007 5.611 54.937 5.538Q54.867 5.464 54.867 5.365Q54.867 5.311 54.881 5.280L54.887 5.266Q54.894 5.246 54.903 5.235Q54.911 5.225 54.915 5.218Q54.559 5.218 54.272 5.441Q54.559 5.734 54.559 6.100Q54.559 6.415 54.375 6.647Q54.190 6.880 53.901 7.008Q53.613 7.136 53.305 7.136Q53.103 7.136 52.912 7.086Q52.721 7.037 52.543 6.927Q52.450 7.054 52.450 7.197Q52.450 7.379 52.579 7.514Q52.707 7.649 52.891 7.649L53.524 7.649Q53.971 7.649 54.341 7.720Q54.710 7.792 54.970 8.021Q55.229 8.250 55.229 8.684Q55.229 9.005 54.934 9.207Q54.638 9.409 54.235 9.498Q53.831 9.587 53.517 9.587Q53.199 9.587 52.796 9.498Q52.392 9.409 52.097 9.207Q51.801 9.005 51.801 8.684M52.256 8.684Q52.256 8.913 52.474 9.062Q52.693 9.211 52.985 9.279Q53.278 9.347 53.517 9.347Q53.681 9.347 53.889 9.311Q54.098 9.276 54.305 9.195Q54.512 9.115 54.643 8.987Q54.775 8.859 54.775 8.684Q54.775 8.332 54.394 8.238Q54.012 8.144 53.510 8.144L52.891 8.144Q52.652 8.144 52.454 8.295Q52.256 8.445 52.256 8.684M53.305 6.897Q53.971 6.897 53.971 6.100Q53.971 5.300 53.305 5.300Q52.635 5.300 52.635 6.100Q52.635 6.897 53.305 6.897M57.707 6.897L55.650 6.897L55.650 6.394L57.707 6.394L57.707 6.897M58.511 8.144L58.511 7.081Q58.511 7.057 58.538 7.030Q58.565 7.003 58.589 7.003L58.699 7.003Q58.763 7.003 58.777 7.061Q58.873 7.495 59.119 7.746Q59.365 7.997 59.779 7.997Q60.120 7.997 60.373 7.864Q60.626 7.731 60.626 7.423Q60.626 7.266 60.532 7.151Q60.438 7.037 60.300 6.968Q60.161 6.900 59.994 6.862L59.413 6.763Q59.057 6.695 58.784 6.474Q58.511 6.254 58.511 5.912Q58.511 5.663 58.622 5.488Q58.733 5.314 58.919 5.215Q59.105 5.116 59.321 5.073Q59.536 5.030 59.779 5.030Q60.192 5.030 60.472 5.212L60.688 5.037Q60.698 5.034 60.705 5.032Q60.712 5.030 60.722 5.030L60.773 5.030Q60.801 5.030 60.825 5.054Q60.848 5.078 60.848 5.106L60.848 5.953Q60.848 5.974 60.825 6.001Q60.801 6.028 60.773 6.028L60.660 6.028Q60.633 6.028 60.607 6.003Q60.582 5.977 60.582 5.953Q60.582 5.717 60.476 5.553Q60.370 5.389 60.187 5.307Q60.004 5.225 59.772 5.225Q59.444 5.225 59.187 5.328Q58.931 5.430 58.931 5.707Q58.931 5.902 59.114 6.011Q59.297 6.121 59.526 6.162L60.100 6.268Q60.346 6.316 60.560 6.444Q60.773 6.572 60.910 6.775Q61.047 6.979 61.047 7.228Q61.047 7.741 60.681 7.980Q60.315 8.219 59.779 8.219Q59.283 8.219 58.951 7.925L58.685 8.199Q58.664 8.219 58.637 8.219L58.589 8.219Q58.565 8.219 58.538 8.192Q58.511 8.165 58.511 8.144M63.292 8.151L61.741 8.151L61.741 7.871Q61.966 7.871 62.115 7.837Q62.263 7.802 62.263 7.662L62.263 5.813Q62.263 5.625 62.216 5.541Q62.168 5.458 62.070 5.439Q61.973 5.420 61.761 5.420L61.761 5.140L62.817 5.065L62.817 7.662Q62.817 7.802 62.949 7.837Q63.080 7.871 63.292 7.871L63.292 8.151M62.021 3.844Q62.021 3.673 62.144 3.554Q62.267 3.434 62.438 3.434Q62.605 3.434 62.728 3.554Q62.851 3.673 62.851 3.844Q62.851 4.019 62.728 4.142Q62.605 4.265 62.438 4.265Q62.267 4.265 62.144 4.142Q62.021 4.019 62.021 3.844M63.897 8.684Q63.897 8.438 64.094 8.254Q64.290 8.069 64.547 7.990Q64.410 7.878 64.338 7.717Q64.266 7.556 64.266 7.375Q64.266 7.054 64.478 6.808Q64.143 6.510 64.143 6.100Q64.143 5.639 64.533 5.352Q64.923 5.065 65.401 5.065Q65.873 5.065 66.208 5.311Q66.382 5.157 66.592 5.075Q66.803 4.993 67.032 4.993Q67.196 4.993 67.317 5.100Q67.438 5.208 67.438 5.372Q67.438 5.468 67.366 5.540Q67.295 5.611 67.202 5.611Q67.103 5.611 67.033 5.538Q66.963 5.464 66.963 5.365Q66.963 5.311 66.977 5.280L66.984 5.266Q66.991 5.246 66.999 5.235Q67.008 5.225 67.011 5.218Q66.656 5.218 66.368 5.441Q66.656 5.734 66.656 6.100Q66.656 6.415 66.471 6.647Q66.286 6.880 65.998 7.008Q65.709 7.136 65.401 7.136Q65.200 7.136 65.008 7.086Q64.817 7.037 64.639 6.927Q64.547 7.054 64.547 7.197Q64.547 7.379 64.675 7.514Q64.803 7.649 64.988 7.649L65.620 7.649Q66.068 7.649 66.437 7.720Q66.806 7.792 67.066 8.021Q67.325 8.250 67.325 8.684Q67.325 9.005 67.030 9.207Q66.734 9.409 66.331 9.498Q65.928 9.587 65.613 9.587Q65.295 9.587 64.892 9.498Q64.489 9.409 64.193 9.207Q63.897 9.005 63.897 8.684M64.352 8.684Q64.352 8.913 64.571 9.062Q64.789 9.211 65.082 9.279Q65.374 9.347 65.613 9.347Q65.777 9.347 65.986 9.311Q66.194 9.276 66.401 9.195Q66.608 9.115 66.739 8.987Q66.871 8.859 66.871 8.684Q66.871 8.332 66.490 8.238Q66.109 8.144 65.606 8.144L64.988 8.144Q64.748 8.144 64.550 8.295Q64.352 8.445 64.352 8.684M65.401 6.897Q66.068 6.897 66.068 6.100Q66.068 5.300 65.401 5.300Q64.731 5.300 64.731 6.100Q64.731 6.897 65.401 6.897M69.602 8.151L67.968 8.151L67.968 7.871Q68.197 7.871 68.346 7.837Q68.494 7.802 68.494 7.662L68.494 5.813Q68.494 5.543 68.387 5.482Q68.279 5.420 67.968 5.420L67.968 5.140L69.028 5.065L69.028 5.714Q69.199 5.406 69.503 5.235Q69.807 5.065 70.152 5.065Q70.552 5.065 70.829 5.205Q71.106 5.345 71.191 5.693Q71.359 5.400 71.658 5.232Q71.957 5.065 72.302 5.065Q72.808 5.065 73.092 5.288Q73.375 5.512 73.375 6.008L73.375 7.662Q73.375 7.799 73.524 7.835Q73.673 7.871 73.898 7.871L73.898 8.151L72.268 8.151L72.268 7.871Q72.493 7.871 72.644 7.835Q72.794 7.799 72.794 7.662L72.794 6.022Q72.794 5.687 72.675 5.487Q72.555 5.287 72.241 5.287Q71.971 5.287 71.736 5.423Q71.502 5.560 71.364 5.794Q71.225 6.028 71.225 6.302L71.225 7.662Q71.225 7.799 71.374 7.835Q71.523 7.871 71.748 7.871L71.748 8.151L70.118 8.151L70.118 7.871Q70.347 7.871 70.496 7.837Q70.644 7.802 70.644 7.662L70.644 6.022Q70.644 5.687 70.525 5.487Q70.405 5.287 70.091 5.287Q69.821 5.287 69.586 5.423Q69.352 5.560 69.214 5.794Q69.075 6.028 69.075 6.302L69.075 7.662Q69.075 7.799 69.226 7.835Q69.376 7.871 69.602 7.871L69.602 8.151M74.445 6.668Q74.445 6.326 74.580 6.027Q74.715 5.728 74.954 5.504Q75.194 5.280 75.512 5.155Q75.829 5.030 76.161 5.030Q76.605 5.030 77.005 5.246Q77.405 5.461 77.639 5.839Q77.873 6.216 77.873 6.668Q77.873 7.009 77.731 7.293Q77.590 7.577 77.345 7.784Q77.101 7.990 76.792 8.105Q76.482 8.219 76.161 8.219Q75.730 8.219 75.329 8.018Q74.927 7.816 74.686 7.464Q74.445 7.112 74.445 6.668M76.161 7.970Q76.762 7.970 76.986 7.592Q77.210 7.214 77.210 6.582Q77.210 5.970 76.976 5.611Q76.742 5.253 76.161 5.253Q75.108 5.253 75.108 6.582Q75.108 7.214 75.334 7.592Q75.559 7.970 76.161 7.970M80.085 8.151L78.533 8.151L78.533 7.871Q78.759 7.871 78.907 7.837Q79.056 7.802 79.056 7.662L79.056 5.813Q79.056 5.625 79.008 5.541Q78.960 5.458 78.863 5.439Q78.765 5.420 78.554 5.420L78.554 5.140L79.610 5.065L79.610 7.662Q79.610 7.802 79.741 7.837Q79.873 7.871 80.085 7.871L80.085 8.151M78.813 3.844Q78.813 3.673 78.936 3.554Q79.059 3.434 79.230 3.434Q79.398 3.434 79.521 3.554Q79.644 3.673 79.644 3.844Q79.644 4.019 79.521 4.142Q79.398 4.265 79.230 4.265Q79.059 4.265 78.936 4.142Q78.813 4.019 78.813 3.844M80.731 6.640Q80.731 6.302 80.871 6.011Q81.011 5.721 81.255 5.507Q81.500 5.294 81.804 5.179Q82.108 5.065 82.433 5.065Q82.703 5.065 82.966 5.164Q83.229 5.263 83.421 5.441L83.421 4.043Q83.421 3.773 83.313 3.711Q83.205 3.650 82.894 3.650L82.894 3.369L83.971 3.294L83.971 7.478Q83.971 7.666 84.026 7.749Q84.080 7.833 84.181 7.852Q84.282 7.871 84.497 7.871L84.497 8.151L83.390 8.219L83.390 7.802Q82.973 8.219 82.347 8.219Q81.917 8.219 81.544 8.007Q81.172 7.796 80.951 7.435Q80.731 7.074 80.731 6.640M82.406 7.997Q82.614 7.997 82.800 7.925Q82.987 7.854 83.140 7.717Q83.294 7.580 83.390 7.402L83.390 5.793Q83.304 5.646 83.159 5.526Q83.014 5.406 82.845 5.347Q82.676 5.287 82.494 5.287Q81.934 5.287 81.666 5.676Q81.397 6.066 81.397 6.647Q81.397 7.218 81.631 7.608Q81.866 7.997 82.406 7.997\" fill=\"var(--tk-accent)\" stroke=\"var(--tk-accent)\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(83.011 50.434)\">\u003Cpath d=\"M89.545 8.151L87.942 8.151L87.942 7.871Q88.168 7.871 88.317 7.837Q88.465 7.802 88.465 7.662L88.465 4.043Q88.465 3.773 88.358 3.711Q88.250 3.650 87.942 3.650L87.942 3.369L89.019 3.294L89.019 7.662Q89.019 7.799 89.169 7.835Q89.320 7.871 89.545 7.871L89.545 8.151M90.099 6.668Q90.099 6.326 90.234 6.027Q90.369 5.728 90.608 5.504Q90.848 5.280 91.166 5.155Q91.483 5.030 91.815 5.030Q92.259 5.030 92.659 5.246Q93.059 5.461 93.293 5.839Q93.527 6.216 93.527 6.668Q93.527 7.009 93.386 7.293Q93.244 7.577 92.999 7.784Q92.755 7.990 92.446 8.105Q92.136 8.219 91.815 8.219Q91.384 8.219 90.983 8.018Q90.581 7.816 90.340 7.464Q90.099 7.112 90.099 6.668M91.815 7.970Q92.417 7.970 92.640 7.592Q92.864 7.214 92.864 6.582Q92.864 5.970 92.630 5.611Q92.396 5.253 91.815 5.253Q90.762 5.253 90.762 6.582Q90.762 7.214 90.988 7.592Q91.213 7.970 91.815 7.970M94.122 8.144L94.122 7.081Q94.122 7.057 94.149 7.030Q94.177 7.003 94.201 7.003L94.310 7.003Q94.375 7.003 94.389 7.061Q94.484 7.495 94.730 7.746Q94.977 7.997 95.390 7.997Q95.732 7.997 95.985 7.864Q96.238 7.731 96.238 7.423Q96.238 7.266 96.144 7.151Q96.050 7.037 95.911 6.968Q95.773 6.900 95.605 6.862L95.024 6.763Q94.669 6.695 94.396 6.474Q94.122 6.254 94.122 5.912Q94.122 5.663 94.233 5.488Q94.344 5.314 94.531 5.215Q94.717 5.116 94.932 5.073Q95.147 5.030 95.390 5.030Q95.804 5.030 96.084 5.212L96.299 5.037Q96.310 5.034 96.316 5.032Q96.323 5.030 96.334 5.030L96.385 5.030Q96.412 5.030 96.436 5.054Q96.460 5.078 96.460 5.106L96.460 5.953Q96.460 5.974 96.436 6.001Q96.412 6.028 96.385 6.028L96.272 6.028Q96.245 6.028 96.219 6.003Q96.193 5.977 96.193 5.953Q96.193 5.717 96.087 5.553Q95.981 5.389 95.799 5.307Q95.616 5.225 95.383 5.225Q95.055 5.225 94.799 5.328Q94.542 5.430 94.542 5.707Q94.542 5.902 94.725 6.011Q94.908 6.121 95.137 6.162L95.711 6.268Q95.958 6.316 96.171 6.444Q96.385 6.572 96.521 6.775Q96.658 6.979 96.658 7.228Q96.658 7.741 96.292 7.980Q95.927 8.219 95.390 8.219Q94.895 8.219 94.563 7.925L94.296 8.199Q94.276 8.219 94.249 8.219L94.201 8.219Q94.177 8.219 94.149 8.192Q94.122 8.165 94.122 8.144M97.287 8.144L97.287 7.081Q97.287 7.057 97.314 7.030Q97.342 7.003 97.366 7.003L97.475 7.003Q97.540 7.003 97.554 7.061Q97.649 7.495 97.896 7.746Q98.142 7.997 98.555 7.997Q98.897 7.997 99.150 7.864Q99.403 7.731 99.403 7.423Q99.403 7.266 99.309 7.151Q99.215 7.037 99.076 6.968Q98.938 6.900 98.771 6.862L98.189 6.763Q97.834 6.695 97.561 6.474Q97.287 6.254 97.287 5.912Q97.287 5.663 97.398 5.488Q97.509 5.314 97.696 5.215Q97.882 5.116 98.097 5.073Q98.313 5.030 98.555 5.030Q98.969 5.030 99.249 5.212L99.464 5.037Q99.475 5.034 99.481 5.032Q99.488 5.030 99.499 5.030L99.550 5.030Q99.577 5.030 99.601 5.054Q99.625 5.078 99.625 5.106L99.625 5.953Q99.625 5.974 99.601 6.001Q99.577 6.028 99.550 6.028L99.437 6.028Q99.410 6.028 99.384 6.003Q99.358 5.977 99.358 5.953Q99.358 5.717 99.252 5.553Q99.146 5.389 98.964 5.307Q98.781 5.225 98.548 5.225Q98.220 5.225 97.964 5.328Q97.708 5.430 97.708 5.707Q97.708 5.902 97.890 6.011Q98.073 6.121 98.302 6.162L98.876 6.268Q99.123 6.316 99.336 6.444Q99.550 6.572 99.687 6.775Q99.823 6.979 99.823 7.228Q99.823 7.741 99.458 7.980Q99.092 8.219 98.555 8.219Q98.060 8.219 97.728 7.925L97.461 8.199Q97.441 8.219 97.414 8.219L97.366 8.219Q97.342 8.219 97.314 8.192Q97.287 8.165 97.287 8.144\" 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(169.442 14.554)\">\u003Cpath d=\"M-29.129 8.151L-30.681 8.151L-30.681 7.871Q-30.455 7.871-30.306 7.837Q-30.158 7.802-30.158 7.662L-30.158 5.813Q-30.158 5.625-30.206 5.541Q-30.253 5.458-30.351 5.439Q-30.448 5.420-30.660 5.420L-30.660 5.140L-29.604 5.065L-29.604 7.662Q-29.604 7.802-29.472 7.837Q-29.341 7.871-29.129 7.871L-29.129 8.151M-30.400 3.844Q-30.400 3.673-30.277 3.554Q-30.154 3.434-29.983 3.434Q-29.816 3.434-29.693 3.554Q-29.570 3.673-29.570 3.844Q-29.570 4.019-29.693 4.142Q-29.816 4.265-29.983 4.265Q-30.154 4.265-30.277 4.142Q-30.400 4.019-30.400 3.844M-26.801 8.151L-28.435 8.151L-28.435 7.871Q-28.206 7.871-28.057 7.837Q-27.909 7.802-27.909 7.662L-27.909 5.813Q-27.909 5.543-28.016 5.482Q-28.124 5.420-28.435 5.420L-28.435 5.140L-27.375 5.065L-27.375 5.714Q-27.205 5.406-26.900 5.235Q-26.596 5.065-26.251 5.065Q-25.851 5.065-25.574 5.205Q-25.297 5.345-25.212 5.693Q-25.044 5.400-24.745 5.232Q-24.446 5.065-24.101 5.065Q-23.595 5.065-23.311 5.288Q-23.028 5.512-23.028 6.008L-23.028 7.662Q-23.028 7.799-22.879 7.835Q-22.730 7.871-22.505 7.871L-22.505 8.151L-24.135 8.151L-24.135 7.871Q-23.910 7.871-23.759 7.835Q-23.609 7.799-23.609 7.662L-23.609 6.022Q-23.609 5.687-23.728 5.487Q-23.848 5.287-24.163 5.287Q-24.433 5.287-24.667 5.423Q-24.901 5.560-25.039 5.794Q-25.178 6.028-25.178 6.302L-25.178 7.662Q-25.178 7.799-25.029 7.835Q-24.880 7.871-24.655 7.871L-24.655 8.151L-26.285 8.151L-26.285 7.871Q-26.056 7.871-25.907 7.837Q-25.759 7.802-25.759 7.662L-25.759 6.022Q-25.759 5.687-25.878 5.487Q-25.998 5.287-26.312 5.287Q-26.582 5.287-26.817 5.423Q-27.051 5.560-27.189 5.794Q-27.328 6.028-27.328 6.302L-27.328 7.662Q-27.328 7.799-27.177 7.835Q-27.027 7.871-26.801 7.871L-26.801 8.151M-20.273 9.508L-21.903 9.508L-21.903 9.228Q-21.674 9.228-21.526 9.193Q-21.377 9.159-21.377 9.019L-21.377 5.673Q-21.377 5.502-21.514 5.461Q-21.650 5.420-21.903 5.420L-21.903 5.140L-20.823 5.065L-20.823 5.471Q-20.601 5.270-20.314 5.167Q-20.027 5.065-19.719 5.065Q-19.292 5.065-18.928 5.278Q-18.564 5.492-18.350 5.856Q-18.137 6.220-18.137 6.640Q-18.137 7.085-18.376 7.449Q-18.615 7.813-19.008 8.016Q-19.401 8.219-19.846 8.219Q-20.112 8.219-20.360 8.119Q-20.608 8.018-20.796 7.837L-20.796 9.019Q-20.796 9.156-20.647 9.192Q-20.498 9.228-20.273 9.228L-20.273 9.508M-20.796 5.820L-20.796 7.430Q-20.663 7.683-20.420 7.840Q-20.177 7.997-19.900 7.997Q-19.572 7.997-19.319 7.796Q-19.066 7.594-18.933 7.276Q-18.800 6.958-18.800 6.640Q-18.800 6.411-18.865 6.182Q-18.930 5.953-19.058 5.755Q-19.186 5.557-19.381 5.437Q-19.576 5.318-19.808 5.318Q-20.102 5.318-20.370 5.447Q-20.639 5.577-20.796 5.820M-15.833 8.151L-17.436 8.151L-17.436 7.871Q-17.210 7.871-17.062 7.837Q-16.913 7.802-16.913 7.662L-16.913 4.043Q-16.913 3.773-17.021 3.711Q-17.128 3.650-17.436 3.650L-17.436 3.369L-16.359 3.294L-16.359 7.662Q-16.359 7.799-16.209 7.835Q-16.059 7.871-15.833 7.871L-15.833 8.151M-13.622 8.151L-15.173 8.151L-15.173 7.871Q-14.948 7.871-14.799 7.837Q-14.650 7.802-14.650 7.662L-14.650 5.813Q-14.650 5.625-14.698 5.541Q-14.746 5.458-14.843 5.439Q-14.941 5.420-15.153 5.420L-15.153 5.140L-14.097 5.065L-14.097 7.662Q-14.097 7.802-13.965 7.837Q-13.833 7.871-13.622 7.871L-13.622 8.151M-14.893 3.844Q-14.893 3.673-14.770 3.554Q-14.647 3.434-14.476 3.434Q-14.309 3.434-14.185 3.554Q-14.062 3.673-14.062 3.844Q-14.062 4.019-14.185 4.142Q-14.309 4.265-14.476 4.265Q-14.647 4.265-14.770 4.142Q-14.893 4.019-14.893 3.844M-12.976 6.640Q-12.976 6.312-12.841 6.011Q-12.706 5.711-12.470 5.490Q-12.234 5.270-11.930 5.150Q-11.625 5.030-11.301 5.030Q-10.795 5.030-10.446 5.133Q-10.098 5.235-10.098 5.611Q-10.098 5.758-10.195 5.859Q-10.292 5.960-10.439 5.960Q-10.593 5.960-10.692 5.861Q-10.791 5.762-10.791 5.611Q-10.791 5.423-10.651 5.331Q-10.853 5.280-11.294 5.280Q-11.649 5.280-11.878 5.476Q-12.107 5.673-12.208 5.982Q-12.309 6.292-12.309 6.640Q-12.309 6.989-12.183 7.295Q-12.056 7.601-11.801 7.785Q-11.547 7.970-11.191 7.970Q-10.969 7.970-10.785 7.886Q-10.600 7.802-10.465 7.647Q-10.330 7.491-10.272 7.283Q-10.258 7.228-10.204 7.228L-10.091 7.228Q-10.060 7.228-10.038 7.252Q-10.016 7.276-10.016 7.310L-10.016 7.331Q-10.101 7.618-10.289 7.816Q-10.477 8.014-10.742 8.117Q-11.007 8.219-11.301 8.219Q-11.731 8.219-12.119 8.013Q-12.507 7.806-12.741 7.443Q-12.976 7.081-12.976 6.640M-7.811 8.151L-9.363 8.151L-9.363 7.871Q-9.137 7.871-8.988 7.837Q-8.840 7.802-8.840 7.662L-8.840 5.813Q-8.840 5.625-8.888 5.541Q-8.935 5.458-9.033 5.439Q-9.130 5.420-9.342 5.420L-9.342 5.140L-8.286 5.065L-8.286 7.662Q-8.286 7.802-8.154 7.837Q-8.023 7.871-7.811 7.871L-7.811 8.151M-9.082 3.844Q-9.082 3.673-8.959 3.554Q-8.836 3.434-8.665 3.434Q-8.498 3.434-8.375 3.554Q-8.252 3.673-8.252 3.844Q-8.252 4.019-8.375 4.142Q-8.498 4.265-8.665 4.265Q-8.836 4.265-8.959 4.142Q-9.082 4.019-9.082 3.844M-6.639 7.310L-6.639 5.413L-7.278 5.413L-7.278 5.191Q-6.960 5.191-6.743 4.981Q-6.526 4.771-6.425 4.461Q-6.324 4.152-6.324 3.844L-6.058 3.844L-6.058 5.133L-4.981 5.133L-4.981 5.413L-6.058 5.413L-6.058 7.297Q-6.058 7.573-5.953 7.772Q-5.849 7.970-5.589 7.970Q-5.432 7.970-5.326 7.866Q-5.220 7.761-5.171 7.608Q-5.121 7.454-5.121 7.297L-5.121 6.883L-4.854 6.883L-4.854 7.310Q-4.854 7.536-4.954 7.746Q-5.053 7.956-5.237 8.088Q-5.422 8.219-5.651 8.219Q-6.088 8.219-6.363 7.982Q-6.639 7.744-6.639 7.310\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.442 14.554)\">\u003Cpath d=\"M0.421 8.151L-1.315 8.151L-1.315 7.871Q-1.086 7.871-0.937 7.837Q-0.789 7.802-0.789 7.662L-0.789 5.813Q-0.789 5.543-0.896 5.482Q-1.004 5.420-1.315 5.420L-1.315 5.140L-0.286 5.065L-0.286 5.772Q-0.156 5.464 0.086 5.265Q0.329 5.065 0.647 5.065Q0.866 5.065 1.037 5.189Q1.208 5.314 1.208 5.526Q1.208 5.663 1.108 5.762Q1.009 5.861 0.876 5.861Q0.739 5.861 0.640 5.762Q0.541 5.663 0.541 5.526Q0.541 5.386 0.640 5.287Q0.350 5.287 0.150 5.483Q-0.050 5.680-0.143 5.974Q-0.235 6.268-0.235 6.548L-0.235 7.662Q-0.235 7.871 0.421 7.871L0.421 8.151M1.751 6.616Q1.751 6.295 1.876 6.006Q2.001 5.717 2.226 5.494Q2.452 5.270 2.747 5.150Q3.043 5.030 3.361 5.030Q3.689 5.030 3.951 5.130Q4.212 5.229 4.388 5.411Q4.564 5.594 4.658 5.852Q4.752 6.110 4.752 6.442Q4.752 6.534 4.670 6.555L2.414 6.555L2.414 6.616Q2.414 7.204 2.698 7.587Q2.982 7.970 3.549 7.970Q3.870 7.970 4.138 7.777Q4.407 7.584 4.496 7.269Q4.503 7.228 4.578 7.214L4.670 7.214Q4.752 7.238 4.752 7.310Q4.752 7.317 4.745 7.344Q4.632 7.741 4.262 7.980Q3.891 8.219 3.467 8.219Q3.029 8.219 2.629 8.011Q2.230 7.802 1.990 7.435Q1.751 7.068 1.751 6.616M2.421 6.346L4.236 6.346Q4.236 6.069 4.138 5.817Q4.041 5.564 3.843 5.408Q3.645 5.253 3.361 5.253Q3.084 5.253 2.870 5.411Q2.657 5.570 2.539 5.825Q2.421 6.080 2.421 6.346M6.728 8.124L5.747 5.625Q5.685 5.482 5.567 5.447Q5.449 5.413 5.234 5.413L5.234 5.133L6.714 5.133L6.714 5.413Q6.335 5.413 6.335 5.574Q6.335 5.584 6.348 5.625L7.063 7.457L7.736 5.752Q7.705 5.680 7.705 5.652Q7.705 5.625 7.678 5.625Q7.616 5.478 7.498 5.446Q7.380 5.413 7.169 5.413L7.169 5.133L8.566 5.133L8.566 5.413Q8.190 5.413 8.190 5.574Q8.190 5.605 8.197 5.625L8.953 7.563L9.640 5.813Q9.660 5.762 9.660 5.707Q9.660 5.567 9.547 5.490Q9.435 5.413 9.294 5.413L9.294 5.133L10.515 5.133L10.515 5.413Q10.310 5.413 10.154 5.519Q9.999 5.625 9.927 5.813L9.021 8.124Q8.987 8.219 8.874 8.219L8.806 8.219Q8.696 8.219 8.659 8.124L7.876 6.121L7.090 8.124Q7.056 8.219 6.943 8.219L6.875 8.219Q6.765 8.219 6.728 8.124\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.442 14.554)\">\u003Cpath d=\"M10.903 7.423Q10.903 7.091 11.126 6.864Q11.350 6.637 11.694 6.509Q12.037 6.380 12.410 6.328Q12.782 6.275 13.087 6.275L13.087 6.022Q13.087 5.817 12.979 5.637Q12.871 5.458 12.690 5.355Q12.509 5.253 12.301 5.253Q11.894 5.253 11.658 5.345Q11.747 5.382 11.793 5.466Q11.839 5.550 11.839 5.652Q11.839 5.748 11.793 5.827Q11.747 5.905 11.666 5.950Q11.586 5.994 11.497 5.994Q11.347 5.994 11.246 5.897Q11.145 5.799 11.145 5.652Q11.145 5.030 12.301 5.030Q12.512 5.030 12.762 5.094Q13.011 5.157 13.213 5.276Q13.415 5.396 13.541 5.581Q13.668 5.765 13.668 6.008L13.668 7.584Q13.668 7.700 13.729 7.796Q13.791 7.891 13.904 7.891Q14.013 7.891 14.078 7.797Q14.143 7.703 14.143 7.584L14.143 7.136L14.409 7.136L14.409 7.584Q14.409 7.854 14.182 8.019Q13.955 8.185 13.675 8.185Q13.466 8.185 13.329 8.031Q13.193 7.878 13.169 7.662Q13.022 7.929 12.740 8.074Q12.458 8.219 12.133 8.219Q11.856 8.219 11.572 8.144Q11.289 8.069 11.096 7.890Q10.903 7.710 10.903 7.423M11.518 7.423Q11.518 7.597 11.619 7.727Q11.719 7.857 11.875 7.927Q12.030 7.997 12.195 7.997Q12.413 7.997 12.622 7.900Q12.830 7.802 12.958 7.621Q13.087 7.440 13.087 7.214L13.087 6.486Q12.762 6.486 12.396 6.577Q12.030 6.668 11.774 6.880Q11.518 7.091 11.518 7.423M16.576 8.151L14.840 8.151L14.840 7.871Q15.069 7.871 15.218 7.837Q15.366 7.802 15.366 7.662L15.366 5.813Q15.366 5.543 15.259 5.482Q15.151 5.420 14.840 5.420L14.840 5.140L15.869 5.065L15.869 5.772Q15.999 5.464 16.241 5.265Q16.484 5.065 16.802 5.065Q17.021 5.065 17.192 5.189Q17.363 5.314 17.363 5.526Q17.363 5.663 17.263 5.762Q17.164 5.861 17.031 5.861Q16.894 5.861 16.795 5.762Q16.696 5.663 16.696 5.526Q16.696 5.386 16.795 5.287Q16.505 5.287 16.305 5.483Q16.105 5.680 16.012 5.974Q15.920 6.268 15.920 6.548L15.920 7.662Q15.920 7.871 16.576 7.871L16.576 8.151M17.947 6.640Q17.947 6.302 18.087 6.011Q18.227 5.721 18.472 5.507Q18.716 5.294 19.020 5.179Q19.324 5.065 19.649 5.065Q19.919 5.065 20.182 5.164Q20.446 5.263 20.637 5.441L20.637 4.043Q20.637 3.773 20.529 3.711Q20.422 3.650 20.111 3.650L20.111 3.369L21.187 3.294L21.187 7.478Q21.187 7.666 21.242 7.749Q21.297 7.833 21.397 7.852Q21.498 7.871 21.714 7.871L21.714 8.151L20.606 8.219L20.606 7.802Q20.189 8.219 19.564 8.219Q19.133 8.219 18.760 8.007Q18.388 7.796 18.167 7.435Q17.947 7.074 17.947 6.640M19.622 7.997Q19.830 7.997 20.017 7.925Q20.203 7.854 20.357 7.717Q20.510 7.580 20.606 7.402L20.606 5.793Q20.521 5.646 20.375 5.526Q20.230 5.406 20.061 5.347Q19.892 5.287 19.711 5.287Q19.150 5.287 18.882 5.676Q18.613 6.066 18.613 6.647Q18.613 7.218 18.848 7.608Q19.082 7.997 19.622 7.997\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.442 14.554)\">\u003Cpath d=\"M30.235 7.344L25.402 7.344Q25.334 7.334 25.288 7.288Q25.242 7.242 25.242 7.170Q25.242 7.105 25.288 7.059Q25.334 7.013 25.402 7.003L30.235 7.003Q30.304 7.013 30.350 7.059Q30.396 7.105 30.396 7.170Q30.396 7.242 30.350 7.288Q30.304 7.334 30.235 7.344M30.235 5.806L25.402 5.806Q25.334 5.796 25.288 5.750Q25.242 5.704 25.242 5.632Q25.242 5.488 25.402 5.464L30.235 5.464Q30.396 5.488 30.396 5.632Q30.396 5.704 30.350 5.750Q30.304 5.796 30.235 5.806\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.442 14.554)\">\u003Cpath d=\"M34.707 8.151L34.440 8.151L34.440 4.043Q34.440 3.773 34.333 3.711Q34.225 3.650 33.914 3.650L33.914 3.369L34.994 3.294L34.994 5.464Q35.203 5.273 35.488 5.169Q35.773 5.065 36.071 5.065Q36.389 5.065 36.686 5.186Q36.983 5.307 37.206 5.523Q37.428 5.738 37.554 6.023Q37.681 6.309 37.681 6.640Q37.681 7.085 37.441 7.449Q37.202 7.813 36.809 8.016Q36.416 8.219 35.972 8.219Q35.777 8.219 35.587 8.163Q35.398 8.107 35.237 8.002Q35.076 7.898 34.936 7.737L34.707 8.151M35.022 5.806L35.022 7.423Q35.158 7.683 35.399 7.840Q35.640 7.997 35.917 7.997Q36.211 7.997 36.423 7.890Q36.635 7.782 36.768 7.590Q36.901 7.399 36.960 7.160Q37.018 6.921 37.018 6.640Q37.018 6.281 36.924 5.977Q36.830 5.673 36.602 5.480Q36.375 5.287 36.009 5.287Q35.709 5.287 35.442 5.423Q35.175 5.560 35.022 5.806\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.442 14.554)\">\u003Cpath d=\"M38.491 6.616Q38.491 6.295 38.616 6.006Q38.741 5.717 38.967 5.494Q39.192 5.270 39.488 5.150Q39.783 5.030 40.101 5.030Q40.429 5.030 40.691 5.130Q40.952 5.229 41.128 5.411Q41.304 5.594 41.398 5.852Q41.492 6.110 41.492 6.442Q41.492 6.534 41.410 6.555L39.155 6.555L39.155 6.616Q39.155 7.204 39.438 7.587Q39.722 7.970 40.289 7.970Q40.611 7.970 40.879 7.777Q41.147 7.584 41.236 7.269Q41.243 7.228 41.318 7.214L41.410 7.214Q41.492 7.238 41.492 7.310Q41.492 7.317 41.486 7.344Q41.373 7.741 41.002 7.980Q40.631 8.219 40.207 8.219Q39.770 8.219 39.370 8.011Q38.970 7.802 38.731 7.435Q38.491 7.068 38.491 6.616M39.161 6.346L40.976 6.346Q40.976 6.069 40.879 5.817Q40.781 5.564 40.583 5.408Q40.385 5.253 40.101 5.253Q39.824 5.253 39.611 5.411Q39.397 5.570 39.279 5.825Q39.161 6.080 39.161 6.346M42.607 7.310L42.607 5.413L41.968 5.413L41.968 5.191Q42.285 5.191 42.502 4.981Q42.719 4.771 42.820 4.461Q42.921 4.152 42.921 3.844L43.188 3.844L43.188 5.133L44.264 5.133L44.264 5.413L43.188 5.413L43.188 7.297Q43.188 7.573 43.292 7.772Q43.396 7.970 43.656 7.970Q43.813 7.970 43.919 7.866Q44.025 7.761 44.075 7.608Q44.124 7.454 44.124 7.297L44.124 6.883L44.391 6.883L44.391 7.310Q44.391 7.536 44.292 7.746Q44.193 7.956 44.008 8.088Q43.823 8.219 43.594 8.219Q43.157 8.219 42.882 7.982Q42.607 7.744 42.607 7.310M45.259 7.423Q45.259 7.091 45.483 6.864Q45.707 6.637 46.050 6.509Q46.394 6.380 46.766 6.328Q47.139 6.275 47.443 6.275L47.443 6.022Q47.443 5.817 47.335 5.637Q47.228 5.458 47.047 5.355Q46.865 5.253 46.657 5.253Q46.250 5.253 46.014 5.345Q46.103 5.382 46.149 5.466Q46.196 5.550 46.196 5.652Q46.196 5.748 46.149 5.827Q46.103 5.905 46.023 5.950Q45.943 5.994 45.854 5.994Q45.703 5.994 45.603 5.897Q45.502 5.799 45.502 5.652Q45.502 5.030 46.657 5.030Q46.869 5.030 47.118 5.094Q47.368 5.157 47.570 5.276Q47.771 5.396 47.898 5.581Q48.024 5.765 48.024 6.008L48.024 7.584Q48.024 7.700 48.086 7.796Q48.147 7.891 48.260 7.891Q48.369 7.891 48.434 7.797Q48.499 7.703 48.499 7.584L48.499 7.136L48.766 7.136L48.766 7.584Q48.766 7.854 48.539 8.019Q48.311 8.185 48.031 8.185Q47.822 8.185 47.686 8.031Q47.549 7.878 47.525 7.662Q47.378 7.929 47.096 8.074Q46.814 8.219 46.489 8.219Q46.213 8.219 45.929 8.144Q45.645 8.069 45.452 7.890Q45.259 7.710 45.259 7.423M45.874 7.423Q45.874 7.597 45.975 7.727Q46.076 7.857 46.231 7.927Q46.387 7.997 46.551 7.997Q46.770 7.997 46.978 7.900Q47.187 7.802 47.315 7.621Q47.443 7.440 47.443 7.214L47.443 6.486Q47.118 6.486 46.753 6.577Q46.387 6.668 46.131 6.880Q45.874 7.091 45.874 7.423\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.442 14.554)\">\u003Cpath d=\"M53.561 8.151L51.958 8.151L51.958 7.871Q52.184 7.871 52.333 7.837Q52.481 7.802 52.481 7.662L52.481 4.043Q52.481 3.773 52.374 3.711Q52.266 3.650 51.958 3.650L51.958 3.369L53.035 3.294L53.035 7.662Q53.035 7.799 53.185 7.835Q53.336 7.871 53.561 7.871L53.561 8.151M54.115 6.668Q54.115 6.326 54.250 6.027Q54.385 5.728 54.624 5.504Q54.864 5.280 55.182 5.155Q55.499 5.030 55.831 5.030Q56.275 5.030 56.675 5.246Q57.075 5.461 57.309 5.839Q57.543 6.216 57.543 6.668Q57.543 7.009 57.402 7.293Q57.260 7.577 57.015 7.784Q56.771 7.990 56.462 8.105Q56.152 8.219 55.831 8.219Q55.400 8.219 54.999 8.018Q54.597 7.816 54.356 7.464Q54.115 7.112 54.115 6.668M55.831 7.970Q56.433 7.970 56.656 7.592Q56.880 7.214 56.880 6.582Q56.880 5.970 56.646 5.611Q56.412 5.253 55.831 5.253Q54.778 5.253 54.778 6.582Q54.778 7.214 55.004 7.592Q55.229 7.970 55.831 7.970M58.097 8.684Q58.097 8.438 58.294 8.254Q58.490 8.069 58.746 7.990Q58.610 7.878 58.538 7.717Q58.466 7.556 58.466 7.375Q58.466 7.054 58.678 6.808Q58.343 6.510 58.343 6.100Q58.343 5.639 58.733 5.352Q59.122 5.065 59.601 5.065Q60.073 5.065 60.408 5.311Q60.582 5.157 60.792 5.075Q61.002 4.993 61.231 4.993Q61.395 4.993 61.517 5.100Q61.638 5.208 61.638 5.372Q61.638 5.468 61.566 5.540Q61.495 5.611 61.402 5.611Q61.303 5.611 61.233 5.538Q61.163 5.464 61.163 5.365Q61.163 5.311 61.177 5.280L61.183 5.266Q61.190 5.246 61.199 5.235Q61.207 5.225 61.211 5.218Q60.855 5.218 60.568 5.441Q60.855 5.734 60.855 6.100Q60.855 6.415 60.671 6.647Q60.486 6.880 60.197 7.008Q59.909 7.136 59.601 7.136Q59.399 7.136 59.208 7.086Q59.017 7.037 58.839 6.927Q58.746 7.054 58.746 7.197Q58.746 7.379 58.875 7.514Q59.003 7.649 59.187 7.649L59.820 7.649Q60.267 7.649 60.637 7.720Q61.006 7.792 61.266 8.021Q61.525 8.250 61.525 8.684Q61.525 9.005 61.230 9.207Q60.934 9.409 60.531 9.498Q60.127 9.587 59.813 9.587Q59.495 9.587 59.092 9.498Q58.688 9.409 58.393 9.207Q58.097 9.005 58.097 8.684M58.552 8.684Q58.552 8.913 58.770 9.062Q58.989 9.211 59.281 9.279Q59.574 9.347 59.813 9.347Q59.977 9.347 60.185 9.311Q60.394 9.276 60.601 9.195Q60.808 9.115 60.939 8.987Q61.071 8.859 61.071 8.684Q61.071 8.332 60.690 8.238Q60.308 8.144 59.806 8.144L59.187 8.144Q58.948 8.144 58.750 8.295Q58.552 8.445 58.552 8.684M59.601 6.897Q60.267 6.897 60.267 6.100Q60.267 5.300 59.601 5.300Q58.931 5.300 58.931 6.100Q58.931 6.897 59.601 6.897M64.249 9.901Q63.699 9.501 63.328 8.946Q62.957 8.390 62.776 7.744Q62.595 7.098 62.595 6.401Q62.595 5.888 62.696 5.393Q62.797 4.897 63.002 4.446Q63.207 3.995 63.520 3.603Q63.832 3.212 64.249 2.908Q64.260 2.904 64.267 2.903Q64.273 2.901 64.284 2.901L64.352 2.901Q64.386 2.901 64.408 2.925Q64.431 2.949 64.431 2.986Q64.431 3.031 64.403 3.048Q64.055 3.349 63.802 3.733Q63.549 4.118 63.397 4.559Q63.245 5 63.173 5.456Q63.101 5.912 63.101 6.401Q63.101 7.402 63.410 8.289Q63.720 9.176 64.403 9.761Q64.431 9.778 64.431 9.822Q64.431 9.860 64.408 9.884Q64.386 9.908 64.352 9.908L64.284 9.908Q64.277 9.904 64.268 9.903Q64.260 9.901 64.249 9.901M66.885 9.508L65.254 9.508L65.254 9.228Q65.483 9.228 65.632 9.193Q65.781 9.159 65.781 9.019L65.781 5.673Q65.781 5.502 65.644 5.461Q65.507 5.420 65.254 5.420L65.254 5.140L66.334 5.065L66.334 5.471Q66.557 5.270 66.844 5.167Q67.131 5.065 67.438 5.065Q67.866 5.065 68.230 5.278Q68.594 5.492 68.807 5.856Q69.021 6.220 69.021 6.640Q69.021 7.085 68.782 7.449Q68.542 7.813 68.149 8.016Q67.756 8.219 67.312 8.219Q67.045 8.219 66.798 8.119Q66.550 8.018 66.362 7.837L66.362 9.019Q66.362 9.156 66.510 9.192Q66.659 9.228 66.885 9.228L66.885 9.508M66.362 5.820L66.362 7.430Q66.495 7.683 66.738 7.840Q66.980 7.997 67.257 7.997Q67.585 7.997 67.838 7.796Q68.091 7.594 68.225 7.276Q68.358 6.958 68.358 6.640Q68.358 6.411 68.293 6.182Q68.228 5.953 68.100 5.755Q67.972 5.557 67.777 5.437Q67.582 5.318 67.350 5.318Q67.056 5.318 66.787 5.447Q66.519 5.577 66.362 5.820\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.442 14.554)\">\u003Cpath d=\"M69.845 6.668Q69.845 6.326 69.980 6.027Q70.115 5.728 70.355 5.504Q70.594 5.280 70.912 5.155Q71.230 5.030 71.561 5.030Q72.006 5.030 72.405 5.246Q72.805 5.461 73.040 5.839Q73.274 6.216 73.274 6.668Q73.274 7.009 73.132 7.293Q72.990 7.577 72.746 7.784Q72.501 7.990 72.192 8.105Q71.883 8.219 71.561 8.219Q71.131 8.219 70.729 8.018Q70.327 7.816 70.086 7.464Q69.845 7.112 69.845 6.668M71.561 7.970Q72.163 7.970 72.387 7.592Q72.611 7.214 72.611 6.582Q72.611 5.970 72.376 5.611Q72.142 5.253 71.561 5.253Q70.509 5.253 70.509 6.582Q70.509 7.214 70.734 7.592Q70.960 7.970 71.561 7.970M75.536 8.151L73.933 8.151L73.933 7.871Q74.159 7.871 74.308 7.837Q74.456 7.802 74.456 7.662L74.456 4.043Q74.456 3.773 74.349 3.711Q74.241 3.650 73.933 3.650L73.933 3.369L75.010 3.294L75.010 7.662Q75.010 7.799 75.160 7.835Q75.311 7.871 75.536 7.871L75.536 8.151M77.748 8.151L76.196 8.151L76.196 7.871Q76.422 7.871 76.570 7.837Q76.719 7.802 76.719 7.662L76.719 5.813Q76.719 5.625 76.671 5.541Q76.623 5.458 76.526 5.439Q76.428 5.420 76.217 5.420L76.217 5.140L77.273 5.065L77.273 7.662Q77.273 7.802 77.404 7.837Q77.536 7.871 77.748 7.871L77.748 8.151M76.476 3.844Q76.476 3.673 76.599 3.554Q76.722 3.434 76.893 3.434Q77.061 3.434 77.184 3.554Q77.307 3.673 77.307 3.844Q77.307 4.019 77.184 4.142Q77.061 4.265 76.893 4.265Q76.722 4.265 76.599 4.142Q76.476 4.019 76.476 3.844M78.394 6.640Q78.394 6.312 78.529 6.011Q78.664 5.711 78.900 5.490Q79.135 5.270 79.440 5.150Q79.744 5.030 80.069 5.030Q80.574 5.030 80.923 5.133Q81.272 5.235 81.272 5.611Q81.272 5.758 81.174 5.859Q81.077 5.960 80.930 5.960Q80.776 5.960 80.677 5.861Q80.578 5.762 80.578 5.611Q80.578 5.423 80.718 5.331Q80.516 5.280 80.075 5.280Q79.720 5.280 79.491 5.476Q79.262 5.673 79.161 5.982Q79.060 6.292 79.060 6.640Q79.060 6.989 79.187 7.295Q79.313 7.601 79.568 7.785Q79.822 7.970 80.178 7.970Q80.400 7.970 80.585 7.886Q80.769 7.802 80.904 7.647Q81.039 7.491 81.097 7.283Q81.111 7.228 81.166 7.228L81.279 7.228Q81.309 7.228 81.332 7.252Q81.354 7.276 81.354 7.310L81.354 7.331Q81.268 7.618 81.080 7.816Q80.892 8.014 80.627 8.117Q80.363 8.219 80.069 8.219Q79.638 8.219 79.250 8.013Q78.862 7.806 78.628 7.443Q78.394 7.081 78.394 6.640M82.277 9.286Q82.406 9.354 82.543 9.354Q82.714 9.354 82.864 9.265Q83.015 9.176 83.126 9.031Q83.237 8.886 83.316 8.718L83.579 8.151L82.410 5.625Q82.335 5.478 82.205 5.446Q82.075 5.413 81.843 5.413L81.843 5.133L83.364 5.133L83.364 5.413Q83.015 5.413 83.015 5.560Q83.018 5.581 83.020 5.598Q83.022 5.615 83.022 5.625L83.880 7.484L84.652 5.813Q84.686 5.745 84.686 5.666Q84.686 5.553 84.603 5.483Q84.519 5.413 84.406 5.413L84.406 5.133L85.602 5.133L85.602 5.413Q85.384 5.413 85.211 5.517Q85.038 5.622 84.946 5.813L83.610 8.718Q83.439 9.088 83.169 9.334Q82.899 9.580 82.543 9.580Q82.273 9.580 82.054 9.414Q81.836 9.248 81.836 8.985Q81.836 8.848 81.928 8.759Q82.020 8.671 82.160 8.671Q82.297 8.671 82.386 8.759Q82.475 8.848 82.475 8.985Q82.475 9.088 82.422 9.166Q82.369 9.245 82.277 9.286\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.442 14.554)\">\u003Cpath d=\"M89.027 9.734Q89.027 9.716 89.040 9.669L91.696 3.007Q91.751 2.901 91.857 2.901Q91.922 2.901 91.973 2.952Q92.024 3.004 92.024 3.068Q92.024 3.092 92.023 3.104Q92.021 3.116 92.017 3.133L89.365 9.795Q89.293 9.901 89.205 9.901Q89.136 9.901 89.081 9.850Q89.027 9.798 89.027 9.734\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.442 14.554)\">\u003Cpath d=\"M97.282 8.151L95.546 8.151L95.546 7.871Q95.775 7.871 95.924 7.837Q96.072 7.802 96.072 7.662L96.072 5.813Q96.072 5.543 95.965 5.482Q95.857 5.420 95.546 5.420L95.546 5.140L96.575 5.065L96.575 5.772Q96.705 5.464 96.947 5.265Q97.190 5.065 97.508 5.065Q97.727 5.065 97.898 5.189Q98.069 5.314 98.069 5.526Q98.069 5.663 97.969 5.762Q97.870 5.861 97.737 5.861Q97.600 5.861 97.501 5.762Q97.402 5.663 97.402 5.526Q97.402 5.386 97.501 5.287Q97.211 5.287 97.011 5.483Q96.811 5.680 96.718 5.974Q96.626 6.268 96.626 6.548L96.626 7.662Q96.626 7.871 97.282 7.871L97.282 8.151M98.612 6.616Q98.612 6.295 98.737 6.006Q98.862 5.717 99.087 5.494Q99.313 5.270 99.608 5.150Q99.904 5.030 100.222 5.030Q100.550 5.030 100.812 5.130Q101.073 5.229 101.249 5.411Q101.425 5.594 101.519 5.852Q101.613 6.110 101.613 6.442Q101.613 6.534 101.531 6.555L99.275 6.555L99.275 6.616Q99.275 7.204 99.559 7.587Q99.843 7.970 100.410 7.970Q100.731 7.970 100.999 7.777Q101.268 7.584 101.357 7.269Q101.364 7.228 101.439 7.214L101.531 7.214Q101.613 7.238 101.613 7.310Q101.613 7.317 101.606 7.344Q101.493 7.741 101.123 7.980Q100.752 8.219 100.328 8.219Q99.890 8.219 99.490 8.011Q99.091 7.802 98.851 7.435Q98.612 7.068 98.612 6.616M99.282 6.346L101.097 6.346Q101.097 6.069 100.999 5.817Q100.902 5.564 100.704 5.408Q100.506 5.253 100.222 5.253Q99.945 5.253 99.731 5.411Q99.518 5.570 99.400 5.825Q99.282 6.080 99.282 6.346M103.999 8.151L102.266 8.151L102.266 7.871Q102.491 7.871 102.640 7.837Q102.789 7.802 102.789 7.662L102.789 5.413L102.201 5.413L102.201 5.133L102.789 5.133L102.789 4.316Q102.789 3.998 102.967 3.750Q103.144 3.503 103.435 3.362Q103.725 3.222 104.036 3.222Q104.293 3.222 104.496 3.364Q104.699 3.506 104.699 3.749Q104.699 3.885 104.600 3.984Q104.501 4.084 104.364 4.084Q104.228 4.084 104.129 3.984Q104.030 3.885 104.030 3.749Q104.030 3.568 104.170 3.475Q104.091 3.448 103.992 3.448Q103.783 3.448 103.630 3.581Q103.476 3.714 103.395 3.918Q103.315 4.121 103.315 4.330L103.315 5.133L104.204 5.133L104.204 5.413L103.343 5.413L103.343 7.662Q103.343 7.871 103.999 7.871\" fill=\"currentColor\" stroke=\"currentColor\" class=\"tikz-text\" style=\"stroke-width:0.210\"\u002F>\u003C\u002Fg>\u003Cg transform=\"translate(169.442 14.554)\">\u003Cpath d=\"M105.573 9.908L105.504 9.908Q105.470 9.908 105.448 9.882Q105.426 9.857 105.426 9.822Q105.426 9.778 105.457 9.761Q105.812 9.457 106.062 9.067Q106.311 8.677 106.463 8.245Q106.615 7.813 106.685 7.344Q106.755 6.876 106.755 6.401Q106.755 5.922 106.685 5.456Q106.615 4.989 106.461 4.554Q106.308 4.118 106.056 3.730Q105.805 3.342 105.457 3.048Q105.426 3.031 105.426 2.986Q105.426 2.952 105.448 2.927Q105.470 2.901 105.504 2.901L105.573 2.901Q105.583 2.901 105.592 2.903Q105.600 2.904 105.610 2.908Q106.154 3.308 106.526 3.861Q106.899 4.415 107.080 5.061Q107.261 5.707 107.261 6.401Q107.261 7.102 107.080 7.749Q106.899 8.397 106.525 8.951Q106.150 9.505 105.610 9.901Q105.600 9.901 105.592 9.903Q105.583 9.904 105.573 9.908\" 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\">RLHF versus DPO. RLHF fits a reward then optimizes through RL; DPO inverts the same optimum to train the policy directly on preferences in one supervised step.\u003C\u002Ffigcaption>",1785117782625]